Fixed Point Results for the Family of Multivalued F-Contractive Mappings on Closed Ball in Complete Dislocated b-Metric Spaces

Abstract

The purpose of this paper is to find out fixed point results for the family of multivalued mappings fulfilling a generalized rational type F-contractive conditions on a closed ball in complete dislocated b-metric space. An application to the system of integral equations is presented to show the novelty of our results. Our results extend several comparable results in the existing literature.

1. Introduction and Preliminaries

Fixed point theory plays a fundamental role in functional analysis. Nadler [1] initiated the study of fixed point theorems for the multivalued mappings. Due to its significance, a large number of authors have proved many interesting multiplications of his result (see [2,3,4,5,6,7,8]).

Rasham et al. [9] proved the multivalued fixed point results for new generalized F-contractive mappings on dislocated metric spaces with application to the system of integral equations. Nazir et al. [10] showed common fixed point results for a family of generalized multivalued F-contraction mappings in ordered metric spaces (see also [11,12,13,14,15,16,17,18,19,20,21]). Recently Shoaib et al. [7] discussed the results for the family of multivalued mappings satisfying contranction on a sequence in a closed ball in Hausdorff fuzzy metric space. For further results on closed ball, see [7,8,22,23,24,25,26,27].

In this paper, we have obtained common fixed point for the family of multivalued mappings satisfying conditions only on a sequence contained in a closed ball. We have used a weaker class of strictly increasing mappings F rather than the class of mappings F used by different authors. An example which supports the proved results is also given. Moreover, we investigate our results in a better framework of dislocated b-metric space (see [28,29,30]). New results in ordered spaces, partial b-metric space, dislocated metric space, partial metric space, b-metric space, and metric space can be obtained as corollaries of our results. We give the following definitions and results which will be needed in the sequel.

Definition 1

([28]). Let X be a nonempty set and let $d_b:X\times X\to [0,\infty )$ be a function, called a dislocated b-metric (or simply $d_b$-metric). If there exists $b\ge 1$ such that for any $x,y,z\in X,$ the following conditions holds:

(i) 

If$d_b(x,y)=0,$then$x=y;$

(ii) 

$d_b(x,y)=d_b(y,x);$

(iii) 

$d_b(x,y)\le b[d_b(x,z)+d_b(z,y)].$

The pair $(X,d_b)$ is called a dislocated b-metric space. It should be noted that every dislocated metric is a dislocated b-metric with $b=1.$

It is clear that if $d_b(x,y)=0$ , then from (i), $x=y$ . But if $x=y$ , $d_b(x,y)$ may not be $0.$ For $x\in X$ and $\epsilon >0,$$\overline{B(x,\epsilon )}=\{y\in X:d_b(x,y)\le \epsilon \}$ is a closed ball in $(X,d_b).$ We will use $D.B.M$ space instead of dislocated b-metric space.

Definition 2

([28]). Let $(X,d_b)$ be a $D.B.M$ space.

(i) 

A sequence$\left\{x_n\right\}$in$(X,d_b)$is called Cauchy sequence if given$\epsilon >0$, there corresponds$n_0\in N$such that for all$n,m\ge$$n_0$we have$d_b(x_m,x_n)$$<\epsilon$or$\underset{n,m\to \infty }{lim}{d}_b(x_n,x_m)=0.$

(ii) 

A sequence$\left\{x_n\right\}$dislocated b-converges (for short$d_b$-converges) to x if$\underset{n\to \infty }{lim}{d}_b(x_n,x)=0.$In this case x is called a$d_b$-limit of$\left\{x_n\right\}.$

(iii) 

$(X,d_b)$is called complete if every Cauchy sequence in X converges to a point$x\in X$such that$d_b(x,x)=0$.

Definition 3.

Let K be a nonempty subset of $D.B.M$ space of X and let $x\in X.$ An element $y_0\in K$ is called a best approximation in K if

$$d_b(x,K)=d_b(x,y_0),where{d}_b(x,K)=\underset{y\in K}{inf}{d}_b(x,y).$$

If each $x\in X$ has at least one best approximation in $K,$ then K is called a proximinal set.

We denote $P\left(X\right)$ be the set of all closed proximinal subsets of $X.$

Definition 4

([8]). The function $H_{d_b}:P\left(X\right)\times P\left(X\right)\to R^{+},$ defined by

$$H_{d_b}(N,M)=max\{\underset{n\in N}{sup}{d}_b(n,M),\underset{m\in M}{sup}{d}_b(N,m)\}$$

is called dislocated Hausdorff b-metric on $P\left(X\right).$

Definition 5

([21]). Let $(X,d)$ be a metric space. A mapping $H:X\to X$ is said to be an F−contraction if there exists $\tau >0$ such that

$$\forall j,k\in X,d(Hj,Hk)>0\Rightarrow \tau +F\left(d(Hj,Hk)\right)\le F\left(d(j,k)\right)$$

where $F:{\mathbb{R}}_{+}\to \mathbb{R}$ is a mapping satisfying the following conditions:

(F1) 

F is strictly increasing, i.e., for all$j,k\in {\mathbb{R}}_{+}$such that$j<k$,$F\left(j\right)<F\left(k\right)$;

(F2) 

For each sequence${\left\{{\alpha }_n\right\}}_{n=1}^{\infty }$of positive numbers,${lim}_{n\to \infty }$${\alpha }_n$$=0$if and only if${lim}_{n\to \infty }F\left({\alpha }_n\right)=-\infty$;

(F3) 

There exists$k\in (0,1)$such that$lim\alpha \to 0^{+}{\alpha }^{k}F\left(\alpha \right)=0$.

Lemma 1.

Let $(X,d_b)$ be a $D.B.M$ space. Let $(P\left(X\right),H_{d_b})$ be a dislocated Hausdorff b-metric space on $P\left(X\right).$ Then, for all $G,H\in P\left(X\right)$ and for each $g\in G$ such that $d_b(g,H)=d_b(g,h_g)$ , where $h_g\in H.$ Then the following holds:

$$H_{d_b}(G,H)\ge d_b(g,h_g).$$

2. Main Result

Let $(Z,d_b)$ be a $D.B.M$ space,$c_0\in Z$ and let $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of multivalued mappings from Z to $P\left(Z\right).$ Then there exist $c_1\in S_{\alpha }{c}_0$ for some $\alpha \in \Omega ,$ such that $d_b(c_0,S_{\alpha }{c}_0)=d_b(c_0,c_1).$ Let $c_2\in S_{\beta }{c}_1$ be such that $d_b(c_1,S_{\beta }{c}_1)=d_b(c_1,c_2).$ Continuing this method, we get a sequence $c_n$ of points in Z such that $c_{n+1}\in S_{\gamma }{c}_n,$ $d_b(c_n,S_{\gamma }{c}_n)=d_b(c_n,c_{n+1})$ for some $\gamma \in \Omega .$ We denote this iterative sequence by $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}.$ We say that $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ is a sequence in Z generated by $c_0.$

Theorem 1.

Let $(Z,d_b)$ be a complete $D.B.M$ space with constant $b\ge 1$ and $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of multivalued mappings from Z to $P\left(Z\right)$ and $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ be a sequence in Z generated by $c_0.$ Assume that the following hold:

(i) 

There exist$\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$satisfying$b{\eta }_1+b{\eta }_2+{\eta }_3+{\eta }_4<1$and a strictly increasing mapping F such that

$$\tau +F\left(H_{d_b}(S_{i}e,S_{j}y)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_b(e,y)+{\eta }_2{d}_b(e,S_{i}e)\\ +{\eta }_3{d}_b(y,S_{j}y)+{\eta }_4{\displaystyle \frac{d_b^2(e,S_{i}e)·d_b(y,S_{j}y)}{1+d_b^2(e,y)}}\end{array}\right)$$

(1)

whenever$e,y\in \overline{B_{d_b}(c_0,r)}\cap \{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$with$e\ne y$,$i,j\in \Omega$with$i\ne j$and$H_{d_b}(S_{i}e,S_{j}y)>0.$

(ii) 

If$\lambda ={\displaystyle \frac{{\eta }_1+{\eta }_2}{1-{\eta }_3-{\eta }_4}},$then

$$d_b(c_0,S_{\alpha }{c}_0)\le \lambda (1-b\lambda )r.$$

(2)

Then $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ is a sequence in $\overline{B_{d_b}(c_0,r)}$ and $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}\to u\in \overline{B_{d_b}(c_0,r)}.$ Also, if the inequality (1) holds for $u,$ then there exist a common fixed point for the family of multivalued mappings $\left\{S_{\beta }:\beta \in \Omega \right\}$ in $\overline{B_{d_b}(c_0,r)}$ and $d_b(u,u)=0.$

Proof. 

Let $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ be a sequence in Z generated by $c_0.$ If $c_0=c_1,$ then $c_0$ is a common fixed point of $S_{\alpha }$ for all $\alpha \in \Omega .$ Let $c_0\ne c_1$. From (2), we get

$$d_b(c_0,c_1)=d_b(c_0,S_{\alpha }{c}_0)\le \lambda (1-b\lambda )r<r.$$

It follows that,

$$c_1\in \overline{B_{d_b}(c_0,r)}.$$

Let $c_2,\cdots ,c_j\in \overline{B_{d_b}(c_0,r)}$ for some $j\in \mathbb{N}$. Now by using Lemma 1, we have

$$\begin{array}{ccc}\hfill \tau +F\left(d_b(c_j,c_{j+1})\right)& \le & \tau +F\left(H_{d_b}(S_{\delta }{c}_{j-1},S_{\eta }{c}_j)\right)\hfill \\ & \le & F[{\eta }_1{d}_b\left(c_{j-1},c_j\right)+{\eta }_2{d}_b\left(c_{j-1},S_{\delta }{c}_{j-1}\right)+{\eta }_3{d}_b\left(c_j,S_{\eta }{c}_j\right)\hfill \\ & & +{\eta }_4\frac{d_b\left(c_{j-1},S_{\delta }{c}_{j-1}\right)·d_b(c_j,S_{\eta }{c}_j)}{1+d_b\left(c_{j-1},c_j\right)}\hfill \\ & \le & F[{\eta }_1{d}_b\left(c_{j-1},c_j\right)+{\eta }_2{d}_b\left(c_{j-1},c_j\right)+{\eta }_3{d}_b\left(c_j,c_{j+1}\right)\hfill \\ & & +{\eta }_4\frac{d_b^2\left(c_{j-1},c_j\right)·d_b(c_j,c_{j+1})}{1+d_b^2\left(c_{j-1},c_j\right)}\hfill \\ & \le & F(({\eta }_1+{\eta }_2)d_b\left(c_{j-1},c_j\right)+({\eta }_3+{\eta }_4)d_b\left(c_j,c_{j+1}\right)).\hfill \end{array}$$

This implies

$$F\left(d_b(c_j,c_{j+1})\right)<F(({\eta }_1+{\eta }_2)d_b\left(c_{j-1},c_j\right)+({\eta }_3+{\eta }_4)d_b\left(c_j,c_{j+1}\right)).$$

As F is strictly increasing. So, we have

$$d_b(c_j,c_{j+1})<({\eta }_1+{\eta }_2)d_b\left(c_{j-1},c_j\right)+({\eta }_3+{\eta }_4)d_b\left(c_j,c_{j+1}\right).$$

Which implies

$$\begin{array}{ccc}\hfill (1-{\eta }_3-{\eta }_4)d_b(c_j,c_{j+1})& <& ({\eta }_1+{\eta }_2)d_b\left(c_{j-1},c_j\right)\hfill \\ \hfill d_b(c_j,c_{j+1})& <& \left(\frac{{\eta }_1+{\eta }_2}{1-{\eta }_3-{\eta }_4}\right)d_b\left(c_{j-1},c_j\right).\hfill \end{array}$$

As $\lambda ={\displaystyle \frac{{\eta }_1+{\eta }_2}{1-{\eta }_3-{\eta }_4}}<1.$ Hence

$$d_b(c_j,c_{j+1})<\lambda d_b\left(c_{j-1},c_j\right)<{\lambda }^2{d}_b\left(c_{j-2},c_j\right)<\cdots <{\lambda }^j{d}_b\left(c_0,c_1\right).$$

Now, we have

$$d_b(c_j,c_{j+1})<{\lambda }^j{d}_b\left(c_0,c_1\right)\mathrm{for}\mathrm{some}j\in \mathbb{N}.$$

(3)

Now,

$$\begin{array}{ccc}\hfill d_b(x_0,c_{j+1})& \le & b{d}_b(c_0,c_1)+b^2{d}_b(c_1,c_2)+\cdots +b^{j+1}{d}_b(c_j,c_{j+1})\hfill \\ & \le & b{d}_b(c_0,c_1)+b^2\lambda \left(d_b(c_0,c_1)\right)+\cdots \hfill \\ & & +b^{j+1}{\lambda }^{j+1}\left(d_b(c_0,c_1)\right),(\mathrm{by}\left(3\right))\hfill \\ \hfill d_b(c_0,c_{j+1})& \le & \frac{b(1-{\left(b\lambda \right)}^{j+1})}{1-b\lambda }\lambda (1-b\lambda )r<r,\hfill \end{array}$$

which implies $c_{j+1}\in \overline{B_{d_b}(c_0,r)}.$ Hence, by induction $c_n\in \overline{B_{d_b}(c_0,r)}$ for all $n\in N$. Now,>

$$d_b(c_n,c_{n+1})<{\lambda }^n{d}_b\left(c_0,c_1\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(4)

Now, for any positive integers $m,n$ $(n>m)$, we have

$$\begin{array}{ccc}\hfill d_b(c_m,c_n)& \le & b\left(d_b(c_m,c_{m+1})\right)+b^2\left(d_b(c_{m+1},c_{m+2})\right)+\cdots \hfill \\ & & +b^{n-m}\left(d_b(c_{n-1},c_n)\right),\hfill \\ & <& b{\lambda }^m{d}_b(c_0,c_1)+b^2{\lambda }^{m+1}{d}_b(c_0,c_1)+\cdots \hfill \\ & & +b^{n-m}{\lambda }^{n-1}{d}_b(c_0,c_1),(\mathrm{by}\left(4\right))\hfill \\ & <& b{\lambda }^m(1+b\lambda +\cdots )d_b(c_0,c_1)\hfill \end{array}$$

As ${\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0,$ $b\ge 1$ and $b{\eta }_1+b{\eta }_2+{\eta }_3+{\eta }_4<1,$ so $\left|b\lambda \right|<1.$ Then, we have

$$d_b(c_m,c_n)<\frac{b{\lambda }^m}{1-b\lambda }{d}_b(c_0,c_1)\to 0\mathrm{as}m\to \infty .$$

Hence $\left\{Z{S}_{\beta }\left(c_n\right)\right\}$ is a Cauchy sequence in $\overline{B_{d_b}(c_0,r)}$. Since $(\overline{B_{d_b}(c_0,r)},d_b)$ is a complete metric space, so there exist $u\in \overline{B_{d_b}(c_0,r)}$ such that $\left\{Z{S}_{\beta }\left(c_n\right)\right\}\to u$ as $n\to \infty ,$ then

$$\underset{n\to \infty }{lim}{d}_b(c_n,u)=0.$$

(5)

Suppose that $d_b(u,S_{q}u)>0,$ then there exist a positive integer k such that $d_b(c_n,S_{q}u)>0$ for all $n\ge k$. For $n\ge k,$ we have

$$\begin{array}{ccc}\hfill d_b(u,S_{q}u)& \le & d_b(u,c_n)+d_b(c_n,S_{q}u)\hfill \\ & \le & d_b(u,c_n)+H_{d_b}(S_{\gamma }{c}_{n-1},S_{q}u)\hfill \\ & <& d_b(u,c_n)+{\eta }_1{d}_b(c_{n-1},u)+{\eta }_2{d}_b(c_{n-1},S_{\gamma }{c}_{n-1})\hfill \\ & & +{\eta }_3{d}_b(u,S_{q}u)+{\eta }_4\frac{d_b^2(c_{n-1},S_{\gamma }{c}_{n-1})·d_b(u,S_{q}u)}{1+d_b^2(c_{n-1},u)}.\hfill \end{array}$$

Letting $n\to \infty ,$ and by using (5) we get

$$d_b(u,S_{q}u)<{\eta }_3{d}_b(u,S_{q}u)<d_b(u,S_{q}u),$$

which is a contradiction. So our supposition is wrong. Hence $d_b(u,S_{q}u)=0$ or $u\in S_{q}u.$ Similarly, by using Lemma 1, inequality (1), we can show that $d_b(u,S_{i}u)=0$ or $u\in S_{i}u$ for all $i\in \Omega .$ Now, for some $i\in \Omega$

$$d_b(u,u)\le b{d}_b(u,S_{i}u)+b{d}_b(S_{i}u,u)\le 0.$$

This implies that $d_b(u,u)=0.$ This completes the proof. □

Example 1.

Let $Z=[0,\infty )$ and $d_b:Z\times Z\to \mathbb{R}$ be a complete $D.B.M$ space defined by

$$d_b(x,y)={(x+y)}^{2}forallx,y\in Z.$$

Consider the family of multivalued mappings $S_{\beta }:Z\to P\left(Z\right)$ where $\beta \in \Omega =\alpha ,1,2,3,\cdots ,$ defined as

$$S_n\left(x\right)=\left\{\begin{array}{c}[\frac{x}{3n},\frac{x}{2n}]ifx\in \overline{B_{d_b}(x_0,r)},\hfill \\ [2nx,3nx]ifx\in (4,\infty )\cap Z,\hfill \end{array}\right.wheren=1,2,3,\cdots ,$$

and

$$S_{\alpha }\left(x\right)=\left\{\begin{array}{c}[\frac{x}{3},\frac{5x}{12}]ifx\in [0,4]\cap Z\hfill \\ [2x,3x]ifx\in (4,\infty )\cap Z.\hfill \end{array}\right.$$

Suppose that, $x_0=1,$$r=25,$ then $\overline{B_{d_b}(x_0,r)}=[0,4]\cap Z.$ Now, $d_b(x_0,S_{\alpha }{x}_0)=d_b(1,S_{\alpha }1)=d_b(1,\frac{1}{3})=\frac{16}{9}.$ So $x_1=\frac{1}{3}.$ Now, $d_b(x_1,S_1{x}_1)=d_b(\frac{1}{3},S_1\frac{1}{3})=d_b(\frac{1}{3},\frac{1}{9}).$ So $x_2=\frac{1}{9}.$ Now, $d_b(x_2,S_2{x}_2)=d_b(\frac{1}{9},S_2\frac{1}{9})=d_b(\frac{1}{9},\frac{1}{54}).$ So $x_3=\frac{1}{54}.$ Continuing in this way, we have $\left\{Z{S}_{\beta }\left(x_n\right)\right\}=\{1,\frac{1}{3},\frac{1}{9},\frac{1}{54}....\}.$ Take ${\eta }_1=\frac{1}{10},$${\eta }_2=\frac{1}{20},$${\eta }_3=\frac{1}{60},$${\eta }_4=\frac{1}{30},$ then $b{\eta }_1+b{\eta }_2+{\eta }_3+{\eta }_4<1$ and $\lambda =\frac{3}{19}.$ Now

$$d_b(x_0,S_{\alpha }{x}_0)=\frac{16}{9}<\frac{3}{19}(1-\frac{6}{19})25=\lambda (1-b\lambda )r.$$

Now, take $S_{\alpha },S_n$ where $n=1,2,3,\cdots .$ Now, if $x,y\in \overline{B_{d_b}(x_0,r)}\cap \left\{Z{S}_{\beta }\left(x_n\right)\right\}$ , then, we have

$$\begin{array}{ccc}\hfill H_{d_b}(S_{n}x,S_{\alpha }y)& =& max\{\underset{a\in S_{n}x}{sup}{d}_b(a,S_{\alpha }y),\underset{y\in S_{\alpha }y}{sup}{d}_b(S_{n}x,b)\}\hfill \\ & =& max\{\underset{a\in S_{n}x}{sup}{d}_b(a,[\frac{y}{3},\frac{5y}{12}]),\underset{b\in S_{\alpha }y}{sup}{d}_b([\frac{x}{3n},\frac{x}{2n}],b)\}\hfill \\ & =& max\{d_b(\frac{x}{2n},\frac{y}{3}),d_b(\frac{x}{3n},\frac{5y}{12})\}\hfill \\ & =& max\left\{{\left(\frac{x}{2n}+\frac{y}{3}\right)}^2,{\left(\frac{x}{3n}+\frac{5y}{12}\right)}^2\right\}\hfill \\ & \le & \frac{1}{10}{d}_b(x,y)+\frac{1}{20}{d}_b(x,[\frac{x}{3n},\frac{x}{2n}])+\frac{1}{60}{d}_b(y,[\frac{y}{3},\frac{5y}{12}])\hfill \\ & & +\frac{1}{30}\frac{d_b^2(x,[\frac{x}{3n},\frac{x}{2n}])·d_b(y,[\frac{y}{3},\frac{5y}{12}])}{1+d_b^2(x,y)}.\hfill \end{array}$$

Thus,

$$H_{d_b}(S_{n}x,S_{\alpha }y)<{\eta }_1{d}_b(x,y)+{\eta }_2{d}_b(x,S_{n}x)+{\eta }_3{d}_b(y,S_{\alpha }y)+{\eta }_4\frac{d_b^2(x,S_{n}x)·d_b(y,S_{\alpha }y)}{1+d_b^2(x,y)},$$

which implies that, for any $\tau \in (0,\frac{12}{95}]$ and for a strictly increasing mapping $F\left(s\right)=lns,$ we have

$$\tau +F\left(H_{d_b}(S_{n}x,S_{\alpha }y)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_b(x,y)+{\eta }_2{d}_b(x,S_{n}x)+{\eta }_3{d}_b(y,S_{\alpha }y)\\ +{\eta }_4{\displaystyle \frac{d_b^2(x,S_{n}x)·d_b(y,S_{\alpha }y)}{1+d_b^2(x,y)}}\end{array}\right).$$

Similarly, for some $i,j\in \Omega$ and $\tau >0,$ we can prove

$$\tau +F\left(H_{d_b}(S_{i}x,S_{j}y)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_b(x,y)+{\eta }_2{d}_b(x,S_{i}x)+{\eta }_3{d}_b(y,S_{j}y)\\ +{\eta }_4{\displaystyle \frac{d_b^2(x,S_{i}x)·d_b(y,S_{j}y)}{1+d_b^2(x,y)}}\end{array}\right).$$

Note that, for $x=6\in Z,$$y=7\in Z,$ then, we have

$$\tau +F\left(H_{d_b}(S_{1}6,S_{\alpha }7)\right)>F\left(\begin{array}{c}{\eta }_1{d}_b(6,7)+{\eta }_2{d}_b(6,S7)+{\eta }_3{d}_b(6,S7)\\ +{\eta }_4{\displaystyle \frac{d_b^2(6,S7)·(7,S7)}{1+d_b^2(6,7)}}\end{array}\right).$$

So condition (1) does not hold on $Z.$ Thus the mappings $S_{\beta }$ satisfying all the conditions of Theorem 1 only for $x,y\in \overline{B_{d_b}(x_0,r)}\cap \left\{Z{S}_{\beta }\left(x_n\right)\right\}$ . Hence there exist a common fixed point for the family of multivalued mappings $\left\{S_{\beta }:\beta \in \Omega \right\}$ in $\overline{B_{d_b}(c_0,r)}.$

If we take ${\eta }_2=0$ in Theorem 1 then we are left with the following result.

Corollary 1.

Let $(Z,d_b)$ be a complete $D.B.M$ space with constant $b\ge 1$ and $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of multivalued mappings from Z to $P\left(Z\right)$ and $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ be a sequence in Z generated by $c_0.$ Assume that the following hold:

(i) 

There exist$\tau ,{\eta }_1,{\eta }_3,{\eta }_4>0$satisfying$b{\eta }_1+{\eta }_3+{\eta }_4<1$and a strictly increasing mapping F such that

$$\tau +F\left(H_{d_b}(S_{i}e,S_{j}y)\right)\le F\left({\eta }_1{d}_b(e,y)+{\eta }_3{d}_b(y,S_{j}y)+{\eta }_4\frac{d_b^2(e,S_{i}e)·d_b(y,S_{j}y)}{1+d_b^2(e,y)}\right)$$

(6)

whenever$e,y\in \overline{B_{d_b}(c_0,r)}\cap \{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$with$e\ne y$,$i,j\in \Omega$with$i\ne j$and$H_{d_b}(S_{i}e,S_{j}y)>0.$

(ii) 

If$\lambda =\frac{{\eta }_1}{1-{\eta }_3-{\eta }_4},$then

$$d_b(c_0,S_{\alpha }{c}_0)\le \lambda (1-b\lambda )r.$$

Then$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$is a sequence in$\overline{B_{d_b}(c_0,r)}$and$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}\to u\in \overline{B_{d_b}(c_0,r)}.$Also, if the inequality (6) holds for$u,$then there exist a common fixed point for the family of multivalued mappings$\left\{S_{\beta }:\beta \in \Omega \right\}$in$\overline{B_{d_b}(c_0,r)}$and$d_b(u,u)=0.$

If we take ${\eta }_3=0$ in Theorem 1 then we are left with the following result.

Corollary 2.

Let $(Z,d_b)$ be a complete $D.B.M$ space with constant $b\ge 1$ and $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of multivalued mappings from Z to $P\left(Z\right)$ and $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ be a sequence in Z generated by $c_0.$ Assume that the following hold:

(i) 

There exist$\tau ,{\eta }_1,{\eta }_2,{\eta }_4>0$satisfying$b{\eta }_1+b{\eta }_2+{\eta }_4<1$and a strictly increasing mapping F such that

$$\tau +F\left(H_{d_b}(S_{i}e,S_{j}y)\right)\le F\left({\eta }_1{d}_b(e,y)+{\eta }_2{d}_b(e,S_{i}e)+{\eta }_4\frac{d_b^2(e,S_{i}e)·d_b(y,S_{j}y)}{1+d_b^2(e,y)}\right)$$

(7)

whenever$e,y\in \overline{B_{d_b}(c_0,r)}\cap \{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$with$e\ne y$,$i,j\in \Omega$with$i\ne j$and$H_{d_b}(S_{i}e,S_{j}y)>0.$

(ii) 

If$\lambda =\frac{{\eta }_1+{\eta }_2}{1-{\eta }_4},$then

$$d_b(c_0,S_{\alpha }{c}_0)\le \lambda (1-b\lambda )r.$$

Then$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$is a sequence in$\overline{B_{d_b}(c_0,r)}$and$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}\to u\in \overline{B_{d_b}(c_0,r)}.$Also, if the inequality (7) holds for$u,$then there exist a common fixed point for the family of multivalued mappings$\left\{S_{\beta }:\beta \in \Omega \right\}$in$\overline{B_{d_b}(c_0,r)}$and$d_b(u,u)=0.$

If we take ${\eta }_4=0$ in Theorem 1 then we are left with the following result.

Corollary 3.

Let $(Z,d_b)$ be a complete $D.B.M$ space with constant $b\ge 1$ and $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of multivalued mappings from Z to $P\left(Z\right)$ and $\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$ be a sequence in Z generated by $c_0.$ Assume that the following hold:

(i) 

There exist$\tau ,{\eta }_1,{\eta }_2,{\eta }_3>0$satisfying$b{\eta }_1+b{\eta }_2+{\eta }_3<1$and a strictly increasing mapping F such that

$$\tau +F\left(H_{d_b}(S_{i}e,S_{j}y)\right)\le F\left({\eta }_1{d}_b(e,y)+{\eta }_2{d}_b(e,S_{i}e)+{\eta }_3{d}_b(y,S_{j}y)\right)$$

(8)

whenever$e,y\in \overline{B_{d_b}(c_0,r)}\cap \{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$with$e\ne y$,$i,j\in \Omega$with$i\ne j$and$H_{d_b}(S_{i}e,S_{j}y)>0.$

(ii) 

If$\lambda =\frac{{\eta }_1+{\eta }_2}{1-{\eta }_3},$then

$$d_b(c_0,S_{\alpha }{c}_0)\le \lambda (1-b\lambda )r.$$

Then$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}$is a sequence in$\overline{B_{d_b}(c_0,r)}$and$\{Z{S}_{\beta }\left(c_n\right):\beta \in \Omega \}\to u\in \overline{B_{d_b}(c_0,r)}.$Also, if the inequality (8) holds for$u,$then there exist a common fixed point for the family of multivalued mappings$\left\{S_{\beta }:\beta \in \Omega \right\}$in$\overline{B_{d_b}(c_0,r)}$and$d_b(u,u)=0.$

3. Application to the Systems of Integral Equations

Theorem 2.

Let $(Z,d_b)$ be a complete $D.B.M$ space with constant $b\ge 1$ . Let $c_0\in Z$ and $\left\{S_{\beta }:\beta \in \Omega \right\}$ be a family of mappings from Z to $Z.$ Assume that, there exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+{\eta }_3+{\eta }_4<1$ and a strictly increasing mapping F such that the following holds:

$$\tau +F\left(d_b(S_{\alpha }e,S_{\beta }y)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_b(e,y)+{\eta }_2{d}_b(e,S_{\alpha }e)\\ +{\eta }_3{d}_b(y,S_{\beta }y)+{\eta }_4{\displaystyle \frac{d_b^2(e,S_{\alpha }e)·d_b(y,S_{\beta }y)}{1+d_b^2(e,y)}}\end{array}\right),$$

(9)

for all $e,y\in Z$ and $d(S_{\alpha }e,S_{\beta }y)>0$ where $\alpha ,\beta \in \Omega$ with $\alpha \ne \beta .$ Also if the inequality (9) holds for u, then the family $\left\{S_{\beta }:\beta \in \Omega \right\}$ has a unique common fixed point u in Z.

Proof. 

The proof of this theorem is similar as Theorem 1. We have to prove the uniqueness only. Let v be another common fixed point of $S.$ Suppose $d_b(S_{\alpha }u,S_{\beta }v)>0$. Then, we have

$$\tau +F\left(d_b(S_{\alpha }u,S_{\beta }v)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_b(u,v)+{\eta }_2{d}_b(u,S_{\alpha }u)\\ +{\eta }_3{d}_b(v,S_{\beta }v)+{\eta }_4{\displaystyle \frac{d_b^2(u,S_{\alpha }u)·d_b(v,S_{\beta }v)}{1+d_b^2(u,v)}}\end{array}\right).$$

This implies that

$$d_b(u,v)<{\eta }_1{d}_b(u,v)<d_b(u,v),$$

which is a contradiction. So $d_b(S_{\alpha }u,S_{\beta }v)=0.$ Hence $u=v.$ □

In this section, we discuss the application of fixed point Theorem 2 in form of Volterra type integral equation.

$$u\left(k\right)=\underset{0}{\overset{k}{\int }}{H}_{\alpha }(k,h,u\left(h\right))dh,$$

(10)

for all $k,h\in [0,1]$ and $\alpha \in \Omega .$ We find the solution of $\left(10\right)$. Let $Z=C\left(\right[0,1],\mathbb{R})$ be the set of all real valued continuous functions on $[0,1]$, endowed with the complete dislocated b-metric. For $u\in C\left(\right[0,1],\mathbb{R}),$ define supremum norm as: ${\parallel u\parallel }_{\tau }=\underset{k\in [0,1]}{sup}\left\{\left|u\left(k\right)\right|e^{-\tau k}\right\}$, where $\tau >0$ is taken arbitrary. Then define

$$d_{\tau }(u,c)={\left[\underset{k\in [0,1]}{sup}\left\{\left|u\left(k\right)+c\left(k\right)\right|e^{-\tau k}\right\}\right]}^2={\parallel u+c\parallel }_{\tau }^2$$

for all $u,c\in C\left(\right[0,1],\mathbb{R}),$ with these settings, $(C([0,1],\mathbb{R}),d_{\tau })$ becomes a complete $D.B.M.S$.

Now we prove the following theorem to ensure the existence of solution of integral equation.

Theorem 3.

Assume the following conditions are satisfied:

(i) 

$H_{\alpha }:[0,1]\times [0,1]\times C([0,1],\mathbb{R})\to \mathbb{R}$;

(ii) 

Define$S_{\alpha }:C([0,1],\mathbb{R})\to C([0,1],\mathbb{R}),$where$\alpha \in \Omega$

$$S_{\alpha }u\left(k\right)=\underset{0}{\overset{k}{\int }}{H}_{\alpha }(k,h,u\left(h\right))dh.$$

Suppose there exist $\tau >0,$ such that

$$\left|H_{\alpha }(k,h,u\left(h\right))+H_{\beta }(k,h,c\left(h\right))\right|\le \frac{\tau N\left(u\right(h),c(h\left)\right)}{{\tau \parallel N(u,c)\parallel }_{\tau }+1}$$

for all $k,h\in [0,1]$ and $u,c\in C\left(\right[0,1],\mathbb{R}),$ where

$$\begin{array}{ccc}\hfill N\left(u\right(h),c(h\left)\right)& =& {\eta }_1{\left[\left|u\left(h\right)+c\left(h\right)\right|\right]}^2+{\eta }_2{\left[\left|u\left(h\right)+S_{\alpha }u\left(h\right)\right|\right]}^2+{\eta }_3{\left[\left|c\left(h\right)+S_{\beta }c\left(h\right)\right|\right]}^2\hfill \\ & & +{\eta }_4\frac{{\left[\left|u\left(h\right)+S_{\alpha }u\left(h\right)\right|\right]}^4.{\left[\left|c\left(h\right)+S_{\beta }c\left(h\right)\right|\right]}^2}{1+{\left[\left|u\left(h\right)+c\left(h\right)\right|\right]}^4},\hfill \end{array}$$

where ${\eta }_1,$${\eta }_2,$${\eta }_{3,}$${\eta }_4\ge 0,$ and $2{\eta }_1+2{\eta }_2+{\eta }_3+{\eta }_4<1.$ Then integral Equation (10) has a solution.

Proof. 

By assumption (ii)

$$\begin{array}{ccc}\hfill \left|S_{\alpha }u\left(k\right)+S_{\beta }c\left(k\right)\right|& =& \underset{0}{\overset{k}{\int }}\left|H_{\alpha }(k,h,u\left(h\right)+H_{\beta }(k,h,c\left(h\right)))\right|dh,\hfill \\ & \le & \underset{0}{\overset{k}{\int }}\frac{\tau }{{\tau \parallel N(u,c)\parallel }_{\tau }+1}\left(\left[N(u\left(h\right),c\left(h\right))\right]e^{-\tau h}\right)e^{\tau h}dh\hfill \\ & \le & \underset{0}{\overset{k}{\int }}\frac{\tau }{{\tau \parallel N(u,c)\parallel }_{\tau }+1}{\parallel N(u,c)\parallel }_{\tau }{e}^{\tau h}dh\hfill \\ & \le & \frac{{\tau \parallel N(u,c)\parallel }_{\tau }}{{\tau \parallel N(u,c)\parallel }_{\tau }+1}\underset{0}{\overset{k}{\int }}{e}^{\tau h}dh,\hfill \\ & \le & \frac{{\parallel N(u,c)\parallel }_{\tau }}{{\tau \parallel N(u,c)\parallel }_{\tau }+1}{e}^{\tau k}.\hfill \end{array}$$

This implies

$$\left|S_{\alpha }u\left(k\right)+S_{\beta }c\left(k\right)\right|e^{-\tau k}\le \frac{{\parallel N(u,c)\parallel }_{\tau }}{{\tau \parallel N(u,c)\parallel }_{\tau }+1}.$$

$${\parallel S_{\alpha }u\left(k\right)+S_{\beta }c\left(k\right)\parallel }_{\tau }\le \frac{{\parallel N(u,c)\parallel }_{\tau }}{{\tau \parallel N(u,c)\parallel }_{\tau }+1}.$$

$$\frac{{\tau \parallel N(u,c)\parallel }_{\tau }+1}{{\parallel N(u,c)\parallel }_{\tau }}\le \frac{1}{\parallel S_{\alpha }u\left(k\right)+S_{\beta }{c\left(k\right)\parallel }_{\tau }}.$$

$$\tau +\frac{1}{{\parallel N(u,c)\parallel }_{\tau }}\le \frac{1}{\parallel S_{\alpha }u\left(k\right)+S_{\beta }{c\left(k\right)\parallel }_{\tau }}.$$

which further implies

$$\tau -\frac{1}{\parallel S_{\alpha }u\left(k\right)+S_{\beta }{c\left(k\right)\parallel }_{\tau }}\le \frac{-1}{{\parallel N(u,c)\parallel }_{\tau }}.$$

So all the conditions of Theorem 3 are satisfied for $F\left(c\right)=\frac{-1}{\sqrt{c}};c$ $>0$ and $d_{\tau }(u,c)={\parallel u+c\parallel }_{\tau }^2,$ $b=2$. Hence integral equations given in (10) have a unique common solution. □

Example 2.

Consider the system of integral equations

$$g\left(k\right)=\frac{1}{\alpha }\underset{0}{\overset{k}{\int }}g\left(h\right)dh,wherek\in [0,1]and\alpha \in \Omega =\mathbb{N}.$$

Define $H_{\alpha }:[0,1]\times [0,1]\times C([0,1],{\mathbb{R}}_{+})\to \mathbb{R}$ by $H_{\alpha }=\frac{1}{\alpha }g\left(h\right),$$\alpha \in \Omega =\mathbb{N}.$ Now,

$$S_{\alpha }g\left(k\right)=\frac{1}{\alpha }\underset{0}{\overset{k}{\int }}g\left(h\right)dh.$$

Take ${\eta }_1=\frac{1}{10},$${\eta }_2=\frac{1}{20},$${\eta }_3=\frac{1}{60},$${\eta }_4=\frac{1}{30},$$\tau =\frac{12}{95},$ then $2{\eta }_1+2{\eta }_2+{\eta }_3+{\eta }_4<1.$ Moreover, all conditions of Theorem 3 are satisfied and $g\left(k\right)=0$ for all $k,$ is a unique common solution to the above equations.

4. Conclusions

In the present paper, we have achieved common fixed point of a family of multivalued mappings satisfying conditions only on a sequence contained in a closed ball. We have used a weaker class of strictly increasing mappings F rather than the class of mappings F used by many potential authors. Examples and an application are given to demonstrate the variety of our results. New results for families of multivalued mappings and singlevalued contractive mappings in ordered spaces, partial b-metric space, dislocated metric space, partial metric space, b-metric space, and metric space can be obtained as corollaries of our results.