A note on ‘ψ-Geraghty type contractions’

Abstract

Very recently, the notion of a ψ-Geraghty type contraction was defined by Gordji et al. (Fixed Point Theory and Applications 2012:74, 2012). In this short note, we realize that the main result via ψ-Geraghty type contraction is equivalent to an existing related result in the literature. Consequently, all results inspired by the paper of Gordji et al. in (Fixed Point Theory and Applications 2012:74, 2012) can be derived in the same way.

MSC:47H10, 54H25, 46J10, 46J15.

1 Introduction and preliminaries

One of the celebrated generalizations of the Banach contraction (mapping) principle was given by Geraghty [1].

Theorem 1.1 (Geraghty [1])

Let $(X,d)$ be a complete metric space and $T:X\to X$ be an operator. Suppose that there exists $\beta :(0,\mathrm{\infty })\to [0,1)$ satisfying the condition

$$\beta (t_n)\to 1\mathit{\text{implies}}{t}_n\to 0.$$

(1)

If T satisfies the following inequality:

$$d(Tx,Ty)\le \beta (d(x,y))d(x,y),\mathit{\text{for any }}x,y\in X,$$

(2)

then T has a unique fixed point.

Let $\mathcal{S}$ denote the set of all functions $\beta :(0,\mathrm{\infty })\to [0,1)$ satisfying (1). This nice result of Geraghty [1] has been studied by a number of authors, see e.g. [2–10] and references therein.

In the following Harandi and Emami [2] reconsidered Theorem 1.1 in the framework of partially ordered metric spaces (see also [11]).

Theorem 1.2 Let $(X,⪯,d)$ be a partially ordered complete metric space. Let $f:X\to X$ be an increasing mapping such that there exists an element $x_0\in X$ with $x_0⪯f{x}_0$. If there exists $\alpha \in \mathcal{S}$ such that

$$d(fx,fy)\le \alpha (d(x,y))d(x,y),$$

(3)

for each $x,y\in X$ with $x⪰y$, then f has a fixed point provided that either f is continuous or X is such that if an increasing sequence $\{x_n\}\to x$ in X; then $x_n⪯x$, for all n. Besides, if for each $x,y\in X$ there exists $z\in X$ which is comparable to x and y, then f has a unique fixed point.

Very recently, Gordji et al. [12] supposedly improved and extended Theorem 1.2 in the following way via the auxiliary function defined below. Let Ψ denote the class of the functions $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ which satisfy the following conditions:

(${\psi }_1$) ψ is nondecreasing;

(${\psi }_2$) ψ is subadditive, that is, $\psi (s+t)\le \psi (s)+\psi (t)$;

(${\psi }_3$) ψ is continuous;

(${\psi }_4$) $\psi (t)=0\iff t=0$.

The following is the main theorem of Gordji et al. [12].

Theorem 1.3 Let $(X,⪯,d)$ be a partially ordered complete metric space. Let $f:X\to X$ be a nondecreasing mapping such that there exists $x_0\in X$ with $x_0⪯f{x}_0$. Suppose that there exist $\alpha \in \mathcal{S}$ and $\psi \in \mathrm{\Psi }$ such that

$$\psi (d(fx,fy))\le \alpha \left(\psi (d(x,y))\right)\psi (d(x,y)),$$

(4)

for all $x,y\in X$ with $x⪯y$. Assume that either f is continuous or X is such that if an increasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for each $n\ge 1$. Then f has a fixed point.

2 Main results

We start this section with the following lemma, which is the skeleton of this note.

Lemma 2.1 Let $(X,d)$ be a metric space and $\psi \in \mathrm{\Psi }$. Then, a function $d_{\psi }:X\times X\to [0,\mathrm{\infty })$ defined by $d_{\psi }(x,y)=\psi (d(x,y))$ forms a metric on X. Moreover, $(X,d)$ is complete if and only if $(X,d_{\psi })$ is complete.

Proof

• (1) If $x=y$, then $d(x,y)=0$. Due to (${\psi }_4$), we have $\psi (d(x,y))=0$. The converse is obtained analogously.

• (2) $d_{\psi }(x,y)=\psi (d(x,y))=\psi (d(y,x))=d_{\psi }(y,x)$.

• (3) Since ψ is nondecreasing, we have $\psi (d(x,y))\le \psi (d(x,z)+d(z,y))$. Regarding the subadditivity of ψ, we derived

  $$\begin{array}{rl}{d}_{\psi }(x,y)& =\psi (d(x,y))\le \psi (d(x,z)+d(z,y))\\ \le \psi (d(x,z))+\psi (d(z,y))\\ =d_{\psi }(x,z)+d_{\psi }(z,y).\end{array}$$

Notice that the completeness of $(X,d_{\psi })$ follows from (${\psi }_3$) and (${\psi }_4$). □

The following is the main result of this note.

Theorem 2.2 Theorem  1.3 is a consequence of Theorem  1.2.

Proof Due to Lemma 2.1, we derived the result that $(X,d_{\psi })$ is a complete metric space. Furthermore, the condition (4) turns into

$$d_{\psi }(fx,fy)\le \alpha (d_{\psi }(x,y))d_{\psi }(x,y).$$

(5)

Hence all conditions of Theorem 1.2 are satisfied. □

3 The best proximity case

Let A and B be two nonempty subsets of a metric space $(X,d)$. We denote by $A_0$ and $B_0$ the following sets:

$$\begin{array}{r}{A}_0=\{x\in A:d(x,y)=d(A,B)\text{ for some }y\in B\},\\ B_0=\{y\in B:d(x,y)=d(A,B)\text{ for some }x\in A\},\end{array}$$

(6)

where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$.

In [13, 14], the author introduces the following definition.

Definition 3.1 Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with $A_0\ne \mathrm{\varnothing }$. Then the pair $(A,B)$ is said to have the P-property if and only if, for any $x_1,x_2\in A_0$ and $y_1,y_2\in B_0$,

$$d(x_1,y_1)=d(A,B)\text{and}d(x_2,y_2)=d(A,B)\Rightarrow d(x_1,x_2)=d(y_1,y_2).$$

(7)

Caballero et al. proved the following result.

Theorem 3.2 (See [8])

Let $(A,B)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $A_0$ is nonempty. Let $T:A\to B$ be a Geraghty contraction, i.e. there exists $\beta \in \mathcal{S}$ such that

$$d(Tx,Ty)\le \beta (d(x,y))d(x,y),\mathit{\text{for any }}x,y\in A.$$

(8)

Suppose that T is continuous and satisfies $T(A_0)\subseteq B_0$. Suppose also that the pair $(A,B)$ has the P-property. Then there exists a unique $x^{\ast }$ in A such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$.

Inspired by Gordji et al. [12] and Caballero et al. [8], Karapinar [7] reported the following result.

Theorem 3.3 Let $(A,B)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $A_0$ is nonempty. Let $T:A\to B$ be ψ-Geraghty contraction, i.e. there exists $\beta \in \mathcal{S}$ such that

$$\psi (d(Tx,Ty))\le \alpha \left(\psi (d(x,y))\right)\psi (d(x,y)),\mathit{\text{for any }}x,y\in A.$$

(9)

Suppose that T is continuous and satisfies $T(A_0)\subseteq B_0$. Suppose also that the pair $(A,B)$ has the P-property. Then there exists a unique $x^{\ast }$ in A such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$.

The following lemmas belong to Akbar and Gabeleh [15].

Lemma 3.4 [15]

Let $(A,B)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $A_0$ is nonempty and $(A,B)$ has the P-property. Then $(A_0,B_0)$ is a closed pair of subsets of X.

Lemma 3.5 [15]

Let $(A,B)$ be a pair of nonempty closed subsets of a metric space $(X,d)$ such that $A_0$ is nonempty. Assume that the pair $(A,B)$ has the P-property. Then there exists a bijective isometry $g:A_0\to B_0$ such that $d(x,gx)=dist(A,B)$.

Very recently, by using Lemma 3.4 and Lemma 3.5, Akbar and Gabeleh [15] proved that the best proximity point results via P-property can be obtained from the associate results in fixed point theory. In particular they proved the following theorem.

Theorem 3.6 Theorem  3.2 is a consequence of Theorem  1.1.

As a consequence of Theorem 2.2 we can observe the following result.

Corollary 3.7 Theorem  3.3 is a consequence of Theorem  3.2.

Regarding the analogy, we omit the proof.

Theorem 3.8 Theorem  3.3 is a consequence of Theorem  1.1.