Some fixed point theorems in b2-metric spaces Текст научной статьи по специальности «Математика»

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DOI: 10.5937/vojtehg67-21221; https://doi.org/10.5937/vojtehg67-21221

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SOME FIXED POINT THEOREMS IN ^-METRIC SPACES

Poom Kumama, Zoran D. Mitrovicb, Mirjana V. Pavlovicc

a King Mongkut's University of Technology Thonburi, Faculty of Science, KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), </>

Science Laboratory Building, Bangkok, Kingdom of Thailand, e-mail: [email protected], o

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ORCID iD: http://orcid.org/0000-0002-5463-4581 University of Banja Luka, Faculty of Electrical Engineering Banja Luka, Republic of Srpska, Bosnia and Herzegovina, °

e-mail: [email protected], -a

ORCID iD: http://orcid.org/0000-0001-9993-9082 J

: University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Kragujevac, Republic of Serbia e-mail: [email protected], w

ORCID iD: http://orcid.org/0000-0001-6257-8666 rö"

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FIELD: Mathematics (Mathematics Subject Classification: primary 47H10, E

secondary 54H25) |

ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English

Abstract:

In this paper, we first prove a result that gives a sufficient condition for the convergence of the sequences in the b2-metric space. Next, we give some fixed point theorems in the b2-metric space. Some of our results are the corresponding generalizations of the known results in the b2-metric space, which is confirmed by some examples.

Keywords: fixed points, common fixed points, 2-metric space, b2-metric space.

ACKNOWLEDGMENT:

The second author is grateful for the financial support from the Ministry for Scientific and Technological Development, Higher Education and Information Society of Republika Srpska (Savremena istrazivanja u teoriji fiksne tacke: metricki i topoloski pristup, 1255007). The third author is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia (Metode numericke i nelinearne analize sa primenama, 174002).

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Introduction and Preliminaries

The applications of fixed point theorems are very important in diverse disciplines of mathematics, engineering and economics. The origin of the fixed theory is dated to the last quarter of the nineteenth century. The work of S. Banach in 1922 known as the Banach contraction principle is the starting point of the metric fixed point theory.

More on fixed point results and contractive conditions, the reader can find in (Ciric, 2003), (Agarwal et al, 2015), (Kirk & Shahzad, 2014).

Theorem 1.1 (Banach contraction principle) Let (X, d ) be a complete metric space. Let T be a contractive mapping on X, that is, one for which there is a As [0,1) satisfying

d (Tx, Ty )< Ad (x, y )

for all x, y s X. Then, there exists a unique fixed point x s X of T.

This theorem is a forceful tool in the nonlinear analysis. It has many applications and has been extended by a great number of authors. Although the famous Banach contraction principle was proved in the metric space, after 1990 many new modifications of the definition of the metric space appeared.

From now on, R and N will denote the set of real numbers and natural numbers, respectively. Let us recall the definitions of the b-metric spaces, the rectangular b-metric spaces, the 2-metric spaces, and the £>2-metric spaces.

In the papers of Bakhtin (Bakhtin, 1989, pp.26-37) and Czerwik (Czerwik, 1993, pp.5-11), the notion of the b-metric space was introduced and some fixed point theorems for single-valued and multivalued mappings in the b-metric spaces were proved.

Definition 1.2 Let X be a nonempty set and s > 1 a given real number. The function d: X x X ^ [0, ro) is said to be a b-metric if for all x, y, z s X the following conditions are satisfied:

• d(x, y) = 0 if and only if x = y;

• d(x, y) = d(y, x);

• d(x, z)< s[d(x, y) + d(y, z)].

A triplet (X,d, s) is called a b-metric space with the coefficient s.

In the paper (George et al, 2015, pp. 1005-1013), the authors introduced the concept of a rectangular b-metric space, which is not necessarily Hausdorff and which generalizes the concept of the metric space, the rectangular metric space (RMS) and the b-metric space.

Definition 1.3 (George et al, 2015, pp.1005-1013) Let X bea nonempty set and s > 1 a given real number. The function d : X x X ^ [0, ro) is said to be a rectangular b-metric if the following conditions are satisfied:

(RbM1) d(x, y)= 0 if and only if x = y;

(RbM2) d(x, y) = d(y, x) for all x, y s X ;

(RbM3) d(x,y)< s[d(x,u) + d(u,v) + d(v,y)] forall x,y s X and all distinct points u, v s X \{x, y}.

A triplet (X, d, s) is called a rectangular b-metric space with the

coefficient s (in short RbMS).

Also in (George et al, 2015, pp.1005-1013), the concept of convergence in such spaces is similar to that of standard metric spaces.

In the paper (Gahler, 1963, pp.115-118), Gahler introduced the concept of the 2-metric space.

Definition 1.4 (Gahler, 1963, pp.115-118) Let X be a nonempty set and the mapping d : X x X x X ^ R satisfies:

(1) For every pair of distinct points x, y s X, there exists a point

z s X such that d(x, y, z) 0.

(2) If at least two of three points x, y, z are the same, then d (x, y, z )= 0

(3) The symmetry:

d(x, y, z) = d(x, z, y ) = d(y, x, z) = d(y, z, x) = d(z, x, y ) = d(z, y, x)

for all x, y, z s X .

(4) The rectangle inequality:

d(x, y, z) < d(x, y, t) + d(y, z, t) + d(z, x, t) for all x, y, z, t s X .

Then d is called a 2-metric on X and (X, d) is called a 2-metric space.

Many generalizations of the concept of metric spaces are established, and several papers are published on the topic of the b-metric spaces (see (Aleksic et al, 2018), (Aydi, 2016, pp.2417-2433), (Czerwik, 1993, pp.5-11), (Czerwik, 1998, pp.263-276), (Dung & Le Hang, 2016, pp.267-284), (Miculescu & Mihail, 2017, pp.2153-2163) and others), of the rectangular b-metric spaces, see (George et al, 2015, pp.1005-1013), (Mitrovic & Radenovic, 2017, pp.3087-3095), (Mitrovic & Radenovic, 2017, pp.401-407) and others) and of the 2-metric spaces, see (Ahmed, 2009, pp.2914-2920), (Aliouche & Simpson, 2012, pp.668-690), (Deshpande & Chouhan, 2011,

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pp.37-55), (Dung & Le Hang, 2013), (Fadail et al, 2015, pp.533-548), (Freese et al, 1992, pp.391-400), (Gahler, 1963, pp.115-118), (Iseki, 1975, 133-136), (Iseki, 1976, pp.127-135), (Lahiri et al, 2011, pp.337-£ 352), (Lal & Singh, 1978, pp.137-143), (Naidu & Prasad, 1986, pp.974-| 993), (Popa et al, 2010, pp.105-120) and others).

In the paper (Mustafa et al, 2014), the notion of the £>2-metric space à was introduced and some fixed point theorems in the 62-metric spaces of proved.

Definition 1.5 (Mustafa et al, 2014) Let X be a nonempty set, s > 1 be a real number and let d : X x X x X ^ R satisfies:

(1) For every pair of distinct points x, y e X, there exists a point

o z e X such that d (x, y, z ) 0 .

HN (2) If at least two of three points x, y,z are the same, then

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d(x, y, z)= 0 . & (3) The symmetry:

d(x, y, z) = d(x, z, y ) = d(y, x, z ) = d(y, z, x) = d(z, x, y ) = d(z, y, x) for all x, y, z e X .

(4) The rectangle inequality: | d(x, y, z) < s[d(x, y, t) + d(y, z, t) + d(z, x, t)]

ej for all x, y, z, t e X.

o Then d is called a £>2-metric on Xand (X,d,s) is called a 62_metric

HN space.

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Remark 1.6 Note that, d(x, y, z) > 0 for all x,y, z e X . Applying the ^ rectangle inequality, we get

d(x, y, y ) < s[d(z, y, y) + d(x, z, y ) + d(x, y, z)].

By (2) and the symmetry of d, we obtain d(x,y, z)> 0. Note that a 2-metric space is included in the class of the £>2-metric spaces with the coefficient s = 1. Example 1 (Mustafa et al, 2014)

1. Let X = [0,+<») and d(x, y, z)=(xy + yz + zx)p if x ^ y ^ z ^ x, and otherwise d(x, y, z) = 0, where p > 1 is a real number.

From convexity of the function f (x)= xp for x > 0, then by Jensen inequality we have

(a + b + c)p < 3p-1 (ap + bp + cp).

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So, {X, d, s) is a b2-metric space with s < 3p-1.

2. Let a mapping d : R3 ^ [0,+ro) be defined by g

d{x, y, z) = min jx - y|, \y - z\, \z - x|}.

Then d is a b2-metric on R, i.e., the following inequality holds: |

d{x, y, z) < d{x, y, t) + d{y, z, t) + d{z, x, t),

for all x, y, z, t e R. From the convexity of the function f {x)= xp on | [0,+ro) for p > 1, we obtain that

dp (x,y, z) = min jx - y|, |y - z|, |z - x|}p

is a b2-metric on R with s < 3p1. °

Definition 1.7 (Mustafa et al, 2014) Let jxn} be a sequence in a 62-metric space {X, d, s).

1. jxn} is said to be 62-convergent to x e X, written as ä

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limxn = x, if for all a e X,lim„^„ d{x„, x, a)= 0 . |

2. jxn} is said to be a 62-Cauchy sequence in X if for all

a e X, limn,md{xn, xm, a) = 0.

3. {X, d) is said to be b2-complete if every b2-Cauchy sequence is a | b2-convergent sequence.

Definition 1.8 (Mustafa et al, 2014) Let {X, d, s) be a b2-metric spaces and let f: X ^ X be a mapping. Then f is said to be b2-continuous at a point z e X if for a given s> 0, there exists 8 > 0 such that x e X and

d {z, x,a)<8 for all a e X imply that d{fz,fx,a)<s. The mapping f is b2-continuous on X if it is b2-continuous at all z e X.

Remark 1.9 Let {X,d) be b2-metric spaces. Then a mapping f : X ^ X is b2-continuous at a point x e X if and only if it is b2-sequentially continuous at x, that is, whenever jxn} is b2-convergent to x, jfxn} is b2-convergent to f(x).

This paper is to derive theorems of Banach, Reich and Jungck in the b2-metric spaces. Also, we obtain some results in partially ordered b2-metric spaces.

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One sequence convergence test in the b2-metric space

The inequality given in the next Lemma 2.1 is key to proving our main results.

3 Lemma 2.1 Let (X, d, s) be a b2-metric space and {xn} a

sequence in X. Suppose that A e (0,1) and let c be a real nonnegative ™ number such that

d(xm, xn, a) < Ad(xm_i, xn_i, a) + c(xm + An ) (2.1)

for all m, n e N and a e X . Then {xn} is a b^Cauchy sequence in X. Proof. Let m,n e N, a e X and p e N fixed such that p > -2logA s. From (2.1) we have that

d(xm+k, xn+k, a)<Akd(xm, xn, a)+ ckAk (r + An) (2.2)

for all m, n, k e N and a e X . We have

d (xm , xn , a)< s[ d (xm+ p , xn , a) + d (xm , xm+p , a) + d (xm , xn , xm+p ) ]

< S[ d(xm+p , xn , a) + Amd(x0 , xp , a)

+ cmAm(1 + Ap) + Amd(x0,xn,xp) + cmAm(1 + Ap) ]

< s[ d(xm+ p , xn , a) + Amd(x0 , xp , a) + Amd(x0 , xn , xp ) ]

+ 4scmAm. Next, as it is

Q d (x0, xn , xp )< s[ d (xn+p , xn , xp ) + d (x0, xn+p , xp ) + d (x0, xn , xn+p ) ]

^ < s[ And(xp,x0,xp) + cnAn(Ap + l)+Apd(x0,xn,x0)

> + cpAp(1 + An) + And(x0,x0,xp) + cnAn(1 + Ap) ]

< 2cs(2nAn + p), and

d(xm+ p , xn , a)< s[ d(xn+ p , xn , a) + d(xm+ p , xn+ p , a) + d(xm+p , xn , xn+ p ) ]

< s[ And(xp, x0, a) + cnAn (1 + Ap)+ Apd(xm, xn, a) + cpAp (Am +An) + Apd(xm, xn, xn) + cpAp (Am +An) ]

< s[ And(xp, xCJ,a) + Apd(xm, xn,a) ] + 2cs[p(Am + An) + nAn ],

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d(xm,xn,as2[ Ad(xp,x0,a) + Apd(xm,xn,a) + sXmd(x0,xp,a) ] £

+ 2cs2 [ [m + An) + nAn ] + 2s2cAm (2nAn + p). £

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So,

(l - s2 Ap )d(xm, xn, a) < s2 And(xp, x0, a) + sAmd(x0, xp, a) (2.3) + 2cs2 [ 2pAm + (p + n)An + 2nAm+n ], (2.4)

how is it p > -2logA s we have 1 - s2Ap > 0, therefore, we obtain that {xn} is a 62-Cauchy.

Then

d {xm, xn, a )< —2—d {xo, xp , a), (25)

1 - s p

for all m,n,p e N, p > -2log— s and a e X .

Proof. It follows directly from Lemma 2.1, if we put c = 0, see (2.4).

Remark 2.3 Note that both Lemma 2.1 and Lemma 2.2 improve the result of Lemma 1.6. in (Fadail et al, 2015, pp.533-548).

A Theorem of Jungck in the b2-metric space

The following Theorem is the version of the Theorem of Jungck (Jungck, 1976, pp.261-263) in b2-metric spaces.

Theorem 3.1 Let T and I be commuting mappings of a b2-complete b2-metric space {X, d, s) into itself satisfying the inequality

d {Tx, Ty, a) < — {Ix, Iy, a) (3.1)

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d {xm, xn, a )<— {xm-1, xn-1 ,a ) for all m, n e N, a e X .

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for all x, y, a e X, where 0 < A < 1. If the range of I contains the

range of T and if I is £>2-continuous, then T and I have a unique common fixed point.

3 Proof. Let x0 e X be arbitrary. Then Tx0 and Ix0 are well defined.

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Since Tx0 e I(X), there is any x1 e X such that Ix1 = Tx0. In general, if xn is chosen, then we choose a point xn+1 in X such that Ixn+l = Txn. We show that {lxn} is a b2-Cauchy sequence. From (3.1) we have

o d (lxm, Ixn, a) = d (Txm_j, Txn_j, a) < Ad (lxm_j, , a).

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d(Ixm,Ixn,a)< Ad(lxm_l,Ixn_l,a),for all m,n e N, a e X . (3.2)

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£ d{lxm,Ixn,a)< S l/(/xn,Ixp,a), (3.3)

1 _ s p

for all m, n e N, a e X and for some p e N such that it is p > _2log1 s. Thus, we obtain that {/xn} is a ö2-Cauchy sequence in X. ^t By the b2-completeness of X, there exists u e X such that

^ lim Ixn = lim Txn_l = u .

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^^ Now, since I is b2-continuous, (3.1) implies that both I and T are

§ b2-continuous. Since T and I commute, we obtain

> Iu = I (lim Txl = lim ITxn = lim TIxn = T (lim Ixl = Tu.

n—^ n^w n^w n^w ^

Let v = Iu = Tu . We get Tv = TIu = ITu = Iv. If Tu * Tv, from (3.1) we obtain

d (Tu, Tv, a ) < Ad (Iu, Iv, a) = Ad (Tu, Tv, a) < d (Tu, Tv, a),

a contradiction. So we have Tu = Tv, and finally we obtain Tv = Iv = v i.e. v is a common fixed point for T and I. Condition (3.1) implies that v is the unique common fixed point.

Theorem 3.2 Let T and I be commuting mappings of a complete 2-metric space (X,d, s) into itself satisfying the inequality

d(Tx, Ty, a)<Ad(/x, Iy, a) (3.4) i-L

for all x, y,a g X, where 0 <A< l. If the range of I contains the â

range of T and I is continuous, then T and I have a unique common fixed point.

From Theorem 3.1, we obtain the following variant of the Banach theorem in 62-metric spaces.

Theorem 3.3 Let (X, d, s) be a ¿^-complete £>2-metric space and T : X ^ X a mapping satisfying:

d (ix, Ty, a )<od (x, y, a ) (3.5)

for all x, y, a g X, where ae[0,l). Then T has a unique fixed

point.

Proof. Put I (x ) = x, x g X in Theorem 3.1.

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Theorem 3.4 Let I be a continuous onto mapping of a b2-complete | b2-metric space (X,d,s).If there exists K > 1 such that

d (Ix, Iy, a) > Kd (x, y, a)

for all x, y, a e X, then I has a unique fixed point.

The Reich theorem in b2-metric spaces

The following theorem is the analogue of the Reich contraction principle (Reich, 1971, pp.121-124) in the b2-metric space.

Theorem 4.1 Let (X, d, s) be a b2-complete b2-metric space and T : X ^ X be a mapping satisfying:

d(7x,Ty,a) < ocd(x,y,a)+ /3d(x, Tx,a)+yd(y, Ty,a) (4.1)

for all x,y,a e X, where a,/,y are nonnegative constants with

a + / + y < 1 and min {/,/}<1 . Then Thas a unique fixed point.

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0 Proof. Let x0 g X be arbitrary. Define the sequence {xn} by

xn+1 = Txn for all n > 0 . From condition (4.1) we have that

d (xn+i, xn, a ) < ad (xn, xn_i, a ) + № (xn, xn+i, a ) + yd (xn-i, xn, a ).

Therefore,

d((, xn , a) < ay d(xn , xn-1, a) . (4 2)

o Put r = . We have that r e [0,l). It follows from (4.2) that

1 -ß

d(x«+1,xn,a)< rnd(,x0,a) for all n > 1. (4.3)

>-a:

< From conditions (4.1) and (4.3), we obtain

d(xm , xn , a) < ad(xm-i, Xn-1, a) + ßd(xm-1 , Xm , a) + Yd(xn-1 , Xn , a)

< ad(xm-1, xn-1, a) + ßrm-1 d, x^ a)+yrnld, x1, a)

ö = ccd (xm-1 , xn-1 , a) + (ßr m-1 + Tn-1 ^(x0,, x1, a )

From this, together with Lemma 2.1 (we can put

X = a, c = max{ß, y)d (x0, xx, a)

for all m,n e N, note that if r = 0 then the proof is trivial) we conclude that {xn} is Cauchy. By the ¿^-completeness of (X, d, s) there exists x* e X such that

lim xn = x*. (4.4)

Now we obtain that x* is the unique fixed point of T. Namely, we

have

d(, Tx*,a) < s[[(x„+1, Tx*,a) + d(x*, x„+1, a)+ d(x*, Tx*, x„+1 )] d ((, Tx", a ) + d (x *, xn+1, a ) + d (, Tx *, Txn )] ad (, x *, a )+fid (, xn+1, a )+yd (x *, Tx *, a ) + d (x *, xn+1, a )+ ad (x *, x *, xn )+ fid (x *, x *, Tx *)

+ yd(x *, xn, xn+1)]

= s < s

and

=s

<s +

d(Tx*, x*,a) < s[d((+1, x*, a)+ d(Tx*, xn+1, a)+ d(Tx*, x*, xn+1)] d (xn+1, x *, a ) + d ((x *, Txn, a ) + d ((x *, x *, Txn )] d (xn+1, x *, a )+ ad (x *, xn, a , Tx *, a )

xn, xn+1, a) + ad(x*, x*, xn)+ fid(x*, x*, Tx*) + yd(x *, xn, xn+1)J

Since lim d (x *, xn, a )= 0, lim d (xn, xn+13 a )= 0,

lim d (xn, xn+13 x *)= 0 and min(fi,f)< —, we have d (x *, Tx*, a)= 0 for

all a g X i. e., Tx = x (Axiom (1) in Definition 1.5).

For uniqueness, let y * be another fixed point of T. Then it follows from (4.1) that

d (x *, y *, a)= d (Tx *, Ty *, a)< ad (x *, y *, a)+ fid (x *, Tx *, a)+ jd (y *, Ty *, a)

7/ * * 1 it * * I

= adyx , y , aj < d[x , y , a)

is a contradiction. Therefore, we must have d (x *, y *, a )= 0, i.e.,

* *

x = y .

From Theorem 4.1, we obtain the following variant of the Kannan theorem (Kannan, 1968, pp.71-76) in 62-metric spaces.

Theorem 4.2 Let (X, d, s) be a incomplete £>2-metric space and T : X ^ X be a mapping satisfying:

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for all x, y,a e X, where /, y nonnegative constants with 3 + y < 1 and mm{/, y} < —. Then T has a unique fixed point.

A result in partial order £>2-metric space

Let Fs denote the class of all functions / : [0, <») ^

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g the following condition:

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< /(t„ 1 as n ^ ro implies tn ^ 0 as n ^ ro.

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In the paper (Mustafa et al, 2014), Mus result in partially ordered £>2-metric spaces.

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o In the paper (Mustafa et al, 2014), Mustafa et al. obtain the following

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| Theorem 5.1 (Mustafa et al, 2014, Theorem 1) Let (X,^)be a

partially ordered set and suppose that there exists a ¿2-metric d on X such that (X, d, s) is a ¿^-complete £>2-metric space. Let f : X ^ X be an increasing mapping with respect to < such that there exists an

in

^ element xQ e X with xQ <fx0. Suppose that

2 sd(fx, fy, a) < p(d(x, y, a))(x, y, a) (5.1)

for all a e X and for all comparable elements x, y e X, where

* sí \ \ jt \d (x, fx, a )d (y, fy, a )]

o M (x, y, a ) = max \d (x, y, a ), v ^

^ 1 V h 1 + d((x, fy, a)

> If fis ¿2-continuous, then f has a fixed point. Moreover, the set of

fixed points of f is well ordered if and only if f has one and only one fixed point.

In the further, we consider that M (x, y, a ) is given as in Theorem

5.1.

In the paper (Fadail et al, 2015, pp.533-548), Fadail et al. generalize, complement and improve Theorem 5.1 in several directions.

Theorem 5.2 (Fadail et al, 2015, pp.533-548, Theorem 2.1) Let (X, be a partially ordered set and suppose that there exists a ¿2-metric d on X such that (X, d, s) is a ¿^-complete ¿2-metric space with s > 1. Let f : X ^ X bean increasing mapping with respect to < such that there exists an element xQ e X with xQ <fxQ. Suppose that

s

ssd(( fy, a) < p(d(x, y, a))M(x, y, a) (5.2)

CM m

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for all a g X and for all comparable elements x, y g X, where

s g (0,1]. Iff is 62-contuniuous then f has a fixed point. Moreover, the set «>

of fixed points of fis well ordered if and only if f has one and only one g fixed point.

Using Lemma 2.1, we get the following result. §

Theorem 5.3 Let (X, ^)be a partially ordered set and suppose that there exists a £>2-metric d on X such that (X, d, s) is a ¿^-complete | 62-metric space with s > 1. Let f : X ^ X be an increasing mapping with respect to < such that there exists an element x0 g X with x0 <fX0. Suppose that exists  g (0,l] such that

d (x, fy, a ) < ÂM (x, y, a), (5.3) ^

for all a g X and for all comparable elements x, y g X. If fis i ¿2-continuous, then f has a fixed point. Moreover, the set of the fixed 10" points of f is well ordered if and only if f has one and only one fixed point.

o cp

T3

<u X

<D □L

S=

Proof. Since x0 <fX0 and f is an increasing function, we obtain that §

Xo <fX 0 <f2 Xo <-<f nXo <fn+1 Xo .

Since Xn<Xn+1, where Xn+1 = fXn,n gN, from (5.3) we obtain

d (Xn, Xn+1, a )< A max id Xn, a ),d (Xn1 x; (a >d X ^ a) }

[ 1 + d (Xn, Xn+1, a) J

<A fo^ Xn , a).

Using induction, we conclude that

d (Xn, Xn+1, a )<And X1, a ), (5.4)

for all n g N, a g X. Further from conditions (5.3), we have

d (, Xn, a )< A max id ( _1, Xn _1, a ),d X -fm( (^)Xn, a) }

I 1 + d(m, a) J

< A max(d(x„_1 , Xn_1, a), d(x„_1 , Xm , a(Xn_1, Xn , a^

Now, from inequality (5.4), it follows

CO

"(5 >

^ d(xm, xn, a) < X max d (xm_i, xn_^ a ),Xm 1 d (xQ, x1, a )Xn ld (xQ, x1, a )}

< Xd (xm_1, xn 1, a ) + Xm+n-1d2 (xq , x, a )

< Xd (xm_1, xn_1, a ) + ( + Xn )2 (xq , x1, a ) = Xd (xm _1, a ) c(xm +Xn )

° where c = d2(xQ,x1,a). Now, because of Lemma 2.1, we get that

yy {xn} is a ¿2-Cauchy sequence in (X, d ). The rest of the proof is the

g same as in (Mustafa et al, 2014) (Steps IV and V). o Note that condition (5.1) implies (5.2) and condition (5.2) implies

< (5.3).

Example 2 Let X = {(a,0) : a e [0,+œ)}^ {(0,2)} and let d(x,y, z) u denote the square of the area of a triangle with the vertices x, y, z e X, e.g.,

< d((a,0), (b,0), (0,2))= (a - b)2. Then d is a ¿2-metric with the parameter s = 2. Introduce an order

in X by

(a,0) -< (b,0)o> a > b,

with all other pairs of distinct points in X incomparable. Consider the mapping f : X ^ X given by

f (a,0) = (Xa,0) for a e [0,+œ) and f (0,2)=(0,2),

w and the function fieF2 given as

0 1 +1

f(t ) =-for t e [0,+œ).

2 + 4t 7

Then f is an increasing mapping with (a,0) - f (a,0) for each a > 0 .

1. If X = 1 then the assumptions of Teoreme 5.1 are satisfied (Mustafa et al, 2014, Example 3).

2. If X = 2 then the assumptions of Theorem 5.2 are satisfied

(Fadail et al, 2015, pp.533-548, Example 2.6), but we cannot apply Theorem 5.1.

3. If Xef 1,1) then the assumptions of Theorem 5.3 are satisfied,

but we cannot apply Theorem 5.2.

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НЕКОТОРЫЕ ТЕОРЕМЫ О НЕПОДВИЖНЫХ ТОЧКАХ В Ь2-МЕТРИЧЕСКИХ ПРОСТРАНСТВАХ

Пум Кумам3, Зоран Д. Митрович6, Мирьяна В. Павлович8 а Технологический университет короля Монгкута в Тхонбури,

вычислительных наук, Здание научной лаборатории,

г. Крагуевац, Республика Сербия

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Е ф (5 ф

о тз

Естественно-математический факультет, Исследовательская группа по ф теории неподвижных точек и их применению, Центр теоретических и

ф Е

г. Бангкок, Королевство Таиланд °

1 Университет в г. Баня-Лука, Электротехнический факультет, г. Баня-Лука, Республика Сербская, Босния и Герцеговина в Университет в г. Крагуевац, Естественно-математический факультет, ^

Е"

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РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА; |

27.39.27 Нелинейный функциональный анализ ВИД СТАТЬИ: оригинальная научная статья ЯЗЫК СТАТЬИ: английский

Резюме:

Во введении данной статьи мы доказали результат, который является достаточным условием для сходимости последовательностей в Ь2-метрическом пространстве. Далее представлены некоторые теоремы о неподвижной точке в Ь2-метрическом пространстве. Некоторые из наших результатов представляют собой соответствующие обобщения известных результатов в Ь2-метрическом пространстве, в качестве доказательства приведены примеры.

Ключевые слова: неподвижные точки, обобщенные неподвижные точки, 2-метрическое пространство, Ь2-метрическое пространство.

« НЕКЕ ТЕОРЕМЕ О ФИКСНОJ ТАЧКИ У Ь?-МЕТРИЧКИМ

<u

ПРОСТОРИМА

Пум Кумам3, Зоран Д. Митрови^, Мир^ана В. Павлович6

^ a Технолошки универзитет Кра^а Монгкаста у Тонбури]у,

о Природно-математички факултет, Истраживачка група теорбе и

> примене фиксних тачака, Центар за теорийке и рачунске науке,

Зграда научне ла6оратори]е, Бангкок, Кра^евина Та]ланд

см 6 Универзитет у Ба^ Луци, Електротехнички факултет,

о. Ба^а Лука, Република Српска, Босна и Херцеговина

^ в Универзитет у Крагу]евцу, Природно-математички факултет,

з Крагу]евац, Република Срби]а О

о ОБЛАСТ: математика (математичка тематска класификаци]а:

^ примарна 47H10, секундарна 54H25)

У ВРСТА ЧЛАНКА: оригинални научни рад

^ иЕЗИК ЧЛАНКА: енглески о

ш Сажетак:

У овом раду прво jе доказан резултат ко\и да]е довоъан услов за ¡i конвергенци]у низова у Ь2-метричком простору. Тако^е,

наведене су неке теореме о фиксно] тачки у Ь2-метричком простору. Неки резултати представъа]у одговара]уПе генерализаци}е познатих резултата у Ь2-метричком простору,

и а примери су презентирани да то потврде.

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Къучне речи: фиксне тачке, за}едничке фиксне тачке, 2 2-метрички простор, Ь2-метрички простор.

о

ш Paper received on / Дата получения работы / Датум приема чланка: 05.04.2019. ^ Manuscript corrections submitted on / Дата получения исправленной версии работы /

Датум достав^а^а исправки рукописа: 10.05.2019. О Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 12.05.2019.

© 2019 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (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/).

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>