0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

Open Access

On new fixed point results for (a, ty, %)-contractive multi-valued mappings on a-complete metric spaces and their consequences

Marwan Amin Kutbi1 and Wutiphol Sintunavarat2

"Correspondence: wutiphol@mathstat.sci.tu.ac.th; poom_teun@hotmail.com 2Department of Mathematics and Statistics, Faculty of Science and Technology,Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand

Fulllist of author information is available at the end of the article

Abstract

The purpose of this paper is to establish new fixed point results for multi-valued mappings satisfying an (a, ty,%)-contractive condition, and via Bianchini-Grandolfi gauge functions, on a-complete metric spaces. Our results unify, generalize, and complement various results from the literature. We also give examples which support our main result while previous results in the literature are not applicable. Some of the fixed point results in metric spaces endowed with an arbitrary binary relation and endowed with a graph are given here to illustrate the usability of the obtained results. MSC: 47H10; 54H25

Keywords: (a, ty,%)-contractive multi-valued mappings; a-complete metric spaces; a-admissible multi-valued mappings; a+-admissible multi-valued mappings; Bianchini-Grandolfi gauge functions

ft Springer

1 Introduction and preliminaries

Throughout this paper, we denote by N, R+, and R the sets of positive integers, nonnegative real numbers, and real numbers, respectively.

We recollect some essential notations, required definitions, and primary results coherent with the literature. For a nonempty set X, we denote by N(X) the class of all nonempty subsets of X. Let (X, d) be a metric space, we denote by CL(X) the class of all nonempty closed subsets of X, by CB(X) the class of all nonempty closed bounded subsets of X. For A,B e CL(X), let the functional H: CL(X) x CL(X) ^ R+ U (cj be defined by

H(A B) I max(suP«eA d(a, B), supbeB d(b, A)}, if the maximum exists, I c, otherwise

for every A, B e CL(X), where d(a, B) = inf(d(a, b): b e Bj is the distance from a to B c X. Such a functional is called the generalized Pompeiu-Hausdorff metric induced by d.

In this paper, we denote by ^ the class of functions ty : [0, c) ^ [0, c) satisfying the following conditions:

(tyi) ty is a nondecreasing function;

(tyi) ^C=i tyn(t) < c, for all t >0, where tyn is the nth iterate of ty.

©2015 Kutbi and Sintunavarat; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (see e.g. [1-3]).

Remark 1.1 For each ty e ^, we see that the following assertions hold:

1. ty n(t) = 0, for all t > 0.

2. ty (t)<t for each t > 0;

3. ty (0) = 0;

Example 1.2 The function ty : [0, to) ^ [0, to) defined by ty (t) = kt, where k e [0,1), is a Bianchini-Grandolfi gauge function.

Example 1.3 The function ty : [0, to) ^ [0, to) defined by

t (t) =

4 t, 0 < t <1,

51, t =1, f t, t >1,

is a Bianchini-Grandolfi gauge function.

In [4], Samet et al. introduced the concepts of an a-admissible mapping and an a-t-contractive mapping as follows.

Definition 1.4 ([4]) Let T be a self mapping on a nonempty set X and a : X x X ^ [0, to) be a mapping. We say that T is a-admissible if the following condition holds:

x,y e X, a(x,y) > 1 a(Tx, Ty) > 1.

Definition 1.5 ([4]) Let (X, d) be a metric space and T : X ^ X be a given mapping. We say that T is an a-t-contractive mapping if there exist two functions a : X x X ^ [0, to) and t e ^ such that

a(x,y)d(Tx, Ty) < ty(d(x,y)) for all x, y e X.

One also proved some fixed point theorems for such mappings on complete metric spaces and showed that these results can be utilized to derive fixed point theorems in partially ordered metric spaces.

Afterwards, Asl et al. [5] introduced the concept of an a^-admissible mapping which is a multi-valued version of the a-admissible mapping provided in [4].

Definition 1.6 ([5]) Let X be a nonempty set, T : X ^ N(X) and a : X x X ^ [0, to) be two mappings. We say that T is a^-admissible if the following condition holds:

x,y e X, a(x,y) > 1 a*(Tx, Ty) > 1,

where a*(Tx, Ty) := inf{a(a, b)\a e Tx,b e Ty}.

They extended the a-t-contractive condition of Samet et al. [4] from a single-valued version to a multi-valued version as follows.

Definition 1.7 ([5]) Let (X, d) be a metric space, T : X ^ CL(X) be a multi-valued mapping and a : X x X ^ [0, to) be a given mapping. We say that T is an a-t-contractive multi-valued mapping if there exists t e ^ such that

a(x,y)H(Tx, Ty) < t (d(x,y))

for all x, y e X.

Asl etal. [5] also established a fixed point result for multi-valued mappings on complete metric spaces satisfying an a-t-contractive condition.

Recently, Ali etal. [6] introduced the notion of (a, t, h )-contractive multi-valued mappings, where h e 3 and 3 is the family of functions h : [0, to) ^ [0, to) satisfying the following conditions:

(hi) h is continuous; h) h is nondecreasing on [0, to); h) h (t) = 0 if and only if t =0; (h4) h is subadditive.

Remark 1.8 From fe) and fe), we have h(t) > 0, for all t e (0, to). Example 1.9 Let h : [0, to) ^ [0, to) be a mapping which is defined by

for each t e [0, to), where $ : [0, to) ^ [0, to) is a Lebesgue integrable mapping which is summable on each compact subset of [0, to) and satisfies the following conditions:

• for each e > 0, we have /J $(t) dt > 0;

• for each a, b >0, we have

Then % e 3.

Lemma 1.10 Let (X, d) be a metric space. If % e 3, then (X, % o d) is a metric space.

Lemma 1.11 ([6]) Let (X, d) be a metric space, % e 3, and B e CL(X). If there exists x e X such that % (d(x, B)) > 0, then there exists y e B such that

% (d(x,y)) < q% (d(x,B)),

where q >1.

Definition 1.12 ([6]) Let (X, d) be a metric space. A multi-valued mapping T: X ^ CL(X) is called an (a, ty, %)-contractive mapping if there exist three functions ty e ^, % e 3, and

a : X x X ^ [0, c) such that

x,y e X, a(x,y) > 1 %(H(Tx, Ty)) < ty(%(M(x,y))),

where M(x, y) = max(d(x, y), d(x, Tx), d(y, Ty),

d(x,Ty)+d(y,Tx) 2

In the case when ty e ^ is strictly increasing, the (a, ty, %)-contractive mapping is called a strictly (a, ty, %)-contractive mapping.

Ali etal. [6] also prove fixed point results for (a, ty, %)-contractive multi-valued mapping on complete metric spaces.

Question 1 Is it possible to prove fixed point results for (a, ty, %)-contractive multi-valued mapping T under some weaker condition for T?

Question 2 Is it possible to prove fixed point results for (a, ty, %)-contractive multi-valued mapping in some space which is more general than complete metric spaces?

Question 3 Is it possible to find some consequences or applications of the fixed point results?

On the other hand, Mohammadi et al. [7] extended the concept of an at-admissible mapping to a-admissible as follows.

Definition 1.13 ([7]) Let X be a nonempty set, T: X ^ N(X) and a : X x X ^ [0, c) be two given mappings. We say that T is a-admissible whenever for each x e X and y e Tx with a(x,y) > 1, we have a(y,z) > 1, for all z e Ty.

Remark 1.14 It is clear that at-admissible mapping is also a-admissible, but the converse may not be true as shown in Example 15 of [8].

Recently, Hussain etal. [9] introduced the concept of a-completeness for metric space which is a weaker than the concept of completeness.

Definition 1.15 ([9]) Let (X, d) be a metric space and a : X x X ^ [0, c) be a mapping. The metric space X is said to be a-complete if and only if every Cauchy sequence (xnj in X with a(xn, xn+1) > 1, for all n e N, converges in X.

Remark 1.16 If X is complete metric space, then X is also a-complete metric space. But the converse is not true.

Example 1.17 Let X = (0, c) and the metric d: X x X ^ R defined by d(x,y) = |x -y|, for all x,y e X. Define a : X x X ^ [0, c)by

ex+y, x,y e [2,5], otherwise.

a(x, y)

It is easy to see that (X, d) is not a complete metric space, but (X, d) is an a-complete metric space. Indeed, if (xnj is a Cauchy sequence in X such that a(xn,xn+1) > 1, for all n e N,

then xn e [2,5], for all n e N. Since [2,5] is a closed subset of R, we see that ([2,5], d) is a complete metric space and then there exists x* e [2,5] such that xn — x* as n — to.

In this paper, we establish new fixed point results for (a, ty, f )-contractive multi-valued mappings on a-complete metric spaces by using the idea of a-admissible multi-valued mapping due to Mohammadi etal. [7]. These results are real generalization of main results of Ali et al. [6] and many results in literature. We furnish some interesting examples which support our main theorems while results of Ali etal. [6] are not applicable. We also obtain fixed point results in metric space endowed with an arbitrary binary relation and fixed point results in metric space endowed with graph.

2 Main results

First, we introduce the concept of a-continuity for multi-valued mappings in metric spaces.

Definition 2.1 Let (X, d) be a metric space, a : X x X — [0, to) and T : X — CL(X) be two given mappings. We say T is an a-continuous multi-valued mapping on (CL(X),H) if, for all sequences {xn} with xn -— x e X as n — to, and a(xn,xn+i) > 1, for all n e N,we have Txn -h Tx as n - to, that is,

lim d(xn,x) = 0 and a(xn,xn+1) > 1 for all n e N lim H(Txn, Tx) = 0.

n—TO n—TO

Note that the continuity of T implies the a-continuity of T, for all mappings a. In general, the converse is not true (see in Example 2.2).

Example 2.2 Let X = [0,to), X e [10,20] andthe metric d: X x X — R defined by d(x, y) = |x -y|, for all x,y e X. Define T: X — CL(X) and a : X x X — [0, to) by

x =J {Xx2}, x e [0,1], I {x}, x >1

a(xy)\cosh(x2 + y2)' xye

I tanh(x + y),

otherwise.

Clearly, T is not a continuous multi-valued mapping on (CL(X),H). Indeed, for sequence |x„} = {1 + n} in X' we see that x„ = 1 + n — 1 but Txn = {1 + n} — {1} = M = T1.

Next, we show that T is an a-continue multi-valued mapping on (CL(X),H). Let {xn} be a sequence in X such that xn -— x e X as n — to and a(xn, xn+1) > 1, for all n e N. Then we have x, xn e [0,1], for all n e N. Therefore, Txn = {Xx;;} {Xx2} = Tx. This shows that T is an a-continuous multi-valued mapping on (CL(X),H).

Now we give first main result in this paper.

Theorem 2.3 Let (X, d) be a metric space and T : X — CL(X) be a strictly (a, ty, f )-contractive mapping. Suppose that the following conditions hold:

(S1) (X, d) is an a-complete metric space;

(52) T is an a-admissible multi-valued mapping;

(53) there exist xo and x1 e Txo such that a(xo, xi) > 1;

(54) T is an a-continuous multi-valued mapping.

Then T has a fixed point.

Proof Starting from xo and x1 e Txo in (S3), we have a(xo,x1) > 1. If xo = x1, then we see that xo is a fixed point of T. Assume that xo = x1. If x1 e Tx1, we obtain that x1 is a fixed point of T. Then we have nothing to prove. So we let x1 e Tx1. From (a, ty, f )-contractive condition, we get

f (H(Txo, Tx!))

< ma^Jd(xo,x1), d(xo, Txo), d(x1, Tx1),

= ty ^f ^max j d(xo, x1), d(x1, Tx1),

< ty\f\ ma« d(xo,x1), d(x1, Tx1),

\ Ji T N Jt T \ d(xo, Tx 1) + d(x1, Txo)

td(xo, X1), ^ Txo), d(xl, TX1),-2-

d(xo, Tx1)

d(xo, x1) + d(x1, Tx 1)

= ty{% (ma^ d(x0, x1), d(x1, Tx^})). (.1)

If max(d(x0,x1), d(x1, Tx1)j = d(x1, Tx1), then we get

0 < % (d(x1, Tx1))

< %(H(Tx0, T*1))

< ^^ma^d(x0,x1),d(x1, Tx1)}))

= ty(%(d(x1, Tx1))), (2.2)

which is a contradiction. Therefore, max(d(x0, x1), d(x1, Tx1)j = d(x0, x1). From (2.1), we get 0 < %(d(x1, Tx1)) < %(H(Tx0, TxO) < ty (% (d(x0,x1))). (2.3)

For fixed q > 1 by using Lemma 1.11, there exists x2 e Tx1 such that

0 < % (d(x1, x2)) < q% (d(x1, Tx 1)). (.4)

From (2.3) and (2.4), we have

0 < %(d(x1, x2^ < qty(%(d(x0, x^)). (.5)

Since ty is strictly increasing function, we have

0 < ty(%(d(x1,x2))) < ty(qty(%(d(x0,x1)))). (.6)

Put /7 - ty(qty(%(d(x0,x1)))) „„J tu„„ „ . 1

Put q1 = ty(% (d(xix2))) and then q1 > 1

If x1 = x2 or x2 e Tx2, then we find that x2 is a fixed point of T and thus we have nothing to prove. Therefore, we may assume that x1 = x2 and x2 e Tx2. Since x1 e Tx0, x2 e Tx 1, a(x0,x1) > 1, and T is an a-admissible multi-valued mapping, we have a(x1,x2) > 1. Applying from (a, t, h)-contractive condition, we have

h (H(Tx1, Tx2))

t_|«(x1,x2), «(x1, Tx1), «(x2, Tx2), ---J

= t ( h( max!d(x1,x2), d(x2, Tx2), d(Xl, Tx2) }

< ^ M max!d(x1,x2), d(x1,2x1), d(x>,2x2), ^ Tx2) + d(x2,Tx 1) 1

S.|d(x1,x2), d(x2, tx2),-2-J

< ti H max] d(x1,x2), d(x2, Tx2), d(Xl,x2) + d(x2 Tx2) I

|M(x1,x2), u(xi, 1x2), ---J

= ty(% (ma^d(x1,x-), d(x-, Tx2)})). (.7)

Suppose that max{d(x1,x2), d(x2, Tx2)} = d(x2, Tx2). From (2.7), we get

0 < % (d(x2, Tx2))

< % (W(Tx1, Tx2))

< ^^max{d(x1,x2), d(x2, Tx2)}))

= ty(%(d(x2, Tx2))), (.8)

which is a contradiction. Therefore, we may let max{d(x1,x2), d(x2, Tx2)} = d(x1,x2). From (2.7), we have

0 < %(d(x2, Tx2)) < % (W(Tx 1, Tx2)) < ty (% (d(x1,x^))). (2.9)

By using Lemma 1.11 with q1, there exists x3 e Tx2 such that

0 < %(d(x2,x3)) < q1% (d(x2, Tx2)). (.10)

From (2.9) and (2.10), we get

0 <%(d(x2,x3)) < q1ty(%(d(x1,x2))) = ty(qty(%(d(x0,x1)))). (2.11)

It follows from ty being a strictly increasing function that

0 < ty(%(d(x2,x3))) < ty2(qty(%(d(xc,xx)))). ((2)

Continuing this process, we can construct a sequence {xn} in X such that xn = xn+1 e Txn, a(xn,xn+1) > 1 (2.13)

0 < %(d(xn+1,xn+2^ < tyn(qty(%(d(x0,x1)))) (2.14)

for all n e N U {0}.

Let m, n e N such that m > n. By the triangle inequality, we have

m-l m-1

f (d(xm,xn)) < Y^f (dfe,xi+i)) < Y^ f i-1(qf (f(d(xo,xO))).

i=n i=n

Since f e ^,we have limn,m—TO f (d(xm,xn)) = 0. Using (f1), we get limn,m—TO d(xm,xn) = 0. This implies that {xn} is a Cauchy sequence in (X, d). From (2.13) and the a-completeness of (X, d), there exists x* e X such that xn — x* as n — to. By a-continuity of the multi-valued mapping T, we get

lim H(Txn, Tx) = 0. (2.15)

n—TO

Now we obtain

d(x*, Tx*) = lim d(xn+1, Tx) < lim H(Txn, Tx) = 0.

V / n—TO n—TO

Therefore, x* e Tx* and hence T has a fixed point. This completes the proof. □

Corollary 2.4 Let (X, d) be a metric space and T : X — CL(X) be a strictly (a, f, f )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an a-complete metric space;

(52) T is an a*-admissible multi-valued mapping;

(53) there exist x0 andx1 e Tx0 such that a(x0,x1) > 1;

(54) T is an a-continuous multi-valued mapping.

Then T has a fixed point.

Corollary 2.5 (Theorem 2.5 in [6]) Let (X,d) be a complete metric space and T : X — CL(X) be a strictly (a, f, f )-contractive mapping. Suppose that the following conditions hold:

(A1) T is an a*-admissible multi-valued mapping; (A2) there exist x0 andx1 e Tx0 such that a(x0,x1) > 1; (A3) T is a continuous multi-valued mapping.

Then T has a fixed point.

Next, we give second main result in this work.

Theorem 2.6 Let (X, d) be a metric space and T : X — CL(X) be a strictly (a, f, f )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an a-complete metric space;

(52) T is an a-admissible multi-valued mapping;

(53) there exist x0 andx1 e Tx0 such that a(x0,x1) > 1;

(54) if {xn} is a sequence in X with xn — x e X as n — to and a(xn,xn+1) > 1, for all n e N, then we have a(xn,x) > 1, for all n e N.

Then T has a fixed point.

Proof Following the proof of Theorem 2.3, we know that {xn} is a Cauchy sequence in X such that xn — x* as n —to and

a(xn, xn+1) > 1 (.16)

for all n e N. From condition (S4), we get

a(xn,x*) > 1 (2!7)

for all n e N.By using the (a, t, h)-contractive condition of T, we have

h (H(Txn, Tx*))

< fifi ma^ d(x„,x*), d(xn, Txn), d(x*, Tx*), d(c„' x ) + d(x ' x„„ I M (2.18)

for all n e N. Suppose that d(x*, Tx*) > 0. Let e := di-x*Tx*\ Since xn — x* as n — to, we can find N1 e N such that

d(x*,xn) < d(x*,2Tx*] (2.19) for all n > N1. Furthermore, we obtain that

d(x*, Txn) < d(x*,xn+1) < d(x ' x ) (2.20)

for all n > N1. Also, as {xn} is a Cauchy sequence, there exists N2 e N such that

. . d(x*, Tx*)

d(xn, Txn) < d(xn,xn+1)<----(2.21)

for all n > N2. It follows from d(xn, Tx*) — d(x*, Tx*) as n —to that we can find N3 e N such that

d(x„, Tx*) < 3d(^' Tx) (2.22)

for all n > N3. Using (2.19)-(2.22), we get max

diy^n,x ^, d(xn, Txn), ^x , Tx*),-—-,—— | = d(x*, Tx*) (2.23)

for all n > N := max^,N2,N3}. For n > N,by (2.18) and the triangle inequality, we have

+ %(H^Txn,' -*^

f (d(x*, Tx*)) < f (d(x*,x„+i)) + f(H(Tx„, Tx*))

< f (d(x*,x„+1)) + ^^ma^Jd(x„,x*) , d(x„, Tx„), d( x*, Tx n, d(x„, Tx *) + d(x*, Tx„)

= f (d(x*,x„+i)) + f (f (d(x*, Tx*))).

Letting n —to in the above inequality, we get

f (d(x*, Tx*)) < f (f (d(x*, Tx*))).

This implies that f (d(x*, Tx*)) = 0, which is a contradiction. Therefore, d(x*, Tx * ) = 0, that is, x* e Tx *. This completes the proof. □

Corollary 2.7 Let (X, d) be a metric space and T : X — CL(X) be a strictly (a, f, f )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an a-complete metric space;

(52) T is an a*-admissible multi-valued mapping;

(53) there exist x0 andxi e Tx0 such that a(x0,xi) > 1;

(54) if {xn} is a sequence in X with xn — x e X as n — to and a(xn,xn+1) > 1, for all n e N, then we have a(xn,x) > 1, for all n e N.

Then T has a fixed point.

Corollary 2.8 (Theorem 2.6 in [6]) Let (X, d) be a complete metric space and T : X — CL(X) be a strictly (a, f, f )-contractive mapping. Suppose that the following conditions hold:

(A1 ) T is an a*-admissible multi-valued mapping; (A2) there exist x0 andx1 e Tx0 such that a(x0,x1) > 1;

(A'3) if {xn} is a sequence in X with xn — x e X as n — to and a(xn,xn+1) > 1, for all n e N, then we have a(xn,x) > 1, for all n e N.

Then T has a fixed point.

Remark 2.9 Theorems 2.3 and 2.6 generalize many results in the following sense:

• The condition (1.1) is weaker than some kinds of the contractive conditions such as Banach's contractive condition [10], Kannan's contractive condition [11], Chatterjea's contractive condition [12], Nadler's contractive condition [13], etc.;

• the condition of being a-admissible of a multi-valued mapping T is weaker than the condition of being a*-admissible of T;

• for the existence of fixed point, we merely require that a-continuity of T and a-completeness of X, whereas other result demands stronger than these conditions.

Consequently, Theorems 2.3 and 2.6 extend and improve the following results:

• Theorem 2.5 and Theorem 2.6 of Ali et al. [6];

• Theorem 2.1 and Theorem 2.2 of Samet et al. [4];

• Theorem 2.3 of Asl et al. [5];

• Theorem 2.1 and Theorem 2.2 of Amiri et al. [14];

• Theorem 2.1 of Salimi et al. [15];

• Theorem 3.1 and Theorem 3.4 of Mohammadi et al. [7].

Next, we give an example to show that our result is more general than the results of Ali et al. [6] and many known results in the literature.

Example 2.10 Let X = (-10,10) and the metric d: X x X — R defined by d(x,y) = |x -y|, for all x,y e X. Define T: X — CL(X) and a : X x X — [0, to) by

[-9, |x|], x e (-10,0),

[0,f], x e [0,2],

[x+10,9], x e (2,10)

. . \exy cosh(x + y), x,y e [0,2], a(x, y) = 4

I tanh(ex - ey), otherwise.

Clearly, (X, d) is not complete metric space. Therefore, the results of Ali et al. [6] are not applicable here.

Next, we show that by Theorem 2.6 can be guaranteed the existence of a fixed point of T. Define functions f, f : [0, to) — [0, to) by f (t) = 2 and f (t) = Vt, for all t e [0, to). It is easy to see that f e ^ and f e 3.

Firstly, we will show that T is a strictly (a, f, f )-contractive mapping. For x,y e X and a(x, y) > 1, we have x, y e [0,2] and then

f (H(Tx, Ty)) =

= |x - y1

< iyM(x, y) = f (f (M(x, y))).

It is to be observed that f is strictly increasing function. Therefore, T is a strictly (a, f, f)-contractive mapping.

Moreover, it is easy to see that T is an a-admissible multi-valued mapping and there exists x0 = 1 e X and x1 = 1/4 e Tx0 such that

a(x0,xi) = a(1,1/4) > 1.

Also, T is an a-continuous mapping.

Finally, for each sequence {xn} in X with xn — x e X as n — to and a(xn, xn+1) > 1, for all n e N,we have a(xn,x) > 1, for all n e N. Thus the condition (S4) in Theorem 2.6 holds.

Therefore, by using Theorem 2.3 or 2.6, we get T has a fixed point in X. In this case, T has infinitely fixed points such as -2, -1, and 0.

3 Consequences

3.1 Fixed point results in metric spaces endowed with an arbitrary binary relation

It has been pointed out in some studies that some results in metric spaces endowed with an arbitrary binary relation can be concluded from the fixed point results related with a-admissible mappings on metric spaces. In this section, we give some fixed point results on metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of the results presented in the previous section. The following notions and definitions are needed.

Let (X, d) be a metric space and R be a binary relation over X. Denote 5 := R U R-1,

x, y e X, xSy ^^ xRy or yRx.

Definition 3.1 Let X be a nonempty set and R be a binary relation over X. A multi-valued mapping T: X — N(X) is saidtobea weakly comparative if for each x e X and y e Tx with xSy, we have ySz, for all z e Ty.

Definition 3.2 Let (X, d) be a metric space and R be a binary relation over X. The metric space X is said to be S-complete if and only if every Cauchy sequence {xn} in X with xnSxn+\, for all n e N, converges in X.

Definition 3.3 Let (X, d) be a metric space and R be a binary relation over X. We say that T: X — CL(X) is a S-continuous mapping to (CL(X), W) if for given x e X and sequence {xn} with

lim d(xn,x) = 0 and xnSxn+1 for all n e N lim W(Txn, Tx) = 0.

n—TO n—TO

Definition 3.4 Let (X, d) be a metric space and R be a binary relation over X. A mapping T: X — CL(X) is called an (S, ty, %)-contractive mapping if there exist two functions ty e ^ and % e a such that

x,y e X, xSy => %(W(Tx, Ty)) < ty(%(M(x,y))), (3.1)

where M(x,y) = max{d(x,y), d(x, Tx), d(y, Ty), d(-x,iyy)+d(!,Tx}.

In the case when ty e ^ is strictly increasing, the (S, ty, %)-contractive mapping is called a strictly (S, ty, %)-contractive mapping.

Theorem 3.5 Let (X, d) be a metric space, R be a binary relation over X and T : X — CL(X) be a strictly (S, ty, %)-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an S-complete metric space;

(52) T is a weakly comparative mapping;

(53) there exist x0 and x1 e Tx0 such that x0Sx1;

(54) T is a S-continuous multi-valued mapping.

Then T has a fixed point.

Proof This result can be obtain from Theorem 2.3 by define a mapping a : X x X — [0, to) by

a(x, y)^ x,y e xSy,

0, otherwise.

This completes the proof. □

By using Theorem 2.6, we get the following result.

Theorem 3.6 Let (X, d) be a metric space, R be a binary relation over X and T : X — CL(X) be a strictly (S, f, f )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an S-complete metric space;

(52) T is a weakly comparative mapping;

(53) there exist x0 and x1 e Tx0 such that x0Sx1;

(54) if {xn} is a sequence in X with xn — x e X as n — to and xnSxn+1, for all n e N, then we have xnSx, for all n e N.

Then T has a fixed point.

3.2 Fixed point results in metric spaces endowed with graph

In 2008, Jachymski [16] obtained a generalization of Banach's contraction principle for mappings on a metric space endowed with a graph. Afterwards, Dinevari and Frigon [17] extended some results of Jachymski [16] to multi-valued mappings. For more fixed point results on a metric space with a graph, one can refer to [18-20].

In this section, we give fixed point results on a metric space endowed with a graph. Before presenting our results, we give the following notions and definitions.

Throughout this section, let (X, d) be a metric space. A set {(x,x): x e X} is called a diagonal of the Cartesian product X x X and is denoted by A. Consider a graph G such that the set V(G) of its vertices coincides with X and the set E(G) of its edges contains all loops, i.e., A c E(G). We assume G has no parallel edges, so we can identify G with the pair (V(G), E(G)). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.

Definition 3.7 Let X be a nonempty set endowed with a graph G and T: X — N(X) be a multi-valued mapping, where X is a nonempty set X. We say that T weakly preserves edges if for each x e X and y e Tx with (x,y) e E(G), we have (y, z) e E(G), for all z e Ty.

Definition 3.8 Let (X, d) be a metric space endowed with a graph G. The metric space X is said to be E(G)-complete if and only if every Cauchy sequence {xn} in X with (xn, xn+1) e E(G), for all n e N, converges in X.

Definition 3.9 Let (X, d) be a metric space endowed with a graph G. We say that T: X — CL(X) is an E(G)-continuous mapping to (CL(X), H) if for given x e X and sequence {xn} with

lim d(xn,x) = 0 and (xn,xn+1) e E(G) for all n e N lim H(Txn, Tx) = 0.

n—TO n—TO

Definition 3.10 Let (X, d) be a metric space endowed with a graph G. A mapping T: X — CL(X) is called an (E(G), f, f )-contractive mapping if there exist two functions f e ^ and f e 3 such that

x,y e X, (x,y) e E(G) f (H(Tx, Ty)) < f (f (M(x,y))), (3.2)

where M(x,y) = max{d(x,y), d(x, Tx), d(y, Ty), d(x,T,)+d(y,Tx)}.

In the case when ty e ^ is strictly increasing, the (E(G), ty, %)-contractive mapping is called a strictly (E(G), ty, %)-contractive mapping.

Theorem 3.11 Let (X, d) be a metric space endowed with a graph G, and T: X — CL(X) be a strictly (E(G), ty, % )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an E(G)-complete metric space;

(52) T weakly preserves edges;

(53) there exist x0 andx1 e Tx0 such that (x0,x1) e E(G);

(54) T is an E(G)-continuous multi-valued mapping.

Then T has a fixed point.

Proof This result can be obtained from Theorem 2.3 by defining a mapping a : X x X — [0, to) by

(1, (x, y) e E(G), a(x, y) = <

0, otherwise.

This completes the proof. □

By using Theorem 2.6, we get the following result.

Theorem 3.12 Let (X, d) be a metric space endowed with a graph G and T: X — CL(X) be a strictly (S, ty, % )-contractive mapping. Suppose that the following conditions hold:

(51) (X, d) is an E(G)-complete metric space;

(52) T weakly preserves edges;

(53) there exist x0 andx1 e Tx0 such that (xo,x1) e E(G);

(54) if {xn} is a sequence in X with xn — x e X as n — to and (xn,xn+1) e E(G), for all n e N, then we have (xn, x) e E(G) for all n e N.

Then T has a fixed point. Remark 3.13

1. If we assume G is such that E(G) := X x X, then clearly G is connected and our Theorems 3.11 and 3.12 improve Nadler's contraction principle [13] and in the case of a single-valued mapping, we improve Banach's contraction principle [10], Kannan's contraction theorem [11], Chatterjea's contraction theorem [12], and Bianchini and Grandolfi's fixed point theorem.

2. Theorems 3.11 and 3.12 are partial some generalized fixed point results endowed with a graph of Jachymski [16] and Dinevari and Frigon [17].

3. Theorems 3.11 and 3.12 are generalizations of fixed point results of Theorem 2.5 and Theorem 2.6 of Ali et al. [6] in a graph version.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors contributed equally and significantly in writing this paper. Allauthors read and approved the finalmanuscript.

Author details

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathumthani, 12121,Thailand.

Acknowledgements

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financialsupport during the preparation of this manuscript.

Received: 24 June 2014 Accepted: 20 November 2014 Published: 16 Jan 2015

References

1. Bianchini, RM, Grandolfi, M: Trasformazioni di tipo contrattivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 45, 212-216 (1968)

2. Proinov, PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal. TMA 67, 2361-2369(2007)

3. Proinov, PD: New generalconvergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 26, 3-42 (2010)

4. Samet, B, Vetro, C, Vetro, P: Fixed-point theorems for a-ty-contractive type mappings. Nonlinear Anal. 75, 2154-2165 (2012)

5. Asl, JH, Rezapour, S, Shahzad, N: On fixed points of a-ty-contractive multifunctions. Fixed Point Theory Appl. 2012, 212 (2012)

6. Ali, MU, Kamran, T, Karapinar, E: (a, ty, Ç)-Contractive multivalued mappings. Fixed Point Theory Appl. 2014, 7 (2014)

7. Mohammadi, B, Rezapour, S, Naseer, S: Some results on fixed points of a-ty-Ciric generalized multifunctions. Fixed Point Theory Appl. 2013, 24 (2013)

8. Minak, G, Acar, 0, Altun, I: Multivalued pseudo-Picard operators and fixed point results. J. Funct. Spaces Appl. 2013, Article ID 827458 (2013)

9. Hussain, N, Kutbi, MA, Salimi, P: Fixed point theory in a-complete metric spaces with applications. Abstr. Appl. Anal. 2014, Article ID 280817 (2014)

10. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181 (1922)

11. Kannan, R: Some results on fixed points. II. Am. Math. Mon. 76,405-408 (1969)

12. Chatterjea, SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 25, 727-730(1972)

13. Nadler, SB Jr.: Multi-valued contraction mappings. Pac.J. Math. 30,475-488(1969)

14. Amiri, P, Rezapour, S, Shahzad, N: Fixed points of generalized a-ty-contractions. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser.AMat. 108,519-526 (2014)

15. Salimi, P, Latif, A, Hussain, N: Modified a-ty-contractive mappings with applications. Fixed Point Theory Appl. 2013, 151 (2013)

16. Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359-1373 (2008)

17. Dinevari, T, Frigon, M: Fixed point results for multivalued contractions on a metric space with a graph. J. Math. Anal. Appl. 405, 507-517(2013)

18. Beg, I, Butt, AR: Fixed point of set-valued graph contractive mappings. J. Inequal. Appl. 2013, 252 (2013)

19. Chifu, C, Petrusel, G: Generalized contractions in metric spaces endowed with a graph. Fixed Point Theory Appl. 2012, 161 (2012)

20. Nicolae, A, O'Regan, D, Petrusel, A: Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph. Georgian Math. J. 18, 307-327 (2011)

10.1186/1687-1812-2015-2

Cite this article as: Kutbi and Sintunavarat: On new fixed point results for (a, ty, ^ )-contractive multi-valued mappings on a-complete metric spaces and their consequences. Fixed Point Theory and Applications 2015, 2015:2

Submit your manuscript to a SpringerOpen journal and benefit from:

► Convenient online submission

► Rigorous peer review

► Immediate publication on acceptance

► Open access: articles freely available online

► High visibility within the field

► Retaining the copyright to your article

Submit your next manuscript at ► springeropen.com