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UlamHyers stability and wellposedness of fixed point problems for αλcontractions on quasi bmetric spaces
Fixed Point Theory and Applications volume 2016, Article number: 1 (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).
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
Let X be a nonempty and \(s\geq1\). Let \(q : X \times X \rightarrow [0,\infty)\) be a function which satisfies:

(q1)
\(q(x, y) = 0\) if and only if \(x=y\),

(q2)
\(q(x, y)\leq s[ q(x, z)+q(z, y)] \).
Then q is called a quasi bmetric and the pair \((X,q)\) is called a quasi bmetric space. The number s is called the coefficient of \((X,q)\).
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
Let \(X=\{1,2,3\}\). Define the function \(q:X\times X\to[0,\infty)\) by
for all \(n,m\in X\), with \((n,m)\neq(1,2)\) and \(q(1,2)=\frac{16}{9}\). Then \((X,q)\) is a quasi bmetric space with coefficient \(s=2\). It is neither a bmetric space since \(q(1,2)=\frac{16}{9}\neq q(2,1)=\frac {1}{4}\), nor a quasi metric space since \(q(1,2)=\frac{16}{9}>\frac{10}{9}=q(1,3)+q(3,2)\).
Example 1.2
Let \(X=\mathbb{R}\). Take the real numbers \(p>1\) and \(a,b>0\) such that \(a\neq b\). Define the function \(q:X\times X\to[0,\infty)\) by
Then \((X,q)\) is a quasi bmetric space with coefficient \(s=2^{p1}\). It is neither a bmetric space since \(q(1,0)=a^{p}\neq q(0,1)=b^{p}\), nor a quasi metric space since \(q(1,1)=(2a)^{p}>2a^{p}=q(1,0)+q(0,1)\).
Example 1.3
Let \(X=\mathbb{R}\). Take the real numbers \(p>1\) and \(a>0\). Define the function \(q:X\times X\to[0,\infty)\) by
Then \((X,q)\) is a quasi bmetric space with coefficient \(s=2^{p1}\). It is neither a bmetric space since \(q(1,0)=1\neq q(0,1)=(1+a)^{p}\), nor a quasi metric space since \(q(1,1)=2^{p}>2=q(1,0)+q(0,1)\).
Some topological aspects of a quasi bmetric space are as follows.
Definition 1.2
Let \((X,q)\) be a quasi bmetric space, \(\{x_{n}\}\) be a sequence in X and \(x\in X\). The sequence \(\{x_{n}\}\) converges to x if and only if
Remark 1.2
In a quasi bmetric space, the limit for a convergent sequence is unique. If \(x_{n}\rightarrow u\), we have (in general) \(\lim_{n\rightarrow \infty} q(x_{n},y)\neq q(u,y)\) for all \(y\in X\). We only mention that
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
Let \((X,q)\) be a quasi bmetric space. We say that:

(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
Let \((X, q)\) be a quasi bmetric space and \(T : X \rightarrow X\) be a given mapping. Suppose that T is continuous at \(u \in X\). Then, for all sequence \(\{x_{n}\}\) in X such that \(x_{n} \rightarrow u\), we have \(Tx_{n} \rightarrow Tu\), that is,
In 2012, Samet et al. [13] introduced the notion of αadmissible maps.
Definition 1.7
[13]
For a nonempty set X, let \(T: X\rightarrow X\) and \(\alpha: X\times X\rightarrow[0,\infty)\) be given mappings. T is said αadmissible if for all \(x,y\in X\), we have
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
Let \((X,d)\) be a metric space and \(f: X \rightarrow X\) be a given mapping. We say that f is an αλcontractive mapping if there exist two functions \(\alpha: X\times X \rightarrow[0,\infty)\) and \(\lambda: X\rightarrow[0,1)\) for which \(\lambda(f(x))\leq\lambda (x)\) for all \(x\in X\), such that
for all \(x, y \in X\).
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
Let \((X,q)\) be a quasi bmetric space and \(T: X\rightarrow X\) be a given mapping. We say that T is an αλcontraction if there exist \(\alpha: X\times X\rightarrow[0,\infty)\) and \(\lambda: X\rightarrow [0,\frac{1}{s})\) satisfying \(\lambda(Tx)\leq\lambda(x)\) for all \(x\in X\), such that
for all \(x, y \in X\).
The following examples illustrate Definition 1.9.
Example 1.4
Going back to Example 1.1 where \(X=\{1,2,3\}\) is endowed with the quasi bmetric \(q:X\times X\to[0,\infty)\) defined by
for all \(n,m\in X\) with \((n,m)\neq(1,2)\) and \(q(1,2)=\frac{16}{9}\).
Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
Since \(q(T2,T1)=q(2,3)=1>\frac{1}{4}=q(2,1)\), T is not a Banach contraction on X. Now, we show that T is an αλcontraction, where \(\lambda: X\rightarrow[0,\frac{1}{2})\) is defined by
To this aim, we distinguish the following cases:
Case 1: If \(n=1\), \(m=2\), then we have
Case 2: If \(n=1\), \(m=3\), then we get
Case 3: If \(n=2\), \(m=3\), then we get
Moreover, (4) is verified for all \(n=m\) and for all \(n,m\in X\) such that \(\alpha(n,m)=0\). Since \(\lambda(T1)=\lambda(3)=\lambda(1)\), \(\lambda(T2)=\lambda(2)\), and \(\lambda(T3)=\lambda(1)=\lambda(3)\), the mapping T is an αλcontraction.
Example 1.5
Let \(X=\{0,1\}\cup[2,\infty)\). Consider the mapping \(q:X\times X\to [0,\infty)\) defined by
for all \(x,y\in X\) with \((x,y)\neq(0,1)\) and \(q(0,1)=9\). We mention that \((X,q)\) is a quasi bmetric space with \(s=2\). Note that q is not a quasi metric since \(q(0,1)=9>5=q(0,2)+q(2,1)\).
Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
We have
that is, T is not a Banach contraction on X. Now, we show that T is an αλcontraction where \(\lambda: X\rightarrow [0,\frac{1}{2})\) is defined by \(\lambda(x)=\frac{1}{3}\) for all \(x\in X\). To this aim, we distinguish the following cases:
Case 1: If \(x,y\in X\) such that \(\alpha(x,y)=1\), then we have \(x>y\geq2\). It follows that
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.
Auxiliary results
We have the following useful lemmas.
Lemma 2.1
Let \(X=\mathbb{R}\) and \(p>1\) be a real number. Consider the function \(q:X\times X\to[0,\infty)\) by
where a and b are positive reals such that \(a\neq b\). Then there exist two positive constants c and d such that
for all \(x,y\in X\).
Proof
Without loss of generality, we suppose that \(a< b\). To this aim, we distinguish the following cases:
Case 1: If \(x,y\in X\) such that \(x>y\), then we have
that is,
Case 2: If \(x,y\in X\) such that \(x\leq y\), then we get
that is,
Consequently, we obtain (6), with \(c=a^{p}\) and \(d=b^{p}\). □
Lemma 2.2
Let \(X=\mathbb{R}\) be endowed with quasi bmetric q given by (5). Take \(T:X\rightarrow X\). We have
where \(\cdot \) is the standard metric on X.
Proof
Assume that T is continuous on \((X,\cdot )\). Consider \(\{x_{n}\}\) in X such that \(x_{n}\rightarrow x\) in \((X,q)\). Then
By (6), we get \(x_{n}\rightarrow x\) in \((X,\cdot )\). We deduce \(Tx_{n}\rightarrow Tx\) in \((X,\cdot )\). Again, by (6)
that is, T is continuous on \((X,q)\).
Similarly, if T is continuous on \((X,q)\), then by (6), T is continuous on \((X, \cdot )\). □
Fixed point theorems
In this section, we shall state and prove our main results.
Theorem 3.1
Let \((X,q)\) be a complete quasi bmetric space and \(T: X\to X\) be an αλcontraction. Suppose that

(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 \((X,q)\).
Then T has a fixed point.
Proof
By assumption (ii), there exists a point \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\). Take \(x_{n}=T^{n} x_{0}\) for all \(n\geq0\). From (i), we have by induction
Applying (4) with \(x=x_{0}\) and \(y=x_{1}\) and using (7), we get
We apply again (4) with \(x=x_{1}\) and \(y=x_{2}\) and using (7) together with the propriety of λ, we get
A similar argument leads to
The same procedure allows us to write
Since \(\lambda(x_{0}) \) and \(\lambda(x_{1})\) are in \([0,1)\),
We shall prove that \(\{x_{n}\}\) is a Cauchy sequence in \((X,q)\).
First, we claim that \(\{x_{n}\}\) is a rightCauchy sequence in the quasi bmetric space \((X,q)\). Using (q2) and (8), we have for all \(n,k\in\mathbb{N}\)
Since \(s\lambda(x_{0})<1\),
It follows that \(\{x_{n}\}\) is a rightCauchy sequence in the quasi bmetric space \((X,q)\). Similarly, using (9), we see that \(\{ x_{n}\}\) is a leftCauchy sequence in the quasi bmetric space \((X,q)\). We deduce that \(\{x_{n}\}\) is a Cauchy sequence in the quasi bmetric space \((X,q)\).
Since \((X,q)\) is complete, the sequence \(\{x_{n}\}\) converges to some \(u\in X\), that is,
The continuity of T yields
By uniqueness of the limit, we get \(Tu=u\). Therefore, u is a fixed point of T. □
Using the same techniques we obtain the following result.
Theorem 3.2
Let \((X,q)\) be a complete bmetric space and \(T: X\to X\) be an αλcontraction. Suppose that

(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 \((X,q)\).
Then T has a fixed point.
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.
Suppose that:

(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 \((X,q)\).
Then there exists \(u\in X\) such that \(u=Tu\).
Corollary 3.2
(Theorem 10, [1])
Let \((X, d)\) be a complete metric space and \(T: X\to X\) be an αλcontraction satisfying the following conditions:

(i)
T is αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);

(iii)
T is continuous on \((X,d)\).
Then there exists \(u\in X\) such that \(u=Tu\).
We may replace the continuity hypothesis of T in Theorem 3.1 (resp. Theorem 3.2) by one of the following hypotheses:
 (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
Let \((X,q)\) be a complete quasi bmetric space and \(T: X\to X\) be an αλcontraction. Suppose that:

(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.
Then there exists \(u\in X\) such that \(u=Tu\).
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 have, for all \(k\geq0\),
Taking \(x=x_{n(k)}\) and \(y=u\) in (4), we obtain
Then we get for all \(k\geq0\)
Letting \(k\rightarrow\infty\) in (15), we have
This yields \(Tu=u\). This completes the proof. □
We also state the following result. Its proof is very immediate.
Theorem 3.4
Let \((X,q)\) be a complete bmetric space and \(T: X\to X\) be an αλcontraction. Suppose that:

(i)
T is αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);

(iii)
(R) holds.
Then there exists \(u\in X\) such that \(u=Tu\).
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.
Suppose that:

(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.
Then there exists \(u\in X\) such that \(u=Tu\).
Corollary 3.4
(Theorem 12, [1])
Let \((X, d)\) be a complete metric space and \(T: X\to X\) be an αλcontraction satisfying the following conditions:

(i)
T is αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);

(iii)
(H) holds.
Then there exists \(u\in X\) such that \(u=Tu\).
We provide the following examples.
Example 3.1
Let \(X=[0,\infty)\). Consider \(q(x,y)=(\max\{(xy),2(yx)\})^{2}\) for all \(x,y\in X\). We mention that \((X,q)\) is a complete quasi bmetric space with \(s=2\). Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
Now, we show that T is an αλcontraction where \(\lambda: X\rightarrow[0,\frac{1}{2})\) is defined by \(\lambda(x)=\frac{1}{4}\) for all \(x\in X\). To this aim, we distinguish the following cases:
Case 1: \(x,y\in X\) such that \(x\geq y\) and \(\alpha (x,y)=1\). We have
Case 2: \(x,y\in X\) such that \(x< y\) and \(\alpha (x,y)=1\). Similarly, we get
Hence, (4) is verified and, since \(\lambda(Tx)=\lambda(x)\) for all \(x\in X\), the mapping T is an αλcontraction.
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
Let \(X=[0,1]\). Consider \(q(x,y)=(\max\{xy,2(yx)\})^{2}\) for all \(x,y\in X\). We mention that \((X,q)\) is a quasi bmetric space with \(s=2\). Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
We have
that is, T is not a Banach contraction on X. Now, we show that T is an αλcontraction where \(\lambda: X\rightarrow [0,\frac{1}{2})\) is defined by
First of all, we show that \(\lambda(Tx)\leq\lambda(x)\) for all \(x\in X\). For \(x=1\), we have \(\lambda(T1)=\lambda(1)\). Also, for \(x\in[0,1)\), we have
Again, we show that (4) is verified. To this aim, we distinguish the following cases:
Case 1: If \(x,y\in[0,1)\) such that \(x\leq y\), then we have
Case 2: If \(x,y\in[0,1)\) such that \(x>y\), then we obtain
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.
To prove uniqueness of the fixed point given in Theorem 3.1 (resp. Theorem 3.2, Theorem 3.3, Theorem 3.4), we need to take one of the following additional hypotheses:
 (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
We argue by contradiction, that is, there exist \(u,v\in X\) such that \(u=Tu\) and \(v=Tv\) with \(u\neq v\). By (4) and the fact that \(\alpha(u,v)\geq1\), we get
which is a contradiction. Hence, \(u=v\). □
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
Suppose that there exist \(u,v\), two fixed points of T. By condition (V), there exists \(z\in X\) such that \(\min\{\alpha(u,z),\alpha(z,v)\} \geq1\). Since T is αadmissible, it follows that
We have
By induction, we obtain
A similar reasoning shows that
On the other side, we have
which yields
Passing to the limit as \(n\to\infty\), we obtain
and so \(u=v\). □
The following examples illustrate Theorem 3.5.
Example 3.3
Let \(X=\{0,1,2,3\}\). Consider the function \(q:X\times X\to[0,\infty)\) defined by
for all \(n,m\in X\) with \((n,m)\neq(0,1)\) and \(q(0,1)=9\). We mention that \((X,q)\) is a complete quasi bmetric space with \(s=2\).
Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
We have
that is, T is not a Banach contraction on X. Now, we show that T is an αλcontraction where \(\lambda: X\rightarrow [0,\frac{1}{2})\) is defined by \(\lambda(n)=\frac{1}{4}\) for all \(n\in X\). To this aim, we distinguish the following cases:
Case 1: If \(n,m\in X\) such that \(\alpha(n,m)=1\), then \(n,m\in\{2,3\}\). So
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
Going back again to Example 3.2 where \(X=[0,1]\) is endowed with the quasi bmetric \(q(x,y)=(\max\{xy,2(yx)\})^{2}\) for all \(x,y\in X\). Define \(T:X\to X\) and \(\alpha:X\times X\to[0,\infty)\) by
We know that T is an αλcontraction where \(\lambda: X\rightarrow[0,\frac{1}{2})\) is defined by
Therefore, all hypotheses of Theorem 3.3 are satisfied. Here, 0 is the unique fixed point of T.
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]
Let X be a nonempty set endowed with a graph G. We say that \(T: X\rightarrow X\) weakly preserves edges of G if for all \(x,y\in X\)
Definition 4.2
Let \((X,q)\) be a quasi bmetric space endowed with a graph G. We say that:

(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
Let \((X,q)\) be a quasi bmetric space endowed with a graph G.

(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
Let \((X,q)\) be a quasi bmetric space endowed with a graph G. A mapping \(T:X\rightarrow X\) is said to be a Gλcontraction if there exists a function \(\lambda: X\rightarrow[0,\frac{1}{s})\) for which \(\lambda(Tx)\leq \lambda(x)\) for all \(x\in X\) such that
for all \(x, y \in X\) satisfying \((x,y)\in E(G)\).
We obtain the following results.
Theorem 4.1
Let \((X,q)\) be a quasi bmetric space endowed with a graph G and \(T: X\to X\) be a Gλcontraction. Suppose that:

(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.
Then T has a fixed point.
Proof
Define the function \(\alpha:X\times X\to[0,\infty)\) by
It is easy to see that all conditions of Theorem 3.1 are satisfied and so T has a fixed point. □
Corollary 4.1
Let \((X,q)\) be a complete quasi bmetric space endowed with a graph G and \(T: X\to X\) be a Gλcontraction. Suppose that:

(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 \((X,q)\).
Then T has a fixed point.
Corollary 4.2
Let \((X,q)\) be a complete quasi bmetric space endowed with a graph G and \(T: X\to X\) be a Gλcontraction. Suppose that:

(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 \((X,q)\).
Then T has a fixed point.
Theorem 4.2
Let \((X,q)\) be a quasi bmetric space endowed with a graph G and \(T: X\to X\) be a Gλcontraction. Suppose that:

(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.
Then T has a fixed point.
Corollary 4.3
Let \((X,q)\) be a complete quasi bmetric space endowed with a graph G and \(T: X\to X\) be a Gλcontraction. Suppose that:

(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.
Then T has a fixed point.
Application
In this section, we apply Theorem 4.2 to the existence of a solution of an integral equation.
Let \(X=C([a,b],\mathbb{R})\) be the set of real continuous functions defined on \([a,b]\). Consider the quasi bmetric \(q_{\infty}:X\times X\to[0,\infty)\) given as follows:
We mention that \((X,q)\) is a complete quasi bmetric space with \(s=2\). Also, suppose that X is endowed with a graph G. Consider the integral equation as follows:
where \(f:[a,b]\to\mathbb{R}\) and \(K:[a,b]\times[a,b]\times\mathbb{R} \to\mathbb{R}\) are given continuous functions. Let \(T: X\to X\) be a mapping defined by
It is clear that x is a solution of (24) if and only if x is a fixed point of T.
We have the following result.
Theorem 5.1
Suppose that there exists \(r\in[0,\frac{1}{\sqrt{2}})\) such that for all \(t,s\in[a,b]\) we have
for all \(x,y\in X\) satisfying \((x,y)\in E(G)\).
Also, suppose that:

(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.
Then the integral equation (24) has a solution in \(C([a,b],\mathbb{R})\).
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 \(x\in X\).
Let \((x,y)\in E(G)\), then we get
where \(\ (xy)^{2}\_{\infty}=\sup_{t\in[a,b]}(x(t)y(t))^{2}\). It follows that
Hence, all conditions of Theorem 4.2 are satisfied and hence T has a fixed point in X. □
UlamHyers stability
Let \((X,q)\) be a quasi bmetric space and \(T:X\rightarrow X\) be a given mapping. Let us consider the fixed point equation
and the inequality (for \(\varepsilon>0\))
We say that the fixed point problem (27) is UlamHyers stable in the framework of a quasi bmetric space if there exists \(c>0\), such that for each \(\varepsilon> 0\) and an εsolution \(v^{*}\in X\), that is, \(v^{*}\) satisfies the inequality (28), there exists a solution \(u^{*}\in X\) of the fixed point equation (27) such that
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
By Theorem 3.5 (resp. Theorem 3.6), we have a unique \(u\in X\) such that \(u=Tu\), that is, \(u\in X\) is a solution of the fixed point equation (27). Let \(\varepsilon>0\) and \(v\in X\) be an εsolution, that is,
Since \(q(u,Tu)=q(u,u)=0\leq\varepsilon\), u and v are εsolutions. By hypothesis, we get \(\alpha(u,v)\geq1\) and so
We deduce
where \(c=\frac{s}{1s\lambda(u)}>0\). Consequently, the fixed point problem of T is UlamHyers stable. □
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
Let \((X,q)\) be a quasi bmetric space and \(T:T\rightarrow X\) be a given mapping. The fixed point problem (27) is said to be well posed if:

(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 \(\lim_{n\rightarrow \infty}q(x_{n},u^{*})= \lim_{n\rightarrow\infty}q(u^{*},x_{n})=0\).
In the following results, we need new conditions to ensure the wellposedness via αadmissibility:
 (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.
Also, suppose that:

(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 \(n\geq N\).
Then the fixed point equation (27) is well posed.
Proof
By Theorem 3.5 (resp. Theorem 3.6), we have a unique \(u\in X\) such that \(u=Tu\), that is, u is a solution of the fixed point equation (27). Let \(\{x_{n}\}\) be a sequence in X such that \(\lim_{n\rightarrow\infty}q(x_{n},Tx_{n})= \lim_{n\rightarrow \infty}q(Tx_{n},x_{n})=0\). From condition (S_{1}), we have \(\alpha (x_{n},u)\geq1\) and \(\alpha(u,x_{n})\geq1\), for all n. Using (q2) and the fact that \(\alpha(x_{n},u)\geq1\) in (4), one writes
By condition (ii) of Theorem 7.1, we get
that is,
Letting \(n\rightarrow\infty\), we obtain
Again, using \(\alpha(u,x_{n})\geq1\)
We deduce
Letting \(n\rightarrow\infty\), we obtain
By (30) and (31), the fixed point problem (27) is well posed. □
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. □
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Acknowledgements
The authors gratefully acknowledge the support from King Abdulaziz City for Science and Technology (KACST), Kingdom of Saudi Arabia, Project Number (SG: 3639).
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MSC
 47H10
 54H25
 46J10
Keywords
 fixed point
 αλcontraction
 UlamHyers stability
 quasi bmetric space