Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications

Abstract

The $H^{\beta }$-Hausdorff–Pompeiu b-metric for $\beta \in [0,1]$ is introduced as a new variant of the Hausdorff–Pompeiu b-metric H. Various types of multi-valued $H^{\beta }$-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued $H^{\beta }$-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction.

1. Introduction

Romanian mathematician D. Pompeiu in [1] initiated the study of distance between two sets and introduced the Pompeiu metric. Hausdorff [2] further studied this concept and thereby introduced the Hausdorff–Pompeiu metric H induced by the metric d of a metric space $(X,d)$, as follows:

For any two subsets A and B of X, the function H given by $H(A,B)=max\{{sup}_{x\in A}$$d(x,B),{sup}_{x\in B}d(x,A)\}$ is a metric for the set of compact subsets of X. Note that

$$\begin{array}{ccc}\hfill H(A,B)& =& max\{\beta \underset{x\in A}{sup}d(x,B)+(1-\beta )\underset{x\in B}{sup}d(x,A),\beta \underset{x\in B}{sup}d(x,A)\hfill \\ & +& (1-\beta )\underset{x\in A}{sup}d(x,B)\}\mathrm{for}\beta =0\mathrm{or}1.\hfill \end{array}$$

(1)

Nadler [3] extending the Banach contraction principle introduced multi-valued contraction principle in a metric space using the Hausdorff–Pompieu metric H. Thereafter many extensions and generalizations of multi-valued contraction appeared (see [4,5,6,7]). In 1998, Czerwik [8] introduced the Hausdorff–Pompeiu b-metric $H_b$ as a generalization of Hausdorff–Pompeiu metric H and proved the b-metric space version of Nadler contraction principle. Czervik’s result drew attention of many researchers who further obtained many generalized multi-valued contractions, named q-quasi contraction [9], Hardy Rogers contraction [10], weak quasi contraction [11], Ciric contraction [12], etc. and proved the existence theorem for such contraction mappings in a b-metric space. The aim of this work is to introduce new variants of the Hausdorff–Pompeiu b-metric and thereby introduce various types of multi-valued $H^{\beta }$-contraction and prove fixed point theorems for such types of contractions in a b-metric space. It is shown that for any b-metric space $(X,d_s)$ and $\beta \in [0,1]$, the function given in (1) defines a b-metric for the set of closed and bounded subsets of X. We call this metric $H^{\beta }$-Hausdorff–Pompeiu b-metric induced by the b-metric $d_s$. Thereafter, using this $H^{\beta }$-Hausdorff–Pompeiu b-metric, we have introduced various types of multi-valued $H^{\beta }$-contraction and proved fixed point theorems for such types of contractions in a b-metric space. The multi-valued Nadler contraction [3], Czervik contraction [8], q-quasi contraction [9], Hardy Rogers contraction [10], Ciric contraction [12], weak quasi contraction [11] existing in literature are all one or the other type of multi-valued $H^{\beta }$-contraction; however, it is shown with proper examples that the converse is not necessarily true. Finally to demonstrate the applications of our results, we prove the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and also an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction.

2. Preliminaries

Bakhtin [13] introduced b-metric space as follows:

Definition 1

([13]). Let X be a nonempty set and $d_s:X\times X\to [0,\infty )$ satisfies:

1. 

$d_s(x,y)=0$ if and only if $x=y$ for all $x,y\in X$;

2. 

$d_s(x,y)=d(y,x)$ for all $x,y\in X$;

3. 

there exist a real number $s\ge 1$ such that $d(x,y)\le s[d_s(x,z)+d_s(z,y)]$ for all $x,y,z\in X.$

Then, $d_s$ is called a b-metric on X and $(X,d_s)$ is called a b-metric space with coefficient s.

Example 1.

Let $X=R$ and $d:X\times X\to [0,\infty )$ be given by $d(x,y)={|x-y|}^2$, for all $x,y\in X$. Then $(X,d)$ is a b-metric space with coefficient $s=2$.

Definition 2

([13]). Let $(X,d_s)$ is a b-metric space with coefficient s.

(i) 

A sequence $\left\{x_n\right\}$ in X, converges to $x\in X$, if $li{m}_{n\to \infty }{d}_s(x_n,x)=0$.

(ii) 

A sequence $\left\{x_n\right\}$ in X is a Cauchy sequence if for all $ϵ>0$, there exist a positive integer $n\left(ϵ\right)$ such that $d_s(x_n,x_m)<ϵ$ for all $n,m\ge n\left(ϵ\right)$.

(iii) 

$(X,d_s)$ is complete if every Cauchy sequence in X is convergent.

For some recent fixed point results of single valued and multi-valued mappings in a b-metric space, see [9,14,15,16,17,18]. Throughout this paper, ($X,d_s$) will denote a complete b-metric space with coefficient s and $C{B}^{d_s}\left(X\right)$ the collection of all nonempty closed and bounded subsets of X with respect to $d_s$.

For $A,B\in C{B}^{d_s}\left(X\right)$, define $d_s(x,A)=inf\{d_s(x,a):a\in A\}$, ${\delta }_{d_s}(A,B)={sup}_{a\in A}{d}_s(a,B)$ and $H_{d_s}(A,B)=max\left\{{\delta }_{d_s}(A,B),{\delta }_{d_s}(B,A)\right\}$. Czerwik [8] has shown that $H_{d_s}$ is a b-metric in the set $C{B}^{d_s}\left(X\right)$ and is called the Hausdorff–Pompeiu b-metric induced by $d_s$.

Motivated by the fact that a b-metric is not necessarily continuous (as ${\displaystyle \frac{1}{s^2}}{d}_s(x,y)\le {\underline{lim}}_{n\to \infty }{d}_s(x_n,y_n)\le {\overline{lim}}_{n\to \infty }{d}_s(x_n,y_n)\le s^2{d}_s(x,y)$ and ${\displaystyle \frac{1}{s}}{d}_s(x,y)\le {\underline{lim}}_{n\to \infty }{d}_s(x_n,y)\le {\overline{lim}}_{n\to \infty }{d}_s(x_n,y)\le s{d}_s(x,y)$ see [19,20,21]), Miculescu and Mihail [12] introduced the following concept of *-continuity.

Definition 3

([12]). The b-metric $d_s$ is called *-continuous if for every $A\in C{B}^{d_s}\left(X\right)$, every $x\in X$ and every sequence $\left\{x_n\right\}$ of elements from X with $li{m}_{n\to \infty }{x}_n=x$, we have $li{m}_{n\to \infty }{d}_s(x_n,A)=d_s(x,A)$.

Proposition 1

([17]). For any $A\subseteq X$,

$$a\in \overline{A}⟺d_s(a,A)=0.$$

Lemma 1

([12]). Let $\left\{x_n\right\}$ be a sequence in ($X,d_s$). If there exists $\lambda \in [0,1)$ such that $d_s(x_n,x_{n+1})\le \lambda d_s(x_{n-1},x_n)$ for all $n\in N$, then $\left\{x_n\right\}$ is a Cauchy sequence.

The following lemma can also be proved using the same technique of proof of the above Lemma.

Lemma 2.

Let $\left\{x_n\right\}$ be a sequence in ($X,d_s$). If there exists $\lambda ,ϵ\in [0,1)$, with $\lambda <ϵ$ such that $d_s(x_n,x_{n+1})\le \lambda d_s(x_{n-1},x_n)+{ϵ}^n$ for all $n\in N$, then $\left\{x_n\right\}$ is a Cauchy sequence.

Czerwik [8] introduced multi-valued contraction in a b-metric space and proved that every multi-valued contraction mapping in a b-metric space has a fixed point.

Definition 4

([8]). A mapping $T:X\to C{B}^{d_s}\left(X\right)$ is a multi-valued contraction if there exists $\alpha \in (0,{\displaystyle \frac{1}{s}})$, such that $g^{ı},g^j\in X$ implies $H_{d_s}(T{g}^{ı},T{g}^j)\le \alpha d_s(g^{ı},g^j)$.

Theorem 1

([8]). Every multi-valued contraction mapping defined on ($X,d_s$) has a fixed point.

Thereafter using Hausdorff–Pompieu b-metric $H_{d_s}$, many authors introduced several generalized multi-valued contractions in a b-metric space (see Definitions 5 to 8 below) and proved the existence of fixed points for such generalized multi-valued contraction mappings.

Definition 5

([9]). A mapping $T:X\to C{B}^{d_s}\left(X\right)$ is a q-multi-valued quasi contraction if there exists $q\in (0,{\displaystyle \frac{1}{s}})$, such that $g^{ı},g^j\in X$ implies

$$\begin{array}{cc}& H_{d_s}(T{g}^{ı},T{g}^j)\le qmax\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j),d_s(g^{ı},T{g}^j),d_s(g^j,T{g}^{ı})\}.\end{array}$$

Definition 6

([12]). A mapping $T:X\to C{B}^{d_s}(X)$ is a q-multi-valued Ciric contraction if there exists $q,c,d\in (0,1)$, such that $g^{ı},g^j\in X$ implies

$$\begin{array}{cc}& H_{d_s}(T{g}^{ı},T{g}^j)\le qmax\{d_s(g^{ı},g^j),c{d}_s(g^{ı},T{g}^{ı}),c{d}_s(g^j,T{g}^j),{\displaystyle \frac{d}{2}}(d_s(g^{ı},T{g}^j)+d_s(g^j,T{g}^{ı}))\}.\end{array}$$

Definition 7

([10]). A mapping $T:X\to C{B}^{d_s}(X)$ is a multi-valued Hardy–Roger’s contraction if there exists $a,b,c,e,f\in (0,1)$, $a+b+c+2(e+f)<1$, such that $g^{ı},g^j\in X$ implies $H_{d_s}(T{g}^{ı},T{g}^j)\le a{d}_s(g^{ı},g^j)+b{d}_s(g^{ı},T{g}^{ı})+c{d}_s(g^j,T{g}^j)+e{d}_s(g^{ı},T{g}^j)+f{d}_s(g^j,T{g}^{ı})$.

Definition 8

([11]). A mapping $T:X\to C{B}^{d_s}(X)$ is a multi-valued weak quasi contraction if there exists $q\in (0,1)$ and $L\ge 0$ such that $g^{ı},g^j\in X$ implies $H_{d_s}(T{g}^{ı},T{g}^j)\le qmax\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j)\}+L{d}_s(g^{ı},T{g}^j)$.

3. Main Results

3.1. The $H^{\beta }$ Hausdorff–Pompieu b-metric

Definition 9.

For $U,V\in C{B}^{d_s}(X)$, $\beta \in [0,1]$, we define

$$R^{\beta }(U,V)=\beta \delta {}_{d_s}(U,V)+(1-\beta )\delta {}_{d_s}(V,U)$$

and

$$H^{\beta }(U,V)=max\{R^{\beta }(U,V),R^{\beta }(V,U)\}.$$

Proposition 2.

Let $U,V,W\in C{B}^{d_s}(X)$, we have

(i) 

$H^{\beta }(U,V)=0$ if and only if $U=V$.

(ii) 

$H^{\beta }(U,V)=H^{\beta }(V,U)$.

(iii) 

$H^{\beta }(U,V)\le s[H^{\beta }(U,W)+H^{\beta }(W,V)].$

Proof. 

(i) By definition, $H^{\beta }(U,V)=0$ implies $max\left\{\beta \delta {}_{d_s}(U,V)+(1-\beta )\delta {}_{d_s}(V,U),(1-\beta )\delta {}_{d_s}(U,V)+\beta \delta {}_{d_s}(V,U)\right\}=0.$ This gives $\delta {}_{d_s}(U,V)=0$ and $\delta {}_{d_s}(V,U)=0$. Now, $\delta {}_{d_s}(U,V)=0$ implies $d_s(u,V)=0$ for all $u\in U$. By Proposition 1, we have $u\in \overline{V}=V$ for all $u\in U$ and so $U\subseteq V$. Similarly, $\delta {}_{d_s}(V,U)=0$ will imply $V\subseteq U$ and so $U=V$. The reverse implication is clear from the definition.

(ii) Follows from the definition of $H^{\beta }(U,V)$.

(iii) Let $u,v,w$ be arbitrary elements of $U,V,W$, respectively. Then we have

$$\begin{array}{c}\hfill d_s(u,V)\le s[d_s(u,w)+d_s(w,V)].\end{array}$$

Since w is arbitrary, we get

$$\begin{array}{c}\hfill d_s(u,V)\le s[d_s(u,w)+{\delta }_{d_s}(W,V)]\le s[d_s(u,W)+{\delta }_{d_s}(W,V)].\end{array}$$

Again, since u is arbitrary, we get

$$\begin{array}{c}\hfill {\delta }_{d_s}(U,V)\le s[{\delta }_{d_s}(U,W)+{\delta }_{d_s}(W,V)].\end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill {\delta }_{d_s}(V,U)\le s[{\delta }_{d_s}(V,W)+{\delta }_{d_s}(W,U)].\end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill R^{\beta }(U,V)& =& \beta \delta {}_{d_s}(U,V)+(1-\beta )\delta {}_{d_s}(V,U)\hfill \\ & \le & \beta s[\delta {}_{d_s}(U,W)+\delta {}_{d_s}(W,V)]+(1-\beta )s[\delta {}_{d_s}(V,W)+\delta {}_{d_s}(W,U)]\hfill \\ & & =s[\beta \delta {}_{d_s}(U,W)+(1-\beta )\delta {}_{d_s}(W,U)]+s[\beta \delta {}_{d_s}(W,V)+(1-\beta )\delta {}_{d_s}(V,W)]\hfill \\ & & =s[R^{\beta }(U,W)+R^{\beta }(W,V)].\hfill \end{array}$$

Similarly

$$\begin{array}{ccc}\hfill R^{\beta }(V,U)& \le & s[R^{\beta }(V,W)+R^{\beta }(W,U)].\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill H^{\beta }(U,V)& =& max\{R^{\beta }(U,V),R^{\beta }(V,U)\}\hfill \\ & \le & max\{s[R^{\beta }(U,W)+R^{\beta }(W,V)],s[R^{\beta }(V,W)+R^{\beta }(W,U)]\}\hfill \\ & \le & max\{s{R}^{\beta }(U,W),s{R}^{\beta }(W,U)\}+max\{s{R}^{\beta }(W,V),s{R}^{\beta }(V,W)\}\hfill \\ & =& s[H^{\beta }(U,W)+H^{\beta }(W,V)].\hfill \end{array}$$

□

Remark 1.

In view of Proposition 2, the function $H^{\beta }:C{B}^{d_s}(X)\times C{B}^{d_s}(X)\to [0,+\infty ),$ is a b-metric in $C{B}^{d_s}(X)$ and we call it the $H^{\beta }$-Hausdorff–Pompeiu b-metric induced by $d_s$.

Remark 2.

For $\beta \in [0,1]$$H^{\beta }(A,B)\le H_{d_s}(A,B)$ and for $\beta =0\vee 1$$H^{\beta }(A,B)=H_{d_s}(A,B)$.

Remark 3.

The Hausdorff–Pompeiu b-metric $H^{\beta }$ is equivalent to the Hausdorff–Pompeiu b-metric $H_{d_s}$ in the sense that for any two sets A and B, $H^{\beta }(A,B)\le H_{d_s}(A,B)\le 2H^{\beta }(A,B)$. However, the examples and applications provided in this paper illustrates the advantages of using $H^{\beta }$-Hausdorff–Pompeiu b-metric in fixed point theory and its applications.

Theorem 2.

For all $u,v\in X$, $U,V\in C{B}^{d_s}(X)$ and $\beta \in [0,1]$, the following relations holds:

(1) 

$d_s(u,v)=H^{\beta }(\{u\},\{v\}),$

(2) 

$U\subset \overline{S}(V,r_1),V\subset \overline{S}(U,r_2)\Rightarrow H^{\beta }(U,V)\le rwherer=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\},$

(3) 

$H^{\beta }(U,V)<r\Rightarrow \exists r_1,r_2>0such thatr=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}and$$U\subset S(V,r_1),V\subset S(U,r_2)$.

Proof. 

(1) This is immediate from the definition of $H^{\beta }$.

(2) Since $U\subset \overline{S}(V,r_1),V\subset \overline{S}(U,r_2)$, we have that

$$\forall u\in U,\exists v_u\in V\mathrm{satisfying}{d}_s(u,v_u)\le r_1$$

and

$$\forall v\in V,\exists u_v\in U\mathrm{satisfying}{d}_s(u_v,v)\le r_2$$

$$\Rightarrow \underset{v\in V}{inf}{d}_s(u,v)\le r_1\mathrm{for}\mathrm{every}u\in U\mathrm{and}\underset{u\in U}{inf}{d}_s(u,v)\le r_2\mathrm{for}\mathrm{every}v\in V.$$

$$\Rightarrow \underset{u\in U}{sup}\left(\underset{v\in V}{inf}{d}_s(u,v)\right)\le r_1\mathrm{and}\underset{v\in V}{sup}\left(\underset{u\in U}{inf}{d}_s(u,v)\right)\le r_2.$$

Then, $H^{\beta }(U,V)\le r$ where $r=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}.$

(3) Let $H^{\beta }(U,V)=k<r.$ Then, there is some $k_1,k_2>0$ satisfying

$$k=max\{\beta k_1+(1-\beta )k_2,\beta k_2+(1-\beta )k_1\},$$

$$\delta (U,V)=\underset{u\in U}{sup}(\underset{v\in V}{inf}{d}_s(u,v))=k_1,\delta (V,U)=\underset{v\in V}{sup}(\underset{u\in U}{inf}{d}_s(u,v))=k_2.$$

Since $0<k<r$, we can find $r_1,r_2>0$ such that $k_1<r_1$, $k_2<r_2$ and $r=max\{\beta r_1+$$(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}.$ Thus,

$$\underset{v\in V}{inf}{d}_s(u,v)\le k_1<r_1\mathrm{for}\mathrm{every}u\in U\mathrm{and}\underset{u\in U}{inf}{d}_s(u,v))\le k_2<r_2\mathrm{for}\mathrm{every}v\in V.$$

Then, for any $u\in U$ there is some $v_u\in V$ satisfying

$$d_s(u,v_u)<\underset{v\in V}{inf}{d}_s(u,v)+r_1-k_1\le r_1.$$

and, for any $v\in V$ there is some $u_v\in U$ satisfying

$$d_s(u_v,v)<\underset{u\in U}{inf}{d}_s(u,v)+r_2-k_2\le r_2.$$

Thus, for any $u\in U$ and $v\in V$ we have

$$u\in \bigcup _{v\in V}S(v;r_1)\mathrm{and}v\in \bigcup _{u\in U}S(u;r_2),$$

which implies

$$U\subset S(V,r_1)\mathrm{and}V\subset S(U,r_2).$$

□

Remark 4.

From Theorem 2 (2) and (3), it follows that the following statements also hold:

$(2^{\prime })U\subset S(V,r_1),V\subset S(U,r_2)\Rightarrow H^{\beta }(U,V)\le rwherer=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}$

and

$(3^{\prime })H^{\beta }(A,B)<r\Rightarrow \exists r_1,r_2>0such thatr=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}and$$U\subset \overline{S}(V,r_1),V\subset \overline{S}(U,r_2).$

Theorem 3.

Let $U,V\in C{B}^{d_s}(X)$ and $\beta \in [0,1]$. Then the following equalities holds:

(4)$H^{\beta }(U,V)=inf\{r>0:U\subset S(V,r_1),V\subset S(U,r_2)\}$;

(5)$H^{\beta }(U,V)=inf\{r>0:U\subset \overline{S}(V,r_1),U\subset \overline{S}(V,r_2)\}$,

where $r=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}$.

Proof. 

By ($2^{\prime }$), we have

$$H^{\beta }(U,V)\le inf\{r>0:U\subset S(V,r_1),U\subset S(V,r_2)\},r=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}.$$

(2)

Now let $H^{\beta }(U,V)=k$, and let $t>0$. Then $H^{\beta }(U,V)<k+t.$ By Condition (3) of Theorem 2 we can find $t_1,t_2>0\mathrm{with}max\{\beta t_1+(1-\beta )t_2,\beta t_2+(1-\beta )t_1\}=t$ such that $U\subset S(V;k+t_1)$ and $V\subset S(U;k+t_2)$. Thus,

$$\{r>0:U\subset S(V,r_1),B\subset S(U,r_2)\}\supset \{k+t:t>0,U\subset S(V,k+t_1),V\subset S(U,k+t_2)\}.$$

This implies that

$$inf\{r>0:U\subset S(V,r_1),V\subset S(U,r_2)\}\le inf\{k+t:t>0\}=k=H^{\beta }(U,V).$$

To conclude,

$$H^{\beta }(U,V)=inf\{r>0:U\subset S(V,r_1),V\subset S(U,r_2)\},r=max\{\beta r_1+(1-\beta )r_2,\beta r_2+(1-\beta )r_1\}.$$

(3)

□

Theorem 4.

If $(X,d_s)$ is a complete b-metric space, then $(C{B}^{d_s}(X),H^{\beta })$ for any $\beta \in [0,1]$ is also complete. Moreover, $C(X)$ is a closed subspace of $(C{B}^{d_s}(X),H^{\beta })$.

Proof. 

Suppose $(X,d_s)$ is complete and the sequence ${\{A_n\}}_{n\in \mathbf{N}}$ in $C{B}^{d_s}(X)$ is a Cauchy sequence. Let $B=\{x\in X:\forall ϵ>0,m\in \mathbf{N},\exists n\ge m$ for which $S(x,ϵ)\cap A_n\ne \varnothing \}.$

Let $ϵ>0$. By definition of Cauchy sequence, we can find $m(ϵ)\in \mathbf{N}$ for which, $n\ge m(ϵ)$ implies $H^{\beta }(A_n,A_{m(ϵ)})<ϵ$. By Theorem 3 (4), $\exists {ϵ}_1,{ϵ}_2>0$ with $ϵ=max\{\beta {ϵ}_1+(1-\beta ){ϵ}_2,\beta {ϵ}_2+(1-\beta ){ϵ}_1\}$ and $m({ϵ}_1),m({ϵ}_2)\in \mathbf{N}$ such that $min\{m({ϵ}_1),m({ϵ}_2)\}\ge m(ϵ)$, $A_n\subset S(A_{m({ϵ}_1)},{ϵ}_1)$ for $n\ge m({ϵ}_1)$ and $A_{m({ϵ}_2)}\subset S(A_n,{ϵ}_2)$ $n\ge m({ϵ}_2)$. Then we have $B\subset \overline{S}(A_{m({ϵ}_1)},{ϵ}_1)$, and so

(i)

$B\subset \overline{S}(A_{m({ϵ}_1)},4{ϵ}_1)$ holds.

Now set ${\overline{ϵ}}_k={\displaystyle \frac{{ϵ}_1}{2^k}},k\in \mathbf{N}$, and choose $n_k=m({\overline{ϵ}}_k)\in \mathbf{N}$ such that sequence ${\{n_k\}}_{k\in \mathbf{N}}$ is strictly increasing and

$$H^{\beta }(A_n,A_{n_k})<{\overline{ϵ}}_k,\forall n\ge n_k.$$

For some $p\in A_{n_0}=A_{m({ϵ}_1)}$, consider the sequence ${\{p_{n_k}\}}_{k\in \mathbf{N}}$ with $p_{n_0}=p$, $p_{n_k}\in A_{n_k}$ and $d_s(p_{n_k},p_{n_{k-1}})<{\displaystyle \frac{{ϵ}_1}{2^{k-2}}}.$ It follows that the sequence ${\{p_{n_k}\}}_{k\in \mathbf{N}}$ is a Cauchy sequence in the complete b-metric space $(X,d_s)$ and so converges to some point $l\in X$.

Additionally, $d_s(p_{n_k},p_{n_0})<4{ϵ}_1$ implies $d_s(l,p)\le 4{ϵ}_1$ and so $\underset{y\in B}{inf}{d}_s(p,y)\le 4{ϵ}_1$, that is, $p\in \overline{S}(B,4{ϵ}_1)$, from which we get

(ii)

$A_{n_0}\subset \overline{S}(B,4{ϵ}_1).$

Now, relations (i), (ii) from above and Theorem 2 (2) yields $H^{\beta }(A_{n_0},B)\le 4{ϵ}_1.$ Since $H^{\beta }$ is a b-metric on $C{B}^{d_s}(X)$, we have

$$H^{\beta }(A_n,B)\le s[H^{\beta }(A_n,A_{n_0})+H^{\beta }(A_{n_0},B)]<5s{ϵ}_1,$$

for any $n\ge m({ϵ}_1)=n_0$. Hence, sequence ${\{A_n\}}_{n\in \mathbf{N}}$ is convergent and $(C{B}^{d_s}(X),H^{\beta })$ is complete. □

For the second part, consider the Cauchy sequence ${\{A_n\}}_{n\in \mathbf{N}}$ in $C(X)$ and consequently in $C{B}^{d_s}(X)$ and converging to some $A\in C{B}^{d_s}(X)$. Thus, if $ϵ>0$ is chosen, we can find $m(ϵ)\in \mathbf{N}$ for which

$$H^{\beta }(A_n,A)<{\displaystyle \frac{ϵ}{2}}\forall n\ge m(ϵ),n\in \mathbf{N}.$$

Using (4) of Theorem 3, we get $\exists {ϵ}_1,{ϵ}_2>0$ with $ϵ=$ max $\{\beta {ϵ}_1+(1-\beta ){ϵ}_2,\beta {ϵ}_2+(1-\beta ){ϵ}_1\}$ and $m({ϵ}_1),m({ϵ}_2)\in \mathbf{N}$ such that $min\{m({ϵ}_1),m({ϵ}_2)\}\ge m(ϵ)$, $A_n\subset S(A,{\displaystyle \frac{{ϵ}_1}{2}})$ for $n\ge m({ϵ}_1)$ and $A\subset S(A_n,{\displaystyle \frac{{ϵ}_2}{2}})$ for $n\ge m({ϵ}_2)$.

For any fixed $n_0\ge m({ϵ}_2)$, we have, $A\subset S(A_{n_0},{\displaystyle \frac{{ϵ}_2}{2}})$ and the compactness of $A_{n_0}$ in X (due to which it is also totally bounded) gives us $x_i^{{ϵ}_2},i\in \overline{1,p}$ such that $A_{n_0}\subset \bigcup _{i=1}^{p}S(x_i^{{ϵ}_2},{\displaystyle \frac{{ϵ}_2}{2}}),$ whence $A\subset \bigcup _{i=1}^{p}S(x_i^{{ϵ}_2},{ϵ}_2).$ Therefore, $A\in C(X)$.

3.2. Applications to Fixed Point Theory

We begin this section by introducing various classes of multi-valued $H^{\beta }$-contractions in a b-metric space:

Definition 10.

$T:X\to C{B}^{d_s}(X)$ is a multi-valued $H^{\beta }$-contraction if we can find $\beta \in [0,1]$ and $k\in (0,1)$, such that

$$\begin{array}{c}\hfill H^{\beta }(T{g}^{ı},T{g}^j)\le k·d_s(g^{ı},g^j)forall{g}^{ı},g^j\in X.\end{array}$$

(4)

Definition 11.

$T:X\to C{B}^{d_s}(X)$ is a multi-valued $H^{\beta }$-Ciric contraction if we can find $\beta \in [0,1]$ and $k\in (0,{\displaystyle \frac{1}{s}})$, such that for all $g^{ı},g^j\in X$,

$$\begin{array}{c}\hfill H^{\beta }(T{g}^{ı},T{g}^j)\le k·max\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j),{\displaystyle \frac{d_s(g^{ı},T{g}^j)+d_s(g^j,T{g}^{ı})}{2s}}\}.\end{array}$$

(5)

Definition 12.

$T:X\to C{B}^{d_s}(X)$ is a multi-valued $H^{\beta }$-Hardy–Rogers contraction if we can find $\beta \in [0,1]$ and $a,b,c,e,f\in (0,1)$ with $a+b+s(c+e)+f<1$, $min\{s(a+e),s(b+c)\}<1$ such that for all $g^{ı},g^j\in X$,

$$\begin{array}{c}\hfill H^{\beta }(T{g}^{ı},T{g}^j)\le a·d_s(g^{ı},T{g}^{ı})+b·d_s(g^j,T{g}^j)+c·d_s(g^{ı},T{g}^j)+e·d_s(g^j,T{g}^{ı})+f·d_s(g^{ı},g^j).\end{array}$$

(6)

Definition 13.

We say that $T:X\to C{B}^{d_s}(X)$ is a multi-valued $H^{\beta }$-quasi contraction if we can find $\beta \in [0,1]$ and $k\in (0,{\displaystyle \frac{1}{s}})$, such that for all $g^{ı},g^j\in X$,

$$\begin{array}{c}\hfill H^{\beta }(T{g}^{ı},T{g}^j)\le k·max\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j),d_s(g^{ı},T{g}^j),d_s(g^j,T{g}^{ı})\}.\end{array}$$

(7)

Definition 14.

We say that $T:X\to C{B}^{d_s}(X)$ is a multi-valued $H^{\beta }$-weak quasi contraction if we can find $\beta \in [0,1]$, $k\in (0,{\displaystyle \frac{1}{s}})$ and $L\ge 0$, such that for all $g^{ı},g^j\in X$,

$$\begin{array}{c}\hfill H^{\beta }(T{g}^{ı},T{g}^j)\le k·max\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j)\}+L{d}_s(g^{ı},T{g}^j).\end{array}$$

(8)

Example 2.

Let $X=[0,{\displaystyle \frac{7}{9}}]\bigcup \{1\}$ and $d_s(g^{ı},g^j)={|g^{ı}-g^j|}^{2}forall{g}^{ı},g^j\in X$.

Then $\{X,d_s\}$ is a b-metric space. Define the mapping $T:X\to C{B}^{d_s}(X)$ by

$$T(g^{ı})=\left\{\begin{array}{c}\{{\displaystyle \frac{g^{ı}}{4}}\},\mathrm{for}{g}^{ı}\in [0,{\displaystyle \frac{7}{9}}]\hfill \\ \\ \{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}}\},\mathrm{for}{g}^{ı}=1.\hfill \end{array}\right.$$

Then T is a multi-valued $H^{\beta }$-contraction with $\beta ={\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{217}{256}}\le k<1$ as shown below.

We will consider the following different cases for the elements of X.

(i)

$g^{ı},g^j\in [0,{\displaystyle \frac{7}{9}}]$.

By Theorem 2(1), we have $H^{\frac{3}{4}}(T{g}^{ı},T{g}^j)=d_s({\displaystyle \frac{g^{ı}}{4}},{\displaystyle \frac{g^j}{4}})\le k{d}_s(g^{ı},g^j),k\ge {\displaystyle \frac{1}{16}}$.

(ii)

$g^{ı}\in [0,{\displaystyle \frac{7}{9}}],g^j=1$.

We have the following sub cases:

(ii)(a)

$g^{ı}\in [0,{\displaystyle \frac{2}{3}}],g^j=1$. Then $T{g}^{ı}=\{{\displaystyle \frac{g^{ı}}{4}}\}$ and $0\le {\displaystyle \frac{g^{ı}}{4}}\le {\displaystyle \frac{1}{6}}$. Therefore, we have ${\delta }_{d_s}(T{g}^{ı},T1)={\delta }_{d_s}(\{{\displaystyle \frac{g^{ı}}{4}}\},\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}}\})$ and ${\delta }_{d_s}(T1,T{g}^{ı})={\delta }_{d_s}(\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}}\},\{{\displaystyle \frac{g^{ı}}{4}}\})$. Note that for $0\le {\displaystyle \frac{g^{ı}}{4}}\le {\displaystyle \frac{1}{6}}$, ${\displaystyle \frac{g^{ı}}{4}}$ is nearest to 0 and farthest from ${\displaystyle \frac{5}{12}}$. Therefore, ${\delta }_{d_s}(T{g}^{ı},T1)={|{\displaystyle \frac{g^{ı}}{4}}-0|}^2={\displaystyle \frac{{g^{ı}}^2}{16}}$ and ${\delta }_{d_s}(T1,T{g}^{ı})={|{\displaystyle \frac{5}{12}}-{\displaystyle \frac{g^{ı}}{4}}|}^2={\displaystyle \frac{9{g^{ı}}^2-30g^{ı}+25}{144}}$

Therefore,

$$\begin{array}{cc}\hfill H^{\frac{3}{4}}(T{g}^{ı},T1)& =max\left\{{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T{g}^{ı},T1)+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T1,T{g}^{ı}),{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T1,T{g}^{ı})+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T{g}^{ı},T1)\right\}\\ & =max\left\{{\displaystyle \frac{25}{576}}-{\displaystyle \frac{10g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}},{\displaystyle \frac{75}{576}}-{\displaystyle \frac{30g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}}\right\}\\ & ={\displaystyle \frac{75}{576}}-{\displaystyle \frac{30g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}}\le k{d}_s(g^{ı},1),k\ge {\displaystyle \frac{279}{576}}.\end{array}$$

(${\displaystyle \frac{279}{576}}$ is the maximum value of k which satisfies the above inequality for different values of $g^{ı}$ in $[0,{\displaystyle \frac{2}{3}}]$.)

(ii)(b)

$g^{ı}\in ({\displaystyle \frac{2}{3}},{\displaystyle \frac{7}{9}}],g^j=1$.

Then $T{g}^{ı}=\{{\displaystyle \frac{g^{ı}}{4}}\}$ and ${\displaystyle \frac{6}{36}}<{\displaystyle \frac{g^{ı}}{4}}\le {\displaystyle \frac{7}{36}}$.

Therefore, we have ${\delta }_{d_s}(T{g}^{ı},T1)={\delta }_{d_s}(\{{\displaystyle \frac{g^{ı}}{4}}\},\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}}\})$ and ${\delta }_{d_s}(T1,T{g}^{ı})={\delta }_{d_s}(\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}}\},\{{\displaystyle \frac{g^{ı}}{4}}\})$. Note that for ${\displaystyle \frac{6}{36}}<{\displaystyle \frac{g^{ı}}{4}}\le {\displaystyle \frac{7}{36}}$, ${\displaystyle \frac{g^{ı}}{4}}$ is nearest to ${\displaystyle \frac{1}{3}}$ and farthest from ${\displaystyle \frac{5}{12}}$. Therefore, ${\delta }_{d_s}(T{g}^{ı},T1)={|{\displaystyle \frac{g^{ı}}{4}}-{\displaystyle \frac{1}{3}}|}^2$ = ${\displaystyle \frac{{g^{ı}}^2}{16}}-{\displaystyle \frac{2g^{ı}}{12}}+{\displaystyle \frac{1}{9}}$ and ${\delta }_{d_s}(T1,T{g}^{ı})={|{\displaystyle \frac{g^{ı}}{4}}-{\displaystyle \frac{5}{12}}|}^2$ = ${\displaystyle \frac{{g^{ı}}^2}{16}}-{\displaystyle \frac{10g^{ı}}{48}}+{\displaystyle \frac{25}{144}}$. Then, we have

$$\begin{array}{cc}\hfill H^{\frac{3}{4}}(T{g}^{ı},T1)& =max\{{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T{g}^{ı},T1)+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T1,T{g}^{ı}),{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T1,T{g}^{ı})+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T{g}^{ı},T1)\}\\ & =max\{{\displaystyle \frac{73}{576}}-{\displaystyle \frac{34g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}},{\displaystyle \frac{91}{576}}-{\displaystyle \frac{38g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}}\}\\ & ={\displaystyle \frac{91}{576}}-{\displaystyle \frac{38g^{ı}}{192}}+{\displaystyle \frac{4{g^{ı}}^2}{64}}\le k{d}_s(g^{ı},1),k\ge {\displaystyle \frac{217}{256}}.\end{array}$$

However, we see that for $g^{ı}={\displaystyle \frac{7}{9}},g^j=1$,

$$H(T({\displaystyle \frac{7}{9}}),T(1))={\displaystyle \frac{4}{81}}=d_s({\displaystyle \frac{7}{9}},1)$$

and hence T does not satisfy the contraction Condition of Nadler [3] and Czervic [8].

Example 3.

Let $X=\{0,{\displaystyle \frac{1}{4}},1\}$, $d_s(g^{ı},g^j)={|g^{ı}-g^j|}^{2}forall{g}^{ı},g^j\in X$ and $T:X\to CB(X)$ be as follows: $T(g^{ı})=\left\{\begin{array}{c}\{0\},for{g}^{ı}\in \{0,{\displaystyle \frac{1}{4}}\}\hfill \\ \{0,1\},for{g}^{ı}=1,\hfill \end{array}\right.$ We will show that T is a multi-valued $H^{\beta }$-contraction mapping with $\beta \in ({\displaystyle \frac{7}{16}},{\displaystyle \frac{9}{16}})$. If $g^{ı},g^j\in \{0,{\displaystyle \frac{1}{4}}\}$, then the result is clear. Suppose $g^{ı}\in \{0,{\displaystyle \frac{1}{4}}\}$ and $g^j=1$. Then ${\delta }_{d_s}(T{g}^{ı},T1)=0$ and ${\delta }_{d_s}(T1,T{g}^{ı})=1$ so that $H^{\beta }(T{g}^{ı},T1)=max\{\beta ,1-\beta \}$. In addition, we have $d_s(g^{ı},1)=1$ or ${\displaystyle \frac{9}{16}}$. If $\beta \in ({\displaystyle \frac{7}{16}},{\displaystyle \frac{1}{2}}]$, then $H^{\beta }(T{g}^{ı},T1)=1-\beta$. Now $1-\beta \in [{\displaystyle \frac{8}{16}},{\displaystyle \frac{9}{16}})$. Therefore, $1-\beta ={\displaystyle \frac{16}{9}}(1-\beta ){\displaystyle \frac{9}{16}}$ and $1-\beta <{\displaystyle \frac{16}{9}}(1-\beta )1$, that is $1-\beta \le {\displaystyle \frac{16}{9}}(1-\beta )d_s(g^{ı},1)$. Thus, we have $H^{\beta }(T{g}^{ı},T1)=1-\beta \le k{d}_s(g^{ı},1)$, where $k={\displaystyle \frac{16}{9}}(1-\beta )<1$. Similarly if $\beta \in [{\displaystyle \frac{1}{2}},{\displaystyle \frac{9}{16}})$, we get $H^{\beta }(T{g}^{ı},T1)=\beta \le k{d}_s(g^{ı},1)$ where $k={\displaystyle \frac{16}{9}}\beta <1$. Thus, T is a multi-valued $H^{\beta }$-contraction. However T is not a multi-valued quasi contraction mapping. Indeed, for $g^{ı}={\displaystyle \frac{1}{4}}$ and $g^j=1$, we have

$$\begin{array}{ccc}\hfill H_{d_s}(T({\displaystyle \frac{1}{4}}),T(1))& =& max\{{\delta }_{d_s}(T({\displaystyle \frac{1}{4}}),T1),{\delta }_{d_s}(T1,T({\displaystyle \frac{1}{4}}))\}=1\hfill \\ & >& k·max\{d_s({\displaystyle \frac{1}{4}},1),d_s({\displaystyle \frac{1}{4}},T({\displaystyle \frac{1}{4}}),d_s(1,T1),d_s({\displaystyle \frac{1}{4}},T1),d_s(1,T({\displaystyle \frac{1}{4}}))\}\hfill \end{array}$$

for any $k\in (0,1)$. Therefore, T does not satisfy the contraction conditions given in Definitions 4–7.

Now we will present our main results in which we establish the existence of fixed points of generalized multi-valued contraction mappings using $H^{\beta }$ Hausdorff–Pompeiu b-metric. Hereafter, $\mathcal{F}\{T\}$ will denote the fixed point set of T.

Theorem 5.

Suppose $d_s$ is *-continuous and $T:X\to C{B}^{d_s}(X)$ is a multi-valued mapping satisfying the following conditions:

(i) 

There exists $\beta \in [0,1]$, $a,b,c,e,f,h,j\ge 0$, $a+b+s(c+e+{\displaystyle \frac{h}{2}})+f+j<1$ and $min\{s(a+e+{\displaystyle \frac{h}{2}}),s(b+c+{\displaystyle \frac{h}{2}})\}<1$ such that for all $g^{ı},g^j\in X$,

$$\begin{array}{ccc}\hfill H^{\beta }(T{g}^{ı},T{g}^j)& \le & a·d_s(g^{ı},T{g}^{ı})+b·d_s(g^j,T{g}^j)+c·d_s(g^{ı},T{g}^j)+e·d_s(g^j,T{g}^{ı})\hfill \\ & +& h·{\displaystyle \frac{d_s(g^{ı},T{g}^j)+d_s(g^j,T{g}^{ı})}{2}}+j·{\displaystyle \frac{d_s(g^{ı},T{g}^{ı})d_s(g^j,T{g}^j)}{1+d_s(g^{ı},g^j)}}+f·d_s(g^{ı},g^j).\hfill \end{array}$$

(9)

(ii) 

or every $g^{ı}$ in $X,g^j$ in $T(g^{ı})$ and $ϵ>0,$ there exists $g^{ø}$ in $T(g^j)$ satisfying

$$\begin{array}{c}\hfill d_s(g^j,g^{ø})\le H^{\beta }(T{g}^{ı},T{g}^j)+ϵ.\end{array}$$

(10)

Then $\mathcal{F}\{T\}\ne \varphi$.

Proof. 

For some arbitrary $g_0^{ı}\in X$, if $g_0^{ı}\in T{g}_0^{ı}$ then $g_0^{ı}\in \mathcal{F}\{T\}$. Suppose $g_0^{ı}\notin T{g}_0^{ı}$. Let $g_1^{ı}\in T{g}_0^{ı}$. Again, if $g_1^{ı}\in T{g}_1^{ı}$ then $g_1^{ı}\in \mathcal{F}\{T\}$. Suppose $g_1^{ı}\notin T{g}_1^{ı}$. By (10), we can find $g_2^{ı}\in T{g}_1^{ı}$ such that

$$d_s(g_1^{ı},g_2^{ı})\le H^{\beta }(T{g}_0^{ı},T{g}_1^{ı})+ϵ.$$

If $g_2^{ı}\in T{g}_2^{ı}$ then $g_2^{ı}\in \mathcal{F}\{T\}$. Suppose $g_2^{ı}\notin T{g}_2^{ı}$. By (10), we can find $g_3^{ı}\in T{g}_2^{ı}$ such that

$$d_s(g_2^{ı},g_3^{ı})\le H^{\beta }(T{g}_1^{ı},T{g}_2^{ı})+{ϵ}^2.$$

In this way we construct the sequence $\{g_n^{ı}\}$ such that $g_n^{ı}\notin T{g}_n^{ı}$, $g_{n+1}^{ı}\in T{g}_n^{ı}$ and

$$d_s(g_n^{ı},g_{n+1}^{ı})\le H^{\beta }(T{g}_{n-1}^{ı},T{g}_n^{ı})+{ϵ}^n.$$

Then, using (9), we have

$$\begin{array}{ccc}\hfill d_s(g_n^{ı},g_{n+1}^{ı})& \le & H^{\beta }(T{g}_{n-1}^{ı},T{g}_n^{ı})+{ϵ}^n\hfill \\ & \le & a·d_s(g_{n-1}^{ı},T{g}_{n-1}^{ı})+b·d_s(g_n^{ı},T{g}_n^{ı})+c·d_s(g_{n-1}^{ı},T{g}_n^{ı})+e·d_s(g_n^{ı},T{g}_{n-1}^{ı})\hfill \\ & +& h·{\displaystyle \frac{d_s(g_{n-1}^{ı},T{g}_n^{ı})+d_s(g_n^{ı},T{g}_{n-1}^{ı})}{2}}+j·{\displaystyle \frac{d_s(g_{n-1}^{ı},T{g}_{n-1}^{ı})d_s(g_n^{ı},T{g}_n^{ı})}{1+d_s(g_{n-1}^{ı},g_n^{ı})}}+f·d_s(g_{n-1}^{ı},g_n^{ı})+{ϵ}^n,\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill (1-b-sc-j)·d_s(g_n^{ı},g_{n+1}^{ı})\le (a+sc+{\displaystyle \frac{sh}{2}}+f)·d_s(g_{n-1}^{ı},g_n^{ı})+{ϵ}^n.\end{array}$$

(11)

Using symmetry of $H^{\beta }$, we also have

$$\begin{array}{c}\hfill (1-a-se-j)·d_s(g_n^{ı},g_{n+1}^{ı})\le (b+se+{\displaystyle \frac{sh}{2}}+f)·d_s(g_{n-1}^{ı},g_n^{ı})+{ϵ}^n.\end{array}$$

(12)

Adding (11) and (12), we get

$$d_s(g_n^{ı},g_{n+1}^{ı})\le (a+b+s(c+e+{\displaystyle \frac{h}{2}})+f+j)·d_s(g_{n-1}^{ı},g_n^{ı})+{ϵ}^n.$$

By Lemma 2, the sequence $\{{g^{ı}}_n\}$ is a Cauchy sequence. Completeness of ($X,d_s$) gives ${lim}_{n\to +\infty }{d}_s(g_n^{ı},{g^{ı}}^{\ast })=0$ for some ${g^{ı}}^{\ast }\in X$. We now show that ${g^{ı}}^{\ast }\in T{g^{ı}}^{\ast }$. Suppose, on the contrary, that ${g^{ı}}^{\ast }\notin T{g^{ı}}^{\ast }$. Then,

$$\begin{array}{ccc}& & \beta ·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })+(1-\beta )·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})\le H^{\beta }(T{g}_n^{ı},T{g^{ı}}^{\ast })\hfill \\ & & \le a·d_s(g_n^{ı},T{g}_n^{ı})+b·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })+c·d_s(g_n^{ı},T{g^{ı}}^{\ast })+e·d_s({g^{ı}}^{\ast },T{g}_n^{ı})\hfill \\ & & +h·{\displaystyle \frac{d_s(g_n^{ı},T{g^{ı}}^{\ast })+d_s({g^{ı}}^{\ast },T{g}_n^{ı})}{2}}+j·{\displaystyle \frac{d_s(g_n^{ı},T{g}_n^{ı})d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })}{1+d_s(g_n^{ı},{g^{ı}}^{\ast })}}+f·d_s(g_n^{ı},{g^{ı}}^{\ast })\hfill \\ & & \le a·d_s(g_n^{ı},g_{n+1}^{ı})+b·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })+c·d_s(g_n^{ı},T{g^{ı}}^{\ast })+e·d_s({g^{ı}}^{\ast },g_{n+1}^{ı})\hfill \\ & & +h·{\displaystyle \frac{d_s(g_n^{ı},T{g^{ı}}^{\ast })+d_s({g^{ı}}^{\ast },g_{n+1}^{ı})}{2}}+{\displaystyle \frac{d_s(g_n^{ı},g_{n+1}^{ı})d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })}{1+d_s(g_n^{ı},{g^{ı}}^{\ast })}}+f·d_s(g_n^{ı},{g^{ı}}^{\ast }).\hfill \end{array}$$

and using the *-continuity of $d_s$, we get

$$\underset{n\to \infty }{lim\; inf}\beta ·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })+(1-\beta )·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})\le (b+c+{\displaystyle \frac{h}{2}})·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast }).$$

Similarly,

$$\underset{n\to \infty }{lim\; inf}\beta ·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})+(1-\beta )·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })\le (a+e+{\displaystyle \frac{h}{2}})·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast }).$$

It follows that

$$\begin{array}{c}{d}_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })=\beta ·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })+(1-\beta )·d_s(T{g^{ı}}^{\ast },{g^{ı}}^{\ast })\le s[\beta ·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })\\ +(1-\beta )·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})]+s.d_s(g_{n+1}^{ı},{g^{ı}}^{\ast })\end{array}$$

that is,

$$\begin{array}{cc}\hfill d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })& \le s[\underset{n\to \infty }{lim\; inf}[\beta \delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })+(1-\beta )\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})]]+s[\underset{n\to \infty }{lim\; inf}{d}_s(g_{n+1}^{ı},{g^{ı}}^{\ast })]\hfill \\ \hfill & \le s(b+c+{\displaystyle \frac{h}{2}})d_s(x^{\ast },T{g^{ı}}^{\ast })\hfill \end{array}$$

and

$$\begin{array}{c}{d}_s(T{g^{ı}}^{\ast },{g^{ı}}^{\ast })=\beta ·d_s(T{g^{ı}}^{\ast },{g^{ı}}^{\ast })+(1-\beta )·d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })\le s[\beta ·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})\\ +(1-\beta )·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })]+s·d_s({g^{ı}}^{\ast },g_{n+1}^{ı})\end{array}$$

that is,

$$\begin{array}{cc}\hfill d_s(T{g^{ı}}^{\ast },{g^{ı}}^{\ast })& \le s[\underset{n\to \infty }{lim\; inf}[\beta ·\delta {}_{d_s}(T{g^{ı}}^{\ast },T{g}_n^{ı})+(1-\beta )·\delta {}_{d_s}(T{g}_n^{ı},T{g^{ı}}^{\ast })]]+s[\underset{n\to \infty }{lim\; inf}{d}_s({g^{ı}}^{\ast },g_{n+1}^{ı})]\hfill \\ \hfill & \le s(a+e+{\displaystyle \frac{h}{2}})·d_s(T{g^{ı}}^{\ast },x^{\ast }).\hfill \end{array}$$

Since $min\{s(a+e+{\displaystyle \frac{h}{2}}),s(c+e+{\displaystyle \frac{h}{2}}\}<1$, we get $d_s({g^{ı}}^{\ast },T{g^{ı}}^{\ast })=0$ which from Proposition 1 implies that ${g^{ı}}^{\ast }\in \overline{T{g^{ı}}^{\ast }}$ and since $T{g^{ı}}^{\ast }$ is closed it follows that ${g^{ı}}^{\ast }\in T{g^{ı}}^{\ast }$. □

Remark 5.

Theorem 5 is true even if we replace (9) by any of the following conditions:

For some $0\le k<{\displaystyle \frac{1}{s}}$,

$$\begin{array}{ccc}\hfill H^{\beta }(T{g}^{ı},T{g}^j)& \le & k·max\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j),{\displaystyle \frac{d_s(g^{ı},T{g}^j)+d_s(g^j,T{g}^{ı})}{2s}},\hfill \\ & & {\displaystyle \frac{d_s(g^{ı},T{g}^{ı})d_s(g^j,T{g}^j)}{1+d_s(g^{ı},g^j)}}\},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill H^{\beta }(T{g}^{ı},T{g}^j)& \le & k·max\{d_s(g^{ı},g^j),d_s(g^{ı},T{g}^{ı}),d_s(g^j,T{g}^j),d_s(g^{ı},T{g}^j),\hfill \\ & & d_s(g^j,T{g}^{ı}),{\displaystyle \frac{d_s(g^{ı},T{g}^{ı})d_s(g^j,T{g}^j)}{1+d_s(g^{ı},g^j)}}\}\}\hfill \end{array}$$

(14)

The following result is a consequence of Theorem 5 and Remark 5:

Corollary 1.

Suppose $d_s$ is *-continuous and $T:X\to C{B}^{d_s}(X)$ satisfy Condition (10) and any of the following conditions:

(i) 

T is a multi-valued H${}^{\beta }$-Ciric contraction.

(ii) 

T is a multi-valued H${}^{\beta }$-Hardy–Roger’s contraction.

(iii) 

T is a multi-valued H${}^{\beta }$-quasi contraction.

(iv) 

T is a multi-valued H${}^{\beta }$-weak quasi contraction.

(v) 

T is a multi-valued H${}^{\beta }$-contraction.

Then $\mathcal{F}\{T\}\ne \varphi$.

Taking $T:X\to X$ in Corollary 1 (ii) and using Theorem 2 (i), we have the following corollary.

Corollary 2.

Suppose $d_s$ is *-continuous and $T:X\to X$. If there exists non-negative real numbers $a,b,c,e,f$ such that $a+b+s(c+e)+f<1$, $min\{s(a+e),s(b+c)\}<1$ and

$$d_s(T{g}^{ı},T^j)\le a·d_s(g^{ı},g^j)+b·d_s(g^{ı},T{g}^{ı})+c·d_s(g^j,T^j)+e·d_s(g^{ı},T^j)+f·d_s(g^j,T{g}^{ı}),forall{g}^{ı},g^j\in X,$$

(15)

then $\mathcal{F}(T)\ne \varphi$.

Remark 6.

For $\beta =1$, Condition (10) is obviously satisfied and hence, (Theorem 5 [3]), (Theorem 2.1 [8]), (Theorem 2.2 [9]), (Theorem 2.11 [10]), (Theorem 3.1 [12]) and (Theorem 3.1 [11]) are all particular cases of Corollary 1. However, the examples which follow illustrate that the converse is not necessarily true.

We now furnish the following examples to validate our results.

Example 4.

Let X, $d_s$ and T be as in Example 2. Then, as shown above, T belongs to the class of multi-valued $H^{\beta }$-contraction with $\beta \in ({\displaystyle \frac{7}{16}},{\displaystyle \frac{9}{16}})$ and consequently T satisfies all the contraction conditions given in Definitions 11–14. We will show that T satisfies (10):

For $g^{ı}\in [0,{\displaystyle \frac{7}{9}}]$, $T{g}^{ı}$ is singleton and so the result is obvious. Now for $g^{ı}=1$, if $g^j=0\in T{g}^{ı}$ then $g^{ø}=0\in T{g}^j$ will satisfy (10). If $g^j={\displaystyle \frac{1}{3}}\in T{g}^{ı}$, then $g^{ø}={\displaystyle \frac{1}{12}}\in T{g}^j$ and if $g^j={\displaystyle \frac{5}{12}}\in T{g}^{ı}$ then $g^{ø}={\displaystyle \frac{5}{48}}\in T^j$ will satisfy (10). Thus, T satisfies conditions of Theorem 5 and Corollary 1 and $0,1\in \mathcal{F}(T)$.

However, as shown in Example 2, T does not satisfy the contraction condition of Nadler [3] and Czervic [8].

Example 5.

Let X, $d_s$ and T be as in Example 3. Then as shown above, T belongs to the class of multi-valued $H^{\beta }$-contraction with $\beta \in ({\displaystyle \frac{7}{16}},{\displaystyle \frac{9}{16}})$ and consequently T satisfies all the contraction conditions given in Definitions 11–14.

We will show that T satisfies (10):

For $g^{ı}\in \{0,{\displaystyle \frac{1}{4}}\}$, $T{g}^{ı}$ is singleton and so the result is obvious. Now for $g^{ı}=1$, if $g^j=0\in T{g}^{ı}$ then $g^{ø}=0\in T{g}^j$ will satisfy (10). If $g^j=1\in T{g}^{ı}$ then $g^{ø}=1\in T{g}^j$ will satisfy (10). Thus, Theorem 5 and Corollary 1 are applicable and $0,1\in \mathcal{F}(T)$. However, we see that T does not satisfy the conditions of (Theorem 2.2 [9]), (Theorem 2.11 [10]) and (Theorem 3.1 [12]).

Example 6.

Let $X=\{0,{\displaystyle \frac{1}{12}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}},{\displaystyle \frac{34}{48}},1\}$, $d_s(g^{ı},g^j)=|g^{ı}-g^j|\mathrm{for}\mathrm{all}{g}^{ı},g^j\in X$ and $T:X\to C{B}^{d_s}(X)$ be as follows:

$$T(0)=T({\displaystyle \frac{1}{12}})=\{0\},T({\displaystyle \frac{1}{3}})=T({\displaystyle \frac{5}{12}})=T({\displaystyle \frac{34}{48}})=\left\{{\displaystyle \frac{1}{12}}\right\},T(1)=\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{34}{48}},1\}.$$

Then, T is a multi-valued $H^{\beta }$-quasi contraction for $\beta ={\displaystyle \frac{3}{4}}$ with ${\displaystyle \frac{34}{44}}\le k<1$ as shown below:

(1) If $g^{ı}={\displaystyle \frac{34}{48}}$ and $g^j=1$, then ${\delta }_{d_s}(T({\displaystyle \frac{34}{48}}),T1)={\delta }_{d_s}(\{{\displaystyle \frac{1}{12}}\},\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{34}{48}},1\})={\displaystyle \frac{1}{12}}$ and ${\delta }_{d_s}(T1,T({\displaystyle \frac{34}{48}}))={\delta }_{d_s}(\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{34}{48}},1\},\{{\displaystyle \frac{1}{12}}\})={\displaystyle \frac{11}{12}}$.

$$\begin{array}{ccc}\hfill H^{\frac{3}{4}}(T({\displaystyle \frac{34}{48}}),T1)& =& max\{{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T({\displaystyle \frac{34}{48}}),T1)+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T1,T({\displaystyle \frac{34}{48}}),{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T1,T({\displaystyle \frac{34}{48}}))+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T({\displaystyle \frac{34}{48}}),T1)\}\hfill \\ & =& max\{{\displaystyle \frac{3}{4}}.{\displaystyle \frac{1}{12}}+{\displaystyle \frac{1}{4}}.{\displaystyle \frac{11}{12}},{\displaystyle \frac{3}{4}}.{\displaystyle \frac{11}{12}}+{\displaystyle \frac{1}{4}}.{\displaystyle \frac{1}{12}}\}={\displaystyle \frac{34}{48}}\hfill \\ & \le & k{\displaystyle \frac{44}{48}},\mathrm{for}\mathrm{any}k\ge {\displaystyle \frac{34}{44}}\hfill \\ & =& k{d}_s(1,T({\displaystyle \frac{34}{48}}))\hfill \\ & \le & kmax\{d_s({\displaystyle \frac{34}{48}},1),d_s({\displaystyle \frac{34}{48}},T({\displaystyle \frac{34}{48}}),d_s(1,T1),d_s({\displaystyle \frac{34}{48}},T1),d_s(1,T({\displaystyle \frac{34}{48}}))\}.\hfill \end{array}$$

(2) If $g^{ı}={\displaystyle \frac{1}{12}}$ and $g^j=1$. ${\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T1)={\delta }_{d_s}(\{0,\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{34}{48}},1\})=0$. ${\delta }_{d_s}(T1,T({\displaystyle \frac{1}{12}}))={\delta }_{d_s}(\{0,{\displaystyle \frac{1}{3}},{\displaystyle \frac{34}{48}},1\},0\})=1$.

$$\begin{array}{ccc}\hfill H^{\frac{3}{4}}(T({\displaystyle \frac{1}{12}}),T1)& =& max\{{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T1)+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T1,T({\displaystyle \frac{1}{12}}),{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T1,T({\displaystyle \frac{1}{12}}))+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T1)\}={\displaystyle \frac{3}{4}}\hfill \\ & \le & k.1,\mathrm{for}\mathrm{any}k\ge {\displaystyle \frac{3}{4}}\hfill \\ & =& k·d_s(1,T({\displaystyle \frac{1}{12}}))\hfill \\ & \le & k·max\{d_s({\displaystyle \frac{1}{12}},1),d_s({\displaystyle \frac{1}{12}},T({\displaystyle \frac{1}{12}}),d_s(1,T1),d_s({\displaystyle \frac{1}{12}},T1),d_s(1,T({\displaystyle \frac{1}{12}}))\}.\hfill \end{array}$$

(3) If $g^{ı}={\displaystyle \frac{1}{12}}$ and $g^j={\displaystyle \frac{1}{3}}$, then ${\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T({\displaystyle \frac{1}{3}}))={\delta }_{d_s}(\{0,\{{\displaystyle \frac{1}{12}}\})={\displaystyle \frac{1}{12}}$ and ${\delta }_{d_s}({\displaystyle \frac{1}{3}},T({\displaystyle \frac{1}{12}}))={\delta }_{d_s}(\{{\displaystyle \frac{1}{12}}\},0\})={\displaystyle \frac{1}{12}}$.

$$\begin{array}{ccc}\hfill H^{\frac{3}{4}}(T({\displaystyle \frac{1}{12}}),T({\displaystyle \frac{1}{3}}))& =& max\{{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T({\displaystyle \frac{1}{3}}))+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{3}}),T({\displaystyle \frac{1}{12}}),{\displaystyle \frac{3}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{3}}),T({\displaystyle \frac{1}{12}})+{\displaystyle \frac{1}{4}}{\delta }_{d_s}(T({\displaystyle \frac{1}{12}}),T({\displaystyle \frac{1}{3}}))\}\hfill \\ & =& {\displaystyle \frac{1}{12}}\le k.{\displaystyle \frac{4}{12}},\mathrm{for}\mathrm{any}k\ge {\displaystyle \frac{1}{4}}\hfill \\ & =& k·d_s({\displaystyle \frac{1}{3}},T({\displaystyle \frac{1}{12}})\hfill \\ & \le & k·max\{d_s({\displaystyle \frac{1}{12}},{\displaystyle \frac{1}{3}}),d_s({\displaystyle \frac{1}{12}},T({\displaystyle \frac{1}{12}}),d_s({\displaystyle \frac{1}{3}},T({\displaystyle \frac{1}{3}})),d_s({\displaystyle \frac{1}{12}},T({\displaystyle \frac{1}{3}})),d_s({\displaystyle \frac{1}{3}},T({\displaystyle \frac{1}{12}}))\}.\hfill \end{array}$$

For all other values of $g^{ı}$ and $g^j$, a similar argument as above follows. Thus, T is a multi-valued $H^{\beta }$-quasi contraction. We will show that T satisfies (10): For $g^{ı}\in \{0,{\displaystyle \frac{1}{12}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{12}},{\displaystyle \frac{34}{48}}\}$, $T{g}^{ı}$ is singleton and so the result is obvious. Now, for $g^{ı}=1$, if $g^j=0\in T{g}^{ı}$ then $g^{ø}=0\in T{g}^j$ will satisfy (10). If $g^j={\displaystyle \frac{1}{3}}$ or ${\displaystyle \frac{34}{48}}\in T{g}^{ı}$ then, $g^{ø}={\displaystyle \frac{1}{12}}\in T{g}^j$ will satisfy (10). Thus, Theorem 5 and Corollary 1 are applicable and $0,1\in \mathcal{F}(T)$. However, we see that $H(T({\displaystyle \frac{34}{48}}),T(1))={\displaystyle \frac{11}{12}}$, where $d({\displaystyle \frac{34}{48}},1)={\displaystyle \frac{14}{48}}$, $d({\displaystyle \frac{34}{48}},T({\displaystyle \frac{34}{48}}))={\displaystyle \frac{30}{48}}$, $d(1,T(1))=0$, $d({\displaystyle \frac{34}{48}},T(1)=0$ and $d(1,T({\displaystyle \frac{34}{48}}))\}={\displaystyle \frac{11}{12}}$ and so T does not satisfy the conditions of (Theorem 2.2 [9]), (Theorem 2.11 [10]), (Theorem 3.1 [12]) and (Theorem 3.1 [11]).

Proposition 3.

Let $T_1,T_2:X\to C{B}^{d_s}(X),$ satisfy the following:

(3.1) 

For all $q,r\in \{1,2\}$, every $g^{ı}$ in $X,g^j$ in $T_q(g^{ı})$ and $ϵ>0,$ there exists $g^{ø}$ in $T_r(g^j)$ satisfying

$$d_s(g^j,g^{ø})\le H^{\beta }(T_q{g}^{ı},T_r{g}^j)+ϵ.$$

(3.2) 

Any of the following conditions holds:

(i) 

$T_{1}and{T}_2$ is a multi-valued H${}^{\beta }$-Ciric contraction;

(ii) 

$T_{1}and{T}_2$ is a multi-valued H${}^{\beta }$-quasi contraction;

(iii) 

$T_{1}and{T}_2$ is a multi-valued H${}^{\beta }$-weak quasi contraction;

Then, for any $u\in \mathcal{F}\{T_q\}$, there exist $w\in \mathcal{F}\{T_r\}$ ($q\ne r$) such that

$$d_s(u,w)\le {\displaystyle \frac{s}{1-k}}\underset{x\in X}{sup}{H}^{\beta }(T_{q}x,T_{r}x),$$

where k is the Lipschitz’s constant.

Proof. 

Let $g_0^{ı}\in \mathcal{F}\{T_1\}$. By (3.1) we can find $g_1^{ı}\in T_2{g}_0^{ı}$ such that

$$d_s(g_0^{ı},g_1^{ı})\le H^{\beta }(T_1{g}_0^{ı},T_2{g}_1^{ı})+ϵ.$$

By (3.1), choose $g_2^{ı}\in T_2{g}_1^{ı}$ such that

$$d_s(g_1^{ı},g_2^{ı})\le H^{\beta }(T_2{g}_0^{ı},T_2{g}_1^{ı}).$$

Inductively, we define sequence $\{g_n^{ı}\}$ such that $g_{n+1}^{ı}\in T_2(g_n^{ı})$ and

$$d_s(g_n^{ı},g_{n+1}^{ı})\le H^{\beta }(T_2{g}_{n-1}^{ı},T_2{g}_n^{ı})+ϵ.$$

(16)

Now, following the same technique as in the proof of Theorem 5, we see that the sequence $\{g_n^{ı}\}$ converges to some $g_{\ast }^{ı}$ in X and $g_{\ast }^{ı}\in \mathcal{F}\{T_2\}$. Since $ϵ$ is arbitrary, taking $ϵ\to 0$ in (16) we get

$$d_s(g_n^{ı},g_{n+1}^{ı})\le H^{\beta }(T_2{g}_{n-1}^{ı},T_2{g}_n^{ı}).$$

Then, using (Section 3.2), we get

$$d_s(g_n^{ı},g_{n+1}^{ı})\le k^n{d}_s(g_0^{ı},g_1^{ı}).$$

Then, we have $d(g_0^{ı},g_{\ast }^{ı})\le {\sum }_{n=0}^{\infty }{s}^{n+1}{d}_s(g_{n+1}^{ı},g_n^{ı})\le s(1+sk+{(sk)}^2+\cdots )d_s(g_1^{ı},g_0^{ı})\le {\displaystyle \frac{s}{1-sk}}(H^{\beta }(T_2{g}_0^{ı},T_1{g}_0^{ı})+ϵ).$ Interchanging the roles of $T_1$ and $T_2$ and proceeding as above, it gives that for each $g_0^j\in \mathcal{F}\{T_2\}$ there exist $g_1^j\in T_1{g}_0^j$ and $g^{\ell }\in F(T_1)$ such that

$$d(g_0^j,g^{\ell })\le {\displaystyle \frac{s}{1-sk}}(H^{\beta }(T_1{g}_0^j,T_2{g}_0^j)+ϵ).$$

Now the result follows as $ϵ>0$ is arbitrary. □

3.3. Application to Multi-Valued Fractals

Inspiring from some recent works in [18,22,23], we provide an application of our result to multi-valued fractals. Let $P_i:X\to C{B}^{d_s}(X)$, $i=1,2,\cdots n$ be upper semi continuous mappings. Then, $P=(P_1,P_2,\cdots P_n)$ is an iterated multifunction system (in short IMS) defined on the b-metric space $(X,d_s)$. The operator $T_P:CB(X)\to CB(X)$ defined by $T_P(Y)={\bigcup }_{i=1}^n{P}_i(Y)$ is called the extended multifractal operator generated by the IMS $P=(P_1,P_2,\cdots P_n)$. Any non empty compact subset of X which is a fixed point of $T_P$ is called a multi-valued fractal of the iterated multifunction system $P=(P_1,P_2,\cdots P_n)$.

Theorem 6.

Let $P_i:X\to CB(X)$, $i=1,2,\cdots n$ be upper semi continuous mappings such that for each $i=1,2,\cdots n$ the following conditions hold:

We can find $\beta \in [0,1]$ and $a,e\in (0,1)$, $a+2se<1$, such that for all $x,y\in X,i=1,2\cdots n$

$$\begin{array}{c}\hfill H^{\beta }(P_{i}x,P_{i}y)\le a{d}_s(x,y)+e[d_s(x,P_{i}y)+d_s(y,P_{i}x)].\end{array}$$

(17)

Then,

(i) 

For all $U_1,U_2\in CB(X)$, $H^{\beta }(T_P(U_1),T_P(U_2))\le a{H}^{\beta }(U_1,U_2)+e[H^{\beta }(U_1,T_P(U_2))+H^{\beta }(U_2,T_P(U_1))]$.

(ii) 

A unique multi-valued fractal $U^{\ast }$ exists for the iterated multifunction system $P=(P_1,P_2,\cdots P_n)$.

Proof. 

Suppose condition (17) holds. Then, for $U_1,U_2\in CB(X)$, we have

$$\begin{array}{ccc}\hfill R^{\beta }(P_i(U_1),P_i(U_2))& =& \beta \delta (P_i(U_1),P_i(U_2))+(1-\beta )\delta (P_i(U_2),P_i(U_1))\hfill \\ & & =\beta \underset{x\in U_1}{sup}(\underset{y\in U_2}{inf}{H}^{\beta }(P_i(x),P_i(y))+\hfill \\ & & (1-\beta )\underset{y\in U_2}{sup}(\underset{x\in U_1}{inf}{H}^{\beta }(P_i(x),P_i(y))\hfill \\ & & \le \beta \underset{x\in U_1}{sup}(\underset{y\in U_2}{inf}\left\{a{d}_s(x,y)+e[d_s(x,P_{i}y)+d_s(y,P_{i}x)]\right\}\hfill \\ & & +(1-\beta )\underset{y\in U_2}{sup}(\underset{x\in U_1}{inf}\left\{a{d}_s(x,y)+e[d_s(x,P_{i}y)+d_s(y,P_{i}x)]\right\}\hfill \\ & & =a{H}^{\beta }(U_1,U_2)+e[H^{\beta }(U_1,P_i(U_2)+H^{\beta }(U_2,P_i(U_1))].\hfill \end{array}$$

Similarly, we get

$$\begin{array}{ccc}\hfill R^{\beta }(P_i(U_2),P_i(U_1))& \le & a{H}^{\beta }(U_2,U_1)+e[H^{\beta }(U_2,P_i(U_1)+H^{\beta }(U_1,P_i(U_2))].\hfill \end{array}$$

Thus, we have, for $i=1,2,\cdots n$,

$$\begin{array}{ccc}\hfill H^{\beta }(P_i(U_1),P_i(U_2))& \le & a{H}^{\beta }(U_1,U_2)+e[H^{\beta }(U_2,P_i(U_1)+H^{\beta }(U_1,P_i(U_2))].\hfill \end{array}$$

Note that

$$\begin{array}{ccc}\hfill H^{\beta }(\bigcup _{i=1}^n{P}_i(U_1),\bigcup _{i=1}^n{P}_i(U_2))& \le & max\{H^{\beta }(P_1(U_1),P_1(U_2)),H^{\beta }(P_2(U_1),P_2(U_2)),\cdots H^{\beta }(P_n(U_1),P_n(U_2))\}\hfill \end{array}$$

and so

$$\begin{array}{ccc}\hfill H^{\beta }(T_P(U_1),T_P(U_2))& \le & a{H}^{\beta }(U_1,U_2)+e[H^{\beta }(U_1,T_P(U_2))+H^{\beta }(U_2,T_P(U_1))].\hfill \end{array}$$

Thus, $T_P:CB(X)\to CB(X)$ satisfies the conditions of Corollary 2 in the metric space $\{CB(X),H^{\beta }\}$, with $b=c=0$ and $e=f$ and hence has a fixed point $U^{\ast }$ in $CB(X)$, which in turn is the unique multi-valued fractal of the iterated multifunction system $P=(P_1,P_2,\cdots P_n)$. □

Remark 7.

Since $H^{\beta }(A,B)\le H(A,B)$, Theorem 6 is a proper improvement and generalization of (Theorem 3.4 [18]), (Theorem3.1 [22]) and (Theorem 3.8 [23]).

3.4. Application to Nonconvex Integral Inclusions

We will begin this section by introducing the following generalized norm on a vector space:

Definition 15.

Let V be a vector space over the field K. For some $\rho >0$ and $\gamma \ge 1$, a real valued function ${\parallel .\parallel }_{\gamma }^{\rho }:V\to R$ is a generalized ($\rho ,\gamma$)-norm if for all $x,y\in V$ and $\lambda \in K$

(1) 

${\parallel x\parallel }_{\gamma }^{\rho }\ge$ 0 and ${\parallel x\parallel }_{\gamma }^{\rho }$ = 0 if and only if $x=0$.

(2) 

${\parallel \lambda x\parallel }_{\gamma }^{\rho }\le {|\lambda |}^{\rho }{\parallel x\parallel }_{\gamma }^{\rho }$.

(3) 

${\parallel x+y\parallel }_{\gamma }^{\rho }\le {\gamma [\parallel x\parallel }_{\gamma }^{\rho }+{\parallel y\parallel }_{\gamma }^{\rho }]$.

We say that $(V,\parallel .{\parallel }_{\gamma }^{\rho }$ is a generalized ($\rho ,\gamma$)-normed linear space.

Remark 8.

The following are immediate consequences of the above definition:

(i) 

Every norm is a generalized ($\rho ,\gamma$)-norm with $\rho =1$ and $\gamma =1$.

(ii) 

Every generalized ($\rho ,\gamma$)-norm induces a b-metric with coefficient γ, given by $d_{\gamma }(x,y)={\parallel x-y\parallel }_{\gamma }^{\rho }$.

Example 7.

Every norm defined on a vector space is a generalized ($\rho ,\gamma$)-norm.

Example 8.

Let $V=R$. Define ${\parallel x\parallel }_{\gamma }^{\rho }={|x|}^2$. Then ${\parallel .\parallel }_{\gamma }^{\rho }$ is a generalized ($2,2$)-norm.

Example 9.

Let $V=R^n$. Define ${\parallel x\parallel }_{\gamma }^{\rho }={\sum }_k{|x_k|}^p$, $1\le p<\infty$. Then ${\parallel .\parallel }_{\gamma }^{\rho }$ is a generalized ($p,2^{p-1}$)-norm.

The convergence, Cauchy sequence and completeness in a generalized ($\rho ,\gamma$)-normed linear space is defined in the same way as that in a normed linear space.

Throughout this section we will use the following notations and functions:

(i)

$A=[0,\tau ],\tau >0$.

(ii)

$\mathcal{L}(A)$: is the $\sigma$-algebra of all Lebesgue measurable subsets of A.

(iii)

Z: is a real separable Banach space with the generalized ($\rho ,\gamma$)-norm ${\parallel .\parallel }_{\gamma }^{\rho }$, for some $\rho >0$ and $\gamma \ge 1$.

(iv)

$\mathcal{P}(Z)$: is the family of all nonempty closed subsets of Z.

(v)

$d_{\gamma }$ is the b-metric induced by the generalized ($\rho ,\gamma$)-norm ${\parallel .\parallel }_{\gamma }^{\rho }$ and $H^{\beta }$ is the $H^{\beta }$-Hausdorff–Pompeiu b-metric on $\mathcal{P}(Z)$, induced by the b-metriv $d_{\gamma }$.

(vi)

$\mathcal{B}(Z)$: is the collection of all Borel subsets of Z.

(vii)

$\mathcal{C}(A,Z)$: is the Banach space of all continuous functions $g(.):A\to Z$ with norm ${\parallel g(.)\parallel }_{\ast }={sup}_{t\in A}{\parallel g(t)\parallel }_{\gamma }^{\rho }$.

(viii)

${\lambda }^{\ell }(.):A\to Z$.

(ix)

$p(.,.):A\times Z\to Z$.

(x)

$Q(.,.):A\times Z\to \mathcal{P}(Z)$.

(xi)

$q(.,.,.):A\times A\times Z\to Z$.

(xii)

$V:\mathcal{C}(A,Z)\to \mathcal{C}(A,Z)$.

(xiii)

${\alpha }_1,{\alpha }_2:A\times A\to (-\infty ,+\infty )$.

(xiv)

$L_{{\lambda }^{\ell },\sigma }(t)=Q(t,V(x_{\sigma ,{\lambda }^{\ell }})(t))$, $x\in Z,{\lambda }^{\ell }\in \mathcal{C}(A,Z),\sigma \in {\mathcal{L}}^1(A,Z)$.

(xv)

$S_{{\lambda }^{\ell }}(\sigma )=\{\psi (.)\in {\mathcal{L}}^1(A,Z):\psi (t)\in L_{{\lambda }^{\ell },\sigma }(t)\}$.

(xvi)

${\mathcal{L}}^1(A,Z)$: is the Banach space of all integrable functions u: $A\to Z$, endowed with the norm

$${\parallel u(.)\parallel }_1={\int }_0^T{e}^{-\alpha (M_4{M}_2+M_5{M}_1)M_{3}m(t)}{\parallel u(t)\parallel }_{\gamma }^{\rho }dt,$$

where $m(t)={\int }_0^{t}k(s)ds,t\in A,M_1,M_2,M_3,M_4,M_5$ are positive real constants.

It is well known (see [24]) that $L_{{\lambda }^{\ell },\sigma }(t)$ is measurable and $S_{\lambda }^{\ell }(\sigma )$ is nonempty with closed values.

We consider the following integral inclusion

$$x^{\ell }(t)={\lambda }^{\ell }(t)+{\int }_0^t[{\alpha }_1(t,s)p(t,u(s))+{\alpha }_2(t,s)q(t,s,u(s))],ds$$

(18)

$$u(t)\in Q(t,V(x^{\ell })(t))a.e.t\in A.$$

(19)

We will analyze the above problem (18) and (19) under the following assumptions: $({\mathbf{AS}}_{\mathbf{1}})$$Q(·,·)$ is $\mathcal{L}$$(I)\otimes \mathcal{B}$$(X)$ measurable.

$({\mathbf{AS}}_{\mathbf{2}}(\mathbf{i}))$ There exists $k(·)\in L^1(A,{\mathbf{R}}_{+})$ such that, for almost all $t\in A,Q(t,·)$ satisfies

$$H^{\beta }(Q(t,x),Q(t,y))\le k(t){\parallel x-y\parallel }_{\gamma }^{\rho }$$

for all $x,y$ in Z.

$({\mathbf{AS}}_{\mathbf{2}}(\mathbf{ii}))$ For all $x,y\in Z$, $ϵ>0$, if $w_1\in Q(t,x)$ then there exists $w_2\in Q(t,y)$ such that

$$\parallel w_1(t)-w_2{(t)\parallel }_{\gamma }^{\rho }\le H^{\beta }(Q(t,x),Q(t,y))+ϵ.$$

$({\mathbf{AS}}_{\mathbf{2}}(\mathbf{iii}))$ For any $\sigma \in {\mathcal{L}}^1(A,Z)$, $ϵ>0$ and ${\sigma }_1\in S_{{\lambda }^{\ell }}(\sigma )$, there exists ${\sigma }_2\in S_{{\lambda }^{\ell }}({\sigma }_1)$ such that

$$\parallel {\sigma }_1-{\sigma }_2{\parallel }_1\le H^{\beta }(S_{{\lambda }^{\ell }}(\sigma ),S_{{\lambda }^{\ell }}({\sigma }_1))+ϵ.$$

$({\mathbf{AS}}_{\mathbf{3}})$ The mappings $f:A\times A\times Z\to Z,g:A\times Z\to Z$ are continuous, $V:C(A,Z)\to C(A,Z)$

and there exist the constants $M_1,M_2,M_3,M_4>0$ such that $(A{S}_3(i))$ and either $(A{S}_3(ii)(a))$

or $(A{S}_3(ii)(b))$ holds $\forall t,s\in A,u_1,u_2\in {\mathcal{L}}^1(A,Z),x_1,x_2\in \mathcal{C}(A,Z)$.

$({\mathbf{AS}}_{\mathbf{3}}(\mathbf{i}))\parallel V(x_1)(t)-V(x_2)(t){\parallel }_{\gamma }^{\rho }\le M_3{\parallel x_1(t)-x_2(t)\parallel }_{\gamma }^{\rho }.$

$({\mathbf{AS}}_{\mathbf{3}}(\mathbf{ii})(\mathbf{a})){\parallel q(t,s,u_1(s))-q(t,s,u_2(s))\parallel }_{\gamma }^{\rho }\le M_{1}N(u_1,u_2),$

$$\begin{array}{c}\hfill \parallel p(s,u_1(s))-p(s,u_2(s)){\parallel }_{\gamma }^{\rho }\le M_{2}N(u_1,u_2).\end{array}$$

$({\mathbf{AS}}_{\mathbf{3}}(\mathbf{ii})(\mathbf{b})){\parallel q(t,s,u_1(s))-q(t,s,u_2(s))\parallel }_{\gamma }^{\rho }\le M_{1}n(u_1,u_2),$

$$\begin{array}{c}\hfill \parallel p(s,u_1(s))-p(s,u_2(s)){\parallel }_{\gamma }^{\rho }\le M_{2}n(u_1,u_2),\end{array}$$

where

$$N(u_1,u_2)=max\{\parallel u_1(s)-u_2(s){\parallel }_{\gamma }^{\rho },\parallel u_1(s)-S_{{\lambda }^{\ell }}(u_1){\parallel }_{\gamma }^{\rho },\parallel u_2(s)-S_{{\lambda }^{\ell }}(u_2){\parallel }_{\gamma }^{\rho },\parallel u_1(s)-S_{{\lambda }^{\ell }}(u_2){\parallel }_{\gamma }^{\rho },\parallel u_2(s)-S_{{\lambda }^{\ell }}(u_1){\parallel }_{\gamma }^{\rho }\},$$

$$n(u_1,u_2)=max\{\parallel u_1(s)-u_2(s){\parallel }_{\gamma }^{\rho },\parallel u_1(s)-S_{{\lambda }^{\ell }}(u_1){\parallel }_{\gamma }^{\rho },\parallel u_2(s)-S_{{\lambda }^{\ell }}(u_2){\parallel }_{\gamma }^{\rho }\}+K\parallel u_1(s)-S_{{\lambda }^{\ell }}(u_2){\parallel }_{\gamma }^{\rho }$$

and

$$\parallel u(s)-S_{\lambda }^{\ell }{(v)\parallel }_{\gamma }^{\rho }=\underset{w\in S_{{\lambda }^{\ell }}(v)}{inf}{\parallel u(s)-w(s)\parallel }_{\gamma }^{\rho }.$$

$(A{S}_4)$${\alpha }_1$, ${\alpha }_2$ are continuous, $|{\alpha }_1{(t,s)|}^{\rho }\le M_4$ and $|{\alpha }_2{(t,s)|}^{\rho }\le M_5$.

Theorem 7.

Suppose assumptions $(A{S}_1)$ to $(A{S}_4)$ hold and let ${\lambda }^{\ell }(·),{\mu }^{\ell }(·)\in \mathcal{C}(A,Z)$, $v(·)\in {\mathcal{L}}^1(A,Z)$ be such that $d(v(t),Q(t,V(y^{\ell })(t))\le l(t)a.e.t\in A,$ where $l(·)\in {\mathcal{L}}^1(A,{\mathbf{R}}_{+})$ and $y^{\ell }(t)={\mu }^{\ell }(t,u(t))+\Phi (u)(t),\forall t\in A$ with $\Phi (u)(t)={\int }_0^t[{\alpha }_1(t,\tau )p(\tau ,u(\tau ))+{\alpha }_{2}q(t,\tau ,u(\tau ))]$$d\tau ,t\in A$. Then, for every $\eta >\gamma$ and $ϵ>0$, we can find a solution $x^{\ell }(·)$ of the problem (18) and (19) such that for every $t\in A$

$$\parallel x^{\ell }(t)-y^{\ell }(t)\parallel \le \parallel {\lambda }^{\ell }-{\mu }^{\ell }{\parallel }_{\ast }\left[1+{\displaystyle \frac{\gamma e^{\eta (M_4{M}_2+M_5{M}_1)M_{3}m(T)}}{\eta -\gamma }}\right]$$

$$+{\displaystyle \frac{\gamma \eta }{\eta -\gamma }}(M_4{M}_2+M_5{M}_1)e^{\eta (M_4{M}_2+M_1)M_{3}m(T)}{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt.$$

Proof. 

For ${\lambda }^{\ell }\in \mathcal{C}(A,Z)$ and $u\in {\mathcal{L}}^1(A,Z)$, define

$${x^{\ell }}_{u,{\lambda }^{\ell }}(t)={\lambda }^{\ell }(t)+{\int }_0^t[{\alpha }_1(t,s)p(t,u(s))+{\alpha }_2(t,s)q(t,s,u(s))]ds.$$

Let ${\sigma }_1,{\sigma }_2\in {\mathcal{L}}^1(A,Z)$, $w_1\in S_{{\lambda }^{\ell }}({\sigma }_1)$ and

$$\mathcal{H}(t):=L_{{\lambda }^{\ell },{\sigma }_2(t)}\cap \left\{z\in Z:\parallel w_1(t)-z\parallel \le (M_4{M}_2+M_5{M}_1)M_{3}k(t){\int }_0^{t}N({\sigma }_1,{\sigma }_2)ds+\delta \right\}.$$

By assumption ($A{S}_2(ii)$), we have

$$d_{\gamma }(w_1(t),L_{{\lambda }^{\ell },{\sigma }_2})\le H^{\beta }\left(Q(t,V(x_{{\sigma }_1,{\lambda }^{\ell }})(t)),Q(t,V(x_{{\sigma }_2,{\lambda }^{\ell }})(t))\right)+ϵ$$

$$\le k(t)\parallel V(x_{{\sigma }_1,{\lambda }^{\ell }})(t))-V(x_{{\sigma }_2,{\lambda }^{\ell }})(t){)\parallel }_{\gamma }^{\rho }+ϵ$$

$$\le M_{3}k(t){\parallel x_{{\sigma }_1,{\lambda }^{\ell }}(t)-x_{{\sigma }_2,{\lambda }^{\ell }}(t)\parallel }_{\gamma }^{\rho }+ϵ$$

$$\le M_{3}k(t)[{\int }_0^t|{\alpha }_1{(t,s)|}^{\rho }{\parallel p(t,{\sigma }_1(s))-p(t,{\sigma }_2(s))\parallel }_{\gamma }^{\rho }ds$$

$$+{\int }_0^t|{\alpha }_2{(t,s)|}^{\rho }{\parallel q(t,s,{\sigma }_1(s))-q(t,s,{\sigma }_2(s))\parallel }_{\gamma }^{\rho }ds]+ϵ$$

$$\le M_{3}k(t)\left[(M_4{M}_2+M_5{M}_1){\int }_0^{t}N({\sigma }_1,{\sigma }_2)ds\right]+ϵ.$$

Since $ϵ$ is arbitrary, we conclude that $\mathcal{H}(·)$ is nonempty, closed, bounded and measurable.

Let $w_2(·)$ be a measurable selector of $\mathcal{H}(·)$. Then, $w_2\in S_{{\lambda }^{\ell }}({\sigma }_2)$. If assumption $A{S}_3(ii)(a)$ is assumed, then we have

$$\parallel w_1-w_2{\parallel }_1={\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}{\parallel w_1(t)-w_2(t)\parallel }_{\gamma }^{\rho }dt$$

$$\le {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}{M}_{3}k(t)\left[(M_4{M}_2+M_5{M}_1){\int }_0^{t}N({\sigma }_1,{\sigma }_2)ds\right]dt$$

$$+\delta {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}dt$$

$$\le {\displaystyle \frac{1}{\eta }}{N}^1({\sigma }_1,{\sigma }_2)+\delta {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}dt,$$

where $N^1({\sigma }_1,{\sigma }_2)=max\{\parallel {\sigma }_1-{\sigma }_2{\parallel }_1,\parallel {\sigma }_1-S_{{\lambda }^{\ell }}({\sigma }_1){\parallel }_1,\parallel {\sigma }_2-S{}_{\lambda }^{\ell }({\sigma }_2){\parallel }_1,\parallel {\sigma }_1-S_{{\lambda }^{\ell }}({\sigma }_2){\parallel }_1,\parallel {\sigma }_2-S_{{\lambda }^{\ell }}({\sigma }_1){\parallel }_1\}$. Since $\delta$ is arbitrary, we have

$$d_{\gamma }(w_1,S_{{\lambda }^{\ell }}({\sigma }_2)=\underset{w_2\in S_{{\lambda }^{\ell }}({\sigma }_2)}{inf}{\parallel w_1-w_2\parallel }_1\le {\displaystyle \frac{1}{\eta }}{N}^1({\sigma }_1,{\sigma }_2).$$

Therefore,

$${\delta }_{\gamma }(S_{{\lambda }^{\ell }}({\sigma }_1),S_{{\lambda }^{\ell }}({\sigma }_2)=\underset{w_1\in S_{{\lambda }^{\ell }}({\sigma }_1)}{sup}{d}_{\gamma }(w_1,S_{{\lambda }^{\ell }}({\sigma }_2)\le {\displaystyle \frac{1}{\eta }}{N}^1({\sigma }_1,{\sigma }_2).$$

(20)

Similarly, we also get

$${\delta }_{\gamma }(S_{{\lambda }^{\ell }}({\sigma }_2),S_{{\lambda }^{\ell }}({\sigma }_1)=\underset{w_1\in S_{{\lambda }^{\ell }}({\sigma }_1)}{sup}{d}_{\gamma }(w_1,S_{{\lambda }^{\ell }}({\sigma }_2)\le {\displaystyle \frac{1}{\eta }}{N}^1({\sigma }_1,{\sigma }_2).$$

(21)

Multiplying (20) by $\beta$ and (21) by $1-\beta$ and adding, we get

$$H^{\beta }(S_{{\lambda }^{\ell }}({\sigma }_1),S_{{\lambda }^{\ell }}({\sigma }_2))\le {\displaystyle \frac{1}{\eta }}{N}^1({\sigma }_1,{\sigma }_2).$$

Thus, $S_{{\lambda }^{\ell }}(·)$ is a $H^{\beta }$-quasi contraction on ${\mathcal{L}}^1(A,Z)$.

Now let

$$\tilde{Q}(t,x):=Q(t,x)+l(t),$$

$${\tilde{M}}_{{\lambda }^{\ell },\sigma }(t):=\tilde{Q}(t,V(x_{\sigma ,{\lambda }^{\ell }})(t)),t\in I,$$

$${\tilde{S}}_{{\mu }^{\ell }}(\sigma ):=\{\psi (·)\in {\mathcal{L}}^1(A,Z);\psi (t)\in {\tilde{L}}_{{\mu }^{\ell },\sigma }(t).$$

It is obvious that $\tilde{Q}(·,·)$ satisfies Hypothesis 5.1.

Let $\varphi \in S_{{\lambda }^{\ell }}(\sigma ),\delta >0$ and define

$$\tilde{\mathcal{H}}(t):={\tilde{L}}_{{\lambda }^{\ell },\sigma (t)}\cap \left\{z\in Z:\parallel \varphi (t)-z\parallel \le M_{3}k(t){\parallel {\lambda }^{\ell }-{\mu }^{\ell }\parallel }_{\ast }+l(t)+\delta \right\}.$$

Proceeding in the same way as in the case of $\mathcal{H}(·)$ above, we see that $\tilde{\mathcal{H}}(·)$ is measurable, nonempty and has closed values.

Let $\omega (·)\in S_{{\mu }^{\ell }}(\sigma )$. Then

$${\parallel \varphi -\omega \parallel }_1\le {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}{\parallel \varphi (t)-\omega (t)\parallel }_{\gamma }^{\rho }dt$$

$$\le {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}[M_{3}k(t)\parallel {\lambda }^{\ell }-{\mu }^{\ell }{\parallel }_{\ast }+l(t)+\delta ]dt$$

$$=\parallel {\lambda }^{\ell }-{\mu }^{\ell }{\parallel }_{\ast }{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}{M}_{3}k(t)dt$$

$$+{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt+\delta {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}dt$$

$$\le {\displaystyle \frac{1}{\eta (M_4{M}_2+M_5{M}_1)}}{\parallel {\lambda }^{\ell }-{\mu }^{\ell }\parallel }_{\ast }$$

$$+{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt+\delta {\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}dt.$$

As $\delta \to 0$ we get

$$\begin{array}{cc}\hfill H^{\beta }(S_{{\lambda }^{\ell }}(\sigma ),{\tilde{S}}_{{\mu }^{\ell }}(\sigma ))\le & {\displaystyle \frac{1}{\eta (M_4{M}_2+M_5{M}_1)}}{\parallel {\lambda }^{\ell }-{\mu }^{\ell }\parallel }_{\ast }\hfill \\ & +{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt.\hfill \end{array}$$

(22)

Since $S_{{\lambda }^{\ell }}(.,.)$ and ${\tilde{S}}_{\mu }^{\ell }(.,.)$ are $H^{\beta }$-quasi contractions with Lipschitz constant ${\displaystyle \frac{1}{\eta }}$ and since $v(·)\in \mathcal{F}\{{\tilde{S}}_{{\mu }^{\ell }}\}$ by Proposition 3 there exists $u(·)\in \mathcal{F}\{S_{{\lambda }^{\ell }}\}$ such that

$${\parallel v-u\parallel }_1\le {\displaystyle \frac{\gamma \eta }{\eta -\gamma }}\underset{x\in X}{sup}{H}^{\beta }({\tilde{S}}_{{\mu }^{\ell }}x,S_{{\lambda }^{\ell }}x).$$

Using (22), we have

$$\begin{array}{cc}\hfill {\parallel v-u\parallel }_1\le & {\displaystyle \frac{\gamma }{(\eta -\gamma )(M_4{M}_2+M_5{M}_1)}}{\parallel {\lambda }^{\ell }-{\mu }^{\ell }\parallel }_{\ast }\hfill \\ & +{\displaystyle \frac{\gamma \eta }{\eta -\gamma }}{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt.\hfill \end{array}$$

(23)

Now let

$$x^{\ell }(t)={\lambda }^{\ell }(t)+{\int }_0^t[{\alpha }_1(t,s)p(t,u(s))+{\alpha }_2(t,s)q(t,s,u(s))]ds.$$

Then, we have

$$\parallel x^{\ell }(t)-y^{(}t)\parallel \le \parallel {\lambda }^{\ell }(t)-{\mu }^{\ell }(t)\parallel +(M_4{M}_2+M_5{M}_1){\int }_0^t\parallel u(s)-v(s)\parallel ds$$

$$\le \parallel {\lambda }^{\ell }-{\mu }^{\ell }{\parallel }_{\ast }+(M_4{M}_2+M_5{M}_1)e^{\eta (M_4{M}_2+M_5{M}_1)M_{3}m(T)}{\parallel u-v\parallel }_1.$$

Using (23) we get

$$\parallel x^{\ell }(t)-y^{\ell }(t)\parallel \le \parallel {\lambda }^{\ell }-{\mu }^{\ell }{\parallel }_{\ast }\left[1+{\displaystyle \frac{\gamma e^{\eta (M_4{M}_2+M_5{M}_1)M_{3}m(T)}}{\eta -\gamma }}\right]$$

$$+{\displaystyle \frac{\gamma \eta }{\eta -\gamma }}(M_4{M}_2+M_5{M}_1)e^{\eta (M_4{M}_2+M_1)M_{3}m(T)}{\int }_0^T{e}^{-\eta (M_4{M}_2+M_5{M}_1)M_{3}m(t)}l(t)dt.$$

This completes the proof. □

Remark 9.

Since $H^{\beta }(A,B)\le H(A,B)$ and the class of generalized ($\rho ,\gamma$)-norms includes the usual norm $\parallel .\parallel$, we note that the hypothesis conditions $A{S}_2(i)$ and $A{S}_3(i),(ii)$ are much weaker than the corresponding hypothesis conditions (Hypothesis 2.1 (ii) and (iii)) of [24]).

3.5. Conclusions

The $H^{\beta }$-Hausdorff–Pompeiu b-metric is introduced as a new tool in metric fixed point theory and new variants of Nadler, Ciric, Hardy–Rogers contraction principles for multi-valued mappings are established in a b-metric space. The examples and applications provided illustrates the advantages of using $H^{\beta }$-Hausdorff–Pompeiu b-metric in fixed point theory and its applications. The new tool of $H^{\beta }$-Hausdorff–Pompeiu b-metric can be utilized by young researchers in extending and generalizing many of the fixed point results for multi-valued mappings existing in literature and investigate how the new tool would enhance, extend and generalize the applications of the fixed-point theory to linear differential and integro-differential equations, nonlinear phenomena, algebraic geometry, game theory, non-zero-sum game theory and the Nash equilibrium in economics.