Existence and convergence theorems of fixed points for multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings in hyperbolic spaces
- Shih-sen Chang^{1},
- Ravi P Agarwal^{2}Email author and
- Lin Wang^{1}
https://doi.org/10.1186/s13663-015-0339-9
© Chang et al. 2015
Received: 2 March 2015
Accepted: 25 May 2015
Published: 11 June 2015
Abstract
The purpose of this paper is to introduce the concepts of multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings and propose a classical Kuhfitting-type iteration (Kuhfitting in Pac. J. Math. 97(1):137-139, 1981) for finding a common fixed point of the SKC-, KSC-, SCS- and C-type multi-valued mappings in the setting of hyperbolic spaces. Under suitable conditions some Δ-convergence theorems and strong convergence theorems for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a finite family of SKC-, KSC-, SCS- and C-type multi-valued mappings are proved. The results presented in the paper extend and improve some recent results announced in the current literature.
Keywords
MSC
1 Introduction and preliminaries
In 2010, Nanjaras et al. [2] gave some characterization of existing fixed points for mappings with condition (C) in the framework of \(\operatorname{CAT}(0)\) spaces. In 2012, Dhompongsa et al. [3] proved some strong convergence theorems for multi-valued nonexpansive mappings in \(\operatorname{CAT}(0)\) space. In 2011, the notion of C-condition was generalized by Karapinar and Tas [4], and some new fixed point theorems were obtained in the setting of Banach spaces.
More recently, in Ghoncheh and Razani [5], the notion of C-condition introduced in [4] was generalized to the case of multi-valued version and some existence theorems of fixed point for these mappings were proved in a Ptolemy metric space.
The purpose of this paper is first to introduce the concepts of multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings and then to propose a classical Kuhfitting-type iteration [6] for finding a common fixed point for such kind of multi-valued mappings in the setting of hyperbolic spaces (see the definition below). Under suitable conditions some Δ-convergence theorem and strong convergence theorems are proved for the iterative sequence generated by the proposed scheme to approximate a common fixed point. The results presented in the paper extend and improve some recent results announced in the current literature [1–15].
For the purpose let us first recall some definitions, notations and conclusions which will be needed in proving our main results.
- (i)
\(d(u,W(x,y,\alpha))\leq\alpha d(u,x)+(1-\alpha)d(u,y)\),
- (ii)
\(d(W(x,y,\alpha),W(x,y,\beta))=|\alpha-\beta|d(x,y)\),
- (iii)
\(W(x,y,\alpha)=W(y,x,(1-\alpha))\),
- (iv)
\(d(W(x,z,\alpha),W(y,w,\alpha))\leq(1-\alpha)d(x,y)+\alpha d(z,w)\),
A nonempty subset K of a hyperbolic space X is said to be convex if \(W(x,y,\alpha)\in K\) for all \(x,y\in K\) and \(\alpha\in[0,1]\). The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [16], Hadamard manifolds as well as \(\operatorname{CAT}(0)\) spaces in the sense of Gromov (see [17]).
A map \(\eta:(0,\infty)\times(0,2]\rightarrow(0,1]\), which provides such \(\delta=\eta(r,\epsilon)\) for given \(r>0\) and \(\epsilon\in (0,2]\), is known as a modulus of uniform convexity of X. η is said to be monotone if it decreases with r (for fixed ϵ), i.e., for any given \(\epsilon>0\) and for any \(r_{2}\geq r_{1}>0\), we have \(\eta(r_{2},\epsilon)\leq\eta(r_{1},\epsilon)\).
In the sequel, let \((X,d)\) be a metric space and K be a nonempty subset of X. We shall denote by \(F(T)=\{x\in K: Tx=x\}\) the fixed point set of a mapping T.
If \(T: K \to2^{K}\) is a multi-valued mapping, we shall use \(F(T) =\{x\in K: x\in Tx\}\) to denote the fixed point set of T in K.
Remark 1.1
It is well known that each weakly compact convex subset of a Banach space is proximal. As well as each closed convex subset of a uniformly convex Banach space is also proximal.
Definition 1.2
- (i)nonexpansive if for all \(x,y\in K\),$$H(Tx,Ty)\leq d(x,y); $$
- (ii)quasi-nonexpansive, if \(F(T)\neq\emptyset\) and$$H(Tx,Tp)\leq d(x,p),\quad \forall p\in F(T), x\in K. $$
Definition 1.3
- (i)T is said to be SCC-type ifwhere$$\frac{1}{2}d(x, Tx)\le d(x, y) \quad \mbox{implies}\quad H(Tx, Ty) \le M(x, y),\quad \forall x, y \in K, $$$$M(x, y) = \max \bigl\{ d(x, y), d(x, Tx), d(y,Ty), d(x, Ty), d(y, Tx)\bigr\} ; $$
- (ii)T is said to be SKC-type ifwhere$$\frac{1}{2}d(x, Tx) \le d(x, y)\quad \mbox{implies}\quad H(Tx, Ty) \le N(x, y),\quad \forall x, y \in K, $$$$N(x, y) = \max \biggl\{ d(x, y), \frac{1}{2}\bigl\{ d(x, Tx)+ d(y,Ty)\bigr\} , \frac {1}{2}\bigl\{ d(x, Ty)+ d(y,Tx)\bigr\} \biggr\} ; $$
- (iii)T is said to be KSC-type if$$\frac{1}{2}d(x, Tx) \le d(x, y) \quad \mbox{implies}\quad H(Tx, Ty) \le \frac{1}{2}\bigl\{ d(x, Tx)+ d(y,Ty)\bigr\} , \quad \forall x, y \in K; $$
- (iv)T is said to be CSC-type if$$\frac{1}{2}d(x, Tx) \le d(x, y) \quad \mbox{implies} \quad H(Tx, Ty) \le \frac{1}{2}\bigl\{ d(x, Ty)+ d(y,Tx)\bigr\} ,\quad \forall x, y \in K; $$
- (v)T is said to be C-type if$$\frac{1}{2}d(x, Tx) \le d(x, y) \quad \mbox{implies}\quad H(Tx, Ty) \le d(x,y),\quad \forall x, y \in K. $$
Remark 1.4
- 1.
It is obvious that each SKC-type, KSC-type, CSC-type and C-type multi-valued mapping is a special case of an SCC-type multi-valued mapping. As well as each KSC-type, CSC-type and C-type multi-valued mapping is a special case of an SKC-type multi-valued mapping.
- 2.
Each multi-valued nonexpansive mapping is a special case of a C-type multi-valued mapping, so it is a special case of SKC-type and SCC-type multi-valued mappings.
Proposition 1.5
If \(T : K \to \mathit{CB}(K)\) is a multi-valued SKC-type mapping with \(F(T) \neq \emptyset\), then T is a multi-valued quasi-nonexpansive mapping.
Proof
(a) If \(N(p,x) = d(p, x)\), then \(H(Tp, Tx) \le d(p, x)\). The conclusion holds.
This completes the proof of Proposition 1.5. □
Remark 1.6
Since each KSC-type, CSC-type and C-type multi-valued mapping is an SKC-type multi-valued mapping, it follows from Proposition 1.5 that if their fixed point set is nonempty, then all of them are multi-valued quasi-nonexpansive mappings.
Example 1.7
(Examples of SKC-type multi-valued mapping [4])
In order to introduce the concept of Δ-convergence in the general setting of hyperbolic spaces [19], we first recall some definition and conclusions.
It is known that each uniformly convex Banach space and each \(\operatorname{CAT}(0)\) space enjoy the property that ‘each bounded sequence has a unique asymptotic center with respect to closed convex subsets.’ This property also holds in a complete uniformly convex hyperbolic space. This can been seen from the following.
Lemma 1.8
[20]
Let \((X,d,W)\) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η. Then every bounded sequence \(\{x_{n}\}\) in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Recall that a sequence \(\{x_{n}\}\) in X is said to ‘Δ-converge to \(x\in X\)’ if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case, we write \(\Delta\mbox{-} \lim_{n\to\infty}x_{n}=x\) and call x the Δ-limit of \(\{x_{n}\}\).
Lemma 1.9
[15]
Lemma 1.10
Proof
In fact, the conclusion of Lemma 1.10 can be obtained from the uniqueness of the asymptotic center for each complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. □
By a similar method as given in [19], we can also prove the following lemma.
Lemma 1.11
Let \((X,d, W)\) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η and \(\{x_{n}\}\) be a bounded sequence in X with \(A(\{x_{n}\})=\{p\}\). Suppose that \(\{u_{n}\}\) is a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\})=\{u\}\), and the sequence \(\{d(x_{n},u)\}\) is convergent, then \(p=u\).
Lemma 1.12
- (1)
\(H(Tx,Tu_{x}) \le d(x,Tx) = d(x,u_{x})\);
- (2)
either \(\frac{1}{2}d(x, Tx) \le d(x, y)\) or \(\frac{1}{2}d(u_{x}, Tu_{x}) \le d(y, u_{x})\);
- (3)either \(H(Tx, Ty) \le N(x,y)\) or \(H(Ty, Tu_{x}) \le N(y, u_{x})\), where$$ \left \{ \textstyle\begin{array}{l} N(x,y) : = \max\{d(x,y), \frac{1}{2}\{d(x, Tx) + d(y, Ty)\}, \frac {1}{2}\{d(x, Ty) + d(y, Tx)\}\}, \\ N(y, u_{x}): = \max\{d(y, u_{x}), \frac{1}{2}\{d(y, Ty) + d(u_{x}, Tu_{x})\}, \\ \hphantom{N(y, u_{x}): =} \frac{1}{2}\{d(y, Tu_{x}) + d(u_{x}, Ty)\}\}. \end{array}\displaystyle \right . $$(1.9)
Proof
(a) If \(N(x,u_{x}) = d(x, u_{x})\), then \(H(Tx,Tu_{x})\leq d(x, u_{x})\). The conclusion of Lemma 1.12 holds.
Conclusion (1) is proved.
It is obvious that conclusion (3) is a consequence of conclusion (2).
Next we prove conclusion (2).
Lemma 1.13
Proof
(I) Now we consider the first case: \(H(Tx, Ty) \le N(x,y)\).
(II) Now we consider the second case, i.e., \(H(Ty, Tu_{x}) \le N(y, u_{x})\).
This completes the proof of Lemma 1.13. □
Lemma 1.14
Let \((X,d, W)\) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η, K be a nonempty closed and convex subset of X and \(T: K\to P(K)\) be an SKC-type multi-valued mapping with convex values. Suppose \(\{x_{n}\}\) is a sequence in K such that \(\Delta\mbox{-} \lim_{n\to\infty}x_{n}=z\) and \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\). Then z is a fixed point of T.
Proof
2 Main results
Now we are in a position to give the following existence and approximation results.
Theorem 2.1
Proof
The proof of Theorem 2.1 is divided into three steps as follows.
Step 1. First we prove that \(\lim_{n\to\infty}d(x_{n},p)\) exists for each \(p\in\mathcal{F}\).
This completes the proof of (2.6).
Step 3. Finally, we prove that the sequence \(\{x_{n}\} \Delta\)-converges to a common fixed point of \(\mathcal{F}\).
Denote by \(W_{\omega}(x_{n})=\bigcup_{\{u_{n}\}\subset \{x_{n}\}}A(\{u_{n}\})\). Firstly, we show that \(W_{\omega}(x_{n})\subset\mathcal{F}\). Indeed, if \(u\in W_{\omega}(x_{n})\), then there exists a subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{u_{n}\})=\{u\}\). By Lemma 1.8, there exists a subsequence \(\{u_{n_{k}}\}\) of \(\{u_{n}\}\) such that \(\Delta\mbox{-} \lim_{k\to\infty}u_{n_{k}}=p\in K\). Since \(\lim_{n\to\infty}d(x_{n},T_{i}x_{n})=0\) (\(i=1,2,\ldots,m\)), it follows from Lemma 1.14 that \(p\in\mathcal{F}\). So \(\lim_{n\to\infty}d(x_{n},p)\) exists. By Lemma 1.11, we have that \(p=u\in\mathcal{F}\). This implies that \(W_{\omega}(x_{n})\subset\mathcal{F}\).
Next, let \(\{u_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\})=\{u\}\) and \(A\{x_{n}\}=\{v\}\). Since \(u\in W_{\omega}(x_{n})\subset\mathcal{F}\), and \(\lim_{n\to\infty}d(x_{n},u)\) exists, by Lemma 1.11 it follows that \(v=u\). This implies that \(W_{\omega}(x_{n})\) contains only one point. Again since \(W_{\omega}(x_{n})\subset\mathcal{F}\) and \(W_{\omega}(x_{n})\) contains only one point and \(\lim_{n\to\infty}d(x_{n},q)\) exists for each \(q\in \mathcal{F}\), we known that \(\{x_{n}\}\) Δ-converges to a common fixed point of \(T_{i}\) (\(i=1,2,\ldots,m\)). The proof is completed. □
Theorem 2.2
Let \((X, d, W)\) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η and K be a nonempty compact convex subset of X. Let \(T_{i}: K\to \mathit{CB}(K)\) (\(i=1,2,\ldots,m\)) be a finite family of SKC-type multi-valued mappings with nonempty convex-values. Suppose that \(\mathcal{F}=\bigcap^{m}_{i=1}F(T_{i})\neq\emptyset\) and \(T_{i}(p)=\{p\}\) for each \(p\in\mathcal{F}\). Let \(\{\alpha_{n,i}\}\subset [a,b]\subset(0,1)\) (\(i=1,2,\ldots,m\)) and \(\{x_{n}\}\) be the sequence as given in Theorem 2.1. Then \(\{x_{n}\}\) converges strongly to a point in \(\mathcal{F}\).
Proof
This completes the proof. □
Lemma 2.3
[14]
- (1)
\(F(T) = F(P_{T})\);
- (2)
\(P_{T} (p) = \{p\}\) for each \(p \in F(T)\);
- (3)
for each \(x \in K\), \(P_{T}(x)\) is a closed subset of \(T(x)\);
- (4)
\(d(x, Tx) = d(x, P_{T} (x))\) for each \(x \in K\);
- (5)
\(P_{T}\) is a multi-valued mapping from K to \(P(K)\).
Theorem 2.4
Proof
By virtue of Lemma 2.3, we know that the mapping \(P_{T_{i}}\), \(i = 1,2, \ldots, m\), defined by (2.25) possesses the following properties: for each \(i =1, 2, \ldots, m\), \(P_{T_{i}}: K \to P(K)\) is an SKC-type multi-valued mapping with \(\mathcal{F} = \bigcap_{i=1}^{m} F(P_{T_{i}}) = \bigcap_{i=1}^{m} F(T_{i})\neq\emptyset\) and for each \(i = 1,2, \ldots, m\), \(P_{T_{i}}(p) = \{p\}\) for each \(p \in \mathcal{F}\). Replacing the mappings \(T_{i}\) by \(P_{T_{i}}\) in Theorem 2.1, \(i = 1, 2, \ldots, m\), then all the conditions in Theorem 2.1 are satisfied. Therefore the conclusion of Theorem 2.4 can be obtained from Theorem 2.1 immediately. □
3 An application to the image recovery
The image recovery problem is formulated as to find the nearest point in the intersection of a family of closed convex subsets from a given point by using corresponding metric projection of each subset. In this section, we consider this problem for two subsets of a complete \(\operatorname{CAT}(0)\) space.
Theorem 3.1
Proof
Since \((X,d)\) is a \(\operatorname{CAT}(0)\) space, it is a uniformly convex hyperbolic space with a monotone modulus of uniform convexity \(\eta= \frac{\varepsilon^{2}}{8}\), and \(W(x, y, \alpha) = \alpha x \oplus(1-\alpha)y\) for all \(x, y \in X\) and \(\alpha\in[0, 1]\). Further, since \(P_{1}\) and \(P_{2}\) are metric projections, they are single-valued SKC-type mappings. Further, we also get \(F(P_{1}) = C_{1}\) and \(F(P_{2}) = C_{2}\). Thus, letting \(T_{1} = P_{1}\) and \(T_{2} = P_{2}\) in Theorem 2.1, we know that all conditions in Theorem 2.1 are satisfied. Therefore the desired result can be obtained from Theorem 2.1 immediately. □
4 A numerical example
Declarations
Acknowledgements
The authors would like to express their thanks to the referees for their helpful comments and advises. 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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