Abstract
Based on the concept of a \(C^{*}\)-algebra-valued b-metric space, this paper establishes some coupled fixed point theorems for mapping satisfying different contractive conditions on such space. As applications, we obtain the existence and uniqueness of a solution for an integral equation.
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1 Introduction and preliminaries
In 1989, Bakhtin [1] introduced b-metric space as a generalization of metric space. Since then, more other generalized b-metric spaces such as b-metric-like spaces [2], quasi-b-metric spaces [3] and quasi-b-metric-like spaces [4] were introduced. Recently, Ma and Jiang [5] initially introduced the concept of a \(C^{*}\)-algebra-valued b-metric space which generalized the concept of b-metric spaces, and they established certain basic fixed point theorems for self-map with contractive condition in this new setting. In 2016, Kamran et al. [6] also introduced the concept of \(C^{*}\)-algebra-valued b-metric space, and they generalized the Banach contraction principle on such spaces.
The notion of coupled fixed point was introduced by Guo and Lakshmikantham [7] in 1987. Since then, many researchers investigated coupled fixed point theorems in ordered metric spaces and have given some applications [8–12]. Recently, Cao [13] first studied some coupled fixed point theorems in the context of complete \(C^{*}\)-algebra-valued metric spaces.
Motivated by the work in [5, 6, 13–17], in this paper, we will establish coupled fixed point theorems in \(C^{*}\)-algebra-valued b-metric space. More precisely, we will prove some coupled fixed point theorems for the mapping with different contractive conditions on such spaces.
For convenience, we now recall some basic definitions, notations, and results of \(C^{*}\)-algebra. The details of \(C^{*}\)-algebras can be found in [18].
Let \(\mathbb{A}\) be an algebra. An involution on \(\mathbb{A}\) is a conjugate linear map \(a \mapsto a^{*}\) such that \((a^{*})^{*} = a\) and \((ab)^{*} = b^{*} a^{*}\) for all \(a, b \in \mathbb{A}\). The pair \((\mathbb{A}, *)\) is called a ∗-algebra. If \(\mathbb{A}\) contains the identity element \(1_{\mathbb{A}}\), then \((\mathbb{A}, *)\) is called a unital ∗-algebra. A ∗-algebra \(\mathbb{A}\) together with a complete submultiplicative norm such that \(\Vert a^{*}\Vert = \Vert a\Vert \) is said to be a Banach ∗-algebra. Moreover, if for all \(a \in \mathbb{A}\), we have \(\Vert a^{*} a\Vert = \Vert a\Vert ^{2}\) in a Banach ∗-algebra, then \(\mathbb{A}\) is known as a \(C^{*}\)-algebra. An element a of a \(C^{*}\)-algebra \(\mathbb{A}\) is positive if \(a = a^{*}\) and its spectrum \(\sigma(a) \subset \mathbb{R}_{+}\), where \(\sigma(a) = \{\lambda \in \mathbb{R}: \lambda 1_{\mathbb{A}}\mbox{-}a\mbox{ is not invertible}\}\). Each positive element a of \(C^{*}\)-algebra \(\mathbb{A}\) has a unique positive square root. The set of all positive elements will be denoted by \(\mathbb{A}_{+}\). There is a natural partial ordering on the elements of \(\mathbb{A}\) given by
If \(a \in \mathbb{A}_{+}\), then we write \(a \succeq 0_{\mathbb{A}}\), where \(0_{\mathbb{A}}\) is the zero element of \(\mathbb{A}\). In the following, we always assume that \(\mathbb{A}\) is a unital \(C^{*}\)-algebra with identity element \(1_{\mathbb{A}}\).
Let \(\mathbb{A}^{\prime} = \{a\in \mathbb{A}: ab=ba, \forall b \in \mathbb{A}\}\), and \(\mathbb{A}_{+}^{\prime} = \mathbb{A}_{+} \cap \mathbb{A}^{\prime}\). From [5, 6], we now give the definition of \(C^{*}\)-algebra-valued b-metric as follows.
Definition 1.1
Let \(\mathbb{A}\) be a \(C^{*}\)-algebra, and X be a nonempty set. Let \(b \in \mathbb{A}_{+}^{\prime}\) be such that \(\Vert b\Vert \ge 1\). A mapping \(d_{b} : X \times X \to \mathbb{A}_{+}\) is said to be a \(C^{*}\)-algebra-valued b-metric on X if the following conditions hold for all \(x, y, z \in \mathbb{A}\):
-
1.
\(d_{b}(x, y) = 0_{\mathbb{A}}\) if and only if \(x=y\);
-
2.
\(d_{b}(x, y) = d_{b}(y, x)\);
-
3.
\(d_{b}(x, y) \preceq b[d_{b}(x, z) + d_{b}(z, y)]\).
The triplet \((X, \mathbb{A}, d_{b})\) is called a \(C^{*}\)-algebra-valued b-metric space with coefficient b.
Remark 1.1
From Example 2.1 in [6], we know that a \(C^{*}\)-algebra-valued metric space is \(C^{*}\)-algebra-valued b-metric space, but the converse is not true.
Definition 1.2
Let \((X, \mathbb{A}, d_{b})\) be a \(C^{*}\)-algebra-valued b-metric space, \(x \in X\), and \(\{x_{n}\}\) a sequence in X. Then:
-
1.
\(\{x_{n}\}\) converges to x with respect to \(\mathbb{A}\) whenever for any \(\varepsilon > 0\) there is an \(N \in \mathbb{N}\) such that \(\Vert d_{b}(x_{n}, x)\Vert < \varepsilon\) for all \(n > N\). We denote this by \(\lim_{n \to \infty} x_{n} = x\) or \(x_{n} \to x\).
-
2.
\(\{x_{n}\}\) is a Cauchy sequence with respect to \(\mathbb{A}\) if for each \(\varepsilon > 0\), there is an \(N \in \mathbb{N}\) such that \(\Vert d_{b}(x_{n}, x_{m})\Vert < \varepsilon\) for all \(n, m > N\).
-
3.
\((X, \mathbb{A}, d_{b})\) is complete if every Cauchy sequence in X is convergent with respect to \(\mathbb{A}\).
Lemma 1.1
Assume that \(\mathbb{A}\) is a unital \(C^{*}\)-algebra with a unit \(1_{\mathbb{A}}\).
-
(1)
For any \(x \in \mathbb{A}_{+}\), we have \(x \preceq 1_{\mathbb{A}} \Leftrightarrow \Vert x\Vert \le 1\);
-
(2)
if \(a \in \mathbb{A}_{+}\) with \(\Vert a\Vert < \frac{1}{2}\), then \(1_{\mathbb{A}} - a\) is invertible and \(\Vert a(1_{\mathbb{A}} - a)^{-1}\Vert < 1\);
-
(3)
assume that \(a, b \in \mathbb{A}\) with \(a, b \succeq 0_{\mathbb{A}}\) and \(ab = ba\), then \(ab \succeq 0_{\mathbb{A}}\);
-
(4)
let \(a \in \mathbb{A}^{\prime}\), if \(b, c \in \mathbb{A}\) with \(b \succeq c \succeq 0_{\mathbb{A}}\), and \(1_{\mathbb{A}} - a \in \mathbb{A}_{+}^{\prime}\) is an invertible operator, then
$$(1_{\mathbb{A}} - a)^{-1} b \succeq (1_{\mathbb{A}} - a)^{-1} c; $$ -
(5)
if \(b, c \in \mathbb{A}_{h} = \{x \in \mathbb{A}: x = x^{*}\}\) and \(a \in \mathbb{A}\), then \(b \preceq c \Longrightarrow a^{*} b a \preceq a^{*} c a\);
-
(6)
if \(0_{\mathbb{A}} \preceq a \preceq b\), then \(\Vert a\Vert \le \Vert b\Vert \).
Lemma 1.2
[18]
The sum of two positive elements in a \(C^{*}\)-algebra is a positive element.
Remark 1.2
From Lemmas 1.1(3) and 1.2, we know that the condition \(b \in \mathbb{A}_{+}^{\prime}\) in Definition 1.1 is necessary, in this case, we see that \(b[d_{b}(x, z) + d_{b}(z, y)]\) is a positive element.
Definition 1.3
Let \((X, \mathbb{A}, d_{b})\) be a \(C^{*}\)-algebra-valued b-metric space. An element \((x, y) \in X \times X\) is said to be a coupled fixed point of the mapping \(T: X \times X \to X\) if \(T(x, y) = x\) and \(T(y, x) = x\).
2 Main results
In this section, we will prove some coupled fixed point theorems for mappings with contractive conditions in the setting of \(C^{*}\)-algebra-valued b-metric space.
Theorem 2.1
Let \((X, \mathbb{A}, d_{b})\) be a complete \(C^{*}\)-valued b-metric space. Assume that the mapping \(T: X \times X \to X\) satisfies the following condition:
where \(a \in \mathbb{A}\) with \(2 \Vert a\Vert ^{2} \Vert b\Vert < 1\). Then T has a unique coupled fixed point in X. Moreover, T has a unique fixed point in X.
Proof
Let \(x_{0}, y_{0} \in X\). Define two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X by the iterative scheme as
By using the condition (2.1), for \(n \in \mathbb{N}\), we obtain
where
Similarly, we get
By (2.2), (2.3), and (2.4), we have
Thus, from (2.5) and Lemma 1.1(5), we have
If \(M_{1} = 0_{\mathbb{A}}\), then from Definition 1.3 we easily know that \((x_{0}, y_{0})\) is a coupled fixed point of the mapping T. Now, let \(0_{\mathbb{A}} \preceq M_{1}\). Let \(n, m \in \mathbb{N}\) with \(m > n\), by using Definition 1.1, it follows that
Similarly, we have
Hence,
by the condition \(2 \Vert a\Vert ^{2} \Vert b\Vert < 1\) and \(\Vert b\Vert \ge 1\). Hence \(\{x_{n}\}\) and \(\{y_{n}\}\) are Cauchy sequences in X. By the completeness of \((X,\mathbb{A},d)\), there exist \(x^{*}, y^{*} \in X\) such that \(x_{n} \to x^{*}\) and \(y_{n} \to y^{*}\) as \(n \to \infty\). We now show that \(T (x^{*}, y^{*}) = x^{*}\) and \(T(y^{*}, x^{*}) = y^{*}\). From Definition 1.1 and (2.1), we get
So, \(T(x^{*}, y^{*}) = x^{*}\). Similarly, we have \(T(y^{*}, x^{*}) = y^{*}\). Thus, \((x^{*}, y^{*})\) is a coupled fixed point of T.
If there exists another coupled fixed point \((u, v)\) of T, then
which implies that
Thus, we have
which is a contradiction. Thus, \((u, v) = (x^{*}, y^{*})\), that is, the coupled fixed point is unique. Finally, we will prove that T has a unique fixed point. Since
we have
It follows from the condition \(2 \Vert a\Vert ^{2} < \frac{1}{\Vert b\Vert } \le 1\) that \(\Vert d_{b}(x^{*}, y^{*})\Vert = 0\). Hence, \(x^{*} = y^{*}\). The proof is completed. □
Remark 2.1
Taking \(b=1_{\mathbb{A}}\), Theorem 2.1 of [13] becomes a special case of Theorem 2.1.
Theorem 2.2
Let \((X, \mathbb{A}, d_{b})\) be a complete \(C^{*}\)-valued b-metric space. Assume that the mapping \(T: X \times X \to X\) satisfies the following condition:
where \(a_{1}, a_{2} \in \mathbb{A}_{+}^{\prime}\) with \(\Vert a_{1} + a_{2}\Vert \Vert b\Vert ^{2} < 1\). Then T has a unique coupled fixed point in X. Moreover, T has a unique fixed point in X.
Proof
From \(a_{1}, a_{2} \in \mathbb{A}_{+}^{\prime}\) and Lemma 1.2, we see that \(a_{1} d_{b}(T(x, y), u) + a_{2} d_{b}(T(u, v), x)\) is a positive element. Choose \(x_{0}, y_{0} \in X\). Set \(x_{n+1} = T(x_{n}, y_{n})\) and \(y_{n+1} = T(y_{n}, x_{n})\) for \(n = 0, 1, \ldots\) . Applying (2.8), we have
which implies that
Moreover, we obtain
which yields
Since \(a_{1}, a_{2}, b \in \mathbb{A}_{+}^{\prime}\), we have \(\frac{(a_{1} + a_{2}) b}{2} \in \mathbb{A}_{+}^{\prime}\) and \(\frac{(a_{1} + a_{2}) b^{2}}{2} \in \mathbb{A}_{+}^{\prime}\). Moreover, from the condition \(\Vert (a_{1} + a_{2})\Vert \Vert b\Vert ^{2} < 1\), we get
and
which implies that \((1_{\mathbb{A}} - \frac{(a_{1} + a_{2}) b}{2})^{-1} \in \mathbb{A}_{+}^{\prime} \) and \((1_{\mathbb{A}} - \frac{(a_{1} + a_{2}) b^{2}}{2})^{-1} \in \mathbb{A}_{+}^{\prime} \) with
by Lemma 1.1(2). Thus, we have by (2.11)
where
with \(\Vert h\Vert \le \Vert h b\Vert < 1\) by (2.12). Inductively, for all \(n \in \mathbb{N}\), we have
where \(m_{0} = d_{b}(x_{1}, x_{0})\). Let \(n, m \in \mathbb{N}\) with \(m > n\), by using Definition 1.1 and (2.12)-(2.14), we have
Hence \(\{x_{n}\}\) is a Cauchy sequence in X. Similarly, we can prove that \(\{y_{n}\}\) is also a Cauchy sequence in X. Since \((X, \mathbb{A}, d_{b})\) is complete, we see that \(\{x_{n}\}\) and \(\{y_{n}\}\) converge to some \(u \in X\) and \(v \in X\), respectively. In the following, we will show that \(T (u, v) = u\) and \(T(v, u) = v\). By (2.8), we get
Thus
Since \(0_{\mathbb{A}} \preceq b a_{2} \preceq (a_{1} + a_{2}) b\), we have \(\Vert a_{2} b\Vert \le \Vert (a_{1} + a_{2})b\Vert < 1\) by Lemma 1.1(6). This and (2.15) imply that \(\Vert d_{b}(T(u, v), u)\Vert = 0\). Hence \(T(u, v) = u\). Similarly, we obtain \(T(v, u) = v\). Thus \((u, v)\) is a coupled fixed point of T.
Now if \((u^{*}, v^{*})\) is another coupled fixed point of T, then
so, we get
which implies that \(\Vert d_{b}(u, u^{*})\Vert = 0\), then we have \(u = u^{*}\). Similarly, we can get \(v = v^{*}\). Hence, the coupled fixed point is unique. Moreover, we will prove the uniqueness of fixed points of T. By (2.8), we have
then
which yields \(u = v\). This completes the proof. □
Remark 2.2
Taking \(b=1_{\mathbb{A}}\), Theorem 2.3 of [13] becomes a special case of Theorem 2.2.
Theorem 2.3
Let \((X, \mathbb{A}, d_{b})\) be a complete \(C^{*}\)-valued b-metric space. Assume that the mapping \(T: X \times X \to X\) satisfies the following condition:
where \(a_{1}, a_{2} \in \mathbb{A}_{+}^{\prime}\) with \((\Vert a_{1}\Vert + \Vert a_{2}\Vert ) \Vert b\Vert < 1\). Then T has a unique coupled fixed point in X. Moreover, T has a unique fixed point.
Proof
Since \(a_{1}, a_{2} \in \mathbb{A}_{+}^{\prime}\), we see that \(a_{1} d_{b}(T(x, y), x) + a_{2} d_{b}(T(u, v), u)\) is a positive element. Similar to the proof of Theorem 2.2, we construct \(\{x_{n}\}\) and \(\{y_{n}\}\) such that \(x_{n+1} = T(x_{n}, y_{n})\) and \(y_{n+1} = T(y_{n}, x_{n})\). By (2.16), we obtain
which implies that
Since \(a_{1}, a_{2} \in \mathbb{A}_{+}^{\prime}\) with \(\Vert a_{1}\Vert + \Vert a_{2}\Vert < \frac{1}{\Vert b\Vert } \le 1\), we have \(1_{\mathbb{A}} - a_{2}\) is invertible and \((1_{\mathbb{A}} - a_{2})^{-1} a_{1} \in \mathbb{A}_{+}^{\prime}\). Hence
Inductively, for all \(n \in \mathbb{N}\), we have
where \(k = (1_{\mathbb{A}} - a_{2})^{-1} a_{1}\) and \(m_{0} = d_{b}(x_{1}, x_{0})\). Since \(\Vert a_{1}\Vert \Vert b\Vert + \Vert a_{2}\Vert \le (\Vert a_{1}\Vert + \Vert a_{2}\Vert ) \Vert b\Vert < 1\), we have
And \(\Vert k\Vert \le \Vert b k\Vert < 1\) by Lemma 1.1(6).
Let \(n, m \in \mathbb{N}\) with \(m > n\), by using Definition 1.1, (2.16), and (2.17), we have
Hence \(\{x_{n}\}\) is a Cauchy sequence. Similarly, we can prove that \(\{y_{n}\}\) is also a Cauchy sequence. Since \((X, \mathbb{A}, d_{b})\) is complete, there are \(u, v \in X\) such that \(x_{n} \to u\) and \(y_{n} \to v\) as \(n \to \infty\). In the following, we will show that \(T (u, v) = u\) and \(T(v, u) = v\). From (2.16), we get
which implies that
Thus \(d_{b}(T(u, v), u) = 0_{\mathbb{A}}\). Equivalently, \(T(u, v) = u\). Similarly, we can obtain \(T(v, u) = v\).
Now if \((u^{*}, v^{*})\) is another coupled fixed point of T, then
so, we get \(d_{b}(u, u^{*}) = 0_{\mathbb{A}}\), which yields \(u^{*} = u\). Similarly, we have \(v^{*} = v\). Thus, \((u, v)\) is the unique coupled fixed point of T. Finally, we will show the uniqueness of fixed points of T. By (2.16), we have
which implies that \(u=v\). □
Remark 2.3
Taking \(b=1_{\mathbb{A}}\) [13], Theorem 2.2 becomes a special case of Theorem 2.3.
3 Application
As an application of coupled fixed point theorems on complete \(C^{*}\)-algebra-valued b-metric spaces, we prove here the existence and uniqueness of a solution for a Fredholm nonlinear integral equation.
Let E be a Lebesgue-measurable set with \(m(E) < \infty\) and \(X = L^{\infty}(E)\) denote the class of essentially bounded measurable functions on E.
Consider the Hilbert space \(L^{2}(E)\). Let the set of all bounded linear operators on \(L^{2}(E)\) be denoted by \(B(L^{2}(E))\). Obviously, \(B(L^{2}(E))\) is a \(C^{*}\)-algebra with usual operator norm.
Let \(K_{1}, K_{2}: E \times E \to \mathbb{R}\), assume that there exist two continuous functions \(f, g: E \times E \to \mathbb{R}\) and a constant \(\alpha \in (0, \frac{1}{4})\) such that for all \(x, y \in X\) and \(u, v \in E\), we have
Example 3.1
Consider the integral equation
Assume that (3.1) and (3.2) hold. Moreover, if
then the integral equation (3.3) has a unique solution in \(L^{\infty}(E)\).
Proof
Define \(d_{b}: X \times X \to B(L^{2}(E))\) as follows:
where \(\pi_{h}: L^{2}(E) \to L^{2}(E)\) is the product operator given by
Working in the same lines as in [5], Example 3.2, we easily see that \((X, B(L^{2}(E)), d_{b})\) is a complete \(C^{*}\)-valued b-metric space with \(b = 2 \cdot 1_{B(L^{2}(E))}\).
Let \(T: X \times X \to X\) be
Then by (3.1), (3.2), and (3.4), we obtain
Set \(a = \sqrt{2} \alpha 1_{B(L^{2}(E))}\), then \(a \in B(L^{2}(E))\) and \(\Vert a\Vert = \sqrt{2} \alpha < \frac{1}{2 \sqrt{2}} = \frac{1}{\sqrt{2} \Vert b\Vert }\). Hence, all the conditions of Theorem 2.1 hold. Applying Theorem 2.1, we see that the integral equation (3.3) has a unique solution in \(L^{\infty}(E)\). □
References
Bakhtin, IA: The contraction principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 26-37 (1989)
Alghamdi, MA, Hussain, N, Salimi, P: Fixed point and coupled fixed point theorems on b-metric-like spaces. J. Inequal. Appl. 2013, 402 (2013)
Shah, MH, Hussain, N: Nonlinear contractions in partially ordered quasi b-metric spaces. Commun. Korean Math. Soc. 27(1), 117-128 (2012)
Zhu, CX, Chen, CF, Zhang, X: Some results in quasi-b-metric-like spaces. J. Inequal. Appl. 2014, 437 (2014)
Ma, Z, Jiang, L: C ∗-Algebra-valued b-metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2015, 222 (2015)
Kamran, T, Postolache, M, Ghiura, A, Batul, S, Ali, R: The Banach contraction principle in C ∗-algebra-valued b-metric spaces with application. Fixed Point Theory Appl. 2016, 10 (2016)
Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11, 623-632 (1987)
Samet, B: Coupled fixed point theorems for s generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 72, 4508-4517 (2010)
Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for \((\psi, \phi)\)-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl. 63, 298-309 (2012)
Berinde, V: Coupled fixed point theorems for ϕ-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 75, 3218-3228 (2012)
Asgari, MS, Mousavi, B: Solving a class of nonlinear matrix equations via the coupled fixed point theorem. Appl. Math. Comput. 259, 364-373 (2015)
Luong, NV, Thuan, NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 74, 983-992 (2011)
Cao, T: Some coupled fixed point theorems in C ∗-algebra-valued metric spaces (2016) arXiv:1601.07168v1
Huang, H, Radenovic, S: Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications. J. Nonlinear Sci. Appl. 8(5), 787-799 (2015)
Latif, A, Kadelburg, Z, Parvaneh, V, Roshan, JR: Some fixed point theorems for G-rational Geraghty contractive mappings in ordered generalized b-metric spaces. J. Nonlinear Sci. Appl. 8(6), 1212-1227 (2015)
Yamaod, O, Sintunavarat, W, Cho, YJ: Common fixed point theorems for generalized cyclic contraction pairs in b-metric spaces with applications. Fixed Point Theory Appl. 2015, 164 (2015)
Kadelburg, Z, Radenovic, S: Pata-type common fixed point results in b-metric and b-rectangular metric spaces. J. Nonlinear Sci. Appl. 8(6), 944-954 (2015)
Murphy, GJ: C ∗-Algebras and Operator Theory. Academic Press, London (1990)
Douglas, RG: Banach Algebra Techniques in Operator Theory. Springer, Berlin (1998)
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The author thanks the editor and reviewers for valuable comments and suggestions. This work is supported by the Natural Science Foundation of China (11571136 and 11271364).
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Bai, C. Coupled fixed point theorems in \(C^{*}\)-algebra-valued b-metric spaces with application. Fixed Point Theory Appl 2016, 70 (2016). https://doi.org/10.1186/s13663-016-0560-1
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DOI: https://doi.org/10.1186/s13663-016-0560-1