Open Access

# Some new fixed point theorems for set-valued contractions in complete metric spaces

Fixed Point Theory and Applications20112011:72

https://doi.org/10.1186/1687-1812-2011-72

Accepted: 31 October 2011

Published: 31 October 2011

## Abstract

In this article, we obtain some new fixed point theorems for set-valued contractions in complete metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

MSC: 47H10, 54C60, 54H25, 55M20.

## Keywords

fixed point theoremset-valued contraction

## 1 Introduction and preliminaries

Let (X, d) be a metric space, D a subset of X and f : DX be a map. We say f is contractive if there exists α [0, 1) such that for all x, y D,
$d\left(fx,fy\right)\le \alpha \cdot d\left(x,y\right).$
The well-known Banach's fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping f : XX is called a quasi-contraction if there exists k < 1 such that
$d\left(fx,fy\right)\le k\cdot max\left\{d\left(x,y\right),d\left(x,fx\right),d\left(y,fy\right),d\left(x,fy\right),d\left(y,fx\right)\right\}$

for any x, y X. In 1974, C'iric' [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

Throughout we denote the family of all nonempty closed and bounded subsets of X by CB(X). The existence of fixed points for various multi-valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [3] extended the famous Banach Contraction Principle from single-valued mapping to multi-valued mapping and proved the below fixed point theorem for multi-valued contraction.

Theorem 1[3]Let (X, d) be a complete metric space and T : XCB(X). Assume that there exists c [0, 1) such that
$\mathcal{H}\left(Tx,Ty\right)\le cd\left(x,y\right)\phantom{\rule{1em}{0ex}}forall\phantom{\rule{2.77695pt}{0ex}}x,y\in X,$

where$\mathcal{H}$denotes the Hausdorff metric on CB(X) induced by d, that is, H(A, B) = max{supxAD(x, B), supyBD(y, A)}, for all A, B CB(X) and D(x, B) = infzBd(x, z). Then, T has a fixed point in X.

In 1989, Mizoguchi-Takahashi [4] proved the following fixed point theorem.

Theorem 2[4]Let (X, d) be a complete metric space and T : XCB(X). Assume that
$\mathcal{H}\left(Tx,Ty\right)\le \xi \left(d\left(x,y\right)\right)\cdot d\left(x,y\right)$

for all x, y X, where ξ : [0, ∞) → [0, 1) satisfies$lim\underset{s\to {t}^{+}}{sup}\xi \left(s\right)<1$for all t [0, ∞). Then, T has a fixed point in X.

In the recent, Amini-Harandi [5] gave the following fixed point theorem for set-valued quasi-contraction maps in metric spaces.

Theorem 3[5]Let (X, d) be a complete metric space. Let T : XCB(X) be a k-set-valued quasi-contraction with $k<\frac{1}{2}$, that is,
$ℋ\left(Tx,Ty\right)\le k\cdot \mathrm{max}\left\{\left(x,y\right),D\left(x,Tx\right),D\left(y,Ty\right),D\left(x,Ty\right)\right),D\left(y,Tx\right)\right\}$

for any x, y X. Then, T has a fixed point in X.

## 2 Fixed point theorem (I)

In this section, we assume that the function ψ : +5+ satisfies the following conditions:

(C1) ψ is a strictly increasing, continuous function in each coordinate, and

(C2) for all t +, ψ(t, t, t, 0, 2t) < t, ψ(t, t, t, 2t, 0) < t, ψ(0, 0, t, t, 0) < t and ψ(t, 0, 0, t, t) < t.

Definition 1 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued ψ-contraction, if
$ℋ\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right),D\left(x,Tx\right),D\left(y,Ty\right),D\left(x,Ty\right)\right),D\left(y,Tx\right)\right)$

for all x, y X.

We now state the main fixed point theorem for a set-valued ψ-contraction in metric spaces, as follows:

Theorem 4 Let (X, d) be a complete metric space. Let T : XCB(X) be a set-valued ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b B such that d(a, b) < γ. Since T : XCB(X) is a set-valued ψ-contraction, we have
$ℋ\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right),D\left(x,Tx\right),D\left(y,Ty\right),D\left(x,Ty\right)\right),D\left(y,Tx\right)\right)$
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
$\begin{array}{ll}\hfill d\left({x}_{n+1},{x}_{n}\right)& \le \psi \left(d\left({x}_{n},{x}_{n-1}\right),D\left({x}_{n},T{x}_{n}\right),D\left({x}_{n-1},T{x}_{n-1}\right),D\left({x}_{n},T{x}_{n-1}\right),D\left({x}_{n-1},T{x}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}_{n-1},{x}_{n}\right),d\left({x}_{n},{x}_{n}\right),d\left({x}_{n-1},{x}_{n+1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}_{n-1},{x}_{n}\right),0,d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right)\right),\phantom{\rule{2em}{0ex}}\end{array}$
and hence, we can deduce that for each n ,
$d\left({x}_{n+1},{x}_{n}\right)\le d\left({x}_{n},{x}_{n-1}\right).$
Let we denote c m = d(xm+1, x m ). Then, c m is a non-increasing sequence and bounded below. Thus, it must converges to some c ≥ 0. If c > 0, then by the above inequalities, we have
$c\le {c}_{n+1}\le \psi \left({c}_{n},{c}_{n},{c}_{n},0,2{c}_{n}\right).$
Passing to the limit, as n → ∞, we have
$c\le c\le \psi \left(c,c,c,0,2c\right)

which is a contradiction. Hence, c = 0.

We next claim that the following result holds:

for each γ > 0, there is n0(γ) such that for all m > n > n0(γ),
$d\left({x}_{m},{x}_{n}\right)<\gamma .\phantom{\rule{1em}{0ex}}\left(*\right)$
We shall prove (*) by contradiction. Suppose that (*)is false. Then, there exists some γ > 0 such that for all k , there exist m k , n k with m k > n k k satisfying:
1. (1)

m k is even and n k is odd;

2. (2)

$d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)\ge \gamma$;

3. (3)

m k is the smallest even number such that the conditions (1), (2) hold.

Since c m 0, by (2), we have $\underset{k\to \infty }{lim}d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)=\gamma$ and
$\begin{array}{c}\gamma \le d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)\le ℋ\left(T{x}_{{m}_{k}-1},T{x}_{{n}_{k}-1}\right)\\ \le \psi \left(d\left({x}_{{m}_{k}-1},{x}_{{n}_{k}-1}\right),d\left({x}_{{m}_{k}-1},{x}_{{m}_{k}}\right),d\left({x}_{{n}_{k}-1},{x}_{{n}_{k}}\right),d\left({x}_{{m}_{k}-1},{x}_{{n}_{k}}\right),d\left({x}_{{n}_{k}-1},{x}_{{m}_{k}}\right)\right)\\ \le \psi \left({c}_{{m}_{k}-1}+d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)+{c}_{{n}_{k}-1},{c}_{{m}_{k}-1},{c}_{{n}_{k}-1},{c}_{{m}_{k}-1}+d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right),d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)+{c}_{{n}_{k}-1}\right)\right).\end{array}$
Letting k → ∞. Then, we get
$\gamma \le \psi \left(\gamma ,0,0,\gamma ,\gamma \right)<\gamma ,$

a contradiction. It follows from (*) that the sequence {x n } must be a Cauchy sequence.

Similarly, we also conclude that for each n ,
$\begin{array}{ll}\hfill d\left({x}_{n},{x}_{n+1}\right)& \le \psi \left(d\left({x}_{n-1},{x}_{n}\right),D\left({x}_{n-1},T{x}_{n-1}\right),D\left({x}_{n},T{x}_{n}\right),D\left({x}_{n-1},T{x}_{n}\right),D\left({x}_{n},T{x}_{n-1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({x}_{n-1},{x}_{n}\right),d\left({x}_{n-1},{x}_{n}\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}_{n-1},{x}_{n+1}\right),d\left({x}_{n},{x}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({x}_{n-1},{x}_{n}\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}_{n-1},{x}_{n}\right),d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right),0\right),\phantom{\rule{2em}{0ex}}\end{array}$
and hence, we have that for each n ,
$d\left({x}_{n},{x}_{n+1}\right)\le d\left({x}_{n-1},{x}_{n}\right).$
Let we denote b m = d(x m , xm+1). Then, b m is a non-increasing sequence and bounded below. Thus, it must converges to some b ≥ 0. If b > 0, then by the above inequalities, we have
$b\le {b}_{n+1}\le \psi \left({b}_{n},{b}_{n},{b}_{n},2{b}_{n},0\right).$
Passing to the limit, as n → ∞, we have
$b\le b\le \psi \left(b,b,b,2b,0\right)

which is a contradiction. Hence, b = 0. By the above argument, we also conclude that {x n } is a Cauchy sequence.

Since X is complete, there exists μ X such that limn→∞x n = μ. Therefore,
$\begin{array}{ll}\hfill D\left(\mu ,T\mu \right)& =\underset{n\to \infty }{lim}D\left({x}_{n+1},T\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_{n},T\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\psi \left(d\left({x}_{n},\mu \right),D\left({x}_{n},T{x}_{n}\right),D\left(\mu ,T\mu \right),D\left({x}_{n},T\mu \right),D\left(\mu ,T{x}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\psi \left(d\left({x}_{n},\mu \right),d\left({x}_{n},{x}_{n+1}\right),D\left(\mu ,T\mu \right),D\left({x}_{n},T\mu \right),d\left(\mu ,{x}_{n+1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(0,0,D\left(\mu ,T\mu \right),D\left(\mu ,T\mu \right),0\right)\phantom{\rule{2em}{0ex}}\\

and hence, D(μ, ) = 0, that is, μ , since is closed.

## 3 Fixed point theorem (II)

In 1972, Chatterjea [6] introduced the following definition.

Definition 2 Let (X, d) be a metric space. A mapping f : XX is said to be a$\mathcal{C}$-contraction if there exists$\alpha \in \left(0,\frac{1}{2}\right)$such that for all x, y X, the following inequality holds:
$d\left(fx,fy\right)\le \alpha \cdot \left(d\left(x,fy\right)+d\left(y,fx\right)\right).$

Choudhury [7] introduced a generalization of $\mathcal{C}$-contraction, as follows:

Definition 3 Let (X, d) be a metric space. A mapping f : XX is said to be a weakly$\mathcal{C}$-contraction if for all x, y X,
$d\left(fx,fy\right)\le \frac{1}{2}\left(d\left(x,fy\right)+d\left(y,fx\right)-\varphi \left(d\left(x,fy\right),d\left(y,fx\right)\right)\right),$

where ϕ : +2+is a continuous function such that ψ(x, y) = 0 if and only if x = y = 0.

In [6, 7], the authors proved some fixed point results for the $\mathcal{C}$-contractions. In this section, we present some fixed point results for the weakly ψ-$\mathcal{C}$-contraction in complete metric spaces.

Definition 4 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued weakly ψ-$\mathcal{C}$-contraction, if for all x, y X
$\mathcal{H}\left(Tx,Ty\right)\le \psi \left(\left[D\left(x,Ty\right)+D\left(y,Tx\right)-\varphi \left(D\left(x,Ty\right),D\left(y,Tx\right)\right)\right]\right),$

where

(1) ψ : ++is a strictly increasing, continuous function with$\psi \left(t\right)\le \frac{1}{2}t$for all t > 0 and ψ(0) = 0;

(2) ϕ : +2+is a strictly decreasing, continuous function in each coordinate, such that ϕ(x, y) = 0 if and only if x = y = 0 and ϕ(x, y) ≤ x + y for all x, y +.

Theorem 5 Let (X, d) be a complete metric space. Let T : XCB(X) be a set-valued weakly$\mathcal{C}$-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b B such that d(a, b) < γ. Since T : XCB(X) be a set-valued weakly ψ-$\mathcal{C}$-contraction, we have that
$\mathcal{H}\left(Tx,Ty\right)\le \psi \left(\left[D\left(x,Ty\right)+D\left(y,Tx\right)-\varphi \left(D\left(x,Ty\right),D\left(y,Tx\right)\right)\right]\right)$
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
$\begin{array}{ll}\hfill d\left({x}_{n+1},{x}_{n}\right)& \le \mathcal{H}\left(T{x}_{n},T{x}_{n-1}\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(\left[D\left({x}_{n},T{x}_{n-1}\right)+D\left({x}_{n-1},T{x}_{n}\right)-\varphi \left(D\left({x}_{n},T{x}_{n-1}\right),D\left({x}_{n-1},T{x}_{n}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(\left[d\left({x}_{n},{x}_{n}\right)+d\left({x}_{n-1},{x}_{n+1}\right)-\varphi \left(d\left({x}_{n},{x}_{n}\right),d\left({x}_{n-1},{x}_{n+1}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ =\psi \left(\left[0+d\left({x}_{n-1},{x}_{n+1}\right)-\varphi \left(0,d\left({x}_{n-1},{x}_{n+1}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(\left[d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{2}\left[d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right)\right],\phantom{\rule{2em}{0ex}}\end{array}$
and hence, we deduce that for each n ,
$d\left({x}_{n+1},{x}_{n}\right)\le d\left({x}_{n},{x}_{n-1}\right).$
Thus, {d(xn+1, x n )} is non-increasing sequence and bounded below and hence it is convergent. Let limn→∞d(xn+1, x n ) = ξ. Letting n → ∞ in (**), we have
$\begin{array}{ll}\hfill \xi =\underset{n\to \infty }{lim}d\left({x}_{n+1},{x}_{n}\right)& \le \underset{n\to \infty }{lim}\psi \left(\left[d\left({x}_{n-1},{x}_{n+1}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\frac{1}{2}\left[d\left({x}_{n-1},{x}_{n+1}\right)\right]\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\frac{1}{2}\left[d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right)\right]\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{2}\left[\xi +\xi \right]=\xi ,\phantom{\rule{2em}{0ex}}\end{array}$
that is,
$\underset{n\to \infty }{lim}d\left({x}_{n-1},{x}_{n+1}\right)=2\xi .$
By the continuity of ψ and ϕ, letting n → ∞ in (**), we have
$\xi \le \psi \left(2\xi -\varphi \left(0,2\xi \right)\right)\le \xi -\frac{1}{2}\cdot \varphi \left(0,2\xi \right)\le \xi .$

Hence, we have ϕ(0, 2ξ) = 0, that is, ξ = 0. Thus, limn→∞d(xn+1, x n ) = 0.

We next claim that the following result holds:

for each γ > 0, there is n0(γ) such that for all m > n > n0(γ),
$d\left({x}_{m},{x}_{n}\right)<\gamma .\phantom{\rule{1em}{0ex}}\left(***\right)$
We shall prove (***) by contradiction. Suppose that (***) is false. Then, there exists some γ > 0 such that for all k , there exist m k , n k with m k > n k k satisfying:
1. (1)

m k is even and n k is odd;

2. (2)

$d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)\ge \gamma$;

3. (3)

m k is the smallest even number such that the conditions (1), (2) hold.

Since d(xn+1, x n ) 0, by (2), we have $\underset{k\to \infty }{lim}d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)=\gamma$ and
$\begin{array}{ll}\hfill \gamma & \le d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)\le \mathcal{H}\left(T{x}_{{m}_{k}-1},T{x}_{{n}_{k}-1}\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(\left[D\left({x}_{{m}_{k}-1},T{x}_{{n}_{k}-1}\right)+D\left({x}_{{n}_{k}-1},T{x}_{{m}_{k}-1}\right)-\varphi \left(D\left({x}_{{m}_{k}-1},T{x}_{{n}_{k}-1}\right),D\left({x}_{{n}_{k}-1},T{x}_{{m}_{k}-1}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(\left[d\left({x}_{{m}_{k}-1},{x}_{{n}_{k}}\right)+d\left({x}_{{n}_{k}-1},{x}_{{m}_{k}}\right)-\varphi \left(d\left({x}_{{m}_{k}-1},{x}_{{n}_{k}}\right),d\left({x}_{{n}_{k}-1},T{x}_{{m}_{k}}\right)\right)\right]\right).\phantom{\rule{2em}{0ex}}\end{array}$
Since
$d\left({x}_{{m}_{k}-1},{x}_{{n}_{k}}\right)+d\left({x}_{{n}_{k}-1},{x}_{{m}_{k}}\right)\le d\left({x}_{{m}_{k}-1},{x}_{{m}_{k}}\right)+d\left({x}_{{m}_{k}},{x}_{{n}_{k}}\right)+d\left({x}_{{n}_{k}},{x}_{{m}_{k}}\right)+d\left({x}_{{n}_{k}-1},{x}_{{n}_{k}}\right),$
letting k → ∞, then we get
$\gamma \le \psi \left(2\gamma -\varphi \left(\gamma ,\gamma \right)\right)\le \gamma ,$

and hence, ϕ(γ, γ)) = 0. By the definition of ϕ, we get γ = 0, a contradiction. This proves that the sequence {x n } must be a Cauchy sequence.

Since X is complete, there exists z X such that limn→∞x n = z. Therefore,
$\begin{array}{ll}\hfill D\left(z,Tz\right)& =\underset{n\to \infty }{lim}D\left({x}_{n+1},Tz\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_{n},Tz\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\psi \left(\left[D\left({x}_{n},Tz\right)+D\left(z,T{x}_{n}\right)-\varphi \left(D\left({x}_{n},Tz\right),D\left(z,T{x}_{n}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\psi \left(\left[D\left({x}_{n},Tz\right)+d\left(z,{x}_{n+1}\right)-\varphi \left(D\left({x}_{n},Tz\right),d\left(z,{x}_{n+1}\right)\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{2}D\left(z,Tz\right)\phantom{\rule{2em}{0ex}}\end{array}$

and hence, D(z, Tz) = 0, that is, z Tz, since Tz is closed.

## 4 Fixed point theorem (III)

In this section, we recall the notion of the Meir-Keeler type function (see [8]). A function φ : ++ is said to be a Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for t + with ηt < η + δ, we have φ(t) < η. We now introduce the new notions of the weaker Meir-Keeler type function φ : ++ in a metric space and the φ-function using the weaker Meir-Keeler type function, as follow:

Definition 5 Let (X, d) be a metric space. We call φ : ++a weaker Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for x, y X with ηd(x, y) < δ + η, there exists n0 such that${\phi }^{{n}_{0}}\left(d\left(x,y\right)\right)<{\gamma }_{\eta }$.

Definition 6 Let (X, d) be a metric space. A weaker Meir-Keeler type function φ ; ++is called a φ-function, if the following conditions hold:

1) φ(0) = 0, 0 < φ(t) < t for all t > 0;

2 ) φ is a strictly increasing function;

3) for each t +, {φ n (t)}nis decreasing;

4) for each${t}_{n}\in {ℝ}^{+}\\left\{0\right\}$, if limn→∞t n = γ > 0, then limn→∞φ(t n ) < γ;

5) for each t n +, if limn→∞t n = 0, then limn→∞φ(t n ) = 0.

Definition 7 Let (X, d) be a metric space. The set-valued map T : XX is said to be a set-valued weaker Meir-Keeler type φ-contraction, if
$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(\frac{1}{2}\left[D\left(x,Ty\right)+D\left(y,Tx\right)\right]\right)$

for all x, y X.

We now state the main fixed point theorem for a set-valued weaker Meir-Keeler type ψ-contraction in metric spaces, as follows:

Theorem 6 Let (X, d) be a complete metric space. Let T : CB(X) be a set-valued weaker Meir-Keeler type ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B CB(X), a A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b B such that d(a, b) < γ. Since T : XCB(X) be a set-valued ψ-contraction, we have that
$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(\frac{1}{2}\left[D\left(x,Ty\right)+D\left(y,Tx\right)\right]\right)$
for all x, y X. Suppose that x0 X and that x1 X. Then, by induction and by the above observation, we can find a sequence {x n } in X such that xn+1 Tx n and for each n ,
$\begin{array}{ll}\hfill d\left({x}_{n+1},{x}_{n}\right)& \le \phi \left(\frac{1}{2}\left[D\left({x}_{n},T{x}_{n-1}\right)+D\left({x}_{n-1},T{x}_{n}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(\frac{1}{2}\left[d\left({x}_{n},{x}_{n}\right)+d\left({x}_{n-1},{x}_{n+1}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(\frac{1}{2}\left[d\left({x}_{n-1},{x}_{n}\right)+d\left({x}_{n},{x}_{n+1}\right)\right]\right),\phantom{\rule{2em}{0ex}}\end{array}$
and by the conditions (φ1) and (φ2), we can deduce that for each n ,
$d\left({x}_{n+1},{x}_{n}\right)\le \phi \left(d\left({x}_{n},{x}_{n-1}\right)\right)
and
$d\left({x}_{n+1},{x}_{n}\right)\le \phi \left(d\left({x}_{n},{x}_{n-1}\right)\right)\le \cdots \le {\phi }^{n}\left(d\left({x}_{1},{x}_{0}\right)\right).$

By the condition (φ3), {φ n (d(x0, x1))}nis decreasing, it must converges to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then, by the definition of the weaker Meir-Keeler type function, there exists δ > 0 such that for x0, x1 X with ηd(x0, x1) < δ + η, there exists n0 such that ${\phi }^{{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$. Since limn→∞φ n (d(x0, x1)) = η, there exists m0 such that ηφ m (d(x0, x1)) < δ + η, for all mm0. Thus, we conclude that ${\phi }^{{m}_{0}+{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$. Hence, we get a contradiction. Hence, limn→∞φ n (d(x0, x1)) = 0, and hence, limn→∞d(x n , xn+1) = 0.

Next, we let c m = d(x m , xm+1), and we claim that the following result holds:

for each ε > 0, there is n0(ε) such that for all m , nn0(ε),
$d\left({x}_{m},{x}_{m+1}\right)<\epsilon .\phantom{\rule{1em}{0ex}}\left(****\right)$
We shall prove (****) by contradiction. Suppose that (****) is false. Then, there exists some ε > 0 such that for all p N, there are m p , n p with m p > n p p satisfying:
1. (i)

m p is even and n p is odd,

2. (ii)

$d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\ge \epsilon$, and

3. (iii)

m p is the smallest even number such that the conditions (i), (ii) hold.

Since c m 0, by (ii), we have $\underset{p\to \infty }{lim}d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)=\epsilon$, and
$\begin{array}{ll}\hfill \epsilon & \le d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\phantom{\rule{2em}{0ex}}\\ \le \mathcal{H}\left(T{x}_{{m}_{p}-1},T{x}_{{n}_{p}-1}\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(\frac{1}{2}\left[D\left({x}_{{m}_{p}-1},T{x}_{{n}_{p}-1}\right)+D\left({x}_{{n}_{p}-1},T{x}_{{m}_{p}-1}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(\frac{1}{2}\left[d\left({x}_{{m}_{p}-1},{x}_{{n}_{p}}\right)+d\left({x}_{{n}_{p}-1},{x}_{{m}_{p}}\right)\right]\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(\frac{1}{2}\left[d\left({x}_{{m}_{p}-1},{x}_{{m}_{p}}\right)+2d\left({x}_{{n}_{p}},{x}_{{m}_{p}}\right)+d\left({x}_{{n}_{p}-1},{x}_{{n}_{p}}\right)\right]\right).\phantom{\rule{2em}{0ex}}\end{array}$
Letting p → ∞. By the condition (φ4), we have
$\epsilon \le \underset{p\to \infty }{lim}\phi \left(\frac{1}{2}\left[d\left({x}_{{m}_{p}-1},{x}_{{m}_{p}}\right)+2d\left({x}_{{n}_{p}},{x}_{{m}_{p}}\right)+d\left({x}_{{n}_{p}-1},{x}_{{n}_{p}}\right)\right]\right)<\epsilon ,$
a contradiction. Hence, {x n } is a Cauchy sequence. Since (X, d) is a complete metric space, there exists μ X such that limn→∞xn+1= μ. Therefore,
$\begin{array}{ll}\hfill D\left(\mu ,T\mu \right)& =\underset{n\to \infty }{lim}D\left({x}_{n+1},T\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_{n},T\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\phi \left(\frac{1}{2}\left[D\left({x}_{n},T\mu \right)\right)+D\left(\mu ,T{x}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{lim}\phi \left(\frac{1}{2}\left[D\left({x}_{n},T\mu \right)\right)+d\left(\mu ,{x}_{n+1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{2}D\left(\mu ,T\mu \right),\phantom{\rule{2em}{0ex}}\end{array}$

and hence, D(μ, ) = 0, that is, μ , since is closed.

## Declarations

### Acknowledgements

This research was supported by the National Science Council of the Republic of China.

## Authors’ Affiliations

(1)
Department of Applied Mathematics, National Hsinchu University of Education, Taiwan

## References

1. Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integerales. Fund Math 1922, 3: 133–181.Google Scholar
2. C'iric' LB: A generalization of Banach's contraction principle. Proc Am Math Soc 1974,45(2):45–181.
3. Nadler SB Jr: Multi-valued contraction mappings. Pacific J Math 1969, 30: 475–488.
4. Mizoguchi N, Takahashi W: Fixed point theorems for multi-valued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X
5. Amini-Harandi A: Fixed point theory for set-valued quasi-contraction maps in metric spaces. Appl Math Lett 2011,24(2):24–1794.
6. Chatterjea SK: Fixed point theorems. C.R Acad Bulgare Sci 1972, 25: 727–730.
7. Choudhury BS:Unique fixed point theorem for weakly $\mathcal{C}$-contractive mappings. Kathmandu Uni J Sci Eng Technol 2009,5(2):5–13.Google Scholar
8. Meir A, Keeler E: A theorem on contraction mappings. J Math Anal Appl 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6