CC BY
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SOLUTIONS AND ULAM-HYERS STABILITY OF DIFFERENTIAL 8 INCLUSIONS INVOLVING SUZUKI TYPE
S MULTIVALUED MAPPINGS ON
g" b-METRIC SPACES
CM
yy Mujahid Abbasa, Basit Alib, Talat Nazir°, Nebojsa M.
■■3 Dedovicd, Bandar Bin-Mohsine, Stojan N. Radenovicf
o a
q Government College University, Department of Mathematics,
-j Lahore, Islamic Republic of Pakistan;
o University of Pretoria, Department of Mathematics and Applied
Mathematics, Pretoria, Republic of South Africa, q e-mail: [email protected],
w ORCID ID: https://orcid.org/0000-0001-5528-1207
>- b University of Management and Technology, Department of
^ Mathematics, Lahore, Islamic Republic of Pakistan,
e-mail: [email protected], ORCID ID: https://orcid.org/0000-0003-4111-5974 : COMSATS University Islamabad, Department of Mathematics, Abbottabad Campus, Islamic Republic of Pakistan; w University of South Africa, Department of Mathematical Science,
^ Science Campus, Johannesburg, Republic of South Africa,
^ e-mail: [email protected],
ORCID ID: https://orcid.org/0000-0001-6516-3212 d University of Novi Sad, Faculty of Agriculture, Department of J Agricultural Engineering, Novi Sad, Republic of Serbia,
e-mail: [email protected], corresponding author, ORCID ID: https://orcid.org/0000-0002-4628-1405 O e King Saud University, College of Science, Department of
Mathematics, Riyadh, Kingdom of Saudi Arabia, ifii e-mail: [email protected],
ORCID ID: https://orcid.org/0000-0002-2160-4159 f University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia, e-mail: [email protected],
ORCID ID: https://orcid.org/0000-0001-8254-6688 DOI: 10.5937/vojtehg68-26718; https://doi.org/10.5937/vojtehg68-26718
O
FIELD: Mathematics
ARTICLE TYPE: Original Scientific Paper
ACKNOWLEDGMENT: The work of the fourth author Nebojsa M. Dedovic is supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, project TR-37017.
Abstract:
00 oo
CO
Introduction/purpose: This paper presents coincidence and common fixed points of Suzuki type (a, ~4>)~ multivalued operators on b-metric spaces. w
Methods: The limit shadowing property was discussed as well as the well-posedness and the Ulam-Hyers stability of the solution for the fixed point problem of such operators.
Results: The upper bound of the Hausdorff distance between the fixed point sets is obtained. Some examples are presented to support the obtained results.
Conclusion: The application of the obtained results establishes the fo existence of differential inclusion. E
Keywords: b-metric space, multi-valued mapping, fixed point problems, Ulam-Hyers stability, initial value problem.
cp
Introduction and preliminaries
Euclidean distance is an important measure of "nearness" between two real or complex numbers. Frechet (1905) introduced the concept of a metric to obtain the distance between two arbitrary objects. Since then, this notion has been generalized further in one to many directions, see (An et al, 2015a), among which one of the most important generalizations is the concept of a b-metric initiated by (Czerwik, 1993). For more details of b-metric spaces see (Aleksic et al, 2018), (Hussain et al, 2012), (Kirk & Shahzad, 2014) and the references therein.
Definition 1.1 Let X be a nonempty set. A mapping d.XxX^ [0, is said to be a b-metric on X if there exists some real constant ¿>1 such that for any x,y,ze X, the following condition holds:
a1): d(x,y) = 0 if and only if x = y;
82): d(x,y) = d(y,x);
a3): d(x,y) < bd(x,z) + bd{z,y)
The pair (X, d) is termed a b-metric space with b-metric constant b. Every metric is b-metric for b = 1 but the converse does not hold in general (Ciric et al, 2012), (Czerwik, 1993), (Singh & Prasad, 2008).
In the sequel, the letters, R+, M, N and Z+ will denote the set of all ™ nonnegative real numbers, the set of all real numbers, the set of all natural numbers and the set of all nonnegative integer numbers, respectively.
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Let (X,d) be a b-metric space and P(X) a collection of all subsets of X. Denote Cl(X),CB(X), and K(X) by the collection of closed, closed and bounded and compact subsets of X, respectively. s Let U,V e P(X). The gap functional D, the excess generalized
£ function p, the Pompeiu-Hausdorff generalized functional H, and the functional S induced by a b-metric d on X are defined as:
oo
(infuEUvEVd(u, v),ifU^V^0^U,
E (1) D(U,V) = \o,ifu'=V = 0,
O vot, otherwise.
o
< (supueuD (u, V),ifU ^ (2) p(U,V) = \o,ifU = 0,
O IOT,ify = 0,
LU I—
£ (max{p{U, V), p(V, U)},if U
pi (3) H(U,V) = \o,ifU = V = 0,
vot, otherwise. fsupueU vevd(u,v),ifU ^ V ^ 0 ^ U, _ S(U,V) = \o,ifU = V = 0,
< vot, otherwise. CD
g An et al (2015b) studied the topological properties of b-metric
spaces and stated that a b-metric is not necessarily continuous in each w variable. If a b-metric is continuous in one variable, then it is continuous o in other variable. A ball B(u0,e) = {v E X:d(u0,v) < e] in a b-metric space (X,d) is not necessarily an open set. A ball is an open set if d is > continuous in one variable.
Let (X,d) be a b-metric space. We call (f,T) a hybrid pair of mappings if f-.X^X and T:X^ CB(X).
A mapping f is called a contraction if there is some real constant re [0,1) such that for any u, v e X, we have d(fu,fv) < rd(u, v).
A point u in X is a fixed point of f, if u = fu, a fixed point of T, if ue Tu, a coincidence point of (f,T) if fu e T, and a common fixed point of (f, T) if u = fue Tu. Denote F(f), F(T) by the fixed points of f and T, respectively, and C(f,T) and F(f,T) by coincidence and common fixed point of (J, T), respectively.
Definition 1.2,compare with (Abbas et al, 2012). A pair (f,T) is w- § compatible if f(Tu) Q T(fu) forall ueC(f,T). The mapping f is T- g weakly commuting at some point ue X if f2(u) e T(fu).
Using an axiom of choice, Haghi et al (2011) proved the following lemma.
Lemma 1.3 (Haghi et al, 2011) Let f-.X^X be a self-mapping of a nonempty set X, then there exists a subset EQX such that f(E) = f(X) and f is one-to-one on E.
Lemma 1.4, compare (Rus et al, 2003). Let (X,d) be a b-metric space, U,V e P(X). If there exists a A> 0 such that for each ue U, there exists a veV such that d(u,v) < A, and for each veV, there exists a ueU such that d(u, v) < A, then H(U, V) < A.
We need following lemmas given in (Czerwik, 1993), (Singh & Prasad, 2008).
Lemma 1.5 Let (X, d) be a b-metric space,u,v e X, {un} a sequence in X and U,V e CB(X). The following statements hold:
b1-: (CB(X),H) is a b-metric space and (CB(X),H) is complete whenever (X, d) is complete;
b2-b3-b4-
D(u,V) < H(U,V) for all ue U; D(u, U~) < bd(u, v) + bD(v, U);
for h> 1 and ue U, there is a veV such that d(u,v) < hH(U,V);
b5-: for every h> 0 and ue U, there is a veV such that d(u, v) < H(U, V) + h;
b6-: D(u, U) = 0 if and only if ueU = U;
b7-: d(u0,un)^bd(u0,u1)+. ..+bn~1d(un_2,un_1) + bn~1d(un_1,un); b8-: {un} is a Cauchy sequence if and only if for e> 0, there exists n(e) e N such that for each n,m> n(e) we have d(un,um)<e;
b9-: {un} is a convergent sequence if and only if there exists ue X such that for all e> 0 there exists n(e) e N such that for all n > n(e), d(un,u) < e.
A sequence {un} is convergent to ue X if and only if
limn_,+md(un,u) = 0.
CP
X
oo
A subset icX is closed if and only if for each sequence {un} in Y that converges to an element u, ueY. A subset YcX is bounded if diam(7) is finite, where diam(7) = sup{d(u,v):u,v e Y}. A b-metric s space (X,d) is said to be complete if every Cauchy sequence in X is convergent in X.
Following are some known classes of mappings given in (Berinde, 1993), (Berinde, 1996), (Berinde, 1997), (Bota et al, 2015), (Rus, 2001). Let then
cE c1-: {Vi = is increasing, lim = 0,for any t > 01.
O
0
^ The elements in this class are called comparison functions. If
^ ^e then the nth iterate of ^ is a comparison function, ^ is continuous
1 at t = 0, and ^(t) < t, for any t> 0.
jJU
>- c2-: = Hn=i < +ot for all t > 0 and^ is nondecreasing}.
< is increasing, there exists a n0 EN, a E(0,1),, c3-: = i a sequence un >0 such that m=i un <+OTand
(^n+1(t) ^aipn(t) + un for all n > n0 , t > 0 is called the class of (c) -comparison functions.
<
(¡3 l^:^ is increasing,there exists a n0EN,oE(0,1)l
SZ c4-: = < b > 1, a sequence un>0 such that £n=i un <+ooand J
[bn+1^jn+1(t) <abn^jn(t) + un for all n > n0, t > 0 is known as the class of (b) -comparison functions. 5 Note that c ^ If b = 1, then =
o >
Lemma 1.6 (Berinde, 1993) If with b>1, then the series
& E£=o bn^jn(t) converges for all te!+, and rö(t) = ^=0 bn^n(t) is increasing and continuous at t = 0.
In the light of the above lemma, c
Lemma 1.7 (Päcurar, 2010) If \peWA with b>1, and {a„}CE+ is then
ZOT
bk~n^k~n(an) = 0.
n=0
such that Um an = 0, then
Example 1.8 Let ^(t) = qt, where qE[0,1) and tEl+. Consider Ek=0 uk(t), where uk(t) = bk^k(t) and b>1. If t=0, then £k=0 uk(t)
converges trivially. If t>0, then by the generalized ratio test (Berinde, fe
1993), Ek=o uk(t) is convergent for any t>0. Hence for some n0 e °°
N,bn+1^n+1(t) < abn^n(t)+ un for all n^n0 and t>0. Consequently, we have ^e «
o
The Banach contraction principle (BCP) (Banach, 1922) states that a contraction mapping on a complete metric space has a unique fixed point.
Let a:X xX ^ R+. A mapping f:X ^ X is called an a-admissible if for all u,v e X,a(u,v) > 1 implies that a(fu,fv) > 1. Samet et al (2012), Theorem 1, obtained the following generalization of BCP.
Theorem 1.9 Let (X,d) be a complete metric space and f:X ^ X an a-admissible mapping. Suppose that there is an element u0 in X with
a(u0,fu0)^1. If forany u,v e X, there exists ^eW2 such that a(u,v)d(fu,fv) < tfj(d(u,v)),then F(f) is nonempty provided that f is continuous.
Suzuki (2008) provided an interesting generalization of BCP that characterizes metric completeness.
For some other important generalizations of BCP, see (Berinde, 1993), (Berinde, 1996), (Berinde, 1997), (Bhaskar & Lakshmikantham,
2006), (Nieto & Rodriguez-Lopez, 2005), (Nieto & Rodriguez-Lopez,
2007), (Ran & Reurings, 2004) and references therein. A number of fixed point theorems have been obtained in b-metric spaces (Aleksic et al, 2019a), (Aleksic et al, 2019b), (Ali & Abbas, 2017), (An et al, 2015a), (An et al, 2015a), (Ciric et al, 2012), (Chifu & Petru§el, 2014), (Czerwik, 1993), (Karapinar et al, 2020), (Latif, 2015), (Mitrovic, 2019), (Pacurar, 2010).
The development of the metric fixed point theory of multivalued mappings was initiated by (Nadler, 1969). He introduced the concept of set-valued contraction mappings and extended the Banach contraction principle to set-valued mappings by using the Hausdorff metric as follows.
Theorem 1.10 Let (X,d) be a complete metric space. If a multivalued mapping T:X ^ CB(X) satisfies H(Tu,Tv) < rd(u,v) for all u,v e X and for some re [0,1), then F(T) is nonempty.
The fixed point theory of multivalued mappings provides a useful machinery to analyze the problems of pure, applied and computational
X
" mathematics which can be reformulated in the form of an inclusion for an
<D
appropriate multivalued mapping.
For more results in this direction, we refer to (Abbas et al, 2012), ® (Abbas et al, 2013), (Asl et al, 2012), (Rus et al, 2003), (Mitrovic et al, 2020).
Khojasteh et al (2014) proved a new type of the fixed point theorem for multivalued mappings in metric spaces as follows.
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Theorem 1.11 Let (X,d) be a complete metric space. If a multivalued mapping T.X ^ CB(X) satisfies
< ( D(u,Tv) + D(v,Tu) ,
for all u,v e X, then F(T) is nonempty.
Recently, Rhoades (2015) improved the result of Khojasteh for two multivalued mappings as follows.
Theorem 1.12 (Rhoads, 2015) Let (X,d) be a complete metric < space. If multivalued mappings S,T.X ^ CB(X) satisfy H(Su,Tv) < d nS T(u, v)mST(u, v) for all u,v e X, then F(T) n F(S) is nonempty, where
/max{d(u, v), D(u, Su) + D(v, Tv), D(u, Tv) + D(v, Su)}\ ns,T{u,v) = ^ 1 + S(u,Su) + S(v,Tv) )
(1.2)
( D(u,Tv) + D(v,Su)) ms,tiu,v) = max \d(u, v), D(u, Su), D(y, Tv),-^-j.
If S = T in above theorem then we get the following result.
Theorem 1.13, (Rhoads, 2015). Let (X,d) be a complete metric space. If a multivalued mapping T.X ^ CB(X) satisfiesfor all u,v e
X, H(Tu, Tv) < nT T(u, v)mT T(u, v), then F(T) is nonempty.
Let a.XxX^R+ and U,VeP(X). Define
a„(U,V) = mfueu,veVa(u,v).
A multivalued mapping T.X^Cl(X) is called - admissible mapping if for any u,v e X,a(u,v) > 1 implies that a„(Tu,Tv) > 1. The concepts of - admissible mapping coincides with a- admissible mapping in case of a single valued mapping.
Asl et al (2012), Theorem 1, defined (a* contractive fe
multifunctions and proved the following result. °°
Definition 1.15, compare with (Rus et al, 2003). Let (X,d) be a b-metric space. A mapping T:X ^ Cl(X) is called a multivalued weakly Picard operator (MWP operator), if for all ueX and for some ve Tu, there exists a sequence {un} satisfying (a1) u0 = u,u1 = v, (a2) un+1 e Tun for all n>0, (a3) {un} converges to some ze F(T). The sequence {un} satisfying (a1) and (a2) is called a sequence of successive approximations (ssa) of T starting from (u,v). If T is a single-valued mapping, then we call it a Picard operator if it satisfies (a1) to (a3).
Recently Bota et al (2015) proved the fixed point theorem for (a* -- contractive multivalued mappings as follows.
Theorem 1.16 Let (X,d) be a complete b-metric space, a:XxX ^ R+ and T:X^Cl(X) an a*- admissible multivalued operator that satisfies
a*(Tu,Tv)H(Tu,Tv) < ip(d(u,v))
for all u,veX and ^e W4. Assume that there exists a u0 eX and ut eTu0 such that a(u0,u1)>1. Then T is a MWP operator provided that if there is a sequence {un} in X such that un ^ u, then a(un,u) > 1 for all
nE Nx.
A mapping T:X ^ Cl(X) is called (a*-/)-admissible mapping if u,v E X,a(fu,fv) > 1 implies that a*(Tu,Tv) > 1.
Let (X,d) be a b-metric space, g:X ^X and T:X ^ Cl(X). Set
Theorem 1.14 Let (X,d) be a complete metric space and T:X^ « Cl(X) an a*- admissible mapping that satisfies a*(Tu, Tv)H(Tu, Tv) < ^(d(u,v)) for all u,veX and ^e W2. Moreover, if there exists a u0eX and ux eTu0 such that a(u0,u1)>1, then F(T) is nonempty provided that if {un} is a sequence in X such that a(un,un+1)>1 for all n and Um un = u, then a(un,u) > 1 for all n.
In what follows, we assume that a b-metric d is continuous in one e variable.
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max{d(u, v),D(u, Tu) + D(v, Tv),D(u, Tv) + D(v, Tu)}
NT(u,v) = b(S(u,Tu) + S(v,Tv) + 1) ,
( D(u,Tv) + D(y,Tu)) MT(u, v) = max \d(u, v), D(u, Tu), D(y, Tv),-—-j,
° and
01 iv,j
max{d(gu, gv), D(gu, Tu) + D(gv, Tv), D(gu, Tv) + D(gv, Tu)} yy N„T (u, v) = ■
a,TK ' J b(S(gu,Tu) + S(gv,Tv) + 1)
o ""a,T
Mn
( D(gu,Tv) +D(gv,Tu)) ~(u, v) = max\d(gu, gv),D(gu, Tu), D(gv, Tv),-—-j.
We now give the following definitions.
Definition 1.17 Let (X,d) be a b-metric space. A mapping T:X^ Cl(X) is called Suzuki type (a* -ty)- multivalued operator if there exists ^ a such that
^Diu.Tu) < bd(u,v) (1.3)
o implies
a*(Tu, Tv)H(Tu, Tv) < max{1, NT(u, v)}^j(Mt(u, v)) (1.4)
o for all u,v E X.
—>
> If in the above definition we replace a mapping T by a single valued
O mapping f:X ^X, then we call it a Suzuki type (a* - operator.
Definition 1.18 Let (X, d) be a b-metric space. A hybrid pair (g, T) is called a Suzuki type (a* -ty)- hybrid pair of operators if there exists a \pE such that
2D(gu,Tu) < bd(gu,gv) (1.5)
implies
a*(Tu, Tv)H(Tu, Tv) < max{1, NgiT(u, v)}^)(MgiT(u, v)) (1.6)
for all u,veX.
In case T is replaced by a single valued mapping f.X ^ X, we call it fe a Suzuki type (a* -ty)- pair of operators. °°
so from (1.4), we obtain
<D(x1,Tx1)^H(Tx0,Tx1)<a*(Tx0,Tx1)H(Tx0,Tx1) <max{1, NT(x0,x1)}^(MT(x0,x1))
{imax{d(x0,x1),D(x0,Tx0) + D(x1,Tx1),D(x0,Tx1)+D(x1,Tx0)} 1V b(1 + S(x0,Tx0) + S(x1,Tx1))
Fixed point of Suzuki type {a* -ty)- multivalued operators on b-metric spaces
In this section, we prove that Suzuki type (a* -ty)- multivalued operators are MWP operators.
Theorem 2.1 Let (X,d) be a complete b-metric space, a.XxX^ R+ and T.X ^ Cl(X) a Suzuki type (a*-^-multivalued operator. | Further, assume that T is a* -admissible mapping and there exists x0eX and x1 eTx0 such that a{x0,x1)'^1. If for any sequence {xn} converging to x in X, we have a(xn,x)^1 for all ne , then
d1-: T is an MWP operator.
d2-: If there is some ue F(T) such that u^z and a(z,u) > 1, then ^ d(z, provided that 1^NT(x, y) for all x,y e X.
Proof. (d1) By the given assumption, there exists x0eX and xteTx0 such that a{x0,x1)'^1. If x0 = x1, then x0eTx0. Define a sequence {xn} in X by xn = x1 = x0 for all ne X+. Thus xn eTxn for all n^0 and {xn} converges to x = x0eF(T) and hence T is an MWP operator. Let x0 Since T is a*- admissible mapping, a*{x0,x1)'^1 implies that a*(Tx0,Tx1)^1. As T is a Suzuki type multivalued operator and
2D(xQ,TxQ)^d(xQ,x1)^bd(xQ,x1), (2.1)
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ï max-i
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C imax{d(x0,x1),d(x0,x1) + D(x1,Tx1),bd(x0,x1) + bD(x1,Tx1)} ' \ b(1 + d(x0,x1)+D(x1,Tx1))
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! ( d(x0,x1) + D(x1,l x1))\ max \d(x0,x1), D (x1,7x1),-^-M
L / d(x0,x1) + D(x1,Tx1) \) maXl ,\1 + d(x0,x1) + D(x1,Tx1)J)
! ( d(x0,x1) + D{x1,l x1))\ max \d (x0,x1), D (x1,7x1),-^-i)
<^(max{d(x0,x1), D(x1,rx1)}).
0<D(x1,rx1)<^(max{d(xo,x1), D(x1,rx1)}). (2.2)
If max{d(x0,x1),D(x1,rx1)} = D(x1,Tx1), then (2.2) implies that | 0<D(x1,rx1)<^(D(x1,rx1)). (2.3)
As D(x1,Tx1)>0 and ^e (2.3) give
£ 0<D(x1,rx1)<^(D(x1,Tx1)) <D(x1,Tx1),
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2 a contradiction. Hence max{d(x0,xi),D(xi,Txi)} = d(x0,x1). From (2.3), it follows that
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0<D(x1,rx1)<^(d(xo,x1)). (2.4)
Let q> 1. We may choose x2 eTxt such that
0<D(x1,Tx1)^d(x1,x2)<qD(x1,Tx1)^qip(d(xo,x1)). That is
0<d(x1,x2)<qD(x1,'Tx1)< qTp(d(xQ,x1)). (2.5)
Since a(x1,x2)>a(7,x0,rx1)>1, we get a(rx1;rx2)>1. Set c0 = d(xo,x1)>0, then from (2.5) we get xx ^x2 and d(x1,x2)<qxp(c0). As ^e so
^(d(x1;x2)) <\p{q\p(c0)). (2.6)
If Rl = then by (2.6) qi >1. Now, if x2 eTx2 then the proof $
d(x1,x2)<bd(x1,x2).
By (1.4)
<D(x2,Tx2)<H(Tx1,Tx2)<a„(Tx1,Tx2)H(Tx1,Tx2) <max{1, NT(x1,x2)}ip(MT(x1,x2))
{fmax{d(x1,x2),D(x1,Tx1) + D(x2,Tx2),D(x1,Tx2) + D(x2,Tx1)}
= '{ b(l + S(x1,Tx1)+ S(x2,Tx2))
/ { s ^ , , ^ D(X1,TX2) +D(x2,TX1) max id(x1,x2),D (x1,Tx1),D {x2,Tx2),-—-
mi1 ( max{d(x1,x2),d(x1,x2) + P (x2,Tx2),b (d(x1,x2) + D (x2 ,Tx2))} <maX{,{ b(1 + d(x1,x2)+D(x2,Tx2))
d(x1,x2) + D(x2,Tx2))
I ( d(x1,x2) +D(x2,I x2))\ ^\max\d(x1,x2),d(x1,x2), D (x2,Tx2),-^-j)
d(x1,x2) + P(x2,Tx2) >
+ d(x1,x2)+D(x2,Tx2)J
L ( d(x1,x2) + P(x2,Tx2) <maXl ,\1 + d(x1,x2)+D(x2,Tx2)J)
^ ( ( d(x1,x2) + D(x2,Tx2))\ ^^max|d(x1,x2), d(x1,x2), D (x2,Tx2),-^-j)
<ifi(max{d(x1,x2),D(x2,Tx2)}). That is
0<D(x2,Tx2)^ip(max{d(x1,x2),D(x2,Tx2)}). (2.7)
If max{d(x1,x2),D(x2,Tx2)} = D(x2,Tx2), then
0<D(x2,TX2)^^(D(X2,TX2)). (2.8)
Now D{x2,Tx2)>0, and (2.8) give
0<D(x2,TX2)^^(D(X2,TX2)) <D(X2,TX2), a contradiction. Hence
0< D(x2,rx2)<^(d(x1,x2)). (2.9)
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is finished. Let x2 £Tx2. Note that ^
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We may choose x3 eTx2 such that 0<D(x2,Tx2)^d(x2,x3)<q1D(x2,Tx2)^q1^(d(x1,x2)) = That is
0<d(x2,x3)<q1D(x2,Tx2)^q1^(d(x1,x2)) = (2.10)
a: As a(_x2,x3)^a(Tx1,Tx2)>1, so a(Tx2,Tx3)^1. From (2.10), we
o get x2 ^x3 and
o xp(d(x2,x3))<xp2(qxp(cQ)). (2.11)
z
X
w Set q2 = >1. If x3 eTx3 then we are done. Suppose that
x3£Tx3. Similarly, we obtain x4 e7x3 such that
0< d(x3,x4)<q1D(x3,rx3)<q2^(d(x2,x3)) = \p2{q\p(c0)). (2.12)
^ Continuing this way, we can obtain a sequence {xn} in X such that
xn+i ^1, and it satisfies.
o
>o 0< D(xn+i,Txn+i)^ \p(d(xn,xn+1)) (2.13)
z
£ and
o
| 0<d(xn+1,xn+2)<^(q^co)) (2.14)
&
for all nE 1L+. From (2.14), for n,m e n with m> n, we have
d(xn,xm) t^.bd(xn,xn+i) + b2d(xn+i,xn+2)+...
d(xm_2,xm_1) + b d(xm_1,xm)
+bm-n\pm-2(q\p(c0)) + bm-n\pm~2(q\p(c0)) +bm-2^m~2(q^(c0)))
m-2 t>
CO
ab CO
= Z ^'(^M
i=n-1
im-2 n-2
That is
i = 0 i = 0
! m-2 n-2
1
i = 0 i = 0
Set sn = !f=0 bi^i(q^j(c0)). Then from (2.15) we obtain that
1
d(xn,Xm)< ¿n-2 ^m-2 2). (216)
By Lemma 1.6, converges for any t>0. Hence
lim Sn_2 =S for some Set+. If ¿ = 1, then from (2.16) we get
lim d(x„,xm)< lim Sm^1 - lim Sn^1 = 0. If b> 1, then from (2.16) we
have
liiri d(x„,xm)< lirti "-1rT(Sm_1 -S^) < lim T^rr = 0
for all m,ne! Hence {x„} is a Cauchy sequence in X. There exists zeX such that
lim d(xn,z) = 0. (2.17)
—i-l- on V '
d(xn,xm)<—I ^ ^(^^-^^(^M). (2.15) II
X
Now we show that ze F(T). If D(z,Tz) > 0, then we claim that one of the following two inequalities
1 ™
2D(xn,Txn)^bd(xn,z) (2.18) *
CO
<u
oo CD
>o
X LU
2D(xn+1,Txn+1)^bd(xn+1,z) (2.19)
holds for all ne TL+. Assume on the contrary that there exists an n0
> such that
o" Ol
0
01
DC -D(xno,Txno)> bd(xno,z) (2.20)
yy 2
QC
O o
and
< o
- 2D(xn0+i,Txno+1)>bd(xno+1,z). (2.21)
O 2
LU
oc Now from (2.13), (2.20) and (2.21), we have
d(x„o,x„o+1)< bd(xno,z) + bd(z,xno+1) <-D(xno,Txno) + -D(xno+1,Txno+1)
< — d(xno,xno+i)+ -d(x„o,x„o+1)
— d(xrlo,xrjo+1)
¡5 a contradiction. Hence either (2.18) or (2.19) holds for an infinite subset of 1+. By the given assumption, it follows that a(xn,z) > 1. As T is Q a* -admissible, a(Txn,Tz) > 1. Now if (2.18) holds for all ne then from (1.4) we get
D(xn+1,Tz) < H(Txn,Tz) < a*(Txn,Tz)H(Txn,Tz) < max{1,NT(xn,z)}^{MT(xn,z))
/ max{d(x„,z), D (xn,Txn) + D(z, Tz), D(xn,Tz) + D (z, Txn)}
max-! 1,
K:
6(1 + 5(xn,rxn) + 5(z,7z))
D(xn,Tz) + D(z, Txn)")
/ r D(xn,Tz) + D(z, Txn))\ ^max |d(xn,z), D (xn,Txn), D (z, Tz),-—-N
/max{d(xn,z),d(x„,xn+1) + D(z,Tz),D(xn,Tz) + d(z,xn+1)}
i1
max < i I -
< l A b(1 + d(xn,xn+1) + D(z,Tz))
( ( D(xn,Tz) + d(z,xn+1))\ max \d(xn,z), d (x„,xn+1), D (z, Tz),-—-M.
On taking limit as m +00, we have fe
lim D(xn+1,Tz)
m fi
< v v 6(1 + S(z, Tz) + S(u, Tu))
^ (max Id(z,u),D(z, Tz), D(u, Tu),
D(z,Tu) + D(u,Tzy
2b
( (max{d (z, u), d (z, z) + d (u, u), d (z, u) + d (u, z) }
< [ \ b(1 + d(z, z) + d(u,u))
^ / r d(z,u) + d(u,z)) ^ (max \d(z, u), d(z, z), d(u, u),-2^-i
( (2d(z,u)\)
< ¿maxjl, (---)j^(d(z,u)).
Now, max{1, NT(x,y)} = NT(x,y) gives
d(z,u) < 2d(z,u)ip(d(z,u)) <2d2(z,u) and hence d(z,u) >
oo
CO
max(1 (max{d(xn,z),d (xn,xn+1)+D(z,Tz),D(xn,Tz)+d(z,xn+1)}Y\ _
<lim I ,V b(l+d(xn,xn+1)+D(z,Tz)) )] "
" (max {d(zn,z), d(xn,xn+1),D(z, Tz), »^tz^x^ .
As ^ is continuous at 0,^ e and D(z,Tz) > 0, we have
( D(z,Tz) ) f
D(z,Tz) < max|1,ft(1^D(z ' J^(D(z,Tz)) = xp(D(z,Tz)) <D(z,Tz). J
a contradiction. Consequently,ze Tz. Similarly, we obtain zeTz when (2.19) holds for an infinite subset Ni of Z+.
To prove part (d 2), let ue F(T) such that u^z and a(z,u) >1. Since T is a* -admissible, a*(Tz,Tu) > 1. Now ^D(z,Tz) = 0 < d(z,u) implies that
d(z,u) < bD(z,Tz) + bD(Tz, u) < bH(Tz,Tu) < ba*(Tz,Tu)H(H(Tz,Tu))
(max{d (z, u), D (z, Tz) + D (u, Tu), D (z, Tu) + D (u, Tz) }
¿max <1,
X
oo CD
"o
>
o" CM O CM
a:
LU
o z
X
o
LU
I—
>-
CC
Corollary 2.2 Let (X,d) be a complete b-metric space, a.XxX^ and T.X ^ Cl(X) an - admissible mapping such that
2D(x,Tx) < bd(x,y)
implies that
a„(Tx,Ty)H(Tx,Ty) < max{1,NT(x,y)}ip(d(x,y))
IT O
o for all x,y e X,\p eWA. Further, assume that there exists x0 eX and < xteTx0 such that a(x0,x1)^1. If for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all neZ+ then
ei-: T is an MWP operator
e2-: If there is some ue F(T) such that u^z and a(z,u) > 1, then ^ d(z,u)> ^ provided that
max{1,NT(x,y)} = NT(x,y) for all x,y e X.
w
^ Corollary 2.3 Let (X,d) be a complete b-metric space, a.XxX^
^ R+ and T.X ^ Cl(X) an - admissible mapping such that
^ ^-D(x,Tx) < bd(x,y)
I— 2
O z
¡3 implies that
>
( d(x,y) )
a.(Tx,Ty)H(Tx,T?) < ^^x\1,b(i + S(XiTx) + S(yiTy))\Mix,?»
for all x,yeX,^e¥4. Further, assume that there exists x0eX and xt eTx0 such that a(x0,x1)^1. If for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all neZ+ then
e3-: T is an MWP operator.
e4-: If there is some ue F(T) such that u^z and a(z,u) > 1, then d(z,u) > 1 provided that
d(x,y) ) d(x,y)
max •"
. L _d(x,y)_)
( ,b(1 + S(x,Tx) + S(y,Ty)))
b{1 + 8(x, Tx) + S{y, Ty))) b{1 + 8(x, Tx) + S{y, Ty)) for all x,ye!
Proof. Follows from Corollary 2.3.
1> +
Corollary 2.4 Let (X,d) be a complete b-metric space, a:XxX and T:X i Cl(X) an a* - admissible mapping such that
i
Tx) < bd{x,y)
implies that
a* (Tx, Ty)H(Tx, Ty) < NT(x, y)^(MT (x, y))
for all x,yeX,^eW4. Further, assume that there exists x0eX and x1 eTx0 such that a(x0,x1)>1. If for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all neZ+ then
e5-: T is an MWP operator.
e6-: If there is some ue F(T) such that u^z and a(z,u) > 1, then d{z,u) >
Proof. Take max{1, NT(x,y)} = NT(x,y) in Theorem 2.1.
Corollary 2.5 Let (X,d) be a complete b-metric space, a:XxX i R+ and T:X i Cl(X) an a* -admissible mapping such that
1
D(x,Tx) < b,
<NT(x,y)^(d(x,y))
Tx) < bd(x, y)impliesthata*(Tx, Ty)H(Tx, Ty) (2 22)
CD
oo
CO
X
for all x,yeX,^eW4. Further, assume that there exists x0eX and x1 eTx0 such that a(x0,x1)>1. If for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all neZ+ then ^
e7-: T is an MWP operator.
00 CD
"o
>
e8-: If there is some ue F(T) such that u^z and a(z,u) > 1, then
co
d{z,u) >
Proof. Take MT(x,y) = d(x,y) in Corollary 2.4.
The following Corollary is a Suzuki type generalization of (Asl et al, 2012), Theorem 2.1 (Bota et al, 2015), Theorem 1 (Mohammadi, 2013), Su Theorem 3.1 (Samet et al, 2012), Theorem 2.2 and references therein in S the context of b-metric spaces.
o
_i Corollary 2.6 Let (X,d) be a complete b-metric space, a:XxX ^
o R+ and T:X ^ Cl(X) an a, - admissible mapping such that
X
O 1
"' <bd(x,y) implies that (2 23)
£ a*(Tx, Ty)H(Tx, Ty) < $(MT(x, y))
for all x,yeX,^eW4. Further, assume that there exists x0eX and xx eTx0 such that a(x0,x1)>1. Then T is an MWP operator provided that for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for
< all ne TL+.
>o Proof. Take max{1, NT(x,y)} = 1 in Theorem 2.1.
z
X
^ Corollary 2.7 Let (X,d) be a complete b-metric space, a:X xX i
° R+ and T:X i Cl(X) an a* - admissible mapping such that
o >
a*(Tx, Ty)H(Tx, Ty) < $(MT(x, y)) (2.24)
for all x,yeX,^eW4. Further, assume that there exists xQ eX and xx eTx0 such that a(x0,x1)>1. Then T is an MWP operator provided that for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all ne TL+.
Corollary 2.8 Let (X,d) be a complete b-metric space, a:XxXi R+ and T:X i Cl(X) an a* - admissible mapping such that
■^D(x,Tx) < bd{x,y)
implies that
a*(Tx, Ty)H(Tx, Ty) < ^(d(x,y)) (2.25)
for all x,y e e W4. Further, assume that there exists x0eX and fe x1 eTx0 such that a(x0,x1)'^1. Then T is an MWP operator provided g that for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all ne TL+.
Proof. Take MT(x,y) = d(x,y) in Corollary 2.6.
Now we state Theorem 2.1 in the context of single valued mapping f.X ^X, where X is a b-metric space. The existence of a fixed point follows immediately from Theorem 2.1. To prove the uniqueness of the fixed point, we need the condition H, given as follows: |
(H): for all x,y e X, there exists a zeX such that a(x,z) > 1 and a(y,z) > 1.
cp
Corollary 2.9 Let (X,d) be a complete b-metric space and a.Xx X^ R+ and f.X ^ X an a - admissible mapping such that ;f
1
^d(x,fx) < bd(x,y)
implies that
a(fx, fy)d(fx, fy) < max{1, Nf(x,y)}rp(Mf(x,y)) (2.26)
for all x,yeX,^e¥4. Further, assume that there exists xQ eX and xi =fxo such that a(x0,x1)~^1 and for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all ne If the condition (H) is satisfied, then f is a Picard operator and for an arbitrary zeX, the sequence {fnz} converges to some we F(f) and
e9-: F(f) = {w} if max{1,Nf(x,y)} = 1,
e10-: d(u,w)>^ for any ueF(f) such that u^w provided that
max{1,Nf(x,y)] = Nf(x,y).
Proof. Theorem 2.1, f is a Picard operator and F(f) is nonempty. Let u,v e F(f) such that u^ v. By the condition (H), there exists a zeX such that a(u,z)^1 and a(v,z)^1. Note that {fnz} is a Picard sequence which converges to some we F(f). As f is an a- admissible mapping, so for all n^1 we have a(u,fnz) > 1 and a(v,fnz) > 1. Since
X
CO 1
<u
oo
ZD
o
o <
o z
X
o
LU
I—
>-
CC <
o >
2d(u,fu) = 0 < bd(u,fn~1z),
<o by (2.26) we have
"o
d(u,fnz) = d(fu,fnz)
g <n(fTI ffn-i^HffT, ffn-1
cm
<a(fu,ffn~1z)d(fu,ffn~1z)
<max{1, Nf (u, fn~1z)}^>{Mf (u, ffn~1z))
yy ( (max{d(u,fn~1z),d(u,fu) + d(fn-1z,ffn-1z),d(u,ffn~1z) + d{fn~1z,fu)}\
cc <max <"
f / max{d(u,fn~1z),d(u,fu) + d(fn-1z,ffn-1z),d(u,ffn~1z) + d(/n"1z,/u)}\] :|1,V ¿(1 + d(u,fu) + d(fn~1z,ffn~1z)) Jj
max j d(u,fn~1z), d(u,fu), d(fn-1z,ffn~1z), -2ft- l)
f /max{d(u,/n"1z),d(u,u) + d(fn~1z,fnz),d(u,fnz) + d(/n"1z,u)}\|
<max I1 ^ ¿(1 + d(u,u) +d(/""1z,/"z)) Jj
/ i d(u,fnz) + d(fn~1z,u))\ max \d(u,fn~1z),d(u,u),d(fn~1z,fnz),^^^—-— M.
On taking limit as m +oo, we obtain that
^ max{d(u, w), d(u, u) + d(w, w), d(u, w) + d(w, u)}
max
m
r(-
3 d(u,w)<~TA t(l+d(u,u) + d(w,w))
CD / r d(u, w) + d(w, u)'
2 ^ \max]d(u,w),d(u,u),d(w,w),-—
o V I 2o
¡¡5 =max
I—
o
That is
d(u,w) < maxj1,(2d^||,W^)j^(d(u,w)). (2.27)
Now, If max{1, A/f(x,y)} = 1, and w^ u, then we have
d(u,w) < Tp(d(u,w)) < d(u,w). (2.28) Also, if max{1, A/f(x,y)} = 1, and v^w, then we have
d(v,w) < ^(d(v,w)). (2.29)
A contradiction in both cases. Thus w = u = v, and hence F(f) is fe singleton. If max{l,Nr(x,y)} = Nf(x,y) and u^w then from (2.27) we ® get
d(u,w) <----—Tp(d(u,w)) < —d2(u,w)
2d(x,fx) < bd(x,y)
implies that
a(fx,fy~)d(fx,fy) < max{l,Nf(x,y)}$(d(x,y))
for all x,y eX,\p e ¥4. Moreover, suppose that there exists x0eX and xi =fxo such that a(x0,x1)~^1 and if there is a sequence {xn} in X such that xn ^x, then a(xn,x) > 1 for all ne Further, assume that there exists x0eX and x1 =fx0 such that a(x0,x1)~^1 and for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all ne If the condition (H) is satisfied, then f is a Picard operator and for an arbitrary zeX, the sequence {fnz} converges to we F(f) and F(f) = {w}. Also, d(u,w)>^ for any ueF(f) such that u^w provided that maxtj
{l,Nf(x,y)} = Nf(x,y).
2d(u,w) 2 s ....... iz(u,w) g
CL
tn o
and d(u,w)>^. f
Corollary 2.10 Let (X,d) be a complete b-metric space and and f.X ^X an a- admissible mapping such that
E
1
^d{x,fx) < bd(x,y) implies that a(fx, fy)d(fx, fy) < ^>(Mf(x,y))
for all x,y e X,-^ e*¥4. Further, assume that there exists x0eX and j| x1 =fx0 such that a(x0,x1)~^1 and for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all ne If the condition (H) is satisfied, then f is a Picard operator and for an arbitrary zeX, the sequence {fnz} converges to we F(J) and F(f) = {w}.
Corollary 2.11 Let (X,d) be a complete b-metric space and a.XxX^ R+. Let f.X ^X be an a- admissible mapping that satisfies
X
Example 2.12 Let X = {x1,x2,x3,x4,xs} and d:XxX^R+ be defined as
£ d(x2,x3) = 9,d(x1,x4.) = d(x1,x5) = 10,
d(x1,x2) = d(x1,x3) = 4,d(x4,x5) = 1, ° d(x, x) = 0 and d(x,y) = d(y, x) for all x,y E X.
oo
0
01
or As 9 = d(x2,x3)^d(x2,x1) + d(x1,x3) = 8, so d is not a metric on
UJ
^ X. Indeed,(X,d) is a b-metric space with b = ->1. Consider a mapping
8
§ T:X —> Cl(X) defined by Tx1 = Tx2 = Tx3 = {x1},TxA = {x2} and < Tx5 = {x3}. If we take \p(t) for tE R+, then (see 1.8). If
mapping is defined as a(xi,x/) = 1 for all i,j e {1,2,3,4,5},
o then T is an -admissible mapping. For x,y e {x1,x2,x3}, we have H(Tx,Ty) = 0 ^ max(1,NT(x,y)}^(MT(x,y)). For (x,y) when xE & {x1,x2,x3} and yE {x4,x5}, we obtain that
a(Tx1,Tx4)H(Tx1,Tx4,) = d(x1,x2) = 4^9 = ip(d(x1,x4,))
< max{1,NT(x1,x4)-^(MT(x1,xA)'),
54
a(Tx2,Tx4)H(Tx2,Tx4) = d(<x1,x2) = 4^ — = ^(d(<x 2,x4))
« < max{1,NT(x2,x4)^(MT(x2,x4)),
X 54
a(Tx3,Tx4)H(Tx3,Tx4) = d(xi,x2) = 4^ — = ty{d(x3,xA))
w < max{1,NT(x3,xA)^(MT(x3,xA)),
O a(Tx1,Txs)H(Tx1,Txs) = d(x1,x3) = 4^9 = ^(d(x1,xs))
¡2 < max{1,NT(x1,x5)ip(MT(x1,x5)),
> 54
^ a(Tx2,Tx5)H(Tx2,Tx5) = d(x1,x3) = 4^ — = ^(d(x2,x5))
< max{1,NT(x2,x5)^(MT(x2,x5)),
54
a(Tx3,Txs)H(Tx3,Txs) = d(x1,x3) = 4^ — = ^(d(x3,x5))
< max{1,NT(x3,x5)^(MT(x3,x5)).
Note that
l
and
— D(xA,TxA)=—d(xA,x2) = 3> — = bd(xA,xs),
2 2 o
2D(x5,Tx5)= 2d(x5,x3) = 3> 4 = bd(x5,x4).
Hence
1
2^d(x,tx) < bd(x,y)) <0
implies
a(Tx,Ty)H(Tx,Ty) < max{1, NT(x,y)}^(MT(x,y))
holds for all x,y£l Thus all the conditions of Theorem 2.1 are satisfied. On the other hand, if we take x = x4,y = x5, then
a(Tx4,Tx5)H(Tx4,Tx5) = d(x2,x3) = 9>^(d(x4,x5)) = ^(1) = ^ and
a(Tx4,Tx5)H(Tx4,Tx5)^^j(d(x4,x5)). Consequently, Theorem 1.16 in (Bota et al, 2015) does not hold in this case.
The following example illustrates an assumption max{1,NT(x,y)} >
1.
Example 2.13 Le№ = {x1,x2,x3}andd:Z xX ^ E+be defined as
d(x1,x2) = 4,d(x1,x3) = 1,d(x2,x3) = 2, d(x, x) = 0 and d(x, y) = d(y, x) for all x,y£l
As 4 = d(x1,x2)^d(x1,x3) + d(x3,x2) = 3, so d is not a metric on X. Indeed,(X,d) is a b-metric space with b = ^>1. Consider a mapping T:X ^ Cl(X) defined by
f{x2} if x = xx, Tx = \ {xx} if x = x2, ({x2} if x = x3.
If we take ^(t) = ft for te E+, then (see Example 1.8). If
is defined as a(xi,xJ) = 1 for all i,j e {1,2,3}, then T is a* - admissible. Note that
3max{d(x1,x2), D(x1,Tx1) + D(x2,Tx2), D(x1,Tx2) + D(x2,Tx1)} nt(x1,x2)=-
4(1 + S(x1,Tx2) + S(x2,Tx1)) 3max{d(x1,x2), d(x1,x2) + d(x2,x1), d(x1,x1) + d(x2,x2)} 4(1 + d(x1,x1) + d(x2,x2))
3max{4,8,0}
=----=6>1.
4(1 + 0)
Hence max{1, NT(x,y)} = 6 > 1. Note that
CD
00 CO
cp
> X
oo CD
"o
>
o~ CM O CM
a:
yy
0£ ZD
o o
LU
N 3max{d(x1,x3),D(x1,Tx1) + D(x3,Tx3),D(x1,Tx3) + D(x3,Tx1)}
3max{d{x1,x3),d{x1,x2) + d{x3,x3),d{x1,x3) + d{x3,x2)} 4(1 + d(x1,x3) + d(x3,x2)) _ 3max{1,4,3} _ 3 _ 4(1 + 1 + 2) _ 4
and
< 3max{d(x2,x3),D(x2,Tx2) + D(x3,Tx3),D(x2,Tx3) + D(x3,Tx2)}
| Nt{X2,X3}- 4(1 + S(x2,Tx3) + S(x3,Tx2)~)
J 3max{d(x2,x3), d(x2,x1) + d(x3,x3), d(x2,x3) + d(x3,x1)}
4(1 + d(x2,x3) + d(x3,x1))
3max{2,4,3} 3 ' _ 4(1 + 2 + 1)_ 4.
Also,
w
< 64
CD a(Tx1,Tx2)H(Tx1,Tx2) = d(x2,x1) = 4< — = NT(x1,x2)y>(d(x1,x2)) >o < max{1, NT(x1,
8
J a(Tx1,Tx3)H(Tx1,Tx3) = d(x2,x3) = 2< 3 = NT(x1,x3)^)(d(x1,Tx1))
O < max{1,NT(x1,x3)^(MT(x1,x3)),
4
> a(Tx2,Tx3)H(Tx2,Tx3) = d(x1,x3) = 1<3 = NT(x2,x3)^(d(x2,x3))
< max{1,WT(x2,x3)^(MT(x2,x3)).
Thus all the conditions of Theorem 2.1 are satisfied. On the other hand, if we take x = x1,y = x2, then a(Tx1,Tx2)H(Tx1,Tx2) = d(x2,x3) =
3 2
4> ^(d(x1,x2)) = ^(4) = —. Hence a(Tx1,Tx2)H(Tx1,Tx2)S
d(x1,x2)). Consequently Theorem 1.16 in (Bota et al, 2015) is not applicable in this case which is a generalization of Theorems 1.14 and 1.9.
For 6 = 1, Theorem 2.1 reduces to the following important Corollary.
Corollary 2.14 Let (X,d) be a complete metric space, a:XxX^ R+ and T:X ^ Cl(X) satisfies the following implication
1
2D(x,Tx) <d(x,y) implies max{1,nTT(x,y)}^i(mTT(x,y))
for all x,y E X and ^ e ¥4 where nTT(x,y) and mT,T(x,y) are the same as given in (1.2), Further, assume that there exists x0 eX and x1 eTx0 such that a(x0,x1)> 1 and for any sequence {xn} converging to x in X, we have a(xn,x) > 1 for all nE X+. Then
e11-: T is an MWP operator.
e12-: If there is some ue F(T) such that u^z and a(z,u) > 1, then d{z,u) > ^ provided that
max{1,nTT(x,y)} = nTT(x,y).
Next we present an example which shows that Corollary 2.14 is a potential generalization of Theorems 1.14, 1.9, 1.11, 1.13.
Example 2.15 LetX = {x1,x2,x3,x4,x5}andd:Z x X ^ R+bedefined
by _ _ _ _
d(x1,x4) = d(x1,x5) = 9, d(x1,x2) = d(x1,x3) = 4,
d(x4,x5) = 2,d(x2,x3) = 8,
d(x, x) = 0 and d(x, y) = d(y, x) for all x,y EX.
Note that d is a metric on X. Consider a mapping T:X — Cl(X) defined by Tx1 = Tx2 = Tx3 = {x1},Tx4 = {x2} and Tx5 = {x3}. If we take ^(t) = for tE R+, then for each i = 1,2,3,4 (see Example 1.8).
If a:XxX is defined as a(xi,xj) = 1 for all i,j E {1,2,3,4,5}, then T is - admissible mapping. For x,y e {x1,x2,x3}, we have H(Tx,Ty) = 0^max{1,NT(x,y)}^(MT(x,y)). For (x,y), when xe{x1,x2,x3} and yE {x4,x5}, we obtain that
a(Tx1,Tx4,)H(a(Tx1,Tx4,)) = d(x1,x2) = 4<8 = ^(d(x1,x4)) < max{1,nTT(x1,x4)Tp(m.TT(x1,x4)),
40
a(Tx2,Tx4)H(Tx2,Tx4) = d(x1,x2) = 4^ — = ^(d(x2,x4))
< max{1,nTT(x2,x4)^(mTT(x2,x4)),
40
a(Tx3,Tx4)H(Tx3,Tx4) = d(xi,x2) = 4^ — = ^(d(x3,x4))
CD
oo
CO
cp
X
oo
<¡2 < max{1, nTT(x3,x4)^(mTT(x3,x4)),
a(Tx1,Tx5)H(Tx1,Tx5) = d(x1,x3) = 4^8 = ip(d(x1,x5))
< max{1,nTiT(x1,xs)^(mTiT(x1,xs)),
«a 40
| a(Tx2,Tx5)H(Tx2,Tx5) = d(xi,x3) = 4^ — = ^(d(x2,x5))
< max{1,nTT(x2,x5)^(mTT(x2,x5)),
40
^ a(Tx3,Tx5)H(Tx3,Tx5) = d(xi,x3) = 4^ — = ^(d(x3,x5))
LU 9
E < max{1,nTT(x3,x5)^(mTT(x3,x5)).
z)
o Note that ^D(x4,Tx4) = ^d(x4,x2) = ^ >1 = d(x4,x5), and
g iD(x5,7x5) = id(x5,x3) = ^>1 = d(x5,x4).
LU Z Z Z
I—
>-
Q1 <
iO <
X LU
to
Hence
2(D(x,Tx) <d(x,y))
cd implies
a(Tx,Ty)H(Tx,Ty) < max{1, NT(x,y)^(MTT(x,y))
o holds for all x,yeX. Thus all the conditions of Corollary 2.14 are satisfied. On the other hand, if we take x = x4,y = x5,
> then
8
a(Tx4,Tx5)H(Tx4,Tx5) = d(<x2,x3) = 8>^(d(x4,x5)) = ^(1) = 9. Hence,
8
a(Tx4,Tx5)H(Tx4,Tx5) = 8S9 = ^j(d(x4,x5)).
Consequently, Theorem 1.14 is not applicable in this case. Note that Theorem 1.14 is a generalization of Theorem 1.9. Now
max{d(x4,,x5),D(x4,,Tx4,) + D(x5,Tx5),D(x4,,Tx5)+D(x5,Tx4,)} nT,Ax 4,x5)= 1 + 5(x4,rx4) + 5(x5,rx5)
max{d(x4,x5), d(x4,x2) + d(x5,x3), d(x4,x3) + d(x5,x2)} 1 + d(x4,x2) + d(x5,x3)
max{1,10,10} 10 =---- = —
11 11
( D(x4,TX5) + D(X5,TX4)) mTT(x4,x5) = max|d(x4,x5),D(x4,rx4),D(x5,rx5),-^-j
= max{1,5,5,5} = 5 implies that a(Tx4,Tx5)H(Tx4,Tx5) = 8%
Hence, Theorem 1.13 which is a generalization of Theorem 1.11 does not hold in this case.
Coincidence and common fixed point results in b-metric spaces
As an application of Theorem 2.1, we obtain the existence of coincidence and common fixed point of Suzuki type (a* -ty)- hybrid pair of operators in b-metric spaces.
Theorem 3.1 Let (X,d) be a b-metric space and (g,T) a Suzuki type (a* - hybrid pair of operators such that T an (a* -g)-admissible mapping. Suppose that there exists x0 eX and gxx eTx0 such that a(gx0,gx1)^1 and for any sequence [xn] in X with gxn ^ gx, we have a(gxn,gx) > 1 for all ne 1+. Then there exists x in X such that gx e Tx provided that T(X) Q g(X) and g(X) is complete. Moreover, if there is some gy e Ty such that gx ^ gy and a(gx,gy) > 1 then d(gx,gy) > ^ Further, F(g,T) is nonempty if any of the following conditions hold:
f1-: The hybrid pair (g,T) is w- compatible, lim gn(x) = w for
some we X and xe C(g, T) and g is continuous at w.
f2-: The mapping g is T — weakly commuting at some xe C(g,T) and g2x = gx.
00 oo
CO
> X
f3-: The mapping g is continuous at at some xeC(g,T) and
lim gn(w) = x for some we X.
Proof. Lemma 1.3, there is a set EQX such that g.E^X is one-to-> one and g(E) = g(X). Define a mapping T. g(E) ^ CB(X) by Tgx = Tx for all g(x) e g(E). The mapping T is well defined because g is one-to-one. Since (,g, T) is a Suzuki type (a* - hybrid pair of operators, ES therefore
o
Q1 ZD
o
o <
o
X
o
LU
I—
>-
CC <
iO <
-J
CD >Q
X LU I—
o
o >
Q
d
^D(gx,Tx) < bd(gx,gy)) implies
a*(Tx,Ty)H(Tx,Ty) < max{1,Wfl,r(z,y)ty(Mfl,r(z,y))
i. max{d(,gx, gy), D(gx, Tx) + Djgy, Ty), D(gx, Ty) + D(gy, Tx)}) = maX I , b(1 + S(gx, Tx) + S(gy, Ty)) J
xp I max \d (gx, gy), D (gx, Tx), D (gy, Ty),-—-i
for all x,y eX for some and ^eO. Thus
'^D(gx,Tgx) < bd(gx,gy)) implies
a*(Tgx,Tgy)H(Tgx,Tgy) < max{1,NT(x,y)}\p(MT(x,y))
r ma x{d(gx,gy),D (gx,Tgx) +D (gy,Tgy),D (gx,Tgy) +D (gy,Tgx)J\
ni ax 11, ■ ■ i
I b(l+S(gx,Tgx)+S(gy,Tgy)) J
" tf (max[d(gx, gy),D(gx, Tgx),D(gy, Tgy), ^W»^*)})
for all gx,gyeg(E). Since g(E) = g(X) is complete. By the given assumption, there exists x0 eX and gxx eTx0 such that a(gx0,gx1)^1. As T is (a* -g)~ admissible, we have a(Tx, Ty) > 1 which implies that a(Tgx,Tgy) > 1. Thus T is a*-admissible. Hence T satisfies all the conditions of Theorem 2.1. Consequently, T is an MWP operator on g(E), and we obtain a point ue g(E) such that ue Tu. Since ue g(E), there is a point x in X such that gx = u. This implies that gx e Tgx = Tx. By Theorem 2.1 if there is some we Tw such that u^w and a(u,w) > 1, then we have d(u,w) > ^ if max{1,NgT(x,y)} = NgT(x,y). For weTw there is a y in X such that such gy = w and gy e Tgy =
Ty. Consequently, d(gx, gy) Now we prove that F(g,T)^0. First ^ consider the case when (C1) holds. Since the pair (g,T) is s? w- compatible and lim gn(x) = u for some ueX, the continuity of g at
n^ + ra <n
u implies that gu = u and lim gn(x) = gu. Now a(gn(x),gu)^ 1 y
and(a* -g) - admissibility of T imply that a(Tgn~1(x),Tu) > 1. By w- compatibility of the pair (g,T), we have gn(x) e T(gn~1(x)), that is j= gn(x) e C(g, T) for all ne N. Note that
1D(gn(x),T(gn-1(x)))^d(gn(x),gn(x)) = 0 < bd{ggn~\x),gu).
<
Since (g, T) is a generalized Suzuki type (a* hybrid pair of
operators, therefore
D(gnx,Tu)
^H(Tgn~1x,Tu) < a(Tgn-1(x),Tu)H(Tgn~1x,Tu)
( . max{d(gnx,gu),D(gnx,Tgn~1x) +D(gy,Tu),D(gnx,Tu) +D(gu,Tgn~1x)\]
max 11 >
< l ' b(l+S(gnx,Tgn~1x)+S(gu,Tu)) J
" ^max [d(gnx, gu),D(gnx, Tgn~1x),D(gu, Tu), j
I , b(l+d(gnx,gnx)+D(gu,Tu)) J
^max [d(gnx, gu), d(gnx, gnx),D(gu, Tu), .
On taking limit as n^ +00, we obtain that
( D(gu, Tu) ) D(gu,Tu) < max|1, + m Tu^^(D(gu,Tu)) = \p(D(gu,Tu)).
If D(gu,Tu) > 0, then we have D(gu,Tu) < D(gu,Tu), a contradiction. Consequently, u = gu e Tu. That is F(g,T) ^ 0. Now let (C2) hold, then g2x = gx for some xe C(g,T). By T- weakly commuting of g, we have gx = g2xe Tgx. Hence gx e F(g,T). In case (C3) holds, lim gn(u) = x for some ueX and xe C(g,T). By continuity of g, we
obtain that x = gx e Tx. Hence F(g,T) * 0.
CL 2?
x
> Q
a*(Tx,Ty)H(Tx,Ty) < max{l,A/gT(x,y)}^(d(x,y))
Corollary 3.2 Let (X, d) be a b-metric space and (g, T) a hybrid pair such that T is an (a* -g) - admissible. If there exists a such that
1
g 2 ■D(ffx,Tx) < bd(gx, gy)
° implies that
0~ CM o CM
C£
E for all x,ye! Suppose that there exists x0eX and gx1 eTx0 such that o a(gxQ,gx1)'^1 and for any sequence {xn} in X such that gxn ^ gx, we ° have a(gxn,gx) > 1 for all ne Then there exists x in X such that gxeTx provided that T(X) Q g(X) and g(X) is complete. Moreover, if there is some gyeTy such that gx ^ gy and a(gx,gy)^1, then ^ d(gx,gy) > Further, F(g,T) is nonempty if the conditions (j1)-(j3) >- Theorem 3.1 hold.
CC
< .
Corollary 3.3 Let (X,d) be a b-metric space and (g,T) a hybrid pair such that T is an (a* -g) - admissible. If there exists a such that
^D(gx,Tx) < bd(gx,gy)
< implies
O a*(Tx, Ty)H(Tx, Ty) < #d(x, y))
g for all x,ye! Suppose that there exists x0eX and gxx eTx0 such that u(gxo,9xi)>1 and for any sequence {xn} in X such that gxn ^ gx, we w have a(gxn,gx) > 1 for all ne Then there exists x in X such that o gxeTx provided that T(X) Q g(X) and g(X) is complete. Moreover, if q there is some gyeTy such that gx ^ gy and a(gx,gy) > 1, then
d(gx, gy) > Further, F(g,T) is nonempty if the conditions (j1)-(j3) in Theorem 3.1 hold.
Data dependence of fixed point sets and Ulam-Hyers stability results
Consider the following class of functions
0 = {<f: R+ ^R+such that <f is increasing and continuous at 0}.
Let (X,d) be a b-metric space and T. X^P(X). The fixed point problem of T is to find an xeX such that
xeTx. (4.1)
Inequality (4.1) is also known as fixed point inclusion. The fixed point Es
inclusion is said to be generalized Ulam-Hyers stable if there exists a °° function (£0 such that for each e> 0 and for each solution u„ of the inequality
CP
D(u,TU)^£ (4.2)
there exists a solution z* of the fixed point problem (4.1) such that d(u*,z*)<£(£).
Further, if there exists a c> 0 such that <f(t) = ct for each te R+, then the fixed point problem (4.1) is said to be Ulam-Hyers stable. Let | F(T) and U be the sets of solutions of (4.1) and (4.2), respectively. For more on Ulam-Hyers stability of fixed point problems, we refer the interested reader to (Ulam, 1964), (Lazar, 2012), (Petru et al, 2011), (Rus, 2009), (Hyers, 1941). Let (X,d) be a b-metric space and T:X ^ Cl(X) be a multivalued mapping then E(T) = {^el:{x} = Tx}. f
Define a multivalued operator Tm:G(T) ^ P(F(T)) by "
Tm(x,y) = {z£ F(T): there is an ssa of Tat (x,y) converging to z]
where G(T) = {(x,y): x e X,y e Tx] is a graph of T.
A selection of T:X ^ P(X) is a single valued mapping t:X ^ X such that tx e Tx for all xel
Definition 4.1 (Rus et al, 2003). Let (X,d) be a metric space and c> 0. An MWP operator T:X ^ P(X) is called ac- multivalued weakly Picard (briefly c-MWP) operator if there exists a selection tm of Tm such that d(x,tro(x,y)) < cd(x,y) for all (x,y) e G(T).
One of the main results concerning c — MWP operators is the following:
Theorem 4.2 (Rus, 2001). Let (X,d) be a metric space and Tx,T2:X^ P(X). If Ti is a Ci -MWP operator for each ie {1,2} and there exists 2> 0 such that H(T1x,T2x) < A, for all xe X. Then H{F(T1),F(T2)) < Amax{ci,c2}.
Now we prove the following result.
X
00
LU
<
O >
Q
Theorem 4.3 Let (X, d) be a complete b-metric space and a.Xx X^ R+. Suppose that
g1-: for each i e {1,2}, T^.X ^ Cl(X) are multivalued operators such
£ that
r < bd(x,y)
implies that
< a„(Tix,Tiy)H(Tix,Tiy) < max{1,NT.(x,y)}^i(d(x,y)) (4.3)
o for all x,y eX,^ie{¥A.
g2-: for each i e {1,2}, Ti is a* - admissible mapping, £ g3-: there exists x0eX and x1 eTtx0 such that a(x0,x1)>1 for
each i e {1,2},
g4-: if there is a sequence {xn} in X such that xn ^x, then a(xn,x) > 1 for all ne TL+,
g5-: there exists X> 0 such that H(7\x, T2x) < X, for all ie!
<t
d Then Fix(Ti) e Cl(X), i e {1,2} and each Tt is an MWP operator such
>§ that
uÜ H(Fix(T1),Fix(T2)) ^ bmax{A1,A2]
O
where Aj = E™=0 for each i e {1,2}.
Proof. From Theorem 2.1, it follows that Fix(T{)^0 for each i e {1,2}. Let {xn} be a sequence in Fîx(7,1) such that xn^z as n ^ +oo. This implies that a(xn,z) > 1. Since T1 is «»-admissible mapping,
a(Tixn,Tiz) > 1. As
20(x„,7,1x„) = 0<ôd(z, x„),
so we get
maxi
(d(z,xn),D(z,T1z) + D(xn,Tixn),Y\ ' [D(z, Tixn)+D (xn,T1z) j|
+bd(z, xn))
max (d^ Xn^,D^Z, TlZ) + d(Xn,Xn),)\
<1 1 H1 + D(Z,T1Z) + d(Xn,Xn)) № (d(z,xn))
+bd(z,xn))
On taking limit as n^+00, we obtain that D(z, 7,1z)<0, that is, ze 7\z and hence F(7\) is closed.
Similarly, F(r2) is a closed subset of X.
From Corollary 2.2, we conclude that for each ¿e {1,2} is an MWP operator.
By a similar process as followed in Theorem 2.1 starting from x1 eF(T1) and x2 eT2x1, we obtain a sequence {xn} such that xn+1 e
^n for all ^1,0
^2(d(xn,xn+i)) and
0<d(xn+1,xn+2)<^l^(q^(co)) (4.4)
for all n>1, where c0 = d(x1,x2).
Following the arguments similar to those in the proof of Theorem 2.1, we conclude that {xn} is a Cauchy sequence and there is an element u in X such that xn ^u as n ^ +00 and ue T2u.
Note that
D(Z,T1Z)
<bd(z, xn) + bD(xn,T1z) oo
< bd(z,xn) + bH(T1z,T1xn)<bd(z,xn) + ba(T1xn,T1z)H(T1z,T1xn) <bd(z,xn) + max{1,NTi(x,y)}Tp1(d(x,y))
X
" d(xn,xn+p)
^bd(xn,xn+1) + b2d(xn+1,xn+2)+. ..+bp 1d(xn+p_2,xn+p_1)
< un
yy bn
Cd n+p-2 n+p-2
ZD
o o
< o
X
o
LU
I—
>-
CC <
iO <
k=n-1 k=n-1
/n+p-2 n-1
fc = 0 fc=0 That is,
'n+p-2 n-1
d(xn,^il+p)< ^ bk№(qxp2{Xj)- ^ bkxpk2(qxp2{Xj)
\ fc = 0 fc = 0 V V (4.5)
On taking limit as p^ +00, we get
(ra n-1
^ bkM>k2{qyp2(X))- ^ bkypk{qyp2(X)) k-° k-° (4.6)
+ I.
By Lemma 1.6, Xfc=o converges for any t>0, there exists
12 >0 such that £fc=0 =12 and hence
n-1
d(xn,u) < - £ + (4.7)
For n = 1, we get d(x1,u) < M2. Thus for x0 eF(7\), there exists ue F(TZ) such that d(x0,u) < öl2. Similarly for each z0 eF(T2), we get ve F(Tt) and A1 >0 such that d(z0,v) < It follows from Lemma 1.4 that
H(F(T1),F(T2)) < ömax^,^}.
<u
Now we discuss the Ulam-Hyers stability results.
Theorem 4.4 Let (X,d) be a b-metric space and T.X _ Cl(X). Assume that all the hypotheses of Corollary 2.3 hold. Then we have
h1-: The fixed point inclusion (4.1) is - generalized Ulam-Hyers & stable for i = 1,2, provided that for each xeF(T) there exists zeU such that a(x,z) > 1, where (i,(2:R+ _R+ defined by (i(t) = t -b2tfj(t),Z2(t) = t — bt^(t) are strictly increasing, onto and continuous at t = 0.
h2-: If E(T) ^ 0, then the fixed point inclusion (4.1) is -generalized Ulam-Hyers stable for i = 3,4, provided that for xe E(T) there exists zeU such that a(x,z) > 1, <,3,<,4.E+ _R+ defined by (3(0 = f _ b$(t), (4(t) = t - t^(t) are strictly increasing, onto and continuous at t = 0.
h3-: (Estimate between the fixed point sets of two multivalued mappings) If S.X _ Cl(X) is such that for xe F(S) there exists ze F(T) with a(x,z) > 1 and for xeF(T) there exists ze F(S) with a(x,z) > 1, and H(S(x),T(x)) < A for all xe Z, then H(F(S),F(T)) < max^^A)
where £ is same as in (h1) for each ¿ = 1,2.
h4-: (Estimate between the fixed point sets of two multivalued mappings) If S.X _ Cl(X) is such that for xe F(S), there exists ze E(T) with a(x,z) > 1 and for xe E(T) there exists ze F(S) with a(x,z) > 1, and H(S(x),T(x)) < A for all xe Z, then H(F(S),F(T)) < max^ibA)
i — 3,4
where £ is same as in (h2) for each i = 3,4.
h5-: (Well-posedness of the fixed point problem with respect to b- ^ metric d)If forany sequence {xn} in X, there exists a unique point x* eE(T)such that a(x„,x*)> 1 and limn_+mD(xn,Txn) = 0, then _^^ ^ ^ 0.
X
<u
h6-: (Well-posedness of the fixed point problem with respect to b-metric H) If for any sequence {xn} in X, there exists a unique point x* eE(T) such that a(x„,x*)>l and limn_+mH({xn},Txn) = 0, then _^^ ^ ^ 0.
-g h7-: (Limit shadowing property of the multivalued operators) If for
o" any sequence {xn} in X, there exists a unique point x* eE(T) with ° a(x„,x*)>l and limn_+mD(xn,Txn) = 0, then there exists a sequence of successive approximations {yn} such that limn_+md(xn,yn) = 0.
cE
g Proof. (h1) From Corollary 2.3, T is an MWP operator and hence
u F(T) is nonempty. If x* e F(T), then by given condition there exists a < y*eU such that a(x*,y*)>l. The a* - admissibility of T gives that
a(Tx*,Ty*)> l. Since y* eU, for any given e> 0, we have D(y*,Ty*)<
o e. Note that
LU
<
X LU
l
>- -D(x*,Tx*) = 0<bd(x*,y*).
on 2
Then
d(x*,y*)< bD(x*,Tx*) + bD(Tx*,y*) = bD(Tx*,y*) < ^b2(H(Tx*,Ty*) +D(Ty*,y*))
« < b2(a(Tx*,Ty*)H(Tx*,Ty*) + e)
>o („ d(x*,y*)
(u-3) (maX{1,6(1 + S(x*,Tx*)y+ S(y*,Ty*))i^(d(X*,y*)) + £
d(x*,y*)
d(x*,y*))
O <62(maxfl,—--—( 'y *-——)^(d(x*,y*)) + e)
V { b(1 + d(x*,x*) + D(y*,Ty*))rv y y " J
O >
<62 (maxjl, jj^(d(x*,y*)) + £
If max{l,d(x*,y*}] = 1, then we have d(x*,y*)<62(^(d(x*,y*)) + e).
If (i(d(x*,y*)) = d{x*,y*)-b2\p(d{x*,y*)), then from the above inequality we get <'1(d(x*,y*)) <62£ and hence d(x*,y*)<<,]"1(62£). Consequently, the fixed point inclusion (4.1) is ^-generalized Ulam-Hyers stable, where <f = (f1.
If max{l,d(x*,y*}} = then d(x*,y*)>b. From (4.1) we obtain
that
d(x*,y*X b2 (^-^(¿(x*,y*)) + e)
j2I
<M(x*,y*)^(d(x*,y*)) + b2s.
d(x*,y*) = D(Tx*,y*)^b(H(Tx*,Ty*) + D(Ty*,y*)).
CO
<bd(x*,y*W(d(x*,y*)) + b2£ §
<u
Now if <2(d(x*,y*)) = d(x*,y*)-bd(x*,y*)^(d(x*,y*)), then from the above inequality we get <"2(d(x*,y*)) <b2s and hence d(x*,y*X (¿_1(ô2£). Consequently, the fixed point inclusion (4.1) is <f- generalized ^ Ulam-Hyers stable, where <f = «^T1.
(h2) Let E(T) * 0, and x* eE(T) then
Following the arguments similar to those in the proof of (hi), the result follows.
(ha) Let x* e F(S), then there exists a y* eF(T) such that a(x*,y*)> t 1. By a* - admissibility of T we get a(Tx*,Ty*)> 1. Note that
2;D(y*,Ty*) = 0<bd(x*,y*).
Then by the given assumption on T, we obtain that
d(x*,y*)< bD(x*,Sx*) + bD(Sx*,y*)
= bD(Sx* ,y*)<ft2(H(Sx* ,Tx*) + D(Tx* ,y*))
< b2(H(Sx*,Tx*) + H(Tx*,Ty*)) <62(A + tf(7V,7y))
<£2 (A + a(Tx*,Ty*)H(Tx*,Ty*))
<£2 (x + max(1,-„ d(**,y?-}^(d(x*,y*)))
^ V I ' b(l+5(x*,Tx*)+5(y*,Ty*))) v >y JJ)
<b2(A + maxf 1,--, f**^ *——) Q(d(x*,y*)j)
^ V I b(l+d(x*,x*)+D(y*,Ty*))J v JJ)
<b2(A + maxf 1,--, f**^ *——) Q(d(x*,y*)j)
^ V l b(l+d(x*,x*)+D(y*,Ty*))) v JJ)
<£2(A + ^(d(x*,y*))).
If max{1,^^} = 1, and
(1(d(x*,y*)) = d(x*,y*)-£2(^(d(x*,y*))), then from the above inequality we get <'1(d(x*,y*)) < b2A. Consequently, ^ for every x*eF(S), there exists a y*eF(T) such that d(x*,y*X Z{1(b2A). Similarly, it can be proved that for every y* e F(T), there exists
X
<D
00 CD
"o
>
a x*eF(T) such that d(x*,y*)< Hence by Lemma 1.4, we
obtain that
H(F(S),F(T))<(^1(b2A).
If max{l,d(x*y*)] = d(x*,y*}, then for <i(d(x*,y*)) = d(x*,y*)-g M(x*,y*)^(d(x*,y*)) we get £ H(F(S),F(T))<(^1(b2A).
yy Consequently,
| H(F(S),F(r))<inax(r1(^2^).
o
(h4) This can be proved on the similar lines as in (h3) using the
X
o
LU
co <
-J
CD 2
definition of E(T).
(h5) If {xn} is a sequence in X, there exists a unique x* eE(T) such
oc that a(x„,x*)> 1 and lim„^+raD(x„,Tx„) = 0. Then there exists un eTx
<
n
such that lim„^+raD(x„,rx„) = lim„^+rad(x„,u„) = 0. Since T is a* - admissible, a(Txn,Tx*)^1. As ^D(x*,Tx*) = 0< bd{xn,x*), by
given assumption we have
d(x„,x*)< b(D(xn,Txn) + D(Txn,x*))
< b(D(xn,Txn)+H(Txn,Tx*))
< blDlxn,Txn) + a(Txn,Tx*)H(Txn,Tx*))
<b(D(xn,Txn)
J ,___ T-, d(xn,x*)
I-
O
y +maxfl,-- dix^p ~U(d(x„,x*))
I b(l+D(xn,Txn)+D(x*,Tx*))) v n JJ
<b(D(xn,Txn)
+maxfl,--}-di(d(xn,x*))
<6 (g(x„,rx„) + max {l, ^(d(x„,x*))^
If max{l,d(^,xT[ = l, then we have
d(xn,x*) — bip(d(xn,x*)))<bD(xn,Txn).
That is, (3(d(x„,x*)) <bD(xn,Txn). Similarly, if max{l/(^x)] =
), we get (4(d(x„,x*)) <bD(xn,Txn). This implies for each ¿e {3,4}
we get d(x„,x*)< ^¡^1(bD(xn,Txn)). On taking limit as n_ +oo and using the continuity of (i at 0, for each ie {3,4} we get the desired result.
(h6) follows from (h5) as D(xn,Txn)^H({xn}, Txn). fe
(h7) From (h5) it is clear that lim„^+rod(x„,x*) = 0. Since x* e E(T), so there exists a sequence of successive approximations defined by
yn =x* for all n such that lim„^+rad(x„,y„) = lim,i^+rad(x„,x*) = 0.
Existence and stability of solutions of differential inclusions
Let Clc(R) be collection of nonempty closed and convex subsets of R and F:R^Clc(R) a lower semicontinuous multivalued mapping. Consider the initial value problem
Note that the solution of problem (4.9) is a solution of problem (4.8). Integrating from at to t, we obtain
L L
J *'«*=//(*«)*.
t
x(t) = xo+ J f(x(s))ds,fort e ;. (4.10)
a1
oo
CO
cp
x'(t) e F(x(i)),fori e j,
x(t) = x0fort = a1, (4.8)
xe C(J), f
where ] = [ai,a2] and C(J) is a Banach space of absolutely continuous real valued functions defined on J. Since R with usual metric is paracompact, F isa lower semicontinuous multivalued mapping with F(u) closed and convex for each ue R, by Michael's Theorem (Michael, 1956), there exists a continuous function f:R^R such that f(u) e F(u) for all «el.
Now consider the following initial value problem
x'(t)=f{x(t)),forte],
x(t) = x0fort = alt (4.9)
xe C(J).
X
a1 a1
that is, a
<u
00 CD
"o
>
O >
On the other hand, if (4.10) holds then (4.9) holds. Thus (4.9) and (4.10) are equivalent.
Suppose that ^satisfies the following hypotheses:
Jf f(x(s))ds = 0, for tej if and only if x(t) = x0(t) for all tej.
There exists a nonnegative real number Lf such that
CM 1
or Lf(a2 -a1)< — where b is b-metric constant and for all the
LLI ^
relation ||/(u) -f(y)\\ <i/^u(t) -v(t)ll holds.
o Define T:X^X, where X: = C(J) by
o <
x T(x(t)) = x0+ i f(x(s))ds,fortEj. (4.11)
o J
LU «1
I—
| Let d: C(J) x C(/) ^ be defined as
d(x,y): = max||x(t) — y(t)|2.
tE/
Then (C(/), d) is complete b-metric space. Define a: C(J) x C(J) ^ ^ E+ by
>Q
t&y) = {f
J CO otherwise,
O
Let E+ be defined as ^(t): = L^(a2 Clearly, ^e
First we show that mapping T is a* - admissible. As a(x,y) > 1 implies that a(x,y) = k. For x^ x0 and y^x0, from (i1-) we have Tx ^ x0 and Ty ^ x0 on /. It follows that a*(Tx, Ty) = k^1 and hence T is a* - admissible. Now, by (i2-) for all x,y E X,
d(Tx, Ty) = max
tEJ
t t
I f(x(s))ds- I f(y(s))ds
a1 a1
t
< max I II f(x(s))ds — f(y(s))ds ^EJ J
a1
t tr "
< max I Lj II x(s) - y(s) II2 ds |
t in
r o
< L^max I max II x(s) — y(s) II2 ds
tej J sej «1
= Lfd(x,y)max I ds = Lf(a2 — a1)d(x,y) = ^(d(x,y)). ^eJ J
a i
By Corollary 2.3, we obtain the solution of problem (4.9) which e provides the solution of problem (4.8) as well. Define the mappings
<,1,<,2:^+ by
(1(t) = t - b2L}(a2 -a1)t = t - b2^j(t),
$2(t) = t-bL2f(<a2-a1)t2 = t-bt\p(t), |
where ^(t) = L}(az -a1)t. Clearly the mapping (1 is strictly increasing and onto. Consequently, all the axioms of Theorem 4.4 hold with mapping Hence the fixed point inclusion (4.8) is (f1 - generalized Ulam-Hyers stable. Now (t) > 0 if 1-2bL2f(a2 -a1)t> 0. As bL2fia2-a1)<1, hence the fixed point inclusion (4.8) is (21 - generalized Ulam-Hyers stable if t<^.
Let F:/x!xR^Cic(I) be a lower semicontinuous multivalued mapping. Consider the initial value problem
x'(t) e F(t, x(t),x(t - h)), fort e J, x(t) = x0fort e — h,ai\, (4.12)
xe C(J),
where h is a positive real number. Then there exists a selection f such that f(s,u,v)eF(s,u,v) for all and se[a1-h,a1], see
(Michael, 1956). Note that any solution of the problem
x'(t) = f(t, x(t),x(t- h)), fort e J, x(t) = x0fort e a1 — h,a1], (4.13)
xeC(J)
is a solution for problem (4.12). Further, (4.13) is equivalent to
X
oo CD
"o
>
o~ CM O CM
a:
LU
X LU I—
O
o >
Q
x(t) = x0 + ja f(s,x(s),x(s — h)ds,fort E J, x(t) = x0fort E a1 — h, %].
We suppose that f satisfies the following hypotheses:
f(s, x(s), x(s - h))ds = 0, for tEj if and only if x = x0 on J. \f(t,ui,vi)-f(t,u2,v2)\ <L/(\u1 -u2\\ + \\v1 -vj),
IT
o
" for all u1,u2,v1,v2 El, where Lf(a2 and b is a metricconstant.
u Define the operator T:Y ^Y, where Y: = C[a1 -h,a2]xRxRby
LU
£ r(x(t)) = |Xo+ j f(s,x(s),x(s~h))ds,fortEJ, (414)
^ I «1
vx(t) = x0fort E a1 — h, a{\.
From the definition of a, the admissibility of T follows. Now by (i4-) for Q all x,yEY, we have CD >Q
L L
d(Tx,Ty) = max I j f(s,x(s),x(s — h))ds — j f(s,x(s),y(s — h))ds
a1 a1
t
<max I II/(s, x(s), x(s — h))ds — f(s, x(s), y(s — h))ds II2 ^EJ J
< max j Lj II x(s) - y(s) II2 (22)ds
tEJ
"1
t
= 4L^niarx J IIx(s) - y(s) II2 ds < 4Lj(a2 — a1)^(d(x,y)),
tEJ
where ^(t) = 4L2f(a2 -a1)t. By Corollary 2.3, we obtain the solution of problem (4.13) which is also being the selection is the solution for (4.12).
If (1(t) = t - 4b2Lj(a2 -a1)t and (2(0 = t - 4bL2f(a2 -a1)t2, then the
fixed point inclusion (4.12) is (11 - generalized Ulam-Hyers stable. Now fe
^ u 11 ^ _ uuLj(a2 -at)t > 0. As 4bLj(a2 -at)< 1, hence the
fixed point inclusion (4.12) is <"21 _ generalized Ulam-Hyers stable if
i 2.
. ^ 1 8
References i
Abbas, M., Ali, B., & Vetro, C. 2013. A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces. Topology and its Applications, 160(3), pp.553-563. Available at: https://doi.org/10.1016/j.topol.2013.01.006.
Abbas, M., Ciric, Lj., Damjanovic, B., & Khan, M.A. 2012. Coupled coincidence and common fixed point theorems for hybrid pair of mappings on a test. Fixed Point Theory and Applications, art.number:4. Available at: https://doi.org/10.1186/1687-1812-2012-4.
Aleksic, S., Dosenovic, T., Mitrovic, D.Z., & Radenovic, S. 2019a. Remarks on common fixed point results for generalized a, ~4>- contraction multivalued mappings in b-metric spaces. Advances in Fixed Point Theory, 9(1), pp.1 -16. Available at: https://doi.org/10.28919/afpt/3731.
Aleksic, S., Huang, H, Mitrovic, D.Z., & Radenovic, S. 2018. Remarks on some fixed point results in b-metric spaces. Journal of Fixed Point Theory and Applications, 20(art.number:147). Available at: https://doi.org/10.1007/s11784-018-0626-2.
Aleksic, S., Mitrovic, Z.D., & Radenovic, S. 2019b. Picard sequences in b-metric spaces. Fixed Point Theory, 21(1), pp.35-46. Available at: https://doi.org/10.24193/fpt-ro.2020.1.03.
Ali, B., & Abbas, M. 2017. Existence Ulam-Hyers stability of fixed point problem of generalized Suzuki type («„^^-contractive multivalued operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 111(4), pp.1129-1146. Available at: https://doi.org/10.1007/s13398-016-0351 -x.
An, T.V., Dung, N.V., Kadelburg, Z., & Radenovic, S. 2015a. Various generalizations of metric spaces and fixed point theorems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 109, pp.175-198. Available at: https://doi.org/10.1007/s13398-014-0173-7.
An, T.V., Tuyen, L.Q., & Dung, N.V. 2015b. Stone-type theorem on b-metric spaces and applications. Topology and its Applications,185-186, pp.5064. Available at: https://doi.org/10.1016Zj.topol.2015.02.005.
Asl, J.H., Rezapour, Sh., & Shahzad, N. 2012. On fixed points of a-xp-contractive multifunctions. Fixed Point Theory and Applications, art.number:212. Available at: https://doi.org/10.1186/1687-1812-2012-212.
<u
00 CD
"o
>
o~ CM o CM
of
UJ
a.
z)
o
o <
o
X
o
UJ
H
>-
OH <
H
<
cd >o
X UJ H
O
O >
Q
Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundaments Mathematicae, 3, pp.133181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181.
Berinde, V. 1993. Generalized contractions in quasimetric spaces. In: Seminar on Fixed Point Theory, Babe§-Bolyai University, Cluj-Napoca, pp.3-9.
Berinde, V. 1996. Sequences of operators and fixed points in quasimetric spaces. Studia Universitatis Babe§-Bolyai Mathematica, 41(4), pp.23-27.
Berinde, V. 1997. Contractu generalizate §i aplicaiii. Baia Mare: Editura Club Press (in Romanian).
Bhaskar, T.G., & Lakshmikantham, V. 2006. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis: Theory, Methods &Applications, 65(7), pp.1379-1393. Available at: https://doi.org/10.1016Zj.na.2005.10.017.
Bota, M.F., Ilea, V., Karapinar, E., & Mleçniî, O. 2015. On a„-tp-contractive multivaalued operators in b-metric spaces and applications. Applied Mathematics & Information Sciences, 9(5), pp.2611-2620 [online]. Available at: http://www.naturalspublishing.com/Article. Asp?ArtcID=9576 [Accessed: 21 May 2020].
Chifu, C., & Petruçel, G. 2014. Fixed points for multivalued contractions in ô-metric spaces with applications to fractals. Taiwanese Journal of Mathematics, 18(5), pp.1365-1375. Available at: https://doi.org/10.11650/tjm.18.2014.4137.
Czerwik, S. 1993. Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1), pp.5-11 [online]. Available at: https://dml.cz/dmlcz/120469[Accessed: 21 May 2020].
Ciric, Lj., Abbas, M., Rajovic, M., & Ali, B. 2012. Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metric. Applied Mathematics and Computation, 219(4), pp.1712-1723. Available at: https://doi.org/10.1016/j. Amc.2012.08.011.
Fréchet, M. 1905. La notion d'écart et le calcul fonctionnel. Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, 140(1), pp.772-774 [online]. Available at: https://www.biodiversitylibrary.org/item/31541#page/778/mode/1up (in French) [Accessed: 21 May 2020].
Haghi, R.H., Rezapour, Sh., & Shahzad, N. 2011. Some fixed point generalisations are not real generalizations. Nonlinear Analysis: Theory, Methods & Applications, 74(5), pp.1799-1803. Available at: https://doi.org/10.1016/j.na.2010.10.052.
Hussain, N., Doric, D., Kadelburg, Z., & Radenovic, S. 2012. Suzuki-type fixed point results in metric type spaces. Fixed Point Theory and Applications, art.number:126. Available at: https://doi.org/10.1186/1687-1812-2012-126.
Hyers,D.H. 1941. On the stability of the linear functional equation. In: Proceedings of the National Academy of Sciences of the USA, 27(4), pp.222224, April 15. Available at: https://doi.org/10.1073/pnas.27.4.222.
Karapinar, E., Mitrovic, Z.D., Ozturk, A., & Radenovic, S. 2020. On a theorem of Ciric in b-metric spaces. Rendiconti del Circolo Matematico di palermo Series 2. Available at: https://doi.org/10.1007/s12215-020-00491-9.
Khojasteh, F., Abbas, M., & Costache, S. 2014. Two new types of fixed fe
point theorems in complete metric spaces. Variational Analysis, Optimization, 06
CO
E
and Fixed Point Theory, art.ID325840. Available 3 at: https://doi.org/10.1155/2014/325840. °
Kirk, W., & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Cham, Switzerland: Springer International Publishing.
Latif, A., Parvaneh, V., Salimi, P., & Al-Mazrooei, A.E. 2015. Various Suzuki type theorems in b-metric spaces. Journal of Nonlinear Sciences and | Applications (JNSA), 8(4), pp.363-377.
Availableat: https://doi.org/10.22436/jnsa.008.04.09.
Lazar, V.L. 2012. Ulam-Hyers stability for partial differential inclusions. Electronic Journal of Qualitative Theory of Differential Equations, 21, pp.1-19. | Available at: https://doi.org/10.14232/ejqtde.2012.1.21.
Michael, E. 1956. Continuous selections. I. Annals of Mathematics, 63(2), pp.361-382. Available at: https://doi.org/10.2307/1969615.
Mitrovic, Z.D. 2019. A note on the results of Suzuki, Miculescu and Mihail. Journal of Fixed Point Theory and Applications, 21(art.number:24). Available at: https://doi.org/10.1007/s11784-019-0663-5.
Mitrovic, Z.D., Parvaneh, V., Mlaiki, N., Hussain, N., & Radenovic, S. 2020. On some new generalizations of Nadler contraction in b-metric spaces. Cogent Mathematics and Statistics, 7(1), art.number:1760189. Available at: https://doi.org/10.1080/25742558.2020.1760189.
Mohammadi, B., Dinu, S., & Rezapour, Sh. 2013. Fixed points of Suzuki type quasi-contractions. U.P.B. Scientific Bulletin, Series A,75(3), pp.3-12 [online]. Available at: https://www.scientificbulletin.upb.ro/rev_docs_arhiva/fulld89_199620.pdf.
[Accessed: 21 May 2020].
Nadler, Jr. S.B. 1969. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2), pp.475-488. Available
at: https://doi.org/10.2140/pjm.1969.30.475.
Nieto, J.J., & Rodriguez-Lopez, R. 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order, 22, pp.223-239. Available at: https://doi.org/10.1007/s11083-005-9018-5.
Nieto, J.J., & Rodriguez-Lopez, R. 2007. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica, English Series, 23, pp.2205-2212. Available at: https://doi.org/10.1007/s10114-005-0769-0.
Pacurar, M. 2010. A fixed point result for ^-contractions on b-metric spaces without the boundedness assumption. Fasciculi Mathematici, 43, pp.127-137 [online]. Available at: http://www.math.put.poznan.pl/artykuly/FM43(2010)-PacurarM.pdf [Accessed: 21 May 2020].
Petru, T.P., Petrusel, A., & Yao, J.-C. 2011. Ulam-Hyers stability for operatorial equations inclusions via nonself operators. Taiwanese Journal of Mathematics, 15(5), pp.2195-2212. Available
at: https://doi.org/10.11650/twjm/1500406430.
00 CD
"o
>
o~ CM o CM
of
UJ
a.
z)
o
o <
o
X
o
LU
H
^
OH <
H
<
cd >Q
X LU H
O
O >
Ran, A.C.M., & Reurings, M.C.B. 2004. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society, 132(5), pp.1435-1443. Available at: https://doi.org/10.1090/S0002-9939-03-07220-4.
Rhoads, B.E. 2015. Two new fixed point theorem. Gen. Math. Notes, 27(2), pp.123-132 [online]. Available
at: https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/12_GMN-7082-V27N2.154194604.pdf[Accessed: 21 May 2020].
Rus, I. A. 2001. Generalized contractions and applications. Cluj: University Press.
Rus, I. A. 2009. Remarks on Ulam stability of the operatorial equations. Fixed Point Theory, 10(2), pp.305-320 [online]. Available
at: http://www.math.ubbcluj.ro/~nodeacj/vol_10(2009)_no_2.php[Accessed: 21
May 2020].
Rus, I. A., Petru§el, A., & STntámárian, A. 2003. Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Analysis: Theory, Methods & Applications, 52(8), pp.1947-1959. Available at: https://doi.org/10.1016/S0362-546X(02)00288-2.
Samet, B., Vetro, C., & Vetro, P. 2012. Fixed point theorems for a-ip-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), pp.2154-2165. Available at: https://doi.org/10.1016Zj.na.2011.10.014.
Singh, S.L., & Prasad, B. 2008. Some coincidence theorems and stability of iterative procedures. Computers & Mathematics with Applications, 55(11), pp.2512-2520. Available at: https://doi.org/10.1016/j.camwa.2007.10.026.
Suzuki, T. 2008. A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(5), pp.1861-1869. Available at: https://doi.org/10.1090/S0002-9939-07-09055-7.
Ulam, S.M. 1964. Problems in Modern Mathematics. New York, NY: John Wiley and Sons.
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д Университет им. Короля Сауда, Колледж физико-математических наук, кафедра математики, Эр-Рияд, Королевство Саудовская Аравия
® Белградский университет, Машиностроительный факультет, г. Белград, Республика Сербия
РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА;
27.25.17 Метрическая теория функций, 27.39.27 Нелинейный функциональный анализ ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: В данной статье представлены совпадения и общие неподвижные точки многозначного отображения типа Судзуки в Ь-метрических пространствах.
Методы: Обсуждаются предельные свойства, корректность и устойчивость решенийзадачс неподвижной точкой таких отображений по методу Улама-Хайерса.
Результаты: Получена верхняя граница расстояния Хаусдорфа между неподвижными точками множеств. В качестве доказательства полученных результатов, в статье приведено несколько примеров.
Выводы: Применение полученных результатов доказывает существование дифференциальных включений.
Ключевые слова: Ь-метрические пространства, многозначные отображения, неподвижная точка и задачи, Улам-Хайерс стабильность, начальная задача.
X
« РЕШЕНА И УЛАМ-ХИЕРОВА СТАБИЛНОСТ
Ф ДИФЕРЕНЦИJАЛНИХ ИНКЛУЗИJА, УК^УЧУ1УЪИ СУЗУКИJЕВЕ
ВРСТЕ ВИШЕЗНАЧНОГ ПРЕСЛИКАВА^А НА Ь-МЕТРИЧКИМ ® ПРОСТОРИМА
° Мууахид Абаса, Басит Алиб, Талат Назирв, Небоша М. Дедови^1",
Бандар Бин-Мохсинд, Сто]ан Н. Раденови^ а Владин колед - Универзитет, Одсек математике, Лахоре, Исламска Република Пакистан; ш Универзитет у Претории, Одсек за математику и приме^ену
Е математику, Претори]а, иужноафричка Република
> Ф
О б Универзитет за менадмент и технологи]у, Одсек за математику,
^ Лахоре, Исламска Република Пакистан
^ в Универзитет СОМЭАТЭ у Исламабаду, Одсек за математику,
Кампус Аботабад, Исламска Република Пакистан; ^ Универзитет иужне Африке, Одсек за математичке науке,
ш Научни кампус, иоханесбург, иужноафричка Република
>_ г Универзитет у Новом Саду, По^опривредни факултет,
Департман за по^опривреду технику, Нови Сад, Република Срби]а, аутор за преписку
д Универзитет кра^а Сауда, Колед природно-математичких наука, Одсек за математику, Ри]ад, Кра^евина Сауди]ска Арабка,
4 Универзитет у Београду, Машински факултет, ^ Београд, Република Срби]а
О
— ОБЛАСТ: математика
V
О ВРСТА ЧЛАНКА: оригинални научни рад
ш Сажетак:
н
§ Увод/цил: У раду су представлене коинцидентне и за]едничке
о фиксне тачке Сузуки]еве врсте вишезначног пресликаваъа на
Ь-метричким просторима.
Методе: Анализирана су гранична своства, добра поставленост и Улам-Хиерова стабилност решена за фиксни проблем вишезначних пресликаваъа.
Резултати: Доби}ена }е горъа граница Хауздорфовог расто]ак>а измену фиксних тачака скупова. Наведени су примери ко\и подржава}у доби}ене резултате.
Заклучак: Применом представлених резултата установлена }е егзистенци}а диференццалне инклузи]е.
Клучне речи: Ь-метрички простори, вишезначно пресликаваъе, фиксна тачка и проблеми, Улам-Хиерова стабилност, почетни проблем.
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Paper received on / Дата получения работы / Датум приема чланка: 25.05.2020. Manuscript corrections submittedon / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 03.06.2020.
Paper accepted for publishingon / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 05.06.2020. ш
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© 2020 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier a.
(www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
© 2020 Авторы. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «CreativeCommons» (http://creativecommons.org/licenses/by/3.0/rs/).
© 2020 Аутори. Обjавио Воjнотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). Ово jе чланак отвореног приступа и дистрибуира се у складу са Creative Commons licencom (http://creativecommons.org/licenses/by/3.0/rs/)..
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