0 Journal of Inequalities and Applications

a SpringerOpen Journal

Some new generalizations of Mizoguchi-Takahashi type fixed point theorem

Gülhan Minakand IshakAltun*

"Correspondence:

ishakaltun@yahoo.com Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, Kirikkale 71540, Turkey

ft Springer

Abstract

In the light of the paper of Hasanzade Asl etal. (Fixed Point Theory Appl. 2012:212, 2012, doi: 10.1186/1687-1812-2012-212), we obtain a fixed point theorem for multivalued mappings on a complete metric space. Our result is a generalized version of some results in the literature, including the famous result of Mizoguchi-Takahashi (J. Math. Anal. Appl. 141:177-188,1989). Also, we give some examples to illustrate our result.

MSC: Primary 54H25; secondary 47H10

Keywords: fixed point; multivalued mappings; Mizoguchi-Takahashi's fixed point theorem; Nadler's fixed point theorem

1 Introduction and preliminaries

Let (X, d) be a metric space, and let CB(X) denote the class of all nonempty, closed and bounded subsets of X. It is well known that H: CB(X) x CB(X) ^ R defined by

is a metric on CB(X), which is called a Hausdorff metric, where d(x, B) = inf{d(x,y) :y e B}. Let T: X ^ CB(X) be a map, then T is called a multivalued contraction if for all x,y e X, there exists X e [0,1) such that

H(Tx, Ty) < Xd(x,y).

In 1969, Nadler [1] proved a fundamental fixed point theorem for multivalued maps: Every multivalued contraction on a complete metric space has a fixed point.

Then, a lot of generalizations of the result of Nadler have been given (see, for example, [2-5]). One of the most important generalizations of it was given by Mizoguchi and Taka-hashi [6]. We can find both a simple proof of Mizoguchi-Takahashi fixed point theorem and an example showing that it is a real generalization of Nadler's result in [7]. We can also find some important results about this direction in [8-12].

Definition 1 [2] A function k: [0, to) ^ [0,1) is said to be an ^T-function if it satisfies limsups^t+ k(s) < 1 for all t e [0, to) (Mizoguchi-Takahashi's condition).

©2013 Minak and Altun; 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.

Lemma 1 [9] Let k: [0, to) ^ [0,1) be an MT-function, then the function h : [0, to) ^ [0,1) defined as h(t) = 1+2t) is also an MT-function.

Lemma 2 [9] k: [0, to) ^ [0,1) is an MT-function if and only if for each t e [0, to), there exist rt e [0,1) and et >0 such that k(s) < rtfor all s e [t, t + et).

Theorem 1 [6] Let (X, d) be a complete metric space, and let T: X ^ CB(X) be a multivalued map. Assume

H(Tx, Ty) < k(d(x,y))d(x,y) (1.1)

for all x, y e X, where k is an MT-function. Then T has a fixed point.

Recently, Samet etal. [13] introduced the notion of a-f-contractive mappings and gave some fixed point results for such mappings. Their results are closely related to some ordered fixed point results. Then, using their idea, some authors presented fixed point results for single and multivalued mappings (see, for example, [13-17]). First, we recall these results. Denote by ^ the family of nondecreasing functions f : [0, to) ^ [0, to) such that ETO=i f n(t)<TO for all t >0.

Definition 2 [13] Let (X, d) be a metric space, T be a self-map on X, f e ^ and a : X x X ^ [0, to) be a function. Then T is called a-f -contractive whenever

a(x,y)d(Tx, Ty) < f (d(x,y))

for all x, y e X.

Note that every Banach contraction mapping is an a-f -contractive mapping with a(x, y) = 1 and f (t) = Xt for some X e [0,1).

Definition 3 [13] T is called a-admissible whenever a(x,y) > 1 implies a(Tx, Ty) > 1.

There exist some examples for a-admissible mappings in [13]. For convenience, we mention in here one of them. Let X = [0, to). Define T : X ^ X and a : X x X ^ [0, to) by Tx = Vx for all x e X and a(x,y) = ex-y for x > y and a(x,y) = 0 for x < y. Then T is a-admissible.

Definition 4 [14] a is said to have (B) property whenever {xn} is a sequence in X such that a(xn,xn+1) > 1 for all n e N and xn ^ x, then a(xn,x) > 1 for all n e N.

Theorem 2 (Theorem 2.1 of [13]) Let (X,d) be a complete metric space and T : X ^ X be an a-admissible and a-f -contractive mapping. If there exists x0 e X such that a(x0, Tx0) > 1 and T is continuous, then T has a fixed point.

Remark 1 If we assume that a has (B) property instead of the continuity of T, then again T has a fixed point (Theorem 2.2 of [13]). If for each x,y e X there exists z e X such that a(x,z) > 1 and a(y,z) > 1, then X is said to have (H) property. Therefore, if X has (H) property in Theorem 2.1 and Theorem 2.2 in [13], then the fixed point of T is unique (Theorem 2.3 of [13]).

Then some generalizations of a-f -contractive mappings are given as follows. Definition 5 [14] T is called a CiriC type a-f -generalized contractive mapping whenever

a(x,y)d(Tx, Ty) < f (m(x,y)) for all x, y e X, where

Note that every Ciric type generalized contraction mapping is a Ciric type a-f-generalized contractive mapping with a(x,y) = 1 and f (t) = Xt for some X e [0,1).

Theorem 3 (Theorem 2.3 of [14]) Let (X, d) be a complete metric space and T: X ^ Xbe an a-admissible and Ciric type a-f -generalized contractive mapping. If there exists x0 e X suchthat a(xo, Tx0) > 1 and T is continuous or a has (B) property, then T has a fixed point. IfX has (H) property, then the fixed point of T is unique.

We can find some fixed point results for single-valued mappings in these directions in [15,17]. Now we recall some multivalued case.

Definition 6 [14,16] Let (X, d) be a metric space and T : X ^ CB(X) be a multivalued mapping. Then T is called multivalued a-f -contractive whenever

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

for all x,y e X and T is called multivalued a^-f -contractive whenever

a^(Tx, Ty)H(Tx, Ty) < f (d(x,y)),

where a^(Tx, Ty) = inf{a(a, b): a e Tx, b e Ty}. Similarly, if we replace d(x,y) with m(x,y), we can obtain Ciric type multivalued a-f -generalized contractive and Ciric type multivalued at-f -generalized contractive mappings on X.

Definition 7 [14,16] Let (X, d) be a metric space and T : X ^ CB(X) be a multivalued mapping.

(a) T is said to be a-admissible whenever for each x e X and y e Tx with a(x,y) > 1 implies a(y,z) > 1 for all z e Ty.

(b) T is said to be a^-admissible whenever for each x e X and y e Tx with a(x,y) > 1 implies a^(Tx, Ty) > 1.

Remark 2 It is clear that at-admissible maps are also a-admissible, but the converse may not be true as shown in the following example.

Example 1 Let X = [-1,1] and a : X x X ^ [0, to) be defined by a(x,x) = 0 and a(x,y) = 1 for x = y. Define T: X ^ CB(X) by

{-x}, x e {-1,0}, {0,1}, x = -1, {1}, x = 0.

Let x = -1 andy = 0 e Tx = {0,1}, then a(x,y) > 1, but a*(Tx, Ty) = a*({0,1}, {1}) = 0. Thus T is not a^-admissible. Now we show that T is a-admissible with the following cases: Case 1. If x = 0, theny = 1 and a(x,y) > 1. Also, a(y,z) > 1 since z = -1 e Ty = {-1}. Case 2. Ifx = -1, theny e {0,1} and a(x,y) > 1. Also, a(y,z) > 1 for all z e Ty. Case 3. If x e {-1,0}, then y = -x and a(x,y) > 1. Also, a(y,z) > 1 since z = x e Ty = {x}.

The purpose of this work is to present some generalizations of Mizoguchi-Takahashi's fixed point theorem using this new idea.

2 Main results

Theorem 4 Let (X, d) be a complete metric space, and let T : X ^ CB(X) be an a-admissible multivalued mapping such that

a(x,y)H(Tx, Ty) < k(d(x,y))d(x,y) (2.1)

for all x,y e X, where k is an MT-function. Suppose that there exist x0 e X and xi e Tx0 such that a(x0, x1) > 1. IfTis continuous or a has (B) property, then T has a fixed point.

Proof Define h(t) = 1±2£), then from Lemma 1, h: [0, to) ^ [0,1) is an MT-function. Let x0 and x1 be as mentioned in the hypothesis. If xo = x1, thenxo is a fixed point of T .Assume x0 = x1, then 1-k(dfc0,x1)) d(x0,x1) > 0. Therefore there exists x2 e Tx 1 such that

d(x1,x2) < H(Tx0, Tx1) +-k(d(x0,x1)) d(x0,x1)

< a(x0,x1)H(Tx0, Tx1) + 1—x1)) d(x0,x1)

< k(d(x0,x1))d(x0,x1) +-c(d(x°,x1)) d(x0,x1)

1 + k(d(x0, x1))

= -2-d(x0, x1)

= ^d(x0, x1)) d(x0, x1).

Since T is a-admissible, x1 e Tx0 and a(x0,x1) > 1, then a(x1, u) > 1 for all u e Ix1. Thus a(x1,x2) > 1 since x2 e Tx1. If x1 = x2, then x1 is a fixed point of T. Assume x1 = x2, then 1-k(d(2x1,x2)) d(x1,x2) > 0. Therefore there exists x3 e Tx2 such that

d(x2,x3) < H(Tx1, Tx2) + 1—kk(d(:i'x2)) d(x1,x2)

< a(x1,x2)H(Tx1, Tx2) +-k(d{x\,x2)) d(x1,x2)

< k(d(xi,x2))d(x1,x2) + -—x2)) ^(xi,x2) l + k(d(xi, x2)) ,, .

= -2-d(xi, x2)

= ^d(x1, x2)) d(x1, x2).

Again, since T is a-admissibie, then a(x2,x3) > 1. In this way, we can construct a sequence {xn} in X such that xn+1 e Txn, a(xn,xn+1) > 1 and

d(xn,xn+1) << hiyd(xn—1,xn d(xn—1,xn)

for aii n e N. Since h(t) < 1 for aii t e [0, to), then {d(xn,xn+1)} is a nonincreasing sequence in [0, to) and so there exists X > 0 such that limn^TO d(xn,xn+1) = X. Now since h is an ^T-function, then limsups^X+ h(s) < 1 and h(X) < 1. Therefore from Lemma 2 there exist r e [0,1) and e >0 such that h(s) < r for aii s e [X, X + e). Since limn^TO d(xn,xn+1) = X, then there exists n0 e N such that X < d(xn,xn+1) < X + e for aii n > n0 and so

d(xn+1, xn+2) < h(d(xn,xn+1))d(x n, xn+1)

< rd(xn, xn+1) for aii n > n0. Thus, we have

TO n0 to

^d(xn, xn+1) — ^ d(xn, xn+1) + ^^ ^ d(xn, xn+1) n=1 n—1 n=n0+1

= ^ ^ d(xn, xn+1) + ^ ^ d(xn+1, xn+2)

n—1 n0

' ^^ ^ d(xn, xn+1) ^^ ^ rd(xn, xn+1)

n=1 n=n0

< ^ d(xn, xn+1) + ^ r d(xn0, xn0+1)

n=1 n=1

and so {xn} is a Cauchy sequence. Since X is compiete, there exists z e X such that

limn^TO xn = z.

If T is continuous, then from the inequaiity d(xn+1, Tz) < H(Txn, Tz), we have d(z, Tz) = 0 and so z e Tz.

Now assume that a has (B) property. Then a(xn, z) > 1 for aii n e N. Therefore

d(xn+1, Tz) < H(Txn, Tz)

< a(xn, z)H(Txn, Tz)

< k(d(xn, z))d(xn, z)

< d(xn, z)

and, taking iimit n ^to, we have d(z, Tz) = 0 and so z e Tz. □

Although a*-admissibility implies a-admissibility of T, we will give the following theorem. However, the contractive condition is slightly different from (2.1).

Theorem 5 Let (X, d) be a complete metric space, and let T : X ^ CB(X) be an a*-admissible multivalued mapping such that

a*(Tx, Ty)H(Tx, Ty) < k(d(x,y))d(x,y)

for all x,y e X, where k is an MT-function. Suppose that there exist xo e X and x1 e Tx0 such that a(xo, x1) > 1. IfTis continuous or a has (B) property, then T has a fixed point.

Proof Define h(t) = 1±2£), then from Lemma 1, h: [0, to) ^ [0,1) is an MT-function. Let x0 and x1 be as mentioned in the hypothesis. If x0 e Tx0, then x0 is a fixed point of T. Let x0 e Tx0. Since x0 = x1, then 1-k(d(x0,x1))d(x0,x1) > 0. If x1 e Tx1, x1 is a fixed point of T. Let x1 e Tx1. Also, since T is a*-admissible, a*(Tx0, Tx1) > 1. Therefore, there exists x2 e Tx1 such that

d(xi,x2) < H(Tx0, Txi) +-k(d(x0,xi)) d(x0,x1)

< a*(Tx0, Tx1)H(Tx0, Tx1) + 1—x1))d(x0,x1)

< k(d(x0,x1))d(x0,x1) + 1—x1)) d(x0,x1) 1 + k(d(x0, x1))

= -2-d(x0, x1)

= h(d(x0, x1)) d(x0, x1).

Since a(x1,x2) > a*(Tx0, Tx1) > 1, then a*(Tx1, Tx2) > 1. Therefore there exists x3 e Tx2 such that

d(x2,x3) < H(Tx1, Tx2) + 1—kk(d(:i'x2)) d(x1,x2)

< a*(Tx1, Tx2)H(Tx1, Tx2) +-k(d{x\,x2))d^,x2)

< k(d(x1,x2))d(x1,x2) + 1—k(d(xl,x2)) d(x1,x2)

1 + k(d(x1, x2)) ,, . = -2-d(x1, x2)

= h(d(x^ x2)) d(x1, x2).

Again, if x2 e Tx2, x2 is a fixed point of T. Let x2 e Tx2. Since a(x2,x3) > a*(Tx1, Tx2) > 1, then a*(Tx2, Tx3) > 1. In this way, we can construct a sequence {xn} in X such that xn+1 e Txn, a(xn,xn+1) > 1 and

d(xn,xn+1) << h(d(xn—1,xnd(xn—1,xn)

for all n e N. As in the proof of Theorem 4, we can show that {xn} is a Cauchy sequence in X. Since X is complete, there exists z e X such that limn^TO xn = z.

If T is continuous, then from the inequaiity d(xn+1, Tz) < H(Txn, Tz), we have d(z, Tz) = 0 and so z e Tz.

Now assume that a has (B) property. Then a(xn, z) > 1 for aii n e N. Since T is a* -admissibie, a*(Txn, Tz) > 1. Therefore

d(xn+1, Tz) < H(Txn, Tz)

< a*(Txn, Tz)H(Txn, Tz)

< k(d(xn, z))d(xn, z)

< d(xn, z)

and, taking iimit n ^to, we have d(z, Tz) = 0 and so z e Tz. □

Now we give an exampie to iiiustrate our main theorems. Note that Theorem 1 cannot be appiied to this exampie.

Example 2 Let X = [—1,1] and d(x,y) = |x — y|. Define T: X ^ CB(X) by

{2x + 1}, x e [-1,-f),

{2x -1},

x e (f, 1], x e [-f,f]

and a : X x X ^ [0, œ)by

a(x,y)=\1, x' e [-^2], I 0, otherwise.

Then T is a*-admissible and

a*(Tx, Ty)H(Tx, Ty) < k(d(x,y))d(x,y)

for all x,y e X, where k is any MT-function. Indeed, first we show that T is a* -admissible. If a(x,y) > 1, then x,y e [-2, 2] and hence

■ 1 1" " 1 1"

2_ , 2_

a*(Tx, Ty) = a*

= in^ a(a, b) :a, b e = 1.

1 1' ~2'2

Therefore T is a*-admissible. Now we consider the following cases:

Case 1. Let x, y e X with {x, y} n {[-1,-f) U (f,1]} = 0, then a*(Tx, Ty) = 0. Thus (2.2) is satisfied.

' 1 1" ' 1 1"

_-2' 2_ _-2' 2_

Case 2. Let x,y e X with x,y e [-4, 4], then

/r 11 IT 1 H(Tx, Ty)=H(

and so again (2.2) is satisfied. Now, if x, ye (4,1] with x = y, we have

H(Tx, Ty) = H({2x -1}, {2y - 1}) = 2d(x, y).

Therefore there is no MT-function satisfying (1.1).

Remark 3 If we take a : X x X ^ [0, to) by a(x, y) = 1, then any multivalued mappings T: X ^ CB(X) are a-admissible as well as a*-admissible. Therefore, Mizoguchi-Takahashi's fixed point theorem is a special case of Theorem 4 and Theorem 5.

We can obtain some ordered fixed point results from our theorems as follows. First we recall some ordered notions. Let X be a nonempty set and ^ be a partial order on X.

Definition 8 [18] Let A, B be two nonempty subsets of X, the relations between A and B are defined as follows:

(r1) If for every a e A there exists b e B such that a < b, then A -<1 B. (r2) If for every b e B there exists a e A such that a ^ b, then A -<2 B. (r3) If A B and A B, then A < B.

Remark 4 [18] -<1 and -<2 are different relations between A and B. For example, let X = R, A = [2,1], B = [0,1], ^ be the usual order on X, then A B but A /2 B; if A = [0,1], B = [0, 2], then A ^2 B while A /1B.

Remark 5 [18] -<1, -<2 and -< are reflexive and transitive, but are not antisymmetric. For instance, let X = R, A = [0,3], B = [0,1] U [2,3], ^ be the usual order on X, then A ^ B and B < A, but A = B. Hence, they are not partial orders.

Corollary 1 Let (X, be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let T : X ^ CB(X) be a multivalued mapping such that

H(Tx, Ty) < k(d(x,y))d(x,y)

for all x,y e X with x < y, where k is an MT-function. Suppose that there exists x0 e X such that {x0} -<1 Tx0. Assume that for each xeX andy e Tx with x < y, we have y < zfor allze Ty. IfT is continuous or X satisfies the following condition:

{{x„} c X is a nondecreasing sequence with xn ^ x in x, , „

then xn < xfor all n,

then T has a fixed point.

Proof Define the mapping a : X x X ^ [0, to) by

a(x,y) = I 1 x _ У'

0, otherwise.

Then we have

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

for aii x,y e X. Aiso, since {x0} -<1 Tx0, then there exists x1 e Tx0 such that x0 < x1 and so a(x0,x1) > 1. Nowiet x e X and y e Tx with a(x,y) > 1, then x ^ y and so, by the hypotheses, we have y < z for aii z e Ty. Therefore, a(y, z) > 1 for aii z e Ty. This shows that T is a-admissibie. Finaiiy, if T is continuous or X satisfies (2.3), then T is continuous or a has (B) property. Therefore, from Theorem 4, T has a fixed point. □

Remark 6 We can give a simiiar coroiiary using -<2 instead of -<1.

Competing interests

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

The authors read and approved the finalmanuscript. Acknowledgements

The authors are gratefulto the referees for their suggestions that contributed to the improvement of the paper.

Received: 19 July 2013 Accepted: 10 October 2013 Published: 07 Nov 2013

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10.1186/1029-242X-2013-493

Cite this article as: Minak and Altun: Some new generalizations of Mizoguchi-Takahashi type fixed point theorem.

Journal of Inequalities and Applications 2013, 2013:493

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