Common fixed point results via implicit contractions for multi-valued mappings on b-metric like spaces

Abstract

In this paper, motivated by the recent work [Journal of Nonlinear Sciences and Applications, 10(4):1544–1537] some generalized nonlinear contractive conditions via implicit functions and α-admissible pairs of multi-valued mappings in the setting of b-metric like spaces have been introduced. Some common fixed point results for such mappings in this framework have been provided. Then, some corollaries and consequences for our obtained results are given. Our results are the multi-valued versions of [Journal of Nonlinear Sciences and Applications, 10(4):1544–1537]. An example also is provided to support our obtained results. The presented results generalize and extend some earlier results in the literature.

Keywords:

• Common fixed point
• implicit contractions
• multi-valued mappings
• b-metric like spaces

MR Subject classifications:

• Primary 47H10
• Secondary 54H45

PUBLIC INTEREST STATEMENT

Fixed point theory is one of mathematical research fields which deals with existence and uniqueness of fixed points, coincidence points and common fixed points of 1 or more than 1 mappings defined on metric spaces and generalized metric spaces such as b-metric spaces, partial metric spaces, metric like spaces etc. The obtained results will apply to find a solution of integral equations, differential equations and matrix equations.

1. Introduction

Fixed point theory is one of applicable mathematical research fields. The obtained results in this field will apply to find solutions for integral equations, differential equations and matrix equations. Before starting the topic of this paper it is worth to recall some recent works in fixed point theory. For instance, Xu, Tang, Yang, and Srivastava (2016) obtained sharp estimates of the classical boundary Schwarz lemma involving the boundary fixed points for holomorphic functions which map the unit disk in C into itself. Also, Srivastava, Bedre, Khairnar, and Desale (2014a, 2014b) obtained some hybrid fixed point theorems of Krasnoselskii type, which involve product of two operators in partially ordered normed linear spaces and applied their results to proving the existence of solutions for fractional integral equations under certain monotonicity conditions.

The concepts of $b$-metric spaces (initiated by Bakhtin, 1989, see also Czerwik, 1998) and partial metric spaces (introduced by Matthews, 1994) are two of the most important generalizations of the notion of a metric space. Amini-Harandi (2014) introduced the metric like space as an extension of the concept of a partial metric space. Earlier this notion was known as dislocated metric space studied by Hitzler and Seda (2000). The concept of b-metric like space which is given by Alghamdi, Hussain, and Salimi (2013) (see also Hussain, Roshan, Parvaneh, & Kadelburg, 2014) generalized the notions of metric space, b-metric space, partial metric space, metric like space and partial b-metric space.

2. Preliminaries

In this section, we give some notions and results that will be needed in the sequel.

Definition 4.1. (Bakhtin, 1989) Let $X$ be a nonempty set and $s\ge 1$ be a constant real number. The function $d:X\times X\to [0,\mathrm{\infty })$is called a $b$-metric on $X$ if the following conditions hold:

1. $d(x,y)=0$ iff $x=y$ for all $x,y\in X$;

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

3. $d(x,y)\le s[d(x,z)+d(z,y)]$ for all $x,y,z\in X$.

In this case, $(X,d)$ is called a $b$-metric space with parameter $s$.

Definition 4.2. (Alghamdi et al., 2013) Let $X$ be a nonempty set and $s\ge 1$ be a constant real number. The function $\sigma :X\times X\to [0,\mathrm{\infty })$ is called a $b$ -metric like on $X$ if the following conditions hold:

1. $\sigma (x,y)=0$ implies that $x=y$ for all $x,y\in X$;

2. $\sigma (x,y)=\sigma (y,x)$ for all $x,y\in X$;

3. $\sigma (x,y)\le s[\sigma (x,z)+\sigma (z,y)]$ for all $x,y,z\in X$ .

In this case, $(X,\sigma )$ is called a $b$-metric like space with parameter $s$.

Definition 4.3. (Alghamdi et al., 2013) Let $(X,\sigma )$ be a $b$-metric like space and let $\left\{x_n\right\}$ be any sequence in $X$ and $x\in X$. Then:

1. The sequence $\left\{x_n\right\}$ is said to be convergent to $x$, if

   $\underset{n\to \mathrm{\infty }}{lim}\sigma (x_n,x)=\sigma (x,x).$

2. The sequence $\left\{x_n\right\}$ is said to be a Cauchy sequence in $(X,\sigma )$ if$\underset{n,m\to \mathrm{\infty }}{lim}\sigma (x_m,x_n)$ exists and is finite.

3. $(X,\sigma )$ is said to be complete if any Cauchy sequence $\left\{x_n\right\}$ in $X$ be convergent to some $x\in X$.

Lemma 4.4. (Hussain et al., 2014) Let $(X,\sigma )$ be a $b$-metric like space and $\left\{x_n\right\}$ be a sequence in $X$ which is converges to some point $u\in X$ with $\sigma (u,u)=0$. Then, for each $y,z\in X$,

$\frac{1}{s}\sigma (u,z)\le \underset{n\to \mathrm{\infty }}{liminf}\sigma (x_n,z)\le s\sigma (u,z)and\sigma (z,z)\le 2s\sigma (z,y).$

Recall that $T:X\to X$ is said to be sequentially continuous at $u\in X$ if for any sequence $\left\{x_n\right\}$ in $X$ converging to $u$, we have $T{x}_n\to Tu$, that is, $\underset{n\to \mathrm{\infty }}{lim}\sigma (T{x}_n,Tu)=\sigma (Tu,Tu)$. Also, $T$ is called sequentially continuous on $X$ if it is sequentially continuous at each $u\in X$.

Now, take $s\ge 1$ and denote by $\mathbb{N}$ the set of all natural numbers. Let ${\mathrm{\Psi }}_s$ be the set of all functions $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ satisfying

(ѱ₁) $\psi$ is strictly increasing,

(ѱ₄) ${\mathrm{\Sigma }}_{n=1}^{\mathrm{\infty }}{s}^n{\psi }^n(t)<\mathrm{\infty }$ for each $t>0$.

It is easy to check that for any $\psi \in {\mathrm{\Psi }}_s$, then $\psi (t)<t$ for all $t>0$.

Now, we introduce the following set of functions.

Definition 4.5. Let $s\ge 1$ be a constant real number. Denote by ${\mathrm{\Gamma }}_s$ the set of all functions $F(t_1,...,t_6):{\mathbb{R}}_{+}^6\to \mathbb{R}$ such that

(F1) $F$ is nondecreasing in variable $t_1$ and nonincreasing in variables $t_3,t_4,t_5,t_6$,

(F4) there exists $h_1\in {\mathrm{\Psi }}_s$ such that for all $u,v\ge 0$, $F(u,v,v,u,s(u+v),2sv)\le 0$ implies that $u\le h_1(v)$ and $F(u,v,u,v,2sv,s(u+v))\le 0$ implies that $u\le h_1(v)$.

In the following examples we give two examples of functions in ${\mathrm{\Gamma }}_s$.

Example 4.6. $F(t_1,t_2,t_3,t_4,t_5,t_6)=t_1-a{t}_2-b(t_3+t_4)-c(t_5+t_6)$, where $a,b,c\ge 0$ such that $as+b(1+s)+cs(1+3s)<1$.

Example 4.7. $F(t_1,t_2,t_3,t_4,t_5,t_6)=t_1-kmax\left\{t_2,...,t_6\right\}$, where $k\in [0,\frac{1}{2s^2})$.

Samet, Vetro, and Vetro (2013) introduced an important notion of $\alpha$-admissible mappings as follows.

Definition 4.8. Let $X$ be a nonempty set and $T:X\to X$ be a given mapping. Then, $T$ is said to be $\alpha$-admissible whenever there exists $\alpha :X\times X\to [0,\mathrm{\infty })$ such that

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

for all $x,y\in X$.

The above notion used to prove many important results in literature (see Ali, Kamran, & Karapnar, 2014; Jleli, Karapnar, & Samet, 2013b, 2013a; Karapnar & Samet, 2014; Pansuwan, Sintunavarat, Parvaneh, & Cho, 2015). Mohammadi, Rezapour, and Shahzad (2013) extended the $\alpha$-admissible notion to multi-valued mappings as follows.

Definition 4.9. Let $X$ be a nonempty set, $T:X\to 2^X$ be a given multi-valued mapping and $\alpha :X\times X\to [0,\mathrm{\infty })$. Then, $T$ is said to be $\alpha$-admissible whenever for any $x\in X$ and $y\in Tx$ with $\alpha (x,y)\ge 1$, one has $\alpha (y,z)\ge 1$, for all $z\in Ty$.

Recently, Aydi (2015) generalized Definition (4.8) for a pair of mappings as follows.

Definition 4.10. Let $X$ be a nonempty set, $A,B:X\to X$ be two given mappings and $\alpha :X\times X\to [0,\mathrm{\infty })$. Then, the pair $(A,B)$ is said to be a generalized $\alpha$-admissible pair whenever

$\alpha (x,y)\ge 1\Rightarrow \alpha (Ax,By)\ge 1\hspace{0.17em},\alpha (By,Ax)\ge 1$

for all $x,y\in X$.

Also, Aydi, Velhi, and Sahmim (2017) introduced a pair of $\alpha$-implicit contractive mappings as follows and proved common fixed point results for such mappings.

Definition 4.11. Let $(X,\sigma )$ be a $b$-metric like space, $A,B:X\to X$ be two given mappings and $\alpha :X\times X\to [0,\mathrm{\infty })$. Then, $(A,B)$ is said to be an $\alpha$-implicit contractive pair of mappings whenever there exists $F\in {\mathrm{\Gamma }}_s$ such that

$F(\alpha (x,y)\sigma (Ax,By),\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax))\le 0,$

for all $x,y\in X$.

In this paper, we introduce $\alpha$-implicit contractive pair of multi-valued mappings on $b$-metric like spaces and establish common fixed point results for our presented mappings.

Denote by $CB(X)$ the set of all nonempty closed bounded subsets of $X$. Define the Pompeiu-Hausdorff metric $H_{\sigma }$ induced by $\sigma$ on $CB(X)$ as follows

$H_{\sigma }(A,B)=max\left\{\underset{x\in A}{sup}\sigma (x,B),\underset{y\in B}{sup}\sigma (y,A)\right\},$

for all $A,B\in CB(X)$ where $\sigma (x,B)=\underset{y\in B}{inf}\sigma (x,y)$. An element $x\in X$ is said to be a fixed point of a multi-valued mapping $T:X\to CB(X)$ whenever $x\in Tx$.

3. Main results

In this section, we provide our main results. Firstly, we give the following definitions.

Definition 5.1. Let $X$ be a nonempty set, $A,B:X\to 2^X$ be two given multi-valued mappings and $\alpha :X\times X\to [0,\mathrm{\infty })$. The pair $(A,B)$ is said to be a generalized $\alpha$-admissible pair whenever

(α1) $\mathrm{\forall }x\in X,y\in Ax(\alpha (x,y)\ge 1\Rightarrow \alpha (y,z)\ge 1,\alpha (z,y)\ge 1,\mathrm{\forall }z\in By).$

(α4) $\mathrm{\forall }x\in X,y\in Bx(\alpha (y,x)\ge 1\Rightarrow \alpha (y,z)\ge 1,\alpha (z,y)\ge 1,\mathrm{\forall }z\in Ay).$

Definition 5.2. Let $(X,\sigma )$ be a $b$-metric like space, $A,B:X\to CB(X)$ be two given multi-valued mappings and $\alpha :X\times X\to [0,\mathrm{\infty })$. Then, $(A,B)$ is said to be an $\alpha$-implicit contractive pair of multi-valued mappings whenever there exists $F\in {\mathrm{\Gamma }}_s$ such that

(5.1) $F(\alpha (x,y)H_{\sigma }(Ax,By),\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax))\le 0,$(5.1)

for all $x,y\in X$.

Theorem 5.3. Let $(X,\sigma )$ be a complete $b$-metric like space and $A,B:X\to CB(X)$ be two given $\alpha$-implicit multi-valued mappings. Suppose that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. there exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $A$ and $B$ are sequentially continuous on $(X,\sigma )$.

Then, there exists $u\in X$ such that

(5.2) $\frac{1}{s}\sigma (u,Au)\le H_{\sigma }(Au,Au),\frac{1}{s}\sigma (u,Bu)\le H_{\sigma }(Bu,Bu)and\sigma (u,u)=0.$(5.2)

Assume in addition that

1. $\alpha (z,z)\ge 1$ for all $z$ satisfying (3.4),

2. $F$ satisfies $F_{\gamma }$, where $F_{\gamma }$ is as follows:

F_γ if $F(u,0,v,w,2s^{2}u,2s^{2}u)\le 0$ for all $u,v,w\ge 0$, then there exists $\gamma \in [0,\frac{1}{2s^2})$ such that $u\le \gamma max\left\{v,w\right\}$.

Then, $u$ is a common fixed point of $A$ and $B$, that is, $u\in Au$ and $u\in Bu$.

Proof. By assumption, there exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$. Applying (5.1), we have

$F(\alpha (x_0,x_1)H_{\sigma }(A{x}_0,B{x}_1),\sigma (x_0,x_1),\sigma (x_0,A{x}_0),\sigma (x_1,B{x}_1),\sigma (x_0,B{x}_1),\sigma (x_1,A{x}_0))\le 0.$

From $(F1)$, we get

$F(\sigma (x_1,B{x}_1),\sigma (x_0,x_1),\sigma (x_0,x_1),\sigma (x_1,B{x}_1),s[\sigma (x_0,x_1)+\sigma (x_1,B{x}_1)],2s\sigma (x_0,x_1))\le 0.$

From $(F2)$, $\sigma (x_1,B{x}_1)\le h_1(\sigma (x_0,x_1))$. If $\sigma (x_0,x_1)=0$, then $x_0=x_1$ and so $x_0$ is a fixed point of $A$ and $\sigma (x_1,B{x}_1)\le h_1(0)=0$ which yields that $x_1\in B{x}_1$. Thus, $x_0$ is a common fixed point of $A$ and $B$. Thus, we may assume that $\sigma (x_0,x_1)>0$. Now, we have $\sigma (x_1,B{x}_1)\le h_1(\sigma (x_0,x_1))<\sigma (x_0,x_1)$. Thus, there exists $x_2\in B{x}_1$ such that $\sigma (x_1,x_2)<\sigma (x_0,x_1)$. From ($\alpha$1), we get $\alpha (x_1,x_2)\ge 1$ and $\alpha (x_2,x_1)\ge 1$. Applying (5.1), with $x=x_2$ and $y=x_1$, we have

$F(\alpha (x_2,x_1)H_{\sigma }(A{x}_2,B{x}_1),\sigma (x_2,x_1),\sigma (x_2,A{x}_2),\sigma (x_1,B{x}_1),\sigma (x_2,B{x}_1),\sigma (x_1,A{x}_2))\le 0.$

From $(F1)$, we get

$F(\sigma (x_2,A{x}_2),\sigma (x_2,x_1),\sigma (x_2,A{x}_2),\sigma (x_1,x_2),2s\sigma (x_1,x_2),s[\sigma (x_1,x_2)+\sigma (x_2,A{x}_2)])\le 0.$

From $(F2)$ and since $h_1$ is strictly increasing, we obtain that $\sigma (x_2,A{x}_2)\le h_1(\sigma (x_1,x_2))<h_1(\sigma (x_0,x_1))$. Thus, there exists $x_3\in A{x}_2$ such that $\sigma (x_2,x_3)<h_1(\sigma (x_0,x_1))$. From ($\alpha$4), we get $\alpha (x_2,x_3)\ge 1$ and $\alpha (x_3,x_2)\ge 1$. Continuing this process, we obtain a sequence $\left\{x_n\right\}$ in $X$ such that

$\alpha (x_n,x_{n+1})\ge 1,\hspace{0.17em}\alpha (x_{n+1},x_n)\ge 1,\hspace{0.17em}{x}_{2n+1}\in A{x}_{2n},\hspace{0.17em}{x}_{2n+2}\in B{x}_{2n+1}$

and

$\sigma (x_{n+1},x_{n+2})<{h_1}^n(\sigma (x_n,x_{n+1})),$

for all $n\ge 0$. Thus, for any $m,n\in \mathbb{N}$ with $m>n$, we have

$\begin{array}{ccc}\sigma (x_n,x_m)& \le & s[\sigma (x_n,x_{n+1})+\sigma (x_{n+1},x_m)]\\ \hspace{0.17em}& \le & s\sigma (x_n,x_{n+1})+s^2\sigma (x_{n+1},x_{n+2})+\dots +s^{m-n}\sigma (x_{m-1},x_m)\\ \hspace{0.17em}& =& {\sum }_{i=n}^{m-1}{s}^{i-n+1}\sigma (x_i,x_{i+1})\le {\sum }_{i=n}^{m-1}{s}^{i-n+1}{h_1}^{i-1}(\sigma (x_0,x_1))\\ \hspace{0.17em}& =& \frac{1}{s^{n-2}}{\sum }_{i=n}^{m-1}{s}^{i-1}{h_1}^{i-1}(\sigma (x_0,x_1))\to 0\end{array}$

as $m,n\to \mathrm{\infty }$. Thus, $\left\{x_n\right\}$ is a Cauchy sequence. Since $(X,\sigma )$ is complete, there exists $u\in X$ such that $x_n\to u$, that is,

(5.3) $\underset{n\to \mathrm{\infty }}{lim}\sigma (x_n,u)=\sigma (u,u)=\underset{n,m\to \mathrm{\infty }}{lim}\sigma (x_n,x_m)=0.$(5.3)

Now, we show that $u$ is a common fixed point of $A$ and $B$. We have

$\underset{n\to \mathrm{\infty }}{lim}\sigma (A{x}_{2n},u)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+1},u)=0$

and

$\underset{n\to \mathrm{\infty }}{lim}\sigma (B{x}_{2n+1},u)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+2},u)=0.$

Using the sequential continuity of $A$, we have

$\underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+1},Au)\le \underset{n\to \mathrm{\infty }}{lim}{H}_{\sigma }(A{x}_{2n},Au)=H_{\sigma }(Au,Au),$

and using the sequential continuity of $B$, we have

$\underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+2},Bu)\le \underset{n\to \mathrm{\infty }}{lim}{H}_{\sigma }(B{x}_{2n+1},Bu)=H_{\sigma }(Bu,Bu).$

Now, we have

(5.4) $\frac{1}{s}\sigma (u,Au)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+1},Au)\le H_{\sigma }(Au,Au)$(5.4)

and

(5.5) $\frac{1}{s}\sigma (u,Bu)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n+1},Bu)\le H_{\sigma }(Bu,Bu).$(5.5)

Using (5.3), (5.4) and (5.5), we see that (5.2) holds. Thus, by hypothesis (iv), $\alpha (u,u)\ge 1$. Applying (5.1) for $x=y=u$, we get

$F(H_{\sigma }(Au,Bu),0,\sigma (u,Au),\sigma (u,Bu),\sigma (u,Bu),\sigma (u,Au))\le 0.$

On the other hand, from (5.4) and (5.5),

$\sigma (u,Au)\le s{H}_{\sigma }(Au,Au)\le 2s^2{H}_{\sigma }(Au,Bu)$

and

$\sigma (u,Bu)\le s{H}_{\sigma }(Bu,Bu)\le 2s^2{H}_{\sigma }(Au,Bu).$

Applying (F1) in the fifth and sixth variables, we obtain that

$F(H_{\sigma }(Au,Bu),0,\sigma (u,Au),\sigma (u,Bu),2s^2{H}_{\sigma }(Au,Bu),2s^2{H}_{\sigma }(Au,Bu))\le 0.$

From $(F_{\gamma })$, we get

$H_{\sigma }(Au,Bu)\le \gamma max\left\{\sigma (u,Au),\sigma (u,Bu)\right\}=\gamma 2s^2{H}_{\sigma }(Au,Bu).$

The above inequality holds unless $H_{\sigma }(Au,Bu)=0$ which yields that $Au=Bu$. Thus, $H_{\sigma }(Au,Au)=H_{\sigma }(Bu,Bu)=0$. From (5.4) and (5.5), we get $\sigma (u,Au)=\sigma (u,Bu)=0$, that is, $u\in Au$ and $u\in Bu$.

Taking $A=B$ in Theorem 3.3, we derive the following result.

Corollary 5.4. Let $(X,\sigma )$ be a complete $b$-metric like space and $A:X\to CB(X)$ be a given multi-valued mapping. Assume that there exists $F\in {\mathrm{\Gamma }}_s$ and $\alpha :X\times X\to [0,\mathrm{\infty })$ such that

$F(\alpha (x,y)H_{\sigma }(Ax,Ay),\sigma (x,y),\sigma (x,Ax),\sigma (y,Ay),\sigma (x,Ay),\sigma (y,Ax))\le 0,$

for all $x,y\in X$. Suppose that

1. $(A,A)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $A$ is sequentially continuous on $(X,\sigma )$

Then, there exists $u\in X$ such that

(5.6) $\frac{1}{s}\sigma (u,Au)\le H_{\sigma }(Au,Au)and\sigma (u,u)=0.$(5.6)

Assume in addition that

1. $\alpha (z,z)\ge 1$ for all $z$ satisfying (5.6),

2. $F$ satisfies $F_{\gamma }$.

Then, $u$ is a fixed point of $A$, that is, $u\in Au$.

Considering the $b$-metric case in Theorem (5.3) we have the following results.

Corollary 5.5. Let $(X,d)$ be a complete $b$-metric space and $A,B:X\to CB(X)$ be two multi-valued mappings such that there exists $F\in {\mathrm{\Gamma }}_s$ such that

$F(\alpha (x,y)H_{\sigma }(Ax,By),d(x,y),d(x,Ax),d(y,By),d(x,By),d(y,Ax))\le 0,$

for all $x,y\in X$. Suppose that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $A$ and $B$ are sequentially continuous on $(X,\sigma )$.

Then, there exists $u\in X$ such that $u\in Au$ and $u\in Bu$.

Proof. Following the proof of Theorem (5.3), we find a sequence $\left\{x_n\right\}$ in $X$ and $u\in X$ such that (5.2) holds. Since in a $b$-metric space, we have $H_{\sigma }(Au,Au)=0$ and $H_{\sigma }(Bu,Bu)=0$, thus, from (5.2), we get $\sigma (u,Au)=\sigma (u,Bu)=0$, that is, $u\in Au$ and $u\in Bu$.

Corollary 5.6. Let $(X,d)$ be a complete $b$-metric space and $A:X\to CB(X)$ be a multi-valued mapping such that there exists $F\in {\mathrm{\Gamma }}_s$ such that

$F(\alpha (x,y)H_{\sigma }(Ax,Ay),d(x,y),d(x,Ax),d(y,Ay),d(x,Ay),d(y,Ax))\le 0,$

for all $x,y\in X$. Suppose that

1. $(A,A)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $A$ is sequentially continuous on $(X,\sigma )$.

Then, there exists $u\in X$ such that $u\in Au$.

Now, let the function $F$ used in Theorem (5.3) be the function presented in Example (4.7). Then, we have the following results.

Corollary 5.7. Let $(X,\sigma )$ be a complete $b$-metric like space and $A,B:X\to CB(X)$ be two multi-valued mappings such that

$\alpha (x,y)H_{\sigma }(Ax,By)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{2s^2})$. Suppose that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $A$ and $B$ are sequentially continuous on $(X,\sigma )$.

Then, there exists $u\in X$ such that (5.2) holds. Assume in addition that

1. $\alpha (z,z)\ge 1$ for all $z$ satisfying (5.2).

Then, $u\in Au$ and $u\in Bu$.

In the following result, we omit the continuity condition of $A$ and $B$ and replace it by the following assertion.

(H) For each sequence $\left\{x_n\right\}$ in $X$ with $\alpha (x_n,x_{n+1})\ge 1$ and $\alpha (x_{n+1},x_n)\ge 1$ for all $n$ and $x_n\to x$, then, we have $\alpha (x_n,x)\ge 1$ and $\alpha (x,x_n)\ge 1$ for all $n$.

in this case, we say that $(X,\sigma )$ is double-sided Regular.

Also, we replace Definition (4.5) by the following.

Definition 5.8. Let $s\ge 1$ be a constant real number. Denote by $G_s$ the set of all continuous functions $F(t_1,...,t_6):{\mathbb{R}}_{+}^6\to \mathbb{R}$ such that

(G1) $F$ is nondecreasing in variable $t_1$ and nonincreasing in variables $t_3,t_4,t_5,t_6$,

(G4) there exists $h_1\in {\mathrm{\Psi }}_s$ such that $F(\frac{u}{s},v,v,u,s(u+v),2sv)\le 0$ for all $u,v\ge 0$, implies that $u\le h_1(v)$, and $F(\frac{u}{s},v,u,v,2sv,s(u+v))\le 0$ implies that $u\le h_1(v)$.

Remark 5.9. In the case that $F\in {\mathrm{\Gamma }}_s$, the continuity condition is not a necessary condition, but in Definition (5.8) the continuity condition of $F$ is essential. Also, the property $(G2)$ implies $(F2)$, but the reverse is not true.

Theorem 5.10. Let $(X,\sigma )$ be a complete $b$-metric like space and $A,B:X\to CB(X)$ be two multi-valued mappings such that there exists $F\in G_s$ such that

$F(\alpha (x,y)H_{\sigma }(Ax,By),\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax))\le 0,$

for all $x,y\in X$. Suppose that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $(X,\sigma )$ be double-sided Regular.

Then, there exists $u\in X$ such that $u\in Au$ and $u\in Bu$. We also have $\sigma (u,u)=0$.

Proof. Following the proof of Theorem (5.3), there exists a sequence $\left\{x_n\right\}$ in $X$ such that

$\alpha (x_n,x_{n+1})\ge 1,\hspace{0.17em}\alpha (x_{n+1},x_n)\ge 1,\hspace{0.17em}{x}_{2n+1}\in A{x}_{2n},\hspace{0.17em}{x}_{2n+2}\in B{x}_{2n+1},$

for all $n\ge 0$, which is convergent to some point $u\in X$ and

(5.7) $\underset{n\to \mathrm{\infty }}{lim}\sigma (x_n,u)=\sigma (u,u)=0.$(5.7)

From double-sided Regularity assumption, we have $\alpha (x_n,u)\ge 1$ and $\alpha (u,x_n)\ge 1$ for all $n$. Taking $x=x_{2n},y=u$ in (5.1), we have

$\underset{n\to \mathrm{\infty }}{limsup}F(\alpha (x_{2n},u)H_{\sigma }(A{x}_{2n},Bu),\sigma (x_{2n},u),\sigma (x_{2n},A{x}_{2n}),\sigma (u,Bu),\sigma (x_{2n},Bu),\sigma (u,A{x}_{2n}))\le 0.$

Since $\alpha (x_{2n},u)\ge 1$, from (F1), we get

$\underset{n\to \mathrm{\infty }}{limsup}F(\sigma (x_{2n+1},Bu),\sigma (x_{2n},u),\sigma (x_{2n},x_{2n+1}),\sigma (u,Bu),\sigma (x_{2n},Bu),\sigma (u,x_{2n+1}))\le 0.$

On the other hand, we have

$\frac{1}{s}\sigma (u,Bu)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n},Bu)\le s\sigma (u,Bu).$

Taking $n\to \mathrm{\infty }$ and using the continuity of $F$, we get

$\underset{n\to \mathrm{\infty }}{limsup}F(\frac{1}{s}\sigma (u,Bu),0,0,\sigma (u,Bu),s\sigma (u,Bu),0)\le 0.$

Using (G4), we obtain that $\sigma (u,Bu)\le h_1(0)=0$ which yields that $u\in Bu$. Similarly, Taking $x=u$ and $y=x_{2n-1}$ in (3.1), we have

$\underset{n\to \mathrm{\infty }}{limsup}F(\alpha (u,x_{2n-1})H_{\sigma }(Au,B{x}_{2n-1}),\sigma (u,x_{2n-1}),\sigma (u,Au),\sigma (x_{2n-1},B{x}_{2n-1})$

$,\sigma (u,B{x}_{2n-1}),\sigma (x_{2n-1},Au))\le 0.$

Since $\alpha (u,x_{2n-1})\ge 1$, from (F1), we get

$\underset{n\to \mathrm{\infty }}{limsup}F(H_{\sigma }(Au,x_{2n}),\sigma (u,x_{2n-1}),\sigma (u,Au),\sigma (x_{2n-1},x_{2n}),\sigma (u,x_{2n}),\sigma (x_{2n-1},Au))\le 0.$

On the other hand,

$\frac{1}{s}\sigma (u,Au)\le \underset{n\to \mathrm{\infty }}{lim}\sigma (x_{2n-1},Au)\le s\sigma (u,Au).$

Taking $n\to \mathrm{\infty }$ and using the continuity of $F$, we get

$\underset{n\to \mathrm{\infty }}{limsup}F(\frac{1}{s}\sigma (u,Au),0,\sigma (u,Au),0,0,s\sigma (u,Au))\le 0.$

Using (G4), we obtain that $\sigma (u,Au)\le h_1(0)=0$ which yields that $u\in Au$.

Corollary 5.11. Let $(X,\sigma )$ be a complete $b$-metric like space and $A:X\to CB(X)$ be a multi-valued mapping such that there exists $F\in G_s$ such that

$F(\alpha (x,y)H_{\sigma }(Ax,Ay),\sigma (x,y),\sigma (x,Ax),\sigma (y,Ay),\sigma (x,Ay),\sigma (y,Ax))\le 0,$

for all $x,y\in X$. Suppose that

1. $(A,A)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $(X,\sigma )$ be double-sided Regular.

Then, there exists $u\in X$ such that $u\in Au$. We also have $\sigma (u,u)=0$.

Example 5.12. Let $X=[0,\mathrm{\infty })$ and $\sigma (x,y)=(x+y{)}^2$. It is easy to check that $(X,\sigma )$ is a complete b-metric like space with $s=2$. Define $A,B:X\to CB(X)$ by

$Ax=\left\{\begin{array}{cc}\left[0,\frac{x}{\sqrt{13}}\right],& ifx\in \left[0,1\right],\\ \left\{{(x-1)}^2+\frac{1}{9}\right\},& if\hspace{0.17em}x>1,\end{array}\right.$

$Bx=\left\{\begin{array}{cc}\left[0,\frac{x}{\sqrt{13}}\right],& ifx\in \left[0,1\right],\\ \left\{\frac{2x^2-1}{9}\right\},& if\hspace{0.17em}x>1,\end{array}\right.$

and

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

Then, for any $x,y\in \left[0,1\right]$, we have

$H_{\sigma }(Ax,By)=H_{\sigma }\left(\left[0,\frac{x}{\sqrt{13}}\right],\left[0,\frac{y}{\sqrt{13}}\right]\right)$

$=max\left\{\frac{x^2}{13},\frac{y^2}{13}\right\}$

$=\frac{1}{13}max\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax)\right\}.$

Therefore, we have

$\alpha (x,y)H_{\sigma }(Ax,By)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k=\frac{1}{13}\in [0,\frac{1}{12})=[0,\frac{1}{s^2+s^3})$. It is easy to check that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. There exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. (H) holds.

Thus, all of the conditions of Corollary (5.7) holds and so $A$ and $B$ have a common fixed point. Here, $0\in \left\{0\right\}=A0$ and $0\in \left\{0\right\}=B0$.

Corollary 5.13. Let $(X,\sigma )$ be a complete $b$-metric like space and $A,B:X\to CB(X)$ be two multi-valued mappings such that

$\alpha (x,y)H_{\sigma }(Ax,By)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{s^2+s^3})$. Suppose that

1. $(A,B)$ is a generalized $\alpha$-admissible pair,

2. there exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $(X,\sigma )$ be double-sided Regular.

Then, there exists $u\in X$ such that $u\in Au$ and $u\in Bu$.

Corollary 5.14. Let $(X,\sigma )$ be a complete $b$-metric like space and $A:X\to CB(X)$ be a multi-valued mapping such that

$\alpha (x,y)H_{\sigma }(Ax,Ay)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,Ay),\sigma (x,Ay),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{s^2+s^3})$. Suppose that

1. $(A,A)$ is a generalized $\alpha$-admissible pair,

2. there exist $x_0\in X$ and $x_1\in A{x}_0$ such that $\alpha (x_0,x_1)\ge 1$ and $\alpha (x_1,x_0)\ge 1$,

3. $(X,\sigma )$ be double-sided Regular.

Then, $A$ has a fixed point.

Corollary 5.15. Let $(X,\sigma )$ be a complete $b$-metric like space and $A,B:X\to CB(X)$ be two multi-valued mappings such that

$H_{\sigma }(Ax,By)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,By),\sigma (x,By),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{s^2+s^3})$. Then, there exists $u\in X$ such that $u\in Au$ and $u\in Bu$.

Corollary 5.16. Let $(X,\sigma )$ be a complete $b$-metric like space and $A:X\to CB(X)$ be a multi-valued mapping such that

$H_{\sigma }(Ax,Ay)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,Ay),\sigma (x,Ay),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{s^2+s^3})$. Then, $A$ has a fixed point.

Corollary 5.17. Let $(X,\sigma )$ be a complete metric like space and $A:X\to CB(X)$ be a multi-valued mapping such that

$H_{\sigma }(Ax,Ay)\le kmax\left\{\sigma (x,y),\sigma (x,Ax),\sigma (y,Ay),\sigma (x,Ay),\sigma (y,Ax)\right\},$

for all $x,y\in X$, where $k\in [0,\frac{1}{2})$. Then, $A$ has a fixed point.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Vahid Parvaneh

The research field of all authors of this paper is the fixed point theory with its applications. The authors have published some papers in this field in recent years. They are assistant professor and faculty member in Islamic Azad university. In this paper, the authors deal with existence theorems for common fixed points of multi-valued mappings in the framework of metric like spaces. Their obtained results generalize and extend some results in this field.