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

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.

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 $d:l_p(\mathbb{R})\times l_p(\mathbb{R})\to [0,\mathrm{\infty })$,

$$d(x,y):={\left(\sum _{n=1}^{\mathrm{\infty }}{|x_n-y_n|}^p\right)}^{\frac{1}{p}},$$

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

$$d(x,y):=\{\begin{array}{cc}0,\hfill & x=y,\hfill \\ 2s,\hfill & x,y\in X_1,\hfill \\ 1,\hfill & \text{otherwise},\hfill \end{array}$$

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

Definition 1.5 (Boriceanu et al. [14])

Let $(X,d)$ be a 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

$$d(x_0,x_n)\le sd(x_0,x_1)+\cdots +s^{n-1}d(x_{n-2},x_{n-1})+s^{n-1}d(x_{n-1},x_n).$$

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])

A function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is said to be a $(c)$-comparison function if

• (1) ψ is increasing;

• (2) there exist $n_0\in \mathbb{N}$, $k\in (0,1)$ and a convergent series of nonnegative terms ${\sum }_{n=1}^{\mathrm{\infty }}{ϵ}_n$ such that ${\psi }^{n+1}(t)\le k{\psi }^n(t)+{ϵ}_n$ 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])

Let $s\ge 1$ be a real number. A mapping $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is called a $(b)$-comparison function if the following conditions are fulfilled:

• (1) ψ is increasing;

• (2) there exist $n_0\in \mathbb{N}$, $k\in (0,1)$ and a convergent series of nonnegative terms ${\sum }_{n=1}^{\mathrm{\infty }}{ϵ}_n$ such that $s^{n+1}{\psi }^{n+1}(t)\le k{s}^n{\psi }^n(t)+{ϵ}_n$ 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

$$u=f(u),u\in X$$

(1.1)

is said to be generalized Ulam-Hyers stable in the framework of a 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,

$$d(v^{\ast },f\left(v^{\ast }\right))\le \epsilon ,$$

(1.2)

there exists a solution $u^{\ast }\in X$ of the fixed point equation (1.1) such that

$$d(v^{\ast },u^{\ast })\le \phi (s\epsilon ).$$

(1.3)

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])

Let 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:

$$\text{for }x,y\in X,\alpha (x,y)\ge 1⟹\alpha (f(x),f(y))\ge 1.$$

Example 1.15 (Samet et al. [27])

Let $X=(0,\mathrm{\infty })$. Define $f:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty })$ by

$$f(x)=ln(x)\text{for all }x\in X$$

and

$$\alpha (x,y)=\{\begin{array}{cc}2,\hfill & \text{if }x\ge y;\hfill \\ 0,\hfill & \text{if }x<y.\hfill \end{array}$$

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

$$f(x)=\{\begin{array}{cc}ln(x^2+1),\hfill & \text{if }x>1;\hfill \\ \frac{x}{3},\hfill & \text{if }0\le x\le 1;\hfill \\ x,\hfill & \text{otherwise}\hfill \end{array}$$

and

$$\alpha (x,y)=\{\begin{array}{cc}1,\hfill & \text{if }x,y\in [0,1];\hfill \\ ln1.5,\hfill & \text{otherwise}.\hfill \end{array}$$

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

$$x_n=f(x_{n-1})\text{for all }n\in \mathbb{N}.$$

Since f is α-admissible and

$$\alpha (x_0,x_1)=\alpha (x_0,f(x_0))\ge 1,$$

we deduce that

$$\alpha (x_1,f(x_1))=\alpha (f(x_0),f(x_1))\ge 1.$$

By induction, we get

$$\alpha (x_{n-1},f(x_{n-1}))\ge 1\text{for all }n\in \mathbb{N}.$$

Next, we will show that $\{x_n\}$ is a Cauchy sequence in X. For each $n\in \mathbb{N}$, we have

$$\begin{array}{rl}d(x_n,x_{n+1})& =d(f(x_{n-1}),f(x_n))\\ \le \alpha (x_{n-1},f(x_{n-1}))\alpha (x_n,f(x_n))d(f(x_{n-1}),f(x_n))\\ \le \psi (d(x_{n-1},x_n)).\end{array}$$

By repeating the process above, we get

$$d(x_n,x_{n+1})\le {\psi }^n(d(x_0,x_1))\text{for all }n\in \mathbb{N}.$$

For $m,n\in \mathbb{N}$ with $m>n$, we have

$$\begin{array}{rcl}d(x_n,x_m)& \le & sd(x_n,x_{n+1})+s^{2}d(x_{n+1},x_{n+2})+\cdots +s^{m-n-2}d(x_{m-3},x_{m-2})\\ +s^{m-n-1}d(x_{m-2},x_{m-1})+s^{m-n}d(x_{m-1},x_m)\\ \le & s{\psi }^n(d(x_0,x_1))+s^2{\psi }^{n+1}(d(x_0,x_1))+\cdots +s^{m-n-2}{\psi }^{m-3}(d(x_0,x_1))\\ +s^{m-n-1}{\psi }^{m-2}(d(x_0,x_1))+s^{m-n}{\psi }^{m-1}(d(x_0,x_1))\\ =& \frac{1}{s^{n-1}}[s^n{\psi }^n(d(x_0,x_1))+s^{n+1}{\psi }^{n+1}(d(x_0,x_1))+\cdots +s^{m-2}{\psi }^{m-2}(d(x_0,x_1))\\ +s^{m-1}{\psi }^{m-1}(d(x_0,x_1))].\end{array}$$

Define $S_n:={\sum }_{i=0}^n{s}^i{\psi }^i(d(x_0,x_1))$ for all $n\in \mathbb{N}$. This implies that

$$d(x_n,x_m)\le \frac{1}{s^{n-1}}[S_{m-1}-S_{n-1}]\text{for all }n,m\in \mathbb{N}\text{ with }m>n.$$

By Lemma 1.11 we know that the series ${\sum }_{i=0}^{\mathrm{\infty }}{s}^i{\psi }^i(d(x_0,x_1))$ converges. Therefore, $\{x_n\}$ is a Cauchy sequence in 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

$$\begin{array}{rcl}d(f\left(x^{\ast }\right),x^{\ast })& \le & s[d(f\left(x^{\ast }\right),x_n)+d(x_n,x^{\ast })]\\ =& s[d(f\left(x^{\ast }\right),f(x_{n-1}))+d(x_n,x^{\ast })]\\ \le & s[\alpha (x^{\ast },f\left(x^{\ast }\right))\alpha (x_{n-1},f(x_{n-1}))d(f\left(x^{\ast }\right),f(x_{n-1}))+d(x_n,x^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },x_{n-1})\right)+d(x_n,x^{\ast })]\end{array}$$

for all $n\in \mathbb{N}$. Letting $n\to \mathrm{\infty }$, since ψ is continuous at 0, we obtain

$$d(f\left(x^{\ast }\right),x^{\ast })=0.$$

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

Next, we prove the uniqueness of the fixed point of f. Let $y^{\ast }$ be another fixed point of f such that

$$\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

It follows that

$$\begin{array}{rl}d(x^{\ast },y^{\ast })& =d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))\\ \le \alpha (x^{\ast },f\left(x^{\ast }\right))\alpha (y^{\ast },f\left(y^{\ast }\right))d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))\\ \le \psi \left(d(x^{\ast },y^{\ast })\right)\\ <d(x^{\ast },y^{\ast }),\end{array}$$

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

$$x_n=f(x_{n-1})\text{for all }n\in \mathbb{N}.$$

Since f is α-admissible and $\alpha (x_0,x_1)=\alpha (x_0,f(x_0))\ge 1$, we get

$$\alpha (f(x_0),f(x_1))=\alpha (x_1,f(x_1))\ge 1.$$

By induction, we get

$$\alpha (x_{n-1},f(x_{n-1}))\ge 1\text{for all }n\in \mathbb{N}.$$

Next, we will show that $\{x_n\}$ is a Cauchy sequence in X. For each $n\in \mathbb{N}$, we have

$$\begin{array}{rcl}d(x_n,x_{n+1})+\xi & =& d(f(x_{n-1}),f(x_n))+\xi \\ \le & {[d(f(x_{n-1}),f(x_n))+\xi ]}^{\alpha (x_{n-1},f(x_{n-1}))\alpha (x_n,f(x_n))}\\ \le & \psi (d(x_{n-1},x_n))+\frac{\xi }{s}\\ \le & \psi (d(x_{n-1},x_n))+\xi .\end{array}$$

Now, we get

$$d(x_n,x_{n+1})\le \psi (d(x_{n-1},x_n))\le {\psi }^n(d(x_0,x_1))\text{for all }n\in \mathbb{N}.$$

Following the proof of Theorem 2.1, we know that $\{x_n\}$ is a Cauchy sequence in 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

$$\begin{array}{rcl}d(f\left(x^{\ast }\right),x^{\ast })+\xi & \le & s[d(f\left(x^{\ast }\right),x_n)+d(x_n,x^{\ast })]+\xi \\ \le & s[d(f\left(x^{\ast }\right),f(x_{n-1}))+\xi +d(x_n,x^{\ast })]\\ \le & s[{(d(f\left(x^{\ast }\right),f(x_{n-1}))+\xi )}^{\alpha (x^{\ast },f(x^{\ast }))\alpha (x_{n-1},f(x_{n-1}))}+d(x_n,x^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },x_{n-1})\right)+\frac{\xi }{s}+d(x_n,x^{\ast })]\end{array}$$

for all $n\in \mathbb{N}$. Letting $n\to \mathrm{\infty }$ and ψ be continuous at 0, we obtain that

$$d(f\left(x^{\ast }\right),x^{\ast })=0.$$

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

Next, we prove the uniqueness of the fixed point of f. Let $y^{\ast }$ be another fixed point of f such that

$$\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

It follows that

$$\begin{array}{rcl}d(x^{\ast },y^{\ast })+\xi & =& d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+\xi \\ \le & {[d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+\xi ]}^{\alpha (x^{\ast },f(x^{\ast }))\alpha (y^{\ast },f(y^{\ast }))}\\ \le & \psi \left(d(x^{\ast },y^{\ast })\right)+\frac{\xi }{s}\\ \le & \psi \left(d(x^{\ast },y^{\ast })\right)+\xi .\end{array}$$

This shows that

$$d(x^{\ast },y^{\ast })\le \psi \left(d(x^{\ast },y^{\ast })\right)<d(x^{\ast },y^{\ast }),$$

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

$$x_n=f(x_{n-1})\text{for all }n\in \mathbb{N}.$$

Since f is α-admissible and $\alpha (x_0,x_1)=\alpha (x_0,f(x_0))\ge 1$, we get

$$\alpha (f(x_0),f(x_1))=\alpha (x_1,f(x_1))\ge 1.$$

By induction, we get

$$\alpha (x_{n-1},f(x_{n-1}))\ge 1\text{for all }n\in \mathbb{N}.$$

Next, we will show that $\{x_n\}$ is a Cauchy sequence in X. For each $n\in \mathbb{N}$, we have

$$\begin{array}{rcl}{\xi }^{d(x_n,x_{n+1})}& =& {\xi }^{d(f(x_{n-1}),f(x_n))}\\ \le & {(\alpha (x_{n-1},f(x_{n-1}))\alpha (x_n,f(x_n))-1+\xi )}^{d(f(x_{n-1}),f(x_n))}\\ \le & {\xi }^{\psi (d(x_{n-1},x_n))}.\end{array}$$

Since $\xi >1$, we get

$$d(x_n,x_{n+1})\le \psi (d(x_{n-1},x_n))\le {\psi }^n(d(x_0,x_1))\text{for all }n\in \mathbb{N}.$$

Following the proof of Theorem 2.1, we know that $\{x_n\}$ is a Cauchy sequence in 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

$$\begin{array}{rcl}{\xi }^{d(f(x^{\ast }),x^{\ast })}& \le & {\xi }^{s[d(f(x^{\ast }),x_n)+d(x_n,x^{\ast })]}\\ =& {\xi }^{s(d(f(x^{\ast }),f(x_{n-1})))}{\xi }^{s(d(x_n,x^{\ast }))}\\ =& {\left({\xi }^{d(f(x^{\ast }),f(x_{n-1}))}\right)}^s{\left({\xi }^{d(x_n,x^{\ast })}\right)}^s\\ \le & {\left({(\alpha (x_{n-1},f(x_{n-1}))\alpha (x^{\ast },f\left(x^{\ast }\right))-1+\xi )}^{d(f(x^{\ast }),f(x_{n-1}))}\right)}^s{\left({\xi }^{d(x_n,x^{\ast })}\right)}^s\\ \le & {\left({\xi }^{\psi (d(x^{\ast },x_{n-1}))}\right)}^s{\left({\xi }^{d(x_n,x^{\ast })}\right)}^s\\ =& {\xi }^{s[\psi (d(x^{\ast },x_{n-1}))+d(x_n,x^{\ast })]}\end{array}$$

for all $n\in \mathbb{N}$. Since $\xi >1$, it implies that

$$d(f\left(x^{\ast }\right),x^{\ast })\le s[\psi \left(d(x^{\ast },x_{n-1})\right)+d(x_n,x^{\ast })].$$

Letting $n\to \mathrm{\infty }$, since ψ is continuous at 0, we obtain that

$$d(f\left(x^{\ast }\right),x^{\ast })=0.$$

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

Next, we prove the uniqueness of the fixed point of f. Let $y^{\ast }$ be another fixed point of f such that

$$\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

It follows that

$$\begin{array}{rcl}{\xi }^{d(x^{\ast },y^{\ast })}& =& {\xi }^{d(f(x^{\ast }),f(y^{\ast }))}\\ \le & {(\alpha (x^{\ast },f\left(x^{\ast }\right))\alpha (y^{\ast },f\left(y^{\ast }\right))-1+\xi )}^{d(f(x^{\ast }),f(y^{\ast }))}\\ \le & {\xi }^{\psi (d(x^{\ast },y^{\ast }))}.\end{array}$$

Since $\xi >1$, we have

$$d(x^{\ast },y^{\ast })\le \psi \left(d(x^{\ast },y^{\ast })\right)<d(x^{\ast },y^{\ast }),$$

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

$$d(f(x),f(y))\le \psi (d(x,y))$$

(2.4)

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

$$d(f(x),f(y))\le \psi (d(x,y))$$

(2.5)

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

$$d(x,y)={|x-y|}^2\text{for all }x,y\in X.$$

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

Let us define $f:X\to X$ by

$$f(x)=\{\begin{array}{cc}\frac{x}{2(x+1)},\hfill & x\in [0,1],\hfill \\ 10x,\hfill & x\in (1,\mathrm{\infty }).\hfill \end{array}$$

Also, define $\alpha :X\times X\to [0,\mathrm{\infty })$ and $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ by

$$\alpha (x,y)=\{\begin{array}{cc}1,\hfill & x,y\in [0,1],\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$

and $\psi (t)=\frac{1}{2}t$ for all $t\ge 0$. Clearly, f is an α-admissible mapping. For all $x,y\in X$, we have

$$\alpha (x,f(x))\alpha (y,f(y))d(f(x),f(y))\le \psi (d(x,y)).$$

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

Next, we show that the contractive condition in Theorem 2.2 cannot be applied to this example. For $x=0$ and $y=1$, we obtain that

$${[d(f(x),f(y))+\xi ]}^{\alpha (x,f(x))\alpha (y,f(y))}>\psi (d(x,y))+\frac{\xi }{s},$$

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,

$$d(y^{\ast },f\left(y^{\ast }\right))\le \epsilon .$$

Since $x^{\ast },y^{\ast }\in X$ are an ε-solution, we have

$$\alpha (x^{\ast },f\left(x^{\ast }\right))\ge 1\text{and}\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

Now, we obtain

$$\begin{array}{rcl}d(x^{\ast },y^{\ast })& =& d(f\left(x^{\ast }\right),y^{\ast })\\ \le & s[d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\alpha (x^{\ast },f\left(x^{\ast }\right))\alpha (y^{\ast },f\left(y^{\ast }\right))d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+\epsilon ].\end{array}$$

It follows that

$$d(x^{\ast },y^{\ast })-s\left(\psi \left(d(x^{\ast },y^{\ast })\right)\right)\le s\epsilon .$$

Since $\phi (t):=t-s\psi (t)$, we have

$$\phi \left(d(x^{\ast },y^{\ast })\right)=d(x^{\ast },y^{\ast })-s\psi \left(d(x^{\ast },y^{\ast })\right).$$

This implies that

$$d(x^{\ast },y^{\ast })\le {\phi }^{-1}(s\epsilon ).$$

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,

$$d(y^{\ast },f\left(y^{\ast }\right))\le \epsilon .$$

Since $x^{\ast },y^{\ast }\in X$ are an ε-solution, we have

$$\alpha (x^{\ast },f\left(x^{\ast }\right))\ge 1\text{and}\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

Now, we obtain

$$\begin{array}{rcl}d(x^{\ast },y^{\ast })+\xi & =& d(f\left(x^{\ast }\right),y^{\ast })+\xi \\ \le & s[d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+d(f\left(y^{\ast }\right),y^{\ast })]+\xi \\ \le & s[d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+d(f\left(y^{\ast }\right),y^{\ast })+\xi ]\\ \le & s[{(d(f\left(x^{\ast }\right),f\left(y^{\ast }\right))+\xi )}^{\alpha (x^{\ast },f(x^{\ast }))\alpha (y^{\ast },f(y^{\ast }))}+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+\frac{\xi }{s}+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+\epsilon ]+\xi .\end{array}$$

It follows that

$$d(x^{\ast },y^{\ast })\le s[\psi \left(d(x^{\ast },y^{\ast })\right)+\epsilon ],$$

and then

$$d(x^{\ast },y^{\ast })-s\left(\psi \left(d(x^{\ast },y^{\ast })\right)\right)\le s\epsilon .$$

Since $\phi (t):=t-s\psi (t)$, we have

$$\phi \left(d(x^{\ast },y^{\ast })\right)=d(x^{\ast },y^{\ast })-s\psi \left(d(x^{\ast },y^{\ast })\right).$$

It implies that

$$d(x^{\ast },y^{\ast })\le {\phi }^{-1}(s\epsilon ).$$

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,

$$d(y^{\ast },f\left(y^{\ast }\right))\le \epsilon .$$

Since $x^{\ast },y^{\ast }\in X$ are an ε-solution, we have

$$\alpha (x^{\ast },f\left(x^{\ast }\right))\ge 1\text{and}\alpha (y^{\ast },f\left(y^{\ast }\right))\ge 1.$$

Now, we obtain

$$\begin{array}{rcl}{\xi }^{d(x^{\ast },y^{\ast })}& =& {\xi }^{d(f(x^{\ast }),y^{\ast })}\\ \le & {\xi }^{s[d(f(x^{\ast }),f(y^{\ast }))+d(f(y^{\ast }),y^{\ast })]}\\ =& {\left({\xi }^{d(f(x^{\ast }),f(y^{\ast }))}\right)}^s{\left({\xi }^{d(f(y^{\ast }),y^{\ast })}\right)}^s\\ \le & {\left({(\alpha (x^{\ast },f\left(x^{\ast }\right))\alpha (y^{\ast },f\left(y^{\ast }\right))-1+\xi )}^{d(f(x^{\ast }),f(y^{\ast }))}\right)}^s{\left({\xi }^{d(f(y^{\ast }),y^{\ast })}\right)}^s\\ \le & {\left({\xi }^{\psi (d(x^{\ast },y^{\ast }))}\right)}^s{\left({\xi }^{d(f(y^{\ast }),y^{\ast })}\right)}^s\\ =& {\xi }^{s[\psi (d(x^{\ast },y^{\ast }))+d(f(y^{\ast }),y^{\ast })]}.\end{array}$$

Since $\xi >1$, we have

$$\begin{array}{rcl}d(x^{\ast },y^{\ast })& \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+d(f\left(y^{\ast }\right),y^{\ast })]\\ \le & s[\psi \left(d(x^{\ast },y^{\ast })\right)+\epsilon ].\end{array}$$

It follows that

$$d(x^{\ast },y^{\ast })-s\left(\psi \left(d(x^{\ast },y^{\ast })\right)\right)\le s\epsilon .$$

Suppose that $\phi (t):=t-s\psi (t)$, we have

$$\phi \left(d(x^{\ast },y^{\ast })\right)=d(x^{\ast },y^{\ast })-s\psi \left(d(x^{\ast },y^{\ast })\right).$$

It implies that

$$d(x^{\ast },y^{\ast })\le {\phi }^{-1}(s\epsilon ).$$

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 $n\to \mathrm{\infty }$.

In the following theorems, we add a new condition to assure the well-posedness via 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

$$\alpha (x_n,f(x_n))\ge 1.$$

Now, we have

$$\begin{array}{rcl}d(x_n,x^{\ast })& =& d(x_n,f\left(x^{\ast }\right))\\ \le & s[d(x_n,f(x_n))+d(f(x_n),f\left(x^{\ast }\right))]\\ \le & s[\alpha (x_n,f(x_n))\alpha (x^{\ast },f\left(x^{\ast }\right))d(f(x_n),f\left(x^{\ast }\right))+d(x_n,f(x_n))]\\ \le & s[\psi \left(d(x_n,x^{\ast })\right)+d(x_n,f(x_n))].\end{array}$$

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

$$\alpha (x_n,f(x_n))\ge 1.$$

Now, we have

$$\begin{array}{rcl}d(x_n,x^{\ast })+\xi & =& d(x_n,f\left(x^{\ast }\right))+\xi \\ \le & s[d(x_n,f(x_n))+d(f(x_n),f\left(x^{\ast }\right))+\xi ]\\ \le & s[{(d(f(x_n),f\left(x^{\ast }\right))+\xi )}^{\alpha (x_n,f(x_n))\alpha (x^{\ast },f(x^{\ast }))}+d(x_n,f(x_n))]\\ \le & s[\psi \left(d(x_n,x^{\ast })\right)+\frac{\xi }{s}+d(x_n,f(x_n))]\\ \le & s[\psi \left(d(x_n,x^{\ast })\right)+d(x_n,f(x_n))]+\xi .\end{array}$$

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

$$\alpha (x_n,f(x_n))\ge 1.$$

Now, we have

$$\begin{array}{rcl}{\xi }^{d(x_n,x^{\ast })}& =& {\xi }^{d(x_n,f(x^{\ast }))}\\ \le & {\xi }^{s[d(x_n,f(x_n))+d(f(x_n),f(x^{\ast }))]}\\ =& {\xi }^{sd(f(x_n),f(x^{\ast }))}{\xi }^{sd(x_n,f(x_n))}\\ =& {\left({\xi }^{d(f(x_n),f(x^{\ast }))}\right)}^s{\left({\xi }^{d(x_n,f(x_n))}\right)}^s\\ \le & {\left({(\alpha (x_n,f(x_n))\alpha (x^{\ast },f\left(x^{\ast }\right))-1+\xi )}^{d(f(x_n),f(x^{\ast }))}\right)}^s{\left({\xi }^{d(x_n,f(x_n))}\right)}^s\\ \le & {\left({\xi }^{\psi (d(x_n,x^{\ast }))}\right)}^s{\left({\xi }^{d(x_n,f(x_n))}\right)}^s\\ =& {\xi }^{s\psi (d(x_n,x^{\ast }))}{\xi }^{sd(x_n,f(x_n))}\\ =& {\xi }^{s[\psi (d(x_n,x^{\ast }))+d(x_n,f(x_n))]}.\end{array}$$

Since $\xi >1$, we have

$$d(x_n,x^{\ast })\le s[\psi \left(d(x_n,x^{\ast })\right)+d(x_n,f(x_n))],$$

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. □