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UlamHyers stability and wellposedness of fixed point problems for αλcontractions on quasi bmetric spaces
 Abdelbasset Felhi^{1},
 Slah Sahmim^{1} and
 Hassen Aydi^{2, 3}Email author
https://doi.org/10.1186/s1366301504912
© Felhi et al. 2015
Received: 12 November 2015
Accepted: 20 December 2015
Published: 4 January 2016
Abstract
In this paper, we establish some fixed point results for αλcontractions in the class of quasi bmetric spaces. To illustrate the obtained results, we provide some examples and an application on a solution of an integral equation. We also study the stability of UlamHyers and wellposedness of a fixed point problem. Our obtained results give an answer to an open problem of Kutbi and Sintunavarat (Abstr. Appl. Anal. 2014:268230, 2014).
Keywords
 fixed point
 αλcontraction
 UlamHyers stability
 quasi bmetric space
MSC
 47H10
 54H25
 46J10
1 Introduction and preliminaries
By replacing the triangular inequality by a rectangular one, Czerwik [2] introduced a generalized metric space, named a bmetric space. Since then, several (common) fixed point papers have been obtained. For example, see [3–7]. Also, by lifting the symmetric condition, a quasi metric space generalizes the concept of a metric space. For some known fixed point results on these spaces, we refer to [8–12]. This paper deals with a combination of a bmetric and a quasi metric.
First, the definition of a quasi bmetric space is given as follows:
Definition 1.1
 (q1)
\(q(x, y) = 0\) if and only if \(x=y\),
 (q2)
\(q(x, y)\leq s[ q(x, z)+q(z, y)] \).
Remark 1.1
Any quasi metric space or any bmetric is a quasi bmetric space, but the converse is not true in general.
We state some examples of quasi bmetrics.
Example 1.1
Example 1.2
Example 1.3
Some topological aspects of a quasi bmetric space are as follows.
Definition 1.2
Remark 1.2
Definition 1.3
Let \((X,q)\) be a quasi bmetric space. A sequence \(\{x_{n}\}\) in X is said leftCauchy if and only if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(q(x_{n},x_{k})<\varepsilon\) for all \(n\geq k>N\).
Definition 1.4
Let \((X,q)\) be a quasi bmetric space. A sequence \(\{x_{n}\}\) in X is said rightCauchy if and only if for every \(\varepsilon>0\) there exists a positive integer \(N=N_{\varepsilon}\) such that \(q(x_{n},x_{k})<\varepsilon\) for all \(k\geq n>N\).
Definition 1.5
Let \((X,q)\) be a quasi bmetric space. A sequence \(\{x_{n}\}\) in X is said Cauchy if and only if for every \(\varepsilon>0\) there exists a positive integer \(N=N_{\varepsilon}\) such that \(q(x_{n},x_{k})<\varepsilon \) for all \(k,n>N\).
Remark 1.3
A sequence \(\{x_{n}\}\) in a quasi bmetric space is Cauchy if and only if it is leftCauchy and rightCauchy.
Definition 1.6
 (1)
\((X,q)\) is leftcomplete if and only if each leftCauchy sequence in X is convergent.
 (2)
\((X,q)\) is rightcomplete if and only if each rightCauchy sequence in X is convergent.
 (3)
\((X,q)\) is complete if and only if each Cauchy sequence in X is convergent.
Lemma 1.1
In 2012, Samet et al. [13] introduced the notion of αadmissible maps.
Definition 1.7
[13]
Using and generalizing the above concept, many authors established some (common) fixed point results. We may cite [14–18].
Very recently, Kutbi and Sintunavarat [1] introduced a new class of contractive mappings known as αλcontractions.
Definition 1.8
Starting from a question of Ulam [19] in 1940, the stability problem of functional equations concerns the stability of group homomorphisms. In 1941, Hyers [20] presented a partial answer for a question of Ulam in the case of Banach spaces. The above type of stability is known as UlamHyers stability. Since then, many researchers extended and generalized the notion of the UlamHyers stability for fixed point problems. For example, see [21–23].
Now, we introduce the concept of an αλcontractive mapping in the setting of quasi bmetric spaces.
Definition 1.9
The following examples illustrate Definition 1.9.
Example 1.4
Example 1.5
Case 2: If \((x,y)\in X\) such that \(\alpha(x,y)=0\), then (4) is verified.
Thus, (4) is satisfied and since \(\lambda(Tx)=\lambda(x)\) for all \(x\in X\), so the mapping T is an αλcontraction.
In this paper, we are interested in UlamHyers stability and the wellposedness of the fixed point problem concerning αλcontraction mappings in the setting of quasi bmetric spaces. Our results are proper extensions and generalizations of results of Kutbi and Sintunavarat [1] on quasi bmetric spaces. Some examples and an application are also considered.
2 Auxiliary results
We have the following useful lemmas.
Lemma 2.1
Proof
Without loss of generality, we suppose that \(a< b\). To this aim, we distinguish the following cases:
Lemma 2.2
Proof
Similarly, if T is continuous on \((X,q)\), then by (6), T is continuous on \((X, \cdot )\). □
3 Fixed point theorems
In this section, we shall state and prove our main results.
Theorem 3.1
 (i)
T is an αadmissible mapping;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous on \((X,q)\).
Proof
Using the same techniques we obtain the following result.
Theorem 3.2
 (i)
T is an αadmissible mapping;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous on \((X,q)\).
Considering \(s=1\) in Theorem 3.1 (resp. Theorem 3.2), we have
Corollary 3.1
Let \((X, q)\) be a complete quasi metric space and \(T: X\to X\) be an αλcontraction.
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous on \((X,q)\).
Corollary 3.2
(Theorem 10, [1])
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous on \((X,d)\).
 (H):

If \(\{x_{n}\}\) is a sequence in X such that \(\alpha(x_{n},x_{n+1})\geq1\) and \(\alpha(x_{n+1},x_{n})\geq1\) for all n and \(x_{n} \rightarrow x\in X\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{ x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x)\geq1\), for all k.
 (R):

If \(\{x_{n}\}\) is a sequence in X such that \(\alpha(x_{n},x_{n+1})\geq 1\) for all n and \(x_{n} \rightarrow x\in X\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x)\geq1\), for all k.
Theorem 3.3
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
(H) holds.
Proof
Following the proof of Theorem 3.1, the sequence \(\{x_{n}\}\) is Cauchy and converges to some \(u\in X\) in \((X,q)\). Remember that (7) holds, so from condition (iii), there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},u)\geq1\), for all k. We shall show that \(u=Tu\).
We also state the following result. Its proof is very immediate.
Theorem 3.4
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);
 (iii)
(R) holds.
Considering \(s=1\) in Theorem 3.3 (resp. Theorem 3.4), we have
Corollary 3.3
Let \((X, q)\) be a complete quasi metric space and \(T: X\to X\) be an αλcontraction.
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
(H) holds.
Corollary 3.4
(Theorem 12, [1])
 (i)
T is αadmissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);
 (iii)
(H) holds.
We provide the following examples.
Example 3.1
Note that T is αadmissible. Since T is continuous on \((X, \cdot )\) where \(\cdot \) is the standard metric on X, by Lemma 2.2, T is continuous on \((X,q)\). We mention that \(\alpha(1,T1)=\alpha (T1,1)=1\) and so condition (ii) of Theorem 3.1 is verified. Hence, all hypotheses of Theorem 3.1 hold. Note that \(u=0\) and \(v=2\ln(\frac{3}{2})\) are the two fixed points of T.
Example 3.2
Case 3: If \((x,y)\notin[0,1)^{2}\), then we have \(\alpha(x,y)=0\), and so (4) is verified.
Thus, (4) is satisfied and the mapping T is an αλcontraction.
Note that T is αadmissible. By Lemma 2.2, T is not continuous on \((X,q)\), then Theorem 3.1 is not applicable. Also, it is easy to see that \(\alpha(\frac{1}{2},T\frac{1}{2})=\alpha(T\frac{1}{2},\frac{1}{2})=1\), and so condition (ii) of Theorem 3.3 is verified. Now, we show that condition (H) holds. Let \(\{x_{n}\}\) be a sequence in X such that \(\alpha(x_{n},x_{n+1})\geq 1\) and \(\alpha(x_{n+1},x_{n+})\geq1\) for all n and \(x_{n} \rightarrow u\) in \((X,q)\). Then \(\{x_{n}\}\subset[0,\frac{1}{2}]\) and \(x_{n} \rightarrow u\) in \((X,\cdot )\). Thus, \(u\in[0,\frac{1}{2}]\) and so \(\alpha(x_{n},u)=\alpha(u,x_{n})=1\) for all n.
Therefore, all hypotheses of Theorem 3.3 are satisfied. Here, \(\{ 0,1\}\) is the set of fixed points of T.
 (U):

For all \(x, y \in F(T)\), we have \(\alpha(x,y) \geq1\), where \(F(T)\) denotes the set of fixed points of T.
 (V):

For all \(x, y \in F(T)\), there exists \(z\in X\) such that \(\min\{\alpha(x,z),\alpha(z,y)\}\geq1\).
Theorem 3.5
Adding condition (U) to the hypotheses of Theorem 3.1 (resp. Theorem 3.2, Theorem 3.3, Theorem 3.4), we see that u is the unique fixed point of T.
Proof
Theorem 3.6
Adding condition (V) to the hypotheses of Theorem 3.1 (resp. Theorem 3.2, Theorem 3.3, Theorem 3.4), we see that u is the unique fixed point of T.
Proof
The following examples illustrate Theorem 3.5.
Example 3.3
Case 2: If \(n,m\in X\) such that \(\alpha(n,m)=0\), then (4) is satisfied.
Thus, (4) holds and since \(\lambda(Tn)=\lambda(n)\) for all \(n\in X\), so the mapping T is an αλcontraction.
Note that T is αadmissible. In fact, let \(n,m\in X\) such that \(\alpha(n,m)\geq1\), then \(n,m\in\{2,3\}\), and so \(\alpha(Tn,Tm)=\alpha(2,2)=1\). Moreover, T is continuous on \((X,q)\). In fact if \(\{x_{n}\}\) is a sequence in X such that \(x_{n} \rightarrow u\) in \((X,q)\), it easy to see that there exists \(N\in\mathbb{N}\) such that \(x_{n}=u\) for all \(n\geq N\) and so \(Tx_{n}=Tu\) for all \(n\geq N\). It follows that \(\lim_{n\to\infty }q(Tx_{n},Tu)=0\), that is, T is continuous on \((X,q)\). Also, since \(\alpha(3,T3)=\alpha(3,2)=1\), and \(\alpha(T3,3)=\alpha(2,3)=1\), condition (ii) of Theorem 3.3 is verified. Therefore, all hypotheses of Theorem 3.3 are satisfied. Here, 2 is the unique fixed point of T.
Example 3.4
4 Fixed point results in quasi bmetric spaces endowed with a graph
Recently, Jachymski [24] introduced the concept of a Gcontraction in the setting of metric spaces endowed with a graph. Using this notion, he proved some fixed point results. In this paragraph, we introduce a new class of contractive mappings in the setting of quasi bmetric spaces endowed with a graph. First, we recall some notations and definitions.
Let \((X,q)\) be a quasi bmetric space and \(\Delta=\{(x,x):x\in X\}\) denote the diagonal of the cartesian product \(X\times X\). Following [24], a directed graph G such that the set \(V(G)\) of its vertices coincides with X and the set \(E(G)\) of its edges contains all loops, i.e., \(\Delta\subset E(G)\). Also, we assume that G has no parallel edges and we can identify G with the pair \((V(G),E(G))\). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
Definition 4.1
[24]
Definition 4.2
 (1)
\((X,q)\) is Gcomplete if \(\{x_{n}\}\) is a Cauchy sequence in X such that \((x_{n},x_{n+1}),(x_{n+1},x_{n})\in E(G)\) for all n, then \(\{ x_{n}\}\) converges in \((X,q)\).
 (2)
\(T:X\rightarrow X\) is Gcontinuous if for each sequence \(\{x_{n}\} \) such that \((x_{n},x_{n+1}),(x_{n+1},x_{n})\in E(G)\) for all n and \(x_{n}\to x\), then \(Tx_{n}\rightarrow Tx\) in \((X,q)\).
Remark 4.1
 (1)
If \((X,q)\) is a complete, then it is Gcomplete.
 (2)
If \(T:X\rightarrow X\) is continuous on \((X,q)\), then it is Gcontinuous.
We introduce the notion of a Gλcontractive mapping in the class of quasi bmetric spaces endowed with a graph G.
Definition 4.3
We obtain the following results.
Theorem 4.1
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
T is Gcontinuous on \((X,q)\);
 (iv)
\((X,q)\) is Gcomplete.
Proof
Corollary 4.1
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
T is Gcontinuous on \((X,q)\).
Corollary 4.2
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
T is continuous on \((X,q)\).
Theorem 4.2
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
if \(\{x_{n}\}\) is a sequence in X such that \((x_{n},x_{n+1}),(x_{n+1},x_{n})\in E(G)\) for all n, then there exists \(\{x_{n(k)}\}\) a subsequence of \(\{x_{n}\}\) such that \((x_{n(k)},u)\in E(G)\) for all k;
 (iv)
\((X,q)\) is Gcomplete.
Corollary 4.3
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
if \(\{x_{n}\}\) is a sequence in X such that \((x_{n},x_{n+1}),(x_{n+1},x_{n})\in E(G)\) for all n, then there exists \(\{x_{n(k)}\}\) a subsequence of \(\{x_{n}\}\) such that \((x_{n(k)},u)\in E(G)\) for all k.
5 Application
In this section, we apply Theorem 4.2 to the existence of a solution of an integral equation.
We have the following result.
Theorem 5.1
 (i)
T weakly preserves edges of G;
 (ii)
there exists \(x_{0}\in X\) such that \((x_{0},Tx_{0}),(Tx_{0},x_{0})\in E(G)\);
 (iii)
if \(\{x_{n}\}\) is a sequence in X such that \((x_{n},x_{n+1}),(x_{n+1},x_{n})\in E(G)\) for all n, then there exists \(\{x_{n(k)}\}\) a subsequence of \(\{x_{n}\}\) such that \((x_{n(k)},u)\in E(G)\) for all k.
Proof
Let \(Tx(t)=f(t)+\int_{a}^{b} K(t,s,x(s))\,ds\). We shall show that it is a Gλcontraction where \(\lambda(x)=r^{2}\) for all \(x\in X\).
6 UlamHyers stability
Theorem 6.1
Let \((X, q)\) be a complete quasi bmetric space with coefficient s. Suppose that all the hypotheses of Theorem 3.5 (resp. Theorem 3.6) hold and \(\alpha(w,z)\geq1\) for all εsolutions w, z, then the fixed point equation (27) is UlamHyers stable.
Proof
7 Well fixed point problem
Many mathematicians are interested in the concept of wellposedness of a fixed point problem. For instance, see [1, 25–27]. As in [9], we start to characterize the concept of the wellposedness in the context of quasi bmetric spaces as follows.
Definition 7.1
 (1)
T has a unique fixed point \(u^{*}\in X\);
 (2)
for any sequence \(\{x_{n}\}\subseteq X\) with \(\lim_{n\rightarrow\infty}q(x_{n},Tx_{n})= \lim_{n\rightarrow \infty}q(Tx_{n},x_{n})=0\), then we have \(\lim_{n\rightarrow \infty}q(x_{n},u^{*})= \lim_{n\rightarrow\infty}q(u^{*},x_{n})=0\).
 (S_{1}):

if \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty} q(x_{n},Tx_{n})= \lim_{n\rightarrow \infty} q(Tx_{n},x_{n})=0\), then \(\alpha(x_{n},u^{*})\geq1\) and \(\alpha (u^{*},x_{n})\geq1\) for all n where \(u^{*}\) is a fixed point of T;
 (S_{2}):

if \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty} q(Tx_{n},x_{n})=0\), then \(\alpha(u^{*},x_{n})\geq1\) for all n where \(u^{*}\) is a fixed point of T.
Theorem 7.1
Let \((X, q)\) be a complete quasi bmetric space with coefficient s and \(T: X\rightarrow X\) be a given mapping. Suppose that all the hypotheses of Theorem 3.5 (resp. Theorem 3.6) hold.
 (i)
(S_{1}) holds;
 (ii)
if \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty} q(x_{n},Tx_{n})= \lim_{n\rightarrow\infty} q(Tx_{n},x_{n})=0\), then there exists \(N\in\mathbb{N}\) such that \(\lambda(x_{n})\leq\lambda (x_{N})\), for all \(n\geq N\).
Proof
Theorem 7.2
Let \((X, q)\) be a complete bmetric space with coefficient s and \(T: X\rightarrow X\) be a given mapping. Suppose that all the hypotheses of Theorem 3.5 (resp. Theorem 3.6) hold. If (S_{2}) holds, then the fixed point equation (27) is well posed.
Proof
The proof is similar to that of Theorem 7.1. □
Declarations
Acknowledgements
The authors gratefully acknowledge the support from King Abdulaziz City for Science and Technology (KACST), Kingdom of Saudi Arabia, Project Number (SG: 3639).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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