Split equality fixed point problem for quasi-pseudo-contractive mappings with applications
- Shih-sen Chang^{1}Email author,
- Lin Wang^{2} and
- Li-Juan Qin^{3}
https://doi.org/10.1186/s13663-015-0458-3
© Chang et al. 2015
Received: 1 September 2015
Accepted: 5 November 2015
Published: 17 November 2015
Abstract
In this paper, we consider a split equality fixed point problem for quasi-pseudo-contractive mappings which includes split feasibility problem, split equality problem, split fixed point problem etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.
Keywords
MSC
1 Introduction
Denote by Γ the solution set of split equality fixed point problem (1.4).
2 Preliminaries
In this section, we collect some definitions, notations, and conclusions, which will be needed in proving our main results.
Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and \(T : C \to C\) be a nonlinear mapping.
Definition 2.1
- (i)
Nonexpansive if \(\|Tx -Ty\| \le\|x - y\|\) \(\forall x, y \in C\).
- (ii)Quasi-nonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and$$\bigl\| Tx - x^{*}\bigr\| \le\bigl\| x - x^{*}\bigr\| \quad\forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
- (iii)Firmly nonexpansive if$$\|Tx -Ty\|^{2} \le\|x - y\|^{2} - \bigl\| (I -T)x - (I-T)y \bigr\| ^{2} \quad \forall x, y \in C. $$
- (iv)Firmly quasi-nonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and$$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} - \bigl\| (I- T)x \bigr\| ^{2} \quad \forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
- (v)Strictly pseudo-contractive if there exists \(k \in[0, 1)\) such that$$\|Tx - Ty\|^{2} \le\|x - y\|^{2} + k\bigl\| (I - T)x - (I - T)y \bigr\| ^{2} \quad \forall x, y \in C. $$
- (vi)
Directed if \(\operatorname{Fix}(T) \neq \emptyset\) and \(\langle Tx - x^{*}, Tx -x \rangle\le0\) \(\forall x \in C\) and \(x^{*} \in \operatorname{Fix}(T)\).
- (vii)Demicontractive if \(\operatorname{Fix}(T) \neq \emptyset\) and there exists \(k \in[0, 1)\) such that$$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} + k \|Tx - x \|^{2} \quad \forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
Remark 2.2
Remark 2.3
From the above definitions, we note that the class of demicontractive mappings is fundamental; it includes many kinds of nonlinear mappings such as the directed mappings, the quasi-nonexpansive mappings, and the strictly pseudo-contractive mappings with fixed points as special cases.
Definition 2.4
Definition 2.5
Definition 2.6
(1) A mapping \(T : C \to C\) is said to be demiclosed at 0 if, for any sequence \(\{x_{n}\} \subset C\) which converges weakly to x and with \(\|x_{n} - T(x_{n})\| \to0\), \(T(x) = x\).
(2) A mapping \(T : H \to H\) is said to be semi-compact if, for any bounded sequence \(\{x_{n}\} \subset H\) with \(\|x_{n} - Tx_{n}\| \to0\), there exists a subsequence \(\{x_{n_{i}}\} \subset\{x_{n}\}\) such that \(\{x_{n_{i}}\}\) converges strongly to some point \(x \in H\).
Lemma 2.7
Lemma 2.8
- (1)
\(\sum_{n=1}^{\infty}\gamma_{n} = \infty\);
- (2)
\(\limsup_{n \to\infty} \frac{\delta_{n}}{\gamma_{n}} \le0 \) or \(\sum_{n=1}^{\infty}|\delta_{n}| < \infty\).
Lemma 2.9
- (1)
\(\operatorname{Fix}(T) = \operatorname{Fix}(T((1 - \eta)I + \eta T)) = \operatorname{Fix}(K)\).
- (2)
If T is demiclosed at 0, then K is also demiclosed at 0.
- (3)In addition, if \(T : H \to H\) is quasi-pseudo-contractive, then the mapping K is quasi-nonexpansive, that is,$$\bigl\| Kx - u^{*}\bigr\| \le\bigl\| x - u^{*}\bigr\| \quad\forall x \in H \textit{ and } u^{*} \in \operatorname{Fix}(T) = \operatorname{Fix}(K). $$
Proof
(1) If \(x^{*} \in \operatorname{Fix}(T)\), it is obvious that \(x^{*} \in \operatorname{Fix}(T((1 - \eta)I + \eta T))\).
It is obvious that \(x \in \operatorname{Fix}(K)\) if and only if \(x \in \operatorname{Fix}(T((1 - \eta )I + \eta T))\).
The conclusion (1) is proved.
The conclusion (3) is obvious (see also [16]). □
3 Main results
- (1)
\(H_{1}\), \(H_{2}\), and \(H_{3}\) are three real Hilbert spaces. \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) are two bounded linear operators with their adjoints \(A^{*}\) and \(B^{*}\), respectively;
- (2)
\(T : H_{1} \to H_{1}\) and \(S : H_{2} \to H_{2}\) are two L-Lipschitzian and quasi-pseudo-contractive mappings with \(L \ge1\), \(\operatorname{Fix}(T) \neq \emptyset\), and \(\operatorname{Fix}(S) \neq \emptyset\).
In the sequel, we denote the strong convergence and weak convergence of a sequence \(\{x_{n}\}\) to a point \(x\in H\) by \(x_{n} \to x\) and \(x_{n} \rightharpoonup x\), respectively.
Now, we present our algorithm for finding \((x^{*}, y^{*}) \in\Gamma\).
Algorithm 3.1
Initialization: Choose \(\{\alpha_{n}\} \subset(0, 1)\). Take arbitrary \(x_{0} \in H_{1}\), \(y_{0} \in H_{2}\).
Theorem 3.2
- (i)
\(\gamma_{n} \in(0, \min(\frac{1}{\|A\|^{2}}, \frac {1}{\|B\|^{2}})) \) \(\forall n \ge1\);
- (ii)
\(0 < a < \xi_{n} < \eta_{n} < b < \frac{1}{1 + \sqrt{1 + L^{2}}} \forall n \ge1\).
Proof
First we prove the conclusion (I).
- (1)
Both K and G are quasi-nonexpansive;
- (2)
\(\operatorname{Fix}(K) = \operatorname{Fix}(T)\) and \(\operatorname{Fix}(G) = \operatorname{Fix}(S)\);
- (3)
K and G demiclosed at 0.
On the other hand, from (3.14), one gets \(u_{n} \rightharpoonup x^{*}\) and \(v_{n} \rightharpoonup y^{*}\). By (3.15) and the demiclosed property of K and G, we have \(Kx^{*} = x^{*}\) and \(Gy^{*} = y^{*}\). This implies that \(x^{*} \in \operatorname{Fix}(T)\) and \(y^{*} \in \operatorname{Fix}(S)\).
4 Applications
4.1 Application to the split equality variational inequality problem
Throughout this section, we assume that \(H_{1}\), \(H_{2}\), and \(H_{3}\) are three real Hilbert spaces. C and Q both are nonempty and closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively and assume that \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) are two bounded linear operator with its adjoint \(A^{*}\) and \(B^{*}\), respectively.
- (A1)
\(h(x,x) = 0\), for all \(x\in C\);
- (A2)
h is monotone, i.e., \(h(x,y) + h(y, x) \le0\) for all \(x, y \in C\);
- (A3)
\(\limsup_{t\downarrow0} h(tz + (1-t)x, y) \le h(x,y)\) for all \(x, y, z \in C \);
- (A4)
for each \(x\in C\), \(y \mapsto h(x,y)\) is convex and lower semi-continuous.
Proposition 4.1
[17]
- (1)
\(R_{\lambda, h}\) is single-valued;
- (2)
\(\operatorname{Fix}(R_{\lambda, h}) = VI(M,C)\), where \(VI(M,C)\) is the solution set of variational inequality (4.1) which is a nonempty closed and convex subset of C;
- (3)
\(R_{\lambda, h}\) is a firmly nonexpansive mapping. Therefore \(R_{\lambda, h}\) is demiclosed at 0.
Since \(R_{\lambda, f}\) and \(R_{\lambda, g}\) are firmly nonexpansive with \(\operatorname{Fix}(R_{\lambda, f}) \neq \emptyset\) and \(\operatorname{Fix}(R_{\lambda, g})\neq \emptyset\), the following theorem can be obtained from Theorem 3.2 immediately.
Theorem 4.2
4.2 Application to the split equality convex minimization problem
Let C be a nonempty closed convex subset of \(H_{1}\) and Q be a nonempty closed convex subset of \(H_{2}\). Let \(\phi: C \to\mathbb{R}\) and \(\psi: Q \to\mathbb{R}\) be two proper and convex and lower semi-continuous functions and \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) be two bounded linear operator with its adjoint \(A^{*}\) and \(B^{*}\), respectively.
Since \(R_{\lambda, j}\) and \(R_{\lambda, k}\) both are firmly nonexpansive with \(\operatorname{Fix}(R_{\lambda, f}) \neq \emptyset\) and \(\operatorname{Fix}(R_{\lambda, g})\neq \emptyset\), the following theorem can be obtained from Theorem 3.2 immediately.
Theorem 4.3
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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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