On a Version of Dontchev and Hager’s Inverse Mapping Theorem

Abstract

By revisiting an open question raised by Kirk and Shahzad, we are able to prove a generalized version of Nadler’s fixed-point theorem in the context of strong b-metric space. Such a result leads us to prove a new version of Dontchev and Hager’s inverse mapping theorem. Some examples are provided to illustrate the results.

1. Introduction and Preliminary Assertions

In recents decades, the concept of b-metric space (bMS) (quasi-metric space) has appeared as a generalization of metric space (MS). This notion was introduced in 1974 by Bourbaki [1] and it was given again in 1989 by Bakhtin [2]. Czerwik extended the concept of an MS to a bMS, aiming to generalize the Banach fixed-point theorem [3,4]. Numerous generalizations of b-metric spaces have appeared in the state-of-the-art literature. Examples include rectangular b-metric spaces ([5]), graphical rectangular b-metric spaces, graphical extended b-metric spaces, bipolar metric spaces, and b-metric spaces. Among these, the b-metric space has played a crucial role in developing generalizations and applications of the Banach contraction principle. Several results have been proven for fixed-point theorems for single-valued and set-valued mappings in bMS; see, for example, [6,7,8,9,10,11,12,13] and references therein.

Definition 1.

Let $\mathrm{\Xi }\ne \varnothing$ be a set and $s\ge 1.$ Let ł$:\mathrm{\Xi }\times \mathrm{\Xi }\to [0,+\infty )$ be a distance function satisfying for each $x,y$ and $z\in \mathrm{\Xi }:$

bM1.

ł$(x,y)\ge 0,$

bM2.

ł$(x,y)=0\iff x=y,$

bM3.

ł$(x,y)=$ ł$(y,x),$

bM4.

ł$(x,z)\le s[$ ł$(x,y)+$ ł$(y,z)]$   ($s–$relaxed_t inequality).

Then, the pair $(\mathrm{\Xi },$ ł) is called a bMS.

The fact that the b-metric function need not be continuous and not all open balls in a b-metric space are open sets (see [14]) prompted Kirk and Shahzad in [6] to enhance $s–$relaxed_t inequality to a stronger inequality as follows: ł$(x,z)\le$ ł$(x,y)+s$ ł$(y,z).$ They introduced a special class of b-metric spaces called strong b-metric space.

Definition 2

([6]). Let $\mathrm{\Xi }\ne \varnothing$ be a set and $s\ge 1.$ Let ł$:\mathrm{\Xi }\times \mathrm{\Xi }\to [0,+\infty )$ be a distance function satisfying for each $x,y$ and $z\in \mathrm{\Xi }:$

SbM1.

ł$(x,y)\ge 0,$

SbM2.

ł$(x,y)=0\iff x=y,$

SbM3.

ł$(x,y)=$ ł$(y,x),$

SbM4.

ł$(x,z)\le$ ł$(x,y)+s$ ł$(y,z).$

Then, the pair $(\mathrm{\Xi },$ ł) is called a strong b-metric space $\left(SbMS\right).$

The class of strong b-metric spaces is intermediate between the class of b-metric spaces and the class of metric spaces. This demonstrates the significance of strong b-metric spaces over b-metric spaces, as many well-known fixed-point results that are valid in strong b-metric space do not fully hold in b-metric space. We mentioned that some fixed-point theorems were established in the context of strong b-metric spaces. For example, Ćirić’s and Kannan’s type contraction for single-valued mappings were proved in [15,16], and Nadler’s and Chatterjea’s fixed-point theorems for set-valued mappings were proved in [17].

Example 1.

Let $\mathrm{\Xi }=\{1,2,3\}$ and the distance function ł$:\mathrm{\Xi }\times \mathrm{\Xi }\to \mathbb{R}$ such that ł$(x,y)=$ ł$(y,x),$ for each $x,y\in \mathrm{\Xi }$ and ł$(2,1)=$ ł$(3,2)=2$ and ł$(3,1)=6.$ Then, $(\mathrm{\Xi },$ ł) is a bMS with $s=\frac{3}{2}$ and an SbMS with $s=2$, but it is not an MS since the triangle inequality does not hold.

Example 2.

Let $\mathrm{\Xi }=[1,+\infty )$ and define for every $x,y\in \mathrm{\Xi },$

ł$(x,y)=max\left\{\right|x-y|,2|x-y|-1\}.$ Then,

SbM1.

ł$(x,y)\ge 0,$

SbM2.

If ł$(x,y)=0$, then $0\le |x-y|\le$ ł$(x,y)=0;$ this implies $x=y.$

Conversely, if $x=y\Rightarrow |x-y|=0\Rightarrow$ ł$(x,y)=0,$

SbM3.

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,y)& =& max\left\{\right|x-y|,2|x-y|-1\}\hfill \\ & =& max\left\{\right|y-x|,2|y-x|-1\}\hfill \\ & =& \mathit{ł}(y,x)\hfill \end{array}$$

SbM4.

Let $x,y,z\in \mathrm{\Xi }$; when ł$(x,z)=|x-z|$, this implies that

$$\begin{array}{ccc}\hfill |x-z|& \le & |x-y|+|y-z|\hfill \\ & \le & \mathsf{ł}(x,y)+\mathsf{ł}(y,z)\hfill \\ & \le & \mathsf{ł}(x,y)+2\mathsf{ł}(y,z).\hfill \end{array}$$

On the other hand, when $\mathsf{ł}(x,z)=2|x-z|-1$, then we have four cases:

i.

If ł$(x,y)=|x-y|$ and ł$(y,z)=|y-z|$ then $|x-y|-1\le 0$ and $|y-z|-1\le 0.$

One obtains

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)=2|x-z|-1& \le & |x-y|+2|y-z|+\left(\right|x-y|-1)\hfill \\ & \le & |x-y|+2|y-z|\hfill \\ & \le & \mathsf{ł}(x,y)+2\mathsf{ł}(y,z).\hfill \end{array}$$

Also,

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)=2|x-z|-1& \le & 2|x-y|+|y-z|+\left(\right|y-z|-1)\hfill \\ & \le & 2|x-y|+|y-z|\hfill \\ & \le & 2\mathsf{ł}(x,y)+\mathsf{ł}(y,z).\hfill \end{array}$$

ii.

If ł$(x,y)=|x-y|$ and ł$(y,z)=2|y-z|-1,$ then we have

$$\mathsf{ł}(x,y)=|x-y|\le 1\le \mathsf{ł}(y,z)=2|y-z|-1.$$

Hence,

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)& \le & 2|x-z|+\left(2\right|y-z|-1)\hfill \\ & \le & 2\mathsf{ł}(x,y)+\mathsf{ł}(y,z).\hfill \end{array}$$

Also,

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)& \le & 2|x-y|+\left(2\right|y-z|-1)\hfill \\ & =& |x-y|+|x-y|+\mathsf{ł}(y,z)\hfill \\ & \le & \mathsf{ł}(x,y)+2\mathsf{ł}(y,z).\hfill \end{array}$$

iii.

If $\mathsf{ł}(x,y)=2|x-y|-1$ and $\mathsf{ł}(y,z)=|y-z|,$ it is similar to case ii.

iv.

If ł$(x,y)=2|x-x|-1$ and ł$(y,z)=2|y-z|-1$, then

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)& \le & \left(2\right|x-y|-1)+\left(2\right|y-z|-1)+1\hfill \\ & \le & 2\mathsf{ł}(x,y)+\mathsf{ł}(y,z).\hfill \end{array}$$

Also,

$$\begin{array}{ccc}\hfill \mathsf{ł}(x,z)& \le & \mathsf{ł}(x,y)+2\mathsf{ł}(y,z).\hfill \end{array}$$

Hence, $(\mathrm{\Xi },\mathsf{ł})$ is an SbMS with $s=2.$ It is not MS since the triangle inequality does not hold when $x=2,y=2.5,$ and $z=6.$

In [18], Dontchev and Hager introduced an important generalization of Nadler’s theorem and used it to prove an inverse mapping theorem. They utilized the concept of the “excess” between two sets in a metric space. We invoke some useful definitions that will be needed in this manuscript.

Definition 3.

Let $(\mathrm{\Omega },\rho )$ be a metric space. Let ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ be subsets of $\mathrm{\Omega },\omega \in \mathrm{\Omega };$ then, the distance from a point ω to a set ${\mathrm{\Lambda }}_1$ and the excess between two sets are given, respectively, as follows:

i.

$D(\omega ,{\mathrm{\Lambda }}_1)=inf\{\rho (\omega ,{\lambda }_1):{\lambda }_1\in {\mathrm{\Lambda }}_1\};$

ii.

The excess δ from ${\mathrm{\Lambda }}_1$ to ${\mathrm{\Lambda }}_2$ is

$\delta ({\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_1)=sup\{D({\lambda }_2,{\mathrm{\Lambda }}_1):{\lambda }_2\in {\mathrm{\Lambda }}_2\}$ where $D({\lambda }_2,{\mathrm{\Lambda }}_1)=inf\{\rho ({\lambda }_2,{\lambda }_1):{\lambda }_1\in {\mathrm{\Lambda }}_1\}.$

Definition 4.

Let $(\mathrm{\Gamma },d)$ be a linear metric space; d is invariant if $d(\mu +\zeta ,\nu +\zeta )=d(\mu ,\nu )$ for every $\mu ,\nu ,\zeta \in \mathrm{\Gamma }.$

Definition 5.

Let $(\mathrm{\Omega },\rho )$ and $(\mathrm{\Gamma },d)$ be metric spaces, then

i.

The single-valued $f:\mathrm{\Omega }\to \mathrm{\Gamma }$, which satisfies: $\forall \epsilon >0,\exists \eta >0$ such that for every ${\omega }_1,{\omega }_2\in B[{\omega }_0;\eta ]$,

$$d(f\left({\omega }_1\right),f\left({\omega }_2\right))\le \epsilon \rho ({\omega }_1,{\omega }_2)$$

is called strictly stationary (ss) at ${\omega }_0,$ where $B[{\omega }_0;\eta ]$ is the closed ball centered at ${\omega }_0$ with radius $\eta ;$

ii.

Let $T:\mathrm{\Omega }\to \mathrm{\Gamma }$ be a set-valued map. The graph T is the set $\left\{\right(\omega ,\zeta )\in \mathrm{\Omega }\times \mathrm{\Gamma }:\zeta \in T(\omega \left)\right\}$ and $T^{-1}\left(\zeta \right)=\{\omega \in \mathrm{\Omega }:\zeta \in T\left(\omega \right)\};$

Finally, we recall that the concept of pseudo-Lipschitz (p-Lz) for multi-valued mapping.

iii.

T is (p-Lz) around $({\omega }_0,{\zeta }_0)\in$ graph T with constant λ if there exist positive constants ε and η such that

$$\delta (T\left({\omega }_1\right)\cap B[{\zeta }_0;\epsilon ],T\left({\omega }_2\right))\le \lambda \rho ({\omega }_1,{\omega }_2)$$

for all ${\omega }_1,{\omega }_2\in B[{\omega }_0;\eta ].$

Similarly, we use the above notions within the realm of $\left(SbMS\right).$

The authors of [7] answered the question given by Kirk and Shahzad [6], that is, whether the following theorem holds under the weaker strong b-metric assumption or not.

Theorem 1

(Extension of Nadler’s theorem [18]). Let $(\mathrm{\Omega },\rho )$ be a complete metric space. Assume $T:\mathrm{\Omega }\to C\left(\mathrm{\Omega }\right)$ (where $C\left(\mathrm{\Omega }\right)$ is the collection of all nonempty closed subsets of Ω). Let ${\omega }_0\in \mathrm{\Omega }$ and suppose $r>0$ and $0\le k<1$ satisfy

i.

$D({\omega }_0,T\left({\omega }_0\right))<r(1-k);$

ii.

$\delta (T\left({\omega }_i\right)\cap B[{\omega }_0;r],T\left({\omega }_j\right))\le k\rho ({\omega }_i,{\omega }_j)$ for all ${\omega }_i,{\omega }_j\in B[{\omega }_0;r].$

Then, T has a fixed point in $B[{\omega }_0;r].$

The following counter-example confirms that their question has a negative answer.

Example 3.

Let $\mathrm{\Xi }=\{a,b,c\},\mathsf{ł}:\mathrm{\Xi }\times \mathrm{\Xi }\to \mathbb{R}$ be defined by ł$(a,a)=0,\forall a\in \mathrm{\Xi }$ and ł$(a,b)=\mathsf{ł}(b,a)=3,\mathsf{ł}(b,c)=\mathsf{ł}(c,b)=1,\mathsf{ł}(a,c)=\mathsf{ł}(c,a)=6$, and a function $T:\mathrm{\Xi }\to \mathrm{\Xi }$ be defined by $Ta=b,Tb=c,Tc=a.$

Then,

1.

$(\mathrm{\Xi },\mathsf{ł})$ is a complete SbMS with $s=3;$

2.

T fulfills all the assumptions of the previous theorem with ${\omega }_0=a,r=5$ and $k=\frac{1}{3};$

3.

T does not have any fixed point.

The concepts of Lipschitz and pseudo-Lipschitz will be abbreviated as (Lz) and (p-Lz), respectively. The following (Lz) properties, denoted by $L_i$ where $i\in \{1,2,3,4\},$ hold for a function T from $\mathrm{\Omega }$ to the subsets of $\mathrm{\Gamma }.$

$L_1$.

T has a (p-Lz) selection with a closed range around $({\omega }_0,{\zeta }_0).$ This means that for a given $({\omega }_0,{\zeta }_0)\in$ graph T, there exists a multi-function $S:\mathrm{\Omega }\to \mathrm{\Gamma }$ and a positive constant $\beta >0$ s.t. ${\zeta }_0\in S\left({\omega }_0\right);$ the set $S\left(\omega \right)$ is a closed subset of $T\left(\omega \right)$ for all $\omega$ within the closed ball $B[{\omega }_0;\beta ],$ and S is (p-Lz) around $({\omega }_0,{\zeta }_0).$

$L_2$.

T is (p-Lz) and locally closed-valued around $({\omega }_0,{\zeta }_0).$ In other words, for a given $({\omega }_0,{\zeta }_0)\in$ graph T, there exist $\theta >0$ and $\beta >0$ such that the set $T\left(\omega \right)\cap B[{\zeta }_0;\theta ]$ is closed for $\omega \in B[{\omega }_0;\beta ],$ and the function $\omega \to T\left(\omega \right)\cap B[{\zeta }_0;\theta ]$ is (p-Lz) around $({\omega }_0,{\zeta }_0).$

$L_3$.

T has a (Lz) selection around $({\omega }_0,{\zeta }_0).$ This means that for a given $({\omega }_0,{\zeta }_0)\in$ graph T, there exists a single function $s:\mathrm{\Omega }\to \mathrm{\Gamma }$ and $\beta >0$ such that ${\zeta }_0=s\left({\omega }_0\right),s\left(\omega \right)\in T\left(\omega \right)$ for $\omega \in B[{\omega }_0;\beta ],$ and s is (Lz) in $B[{\omega }_0;\beta ].$

$L_4$.

T is (Lz) and locally single-valued around $({\omega }_0,{\zeta }_0).$ This means that for a given $({\omega }_0,{\zeta }_0)\in$ graph $T,$ there exist $\theta >0$ and $\beta >0$ such that the map $\omega \to T\left(\omega \right)\cap B[{\zeta }_0;\theta ]$ is single-valued and (Lz) in $B[{\omega }_0;\beta ].$

In 1994, Dontchev and Hager proved the following inverse mapping theorem.

Theorem 2.

Let Ω be a complete MS and let $T:\mathrm{\Omega }\to P_{*}\left(\mathrm{\Gamma }\right)$ be a multi-function (where $P_{*}\left(\mathrm{\Gamma }\right)$ denotes subsets of a linear space Γ with an invariant metric). Let ${\zeta }_0\in T\left({\omega }_0\right).$ The single-valued function $f:\mathrm{\Omega }\to \mathrm{\Gamma }$ is (ss) at ${\omega }_0.$ Hence, for $i=1,2,3,4,$ the following statements are equivalent:

i.

The function $T^{-1}$ has the characteristic $L_i$ around $({\zeta }_0,{\omega }_0);$

ii.

The function ${(T+f)}^{-1}$ has the characteristic $L_i$ around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0).$

2. Main Results

The aim of this paper is to revisit Kirk and Shahzad’s question (see [6]), and by modifying their question, we establish some results within the context of strong b-metric spaces. We introduce Theorem 3, which is a generalization of Nadler’s theorem in the context of strong b-metric. Then show that our result does not hold in the b-metric space. As an application, we use Theorem 3 to prove Theorem 4, the inverse mapping theorem in strong b-metric space.

Theorem 3.

Let $(\mathrm{\Omega },\mathsf{ł})$ be a complete strong b-metric space with $s\ge 1.$ Suppose T maps Ω into the nonempty closed subsets of $\mathrm{\Omega }.$ Let ${\omega }_0\in \mathrm{\Omega }$ and suppose $r>0$ and $k\in [0,1)$ satisfy

i.

$D({\omega }_0,T\left({\omega }_0\right))<\frac{r}{s}(1-k);$

ii.

$\delta (T\left({\omega }_i\right)\cap B[{\omega }_0;r],T\left({\omega }_j\right))\le k\mathsf{ł}({\omega }_i,{\omega }_j)$   for all ${\omega }_i,{\omega }_j\in B[{\omega }_0;r].$

Then, T has a fixed point in $B[{\omega }_0;r].$

Proof. 

By condition (i) $\exists {\omega }_1\in T\left({\omega }_0\right)$ s.t. ł$({\omega }_1,{\omega }_0)<\frac{r}{s}(1-k).$ Continuing by induction, assume that there exists ${\omega }_{i+1}\in T\left({\omega }_i\right)\cap B[{\omega }_0;r]$ such that

$$\mathsf{ł}({\omega }_{i+1},{\omega }_i)<\frac{r}{s}(1-k)k^i,\mathrm{where}i\in \{1,2,\cdots ,n-1\}.$$

(1)

Assumption ($ii$) implies that

$$\begin{array}{ccc}\hfill D({\omega }_n,T\left({\omega }_n\right))& \le & \delta (T\left({\omega }_{n-1}\right)\cap B[{\omega }_0;r],T\left({\omega }_n\right))\hfill \\ & \le & k\mathsf{ł}({\omega }_n,{\omega }_{n-1})\hfill \\ & <& \frac{r}{s}(1-k)k^n;\hfill \end{array}$$

then, $\exists {\omega }_{n+1}\in T\left({\omega }_n\right)$ s.t. $\mathsf{ł}({\omega }_{n+1},{\omega }_n)<\frac{r}{s}(1-k)k^n.$ Since $\mathrm{\Omega }$ is an $\left(SbMS\right)$, we have

$$\begin{array}{ccc}\hfill \mathsf{ł}({\omega }_{n+1},{\omega }_0)& \le & s\mathsf{ł}({\omega }_{n+1},{\omega }_n)+\mathsf{ł}({\omega }_n,{\omega }_0)\hfill \\ & \le & s\mathsf{ł}({\omega }_{n+1},{\omega }_n)+s\mathsf{ł}({\omega }_n,{\omega }_{n-1})+\mathsf{ł}({\omega }_{n-1},{\omega }_0)\hfill \\ & \le & \sum _{i=1}^{n}s\mathsf{ł}({\omega }_{i+1},{\omega }_i)+\mathsf{ł}({\omega }_1,{\omega }_0)\hfill \\ & \le & \sum _{i=0}^{n}s\mathsf{ł}({\omega }_{i+1},{\omega }_i)\sin \mathrm{ce}s\ge 1\hfill \\ & \le & \frac{r}{s}(1-k)\sum _{i=0}^{n}s{k}^i\hfill \\ & <& r(1-k)\sum _{i=0}^{\infty }{k}^i=r.\hfill \end{array}$$

Hence, we obtain ${\omega }_{n+1}\in T\left({\omega }_n\right)\cap B[{\omega }_0;r].$ By this, the induction is completed.

For $n>m$, we have

$$\begin{array}{ccc}\hfill \mathsf{ł}({\omega }_n,{\omega }_m)& \le & \sum _{i=m}^{n-1}s\mathsf{ł}({\omega }_{i+1},{\omega }_i)\hfill \\ & \le & \frac{r}{s}(1-k)\sum _{i=m}^{n-1}s{k}^i\hfill \\ & <& r{k}^{m-1}.\hfill \end{array}$$

Thus, ${\left({\omega }_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence which converges to some ${\omega }^{*}\in B[{\omega }_0;r].$ By ($ii$),

$$\begin{array}{ccc}\hfill \mathit{D}({\omega }_n,T\left({\omega }^{*}\right))& \le & \delta (T\left({\omega }_{n-1}\right)\cap B[{\omega }_0;r],T\left({\omega }^{*}\right))\hfill \\ & \le & k\mathsf{ł}({\omega }_{n-1},{\omega }^{*}).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill D({\omega }^{*},T\left({\omega }^{*}\right))& \le & s\mathsf{ł}({\omega }^{*},{\omega }_n)+D({\omega }_n,T\left({\omega }^{*}\right))\hfill \\ & \le & s\mathsf{ł}({\omega }^{*},{\omega }_n)+k\mathsf{ł}({\omega }_{n-1},{\omega }^{*}).\hfill \end{array}$$

Thus, ${lim}_{n⟶\infty }D({\omega }^{*},T\left({\omega }^{*}\right))=0,$ and since $T\left({\omega }^{*}\right)$ is closed, then ${\omega }^{*}\in T\left({\omega }^{*}\right).$ So, ${\omega }^{*}$ is a fixed point of T in $B[{\omega }_0;r].$ □

Corollary 1.

If the mapping T in Theorem 3 is single-valued, then T has a unique fixed point in $B[{\omega }_0;r].$

Proof. 

Suppose that ${\omega }^{*}$ and ${\zeta }^{*}$ are distinct fixed points of T in $B[{\omega }_0;r].$

Since T is a single-valued function, then we have

$$\begin{array}{ccc}\hfill \mathsf{ł}(T\left({\omega }^{*}\right),T\left({\zeta }^{*}\right))& =& \delta (T\left({\omega }^{*}\right)\cap B[{\omega }_0;r],T\left({\zeta }^{*}\right))\hfill \\ & \le & k\mathsf{ł}({\omega }^{*},{\zeta }^{*})\hfill \\ & <& \mathsf{ł}({\omega }^{*},{\zeta }^{*}).\hfill \end{array}$$

Hence,

$$\mathsf{ł}({\omega }^{*},{\zeta }^{*})=\mathsf{ł}(T\left({\omega }^{*}\right),T\left({\zeta }^{*}\right))\le k\mathsf{ł}({\omega }^{*},{\zeta }^{*})<\mathsf{ł}({\omega }^{*},{\zeta }^{*}).$$

Which is a contradiction. □

Theorem 4.

Let $(\mathrm{\Omega },\rho )$ be a complete strong b-metric space with $s_0\ge 1$ and let $T:\mathrm{\Omega }\to P_{*}\left(Y\right)$ be a multi-function (where $P_{*}\left(Y\right)$ denotes subsets of a linear space Y with an invariant strong b-metric (ł) with $s\ge s_0\ge 1$ ); let ${\zeta }_0\in T\left({\omega }_0\right).$ The single-valued function $f:\mathrm{\Omega }\to Y$ is (ss) at ${\omega }_0.$ Hence, for $i=1,2,3,4,$ the following statements are equivalent:

i.

The function ${(T+f)}^{-1}$ has the characteristic $L_j$ around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0);$

ii.

The function $T^{-1}$ has the characteristic $L_j$ around $({\zeta }_0,{\omega }_0).$

Proof. 

First, let i be given

$L_1$.

Let ${(T+f)}^{-1}$ have a (p-Lz) selection with a closed range around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)$. There exists a multi-function $\mathrm{\Psi }$ s.t. For $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)\in graph\mathrm{\Psi };$ $\exists \beta >0$ s.t. $\mathrm{\Psi }\left(\zeta \right)$ is a closed subset of ${(T+f)}^{-1}\left(\zeta \right)$ for each $\zeta \in B[{\zeta }_0+f\left({\omega }_0\right);\beta ],$ and for some positive constants $\alpha$ and $\mu ,$

$$\delta (\mathrm{\Psi }\left({\zeta }_1\right)\cap B[{\omega }_0;\alpha ],\mathrm{\Psi }\left({\zeta }_2\right))\le \mu \mathsf{ł}({\zeta }_1,{\zeta }_2)\mathrm{for}\mathrm{all}{\zeta }_1,{\zeta }_2\in B[{\zeta }_0+f\left({\omega }_0\right);\beta ].$$

(2)

We pick any ${\mu }^{*}>\mu ,$ and let $\epsilon >0$ be such that

$$\mu \epsilon <1\mathrm{and}\mu <\frac{(1-\mu \epsilon ){\mu }^{*}}{s(1+s)}.$$

(3)

Since f is (ss) at ${\omega }_0,$ we select $\alpha$ small enough to satisfy

$$\mathsf{ł}(f\left({\omega }_1\right),f\left({\omega }_2\right))\le \epsilon \rho ({\omega }_1,{\omega }_2)\mathrm{for}\mathrm{all}{\omega }_1,{\omega }_2\in B[{\omega }_0;\alpha ].$$

(4)

We choose a and b such that

$$0<a\le \frac{1}{s}min\{\alpha ,\frac{\beta }{4\epsilon }\},0<b<\frac{1}{s}min\{\frac{\beta }{2},\frac{a}{4{\mu }^{*}}\}.$$

(5)

Since ł is an invariant strong b-metric on Y for $\omega \in B[{\omega }_0;a]$ and $\zeta \in B[{\zeta }_0;b]$, (4) and (5) imply that

$$\begin{array}{ccc}\hfill \mathsf{ł}(\zeta +f\left(\omega \right),{\zeta }_0+f\left({\omega }_0\right))& \le & \mathsf{ł}(\zeta +f\left(\omega \right),{\zeta }_0+f\left(\omega \right))+s\mathsf{ł}({\zeta }_0+f\left(\omega \right),{\zeta }_0+f\left({\omega }_0\right))\hfill \\ & =& \mathsf{ł}(\zeta ,{\zeta }_0)+s\mathsf{ł}(f\left(\omega \right),f\left({\omega }_0\right))\hfill \\ & \le & b+s\epsilon \rho (\omega ,{\omega }_0)\hfill \\ & <& \beta .\hfill \end{array}$$

Hence, $\zeta +f\left(\omega \right)\in B[{\zeta }_0+f\left({\omega }_0\right);\beta ]$ and $\mathrm{\Psi }(\zeta +f(\omega \left)\right)$ is a closed subset of ${(T+f)}^{-1}(\zeta +f\left(\omega \right))$ whenever $\omega \in B[{\omega }_0;a]$ and $\zeta \in B[{\zeta }_0;b].$

Let $\mathrm{\Phi }$ be the function given as $\mathrm{\Phi }(\omega ,\zeta )=\mathrm{\Psi }(\zeta +f(\omega \left)\right).$ If $\omega \in \mathrm{\Phi }(\omega ,\zeta )$ for some $\omega \in B[{\omega }_0;a]$ and $\zeta \in B[{\zeta }_0;b],$ then $\omega \in \mathrm{\Psi }(\zeta +f\left(\omega \right))\subset {(T+f)}^{-1}(\zeta +f\left(\omega \right))$; hence, $\omega \in T^{-1}\left(\zeta \right).$ Let $\mathrm{\Gamma }\left(\zeta \right)$ be the set of fixed points of $\mathrm{\Phi }(·,\zeta )=\mathrm{\Psi }(\zeta +f(·\left)\right)$ in $B[{\omega }_0;a];$ we will show that $\mathrm{\Gamma }$ is a (p-Lz) selection with a closed range of $T^{-1}$ around $({\zeta }_0,{\omega }_0).$ Obviously, ${\omega }_0\in \mathrm{\Gamma }\left({\zeta }_0\right);$ we have already observed that $\mathrm{\Gamma }\left(\zeta \right)\subset T^{-1}\left(\zeta \right)$ for every $\zeta \in B[{\zeta }_0;b].$

To prove the closedness of $\mathrm{\Gamma }\left(\zeta \right)$ for any $\zeta \in B[{\zeta }_0;b],$ we suppose that ${\omega }_n\in \mathrm{\Gamma }\left(\zeta \right)$ and ${lim}_{n\to \infty }\rho ({\omega }_n,\omega )=0.$ Since ł is an invariant strong b-metric on Y, and f is (ss) at ${\omega }_0,$ we have

$$\begin{array}{ccc}\hfill D({\omega }_n,\mathrm{\Phi }(\omega ,\zeta ))& =& D({\omega }_n,\mathrm{\Psi }(\zeta +f\left(\omega \right)))\hfill \\ & \le & \delta (\mathrm{\Psi }(\zeta +f\left({\omega }_n\right))\cap B[{\omega }_0;a],\mathrm{\Psi }(\zeta +f\left(\omega \right))\hfill \\ & \le & \mu \mathsf{ł}(f\left({\omega }_n\right),f\left(\omega \right))\hfill \\ & \le & \mu \epsilon \rho ({\omega }_n,\omega ).\hfill \end{array}$$

As $n\to \infty ,D({\omega }_n,\mathrm{\Psi }(y+f\left(\omega \right)))\to 0.$ By closedness of $\mathrm{\Psi }(\zeta +f(\omega \left)\right),$ $\omega \in \mathrm{\Gamma }\left(\zeta \right)$, so $\mathrm{\Gamma }\left(\zeta \right)$ is closed set for any $\zeta \in B[{\zeta }_0;b].$

On the other hand, to show that the function $\mathrm{\Gamma }$ is (p-Lz) around $({\zeta }_0,{\omega }_0)\in$ graph $\mathrm{\Gamma },$ we take any ${\zeta }^{\prime },{\zeta }^{″}\in B[{\zeta }_0;b],$ where ${\zeta }^{\prime }\ne {\zeta }^{″},$ and show that, for every ${\omega }^{\prime }\in \mathrm{\Gamma }\left({\zeta }^{\prime }\right)\cap B[{\omega }_0;\frac{a}{2s}],$ one finds ${\omega }^{″}\in \mathrm{\Gamma }\left({\zeta }^{″}\right)$ such that

$$\rho ({\omega }^{\prime },{\omega }^{″})\le \frac{{\mu }^{*}}{1+s}\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″}).$$

(6)

We show that by proving that the function $\mathrm{\Phi }(·,{\zeta }^{″})=\mathrm{\Psi }({\zeta }^{″}+f(·))$ has a fixed point in the closed ball of radius $h=\frac{{\mu }^{*}}{1+s}\mathsf{ł}({\zeta }^{\prime },{\zeta }^{\prime \prime })$, centered at ${\omega }^{\prime }.$ To utilize the extended Nadler’s theorem, first, we observe that

$$\begin{array}{ccc}\hfill D({\omega }^{\prime },\mathrm{\Phi }({\omega }^{\prime },{\zeta }^{″}))& \le & D({\omega }^{\prime },\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }^{\prime }\right)))\hfill \\ & \le & \delta (\mathrm{\Psi }({\zeta }^{\prime }+f\left({\omega }^{\prime }\right))\cap B[{\omega }_0;\frac{a}{2s}],\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }^{\prime }\right)))\hfill \\ & \le & \mu \mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})\hfill \\ & <& \frac{h}{s}(1-\mu \epsilon ).\hfill \end{array}$$

For ${\zeta }^{\prime },{\zeta }^{″}\in B[{\zeta }_0;b],$ we have

$$\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})\le (1+s)b<\frac{(1+s)a}{4s{\mu }^{*}}.$$

Hence, $h<a/4s;$ then, for each ${\omega }_1,{\omega }_2\in B[{\omega }^{\prime };h],$ and by (2) and (4), we have

$$\begin{array}{ccc}\hfill \delta (\mathrm{\Phi }({\omega }_1,{\zeta }^{″})\cap B[{\omega }^{\prime };h],\mathrm{\Phi }({\omega }_2,{\zeta }^{″}))& \le & \delta (\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }_1\right))\cap B[{\omega }^{\prime };h],\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }_2\right)))\hfill \\ & \le & \delta (\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }_1\right))\cap B[{\omega }_0;a],\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }_2\right)))\hfill \\ & \le & \mu \mathsf{ł}(f\left({\omega }_1\right),f\left({\omega }_2\right))\hfill \\ & \le & \mu \epsilon \rho ({\omega }_1,{\omega }_2).\hfill \end{array}$$

Since $s\ge s_0\ge 1,$ all the assumptions of extended Nadler’s Theorem 3 hold with ${\omega }_0={\omega }^{\prime },k=\mu \epsilon$, and $r=h$, which yields the existence of ${\omega }^{″}\in \mathrm{\Gamma }\left({\zeta }^{″}\right)$, satisfying (6). Finally, we have

$$\begin{array}{ccc}\hfill \delta (\mathrm{\Gamma }\left({\zeta }^{\prime }\right)\cap B[{\omega }_0;h],\mathrm{\Gamma }\left({\zeta }^{″}\right))& \le & \delta (\mathrm{\Phi }({\omega }^{\prime },{\zeta }^{\prime })\cap B[{\omega }_0;h],\mathrm{\Psi }({\omega }^{″},{\zeta }^{″}))\hfill \\ & \le & \delta (\mathrm{\Psi }({\zeta }^{\prime }+f\left({\omega }^{\prime }\right))\cap B[{\omega }_0;h],\mathrm{\Psi }({\zeta }^{″}+f\left({\omega }^{″}\right)))\hfill \\ & \le & \mu \mathsf{ł}({\zeta }^{\prime }+f\left({\omega }^{\prime }\right),{\zeta }^{″}+f\left({\omega }^{″}\right))\hfill \\ & \le & \mu [\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})+s\mathsf{ł}(f\left({\omega }^{\prime }\right),f\left({\omega }^{″}\right))]\hfill \\ & \le & \mu [\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})+s\epsilon \rho ({\omega }^{\prime },{\omega }^{″})]\hfill \\ & \le & c\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})\mathrm{where}c=\mu (1+s{\mu }^{*}\epsilon ).\hfill \end{array}$$

Then, $L_1$ holds.

$L_2$.

Let ${(T+f)}^{-1}$ be locally closed-valued and (p-Lz) around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)$. There exist $\theta \in (0,+\infty ]$ and $\beta >0$ such that the set ${(T+f)}^{-1}\left(\zeta \right)\cap B[{\omega }_0;\theta ]$ is closed for every $\zeta \in B[{\zeta }_0+f\left({\omega }_0\right);\beta ],$ and the function $\mathrm{\Psi }$ defined by $\mathrm{\Psi }\left(\zeta \right)={(T+f)}^{-1}\left(\zeta \right)\cap B[{\omega }_0;\theta ]$ is (p-Lz) around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0).$ By reiterating the proof for case $\left(L_1\right)$ with a and b as in (5) and $a\le \theta ,$ we determine that if $\mathrm{\Gamma }\left(\zeta \right)$ is the set of fixed points of $\mathrm{\Phi }(·,\zeta )=\mathrm{\Psi }(\zeta +f(·\left)\right)$ in $B[{\omega }_0;a]$, then $\mathrm{\Gamma }\left(\zeta \right)$ is closed for every $\zeta \in B[{\zeta }_0;b]$ and (p-Lz) around $({\zeta }_0,{\omega }_0).$ It can be confirmed that for any $\zeta \in B[{\zeta }_0;b],\omega \in \mathrm{\Gamma }\left(\zeta \right)$ if and only if $\omega \in T^{-1}\left(\zeta \right)\cap B[{\omega }_0;a].$ Hence, the map $y\to T^{-1}\left(\zeta \right)\cap B[{\omega }_0;a]$ is closed-valued and (p-Lz) around $({\zeta }_0,{\omega }_0).$

$L_3$.

Let ${(T+f)}^{-1}$ have a (Lz) selection around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)$. Suppose that ${(T+f)}^{-1}$ has a single-valued (Lz) selection $\mathrm{\Psi }$ around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)$ with constant $\kappa ,$ that is, $\mathrm{\Psi }\left(\zeta \right)\in {(T+f)}^{-1}\left(\zeta \right)$ for every $\zeta \in B[{\zeta }_0+f\left({\omega }_0\right);\beta ],$ and $\mathrm{\Psi }$ is (Lz) continuous in $B[{\zeta }_0+f\left({\omega }_0\right);\beta ]$ with constant $\kappa >0.$ Pick any ${\kappa }^{*}>\kappa ,$ and let $\epsilon >0$ be such that

$$s\kappa \epsilon <1\mathrm{and}\kappa <\frac{(1-s\kappa \epsilon ){\kappa }^{*}}{s_0}.$$

(7)

Select $\alpha >0$ s.t. (4) holds for every ${\omega }_1,{\omega }_2\in B[{\omega }_0;\alpha ].$ Let a and b be as in (5) and let $r={\kappa }^{*}b$ and $\lambda =\kappa \epsilon$, let $\mathrm{\Phi }(\omega ,\zeta )=\mathrm{\Psi }(\zeta +f(\omega \left)\right).$ For $\zeta \in B[{\zeta }_0;b],$ we have

$$\begin{array}{ccc}\hfill \rho ({\omega }_0,\mathrm{\Phi }({\omega }_0,\zeta ))& =& \rho ({\omega }_0,\mathrm{\Psi }(\zeta +f\left({\omega }_0\right)))\hfill \\ & =& \rho (\mathrm{\Psi }({\zeta }_0+f\left({\omega }_0\right)),\mathrm{\Psi }(\zeta +f\left({\omega }_0\right)))\hfill \\ & \le & \kappa \mathsf{ł}(\zeta ,{\zeta }_0)\hfill \\ & \le & \kappa b\hfill \\ & <& \frac{(1-s\kappa \epsilon ){\kappa }^{*}}{s_0}b\hfill \\ & \le & \frac{(1-\kappa \epsilon ){\kappa }^{*}}{s_0}b\hfill \\ & =& \frac{r}{s_0}(1-\lambda ),\mathrm{where}\lambda =\mu \epsilon \hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \rho (\mathrm{\Psi }(\zeta +f\left({\omega }_1\right)),\mathrm{\Psi }(\zeta +f\left({\omega }_2\right)))& \le & \kappa \mathsf{ł}(f\left({\omega }_1\right),f\left({\omega }_2\right))\hfill \\ & \le & \lambda \rho ({\omega }_1,{\omega }_2),\mathrm{for}\mathrm{all}{\omega }_1,{\omega }_2\in B[{\omega }_0;r].\hfill \end{array}$$

For every $\zeta \in B[{\zeta }_0;b]$ and by extended Nadler’s theorem, $\mathrm{\Psi }(\zeta +f(·\left)\right)$ has a unique fixed point in $B[{\omega }_0;r],$ say $x\left(\zeta \right).$ Since $\mathrm{\Psi }(\zeta +f\left(x\left(\zeta \right)\right))\in T^{-1}\left(\zeta \right),x(·)$ is a single-valued of $T^{-1}$ in $B[{\zeta }_0;b].$ For each ${\zeta }^{\prime },{\zeta }^{″}\in B[{\zeta }_0;b],$ we have

$$\begin{array}{ccc}\hfill \rho (x\left({\zeta }^{\prime }\right),x\left({\zeta }^{″}\right))& =& \rho (\mathrm{\Psi }({\zeta }^{\prime }+f\left(x\left({\zeta }^{\prime }\right)\right)),\mathrm{\Psi }({\zeta }^{″}+f\left(x\left({\zeta }^{″}\right)\right)))\hfill \\ & \le & \kappa \mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})+s\kappa \mathsf{ł}(f\left(x\left({\zeta }^{\prime }\right)\right),f\left(x\left({\zeta }^{″}\right)\right))\hfill \\ & \le & \kappa \mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})+s\kappa \epsilon \rho (x\left({\zeta }^{\prime }\right),x\left({\zeta }^{″}\right)).\hfill \end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill (1-s\kappa \epsilon )\rho (x\left({\zeta }^{\prime }\right),x\left({\zeta }^{″}\right))& \le & \kappa \mathsf{ł}({\zeta }^{\prime },{\zeta }^{″})\hfill \\ & <& \frac{(1-s\kappa \epsilon ){\kappa }^{*}}{s_0}\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″}).\hfill \end{array}$$

As a result,

$$\begin{array}{ccc}\hfill \rho (x\left({\zeta }^{\prime }\right),x\left({\zeta }^{″}\right))& \le & \frac{{\kappa }^{*}}{s_0}\mathsf{ł}({\zeta }^{\prime },{\zeta }^{″}).\hfill \end{array}$$

Therefore, $x(·)$ is a (Lz) selection of $T^{-1}$ around $({\zeta }_0,{\omega }_0).$ And case $\left(L_3\right)$ is proved.

$L_4$.

Finally, let ${(T+f)}^{-1}$ be locally single-valued and (Lz) around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0)$. Assume that $\mathrm{\Psi }\left(\zeta \right)={(T+f)}^{-1}\left(\zeta \right)\cap B[{\omega }_0;\theta ]$ is single-valued and (Lz) near ${\zeta }_0$ for some $\theta >0.$ Select a and b as in (5) and $a<\theta$; reiterating the reasoning and using a similar argument as in case $\left(L_3\right),$ for $r={\kappa }^{*}b$ there exists a unique fixed point $x\left(\zeta \right)$ of $\mathrm{\Psi }(\zeta +f(·\left)\right)$ in $B[{\omega }_0;r]$ for every $\zeta \in B[{\zeta }_0;b],$ and $x(·)$ is (Lz) continuous on $B[{\zeta }_0;b].$ Since $r<a,$ it follows that $x\left(\zeta \right)=T^{-1}\left(\zeta \right)\cap B[{\omega }_0;r].$ Hence, $T^{-1}$ is locally single-valued and (Lz) around $({\zeta }_0,{\omega }_0).$

Conversely, let $ii$ be given, let f be an arbitrary map which is (ss) at ${\omega }_0$, then $-f$ is (ss) at ${\omega }_0$ also. Now, let $T^{-1}$ have the characteristic $\left(L_j\right)$ around $({\zeta }_0,{\omega }_0).$ Then, ${(T+f-f)}^{-1}$ has the characteristic $\left(L_j\right)$ around $({\zeta }_0+f\left({\omega }_0\right)-f\left({\omega }_0\right),{\omega }_0).$ From the initial part of the proof, we deduce that the function ${(T+f)}^{-1}$ has the characteristic $\left(L_j\right)$ around $({\zeta }_0+f\left({\omega }_0\right),{\omega }_0).$ The proof of the theorem is completed. □

Example 4.

Let $\mathrm{\Omega }=\mathbb{R}$ and ł$({\omega }_1,{\omega }_2)=max\left\{\right|{\omega }_1-{\omega }_2|,2|{\omega }_1-{\omega }_2|-1\}$ for all ${\omega }_1,{\omega }_2\in \mathbb{R}$ and let $T:\mathbb{R}\to \mathcal{CB}\left(\mathbb{R}\right)$ be defined by $T\omega =\left\{\frac{\omega }{4}\right\}\cup \left\{10\right\}$ for all $\omega \in \mathbb{R}.$ Hence,

1.

$(\mathbb{R},\mathsf{ł})$ is a SbMS with $s=2;$

2.

$(\mathbb{R},\mathsf{ł})$ is complete;

3.

T and $(\mathbb{R},\mathsf{ł})$ satisfy all hypothesis of Theorem 3 with ${\omega }_0=1,r=5$ and $k=\frac{1}{2}.$

Proof. 1. Similar to Example 2.

2.

Let ${\left({\omega }_n\right)}_{n\in \mathbb{N}}$ be a Cauchy sequence in $(\mathbb{R},\mathsf{ł})$, that is, $\forall \epsilon >0,\exists N\in \mathbb{N}$ such that for all $n,m\ge N,\mathsf{ł}({\omega }_n,{\omega }_m)=|{\omega }_n-{\omega }_m|<\epsilon .$ Otherwise, $\mathsf{ł}({\omega }_n,{\omega }_m)=2|{\omega }_n-{\omega }_m|-1$ and one obtains $|{\omega }_n-{\omega }_m|\ge 1.$ Hence, ${\left({\omega }_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(\mathbb{R},|·\left|\right).$

By the completeness of $\mathbb{R}$ with the usual metric, ${\left({\omega }_n\right)}_{n\in \mathbb{N}}$ is a convergent sequence. Then, we conclude that ${\left({\omega }_n\right)}_{n\in \mathbb{N}}$ is a convergent sequence in $(\mathbb{R},\mathsf{ł})$, and $(\mathbb{R},\mathsf{ł})$ is complete.

3.

With ${\omega }_0=1,r=5$ and $k=\frac{1}{2},$ we have

i.

$D({\omega }_0,T{\omega }_0)=inf\{\mathsf{ł}(1,\frac{1}{4}),\mathsf{ł}(1,10)\}=\frac{3}{4}<\frac{r}{s}(1-k)=\frac{5}{4};$

ii.

For any ${\omega }_1,{\omega }_2\in B[1;5]=[-2,4],$ we have

$$\begin{array}{ccc}\hfill \delta (T{\omega }_1\cap [-2,4],T{\omega }_2)& =& \delta (\left\{\frac{{\omega }_1}{4}\right\},\{\frac{{\omega }_2}{4},10\})\hfill \\ & =& sup\{D(\frac{{\omega }_1}{4},\{\frac{-1}{2},\cdots ,0,\cdots ,1,10\}),{\omega }_1\in [-2,4]\}\hfill \\ & =& 0\hfill \\ & \le & \frac{1}{2}\mathsf{ł}({\omega }_1,{\omega }_2).\hfill \end{array}$$

Hence, by Theorem 3, T has a fixed point in $B[1;5]$; that is, $0.$ □

Example 5.

Let $(\mathrm{\Omega },\mathsf{ł})$ be defined as in Example 4. Let $T:\mathbb{R}\to \mathbb{R}$ be defined by $T\omega =\frac{-1}{4}\omega +2$ for all $\omega \in \mathbb{R}.$ Then, T and $(\mathbb{R},\mathsf{ł})$ satisfy all the hypotheses of Corollary 1 with ${\omega }_0=1,r=9$, and $k=\frac{1}{3}.$

Proof. 

With ${\omega }_0=1,r=9$ and $k=\frac{1}{3},$ we have

i.

$D({\omega }_0,T{\omega }_0)=\mathsf{ł}(1,1\frac{3}{4})=\frac{3}{4}<\frac{r}{s}(1-k)=3;$

ii.

For any ${\omega }_1,{\omega }_2\in B[1;9]=[-4,6],$ we have

$$\frac{1}{2}\le T{\omega }_1\le 3$$

and $\delta (T{\omega }_1\cap B[1;9],T{\omega }_2)=\mathsf{ł}(T{\omega }_1,T{\omega }_2);$ then, we have two cases:

1.

$\mathsf{ł}(T{\omega }_1,T{\omega }_2)=|T{\omega }_1-T{\omega }_2|$

$$\begin{array}{ccc}\hfill \mathsf{ł}(T{\omega }_1,T{\omega }_2)=|T{\omega }_1-T{\omega }_2|=\frac{1}{4}|{\omega }_1-{\omega }_2|& \le & \frac{1}{3}\mathsf{ł}({\omega }_1,{\omega }_2);\hfill \end{array}$$

2.

$\mathsf{ł}(T{\omega }_1,T{\omega }_2)=2|T{\omega }_1-T{\omega }_2|-1$; then,

$$\begin{array}{ccc}\hfill \mathsf{ł}(T{\omega }_1,T{\omega }_2)=2|T{\omega }_1-T{\omega }_2|-1& =& \frac{1}{4}\left(2|{\omega }_1-{\omega }_2|\right)-1\hfill \\ & =& \frac{1}{4}\left(\mathsf{ł}({\omega }_1,{\omega }_2)-3\right)\hfill \\ & \le & \frac{1}{3}\mathsf{ł}({\omega }_1,{\omega }_2).\hfill \end{array}$$

Hence, we conclude that

$$\begin{array}{ccc}\hfill \delta (Tx\cap B[1;9],Ty)& \le & \frac{1}{3}\mathsf{ł}({\omega }_1,{\omega }_2).\hfill \end{array}$$

Hence, by Corollary 1, T contains a singular fixed point within $B[1;9]$; that is, $1.6$. □

Finally, the following example shows that the previous Corollary 1 does not hold under the b-metric assumption.

Example 6.

Let $\mathrm{\Omega }=[1,\infty ),\mathsf{ł}({\omega }_1,{\omega }_2)={({\omega }_1-{\omega }_2)}^4$ for all ${\omega }_1,{\omega }_2\in \mathrm{\Omega }$ and let $T:\mathrm{\Omega }\to \mathrm{\Omega }$ be defined by $T\omega =\omega +\frac{1}{\omega }$ for all $\omega \in \mathrm{\Omega }.$ Then,

1.

ł is continuous b-metric on Ω with $s=8;$

2.

$(\mathrm{\Omega },\mathsf{ł})$ is complete;

3.

In Corollary 1, all assumptions on the mapping T are satisfied with ${\omega }_0=4,r=16$ and $k=\frac{1}{3};$

4.

T has no fixed point.

Proof. 

The proof of 1 and 2 was explained clearly in [9].

To see 3, $T\omega$ is a nonempty closed subset of $\mathrm{\Omega }.$ With ${\omega }_0=4,r=16$, and $k=\frac{1}{3},$ we have

i.

$D({\omega }_0,T{\omega }_0)=\mathsf{ł}(4,4\frac{1}{4})={\left(\frac{1}{4}\right)}^4<\frac{16}{8}(1-\frac{1}{3})=\frac{4}{3};$

ii.

$$\begin{array}{ccc}\hfill B[{\omega }_0;r]=B[4;16]& =& \{\omega \in \mathrm{\Omega }:\mathsf{ł}(4,\omega )\le 16\}\hfill \\ & =& \{\omega \in \mathrm{\Omega }:2\le \omega \le 6\}=[2,6].\hfill \end{array}$$

Since T is increasing, for any $\omega \in B[4;16],$

$$\frac{5}{2}\le T\omega \le \frac{37}{6}.$$

This implies, $\frac{5}{2}\le T\omega \cap B[4;16]\le 6.$

For any ${\omega }_1,{\omega }_2\in [2,6]$, we have

$$\begin{array}{ccc}\hfill \delta (T{\omega }_1\cap B[4;16],T{\omega }_2)& =& \delta (\{{\omega }_1+\frac{1}{{\omega }_1}\},\{{\omega }_2+\frac{1}{{\omega }_2}\})\hfill \\ & & \mathrm{where}{\omega }_1\in [2,\frac{6+\sqrt{32}}{2}],{\omega }_2\in [2,6].\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \delta (T{\omega }_1\cap B[4;16],T{\omega }_2)& =& |{\omega }_1+\frac{1}{{\omega }_1}-{\omega }_2-\frac{1}{{\omega }_2}{|}^4\hfill \\ & =& |{\omega }_1-{\omega }_2{|}^4{\left(1-\frac{1}{{\omega }_1{\omega }_2}\right)}^4\hfill \\ & \le & {\left(\frac{3}{4}\right)}^4\mathsf{ł}({\omega }_1,{\omega }_2)\hfill \\ & \le & \frac{1}{3}\mathsf{ł}({\omega }_1,{\omega }_2).\hfill \end{array}$$

According to the above, the assumptions on the mapping T hold.

To see 4, T does not have any fixed point.

□

Remark 1.

Note that in Example 6, all the conditions of Corollary 1 are met except for the strong b-metric assumption. However, T has no fixed points. Therefore, we can conclude that the b-metric space $(\mathrm{\Omega },\mathsf{ł})$ in Example 6 is not a strong b-metric space.

3. Conclusions

In this manuscript, an extension of Nadler’s theorem was generalized in the context of strong b-metric space. This result shows the importance of the strong b-metric space since it does not hold in b-metric space. Moreover, we proved an inverse mapping theorem for set-valued maps in the strong b-metric space. Furthermore, several examples have been provided. As future research proposals and contributions, we suggest that researchers in this field study the convergence of numerical methods for solving variational problems as an application of our result (Theorem 3). See, for example, [19], a study in metric spaces.