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Fixed point results for the αMeirKeeler contraction on partial Hausdorff metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 410 (2013)
Abstract
The purpose of this paper is to study fixed point theorems for a multivalued mapping satisfying the αMeirKeeler contraction with respect to the partial Hausdorff metric ℋ in complete partial metric spaces. Our result generalizes and extends some results in the literature.
MSC:47H10, 54C60, 54H25, 55M20.
1 Introduction and preliminaries
Fixed point theory is one of the most crucial tools in nonlinear functional analysis, and it has application in distinct branches of mathematics and in various sciences, such as economics, engineering and computer science. The most impressed fixed point result was given by Banach [1] in 1922. He concluded that each contraction has a unique fixed point in the complete metric space. Since then, this pioneer work has been generalized and extended in different abstract spaces. One of the interesting generalization of Banach fixed point theorem was given by Matthews [2] in 1994. In this remarkable paper, the author introduced the following notion of partial metric spaces and proved the Banach fixed point theorem in the context of complete partial metric space.
For the sake of completeness, we recall basic definitions and fundamental results from the literature.
Throughout this paper, by {\mathbb{R}}^{+}, we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers.
Definition 1 [2]
A partial metric on a nonempty set X is a function p:X\times X\to {\mathbb{R}}^{+} such that for all x,y,z\in X
(p_{1}) x=y if and only if p(x,x)=p(x,y)=p(y,y);
(p_{2}) p(x,x)\le p(x,y);
(p_{3}) p(x,y)=p(y,x);
(p_{4}) p(x,y)\le p(x,z)+p(z,y)p(z,z).
A partial metric space is a pair (X,p) such that X is a nonempty set, and p is a partial metric on X.
Remark 1 It is clear that if p(x,y)=0, then from (p_{1}) and (p_{2}), we have x=y. But if x=y, the expression p(x,y) may not be 0.
Each partial metric p on X generates a {\mathcal{T}}_{0} topology {\tau}_{p} on X, which has as a base the family of open pballs \{{B}_{p}(x,\gamma ):x\in X,\gamma >0\}, where {B}_{p}(x,\gamma )=\{y\in X:p(x,y)<p(x,x)+\gamma \} for all x\in X and \gamma >0. If p is a partial metric on X, then the function {d}_{p}:X\times X\to {\mathbb{R}}^{+} given by
is a metric on X.
We recall some definitions of a partial metric space, as follows.
Definition 2 [2]
Let (X,p) be a partial metric space. Then

(1)
a sequence \{{x}_{n}\} in a partial metric space (X,p) converges to x\in X if and only if p(x,x)={lim}_{n\to \mathrm{\infty}}p(x,{x}_{n});

(2)
a sequence \{{x}_{n}\} in a partial metric space (X,p) is called a Cauchy sequence if and only if {lim}_{m,n\to \mathrm{\infty}}p({x}_{m},{x}_{n}) exists (and is finite);

(3)
a partial metric space (X,p) is said to be complete if every Cauchy sequence \{{x}_{n}\} in X converges, with respect to {\tau}_{p}, to a point x\in X such that p(x,x)={lim}_{m,n\to \mathrm{\infty}}p({x}_{m},{x}_{n});

(4)
a subset A of a partial metric space (X,p) is closed if whenever \{{x}_{n}\} is a sequence in A such that \{{x}_{n}\} converges to some x\in X, then x\in A.
Remark 2 The limit in a partial metric space is not unique.

(1)
\{{x}_{n}\} is a Cauchy sequence in a partial metric space (X,p) if and only if it is a Cauchy sequence in the metric space (X,{d}_{p});

(2)
a partial metric space (X,p) is complete if and only if the metric space (X,{d}_{p}) is complete. Furthermore, {lim}_{n\to \mathrm{\infty}}{d}_{p}({x}_{n},x)=0 if and only if p(x,x)={lim}_{n\to \mathrm{\infty}}p({x}_{n},x)={lim}_{n\to \mathrm{\infty}}p({x}_{n},{x}_{m}).
Recently, fixed point theory has developed rapidly on partial metric spaces, see, e.g., [3–12] and the reference therein. Very recently, Haghi et al. [13] proved that some fixed point results in partial metric space results are equivalent to the results in the context of a usual metric space. On the other hand, this case is not valid for our main results, that is, the recent result of Haghi et al. [13] is not applicable to the main theorems.
Let (X,d) be a metric space, and let \mathit{CB}(X) denote the collection of all nonempty, closed and bounded subsets of X. For A,B\in \mathit{CB}(X), we define
where d(x,B):=inf\{d(x,b):b\in B\}, and it is well known that ℋ is called the Hausdorff metric induced the metric d. A multivalued mapping T:X\to \mathit{CB}(X) is called a contraction if
for all x,y\in X and k\in [0,1). The study of fixed points for multivalued contractions using the Hausdorff metric was introduced in Nadler [14].
Theorem 1 [14]
Let (X,d) be a complete metric space, and let T:X\to \mathit{CB}(X) be a multivalued contraction. Then there exists x\in X such that x\in Tx.
Very recently, Aydi et al. [15] established the notion of partial Hausdorff metric {\mathcal{H}}_{p} induced by the partial metric p. Let (X,p) be a partial metric space, and let {\mathit{CB}}^{p}(X) be the collection of all nonempty, closed and bounded subset of the partial metric space (X,p). Note that closedness is taken from (X,{\tau}_{p}), and boundedness is given as follows: A is a bounded subset in (X,p) if there exist {x}_{0}\in X and M\in \mathbb{R} such that for all a\in A, we have a\in {B}_{p}({x}_{0},M), that is, p({x}_{0},a)<p(a,a)+M. For A,B\in {\mathit{CB}}^{p}(X) and x\in X, they define
It is immediate to get that if p(x,A)=0, then {d}_{p}(x,A)=0, where {d}_{p}(x,A)=inf\{{d}_{p}(x,a):a\in A\}.
Remark 3 [15]
Let (X,p) be a partial metric space, and let A be a nonempty subset of X. Then
Aydi et al. [15] also introduced the following properties of mappings {\delta}_{p}:{\mathit{CB}}^{p}(X)\times {\mathit{CB}}^{p}(X)\to \mathbb{R} and {\mathcal{H}}_{p}:{\mathit{CB}}^{p}(X)\times {\mathit{CB}}^{p}(X)\to \mathbb{R}.
Proposition 1 [15]
Let (X,p) be a partial metric space. For A,B\in {\mathit{CB}}^{p}(X), the following properties hold:

(1)
{\delta}_{p}(A,A)=sup\{p(a,a):a\in A\};

(2)
{\delta}_{p}(A,A)\le {\delta}_{p}(A,B);

(3)
{\delta}_{p}(A,B)=0 implies that A\subset B;

(4)
{\delta}_{p}(A,B)\le {\delta}_{p}(A,C)+{\delta}_{p}(C,B){inf}_{c\in C}p(c,c).
Proposition 2 [15]
Let (X,p) be a partial metric space. For A,B\in {\mathit{CB}}^{p}(X), the following properties hold:

(1)
{\mathcal{H}}_{p}(A,A)\le {\mathcal{H}}_{p}(A,B);

(2)
{\mathcal{H}}_{p}(A,B)={\mathcal{H}}_{p}(B,A);

(3)
{\mathcal{H}}_{p}(A,B)\le {\mathcal{H}}_{p}(A,C)+{\mathcal{H}}_{p}(C,B){inf}_{c\in C}p(c,c);

(4)
{\mathcal{H}}_{p}(A,B)=0 implies that A=B.
Aydi et al. [15] proved the following important result.
Lemma 2 Let (X,p) be a partial metric space, A,B\in {\mathit{CB}}^{p}(X) and h>1. For any a\in A, there exists b=b(a)\in B such that
In this study, we also recall the MeirKeelertype contraction [16] and αadmissible [17]. In 1969, Meir and Keeler [16] introduced the following notion of MeirKeelertype contraction in a metric space (X,d).
Definition 3 Let (X,p) be a metric space, f:X\to X. Then f is called a MeirKeelertype contraction whenever for each \eta >0, there exists \gamma >0 such that
The following definition was introduced in [17].
Definition 4 Let f:X\to X be a selfmapping of a set X and \alpha :X\times X\to {\mathbb{R}}^{+}. Then f is called an αadmissible if
2 Main results
We first introduce the following notions of a strictly αadmissible and and an αMeirKeeler contraction with respect to the partial Hausdorff metric {\mathcal{H}}_{p}.
Definition 5 Let (X,p) be a partial metric space, T:X\to {\mathit{CB}}^{p}(X) and \alpha :X\times X\to {\mathbb{R}}^{+}\mathrm{\u2572}\{0\}. We say that T is strictly αadmissible if
Definition 6 Let (X,p) be a partial metric space and \alpha :X\times X\to {\mathbb{R}}^{+}\mathrm{\u2572}\{0\}. We call T:X\to {\mathit{CB}}^{p}(X) an αMeirKeeler contraction with respect to the partial Hausdorff metric {\mathcal{H}}_{p} if the following conditions hold:
(c_{1}) T is strictly αadmissible;
(c_{2}) for each \eta >0, there exists \gamma >0 such that
Remark 4 Note that if T:X\to {\mathit{CB}}^{p}(X) is a αMeirKeeler contraction with respect to the partial Hausdorff metric {\mathcal{H}}_{p}, then we have that for all x,y\in X
Further, if p(x,y)=0, then {\mathcal{H}}_{p}(Tx,Ty)=0. On the other hand, if p(x,y)=0, then \alpha (x,y){\mathcal{H}}_{p}(Tx,Ty)<p(x,y).
We now state and prove our main result.
Theorem 2 Let (X,p) be a complete partial metric space. Suppose that T:X\to {\mathit{CB}}^{p}(X) is an αMeirKeeler contraction with respect to the partial Hausdorff metric ℋ and that there exists {x}_{0}\in X such that \alpha ({x}_{0},y)>1 for all y\in T{x}_{0}. Then T has a fixed point in X (that is, there exists {x}^{\ast}\in X such that {x}^{\ast}\in T{x}^{\ast}).
Proof Let {x}_{1}\in T{x}_{0}. Since T:X\to {\mathit{CB}}^{p}(X) is an αMeirKeeler contraction with respect to the partial Hausdorff metric {\mathcal{H}}_{p}, by Remark 4, we have that
Put \alpha ({x}_{0},{x}_{1})={k}_{0}>1, and let {x}_{2}\in T{x}_{1}. From Lemma 2 with h=\sqrt{{k}_{0}}, we have that
Using (1) and (2), we obtain
So, we can obtain a sequence {x}_{n}\in X recursively as follows:
Since T is strictly αadmissible, we deduce that \alpha ({x}_{1},{x}_{2})={k}_{1}>1. Continuing this process, we have that
Since T:X\to {\mathit{CB}}^{p}(X) is an αMeirKeeler contraction with respect to the partial Hausdorff metric {\mathcal{H}}_{p}, by Remark 4, we have that
From Lemma 2 with h=\sqrt{{k}_{n}}, we have that
Using (5) and (6), we obtain
Now, from (7) and by the mathematical induction, we obtain
Since {k}_{n}>1 for all n\in \mathbb{N}\cup \{0\}, we get
Put
Using (8) and (9), we obtain
Letting n\to \mathrm{\infty} in (10). Then
By the property (p_{2}) of a partial metric and using (11), we have
Using (10) and the property (p_{4}) of a partial metric, for any m\in \mathbb{N}, we have
Using (12) and (13), we get
By the definition of {d}_{p}, we get that for any m\in \mathbb{N},
This yields that \{{x}_{n}\} is a Cauchy sequence in (X,{d}_{p}). Since (X,p) is complete, from Lemma 1, (X,{d}_{p}) is a complete metric space. Therefore, \{{x}_{n}\} converges to some {x}^{\ast}\in X with respect to the metric {d}_{p}, and we also have
Since T:X\to {\mathit{CB}}^{p}(X) is an αMeirKeeler contraction with respect to the partial Hausdorff metric ℋ, by Remark 4, we have that
By the definition of the mapping α, we have that \alpha ({x}_{n},{x}^{\ast})>0. Using (15), we get
Now {x}_{n+1}\in T{x}_{n} gives that
Using (16), we get
By the property (p_{4}) of a partial metric, we have
Taking limit as n\to \mathrm{\infty}, and using (12), (15) and (17), we obtain
Therefore, from (15), p({x}^{\ast},{x}^{\ast})=0, we obtain
which implies that {x}^{\ast}\in T{x}^{\ast} by Remark 3. □
The following theorem, the main result of [15], is a consequence of Theorem 2 by taking \alpha (x,y)=\frac{1}{k} for k\in (0,1).
Theorem 3 [15]
Let (X,p) be a complete partial metric space. If T:X\to {\mathit{CB}}^{p}(X) is a multivalued mapping such that for all x,y\in X, we have
where k\in (0,1). Then T has a fixed point.
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Chen, CM., Karapınar, E. Fixed point results for the αMeirKeeler contraction on partial Hausdorff metric spaces. J Inequal Appl 2013, 410 (2013). https://doi.org/10.1186/1029242X2013410
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DOI: https://doi.org/10.1186/1029242X2013410