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Fixed point results, generalized UlamHyers stability and wellposedness via αadmissible mappings in bmetric spaces
Fixed Point Theory and Applications volume 2014, Article number: 188 (2014)
Abstract
In this paper, we prove the existence and uniqueness of a fixed point for some new classes of contractive mappings via αadmissible mappings in the framework of bmetric spaces. We also present an example to illustrate the usability of the obtained results. The generalized UlamHyers stability and wellposedness of a fixed point equation via αadmissible mappings in bmetric spaces are given.
MSC:46S40, 47S40, 47H10.
1 Introduction and preliminaries
1.1 The bmetric 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 selfmapping 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 bmetric spaces as a generalization of metric spaces and proved the contraction mapping principle in bmetric 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 bmetric 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 bmetric on X and a pair (X,d) is called a bmetric space with coefficient s.
Remark 1.2 If we take s=1 in the above definition, then bmetric spaces turn into ordinary metric spaces. Hence, the class of bmetric spaces is larger than the class of metric spaces.
For examples of bmetric 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 bmetric 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 bmetric on X with coefficient s>1.
Definition 1.5 (Boriceanu et al. [14])
Let (X,d) be a bmetric 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 bmetric 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])
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 functions as given by Berinde [17] in order to extend some fixed point results to the class of bmetric 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}}{\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 UlamHyers 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 UlamHyers stability. Several authors have considered UlamHyers 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 bmetric spaces.
Definition 1.12 Let (X,d) be a bmetric space with coefficient s, and let f:X\to X be an operator. By definition, the fixed point equation
is said to be generalized UlamHyers stable in the framework of a bmetric 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 εsolution {v}^{\ast}\in X, that is,
there exists a solution {u}^{\ast}\in X of the fixed point equation (1.1) such that
If \phi (t):=ct for all t\in [0,\mathrm{\infty}), where c>0, then (1.1) is said to be UlamHyers stable in the framework of a bmetric space.
Remark 1.13 If s=1, then Definition 1.12 reduces to the generalized UlamHyers 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 UlamHyers stability.
1.3 αAdmissible mappings
In 2012, Samet et al. [27] introduced the concept of αadmissible 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 αadmissible mapping for singlevalued 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 αadmissible mapping if it satisfies the following condition:
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
and
Then f is αadmissible.
Example 1.16 Let X=\mathbb{R}. Define f:X\to X and \alpha :X\times X\to [0,\mathrm{\infty}) by
and
Then f is αadmissible.
Recently Bota et al. [34] proved the existence and uniqueness of fixed point theorems. They also studied the generalized UlamHyers stability results via an αadmissible mapping in a bmetric 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 αadmissible mappings. We also give some examples to show that our fixed point theorems for new types of contractive mappings are independent. The generalized UlamHyers stability and wellposedness of a fixed point equation for these classes in the framework of bmetric spaces are proved.
2 Fixed point results in bmetric spaces
In this section, we prove the existence and uniqueness of fixed point theorems in a bmetric space.
Theorem 2.1 Let (X,d) be a complete bmetric 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 αadmissible;

(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
Since f is αadmissible and
we deduce that
By induction, we get
Next, we will show that \{{x}_{n}\} is a Cauchy sequence in X. For each n\in \mathbb{N}, we have
By repeating the process above, we get
For m,n\in \mathbb{N} with m>n, we have
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
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
for all n\in \mathbb{N}. Letting n\to \mathrm{\infty}, since 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.
Next, we prove the uniqueness of the fixed point of f. Let {y}^{\ast} be another fixed point of f such that
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. □
Theorem 2.2 Let (X,d) be a complete bmetric 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 αadmissible;

(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
Since f is αadmissible and \alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f({x}_{0}))\ge 1, we get
By induction, we get
Next, we will show that \{{x}_{n}\} is a Cauchy sequence in X. For each n\in \mathbb{N}, we have
Now, we get
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
for all n\in \mathbb{N}. Letting n\to \mathrm{\infty} and 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.
Next, we prove the uniqueness of the fixed point of f. Let {y}^{\ast} be another fixed point of f such that
It follows that
This shows 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 bmetric 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 αadmissible;

(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
Since f is αadmissible and \alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f({x}_{0}))\ge 1, we get
By induction, we get
Next, we will show that \{{x}_{n}\} is a Cauchy sequence in X. For each n\in \mathbb{N}, we have
Since \xi >1, we get
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
for all n\in \mathbb{N}. Since \xi >1, it implies that
Letting n\to \mathrm{\infty}, since 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.
Next, we prove the uniqueness of the fixed point of f. Let {y}^{\ast} be another fixed point of f such that
It follows that
Since \xi >1, we have
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 bmetric 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 bmetric space with coefficient s=2>1, but it is not a usual metric space.
Let us 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. Clearly, f is an αadmissible 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.
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
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 UlamHyers stability in bmetric spaces
In this section, we prove the generalized UlamHyers stability in bmetric spaces which corresponds to Theorems 2.1, 2.2 and 2.3.
Theorem 3.1 Let (X,d) be a complete bmetric 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):=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 (1.1) is generalized UlamHyers 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 εsolution, that is,
Since {x}^{\ast},{y}^{\ast}\in X are an εsolution, we have
Now, we obtain
It follows that
Since \phi (t):=ts\psi (t), we have
This 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 (1.1) is generalized UlamHyers stable. □
Theorem 3.2 Let (X,d) be a complete bmetric 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):=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 (1.1) is generalized UlamHyers 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 εsolution, that is,
Since {x}^{\ast},{y}^{\ast}\in X are an εsolution, we have
Now, we obtain
It follows that
and then
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 (1.1) is generalized UlamHyers stable. □
Theorem 3.3 Let (X,d) be a complete bmetric 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):=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 (1.1) is generalized UlamHyers 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 εsolution, that is,
Since {x}^{\ast},{y}^{\ast}\in X are an εsolution, we have
Now, we obtain
Since \xi >1, we have
It follows that
Suppose that \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 (1.1) is generalized UlamHyers stable. □
4 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 4.1 Let (X,d) be a bmetric 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 wellposedness via αadmissibility.

(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 bmetric 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
Now, we have
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 bmetric 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
Now, we have
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 bmetric 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
Now, we have
Since \xi >1, we have
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. □
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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.
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Phiangsungnoen, S., Sintunavarat, W. & Kumam, P. Fixed point results, generalized UlamHyers stability and wellposedness via αadmissible mappings in bmetric spaces. Fixed Point Theory Appl 2014, 188 (2014). https://doi.org/10.1186/168718122014188
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DOI: https://doi.org/10.1186/168718122014188