CC BY
79
ANSARI'S METHOD IN GENERALIZATIONS OF SOME RESULTS IN THE FIXED POINT THEORY: SURVEY
http://dx.doi.org/10.5937/vojtehg66-16045
Introduction
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Tatjana M. Dosenovic3, Stojan N. Radenovicb
a University of Novi Sad, Faculty of Technology, Novi Sad, Republic of Serbia, e-mail: [email protected], '§_
ORCID ID: (E http://orcid.org/0000-0002-3236-4410 -o
b University of Belgrade, Faculty of Mechanical Engineering, x
Belgrade, Republic of Serbia, e-mail: [email protected],
ORCID ID: http://orcid.org/0000-0002-7417-1342
e
e
FIELD: Mathematics
ARTICLE TYPE: Original Scientific Paper |
ARTICLE LANGUAGE: English
o
Abstract: £
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The aim of this paper is to show that the C-class function introduced by "S A. H. Ansari is a powerful weapon for the generalization of many ü
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n e
important results in the theory of fixed points. Keywords: C class function, Metric space, Cauchy sequence, Common & fixed point, Fixed point.
d o h t e
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With the advent of S. Banach paper (Banach, 1922), the development of the theory of fixed point is moved upwards. A huge number of scientists, for more than 90 years, have managed to generalize Banach contraction principle (Abbas & Jungck, 2008, pp.416420), (Altun et al, 2010, pp.2238-2242), (Boyd & Wong, 1969, pp.458464), (Doric, 2009, 1896-1990), (Geraghty, 1973, pp.604-608), (Amini-Harandi & Emami, 2010, pp.2238-2242), (Hussain et al, 2013), £
ACKNOWLEDGMENT: The first author is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia (Matematicki modeli nelinearnosti, neodredenosti i odlucivanja, 174009) and from the Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina, Republic of Serbia, project no. 142-451-2838/2017-01.
CM <1>
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(Harjani & Sadarangani, 2009, 3403-3410), (Jachymski, 2011, pp.768774), (Jungck, 1976, pp.261-263), (Karapinar & Salimi, 2012), (Khan et al, 1984, pp.1-9), (Liu et al, 2015), (Rhoades, 1977, pp.257-290), ^ (Rhoades, 2001, pp.2683-2693), (Radenovic & Kadelburg, 2010, pp.1776-1783), (Radenovic et al, 2012, pp.625-645), (Salimi et al, 2013), (Samet et al, 2012, pp.2154-2165). In 2014 A. H. Ansari (Ansari, 2014a, pp.373-376), (Ansari, 2014b, pp.377-380) introduced the concept of C -qc class functions which cover a large class of contractive conditions, see E also (Ansari, 2014a, pp.373-376), (Ansari, 2014b, pp.377-380), (Ansari
§ et al, 2017, pp.2657-2673), (Ansari & Chandok, 2016, pp.65-71).
o
£ Definition 1 (Ansari et al, 2017, pp.2657-2673) A C -class function
x is a continuous function F : [0, œ)x[0, œ)^ R such that for any UJ X, y e [0, œ), the following conditions hold:
£ ( C1) F(x, y) < x ;
^ (C2) F(x,y) = x implies that either x = 0 or y = 0 .
An extra condition on F that F(0,0) = 0 could be imposed in some cases if required. By C we will denote the class of all C - functions.
^ Example 1 (Ansari et al, 2017, pp.2657-2673) The following
KI functions belong to the class C :
>o
1. F (x, y) = x - y. o 2. F(x, y) = mx, for some m e (0,1).
—> x
> 3. F(x, y) =- for some r e (0,w).
v (i + y)r ^ , )
4. F(x, y)=log(y + ), for some a >1.
1 + y
5. F(x, y) = (x + /)(1/(1+y)r) -1, l >1, for r e (0, w).
6. F (x, y ) = x — y
k + y
7. F(x, y) = x-^>(x), where cp :[0, w) ^ [0, w) is a continuous function such that p(t) = 0 if and only if t = 0.
8. F (x, y) = ^ ln(1 + xn).
We start this section with the following definitions and notions:
Definition 2 (Ansari, 2014b, pp.377-380) A mapping F: [0,+w)2 ^ R has a property CF, if there exists an CF > 0 such that § (Cp 1) F(x, y)> Cp implies x > y; S
(Cp2) F(y, y)< Cp, for all y e [0,+w). For more examples of C - class functions that have the property CF •> see (Ansari, 2014b, pp.377-380) Here we announce the following three w examples
a) F(x, y) = x - y,CF = r, r e [0,+w);
b) F (x, y )= x-i2-^ ,Cf =0; &
1 + y
(i) y is non-decreasing and continuous;
(ii) < is non-decreasing and continuous;
(iii) y(t) - F(y (s), <(s)) >0 for all t >0 and s = t or s = 0.
cm
Ci
c) F(x,y) = —^—,k > 1,CF = —,r > 2. V 7 1 + ky F 1 + k
Let ¥ denote the class of all functions y:[0, w) ^ [0, w), O
denote the class of all functions <: [0, w) ^ [0, w) and F elements of C s
<u
satisfying the following conditions: e
o tn
M—
O </) c
o +-<
ro
.n
1Ë <u c <u
The condition (iii) generalizes (2.3) from (Karapinar & Salimi, 2012,
p.9).
Definition 3 (Ansari et al, 2017, pp.2657-2673j A subclass of type e I is a function H : R x [0, w) ^ R if it is continuous and
s > 1 implies H(1, t)< H(s, t) for all t e [0, w).
Example 2 (Ansari et al, 2017, pp.2657-2673) We have the }o following functions of the subclass of the type I: °
<u
• H(s,t)= (t + l)s, l >1, °
H (s, t)= (s +1), l >1, H(s, t)= stn, n e N,
CM <1>
CD CD
O >
O CM
QC LLJ
QC ZD
o
o <
o 2:
X
o
LU
I— >-
CC <
H (5, t ) = st, H (s, t ) = t,
H (s, t ) = i±l t, H (s, t ) = ^S±! t,
H (s, t ) =
3
n
Z
i=0
n +1
v f
H (s, t ) =
n +1
-+1
l >1.
< -J
CD >Q
X LU I— O z
—3
O >
Definition 4 (Ansari et al, 2017, pp.2657-2673) We say that the pair (F, H) is an upclass of the type I if F : [0,») x [0,») ^ R is continuous, H is a function of the subclass of the type I and satisfies:
(1) 0 < x < 1 implies F(x,y)< F(l,y)
(2) H(l,yi)< F(x,y2) implies y < xy2, for all x,y,yi,y2 e[0,rc).
Example 3 (Ansari et al, 2017, pp.2657-2673) Below are listed the functions of the upclass of the type I, for all s e R, t, y e [0, rc),
x e [0,l]:
ni
t
n-i
i=0
• H(s,t)= (t + l)s, l >1,F(x,y) = xy +1,
• H(s,t)= (t + l)s, l >1,F(x,y) = (1 + lf,
• H (s, t) = stn, F (x, y)= xnyn,
• H (s, t ) = st, F (x, y ) = xy.
Definition 5 (Ansari et al, 2017, pp.2657-2673) We say that the pair (F, H) is a special upclass of the type I if F : [0,») x [0,») ^ R is continuous, H is a function of the subclass of the type I and satisfies:
(1) 0<x < 1 implies F(x,y)< F(l,y)
(2) H(1,t)< F( y) implies t <,
Example 4 (Ansari et al, 2017, pp.2657-2673) The following functions are a special upclass of the type I, for all s e R, t, x, y e [0,w):
• H(s, t)=(( +1Y, l > 1, F(x, y)= xmyk +l,
• H (s, t ) =(( +1 )tk, l > 1, F (x, y )=(1 + lfyk,
• H(s, t) = sntk, F(x, y) = xpyk,
• H (s, t) = st, F(x, y) = xy.
Remark 1 (Ansari et al, 2017, pp.2657-2673) Every pair (F, H) of the upclass of the type I also belongs to the class of a special upclass of the type I, but converse is not true.
Assertions similar to the following lemma were used (and proved) in the course of proofs of several fixed point results in various papers (Radenovic et al, 2012, pp.625-645).
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01
CD Ol
x < 1 implies Fx, yj < F (1, yj g-
y,
£
for all y, t e [0,»).
w
o <u
o cp
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CD
<u E o
o
ro
c <u
CT
T3
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Lemma 1 (Radenovic et al, 2012, pp.625-645) Let (X,d) be a metric space and let {xn} be a sequence in X such that
lim d (x„, x„+1) = 0.
If {xn} is not a Cauchy sequence, then there exist s>0 and two
sequences {mk} and {nk} of positive integers such that nk > mk > k and s
the following sequences tend to s+ when k ^ w: q
d(xmk , xnt J, d(xmk , xnk +1 ) d(xmk-!,xnk Jl d(xmk-1, \, +1 ^
™ Definition 6 (Abbas & Jungck, 2008, pp.416-420) Let f and g be
self maps of a set X. If c = fx = gx for some x e X, then x is called a § coincidence point of f and g, and c is called a point of coincidence of I f and g. The pair f, g of self maps is weakly compatible if they commute at their coincidence points.
o CM
DC Proposition 1 (Abbas & Jungck, 2008, pp.416-420) Let f and g
Ôç be weakly compatible self maps of a set X. If f and g have a unique § point of coincidence c = fx = gx, then c is the unique common fixed < point of f and g.
X
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>-
CO
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Main results
Previously described functions attracted the attention of authors and now there are various generalizations of the results from the fixed point theory, not only in a metric space, but also in the partial metric spaces, metric like-spaces, G-metric spaces,...(Ansari, 2014a, pp.373-376), (Ansari, 2014b, pp.377-380), (Ansari et al, 2017, pp.2657-2673), (Isik et al, 2015, pp.703-708), (Ansari & Chandok, 2016, pp.65-71). In this paper, fi we will present some of these results. Also, we shall prove some new ^ results, which generalize already known ones, by using the C - class y functions introduced recently by A.H. Ansari (Ansari, 2014a, pp.373-376), (Ansari, 2014b, pp.377-380). In this review paper, we will use only C-class functions.
Our first (probably new) result is the following:
Theorem 1 Let (X, d ) be a complete metric space. Suppose that the mappings f, g : X ^ X satisfy the following condition
W(d (fx, fy )) < F {¥(d (gx, gy )), ¿(d (gx, gy ))) (1)
for all x, y e X where and F e C. If the range of g
contains the range of f and f (X) or g(X) is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.
Remark 2 Putting in (1): y(t) = t,#(t) = (l -k)t, k e(0,1) and F(5, t) = s -1 for all s, t e[0,») we get a well-known Jungck's result from (Jungck, 1976, pp.261-263). Hence, Theorem 1 is a genuine generalization of the old and important Jungck's result in several $ directions (see all assumptions in (Jungck, 1976, pp.261-263).
Further, putting in (1): y(t) = t,F(s,t) = fi(s)-s for all s, t e [0,»)
where ^:[0,») ^ [0,1) such that tn ^ 0+ whether f3(tn l-, g = IX ^
(identity mapping on X), we get a well-known Geraghty type result (Geraghty, 1973, pp.604-608). Hence, Theorem 1 is a new generalization of this old and important result in the fixed point theory in the framework of complete metric spaces.
Proof. Let us prove that the point of coincidence of f and g is unique in the case that it exists. Suppose that o and S are two distinct points of coincidence of f and g. From this follows that there exist two points uo and uS such that fuo = guo = o^S = fuS = guS. Now, (1) implies
y(d (o S)) = y (d (fuo, fus))
F (y(d (gu
o, gu S
Md (guo, gus))) = F (y(d (o, S)), <t>(d (o, S))) < y(d (o, S)),
that is
F (y/(d (a, 8% 4{d (0, S))) = w{d (0, ô\ (2)
o
Ci T3
<u X
< Fdwdddgu ... gu S ) lédddgu ...guS))) °
ro
c <u
From (2) according to the property of the function F follows that "§ either y/(d(a,S)) = 0 or <f>(d(a,S)) = 0. In both cases, we get a contradiction. y>
Further, let x0 be an arbitrary point in X. Let us choose a point
x1 g X such that _y0 = fx0= gxv This can be done, since the range of g
contains the range of f. Continuing this process, having chosen xn in
o
X, we obtain xn+1 in X such that yn = fxn = gxn+1. Now consider the following two possible cases: q
10 yk = yk+1 for some k e N. Hence, gxk+1 = jXk+1 is a (unique) point of coincidence and then the proof of Theorem 1 is finished.
20 Thus, suppose that yn ^ yn+i for all n e N^{o}. In this case, we have
l(d (yn+i, yn+J) = l(d (.An+i, A+2 )) < F (yj(d (gx
n+1 gxn+2 Md (gx n+1 gxn+2 )))
= F ((d (yn, yn+i Md (yn, yn+i))) <l(d (yn, yn+i)).
g Since yeW we get that d (jn^ yn+2 )<d (yn, yn+i), ie-
5 d(yn, yn+i r > 0. We prove now that r = 0. Indeed, if r >0, then passing to the limit in the previous relation when n ^ œ, we obtain that
l(r )< F (l(r Mr ))<l(r ),
that is F (i(r ), ¿(r )) = i(r ). This implies that either y(r ) = 0 or ¿(r ) = 0. In both cases we get a contradiction. Hence, limn^œ d (yn, yn+i) = 0.
We next prove that {yn} is a Cauchy sequence in a complete metric space (X,d). If that is not case, then by using Lemma 1 we get that (rj there exist s>0 and two sequences {mk} and {nk} of positive integers q and sequences
| d(ymkynk ) diymkynk +i ) diymk) d^mk+i )
I-
0 all tend to s+ when k ^œ. By applying condition (1) to the elements
o x = xmk and y = xn,+i and since yn = jxn = gxn+i for each n > 0, we get
that
l(d (Vmkynk +i ))< F l(d (ymk_i ynk )Ji ¿{d\ymkI (3)
Letting k ^ œ in (3), we obtain
l(e)< F(i(^), ¿(^)), which is a contradiction because e > 0. This shows that {yn}= {fx„}= {gxn+i} is a Cauchy sequence in a complete metric space
(X, d )
Since g (X) is closed in a complete metric space (X, d), then it is a complete metric space. Therefore, there exists u, v e X such that v = gu
and lim«^»gxn = gu = v. We shall show that also fu = v = gu. Indeed, putting x = xn, y = u in (1) we get o
W(d {fa«,fu)) < F (y(d (gxn, gu),<(d (gxn, gu))). (4)
Letting n in (4) and applying the properties of all three functions F,y and <, we get
w(d (gu, fu)) < F (y(d (gu, gu)), <(d (gu, gu))) < y (d (gu, gu)) = y (o) = 0, i.e., fu = gu is a (unique) point of coincidence of the functions f and g. o
By the Proposition 1 f and g have the unique common fixed point.
In the case when f (X) is a closed subset in (X,d), the proof is
^ 2t
Putting y(t) = <(() = t,F(s,t) =-,g = IX the identity mapping of
1 + st
X in Theorem 1, we get the following result:
CD CM ci
Ci
<D £
if)
similar. 1
X
<u
Corollary 1 Let (X, d) be a complete metric space. Suppose e mappings f: X ^ X satisfies
d {fa fa) < (5) ta
1 + d {x, y) ^
for all x, y e X. Then f has a unique fixed point in X.
Putting y(t) = <(t) = t, F(s, t) = (p(s), where p :[0,») ^ [0,») is upper semicontinuous from the right, satisfying p(t)< t for t >0 as well o as p(0) = 0, g = IX the identity mapping of X in Theorem 1 we get the following well-known Boyd and Wong result (Boyd & Wong, 1969, pp.458-464).
Corollary 2 Let {X,d) be a complete metric space. Suppose that a mapping f : X ^ X satisfies the following condition o
d ffx, fy) <çfd fx, y)) (6)
for all x, y e X. Then f has a unique fixed point, say u e X and fnx ^ u as n ^ œ for each x e X.
Putting y/(f) = <(() = t, F(5, t) = s -p(s), where p:[0,») ^ [0,») is a continuous function such that p(t) = 0 if and only if t = 0, g = IX the IB identity mapping of X in Theorem 1 we get the following well-known
| B.E. Rhoades result (Rhoades, 2001, pp.2683-2693).
Corollary 3 Let (X,d) be a complete metric space. Suppose that oc the mappings f: X ^ X satisfies the following condition
df fy) < d(x, y)-p(d(x, y)) (7)
for all x, y e X. Then f has a unique fixed point, say u e X and y fnx ^ u as n ^^ for each x e X.
In the sequel of this section we shall consider two results which provide the existence of a coincidence point and a common fixed point £ for three mappings satisfying the generalized (F,^,<)- contractive ^ condition. These results are addressed in the following theorems.
Theorem 2 Let (X, d) be a metric space, and let f, g, S: X ^ X
ot be three mappings such that for all x, y e X
<
0 w(d ((x, gy)) < F {w(m(x, y)), <(m (x, y))), (8) for some < e O and F e C, where
1
o m(x, y) = max-jd (Sx, Sy), d (Sx, fx), d (Sy, gy), ^(((Sx, gy)+d (Sy, fx))\ O
> and
m (x, y) = max{d (Sx, Sy), d (fx, Sx), d (gy, Sy)}. If fX u gX c SX and S(X) is a complete subspace of (X, d) , then f, g and S have a unique point of coincidence. Moreover, if (f, S) and (g, S) are weakly compatible, then f, g and S havea unique common fixed point.
The proof of the following theorem is similar to that of Theorem 1.
Theorem 3 Let (X, d) be a complete metric space, and let f, g, S: X ^ X be three mappings such that for all x, y e X
wfd {fx, gy)) < F {y{m{x, y)), ({m- fx, y))), (9)
for some y e Y, p e O and F e C, where §
u{ y) e|d {x, y\ d {x, M d {y, gy), 2 {d {x, gy) + d fy, fx)) j
CO CM ci
Ci ^
CD
s
m{x, y ) = max jd {Sx, Sy ), 2 {{Sx, fx) + d {Sy, gy )), 2 {{Sx, gy ) + d {Sy, >))J
and
m1 {x, y ) = max{d {Sx, Sy ), d {fx, Sx), d {gy, Sy)} If fX u gX c SX and S(X) is a complete subspace of {X, d) , then f, g and S have a unique point of coincidence. Moreover, if {f, S) and {g, S) are weakly compatible, then f, g and S havea unique common fixed point. ë
The following results represent one other version of Altun Theorem (Altun et al, 2010, pp.310-316) in the terms of {F,y,p)-contractive mappings.
Theorem 4 Let (X, d) be a complete metric space, and let | f, g: X ^ X be two mappings such that for some yet, < e O and F e C and for all x, y e X there exists
ro .N
1Ë CD c CD
such that
y(d ((x, gy)) < F {¥ (u(x, y)), <(u(x, y))), (10)
then f and g have a unique fixed point.
Let (X ,4) be a partially ordered set. A pair (f,g) of self-maps of
X is said to be weakly increasing iff <gfx and gx 4 fgx for all x e X. There are examples (see Altun et al, 2010, pp.310-316) when neither of such mappings f,g is nondecreasing w.r.t < In particular the pair
(f,iX), (iX is the identity mapping on X) is weakly increasing if and
only if x 4jx for each x e X. Q
Theorem 5 Let (X, 4) be a partially ordered set and let there exist a metric d on X such that (X, d) is a complete metric space. Let (f, g)
OI <D
CD CD
O >
0
01
Q¿ LLJ
C£
z> O
—3
O >
be a weakly increasing pair of self-maps on X. Suppose that there exists and F g C such that for every two comparable elements
x, y g X,
yy(d Cfx, gy)) < F {¥(M (x, y)), 4M (x, y))), (11)
where
M (x, y) = max J d (x, y\d (x, fa\ d (y, gy) 2 (f (x, gy) + d fy, Jx)) I.
o Then in each of the following two cases the mappings f and g
< have at least one common fixed point:
o z
g (i) f or g is continuous, or
LLI ( )
£ (ii) if a nondecreasing sequence {xn} converges to x e X, then xn4x
for all n.
Proof. Using that the pair of functions (f,g) is weakly increasing, we can construct inductively, starting with an arbitrary x0 e X, a sequence {xn} such that xn 4 xn+1. Namely, denote:
o xi = fxo 4 gfxo = gxi,
o x2 = gxi 4 fgxi = x,
I X3 = jX2 4 gfX2 = gX3,
I-
o
and in genera|, x2n+1 = fx2n and x2n+2 = gx2n+1 .
Suppose first that xk = xk+1 for some k. Then, the sequence {xn} is constant for n > k. Indeed, let k = 2m. Then x2m = x2m+1 and we obtain from (11) that
¥(d(x2m+1, x2m+J) = (f^m , Sx2m+1 )) < (12)
< F((M(x2m , x2m+1 )), (m , x2m+J)), ( )
where
M(x2m , x2m+J = max{d(x2m , x2m+1 )> d(x2m , Am )> dfem^ Sx2m+1 )>
2 (d(x2m , gx2m+1 ) + d(x2m+1, fx2m ))}
= max-
0,0, d ( m+1, x2 m+2),1 {d (x2m, x2m+2) + 0) 1
= max|d(x2m+1, X2m+2 I ^ {((x2m , X2m+ 2 ) + 0)j
= d(x2m+1, x2m+2 )
Now further from (12) we get that
y/(d (x
2m+1, x2m+2 )) < F ((x
2m+1, x2m+2 Md (x 2m+1, x2m+2 ffi Md(x2m+1, x2m+2 ft
that is,
y/(d (x 2 m+1 , x2m+2 )) = F {y(d (x 2m+1 , x2m+2 )\Ád(x2m+1, x2m+ 2 E or equiValent;ly either W(d (x2m+1, x2m+2^=0 0r 4>(d (x2m+1, x2m+2 ^ = 0
i.e., d(x2m+1,x2m+2) = 0. Hence, if xk = xk+1,k = 2m we obtain that xk+1 = xk+2. Similarly, if k = 2m +1, one easily obtains that xk+1 = xk+2, and so the sequence {xn} is constant (starting from some k) and xk is a common fixed point of f and g.
Suppose now that xn ^ xn+1 for each n. We shall prove that d(xn, xn+1) ^ 0 as n ^o. Using condition (11) (which is possible since xn and xn+1 are comparable for all n), we obtain y/(d (x
2n+2,x2n+1
^ =^(d (gx
2n+1 fx2n))
= w(d (2n , gx2n+1 )) < F{y(M(x2n , x2n+1 ft 0(M(x2n , x2n+1 E
where
M(x2n , x2n+J
max{d(x2n , x2n+1 ^ d(x2n , fx2n )> d(x2n+1, gx2n+1\ 1 (d(x2n , gx2n+J + d(x2n+1, fx2n ))}
= max-|d(x2n , x2n+1 \ d(x2n , x2n+11 d(x2n+1, x2n+2 \ 1 d(x2n , x2n+2 )}
< max-|d(x2n , x2n+1 \ d(x2n+1, x2n+2 \ ^ (d(x2n , x2n+1 ) + d(x2n+1, x2n+2 ^ = max{d(x2n, x2n+1 ), d(x2n+1, x2n+2 )}
O
00
01
CO CM !± cp
cu £
o cu
o ü
T3
cu X
cu
cu E o
o
ro
.n lE
cu c cu
T3
o
-C
cu E №
■O >
o c cu >(« o Q
If d(x2n+1 , X2n+2) > d(X2n , X2nj>0, then ¡t f0ll0WS
y/(d (x
2n+2,X2n+1
)) < F ((d (x
2n+1, X2n+2 )\4dd (x 2n+1,X2n+2 № <V(d(x2n+1 , X2n+2)), or equiva|ent|y, either V{d(^m^^mj = 0 or $(dfem^X2m+2))=0-
This is a contradiction, because d(x2n+1, x2n+2)>0. Hence, M (x2n, x2n+1) = d (x2n, x2n+1). Now, we further easily get that
I d(xn+l, xn+2) <d(xn, Xn+J for al1 n G # ^(0). This means that o d(xn, xn+1) i r > 0 as n ^œ. Let r >0. Passing to the limit in the last inequality, when n ^ œ, we get
y/(r) < F (^(r) ,^(r))<^(r),
w i.e., F ((r ),^(r)) = ^(r), or equivalently, either ^(r) = 0 or ^(r) = 0. A contradiction. Hence, d(xn, xn+1) ^ 0 as n ^œ.
In order to prove that {xn} is a Cauchy sequence in (X, d) we shall use the ideas from (Radenovic et al, 2012, pp.625-645).
It is enough to prove that {x2n} is a Cauchy sequence. Suppose the
< contrary. Then, for some s>0 there exist subsequences {x2m(k)} and {x2n} of {x2n} such that n(k) is the smallest index satisfying
n(k)> m(k)> k and d{xm(k),xn(k))>e. |n Particu|ar, d(xm(k),xn(k)-2)< Now, using Lemma 1 and putting in (11) x = x2n(k ) , y = x2m(k )_1 (x and y are obviously comparable) we have
W(d (x2 n (k >1, X2m (k) ))=^(d (fX2n (k), gx2m(k )_1 )) <
< F [y(M (x2n(k), X2m(k hMM (x2n(k ), X2m(k )_1 ^ (13)
where
M (x2 n(k), X2m(k )_J 2n(k ) ' "2m(k H/'" \X2n(k) , fa 2n(k) P" ' 2m
CD
2 \X2n(k)
>o
X LU I— O
z —>
O >
= maX{d(x2n(k), X2m(k)_J, d(x2n(k), ffX2n(k)) d(x2m(k)_1, gX2m(k)_1 )
"2 ((x2n(k), gX2m(k)_1 ) + d(x2m(k)_1, fX2n(k)))} = max{d(x2n(k), X2m(k)_1 ) d(x2n(k), X2n(k)+1 ) d(x2m(k)_1, X2m(k))
1 {{{x2m{k) )+ d { m» J]
^ max j £,0,0,-2 { + £ I = s.
M{x2n , u) =max j d{x2n , uI d{x2n, fx2n I d({ gu{{{x2n , gu) + d{ f^nDi
O
00
01
CD CM ci
Ci
eu £
CO
Passing to the limit in (13) when k ^ œ, we obtain that
< F {y{£),j>{s))<¥{s), that is £ = 0, which is a contradiction. Hence, the sequence {xn} is a ° Cauchy sequence. Since {X, d) is a complete metric space it follows
o
that x ^ u for some element u e X. ^
n "O
cu
(i) Suppose that the mapping g is continuous. Since x2n+1 ^ u, * we obtain that x2n+2 = gx2n+1 ^ gu. On the other hand, x2n+2 ^ u (as a subsequence of {xn}). It follows that gu = u. To prove that fu = u, using u ^u we can put x = y = u in (11) and obtain that
y{d {>, gu)) < F {{M {u, u)), (M {u, u))) = F {y {d {u, fu)), p{d {u, fu))),
because M {u, u) = d {u, fu) We further have,
y{d {{u, u)) < F {y {d {u, fu)), p{d {u, fu))),
from which follows (as in the previously cases) that u = fu. The proof is similar if f is continuous.
(ii) Suppose now that the condition (ii) of the theorem holds. The sequence {xn} is nondecreasing w.r.t < and it follows that xn ^ u. Take x = x2n, and y = u (which are comparable) in (11) to obtain that e
W{d fe n, gu)) < F {y{M {x2n, u)) p{M {x2n, u)))
where
■O >
o c cu >№ o
^ max jo,0, d {u, gu ),2 d {u, gu )j = d {u, gu ). Q
Hence, we further obtain
¥{d , gu)) < F {y{d {u, gu)) p{d {u, gu)))
or passing to the limit, we get
y/(d (u, gu)) < F (p(d (u, gu)), </>(d (u, gu))). «3 From the last relation we get that gu = u.
The fact that u = /u is now derived in the same way as in the case (i). The theorem is proved.
Finally, we address the following definitions as well as maybe a new
CD
result.
oc Consider the following classes of functions from [0,») into itself:
Z)
o ® Y = {y : Y | is nondecreasing and lower semi - continuous },
O a = {a : a | is upper semi - continuous}, O p = {/ : / | is lower semi - continuous},
Also by using Ansari's method, one can prove that the following Theorem genuinely generalizes recent results from (Karapinar & Salimi, ¡5 2012) in several directions. Its proof is omitted.
Theorem 6 Let (X, 4) be a partially ordered set and let there exist a metric d on X such that (X, d) is a complete metric space and let < /: X ^ X be a nondecreasing selfmap. Assume that there exist ^ / eOr,a eOa, P e ® p and F e C such that for all s, t > 0,
t > 0 and (s = t or s = 0) implies y(t) - F(a(s\ p(s)) > 0,
CO
>o
w and
o
z
o for all comparable x, y e X, where
Y{d (fx, fy)) < F (a(M (x, y)), /(M (x, y)))
M (x, y) = max jd (x, y ), d (x, /X), d (y, fy ), -2 (d (x, fy ) + d (y, /X))j. Suppose that, either
(i) / is continuous, or
(ii) if a nondecreasing sequence {xn} converges to x e X, then xn=4x for all n e N^{0}
If there exists x0 e X such that x0^fx0, then / has a fixed point.
References
Abbas, M., & Jungck, G. 2008. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications, 341(1), pp.416-420. Available at: http://dx.doi.org/10.1016/j.jmaa.2007.09.070.
Ansari, A.H. 2014a. Note on "a - admissible mappings and related fixed point theorems", pp.373-376. In: The 2nd Regional Conference on Mathematics And Applications, PNU, September.
Ansari, A.H. 2014b. Note on "p-y - contractive type mappings and related fixed point", pp.377-38. In: The 2nd Regional Conference on Mathematics And Applications, PNU, September.
Ansari, A.H., & Chandok, S. 2016. Make unification on a - admissible mappings and related fixed point theorems. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 8(1), pp.65-71. Available at:
http://dx.doi.org/10.5937/spsunp1601065a.
Ansari, A.H., Dolicanin-Djekic, D., Dosenovic, T., & Radenovic, S. 2017. Coupled coincidence point theorems for (a-^-y-H-F) two sided-contractive type mappings in partially ordered metric spaces using compatible mappings. Filomat, 31(9), pp.2657-2673. Available at:
http://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/3628/2113.
Altun, I., Damjanovic, B., & Djoric, D. 2010. Fixed point and common fixed point theorems on ordered cone metric spaces. Applied Mathematics Letters, 23(3), pp.310-316. Available at:
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Amini-Harandi, A., & Emami, H. 2010. A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Analysis: Theory, Methods and Applications, 72(5), pp.2238-2242. Available at:
http://dx.doi.org/10.1016Zj.na.2009.10.023.
Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, pp.133-181. Available at: http://eudml.org/doc/213289.
Boyd, D.W., & Wong, J.S.W. 1969. On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), pp.458-464. Available at: http://dx.doi.org/10.1090/s0002-9939-1969-0239559-9.
Doric, D. 2009. Common fixed point for generalized (y,p) - weak contractions. Applied Mathematics Letters, 22(12), pp.1896-1990. Available at: https://doi.org/10.1016/j.aml.2009.08.001.
Geraghty, M.A. 1973. On contractive mappings. Proceedings of the American Mathematical Society, 40(2), pp.604-608. Available at: http://dx.doi.org/10.1090/s0002-9939-1973-0334176-5.
CM CD
CD CD
"o >
03
o CM
of
UJ
a.
Z) O
o <
o
X
o
LU
H ^
a. <
H
<
cd >o
X LU H O
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Harjani, J., & Sadarangani, K. 2009. Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis: Theory, Methods and Applications, 71(7-8), pp.3403-3410. Available at: http://dx.doi.org/10.1016Zj.na.2009.01.240.
Hussain, N., Karapinar, E., Salimi, P., & Akbar, F. 2013. a - admissible mappings and related fixed point theorems. Journal of Inequalities and Applications, 2013(1), p.114. Available at: http://dx.doi.org/10.1186/1029-242x-2013-114.
Jachymski, J. 2011. Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Analysis: Theory, Methods and Applications, 74(3), pp.768-774. Available at:
http://dx.doi.org/10.1016/j.na.2010.09.025.
Jungck, G. 1976. Commuting Mappings and Fixed Points. The American Mathematical Monthly, 83(4), pp.261-263. Available at: http://dx.doi.org/10.2307/2318216.
Karapinar, E., & Salimi, P. 2012. Fixed Point Theorems via Auxiliary Functions. Journal of Applied Mathematics, 2012, pp.1-9. Available at: http://dx.doi.org/10.1155/2012/792174.
Khan, M.S., Swaleh, M., & Sessa, S. 1984. Fixed point theorems by altering distances between the points. Bulletin of the Australian Mathematical Society, 30(1), pp.1-9. Available at:
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Liu, Z., Zhang, X., Ume, J.S., & Kang, S.M. 2015. Common fixed point theorems for four mappings satisfying y-weakly contractive conditions. Fixed Point Theory and Applications, 20(1). Available at: http://dx.doi.org/10.1186/s13663-015-0271-z.
Radenovic, S., & Kadelburg, Z. 2010. Generalized weak contractions in partially ordered metric spaces. Computers & Mathematics with Applications, 60(6), pp.1776-1783. Available at:
http://dx.doi.org/10.1016/j.camwa.2010.07.008.
Radenovic, S., Kadelburg, Z., Jandrlic, D., & Jandrlic, A. 2012. Some results on weakly contractive maps. Bull. Iran. Math. Soc, 38(3), pp.625-645. Available at: http://bims.iranjournals.ir/article_229.html.
Rhoades, B.E. 1977. A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society, 226, pp.257290. Available at: http://dx.doi.org/10.1090/s0002-9947-1977-0433430-4.
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Salimi, P., Latif, A., & Hussain, N. 2013. Modified a-y-contractive mappings with applications. Fixed Point Theory and Applications, 1, p.151. Available at: http://dx.doi.org/10.1186/1687-1812-2013-151.
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РЕЗУЛЬТАТОВ В РАМКАХ ТЕОРИИ НЕПОДВИЖНОЙ ТОЧКИ: ^
ОБЗОР Б
Татьяна М. Дошеновича, Стоян Н. Раденович6 ^
' Университет в г. Нови-Сад, Технологический факультет, ф
г. Нови-Сад, Республика Сербия, 3
6 Белградский университет, Машиностроительный факультет, м г. Белград, Республика Сербия
ОБЛАСТЬ: математика ВИД СТАТЬИ: оригинальная научная статья
ЯЗЫК СТАТЬИ: английский ^
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Цель данной статьи заключается в намерении показать, насколько функция С-класса, введенная А. Х. Ансари, является мощным средством для обобщения многих важных результатов в теории неподвижной точки.
Ключевые слова: «С» класс функций, метрическое § пространство, последовательность Коши, общая неподвижная точка, неподвижная точка.
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ПРИМЕНА АНСАРШЕВОГ МЕТОДА У ГЕНЕРАЛИЗАЦИИ НЕКИХ !
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РЕЗУЛТАТА ИЗ ТЕОРШЕ НЕПОКРЕТНЕ ТАЧКЕ (ПРЕГЛЕД)
Тат]ана М. Дошенови^8, Сто^ан Н. Раденовийб ^
а Универзитет у Новом Саду, Технолошки факултет,
Нови Сад, Република Срби]а, о
б Универзитет у Београду, Машински факултет; Београд, Република Срби]а
ОБЛАСТ: математика
ВРСТА ЧЛАНКА: оригинални научни чланак иЕЗИК ЧЛАНКА: енглески
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Къучне речи: С класа функци]а, метрички простор, Коши]ев низ, за}едничка непокретна тачка, непокретна тачка.
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