α _H-ψ_H-Multivalued Contractive Mappings and Related Results in Complete Metric Spaces with an Application

Abstract

In the present article, the notion of $\alpha$_H-$\psi$_H-multivalued contractive type mappings is introduced and some fixed point results in complete metric spaces are studied. These theorems generalize Nadler’s (Multivalued contraction mappings, Pac. J. Math., 30, 475–488, 1969) and Suzuki-Kikkawa’s (Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69, 2942–2949, 2008) results that exist in the literature. The effectiveness of the obtained results has been verified with the help of some comparative examples. Moreover, a homotopy result has been presented as an application.

1. Introduction

In 1905, D. Pompeiu [1] introduced the concept of distance between two closed sets in his thesis under the guidance of H. Poincare. According to his definition,

If A and B are two closed and bounded sets, the distance between the point $a\in A$ and the set $B$ is defined as

$$d(a, B)=min\left\{d(a, b):b\in B\right\}$$

where $d(a,b)$ is the (Euclidean) distance between the points $a$ and $b$.

After that, Pompeiu defined the notion of asymmetric distance between two sets A and B as

$$D(A,B)=max\left\{d(a, B):a\in A\right\}$$

In 1914, by using Pompeiu’s distance, German mathematician Felix Hausdorff defined a new distance notion denoted by $H(A,B)$ known as Hausdorff metric where

$$H(A, B)=max\left\{D(A,B),D(B,A)\right\}$$

In other words, one can say that Hausdorff distance is the greatest of all the distances measured from a point in one set to some point in the other set.

The list of research fields that use Pompeiu-Hausdorff distance (metric) is quite impressive. Berinde and Pacurar highlighted the role of Pompeiu-Hausdorff metric in fixed point theory in [2] (Berinde, V., Pacurar, M.; The role of the Pompeiou Hausdorff metric in fixed point theory, Creative Mathematics and Informatics, 2013, 22(13), 143–150).

Using the concept of the Hausdorff metric, in 1969, Nadler [3] introduced the notion of multivalued contraction mapping and proved a multivalued version of the well known Banach Contraction Principle. This result led to some important contributions in the field of metric fixed point theory (see definitions in the next section).

Theorem 1

([3]). Let $(X, d)$ be a complete metric space and $T:X\to P_{cb}(X)$ be a multivalued mapping satisfying

$$H(Tx, Ty)\le l d(x, y),$$

for all $x, y\in X$ and for some $l\in$ [0, 1). Then T possesses a fixed point.

Inspired by his result, various fixed point theorems concerning multivalued contractions appeared in the last few decades. For instance, in 1983, S. Reich [4] formulated the following problem:

Theorem 2

([4]). Let $(X, d)$ be a complete metric space and $T:X\to P_{cb}(X)$ be a multivalued mapping satisfying

$$H(Tx, Ty)\le \alpha (d(x, y))d(x,y),$$

for all $x, y\in X, x\ne y,$ where $\alpha :(0,\infty )\to [0, 1)$ fulfills

$$\underset{s\to t^{+}}{\lim \sup }\alpha (s)<1forallt\in (0, \infty ).$$

Then T possesses a fixed point.

After that, in 2008, Kikkawa and Suzuki [5] presented the following version of Nadler’s result by introducing a strictly decreasing function to modify the contractive condition.

Theorem 3

([5]). Let $(X, d)$ be a complete metric space and $T:X\to P_{cb}(X)$ be a multivalued mapping satisfying

$$\eta (r) d(x, Tx)\le d(x, y)⟹H(Tx, Ty)\le r d(x, y),$$

for all $x, y\in X$ and for some $r\in$ [0, 1) where $\eta$: $[0, 1)\to \left(\frac{1}{2}, 1\right]$ is a mapping defined by $\eta (r)=\frac{1}{1+r}$. Then T possesses a fixed point.

Then, Mot and Petrusel [6] gave another result concerning special multivalued generalized contractions, which extended the result of Kikkawa and Suzuki.

Theorem 4

([6]). Let $(X, d)$ be a complete metric space and $T:X\to P_{cb}(X)$ be a multifunction satisfying

$$\frac{1-\beta -\gamma }{1+\alpha } d(x, Tx)\le d(x, y)⟹ H(Tx, Ty)\le \alpha d(x, y)+\beta d(x,Tx)+\gamma d(y,Ty),$$

for all $x, y\in X$ and for $\alpha , \beta , \gamma \in$ [0, 1) such that $\alpha +\beta +\gamma <1$. Then T has a fixed point.

In 2010, S. Reich and A.J. Zaslavski [7,8] studied the existence and approximation of fixed points for certain set-valued contractive type mappings and the existence of convergent iterations for these mappings. Thereafter, C. Chifu, G. Petrusel and M.F. Bota [9] presented some strict fixed point theorems for multivalued operators satisfying Reich-type conditions on a metric space.

Recently, M. Jleli, H.K. Nashine, B. Samet and C. Vetro [10] studied the existence of multivalued weakly Picard operators in partial Hausdorff metric spaces. On the other hand, Sen et al. [11] investigated some results for the existence of fixed points and best proximity points of multivalued cyclic self-mappings by using a generalized contractive condition involving Hausdorff distances. This work is a nice contribution to the literature on fixed point theory.

In 2012, Samet et al. [12] introduced the notion of $\alpha$-$\psi$-contractive and $\alpha$-admissible mappings in complete metric spaces and proved the following result. (see related definitions in next section).

Theorem 5

([12]). Let $(X, d)$ be a complete metric space and $T:X\to X$ be an $\alpha$-$\psi$-contractive mapping satisfying the following conditions:

1. 

$T$is$\alpha$-admissible;

2. 

there exists$x_0\in X$such that ($x_0,$$T{x}_0)\ge$1;

3. 

T is continuous;

4. 

for every${\left\{x_n\right\}}_{n\in N}\subset X$such that$x_n$→ x ∈ X and α($x_n,$$x_{n+1}$) ≥ 1 for n ∈ N, we have α($x_n,$x) ≥ 1 for n ∈ N.

Then T has a fixed point. Moreover, if in addition we suppose that for every pair (u, v) ∈ X × X there exists w ∈ X such that α(u, w) ≥ 1 and α(v, w) ≥ 1, we have a unique fixed point.

In this paper, we introduce the notion of ${\alpha }_H$-${\psi }_H$-multivalued contractive type mappings in Hausdorff metric spaces by modifying assumptions of the above mentioned results. Some fixed point theorems are obtained which generalize some well-known results existing in the literature. Examples are also given to check the efficacy of the results. Moreover, a result, in homotopy theory is presented as an application in the last section.

2. Preliminaries

This section presents some elementary definitions, notations and results that will be used in later sections and introduce some new terminologies.

The following notations will be used throughout the paper.

• R—the set of real numbers,
• R⁺—the set of non-negative real numbers,
• N—the set of positive integers,
• $(X,d)$—metric space,
• $P_{cb}(X)$—the collection of all nonempty closed and bounded subsets of X.

Also, $H(A,B)=\max \left\{D(A,B),D(B,A)\right\}forA,B\in$ $P_{cb}(X),$ where $D(A,B)=sup\left\{d(x,B):x\in A\right\},d(x,B)=inf\left\{d(x,y):y\in B\right\}.$

Here, H is the Hausdorff metric (induced by d) on $P_{cb}(X).$

Samet et al. defined a new category of functions in [12] as follows:

$\mathsf{\Psi }$ denotes the family of functions $\psi :[0, \infty )\to [0, \infty )$ with the following properties:

• ${{\displaystyle \sum }}_{n=1}^{\infty }{\psi }^n(t)<\infty$ for every $t>0$, where ${\psi }^n$ is $n^{th}$ iterate of $\psi ;$
• $\psi$ is nondecreasing.

By [12], if $(X, d)$ is a metric space and $T$ is a self mapping defined on $X$, then $T:X\to X$ is called an $\alpha -\psi$-contractive mapping if there exists $\psi \in \mathsf{\Psi }$ and $\alpha$: $X\times X\to [0, \infty )$ such that

$$\alpha (x, y) d(Tx, Ty)\le \psi (d(x, y))\mathrm{for}\mathrm{all}x, y\in X$$

and the mapping $T$ is called $\alpha$-admissible if

$$\alpha (x, y)\ge 1⟹\alpha (Tx, Ty)\ge 1.$$

3. Main Results

In this section, the notion of ${\alpha }_H$-${\psi }_H$-multivalued contractive type mappings is introduced and related fixed point results in the framework of Hausdorff metric spaces are studied.

Definition 1.

Let ${\Psi }_H$ be the family of non-decreasing functions ${\psi }_H:[0, +\infty )\to [0, +\infty )$ with the following properties:

1. 

${{\displaystyle \sum }}_{n=1}^{+\infty }{\psi }_H{}^n(x)<+\infty$for every$x>0,$where${\psi }_H{}^n$is$n^{th}$iterate of${\psi }_H;$

2. 

${\psi }_H(x)+{\psi }_H(y)={\psi }_H(x+y)$for all$x, y\in [0, +\infty ).$

Definition 2.

If$(X, d)$is a metric space and$T$is a self mapping defined on$X,$then$T:X\to X$is called an${\alpha }_H$-${\psi }_H$-contractive mapping if there exists${\psi }_H\in {\Psi }_H$and${\alpha }_H$:$X\times X\to [0, \infty )$such that

$${\alpha }_H(x, y) d(Tx, Ty)\le {\psi }_H(d(x, y))\mathrm{for}\mathrm{all}x, y\in X$$

(1)

and the mapping$T$is called${\alpha }_H$-admissible if

$${\alpha }_H(x, y)\ge 1⟹{\alpha }_H(s, t)\ge 1$$

for every$s\in Tx$and$t\in Ty$.

Remark 1.

If$T$is a contraction mapping defined on$X,$then$T:X\to X$is called an${\alpha }_H$-${\psi }_H$-contractive mapping where${\alpha }_H(x,y)=1$for all$x, y\in X$and${\psi }_H(t)=kt$for all$t\ge 0$and some$k\in [0,1)$.

Now we prove the main results.

Theorem 1.

Let $(X, d)$ be a complete metric space and $T :X\to P_{cb}(X)$ be a multivalued ${\alpha }_H$ - ${\psi }_H$ -contractive mapping such that

1. 

T is${\alpha }_H$-admissible;

2. 

there exists$x_0\in X$such that${\alpha }_H(x_0, s)\ge$1 for every $s\in T{x}_0$;

3. 

if {$x_n$} is a sequence in $X$ such that ${\alpha }_H(x_n,x_{n+1}$) $\ge$ 1 for all $n$ and {$x_n$} $\to$ $x\in X$ as $n\to \infty ,$ then ${\alpha }_H(x_n,x$) $\ge$ 1 $\forall n\in N.$

Then T has a fixed point.

Proof. 

Let $x_0\in X$. Let $x_1\in T{x}_0$ with $x_0\ne x_1$. Then, by assumption $(ii),$ ${\alpha }_H(x_{0, } x_1)\ge 1$. Let us suppose that there exists $x_2\in T{x}_1$ such that

$$d(x_1, x_2)\le H(T x_0, T x_1)+{\psi }_H(d(x_0, x_1)).$$

Similarly, there exists $x_3\in T{x}_2$ such that

$$d(x_2, x_3)\le H(T x_1, T x_2)+{\psi }_H{}^2(d(x_0, x_1)).$$

Following the same pattern, we get a sequence {$x_n$} in $X$ such that there exists $x_{n+1}\in T{x}_n$ and

$$d(x_n, x_{n+1})\le H(T x_{n-1}, T x_n)+{\psi }_H{}^n(d(x_0, x_1)).$$

Since T is ${\alpha }_H$-admissible, therefore ${\alpha }_H(x_0,x_1)\ge$ 1 $⟹{\alpha }_H(x_1,x_2)$ $\ge$ 1.

By mathematical induction, we obtain

$${\alpha }_H(x_n, x_{n+1})\ge 1 \forall n\in N.$$

(2)

Now, using Equations (1) and (2)

$$\begin{array}{ll}d(x_n, x_{n+1})& \le H(T x_{n-1}, T x_n)+{\psi }_H{}^n(d(x_0, x_1))\\ & \le {\alpha }_H(x_{n-1},x_n) H(T x_{n-1}, T x_n)+{\psi }_H{}^n(d(x_0, x_1))\\ & \le {\psi }_H(d(x_{n-1},x_n))+{\psi }_H{}^n(d( x_0, x_1))\\ & ={\psi }_H[d(x_{n-1},x_n)+{\psi }_H{}^{n-1}(d(x_0, x_1))]\\ & \le {\psi }_H[{\psi }_H(d(x_{n-2},x_{n-1}))+{\psi }_H{}^{n-1}(d( x_0, x_1))+{\psi }_H{}^{n-1}(d( x_0, x_1))]\\ & ={\psi }_H[{\psi }_H(d(x_{n-2},x_{n-1}))+2{\psi }_H{}^{n-1}(d(x_0, x_1))]\\ & ={\psi }_H{}^2(d(x_{n-2},x_{n-1}))+2{\psi }_H{}^n(d( x_0,x_1))\\ & \dots \\ & \dots \\ & \le {\psi }_H{}^n(d(x_0,x_1))+n{\psi }_H{}^n(d( x_0,x_1)).\end{array}$$

Let $\epsilon >0$ be given. Then there exists $n_0\in N$ such that

$${\displaystyle \sum }_{n\ge n_0}(n+1){\psi }_H{}^n(d( x_0, x_1))<\epsilon$$

For $m,n\in N$ with $m>n>n_0$, we have

$$\begin{array}{ccc}\hfill d(x_n, x_m)& \le {{\displaystyle \sum }}_{k=n}^{m-1}d(x_k, x_{k+1})\hfill & \\ & \le {\displaystyle \sum _{k=n}^{m-1}}(k+1){\psi }_H{}^k(d(x_0, x_1))\hfill & \\ & \le {\displaystyle \sum _{k\ge n}}(k+1){\psi }_H{}^k(d(x_0, x_1))\hfill & \\ \le {{\displaystyle \sum }}_{k\ge n_0}(k+1){\psi }_H{}^k(d(x_0, x_1))\hfill & & <\epsilon .\end{array}$$

Thus, {$x_n$} is a Cauchy sequence in X. But ($X$, d) is a complete metric space. Therefore, there exists $x$ $\in$ $X$ such that $x_n\to$ $x\in X$ as $n\to \infty$.

Now we show that $x$ is a fixed point.

$$\begin{array}{cc}\hfill d(x, T x)& \le d(x, x_n)+d(x_n, T x)\hfill \\ & \le d(x, x_n)+H(T x_{n-1},T x)\hfill \\ & \le d(x, x_n)+{\alpha }_H(x_{n-1},x)H(T x_{n-1},T x)\hfill \\ & \le d(x, x_n)+{\psi }_H(d(x_{n-1}, x))\hfill \\ & <d(x, x_n)+d( x_{n-1}, x)\hfill \\ & \to 0\mathrm{as} n\to \infty .\hfill \end{array}$$

Thus, $d(x, T x)=0$. As $T x$ is closed, therefore, $x\in T x$. □

Example 1.

Let $X=R$ equipped with standard metric d and multivalued metric H. Let the mapping $T:X\to P_{cb}(X)$ be defined by

$$Tx=\left\{ \begin{array}{cc}\left\{2x-\frac{3}{2}, 3\right\},\hfill & \hfill if x>1\\ \left\{\frac{x}{2}, \frac{x}{4}\right\},\hfill & \hfill if 0\le x\le 1\\ \left\{0\right\},\hfill & \hfill if x<0.\end{array}\right\}$$

Now, $H(T_1, T_2)=\frac{5}{2 }>1=d(1, 2).$ Therefore, Theorem A is not applicable.

Define ${\alpha }_H$: $X\times X\to [0,\infty )$ by

$${\alpha }_H(x,y)=\left\{\begin{array}{cc}1,& \hfill if x,y\in \left[0, 1\right]\\ 0,& \hfill otherwise.\end{array}\right\}$$

Let ${\psi }_H(t)=\frac{t}{2}$ for all $t\ge 0.$

Then, it can be easily checked that

$${\alpha }_H(x,y)H(T x, T y)\le {\psi }_H(d(x, y)).$$

Thus, T is an ${\alpha }_H$-${\psi }_H$-contractive mapping.

Let $x_0=1$, then T$x_0$ = $\left\{\frac{1}{2},\frac{1}{4}\right\}$. Therefore, ${\alpha }_H(x_0,s)\ge 1$ for every $s\in T{x}_0.$ Let $x,y\in X$ be such that ${\alpha }_H(x,y)\ge 1$, then, $x,y\in \left[0,1\right]$ and therefore, $Tx=\left\{\frac{x}{2},\frac{x}{4}\right\}$ and $Ty=\left\{\frac{y}{2},\frac{y}{4}\right\}$ which implies ${\alpha }_H(s,t)\ge 1$ for every $s\in Tx$ and $t\in Ty$. Thus, T is ${\alpha }_H$-admissible.

Let {$x_n$} be a sequence in X such that ${\alpha }_H(x_n,x_{n+1})\ge 1$ for all $n\in N$ and $x_n\to x\in X$. Then, $x_n\in$ [0, 1] by definition of ${\alpha }_H.$ Therefore, $x\in \left[0,1\right]$ and hence ${\alpha }_H(x_n,x)\ge 1$.

Thus, all of the hypotheses of Theorem 1 are satisfied. Therefore, T has a fixed point; namely, 0.

Theorem 2.

Let $(X,d)$ be a complete metric space and $T:X\to P_{cb}(X)$ be a multivalued mapping such that

1. 

there exists a continuous function${\psi }_H\in {\Psi }_H$and${\alpha }_H$:$X\times X\to [0, \infty )$such that for all$x, y\in X$;

$$d(x, T x)-{\psi }_H(d(x, y))\le d(x, y)) ⟹{\alpha }_H(x, y)H(T x, Ty)\le {\psi }_H(d(x, y)),$$

2. 

T is${\alpha }_H$-admissible;

3. 

there exists$x_0\in X$such that${\alpha }_H(x_0, s) \ge$1 for every$s\in T{x}_0$;

4. 

if {$x_n$} is a sequence in $X$ such that ${\alpha }_H(x_n,x_{n+1})$ $\ge$ 1 for all $n$ and {$x_n$}$\to$ $z\in X$ as $n\to \infty ,$ then ${\alpha }_H(x_n,x$) $\ge$ 1 $\forall x\in \left\{x_k:k\in N\right\}\cup \left\{z\right\}.$

Proof. 

Let $x_0\in X$. Let $x_1\in T{x}_0$ with $x_0\ne x_1$. Then, by assumption $(ii),$ ${\alpha }_H(x_0, x_1)\ge 1$. Now,

$$d(x_0, T x_0)-{\psi }_H(d( x_0, x_1))\le d(x_0, T x_0)\le d(x_0, x_1).$$

By assumption $(i),$ we have

$${\alpha }_H(x_0, x_1)H(T{x}_0, T{x}_1)\le {\psi }_H(d( x_0, x_1)).$$

This implies

$$d(x_1, T{x}_1)\le {\alpha }_H(x_0, x_1)H(T{x}_0, T{x}_1)\le {\psi }_H(d( x_0, x_1))$$

Let $x_2\in T{x}_1$ be such that

$$d(x_1, x_2)<{\psi }_H(d(x_0, x_1))+{\psi }_H(d( x_0, x_1)).$$

Again, we have

$$d(x_1,T x_1)-{\psi }_H(d(x_1, x_2))\le d(x_1,T x_1)\le d(x_1, x_2)$$

By assumption, we have

$${\alpha }_H(x_1, x_2)H(T{x}_1, T{x}_2)\le {\psi }_H(d( x_1, x_2)).$$

which implies

$$d(x_2, T x_2)\le {\alpha }_H(x_1, x_2)H(T{x}_1, T{x}_2)\le {\psi }_H(d( x_1, x_2))$$

Let $x_3\in T{x}_2$ be such that

$$d(x_2, x_3)<{\psi }_H(d(x_1, x_2))+{\psi }_H{}^2(d(x_0, x_1)).$$

Thus, we get a sequence $x_{n+1}\in T{x}_n$ such that

$$\begin{array}{c}d(x_n,x_{n+1})<{\psi }_H(d( x_{n-1},x_n))+ {\psi }_H{}^n(d(x_0, x_1))\\ \le {\psi }_H[d(x_{n-1}, x_n)+{\psi }_H{}^{n-1}(d(x_0, x_1))]\\ \le {\psi }_H[{\psi }_H(d(x_{n-2}, x_{n-1}))+{\psi }_H{}^{n-1}(d( x_0, x_1))+{\psi }_H{}^{n-1}(d( x_0, x_1))]\\ ={\psi }_H[{\psi }_H(d(x_{n-2}, x_{n-1}))+2{\psi }_H{}^{n-1}(d( x_0, x_1))]\\ ={\psi }_H{}^2(d(x_{n-2}, x_{n-1}))+2{\psi }_H{}^n(d( x_0, x_1))\\ \dots \\ \dots \\ \le {\psi }_H{}^n(d(x_0, x_1))+n{\psi }_H{}^n(d( x_0, x_1)).\end{array}$$

Let $\epsilon >0$ be given. Then there exists $n_0\in N$ such that

$${\displaystyle {\sum }_{n\ge n_0}}(n+1){\psi }_H{}^n(d(x_0, x_1))<\epsilon$$

For $m,n\in N$ with $m>n>n_0$, we have

$$\begin{array}{ll}d(x_n, x_m)& \le {\displaystyle \sum _{k=n}^{m-1}}d(x_k, x_{k+1})\\ & \le {\displaystyle {\sum }_{k=n}^{m-1}}(k+1){\psi }_H{}^k(d(x_0, x_1))\\ & \le {\displaystyle {\sum }_{k\ge n}}(k+1){\psi }_H{}^k(d(x_0, x_1))\\ & \le {\displaystyle {\sum }_{k\ge n_0}}(k+1){\psi }_H{}^k(d( x_0, x_1))\\ & \hspace{1em}<\epsilon .\end{array}$$

Thus, {$x_n$} is a Cauchy sequence in X. But ($X$, d) is a complete metric space. Therefore, there exists z $\in$ $X$ such that $x_n\to$ $z\in X$ as $n\to \infty$.

Now we show that $d(z,Tx)\le {\psi }_H(d(z,x))\forall x\in \left\{x_k:k\in N\right\}-\left\{z\right\}.$ Since $x_n\to z$, there exists $n_0\in N$ such that $d(z,x_n)\le \frac{1}{3}d(z,x)$ for all $n\ge n_0.$

Then,

$$\begin{array}{cc}\hfill d(x_n, T{x}_n)-{\psi }_H(d(x, x_n))& \le d(x_n, T{x}_n)\hfill \\ & \le d(x_n, x_{n+1})\hfill \\ & \le d(x_n, z)+d(z, x_{n+1})\hfill \\ & \hspace{1em}\le \frac{2}{3}d(x, z)\hfill \\ & \hspace{1em}=d(x,z)-(1/3)d(x,z)\hfill \\ & \le d(x,z)-d(z,x_n)\hfill \\ & \le d(x_n,x).\hfill \end{array}$$

Therefore, by assumption,

$${\alpha }_H(x_n, x)H(T{x}_n,Tx)\le {\psi }_H(d(x_n, x))$$

$$⟹H(T{x}_n,Tx)\le {\psi }_H(d(x_n,x))$$

$$⟹d(x_{n+1},Tx)\le {\psi }_H(d(x_n, x)) for n\ge n_0$$

Letting $n\to \infty$,

$$d(z, Tx)\le {\psi }_H(d(z, x))$$

Next, we prove that

$${\alpha }_H(x, z)H(T x, Tz)\le {\psi }_H(d(z, x)) \forall x\in \left\{x_k:k\in N\right\}$$

If x = z, then it is obvious. So, let x $\ne$ z. Then for every n $\in$ N, there exists $y_n\in Tx$ such that

$$d(z,y_n)\le d(z,Tx)+\frac{1}{n}d(x,z)$$

Now,

$$\begin{array}{ll}d(x, Tx)& \le d(x,y_n)\\ & \le d(x, z)+d(z,y_n)\\ & \le d(x, z)+d(z,Tx)+\frac{1}{n}d(x,z)\\ & \le d(x, z)+{\psi }_H(d(z,Tx))+\frac{1}{n}d(x,z)\end{array}$$

By taking $n\to \infty$, we get

$$d(x, Tx)-{\psi }_H(d(z, x))\le d(x,z).$$

Thus, by assumption,

$${\alpha }_H(x, z)H(Tx, Tz)\le {\psi }_H(d(z, x)) \forall x\in \left\{x_k:k\in N\right\}$$

Finally,

$$\begin{array}{ll}d(z, Tz)& =\underset{n\to \infty }{\lim }d(x_{n+1},\hspace{1em}Tz)\\ & \hspace{1em}\le \underset{n\to \infty }{\lim \sup } H(T{x}_n, Tz)\\ & \le \underset{n\to \infty }{\mathrm{lims} \sup } \frac{{\psi }_H(d(x_n, z))}{{\alpha }_H((x_n, z))}\\ & =0.\end{array}$$

and $Tz$ is closed, therefore, $z\in Tz$. □

Example 2.

Let X = {a, b, c, d, e}. Define$d :X\times X\to R$as follows:$(a,b)=d(a, c)=3.5, d(a, d)=d(a, c)=8, d(b, c)=7, d(b, d)=d(b, e)=d(c, d)=d(c, e)=4.5, d(d, e)=2$and$d(x, x)=0, d(x, y)=d(y, x) for all x, y\in X$. Then$(X, d)$is a complete metric space.

Define $T:X\to P_{cb}(X)$ as follows:

$$T (a)=T (b)=T (c)=\left\{a\right\}, T (d)=\left\{a, b\right\}, T (e)=\left\{c\right\}.$$

Define ${\psi }_H$: [0, $\infty$) $\to$ [0, $\infty$) as ${\psi }_H(t)=\frac{4}{5}t$ and ${\alpha }_H$: $X\times X\to [0,\infty )$ as

$${\alpha }_H(x, y)=\left\{\begin{array}{cccc}0& if (x=e,& y\ne d) or (x\ne d,& y=e),\\ \frac{1}{2}& if (x=e,& y=d) or (x=d,& y=e),\\ 1& & & otherwise.\end{array}\right\}$$

Now three cases arise:

Case I: If $x,y\in \left\{a,b,c\right\},$ then $H(Tx,Ty)=0$. Therefore,

$${\alpha }_H(x, y)H(Tx,\hspace{1em}Ty)\le {\psi }_H(d(x, y)).$$

Case II: If x $\in \left\{a,b,c\right\},y\in \left\{d,e\right\},$ then $(Tx,Ty)=3.5$. Therefore,

$${\alpha }_H(x,y)H(Tx,Ty)\le {\psi }_H(d(x,y)).$$

Case III: If $x=d,y=eorx=e,y=d$ then $H(Tx,Ty)=7$ and $d(x,Tx)=4.5$.

But in this case, ${\alpha }_H(x,y)H(Tx,Ty)=\frac{1}{2}\times 7\nleqq \frac{8}{5}$ $={\psi }_H(d(x,y)).$

Also, $d(x,Tx)-{\psi }_H(d(x,y))=4.5-\frac{8}{5}=2.9\nleqq 2=d(x,y).$

Thus, combining all cases, we have

$$d(x,Tx)-{\psi }_H(d(x, y))\le d(x,y)⟹{\alpha }_H(x, y)H(Tx, Ty)\le {\psi }_H(d(x, y)) \mathrm{for}\mathrm{all}x,y\in X.$$

Also, ${\alpha }_H(x,y)\ge 1⟹x,y\in \left\{a,b,c,d\right\}.$ Therefore, for every $u\in Tx$ and $v\in Ty$, we have ${\alpha }_H(u,v)\ge 1$. Thus, T is ${\alpha }_H$-admissible.

Now, take $x_0\in \left\{a,b,c,d\right\},$ then, we have, ${\alpha }_H(x_0,s)\ge 1$ for every $s\in T{x}_0$. Also, if {x_n} is a sequence in X such that ${\alpha }_H(x_0,s)\ge 1$ for all n and $x_n\to z,$ then ${\alpha }_H(x_n,x)\ge 1$ for all $x\in \left\{x_n\right\}\cup \left\{z\right\}.$

Hence, all the conditions of Theorem 2 are satisfied. Therefore, T has a fixed point. In this example, $a$ is a fixed point.

Theorem 3.

Let (X, d) be a complete metric space and$T:X\to P_{cb}(X)$be a multivalued$\alpha$_H-$\psi$_H-contractive mapping such that

1. 

there exists a continuous function${\psi }_H\in {\Psi }_H$and${\alpha }_H$:$X\times X\to [0, \infty )$such that for all$x,y\in X$;

$$d(x, Tx)-{\psi }_H(d(x, y))\le d(x, y))⟹{\alpha }_H(x, y)H(Tx, Ty)\le {\psi }_H(M(x, y)),$$

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

2. 

T is${\alpha }_H$-admissible;

3. 

there exists$x_0\in X$such that${\alpha }_H(x_0, s)\ge$1 for every$s\in T{x}_0$;

4. 

if {$x_n$} is a sequence in $X$ such that ${\alpha }_H(x_n,x_{n+1})\ge$ 1 for all $n$ and {$x_n$}$\to$ $z\in X$ as $n\to \infty ,$ then ${\alpha }_H(x_n,x)\ge$ 1 $\forall x\in \left\{x_k:k\in N\right\}\cup \left\{z\right\}.$

Then T has a fixed point.

Proof. 

Let $x_0\in X$. Let $x_1\in T{x}_0$ with $x_0\ne x_1$. Then, by assumption $(ii),$ ${\alpha }_H(x_{0 , } x_1)\ge$ 1.

Now,

$$d(x_0,T{x}_0)-{\psi }_H(d( x_0, x_1))\le d(x_0,T{x}_0)\le d(x_0, x_1).$$

By assumption $(i),$ we have

$${\alpha }_H(x_0, x_1)H(T{x}_0, T{x}_1)\le {\psi }_H(M(x_0, x_1)).$$

This implies

$$d(x_1, T x_1)\le {\alpha }_H( x_0, x_1)H(T{x}_0, T{x}_1)\le {\psi }_H(M( x_0, x_1))$$

Let $x_2\in T{x}_1$ be such that

$$d(x_1, x_2)<{\psi }_H(M(x_0, x_1))+{\psi }_H(M(x_0, x_1)).$$

Again, we have

$$d(x_1,T{x}_1)-{\psi }_H(d(x_1, x_2))\le d(x_1,T{x}_1)\le d(x_1, x_2)$$

By assumption, we have

$${\alpha }_H(x_1, x_2)H(T{x}_1, T{x}_2)\le {\psi }_H(M( x_1, x_2)).$$

which implies

$$d(x_2, T{x}_2)\le {\alpha }_H(x_1, x_2)H(T{x}_1, T{x}_2)\le {\psi }_H(M( x_1, x_2))$$

Let $x_3\in T{x}_2$ be such that

$$d(x_2, x_3)<{\psi }_H(M(x_1, x_2))+{\psi }_H{}^2(M(x_0, x_1)).$$

Thus, we get a sequence $x_{n+1}\in T{x}_n$ such that

$$\begin{array}{cc}d(x_n, x_{n+1})\hfill & <{\psi }_H(M(x_{n-1}, x_n))+{\psi }_H{}^n(M(x_0, x_1)).\hfill \\ & \le {\psi }_H[M(x_{n-1}, x_n)+{\psi }_H{}^{n-1}(M(x_0, x_1))]\hfill \\ & \le {\psi }_H[{\psi }_H(M(x_{n-2}, x_{n-1}))+{\psi }_H{}^{n-1}(M(x_0, x_1))+\hfill \\ & \hspace{1em}{\psi }_H{}^{n-1}(M(x_0, x_1))]\hfill \\ & ={\psi }_H[{\psi }_H(M(x_{n-2},x_{n-1}))+2{\psi }_H{}^{n-1}(M(x_0, x_1))]\hfill \\ & \hspace{1em}={\psi }_H{}^2(M(x_{n-2}, x_{n-1}))+2{\psi }_H{}^n(M(x_0, x_1))\hfill \\ & \dots \\ & \dots \\ & \le {\psi }_H{}^n(M(x_0, x_1))+n{\psi }_H{}^n(M( x_0,x_1))\hfill \\ & =(n+1){\psi }_H{}^n(M(x_0, x_1))\hfill \end{array}$$

Let $\epsilon >0$ be given. Then there exists $n_0\in N$ such that

$${{\displaystyle \sum }}_{n\ge n_0}(n+1){\psi }_H{}^n(M(x_0, x_1))<\epsilon .$$

For $m,n\in N$ with $m>n>n_0$, we have

$$\begin{array}{cc}d(x_n, x_m)& \le {\displaystyle {\sum }_{k=n}^{m-1}}d(x_k, x_{k+1})\hfill \\ & \le {\displaystyle \sum _{k=n}^{m-1}}(k+1){\psi }_H{}^k(M(x_0, x_1))\hfill \\ & \le {\displaystyle \sum _{k\ge n}}(k+1){\psi }_H{}^k(M( x_0, x_1))\hfill \\ & \le {\displaystyle \sum _{k\ge n_0}}(k+1){\psi }_H{}^k(M( x_0, x_1))\hfill \\ & \hspace{1em}<\epsilon .\hfill \end{array}$$

Thus, {$x_n$} is a Cauchy sequence in X. But ($X$, d) is a complete metric space.

Therefore, there exists $z$ $\in$ $X$ such that $x_n\to$ $z\in X$ as $n\to \infty$.

Now we show that $d(z,Tx)\le {\psi }_H(d(z,x))\forall x\in \left\{x_k:k\in N\right\}-\left\{z\right\}.$ Since $x_n\to$ $z$, there exists $n_0\in N$ such that $d(z,x_n)\le \frac{1}{3}d(z,x)$ for all $n\ge n_0$. Then,

$$\begin{array}{ll}d(x_n, T{x}_n)-{\psi }_H(d(x,x_n))& \le d(x_n, T{x}_n)\\ & \le d(x_n, x_{n+1})\\ & \le d(x_n, z)+ d(z,x_{n+1})\\ & \le \frac{2}{3}d(x,z)\\ & \hspace{1em}\hspace{1em}=d(x,z)– (1/3)d(x, z)\\ & \le d(x,z)-d(z,x_n)\\ & \le d(x_n,x).\end{array}$$

Therefore, by assumption,

$$\begin{array}{c}{\alpha }_H(x_n,x)H(T{x}_n,Tx)\le {\psi }_H(M(x_n,x))\\ ⟹H(T{x}_n,Tx)\le {\psi }_H(M(x_n,x))\\ ⟹d(x_{n+1},Tx)\le {\psi }_H(M(x_n, x)) for n \ge n_0\end{array}$$

(3)

Now, $M(x_n,x)=\max \left\{d(x_n,Tx),d(x,T{x}_n)\right\}$. If $M(x_n,x)=d(x,T{x}_n)$ for infinitely many n, then by Equation (3.4), $d(x_{n+1}, Tx)\le {\psi }_H((d(x, T{x}_n))\le H(d(x, x_{n+1}$)) for infinitely many n.

Letting $n\to \infty$, $d(z,Tx)\le {\psi }_H(d(x,z)).$

If $M(x_n,x)=d(x_n,Tx)$ for infinitely many n, then by (3.4), $d(x_{n+1},Tx)\le {\psi }_H$ (d($x_n,Tx$)) for infinitely many n.

Letting $n\to \infty$,

$$d(z, Tx)\le {\psi }_H(d(z, Tx)).$$

(4)

Suppose that $d(z, Tx)$ > 0, then

$${\psi }_H(d(z, Tx))<d(z, Tx)$$

which is contradictory to (3.5). Therefore, $d(z, Tx)$ = 0.

Hence $d(z,Tx)\le {\psi }_H(d(x,z)).$

Next we prove that

$${\alpha }_H(x, z)H(Tx, Tz)\le {\psi }_H(M(z, x)) \forall x\in \left\{x_k:k\in N\right\}$$

If x = z, then it is obvious. So, let x $\ne$ z. Then for every n $\in$ N, there exists $y_n\in Tx$ such that $d(z,y_n)\le d(z,Tx)+\frac{1}{n}d(x,z)$

Now,

$$\begin{array}{cc}\hfill d(x, Tx)& \le d(x,y_n)\hfill \\ & \le d(x, z) + d(z, y_n)\hfill \\ & \le d(x, z) + d(z,Tx)+\frac{1}{n}d(x,z)\hfill \\ & \le d(x, z) + {\psi }_H(d(z,x))+\frac{1}{n}d(x,z)\hfill \end{array}$$

By taking $n\to \infty$, we get

$$d(x, Tx)-{\psi }_H(d(z, x))\le d(x,z).$$

Thus, by assumption,

$${\alpha }_H(x, z)H(Tx, Tz)\le {\psi }_H(M(z, x)) \forall x\in \left\{x_k:k\in N\right\}$$

Finally, we prove that d$(z, Tz)=0$. Let us suppose the contrary to this statement, i.e., d$(z, Tz)$ > 0. Then,

$$\begin{array}{c}\begin{array}{cc}\hfill (z, Tz)& =\underset{n\to \infty }{\lim }d(x_{n+1},Tz)\hfill \\ & \hspace{1em}\le \underset{n \to \infty }{\lim \sup } H(T{x}_n, Tz)\hfill \\ & \le \underset{n \to \infty }{\lim \sup } \frac{{\psi }_H(M(x_n, z))}{{\alpha }_H((x_n, z))}\hfill \\ & = \underset{n \to \infty }{\lim \sup } \frac{{\psi }_H( \max \left\{d(x_n, Tx), d(x, T{x}_n)\right\})}{{\alpha }_H((x_n, z))}\hfill \\ & \hspace{1em}= \underset{n \to \infty }{\lim \sup } \frac{{\psi }_H( \max \left\{d(x_n, Tx), d(x, x_{n+1})\right\})}{{\alpha }_H((x_n, z))}\hfill \end{array}\\ ⟹d(z, T z)\le \frac{{\psi }_H(d(z, Tz))}{\underset{n \to \infty }{\lim \inf } {\alpha }_H((x_n, z))}< \frac{d(z, Tz)}{\underset{n \to \infty }{\lim \inf } {\alpha }_H((x_n, z))} \end{array}$$

which is not possible as $\underset{n \to \infty }{\lim \inf } {\alpha }_H((x_n, z))\ge 1$. Thus, d(z, Tz) = 0. Since $Tz$ is closed, therefore, $z\in Tz$. □

4. Consequences

Now, we will show that many existing results can be deduced from our main results.

4.1. Nadler’s Fixed Point Theorem

Theorem 1

(Nadler [3]). Let (X, d) be a complete metric space and T: X$\to$ P_cb(X) be a multivalued mapping satisfying

$$H(Tx, Ty)\le l d(x, y),$$

for all $x, y\in X$ and $l\in [0, 1).$ Then T possesses a fixed point.

Proof. 

Let $\alpha$_H: $X\times X\to [0, \infty )$ be the mapping defined by $\alpha$_H$(x, y)= 1$ for all $x, y \in X$ and ${\psi }_H$: [0, +$\infty$) $\to$ [0, +$\infty$) defined by ${\psi }_H(x)=lx$ where $l\in [0, 1).$ It is easy to show that all the hypotheses of Theorem 1 are satisfied. Consequently, T has a fixed point. □

4.2. Suzuki-Kikkawa’s Fixed Point Theorem

Theorem 2

(Suzuki, Kikkawa [5]). Let (X, d) be a complete metric space and T: X $\to$ P_cb(X) be a multivalued mapping satisfying

$$\eta (r)d(x, Tx)\le d(x, y))⟹ H(Tx, Ty) r d(x, y),$$

for all $x, y\in X$ and for some $r\in$[0, 1) where $\eta$: $[0, 1)\to (\frac{1}{2}, 1]$ is a mapping defined by $\eta (r)=\frac{1}{1+r}$.

Then T possesses a fixed point.

Proof. 

Let $\alpha$_H: $X\times X\to [0, \infty )$ be the mapping defined by $\alpha$_H$(x, y)= 1$ for all $x, y\in X$ and ${\psi }_H$: [0, +$\infty$) $\to$ [0, +$\infty$) defined by ${\psi }_H(x)=rx$ where $r\in [0, 1).$ It is easy to show that all the hypotheses of Theorem 1 are satisfied. Consequently, T has a fixed point. □

5. Application

In this section, a result, in homotopy theory in the context of fixed points, is presented through our functions used in Theorem 1.

Theorem 1.

Let (X, d) be a complete metric space, M and N are open and closed subsets of X respectively, such that M $\subset$ N. For $a, b\in R,$ let $T:N \left[a, b\right]\to P_{cb}(X)$ be an operator which satisfies the conditions given below:

1. 

$y \notin T(y, t)$for every$y\in N/M$and$t\in \left[a, b\right];$

2. 

there exists${\psi }_H\in {\Psi }_H$and$\alpha$_H:$X\times X \to [0, \infty )$such that

$${\alpha }_H(x,y)H(T(x,t),T(y,t)\le {\psi }_H(d(x,y))$$

for each pair$(x, y)\in N\times N$and t$\in$[a, b];

3. 

there exists a continuous function$\varsigma :\left[a,b\right]\to R$such that for every$s,t\in \left[a,b\right]$and$x\in N,$we have,

$$H(T(x, s), T(x, t))\le {\psi }_H(|\varsigma (s)-\varsigma (t)|);$$

4. 

if$x^{*}\in T(x^{*}, t),$then$T (x^{*}, t)=\left\{x^{*}\right\}$;

5. 

there exists$x_0\in X$such that$x_0\in$T ($x_0,$t);

6. 

the function ${\xi }_H$: [0$,\infty )\to$ [0$,\infty$) is continuous and strictly non-decreasing defined by ${\xi }_H$(x) = x − ${\psi }_H$(x). If T (.$,t^{*}$) has a fixed point in N for some $t^{*}\in \left[a,b\right],$ then T (., t) has a fixed point in M for all $t\in \left[a,b\right].$ Furthermore, for fixed $t\in \left[a,b\right]$ this fixed point is unique if ${\psi }_H$(t) = $\frac{t}{2}$.

Proof. 

Define a mapping $\alpha$_H: $X\times X\to [0, \infty )$ by

$${\alpha }_H(x,y)=\left\{\begin{array}{lll}3,& if x\in T(x, t),& y\in T(y, t),\\ 0,& & otherwise.\end{array}\right\}$$

for $t\in \left[a, b\right]$. We show that T is $\alpha$_H-admissible. Let $\alpha$_H(x, y) $\ge$ 1 which implies that $x\in T(x, t)$ and $y\in T(y, t)$ for all $t\in \left[a, b\right]$. By assumption (iv), T (x, t) = {x} and T (y, t) = {y}. It follows that T is $\alpha$_H-admissible.

By assumption (v), there exists $x_0\in X$ such that $x_0\in$ T ($x_0,$ t) for all t i.e., $\alpha$_H($x_0$, $x_0) \ge$ 1. Assume that $\alpha$_H(x_n, x_n+1) $\ge$ 1 for all n and x_n $\to$ z as n $\to$ $\infty$ for a sequence {x_n} in X, then x_n $\in$ T (x_n, t) and x_n+1 $\in$ T (x_n+1, t) for all n and $t\in \left[a, b\right]$. This implies that z $\in$ T (z, t) and thus $\alpha$_H(x_n, z) $\ge$ 1.

Let

$$\mathrm{U}=\{t\in \left[a, b\right]:x\in T(x,t)\mathrm{for}x \in \mathrm{M}\}.$$

As T (., $t^{*}$) has a fixed point in N for some $t^{*}\in \left[a, b\right]$, therefore, there exists x ∈ N such that x $\in$ T (x, t) and by assumption (i), x $\in$ T (x, t) for t $\in$ $\left[a, b\right]$ and x $\in$ M. So, U is a non-empty set. Now we will prove that U is open as well as closed in $\left[a, b\right].$

Let x₀ $\in$ T (x₀, t₀) for x₀ $\in$ M and t₀ $\in$ U. The set M being open in (X, d) implies the existence of r > 0 such that B_d(x₀, r) $\subseteq$ M. Let $\epsilon$ = r − ${\psi }_H$(r) > 0. As $\varsigma$ is continuous on t₀ $\in$ $\left[a, b\right]$, there exists ${\alpha }_{\epsilon }$ > 0 such that $|\varsigma (t)-\varsigma (t_0)|<\epsilon$ for all t $\in (t_0-{\alpha }_{\epsilon }, t_0+{\alpha }_{\epsilon })$.

Let t $\in (t_0-{\alpha }_{\epsilon },t_0+{\alpha }_{\epsilon })$ for x $\in$ $\overline{B_d(x_0,r)}$ = {x $\in$ X: d(x₀, x) $\le$ r} and u $\in$ T (x, t), we have

$$\begin{array}{cc}\hfill d(u,x_0)& =d(T(x,t),x_0)\hfill \\ & \le H(T(x,t),T(x_0,t_0))+H(T(x,t_0),T(x_0,t_0))\hfill \end{array}$$

Since x₀ $\in$ T(x₀, t₀) and x $\in \overline{B_d(x_0, r)}\subseteq$ M $\subset$ N with t₀ $\in \left[a, b\right],$ therefore, $\alpha$_H(x, x₀) $\ge$ 1. So, by assumption (ii) and (iii), we obtain

$$\begin{array}{cc}\hfill d(u,x_0)& \le {\psi }_H((|\varsigma (t)-\varsigma (t_0)|)+{\alpha }_H(x, x_0) H(T(x, t_0), T(x_0, t_0))\\ & \le {\psi }_H((\varsigma (t)-\varsigma (t_0))+{\psi }_H(d(x,x_0))\hfill \\ & \le {\psi }_H(\epsilon )+{\psi }_H(d(x,x_0))\hfill \\ & \le {\psi }_H(\epsilon )+{\psi }_H(r)\hfill \\ & \le {\psi }_H(r-{\psi }_H(r))+{\psi }_H(r)\hfill \\ & <r-{\psi }_H(r)+{\psi }_H(r)=r.\hfill \end{array}$$

Thus, u $\in \overline{B_d(x_0,r)}$ and hence for every fixed t $\in (t_0-{\alpha }_{\epsilon },t_0+{\alpha }_{\epsilon })$, we have, T(x, t) $\subset \overline{B_d(x_0,r)}.$ So, T: $\overline{B_d(x_0,r)}\to$ P_cb$\overline{(B_d(x_0,r))}$ fulfills all the hypotheses of Theorem 1 and therefore, T (., t) has a fixed point in $\overline{B_d(x_0,r)}$ = B_d(x₀, r) $\subset$ N. But, by assumption (i), this fixed point belongs to M. Therefore, $(t_0-{\alpha }_{\epsilon },t_0+{\alpha }_{\epsilon })$ $\subseteq$ U and thus U is open in $\left[a,b\right].$

Next, we show that the set U is closed. Let {t_n} be a sequence in U with $t_n\to t_0\in \left[a, b\right]$ as n $\to \infty$. We prove that $t_0\in$ U.

By definition of U, there exists x_n $\in$ M such that x_n $\in$ T (x_n, t_n) for all n. For n, m $\in$ N with m > n,

$$\begin{array}{c}\begin{array}{cc}\hfill d(x_n,x_m)& \le H(T(x_n,t_n),T(x_m,t_m))\hfill \\ & \le H(T(x_n,t_n),T(x_n,t_m))+H(T(x_n,t_m),T(x_m,t_m))\hfill \\ & \le {\psi }_H(|(\varsigma (t_n)-\varsigma (t_m))|)+{\alpha }_H(x_n,x_m)H(T(x_n,t_m),T(x_m,t_m))\hfill \\ & \le {\psi }_H(|(\varsigma (t_n)-\varsigma (t_m))|)+{\psi }_H(d(x_n,x_m)⟹d(x_n,x_m)-{\psi }_H(d(x_n,x_m))\hfill \end{array}\\ \begin{array}{cc}\hfill \le {\psi }_H(|(\varsigma (t_n)& -\varsigma (t_m))|)⟹{\xi }_H(d(x_n,x_m))\le {\psi }_H(|(\varsigma (t_n)-\varsigma (t_m))|)\hfill \\ & <|(\varsigma (t_n)-\varsigma (t_m))|\hspace{1em}\hspace{1em}⟹(d(x_n,x_m))<{\xi }_H{}^{-1}|(\varsigma (t_n)-\varsigma (t_m))|\hfill \end{array}\end{array}$$

Continuity of functions ${\xi }_H{}^{-1}$, $\varsigma$ and convergence of the sequence {t_n} implies that $\underset{m, n\to \infty }{\lim }d(x_n,x_m)=0$ i.e., {x_n} is a Cauchy sequence in X and since (X, d) is complete, therefore, there exists $\acute{x}\in$ N such that lim x_n = $\acute{x}$.

Now,

$$\begin{array}{cc}\hfill d(x_n,T(\acute{x},\acute{t}))& \le H(T(x_n,t_n),T(\acute{x},\acute{t}))\hfill \\ & \le H(T(x_n,t_n),T(x_n,\acute{t}))+H(T(x_n, \acute{t}),T(\acute{x},\acute{t}))\hfill \\ & \le {\psi }_H(|(\varsigma (t_n)-\varsigma (\acute{t}))|)+{\alpha }_H(x_n,\acute{x})H(T(x_n,\acute{t}),T(\acute{x},\acute{t}))\hfill \\ & \le {\psi }_H(|(\varsigma (t_n)-\varsigma (t_m))|)+{\psi }_H(d(\acute{x},)\hfill \end{array}$$

Taking limit n $\to \infty$, we get $\underset{ n\to \infty }{\lim }d(x_{\mathrm{n}},T(\acute{x},\acute{t}))$ = 0 and therefore,

$$d(\acute{x},T(\acute{x},\acute{t}))=\underset{ n\to \infty }{\lim }d(x_n,T(\acute{x},\acute{t}))=0.$$

This implies that $\acute{x}$ $\in T(\acute{x},\acute{t})$ and by assumption (i), $\acute{x}$ $\in$ M. So, $\acute{t}$ $\in$ U and hence U is closed in $\left[a, b\right].$

Thus, U = $\left[a, b\right]$ and T (., t) has a fixed point in M for all t in $\left[a, b\right].$

For uniqueness of this fixed point, let us consider another fixed point $\acute{y}$ $\in T(\acute{y},\acute{t})$ for fixed t in $\left[a, b\right].$

$$\begin{array}{ccc}& d(\acute{x},\acute{ y})\le H(T(\acute{x},t), T(\acute{y},t))& \\ \le {\alpha }_H(\acute{x},\acute{ y}) H(T(\acute{x},t),T(\acute{y},t))& & \le {\psi }_H(d(\acute{x},\acute{y})).\end{array}$$

If $\psi$_H(t) = $\frac{t}{2}$ for all t ≥ 0, the uniqueness follows. □