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Generalized UlamHyers stability and wellposedness for fixed point equation via αadmissibility
Journal of Inequalities and Applications volume 2014, Article number: 418 (2014)
Abstract
In this paper, we extend the concept of contraction mappings in bmetric spaces and utilize this concept to prove the existence and uniqueness of fixed point theorems for such mappings in such a space. We also prove the generalized UlamHyers stability and wellposed results for a fixed point equation employing the concept of αadmissibility in bmetric spaces. We shall construct some examples to support our novel results.
MSC:46S40, 47S40, 47H10.
1 Introduction
The classical Banach contraction principle is a very important tool in solving existence problems in many branches of mathematics. Over the years, it has been generalized in several different directions by several mathematicians (see [1–7]). In 1993, Czerwik [8] introduced and proved the contraction mapping principle in bmetric spaces that generalized the famous Banach contraction principle in such spaces. Subsequently several other authors [9–15] have studied and established the existence of fixed points of a contractive mapping in bmetric spaces.
The study of stability problems for various functional equations play the most important role in mathematical analysis. In the fall of 1940, Ulam [16] discussed a number of important unsolved mathematical problems. Among them, a question concerning the stability of homomorphisms seemed too abstract for anyone to reach any conclusion. In the following year, Hyers [17] gave a first affirmative partial answer to Ulam’s question for Banach spaces, this type of stability is called UlamHyers stability. A large number of papers have been published in connection with various generalizations of UlamHyers stability results in fixed point theory and remarkable result on the stability of certain classes of functional equations via fixed point approach (see [18–29] and references therein).
On the other hand, Samet et al. [30] introduced the concepts of αψcontractive mapping and αadmissible selfmappings. Also, they proved some fixed point results for such mappings in complete metric spaces. Naturally, many authors have started to investigate the existence of a fixed point theorem via αadmissible mappings for single valued and multivalued mappings (see [31–38]). Recently Bota et al. [39] considered the existence and the uniqueness of fixed point theorems and generalized UlamHyers stability results via αadmissible mappings in bmetric spaces.
In this paper, we extend the concept of αψcontractive mapping in bmetric spaces. By using this concept, we establish the existence and uniqueness of fixed point for some new types of contractive mappings in bmetric spaces and give an example to illustrate our main results. Moreover, we study and prove the generalized UlamHyers stability and wellposed results by using fixed point method via αadmissible mappings in bmetric spaces.
2 Preliminaries
Throughout this paper, we shall use the following notation.
Let X be a nonempty set and 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 bmetric on X and a pair $(X,d)$ is called a bmetric space with coefficient s.
Remark 2.2 If we take $s=1$ in above definition then bmetric spaces turns into usual metric spaces. Hence, the class of bmetric spaces is larger than the class of usual metric spaces.
Examples of bmetric spaces were given in [8, 40–43].
Example 2.3 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 bmetric spaces 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 spaces.
Example 2.4 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 bmetric on X with coefficient $s>1$.
Definition 2.5 ([42])
Let $(X,d)$ be a bmetric spaces. 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 2.6 ([41])
Let $(X,d)$ be a bmetric spaces and let ${\{{x}_{k}\}}_{k=0}^{n}\subset X$. Then
Definition 2.7 ([21])
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 $t\in [0,\mathrm{\infty})$.
If $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is a comparison function, then

(1)
${\psi}^{n}$ is also a comparison function, where ${\psi}^{n}$ is nth iterate of ψ;

(2)
ψ is continuous at 0;

(3)
$\psi (t)<t$, for any $t>0$.
The concept of $(c)$comparison function was introduced by Berinde [44] in the following definition.
Definition 2.9 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 ${\sum}_{n=1}^{\mathrm{\infty}}{\u03f5}_{n}$ 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 function as given by Berinde [45] in order to extend some fixed point results to the class of bmetric spaces.
Definition 2.10 ([45])
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 ${\sum}_{n=1}^{\mathrm{\infty}}{\u03f5}_{n}$ 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 denote by ${\mathrm{\Psi}}_{b}$ the class of $(b)$comparison function $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$. It is evident that the concept of $(b)$comparison function reduces to that of $(c)$comparison function when $s=1$.
Lemma 2.11 ([43])
If $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is a $(b)$comparison function, then we have the following:

(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 $S(t)={\sum}_{n=0}^{\mathrm{\infty}}{s}^{n}{\psi}^{n}(t)$, $t\in [0,\mathrm{\infty})$, is increasing and continuous at 0.
Next, we will present the concept of αadmissible mappings introduced by Samet et al. [30].
Definition 2.12 ([30])
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 αadmissible mapping if it satisfies the following condition:
Example 2.13 Let $X=(0,\mathrm{\infty})$. Define $f:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by
and
Then f is αadmissible.
Example 2.14 Let $X=\mathbb{R}$. Define $f:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by
and
Then f is αadmissible.
3 Fixed point theorems for αadmissible mapping in bmetric spaces
In this section, we prove the existence and uniqueness of fixed point theorems in a bmetric space.
Definition 3.1 Let $(X,d)$ be a bmetric space with coefficient s. A mapping $f:X\to X$ is said to be a generalized αψcontraction in bmetric space if there exist functions $\psi \in {\mathrm{\Psi}}_{b}$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ such that the following condition holds:
Theorem 3.2 Let $(X,d)$ be a complete bmetric space with coefficient s and f be a generalized αψcontraction. Suppose that the following conditions hold:

(a)
f is an αadmissible;

(b)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$;

(c)
if $\{{x}_{n}\}$ is 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
Since f is an αadmissible and
we deduce that
By continuing this process, we get $\alpha ({x}_{n1},f({x}_{n1}))\ge 1$ for all $n\in \mathbb{N}$. This implies that
for all $n\in \mathbb{N}$. From (3.1), we obtain
for all $n\in \mathbb{N}$. By repeating the above process, we get
for all $n\in \mathbb{N}$. Next, we show that $\{{x}_{n}\}$ is a Cauchy sequence in X. For $m,n\in \mathbb{N}$ with $m>n$, we have
Denote ${S}_{n}={\sum}_{i=0}^{n}{s}^{i}{\psi}^{i}(d({x}_{0},{x}_{1}))$ for all $n\in \mathbb{N}$. This implies that
By Lemma 2.11 we know that the series ${\sum}_{i=0}^{\mathrm{\infty}}{s}^{i}{\psi}^{i}(d({x}_{0},{x}_{1}))$ converges. Therefore, $\{{x}_{n}\}$ is 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 (c), we get $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. Also, we have $\alpha ({x}_{n1},f({x}_{n1}))\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$ for all $n\in \mathbb{N}$. From the assumption (3.1), we have
Letting $n\to \mathrm{\infty}$, it follows that $d(f({x}^{\ast}),{x}^{\ast})=0$, 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
Therefore, we get
It follows 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. □
In view of Theorem 3.2, we have the following corollary.
Corollary 3.3 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$ be three mappings. Suppose that the following conditions hold:

(a)
f is an αadmissible;

(b)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$;

(c)
if $\{{x}_{n}\}$ is 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 $\alpha (x,f(x))\ge 1$;

(d)
f satisfies the following condition:
$$\alpha (x,f(x))\alpha (y,f(y))d(f(x),f(y))\le \psi (d(x,y))$$(3.4)
for all $x,y\in X$.
Then f has a unique fixed point ${x}^{\ast}$ in X such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.
Corollary 3.4 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$ be three mappings. Suppose that the following conditions hold:

(a)
f is an αadmissible;

(b)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$;

(c)
if $\{{x}_{n}\}$ is 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 $\alpha (x,f(x))\ge 1$;

(d)
f satisfies the following condition:
$${[d(f(x),f(y))+\xi ]}^{\alpha (x,f(x))\alpha (y,f(y))}\le \psi (d(x,y))+\frac{\xi}{s}$$(3.5)
for all $x,y\in X$, where $\xi \ge 1$.
Then f has a unique fixed point ${x}^{\ast}$ in X such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.
Corollary 3.5 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$ be three mappings. Suppose that the following conditions hold:

(a)
f is an αadmissible;

(b)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f({x}_{0}))\ge 1$;

(c)
if $\{{x}_{n}\}$ is 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 $\alpha (x,f(x))\ge 1$;

(d)
f satisfies the following condition:
$${(\alpha (x,f(x))\alpha (y,f(y))1+\xi )}^{d(f(x),f(y))}\le {\xi}^{\psi (d(x,y))}$$(3.6)
for all $x,y\in X$, where $\xi >1$.
Then f has a unique fixed point ${x}^{\ast}$ in X such that $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$.
If we set $\alpha (x,y)=1$ for all $x,y\in X$ in Theorem 3.2, we get the following results.
Corollary 3.6 Let $(X,d)$ be a complete bmetric space with coefficient s and $f:X\to X$ be a mapping. Suppose that f satisfies
for all $x,y\in X$, where $\psi \in {\mathrm{\Psi}}_{b}$. Then f has a unique fixed point in X.
If the coefficient $s=1$ in Corollary 3.6, we immediately get the following result.
Corollary 3.7 [46]
Let $(X,d)$ be a complete metric space and $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ be $(c)$comparison function. Suppose that $f:X\to X$ be a mapping satisfies
for all $x,y\in X$. Then f has a unique fixed point in X.
Remark 3.8 If $\psi (t)=kt$, where $k\in (0,1)$ in Corollary 3.7, we get the Banach contraction principle.
Next, we give an example showing that the contractive conditions in our results are independent. Also, our results are real generalizations of the Banach contraction principle in bmetric spaces and several results in literature.
Example 3.9 Let $X=[0,\mathrm{\infty})$ and $d(x,y)={xy}^{2}$ for all $x,y\in X$. Then d is a complete bmetric space on X with coefficient $s=2$. Define $f:X\to X$ by
Also, define $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by
and $\psi (t)=\frac{1}{2}t$ for all $t\ge 0$.
Now, we show that f is a generalized αψcontraction mapping. For $x,y\in X$ with
we get $x,y\in [1,\mathrm{\infty})$. Then we have
It is easy to see that f is an αadmissible mapping. There exists ${x}_{0}=2\in X$ such that
Also, we can easily to prove that condition (c) in Theorem 3.2 holds. Therefore, all of conditions in Theorem 3.2 hold. In this example, we have 1 is a unique fixed point of f and $\alpha (1,f(1))\ge 1$.
Remark 3.10 We observe that the contractive condition in Corollary 3.4 cannot be applied to this example. Indeed, for $x=1$ and $y=2$, we obtain
where $\xi =1$ and $s=2$. Therefore, Corollary 3.4 cannot be applied to this case. Also, by a similar method, we can show that Corollary 3.5 cannot be applied to this case.
Also, we can see that the fixed point result for Banach contraction principle in bmetric spaces cannot be applied to this case. Indeed, for $x=0.4$ and $y=0.5$, we get
4 The generalized UlamHyers stability in bmetric spaces
In this section, we prove the generalized UlamHyers stability in bmetric spaces for which Theorem 3.2 holds.
Let $(X,d)$ be a bmetric spaces with coefficient s and $f:X\to X$ be an operator. Let us consider the fixed point equation
and the inequality
Theorem 4.1 Let $(X,d)$ be a complete bmetric space with coefficient s. Suppose that all the hypotheses of Theorem 3.2 hold and also that the function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\phi (t):=ts\psi (t)$ is strictly increasing and onto. If $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ for all ${u}^{\ast}\in X$ which is an εsolution, then the fixed point equation (4.1) is generalized UlamHyers stable.
Proof By Theorem 3.2, we have $f({x}^{\ast})={x}^{\ast}$, that is, ${x}^{\ast}\in X$ is a solution of the fixed point equation (4.1). Let $\epsilon >0$ and ${y}^{\ast}\in X$ is an εsolution, that is,
Since ${x}^{\ast},{y}^{\ast}\in X$ are εsolution, we have
Also, we have
Now, we obtain
It follows that
Since $\phi (t):=ts\psi (t)$, we have
It implies that
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 (4.1) is generalized UlamHyers stable. □
Corollary 4.2 Let $(X,d)$ be a complete bmetric space with coefficient s. Suppose that all the hypotheses of Corollary 3.3 hold and also that the function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\phi (t):=ts\psi (t)$ is strictly increasing and onto. If $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ for all ${u}^{\ast}\in X$ which is an εsolution, then the fixed point equation (4.1) is generalized UlamHyers stable.
Corollary 4.3 Let $(X,d)$ be a complete bmetric space with coefficient s. Suppose that all the hypotheses of Corollary 3.4 hold and also that the function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\phi (t):=ts\psi (t)$ is strictly increasing and onto. If $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ for all ${u}^{\ast}\in X$ which is an εsolution, then the fixed point equation (4.1) is generalized UlamHyers stable.
Corollary 4.4 Let $(X,d)$ be a complete bmetric space with coefficient s. Suppose that all the hypotheses of Corollary 3.5 hold and also that the function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\phi (t):=ts\psi (t)$ is strictly increasing and onto. If $\alpha ({u}^{\ast},f({u}^{\ast}))\ge 1$ for all ${u}^{\ast}\in X$ which is an εsolution, then the fixed point equation (4.1) is generalized UlamHyers stable.
5 Wellposedness of a function with respect to αadmissibility in bmetric spaces
In this section, we present and prove wellposedness of a function with respect to an αadmissible mapping in bmetric spaces.
Definition 5.1 Let $(X,d)$ be a complete bmetric spaces with coefficient s and $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$. The fixed point problem of f is said to be wellposed 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 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 next theorems, we add a new condition to assure the wellposedness via αadmissibility.

(S)
If $\{{x}_{n}\}$ is 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 5.2 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$. Suppose that all the hypotheses of Theorem 3.2 and condition (S) hold. Then the fixed point equation (4.1) is wellposed with respect to α.
Proof By Theorem 3.2, there unique exists ${x}^{\ast}\in X$ such that $f({x}^{\ast})={x}^{\ast}$ and $\alpha ({x}^{\ast},f({x}^{\ast}))\ge 1$. Let $\{{x}_{n}\}$ be sequence in X such that $d({x}_{n},f({x}_{n}))\to 0$, as $n\to \mathrm{\infty}$. By condition (S), we get
Also, we get
Now, we have
ψ is continuous at 0 and $d({x}_{n},f({x}_{n}))\to 0$ as $n\to \mathrm{\infty}$. It implies that $d({x}_{n},{x}^{\ast})\to 0$ as $n\to \mathrm{\infty}$, that is, ${x}_{n}\to {x}^{\ast}$, as $n\to \mathrm{\infty}$. Therefore, the fixed point equation (4.1) is wellposed with respect to α. □
Corollary 5.3 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$. Suppose that all the hypotheses of Corollary 3.3 and condition (S) hold. Then the fixed point equation (4.1) is wellposed with respect to α.
Corollary 5.4 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$. Suppose that all the hypotheses of Corollary 3.4 and condition (S) hold. Then the fixed point equation (4.1) is wellposed with respect to α.
Corollary 5.5 Let $(X,d)$ be a complete bmetric space with coefficient s, $f:X\to X$, $\alpha :X\times X\to [0,\mathrm{\infty})$, and $\psi \in {\mathrm{\Psi}}_{b}$. Suppose that all the hypotheses of Corollary 3.5 and condition (S) hold. Then the fixed point equation (4.1) is wellposed with respect to α.
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Acknowledgements
The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU57000621). Moreover, the authors are grateful Dr. Wutiphol Sintunavarat and the reviewers for careful reading of the paper and for the suggestion, which improved the quality of this work.
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Phiangsungnoen, S., Kumam, P. Generalized UlamHyers stability and wellposedness for fixed point equation via αadmissibility. J Inequal Appl 2014, 418 (2014). https://doi.org/10.1186/1029242X2014418
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Keywords
 αadmissible mappings
 bmetric space
 fixed point
 generalized UlamHyers stability
 wellposedness