# Fixed point results, generalized Ulam-Hyers stability and well-posedness via *α*-admissible mappings in *b*-metric spaces

- Supak Phiangsungnoen
^{1, 2}, - Wutiphol Sintunavarat
^{3}Email author and - Poom Kumam
^{1, 2}Email author

**2014**:188

https://doi.org/10.1186/1687-1812-2014-188

© Phiangsungnoen et al.; licensee Springer. 2014

**Received: **31 May 2014

**Accepted: **7 August 2014

**Published: **2 September 2014

## Abstract

In this paper, we prove the existence and uniqueness of a fixed point for some new classes of contractive mappings via *α*-admissible mappings in the framework of *b*-metric spaces. We also present an example to illustrate the usability of the obtained results. The generalized Ulam-Hyers stability and well-posedness of a fixed point equation via *α*-admissible mappings in *b*-metric spaces are given.

**MSC:**46S40, 47S40, 47H10.

### Keywords

*α*-admissible mappings

*b*-metric spaces fixed points generalized Ulam-Hyers stability well-posedness

## 1 Introduction and preliminaries

### 1.1 The *b*-metric space

The Banach contraction mapping principle is the most important in mathematics analysis, it guarantees the existence and uniqueness of a fixed point for certain self-mapping in metric spaces and provides a constructive method to find this fixed point. Several authors have obtained fixed point and common fixed point results for various classes of mappings in the setting of several spaces (see [1–6] and the references therein).

In 1993, Czerwik [7] introduced *b*-metric spaces as a generalization of metric spaces and proved the contraction mapping principle in *b*-metric spaces that is an extension of the famous Banach contraction principle in metric spaces. Since then, a number of authors have investigated fixed point theorems in *b*-metric spaces (see [8–11] and the references therein).

**Definition 1.1** (Bakhtin [8], Czerwik [12])

Let *X* be a nonempty set, and let the functional $d:X\times X\to [0,\mathrm{\infty})$ satisfy:

(b1) $d(x,y)=0$ if and only if $x=y$;

(b2) $d(x,y)=d(y,x)$ for all $x,y\in X$;

(b3) there exists a real number $s\ge 1$ such that $d(x,z)\le s[d(x,y)+d(y,z)]$ for all $x,y,z\in X$.

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

**Remark 1.2** If we take $s=1$ in the above definition, then *b*-metric spaces turn into ordinary metric spaces. Hence, the class of *b*-metric spaces is larger than the class of metric spaces.

For examples of *b*-metric spaces, see [7, 8, 12–14].

**Example 1.3**The set ${l}_{p}(\mathbb{R})$ with $0<p<1$, where ${l}_{p}(\mathbb{R}):=\{\{{x}_{n}\}\subset \mathbb{R}\mid {\sum}_{n=1}^{\mathrm{\infty}}{|{x}_{n}|}^{p}<\mathrm{\infty}\}$, together with the functional $d:{l}_{p}(\mathbb{R})\times {l}_{p}(\mathbb{R})\to [0,\mathrm{\infty})$,

where $x=\{{x}_{n}\},y=\{{y}_{n}\}\in {l}_{p}(\mathbb{R})$, is a *b*-metric space with coefficient $s={2}^{\frac{1}{p}}>1$. Notice that the above result holds for the general case ${l}_{p}(X)$ with $0<p<1$, where *X* is a Banach space.

**Example 1.4**Let

*X*be a set with the cardinal $card(X)\ge 3$. Suppose that $X={X}_{1}\cup {X}_{2}$ is a partition of

*X*such that $card({X}_{1})\ge 2$. Let $s>1$ be arbitrary. Then the functional $d:X\times X\to [0,\mathrm{\infty})$ defined by

is a *b*-metric on *X* with coefficient $s>1$.

**Definition 1.5** (Boriceanu *et al.* [14])

*b*-metric space. Then a sequence $\{{x}_{n}\}$ in

*X*is called:

- (a)
convergent if and only if there exists $x\in X$ such that $d({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$;

- (b)
Cauchy if and only if $d({x}_{n},{x}_{m})\to 0$ as $m,n\to \mathrm{\infty}$.

**Lemma 1.6** (Czerwik [12])

*Let*$(X,d)$

*be a*

*b*-

*metric space*,

*and let*${\{{x}_{k}\}}_{k=0}^{n}\subset X$.

*Then*

**Definition 1.7** (Rus [15])

A mapping $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is called a comparison function if it is increasing and ${\psi}^{n}(t)\to 0$ as $n\to \mathrm{\infty}$ for any $t\in [0,\mathrm{\infty})$, where ${\psi}^{n}$ is the *n* th iterate of *ψ*.

**Lemma 1.8** (Rus [15], Berinde [16])

*If*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*is a comparison function*,

*then*

- (1)
${\psi}^{n}$

*is also a comparison function*; - (2)
*ψ**is continuous at*0; - (3)
$\psi (t)<t$

*for any*$t>0$.

The concept of $(c)$-comparison function was introduced by Berinde [16] in the following definition.

**Definition 1.9** (Berinde [16])

- (1)
*ψ*is increasing; - (2)
there exist ${n}_{0}\in \mathbb{N}$, $k\in (0,1)$ and a convergent series of nonnegative terms ${\sum}_{n=1}^{\mathrm{\infty}}{\u03f5}_{n}$ such that ${\psi}^{n+1}(t)\le k{\psi}^{n}(t)+{\u03f5}_{n}$ for $n\ge {n}_{0}$ and any $t\in [0,\mathrm{\infty})$.

Here we recall the definitions of the following class of $(b)$-comparison functions as given by Berinde [17] in order to extend some fixed point results to the class of *b*-metric spaces.

**Definition 1.10** (Berinde [17])

- (1)
*ψ*is increasing; - (2)
there exist ${n}_{0}\in \mathbb{N}$, $k\in (0,1)$ and a convergent series of nonnegative terms ${\sum}_{n=1}^{\mathrm{\infty}}{\u03f5}_{n}$ such that ${s}^{n+1}{\psi}^{n+1}(t)\le k{s}^{n}{\psi}^{n}(t)+{\u03f5}_{n}$ for $n\ge {n}_{0}$ and any $t\in [0,\mathrm{\infty})$.

In this work, we use ${\mathrm{\Psi}}_{b}$ to denote the class of all $(b)$-comparison functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ unless and until it is stated otherwise. It is evident that the concept of $(b)$-comparison function reduces to that of $(c)$-comparison function when $s=1$.

**Lemma 1.11** (Berinde [13])

*If*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*is a*$(b)$-

*comparison function*,

*then the following assertions hold*:

- (i)
*the series*${\sum}_{n=0}^{\mathrm{\infty}}{s}^{n}{\psi}^{n}(t)$*converges for any*$t\in [0,\mathrm{\infty})$; - (ii)
*the function*$S:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$*defined by*$S(t)={\sum}_{n=0}^{\mathrm{\infty}}{s}^{n}{\psi}^{n}(t)$*for*$t\in [0,\mathrm{\infty})$*is increasing and continuous at*0.

### 1.2 The generalized Ulam-Hyers stability

Stability problems of functional analysis play the most important role in mathematics analysis. They were introduced by Ulam [18], he was concerned with the stability of group homomorphisms. Afterward, Hyers [19] gave a first affirmative partial answer to the question of Ulam for a Banach space, this type of stability is called Ulam-Hyers stability. Several authors have considered Ulam-Hyers stability results in fixed point theory, and remarkable results on the stability of certain classes of functional equations via fixed point approach have been obtained (see [20–26] and the references therein).

We recall the following definitions in the class of *b*-metric spaces.

**Definition 1.12**Let $(X,d)$ be a

*b*-metric space with coefficient

*s*, and let $f:X\to X$ be an operator. By definition, the fixed point equation

*b*-metric space if there exists an increasing operator $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, continuous at 0 and $\phi (0)=0$, such that for each $\epsilon >0$ and an

*ε*-solution ${v}^{\ast}\in X$, that is,

If $\phi (t):=ct$ for all $t\in [0,\mathrm{\infty})$, where $c>0$, then (1.1) is said to be Ulam-Hyers stable in the framework of a *b*-metric space.

**Remark 1.13** If $s=1$, then Definition 1.12 reduces to the generalized Ulam-Hyers stability in metric spaces. Also, if $\phi (t):=ct$ for all $t\in [0,\mathrm{\infty})$, where $c>0$, then it reduces to the classical Ulam-Hyers stability.

### 1.3 *α*-Admissible mappings

In 2012, Samet *et al.* [27] introduced the concept of *α*-admissible mappings and established fixed point theorems for such mappings in complete metric spaces. Moreover, they showed some examples and applications to ordinary differential equations. There are many researchers who improved and generalized fixed point results by using the concept of *α*-admissible mapping for single-valued and multivalued mappings (see [28–33]).

**Definition 1.14** (Samet *et al.* [27])

*X*be a nonempty set, $f:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$. We say that

*f*is an

*α*-admissible mapping if it satisfies the following condition:

**Example 1.15** (Samet *et al.* [27])

Then *f* is *α*-admissible.

**Example 1.16**Let $X=\mathbb{R}$. Define $f:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Then *f* is *α*-admissible.

Recently Bota *et al.* [34] proved the existence and uniqueness of fixed point theorems. They also studied the generalized Ulam-Hyers stability results via an *α*-admissible mapping in a *b*-metric space. The purpose of this paper is to establish the existence and uniqueness of fixed point theorems for some new types of contractive mappings via *α*-admissible mappings. We also give some examples to show that our fixed point theorems for new types of contractive mappings are independent. The generalized Ulam-Hyers stability and well-posedness of a fixed point equation for these classes in the framework of *b*-metric spaces are proved.

## 2 Fixed point results in *b*-metric spaces

In this section, we prove the existence and uniqueness of fixed point theorems in a *b*-metric space.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*

*s*,

*let*$f:X\to X$

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$

*be two mappings and*$\psi \in {\mathrm{\Psi}}_{b}$.

*Suppose that the following conditions hold*:

- (a)
*f**is**α*-*admissible*; - (b)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},f({x}_{0}))\ge 1$; - (c)
*for all*$x,y\in X$,*we have*$\alpha (x,f(x))\alpha (y,f(y))d(f(x),f(y))\le \psi (d(x,y));$(2.1) - (d)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},f({x}_{n}))\ge 1$*for all*$n\in \mathbb{N}$,*then*$\alpha (x,f(x))\ge 1$.

*Then* *f* *has a unique fixed point* ${x}^{\ast}$ *in* *X* *such that* $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*Proof*Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$ (from condition (b)). We define the sequence $\{{x}_{n}\}$ in

*X*such that

*f*is

*α*-admissible and

*X*. For each $n\in \mathbb{N}$, we have

*X*. By the completeness of

*X*, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Using condition (d), we get $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. From (2.1), we have

*ψ*is continuous at 0, we obtain

It implies that $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}$ is a fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*f*. Let ${y}^{\ast}$ be another fixed point of

*f*such that

which is a contradiction. Therefore, ${x}^{\ast}$ is the unique fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. This completes the proof. □

**Theorem 2.2**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*

*s*,

*let*$f:X\to X$

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$

*be two mappings and*$\psi \in {\mathrm{\Psi}}_{b}$.

*Suppose that the following conditions hold*:

- (a)
*f**is**α*-*admissible*; - (b)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},f({x}_{0}))\ge 1$; - (c)
*there exists*$\xi \ge 1$*such that*${[d(f(x),f(y))+\xi ]}^{\alpha (x,f(x))\alpha (y,f(y))}\le \psi (d(x,y))+\frac{\xi}{s}$(2.2)

*for all*$x,y\in X$;

- (d)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},f({x}_{n}))\ge 1$*for all*$n\in \mathbb{N}$,*then*$\alpha (x,f(x))\ge 1$.

*Then* *f* *has a unique fixed point* ${x}^{\ast}$ *in* *X* *such that* $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*Proof*Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$ (from condition (b)). We define the sequence $\{{x}_{n}\}$ in

*X*such that

*f*is

*α*-admissible and $\alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f({x}_{0}))\ge 1$, we get

*X*. For each $n\in \mathbb{N}$, we have

*X*. Since $(X,d)$ is complete, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. By condition (d), we have $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$ for all $n\in \mathbb{N}$. From (2.2), we get

*ψ*be continuous at 0, we obtain that

This implies that $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}$ is a fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*f*. Let ${y}^{\ast}$ be another fixed point of

*f*such that

which is a contradiction. Therefore, ${x}^{\ast}$ is the unique fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. This completes the proof. □

**Theorem 2.3**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*

*s*,

*let*$f:X\to X$

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$

*be two mappings and*$\psi \in {\mathrm{\Psi}}_{b}$.

*Suppose that the following conditions hold*:

- (a)
*f**is**α*-*admissible*; - (b)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},f({x}_{0}))\ge 1$; - (c)
*there exists*$\xi >1$*such that*${(\alpha (x,f(x))\alpha (y,f(y))-1+\xi )}^{d(f(x),f(y))}\le {\xi}^{\psi (d(x,y))}$(2.3)

*for all*$x,y\in X$;

- (d)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},f({x}_{n}))\ge 1$*for all*$n\in \mathbb{N}$,*then*$\alpha (x,f(x))\ge 1$.

*Then* *f* *has a unique fixed point* ${x}^{\ast}$ *in* *X* *such that* $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*Proof*Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$ (from condition (b)). We define the sequence $\{{x}_{n}\}$ in

*X*such that

*f*is

*α*-admissible and $\alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f({x}_{0}))\ge 1$, we get

*X*. For each $n\in \mathbb{N}$, we have

*X*. Since $(X,d)$ is complete, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. By condition (d), we have $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$ for all $n\in \mathbb{N}$. From assumption (2.3), we get

*ψ*is continuous at 0, we obtain that

It implies that $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}$ is a fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.

*f*. Let ${y}^{\ast}$ be another fixed point of

*f*such that

which is a contradiction. Therefore, ${x}^{\ast}$ is a unique fixed point of *f* such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. This completes the proof. □

If we set $\alpha (x,y)=1$ for all $x,y\in X$ in Theorems 2.1 or 2.2 or 2.3, we get the following results.

**Corollary 2.4**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*

*s*,

*let*$f:X\to X$

*and*$\psi \in {\mathrm{\Psi}}_{b}$,

*we have*

*for all* $x,y\in X$. *Then* *f* *has a unique fixed point in* *X*.

If the coefficient $s=1$ in Corollary 2.4, we obtain immediately the following fixed point theorems in metric spaces.

**Corollary 2.5** (Berinde [35])

*Let*$(X,d)$

*be a complete metric space*, $f:X\to X$

*be a mapping*, $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*be a*$(c)$-

*comparison function such that*

*for all* $x,y\in X$. *Then* *f* *has a unique fixed point in* *X*.

**Remark 2.6** If $\psi (t)=kt$, where $k\in (0,1)$ in Corollary 2.5, we obtain the Banach contraction mapping principle.

Next, we give some examples to show that the contractive conditions in our results are independent. Also, our results are real generalizations of the Banach contraction principle and several results in literature.

**Example 2.7**Let $X=[0,\mathrm{\infty})$ and define $d:X\times X\to [0,\mathrm{\infty})$ as

Then $(X,d)$ is a complete *b*-metric space with coefficient $s=2>1$, but it is not a usual metric space.

*f*is an

*α*-admissible mapping. For all $x,y\in X$, we have

Moreover, all the conditions of Theorem 2.1 hold. In this example, 0 is a unique fixed point of *f*.

where $\xi =1$ and $s=2$. This claims that Theorem 2.2 cannot be applied to *f*. Also, by a similar method, we can show that Theorem 2.3 cannot be applied to *f*.

Moreover, results from usual metric spaces and the Banach contraction principle are not applicable while Theorem 2.1 is applicable.

## 3 The generalized Ulam-Hyers stability in *b*-metric spaces

In this section, we prove the generalized Ulam-Hyers stability in *b*-metric spaces which corresponds to Theorems 2.1, 2.2 and 2.3.

**Theorem 3.1** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*. *Suppose that all the hypotheses of Theorem * 2.1 *hold and also that the function* $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *defined by* $\phi (t):=t-s\psi (t)$ *is strictly increasing and onto*. *If* $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ *for all* ${u}^{\ast}\in X$, *which is an* *ε*-*solution*, *then the fixed point equation* (1.1) *is generalized Ulam*-*Hyers stable*.

*Proof*By Theorem 2.1, we have $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}\in X$ is a solution of the fixed point equation (1.1). Let $\epsilon >0$ and ${y}^{\ast}\in X$ be an

*ε*-solution, that is,

*ε*-solution, we have

Notice that ${\phi}^{-1}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ exists, is increasing, continuous at 0 and ${\phi}^{-1}(0)=0$. Therefore, the fixed point equation (1.1) is generalized Ulam-Hyers stable. □

**Theorem 3.2** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*. *Suppose that all the hypotheses of Theorem * 2.2 *hold and also that the function* $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *defined by* $\phi (t):=t-s\psi (t)$ *is strictly increasing and onto*. *If* $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ *for all* ${u}^{\ast}\in X$, *which is an* *ε*-*solution*, *then the fixed point equation* (1.1) *is generalized Ulam*-*Hyers stable*.

*Proof*By Theorem 2.2, we have $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}\in X$ is a solution of the fixed point equation (1.1). Let $\epsilon >0$ and ${y}^{\ast}\in X$ be an

*ε*-solution, that is,

*ε*-solution, we have

Notice that ${\phi}^{-1}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ exists, is increasing, continuous at 0 and ${\phi}^{-1}(0)=0$. Therefore, the fixed point equation (1.1) is generalized Ulam-Hyers stable. □

**Theorem 3.3** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*. *Suppose that all the hypotheses of Theorem * 2.3 *hold and also that the function* $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *defined by* $\phi (t):=t-s\psi (t)$ *is strictly increasing and onto*. *If* $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ *for all* ${u}^{\ast}\in X$, *which is an* *ε*-*solution*, *then the fixed point equation* (1.1) *is generalized Ulam*-*Hyers stable*.

*Proof*By Theorem 2.3, we have $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}\in X$ is a solution of the fixed point equation (1.1). Let $\epsilon >0$ and ${y}^{\ast}\in X$ be an

*ε*-solution, that is,

*ε*-solution, we have

Notice that ${\phi}^{-1}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ exists, is increasing, continuous at 0 and ${\phi}^{-1}(0)=0$. Therefore, the fixed point equation (1.1) is generalized Ulam-Hyers stable. □

## 4 Well-posedness of a function with respect to *α*-admissibility in *b*-metric spaces

In this section, we present and prove well-posedness of a function with respect to an *α*-admissible mapping in *b*-metric spaces.

**Definition 4.1**Let $(X,d)$ be a

*b*-metric space with coefficient

*s*, and let $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$ be two mappings. The fixed point problem of

*f*is said to be well posed with respect to

*α*if:

- (i)
*f*has a unique fixed point ${x}^{\ast}$ in*X*such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$; - (ii)
for a sequence $\{{x}_{n}\}$ in

*X*such that $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$, then ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$.

*α*-admissibility.

- (S)
If $\{{x}_{n}\}$ is a sequence in

*X*such that $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},f({x}_{n}))\ge 1$ for all $n\in \mathbb{N}$.

**Theorem 4.2** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*, *let* $f:X\to X$ *and* $\alpha :X\times X\to [0,\mathrm{\infty})$ *be two mappings and* $\psi \in {\mathrm{\Psi}}_{b}$. *Suppose that all the hypotheses of Theorem * 2.1 *and condition* (S) *hold*. *Then the fixed point equation* (1.1) *is well posed with respect to* *α*.

*Proof*By Theorem 2.1, there is a unique point ${x}^{\ast}\in X$ such that $f({x}^{\ast})={x}^{\ast}$ and $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. Let $\{{x}_{n}\}$ be a sequence in

*X*such that $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$. By condition (S), we get

Since *ψ* is continuous at 0 and $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$, it implies that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Therefore, the fixed point equation (1.1) is well posed with respect to *α*. □

**Theorem 4.3** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*, *let* $f:X\to X$ *and* $\alpha :X\times X\to [0,\mathrm{\infty})$ *be two mappings and* $\psi \in {\mathrm{\Psi}}_{b}$. *Suppose that all the hypotheses of Theorem * 2.2 *and condition* (S) *hold*. *Then the fixed point equation* (1.1) *is well posed with respect to* *α*.

*Proof*By Theorem 2.2, there is a unique point ${x}^{\ast}\in X$ such that $f({x}^{\ast})={x}^{\ast}$ and $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. Let $\{{x}_{n}\}$ be a sequence in

*X*such that $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$. By condition (S), we get

Since *ψ* is continuous at 0 and $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$, it implies that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Therefore, the fixed point equation (1.1) is well posed with respect to *α*. □

**Theorem 4.4** *Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* *s*, *let* $f:X\to X$ *and* $\alpha :X\times X\to [0,\mathrm{\infty})$ *be two mappings and* $\psi \in {\mathrm{\Psi}}_{b}$. *Suppose that all the hypotheses of Theorem * 2.3 *and condition* (S) *hold*. *Then the fixed point equation* (1.1) *is well posed with respect to* *α*.

*Proof*By Theorem 2.3, there is a unique point ${x}^{\ast}\in X$ such that $f({x}^{\ast})={x}^{\ast}$ and $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. Let $\{{x}_{n}\}$ be a sequence in

*X*such that $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$. By condition (S), we get

and *ψ* is continuous at 0 and $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$. It implies that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Therefore, the fixed point equation (1.1) is well posed with respect to *α*. □

## Declarations

### Acknowledgements

This research is supported by the Centre of Excellence in Mathematics, the Commission on High Education, Thailand, and also Miss Supak Phiangsungnoen is supported by the Centre of Excellence in Mathematics, the Commission on High Education, Thailand for Ph.D. at KMUTT. The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript. Moreover, the authors are grateful for the reviewers for careful reading of the paper and for suggestions which improved the quality of this work.

## Authors’ Affiliations

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