On some new generalizations of Nadler contraction in b-metric spaces

Abstract

We introduce some generalizations of contractions for multi-valued mappings and establish some fixed point theorems for multi-valued mappings in b -metric spaces. Our results generalize and extend several known results in b-metric spaces. Some examples are included which illustrate the cases when new results can be applied while old ones cannot.

Keywords:

• Fixed points
• b-metric spaces
• multi-valued mapping
• generalized Pompeiu-Hausdorff b -metric
• Nadler contraction

SUBJECTS:

• 47H10

PUBLIC INTEREST STATEMENT

Fixed point theory is one of applicable fields in nonlinear analysis. In this theory, researchers try to state and prove the existence and uniqueness of fixed points of mappings defined on different metric spaces, Banach spaces etc. One of the most famous results in this direction is Banach contraction principle which has been generalized in many directions. Nadler generalized it to a multi-valued mapping. His interesting work followed by many researchers.

1. Introduction and preliminaries

In the papers of Bakhtin (Bakhtin, 1989) and Czerwik (Czerwik, 1993, 1998), the notion of $b$-metric space has been introduced and some fixed point theorems for single-valued and multi-valued mappings in $b$-metric spaces have been proved. Successively, this notion has been reintroduced by Khamsi (Khamsi, 2010) and Khamsi and Hussain (Khamsi & Hussain, 2010), with the name of metric-type space. For several results in metric-type spaces or $b$-metric spaces, we refer the reader to (Aydi et al., 2012; Cosentino et al., 2014; Czerwik, 1993, 1998; Hussain & Mitrović, 2017; Jleli et al., 2015; Khamsi, 2010; Khamsi & Hussain, 2010; Latif et al., 2015; Miculescu & Mihail, 2017; Mitrović, 2019; Parvaneh et al., 2016; Phiangsungnoen & Kumam, 2015) and for more details on fixed point theory in other related spaces we refer the reader to (F. Vetro & Vetro, 2017)-(Vetro, 2018).

Definition 1.1. Let X be a nonempty set and $s\ge 1$ be a given real number. A function $d:X\times X\to [0,\mathrm{\infty })$ is said to be a $b$-metric with coefficient s if and only if for all $x,y,z\in X$ the following conditions are satisfied:

(1) $d(x,y)=0$ if and only if $x=y$;

(2) $d(x,y)=d(y,x)$;

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

A triplet $(X,d,s)$ is called a $b$-metric space.

Note that a metric space is included in the class of $b$-metric spaces.

The concept of convergence in such spaces is similar to that of standard metric spaces. The $b$-metric space $(X,d,s)$ is called complete if every Cauchy sequence of elements from $(X,d,s)$ is convergent. Some examples of $b$-metric spaces can be seen in (Aydi et al., 2012; Cosentino et al., 2014; Czerwik, 1993, 1998; Hussain & Mitrović, 2017; Jleli et al., 2015; Khamsi, 2010; Khamsi & Hussain, 2010; Miculescu & Mihail, 2017; Mitrović, 2019; Phiangsungnoen & Kumam, 2015).

Let $(X,d,s)$ be a $b$-metric space. Let $CB(X)$ be the collection of al nonempty closed bounded subsets of $X$ and $CL(X)$ be the class of all nonempty closed subsets of $X$. For each $x\in X$ and all $A,B\in CL(X)$, we define

$d(x,A)=\underset{a\in A}{inf}d(x,a)$

and

$D(A,B)=sup\{d(a,B):a\in A\}.$

Then the generalized Pompeiu-Hausdorff $b$-metric $H$ on $CL(X)$ induced by $d$ is defined as

$H(A,B)=\left\{\begin{array}{cc}max\{D(A,B),D(B,A)\},& \mathrm{i}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s};\\ +\mathrm{\infty },& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\right.$

for all $A,B\in CL(X)$. The following results are useful for some of the proofs in the paper.

Theorem 1.1. (Czerwik, 1998) If $(X,d,s)$ be a complete $b$-metric space, then $(CL(X),H)$, where $H$ means the Pompeiu-Hausdorff $b$-metric induced by $d$, is also a complete $b$-metric space.

Lemma 1.2. (Czerwik, 1998) Let $(X,d,s)$ be a $b$-metric space and $A,B\in CB(X)$. Then, for each $a\in A$ and $ϵ>0$ there exists $b\in B$ such that

$d(a,b)\le H(A,B)+ϵ.$

Lemma 1.3. (Czerwik, 1998) Let $(X,d,s)$ be a $b$-metric space. For any $A,B,C\in CL(X)$ and all $x,y\in X$, we have the following:

(1) $d(x,A)\le d(x,a)$ for all $a\in A$,

(2) $d(x,B)\le H(A,B)$ for all $x\in A,$

(3) $d(x,A)\le s[d(x,y)+d(y,A)]$,

(4) $H(A,C)\le s[H(A,B)+H(B,C)],$

(5) $d(x,A)=0$ if and only if $x\in A.$

In (Miculescu & Mihail, 2017), the following result was obtained (see also (Mitrović, 2019)).

Lemma 1.4. (Miculescu & Mihail, 2017) Each sequence $(x_n{)}_{n\in \mathbb{N}}$ of elements from a $b$-metric space$(X,d,s)$ having the property that there exists $\gamma \in [0,1)$ such that

$d(x_{n+1},x_n)\le \gamma d(x_n,x_{n-1}),$

for every $n\in \mathbb{N}$, is Cauchy.

Lemma 1.5. (Aghajani et al., 2014) Let $(X,d,s)$ be a $b$-metric space and suppose that $(x_n{)}_{n\in \mathbb{N}}$ and $(y_n{)}_{n\in \mathbb{N}}$ converge to $x,y\in X$, respectively. Then, we have

$\frac{1}{s^2}d(x,y)\le \underset{n\to \mathrm{\infty }}{lim}infd(x_n,y_n)\le \underset{n\to \mathrm{\infty }}{lim}supd(x_n,y_n)\le s^{2}d(x,y).$

In particular, if $x=y$, then $\underset{n\to \mathrm{\infty }}{lim}d(x_n,y_n)=0$. Moreover, for each $z\in X$, we have

$\frac{1}{s}d(x,z)\le \underset{n\to \mathrm{\infty }}{lim}infd(x_n,z)\le \underset{n\to \mathrm{\infty }}{lim}supd(x_n,z)\le sd(x,z).$

Definition 1.2. (Miculescu & Mihail, 2017) Let $(X,d,s)$ be a $b$-metric space. A mapping $T:X\to CB(X)$ is called closed if for all sequences $(x_n{)}_{n\in \mathbb{N}}$ and $(y_n{)}_{n\in \mathbb{N}}$ of elements from $X$ and $x,y\in X$ such that $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x$, $\underset{n\to \mathrm{\infty }}{lim}{y}_n=y$ and $y_n\in T(x_n)$ for each $n\in \mathbb{N}$, we have $y\in T(x).$

Definition 1.3. (Miculescu & Mihail, 2017) Let $(X,d,s)$ be a $b$-metric space. The $b$-metric $d$ is called $\ast$-continuous if for all $A\in CB(X)$, all $x\in X$ and each sequence $(x_n{)}_{n\in \mathbb{N}}$ of elements from $X$ such that $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x$, we have $\underset{n\to \mathrm{\infty }}{lim}d(x_n,A)=d(x,A)$.

2. The generalizations of Nadler contraction for multi-valued mappings

In this section, we introduce the following condition of contractions for multi-valued mappings in metric spaces:

(1) $H(Tx,Ty)\le N_{a,k,b,c}(x,y)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in X,$(1)

where

$N_{a,k,b,c}(x,y)=amax\{d(x,y),kd(x,Tx),kd(y,Ty)\}+bd(x,Ty)+cd(y,Tx),$

and $a,b,c,k\ge 0$ such that $k\in [0,1]$ and $a+2min\{b,c\}<1$.

We shall present some most famous examples of multi-valued contraction mappings in metric spaces:

(1) Banach type contraction, Nadler (Nadler, 1969):

(2) $H(Tx,Ty)\le \alpha d(x,y),\alpha \in [0,1);$(2)

(2) Multi-valued weak contraction, Berinde (Berinde & Berinde, 2007) (see also (Hussain & Mitrović, 2017)):

(3) $H(Tx,Ty)\le \theta d(x,y)+Ld(y,Tx),\theta \in [0,1),L\ge 0;$(3)

(3) Kannan type contraction (Kannan, 1968):

(4) $H(Tx,Ty)\le \lambda [d(x,Tx)+Ld(y,Ty)],\lambda \in [0,\frac{1}{2});$(4)

(4) Reich type contraction (Reich, 1971):

(5) $H(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty),$(5)

$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma <1;$

(5) Chatterjea type contraction (Chatterjea, 1972):

(6) $H(Tx,Ty)\le \lambda [d(x,Ty)+d(y,Tx)],\lambda \in [0,\frac{1}{2});$(6)

(6) Ćirić type contraction (L. B. Ćirić, 1871):

(7) $H(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty)+\delta d(x,Ty)+\delta d(y,Tx),$(7)

$\alpha ,\beta ,\gamma ,\delta \ge 0,\alpha +\beta +\gamma +2\delta <1;$

(7) Hardy-Rogers type contraction (Hardy & Rogers, 1973):

(8) $H(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty)+\delta d(x,Ty)+\mu d(y,Tx),$(8)

$\alpha ,\beta ,\gamma ,\delta ,\mu \ge 0,\alpha +\beta +\gamma +\delta +\mu <1;$

(8) Ćirić quasi-contraction (Lj. B. Ćirić, 1974):

(9) $H(Tx,Ty)\le kmax\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\},$(9)

$k\in [0,1).$

We have the following diagram where arrows stand for inclusions in metric spaces.

$\begin{array}{ccccccccc}\hspace{0.17em}& \hspace{0.17em}& (1)& \to & (3)& \to & (2)& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \downarrow & \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \uparrow & \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& (8)& \to & (7)& \to & (5)& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \uparrow & \hspace{0.17em}& \downarrow & \hspace{0.17em}& \downarrow & \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& (9)& \hspace{0.17em}& (6)& \hspace{0.17em}& (4)& \hspace{0.17em}& \hspace{0.17em}\end{array}$

Remark 2.1. For a presentation and comparison of such kind of contractive conditions and fixed point theorems, see (Rhoades, 1977, 1983).

Again as in (Lj. B. Ćirić, 1974), Aydi et all (Aydi et al., 2012) (Theorem 2.2.) introduced the q-set-valued quasi-contraction in complete $b$-metric space. The multi-valued map $T:X\to CB(X)$ is said to be a q-multi-valued quasi-contraction if

(10) $H(Tx,Ty)\le kM(x,y),$(10)

for any $x,y\in X$, where $0\le k<\frac{1}{s^2+s}$ and

$M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}.$

Recently, Miculescu and Mihail (Miculescu & Mihail, 2017) (Theorem 3.3.) used the following version of q-set-valued quasi-contraction in complete $b$-metric spaces. Let $T:X\to CB(X)$ having the property that there exist $c,d\in [0,1]$ and $\alpha \in [0,1)$ such that $max\{\alpha cs,\alpha ds\}<1$ and

(11) $H(T(x),T(y))\le \alpha N_{c,d}(x,y),$(11)

for all $x,y\in X$, where,

$N_{c,d}(x,y)=max\{d(x,y),cd(x,T(x)),cd(y,T(y)),\frac{d}{2}(d(x,T(y))+d(y,T(x)))\}.$

We introduced the following condition for mapping $T:X\to CB(X)$ in $b$-metric space with coefficient s.

(12) $H(Tx,Ty)\le N_{a,k,b,c}(x,y),$(12)

for all $x,y\in X$, where $a,b,c,k\ge 0$ such that $k\in [0,1]$ and $a+2smin\{b,c\}<1$.

The following example shows that in $b$-metric spaces a condition of contraction (12) may not be a q-quasi-contraction in the sense of Aydi et all and may not be a q-quasi-contraction in the sense of Miculesu and Mihail.

Example 2.1. Let $X=\mathbb{R}$, $d(x,y)=(x-y{)}^2$ for all $x,y\in X$ and $T:X\to X$ be defined by $Tx=\{x\}.$ We obtain that $d$ is a $b$-metric (with $s=2$), but $(X,d)$ is not a metric space. For $x=0,y=1$ and $z=2$, we have

$d(x,z)=4>2=d(x,y)+d(y,z).$

Recall that, for all $x,y\in X$,

$(x-y{)}^2=H(Tx,Ty)$

$\le amax\{d(x,y),kd(x,Tx),kd(y,Ty)\}+bd(x,Ty)+cd(y,Tx)$

$=(a+b+c)(x-y{)}^2$

$=(x-y{)}^2.$

For $a=b=\frac{1}{6}$ and $c=\frac{2}{3},$ we have $a+2smin\{b,c\}<1$ and $T$ satisfied condition (12). Suppose that $T$ is a q-quasi-contraction in the sense of Aydi et al. Thus there exists $\alpha \in [0,1)$ such that for all $x,y\in X$,

$(x-y{)}^2=H(Tx,Ty)$

$\le \alpha max\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}$

$=\alpha (x-y{)}^2$

$<(x-y{)}^{2}ifx\ne y.$

This is a contradiction. Similarly, suppose that $T$ is a q-quasi-contraction in the sense of Miculescu and Mihail. Thus there exist $c,d\in [0,1]$ and $\alpha \in [0,1)$ such that

(i) $\alpha ds<1$;

(ii) $H(T(x),T(y))\le \alpha N_{c,d}(x,y)$ for all $x,y\in X$. So,

$(x-y{)}^2=H(Tx,Ty)$

$\le \alpha max\{d(x,y),cd(x,Tx),cd(y,Ty),\frac{d}{2}(d(x,Ty)+d(y,Tx))\}$

$=\alpha max\{(x-y{)}^2,d(x-y{)}^2\}$

$<(x-y{)}^{2}ifx\ne y.$

This is a contradiction.

The aim of this paper is to obtain sufficient conditions for the existence of fixed point for the multi-valued mappings satisfied condition (12) in $b$-metric spaces.

To prove the main results of this paper, we shall need the following easy lemma.

Lemma 2.1. Let

$d_2\le amax\{d_1,d_2\}+bs(d_1+d_2),$

where $a,b,s,d_1$ and $d_2$ are nonnegative real numbers such that $a+2bs<1$. Then we have

$d_2\le max\left\{\frac{bs}{1-(a+bs)},\frac{a+bs}{1-bs}\right\}{d}_1.$

3. Main results

The following theorem is our main result, which can be regarded as an extension of Nadler’s fixed point theorem in $b$-metric spaces.

Theorem 3.1. Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping satisfying:

(13) $H(Tx,Ty)\le N_{a,k,b,c}(x,y)$(13)

for all $x,y\in X$, where $k\in [0,1]$, $a,b,c\ge 0,$ with $a+2smin\{b,c\}<1$.

Then there exists a sequence $(x_n{)}_{n\in \mathbb{N}}$ in $X$ converges to some point $x^{\ast }\in X$ such that $x_{n+1}\in T(x_n)$ for every $n\in \mathbb{N}$. Also, $x^{\ast }$ is a fixed point of $T$ if any of the following conditions are satisfied:

(i) $T$ is closed;

(ii) $d$ is $\ast$-continuous;

(iii) $s(ak+min\{b,c\})<1.$

Proof. Let $x_0\in X.$ Choose $x_1\in T{x}_0.$ Let

$ϵ=\frac{1-q}{1+q}H(T{x}_1,T{x}_0),$

where $q=a+2smin\{b,c\}.$ If $H(T{x}_1,T{x}_0)=0$, then we obtain $T{x}_1=T{x}_0$ and $x_1\in T{x}_1$. In this case proof is hold. So, we may assume $ϵ>0$. From Lemma 1.2, then there is a point $x_2\in T{x}_1$ such that

$d(x_1,x_2)\le H(T{x}_0,T{x}_1)+ϵ=\frac{2}{1+q}H(T{x}_0,T{x}_1).$

Similarly, there is a point $x_3\in T{x}_2$ such that

$d(x_2,x_3)\le H(T{x}_1,T{x}_2)+ϵ,$

where

$ϵ=\frac{1-q}{1+q}H(T{x}_2,T{x}_1).$

If $H(T{x}_2,T{x}_1)=0$, then we infer that $T{x}_2=T{x}_1$ and $x_2\in T{x}_2$. In this case proof is hold. So, we may assume that $ϵ>0$. Hence,

$d(x_2,x_3)\le \frac{2}{1+q}H(T{x}_1,T{x}_2).$

Continuing in this process, we produce a sequence $(x_n{)}_{n\in \mathbb{N}}$ of points of $X$ such that

(14) $x_{n+1}\in T{x}_n,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{N},$(14)

and

(15) $d(x_n,x_{n+1})\le \frac{2}{1+q}H(T{x}_{n-1},T{x}_n)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}n\in \mathbb{N}.$(15)

1. Suppose that $c=min\{b,c\}$. From condition (13), we obtain

$H(T{x}_n,T{x}_{n-1})\le amax\{d(x_n,x_{n-1}),kd(x_n,T{x}_n),kd(x_{n-1},T{x}_{n-1})\}$

$+bd(x_n,T{x}_{n-1})+cd(x_{n-1},T{x}_n)$

$\le amax\{d(x_n,x_{n-1}),kd(x_n,x_{n+1}),kd(x_{n-1},x_n)\}$

$+bd(x_n,x_n)+cd(x_{n-1},x_{n+1})$

$=amax\{d(x_n,x_{n-1}),kd(x_n,x_{n+1})\}$

$+cd(x_{n-1},x_{n+1})$

$\le amax\{d(x_n,x_{n-1}),kd(x_n,x_{n+1})\}$

$+cs[d(x_{n-1},x_n)+d(x_n,x_{n+1})]$

It follows from (15) that

$d(x_n,x_{n+1})\le \frac{2}{1+q}[amax\{d(x_n,x_{n-1}),d(x_n,x_{n+1})\}$

$+cs(d(x_{n-1},x_n)+d(x_n,x_{n+1}))].$

Now, since

$\frac{2}{1+q}(a+2cs)=\frac{2q}{1+q}<1,$

and from Lemma 2.1, we obtain

(16) $d(x_{n+1},x_n)\le q_{1}d(x_n,x_{n-1}),$(16)

where $q_1=max\left\{\frac{q+a}{1+a},\frac{q-a}{1-a}\right\}<1.$

2. Suppose that $b=min\{b,c\}$. Similarly, as in 1, we obtain

$H(T{x}_{n-1},T{x}_n)\le amax\{d(x_{n-1},x_n),kd(x_n,x_{n+1})\}+bs[d(x_{n-1},x_n)+d(x_n,x_{n+1})].$

Again, it follows from (15) that

$d(x_n,x_{n+1})\le \frac{2}{1+q}[amax\{d(x_{n-1},x_n),d(x_n,x_{n+1})\}+bs(d(x_{n-1},x_n)+d(x_n,x_{n+1}))].$

Now, since

$\frac{2}{1+q}(a+2bs)=\frac{2q}{1+q}<1$

and from Lemma 2.1, we obtain

(17) $d(x_{n+1},x_n)\le q_{2}d(x_n,x_{n-1}),$(17)

where $q_2=max\left\{\frac{q+a}{1+a},\frac{q-a}{1-a}\right\}<1.$

Now, since $q_1<1$ and $q_2<1$, Lemma 1.4 together with (16) and (17) imply that the sequence $(x_n{)}_{n\in \mathbb{N}}$ is a Cauchy sequence. Since $(X,d,s)$ is complete, the sequence $(x_n{)}_{n\in \mathbb{N}}$ converges to some point $x^{\ast }\in X$.

(i) Suppose that $T$ is closed. From Definition 1.2 and (14) we have $x^{\ast }\in T{x}^{\ast }.$

(ii) Suppose that $d$ is $\ast$-continuous. Then, we have

(18) $\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}^{\ast })=d(x^{\ast },T{x}^{\ast }).$(18)

From Lemma 1.3 (2.) and (13) we have

$d(x_{n+1},T{x}^{\ast })\le H(T{x}_n,T{x}^{\ast })$

$\le amax\{d(x_n,x^{\ast }),kd(x_n,T{x}_n),kd(x^{\ast },T{x}^{\ast })\}$

$+bd(x_n,T{x}^{\ast })+cd(x^{\ast },T{x}_n)$

$\le amax\{d(x_n,x^{\ast }),kd(x_n,x_{n+1}),kd(x^{\ast },T{x}^{\ast })\}$

$+bd(x_n,T{x}^{\ast })+cd(x^{\ast },x_{n+1})$

and

$d(T{x}^{\ast },x_{n+1})\le H(T{x}^{\ast },T{x}_n)$

$\le amax\{d(x^{\ast },x_n),kd(x^{\ast },T{x}^{\ast }),kd(x_n,T{x}_n)\}$

$+bd(x^{\ast },T{x}_n)+cd(x_n,T{x}^{\ast })$

$\le amax\{d(x^{\ast },x_n),kd(x^{\ast },T{x}^{\ast }),kd(x_n,x_{n+1})\}+$

$+bd(x^{\ast },x_{n+1})+cd(x_n,T{x}^{\ast }).$

Hence, using (18), we obtain

(19) $d(x^{\ast },T{x}^{\ast })\le (ak+b)d(x^{\ast },T{x}^{\ast })$(19)

and

(20) $d(T{x}^{\ast },x^{\ast })\le (ak+c)d(x^{\ast },T{x}^{\ast }).$(20)

Since $ak+b<a+2smin\{b,c\}$, or, $ak+c<a+2smin\{b,c\}$ we conclude that $d(x^{\ast },T{x}^{\ast })=0$ and from Lemma 1.3 (5.) we obtain $x^{\ast }\in T{x}^{\ast }.$

(iii) (a) Let $s(ak+b)<1.$

Since

(21) $\begin{array}{cc}d(x_{n+1},T{x}^{\ast })& \le amax\{d(x_n,x^{\ast }),kd(x_n,x_{n+1}),kd(x^{\ast },T{x}^{\ast })\}\\ \hspace{0.17em}& +bd(x_n,T{x}^{\ast })+cd(x^{\ast },x_{n+1})\end{array}$(21)

and

(22) $d(x_n,T{x}^{\ast })\le s[d(x_n,x_{n+1})+d(x_{n+1},T{x}^{\ast })],$(22)

we have

(23) $\begin{array}{cc}(1-sb)d(x_{n+1},T{x}^{\ast })& \le amax\{d(x_n,x^{\ast }),kd(x_n,x_{n+1}),kd(x^{\ast },T{x}^{\ast })\}\\ \hspace{0.17em}& +bsd(x_n,x_{n+1})+cd(x^{\ast },x_{n+1}).\end{array}$(23)

Since $1-sb>0$, from (22), (23) and triangle inequalities we obtain

$d(x^{\ast },T{x}^{\ast })$

$\le sd(x^{\ast },x_{n+1})+\frac{s}{1-sb}[amax\{d(x_n,x^{\ast }),kd(x_n,x_{n+1}),kd(x^{\ast },T{x}^{\ast })\}$

$+sbd(x_n,x_{n+1})+c(x^{\ast },x_{n+1})].$

Taking limit as $n\to \mathrm{\infty }$, we obtain

(24) $d(x^{\ast },T{x}^{\ast })\le \frac{ask}{1-sb}d(x^{\ast },T{x}^{\ast }).$(24)

Since $\frac{ask}{1-sb}<1$ (see (iii)) and from (24), we conclude that $d(x^{\ast },T{x}^{\ast })=0,$ i.e. $x^{\ast }\in T{x}^{\ast }$.

(b) Let $s(ak+c)<1.$ Then proof is done similarly as in (a).

4. Some consequences

We shall present some applications of Theorem 3.1 in $b$-metric spaces.

Corollary 4.1. (Version of Nadler’s fixed point theorem in $b$-metric spaces, (Nadler, 1969)) Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping satisfying:

(25) $H(Tx,Ty)\le \alpha d(x,y)$(25)

for all $x,y\in X$, where $\alpha \in [0,1).$ Then $T$ has a fixed point.

Proof. In Theorem 3.1, set $k=b=c=0.$

Theorem 4.1 improves the next result by Czerwik (Czerwik, 1998) and by Phiangsungnoen and Kumam (Phiangsungnoen & Kumam, 2015).

Corollary 4.2. Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping satisfying:

(26) $H(Tx,Ty)\le \lambda d(x,y)$(26)

for all $x,y\in X$, where $\lambda \in [0,\frac{1}{s}).$ Then $T$ has a fixed point.

Example 4.1. Let $X=[1,+\mathrm{\infty })$ be equipped with the complete $b$-metric $d(x,y)=(x-y{)}^2$ for all $x,y\in X$ (with coefficient $s=2$). Define $T:X\to CB(X)$ by $Tx=[1,1+\frac{4x}{5}]$ for all $x\in X$. Also, take $\alpha =\frac{9}{16}.$ We have,

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

for all $x,y\in X$, that is (25) holds. All hypotheses of Theorem 4.1 are satisfied and $x=1$ is a fixed point of $T$.

On the other hand, Corollary 4.2 is not applicable. For $x=2$ and $y=1$, we have $H(Tx,Ty)=\frac{16}{25},d(x,y)=1$. So,

$H(Tx,Ty)>\lambda d(x,y)forall\lambda \in [0,\frac{1}{2}).$

Also, we could not apply the main result of Aydi et al (Aydi et al., 2012) (Theorem 2.2). Again, for $x=2$ and $y=1$, we have

$d(x,Tx)=0,d(y,Ty)=0,d(x,Ty)=0,d(y,Tx)=0.$

So,

$H(Tx,Ty)>\lambda max\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}$

for all $\lambda \in [0,\frac{1}{s^2+s}).$

Corollary 4.3. (Version of fixed point theorem of Berinde (Berinde & Berinde, 2007) in $b$-metric spaces) Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping for which there exist $\theta \in (0,1)$ and $L\ge 0$ such that

(27) $H(Tx,Ty)\le \theta d(x,y)+Ld(y,Tx)$(27)

for all $x,y\in X.$ Then $T$ has a fixed point.

Proof. In Theorem 3.1, set $a=\theta ,k=0,b=0,c=L.$

Example 4.2. Let $T:[0,1]\to [0,1]$ defined by $Tx=\{x\}$ for all $x\in [0,1]$. Then

(i) $T$ does not satisfy the contractive condition (10) of Aydi et all (Aydi et al., 2012).

(ii) $T$ does not satisfy the contractive condition (11) of Miculescu and Mihail (Miculescu & Mihail, 2017).

(iii) $T$ satisfies condition (28) with arbitrary $\theta \in (0,1)$ and $L\ge 1-\theta .$

Corollary 4.4.(Version of fixed point theorem of Hussain and Mitrović (Hussain & Mitrović, 2017) in $b$-metric spaces) Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping for which there exist $\theta \in (0,1),$ $k\in [0,1)$ and $L\ge 0$ such that

(28) $H(Tx,Ty)\le \theta max\{d(x,y),kd(x,Tx),kd(y,Ty)\}+Ld(y,Tx),$(28)

for all $x,y\in X$. Then $T$ has a fixed point if any of the following conditions are satisfied:

(i) T is closed;

(ii) d is $\ast$-continuous;

(iii) $s\theta k<1$.

Proof. In Theorem 3.1, set $a=\theta ,b=0,c=L.$

Corollary 4.5. (Version of fixed point theorem of Kannan (Kannan, 1968) in $b$-metric spaces) Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping for which there exist $\lambda \in (0,\frac{1}{2s})$ such that

(29) $H(Tx,Ty)\le \lambda [d(x,Tx)+d(y,Ty)],forallx,y\in X.$(29)

Then $T$ has a fixed point.

Proof. By condition (29) and triangle rule, we get

$H(Tx,Ty)\le \lambda [d(x,Tx)+d(y,Ty)]$

$\le \lambda [s(d(x,y)+d(y,Tx))+s(d(y,Tx)+H(Tx,Ty)]$

which yields

$(1-\lambda s)H(Tx,Ty)\le \lambda sd(x,y)+2\lambda sd(y,Tx)$

which implies

$H(Tx,Ty)\le \frac{\lambda s}{1-\lambda s}d(x,y)+\frac{2\lambda s}{1-\lambda s}d(y,Tx),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in X,$

i.e., in view of $0\le \lambda <\frac{1}{2s}$, (28) holds with $\theta =\frac{\lambda s}{1-\lambda s}$ and $L=\frac{2\lambda s}{1-\lambda s}.$

Corollary 4.6. (Version of fixed point theorem of Chatterjea (Chatterjea, 1972) in $b$-metric spaces) Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping for which there exists $\lambda \in (0,\frac{1}{s^2+s})$ such that

(30) $H(Tx,Ty)\le \lambda [d(x,Ty)+d(y,Tx)]$(30)

for all $x,y\in X.$ Then $T$ has a fixed point.

Proof. By (30) and triangle rule we have

$H(Tx,Ty)\le \lambda \{s[d(x,y)+s(d(y,Tx)+H(Tx,Ty))]+\lambda d(y,Tx)\}$

$\le \lambda \{sd(x,y)+\lambda (s^2+1)d(y,Tx)+\lambda s^{2}H(Tx,Ty)\}.$

After simple computations we get,

$H(Tx,T,y)\le \frac{\lambda s}{1-\lambda s^2}d(x,y)+\frac{\lambda (s^2+1)}{1-\lambda s^2}d(y,Tx),$

which is (28)), with $\theta =\frac{\lambda s}{1-\lambda s^2}$ (since $\lambda <\frac{1}{s+s^2})$ and $L=\frac{\lambda (s^2+1)}{1-\lambda s^2}\ge 0.$

Corollary 4.7 .(Generalizations of Reich’s theorem in $b$-metric spaces)Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping satisfying:

(31) $H(Tx,Ty)\le \alpha max\{d(x,y),d(x,Tx),d(y,Ty)\},$(31)

for all $x,y\in X$, where $\alpha \in [0,\frac{1}{s}).$ Then $T$ has a fixed point.

Proof. In Theorem 3.1, set $a=\alpha ,k=1,b=c=0.$

Corollary 4.8. (Version of fixed point theorem of Gordji et al (Gordji et al., 2010) in $b$-metric spaces)Let $(X,d,s)$ be a complete $b$-metric space and $T:X\to CB(X)$ be a mapping satisfying:

(32) $H(Tx,Ty)\le \alpha d(x,y)+\beta [d(x,Tx)+d(y,Ty)]+\gamma [d(x,Ty)+d(y,Tx)]$(32)

for all $x,y\in X$, where $\alpha ,\beta ,\gamma \ge 0$, $\alpha +2smin\{\beta ,\gamma \}<1$ and $\alpha +2\beta +2\gamma <\frac{1}{s}.$ Then $T$ has a fixed point.

Proof. Put in Theorem 3.1 $a=\alpha +2\beta ,k=1,b=c=\gamma .$

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgements

The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Disclosure statement

The authors declare that they have no competing interests regarding the publication of this paper.

Additional information

Funding

This research received no external funding.

Notes on contributors

Vahid Parvaneh

Vahid Parvaneh is graduated in mathematical analysis. He works as e faculty member in Islamic Azad University. He started the research in fixed point theory and its applications from 2011. Also, he interests in measure of noncompactness and geometry of Banach spaces. Until now, he published almost 100 papers in quality journals.