Best proximity point for the proximal nonexpansive mapping on the starshaped sets
- Jianren Chen^{1}Email author,
- Song Xiao^{1},
- Hui Wang^{1} and
- Shiwen Deng^{1}
https://doi.org/10.1186/s13663-015-0268-7
© Chen et al.; licensee Springer. 2015
Received: 18 August 2014
Accepted: 19 January 2015
Published: 5 February 2015
Abstract
The existence of the best proximity point for the proximal nonexpansive mapping on starshaped sets is studied. Our results are established without the assumptions of continuity, affinity or the P-property. Finally, as applications of the theorems, analogs for the nonexpansive mappings are also given.
Keywords
best proximity point proximal nonexpansive mapping starshaped set nonexpansive map1 Introduction
We can find an early classical work in Ky Fan [1], and afterward, there have been many interesting results such as in Reich [2], Prolla [3], Sehgal and Singh [4, 5], Vetrivel and Veeramani [6], Sadiq and Veeramani [7], Kirk, Reich and Veeramani [8], Eldred, Kirk and Veeramani [9], Eldred and Veeramani [10], and many others.
Recently, M. Gabeleh introduced a new notion which is called the proximal nonexpansive mapping in [11].
There are two proximity point theorems for the proximal nonexpansive mapping, proved in [11].
The first theorem is as follows.
Theorem 1.1
[11]
- (a)
T is continuous affine proximal nonexpansive.
- (b)
\(T(A_{0})\) is contained in \(B_{0}\).
- (c)
g is an isometry.
- (d)
\(A_{0}\) is contained in \(g(A_{0})\).
In this theorem, T is assumed to be continuous affine. To remove this assumption, the second theorem is given to replace it with another assumption: that the pair \((A,B)\) has the P-property.
Theorem 1.2
[11]
- (a)
T is a proximal nonexpansive.
- (b)
\(T(A_{0})\) is contained in \(B_{0}\).
- (c)
g is an isometry.
- (d)
\(A_{0}\) is contained in \(g(A_{0})\).
In this paper we focus on the sufficient conditions to ensure the existence of the proximity point for the proximal nonexpansive mapping. In our results, sets are not necessarily to be convex or to satisfy the P-property, and the mapping is not necessarily to be continuous or to be affine. The two cases that the sets are compact or weakly compact are considered, respectively. Finally, as applications of the theorems, analogs for the nonexpansive mappings are also given.
2 Preliminaries
Definition 2.1
[12]
It is easy to observe that a proximal contraction for a self-mapping reduces to a contraction.
Definition 2.2
Let A be a nonempty subset of a normed space X. A mapping \(T: A\rightarrow X\) is called nonexpansive if \(\Vert Tx-Ty\Vert \leq \Vert x-y\Vert \) for all \(x, y\in A\).
Definition 2.3
[11]
A nonempty subset A of a linear space X is called a p-starshaped set if there exists a point \(p\in A\) such that \(\alpha p+(1-\alpha)x\in A\), \(\forall x\in A\), \(\alpha\in[0,1]\), and p is called the center of A.
Each convex set C is a p-starshaped set for each \(p\in C\).
It is easy to see that in a normed space \((X, \Vert \cdot \Vert )\), if A is a p-starshaped set and B is a q-starshaped set and \(\Vert p-q\Vert =\operatorname {dist}(A, B)\), \(A_{0}\) is a p-starshaped set, and \(B_{0}\) is a q-starshaped set, respectively. If both of A and B are closed and \(A_{0}\) is nonempty, \(A_{0}\) is closed.
Definition 2.4
[13]
By using the P-property, some best proximity point results were proved for various classes of non-self-mappings. But in [14] the authors have shown that some recent results with the P-property concerning the existence of best proximity points can be obtained from the same results in fixed point theory.
Definition 2.5
[15]
Let \((A,B)\) be a pair of nonempty subsets of a metric space \((X,d)\). The pair \((A, B)\) is said to be a semi-sharp proximinal pair if for each x in A (respectively, in B) there exists at most one \(x^{*}\) in B (respectively, in A) such that \(d(x, x^{*})=\operatorname {dist}(A, B)\).
Notice that if \((A,B)\) is a semi-sharp proximinal pair, the pair \((B, A)\) may be not be.
It is easy to see that if \((A, B)\) has the P-property, then both of \((A, B)\) and \((B, A)\) are semi-sharp proximinal pairs. Obviously a semi-sharp proximinal pair \((A, B)\) is not necessarily to have the P-property.
Next let us introduce a new notion which is important in dealing with weak convergence. In the sequel, let us write ‘⇀’ to denote ‘weak convergence’.
Definition 2.6
Let A and B be two nonempty subsets of a Banach space X. The pair \((A, B)\) is said to have the H-property if for any sequences \(\{x_{n}\} \subseteq A\) and \(\{y_{n}\}\subseteq B\), \(x_{n}\rightharpoonup x_{0}\in A\), \(y_{n}\rightharpoonup y_{0}\in B\), and \(\Vert x_{n}-y_{n}\Vert \rightarrow \operatorname {dist}(A, B)\) imply that \(x_{n}-y_{n}\rightarrow x_{0}-y_{0}\).
Remark 1
Remark 2
If \(B=A\), \(\operatorname {dist}(A, B)=0\). For any \(\{x_{n}\}\subseteq A\), \(\{y_{n}\} \subseteq B\) satisfying \(\Vert x_{n}-y_{n}\Vert \rightarrow \operatorname {dist}(A, B)=0\) must have \(x_{n}-y_{n}\rightarrow0\). So for any nonempty subset A, the pair \((A, A)\) must have the H-property.
Recall that a Banach space X is said to have the H-property if for any sequence \(\{x_{n}\}\subset X\), \(x_{n}\rightharpoonup x_{0}\), and \(\Vert x_{n}\Vert \rightarrow \Vert x_{0}\Vert \) imply that \(x_{n}\rightarrow x_{0}\).
- (i)
Uniform convexity implies locally uniform convexity, and locally uniform convexity implies strict convexity. But a locally uniformly convex space is not necessarily a reflexive space.
- (ii)
Locally uniform convexity is different from uniform convexity or strict convexity; see [16].
- (iii)X is a locally uniformly convex space if and only if$$\forall x, x_{n}\in S(X),\quad x_{n}\rightarrow x\quad \mbox{whenever } \Vert x_{n}+x\Vert \rightarrow2. $$
It is well known that if X is a locally uniformly convex space, X has the H-property. In fact, \(\forall x_{n}\), \(x\in X\), if \(x_{n}\rightharpoonup x\) and \(\Vert x_{n}\Vert \rightarrow \Vert x\Vert \), let us show \(x_{n}\rightarrow x\). Without loss of generality, we may assume that \(\Vert x_{n}\Vert =\Vert x\Vert =1\) and show \(\Vert x_{n}+x\Vert \rightarrow2\) by use of (iii). It is easy to see that \(\lim_{n\rightarrow\infty}\sup \Vert x_{n}+x\Vert \leq2\) since ∀n, \(\Vert x_{n}+x\Vert \leq \Vert x_{n}\Vert +\Vert x\Vert =2\). We also have \(\lim_{n\rightarrow\infty}\sup \Vert x_{n}+x\Vert \geq2\) since \(x_{n}+x\rightharpoonup2x\) and \(2=\Vert 2x\Vert \leq\lim_{n\rightarrow \infty}\inf \Vert x_{n}+x\Vert \). Hence \(\Vert x_{n}+x\Vert \rightarrow2\) and thus \(x_{n}\rightarrow x\).
It is not difficult to see that if X has the H-property, for any sets \(A, B\subset X\), the pair \((A, B)\) satisfies the H-property.
Definition 2.7
[17]
A Banach space X satisfies the Opial condition for the weak topology if \(x_{n}\rightharpoonup x\in X\) implies that \(\lim_{n}\inf \Vert x_{n}-x\Vert <\lim_{n}\inf \Vert x_{n}-y\Vert \) for all \(y\neq x\).
All Hilbert spaces, all finite dimensional Banach spaces, and \(l_{p}\) (\(1< p<\infty\)) have the Opial property.
Definition 2.8
A mapping \(T: A\rightarrow X\) is called demiclosed if for any sequence \(\{x_{n}\}\subseteq A\) which converges weakly to \(x_{0}\in A\), the strong convergence of the sequence \(\{Tx_{n}\}\) to \(y_{0}\) in X implies that \(Tx_{0} = y_{0}\).
It can be shown that in a Banach space X with the Opial property, \((I-T)\) must be demiclosed if T is a nonexpansive mapping (see Lemma 2 in [17]).
Definition 2.9
[18]
Let A and B be nonempty subsets of a normed space X, \(T: A\cup B\rightarrow A\cup B\), \(T(A)\subseteq B\) and \(T(B)\subseteq A\). We say that T satisfies the proximal property if \(x_{n}\rightharpoonup x\in A\cup B\) and \(\Vert x_{n}-Tx_{n}\Vert \rightarrow \operatorname {dist}(A, B)\) imply that \(\Vert x-Tx\Vert =\operatorname {dist}(A, B)\).
If \(\operatorname {dist}(A, B)=0\), the proximal property reduces to the usual demiclosedness property of \(I-T\) at 0.
3 Proximity point for the proximal nonexpansive
First let us prove the following lemma, which plays an important role in our main results.
Lemma 3.1
- (a)
T is a proximal contraction;
- (b)
\(T(A_{0})\subseteq B_{0}\).
Proof
Next, let us show \(\{x_{n}\}\) is a Cauchy sequence and its limit just is the unique best proximity point of T.
Let T satisfy (a) and (b) in the above lemma and \(g: A\rightarrow A\) be an isometry satisfying \(A_{0}\subseteq g(A_{0})\). Denote \(G=g(A)\subseteq A\) and \(G_{0}=\{z\in G:d(z,y)=\operatorname {dist}(A,B) \mbox{ for some }y\in B\}\), then \(G_{0}=A_{0}\) and \(Tg^{-1}: G\rightarrow B\) is a proximal contraction. So we have the following corollary.
Corollary 3.2
- (a)
T is a proximal contraction.
- (b)
\(T(A_{0})\subseteq B_{0}\).
- (c)
g is an isometry.
- (d)
\(A_{0}\) is contained in \(g(A_{0})\).
Let g in the above corollary be the identity, we return to Lemma 3.1. So Lemma 3.1 and Corollary 3.2 are equivalent.
If we contrast Corollary 3.2 with Theorem 3.1 in [12] and Theorem 3.3 in [19], we may find that they have the same assertion, but in Theorem 3.1 in [12] the set B was restricted to be approximatively compact with respect to A, and in Theorem 3.3 in [19] the mapping T was restricted to be continuous.
Now let us present the first main theorem in this section.
Theorem 3.3
- (a)
T is a proximal nonexpansive.
- (b)
\(T(A_{0})\subseteq B_{0}\).
Proof
By the same method we also have \(u_{2}'=u_{2}\).
Next, let us show \(x^{*}\) is the proximity point of T to finish the proof.
So \(\{v_{k}\}\) is also a convergent sequence and \(\lim_{k\rightarrow\infty}v_{k}=\lim_{k\rightarrow\infty}u_{k}=x^{*}\).
The following theorem is the equivalent of Theorem 3.3; we omit the proof.
Theorem 3.4
- (a)
T is a proximal nonexpansive.
- (b)
\(T(A_{0})\subseteq B_{0}\).
- (c)
\(A_{0}\subseteq g(A_{0})\)
- (d)
g is an isometry.
In Theorem 3.4, T is not necessarily continuous affine, \((A, B)\) does not have the P-property and each of A and B is not necessarily convex.
It is easy to see that Theorem 1.2 is the corollary of Theorem 3.4.
Now let us begin to consider the case that the sets are weakly compact.
Lemma 3.5
- (a)
T is weakly continuous.
- (b)
T satisfies the proximal property.
Proof
Since \(A_{0}\) is weakly compact, without loss of generality, we may assume that \(u_{k}\rightharpoonup x^{*}\in A_{0}\).
Let us present the second main result in this section.
Theorem 3.6
Let \((A,B)\) be a pair of nonempty, closed subsets of a Banach space X such that A is a p-starshaped set, B is a q-starshaped set and \(\Vert p-q\Vert =\operatorname {dist}(A,B)\). Let \(T:A\rightarrow B\) be a proximal nonexpansive and \(T(A_{0})\subseteq B_{0}\). Suppose that \((A,B)\) is a semi-sharp proximinal pair, \((A_{0}, B_{0})\) is a weakly compact pair and satisfies the H-property. Then T has at least one best proximity point in A provided that \((I-T)\) is demiclosed.
Proof
Invoking Lemma 3.5, it suffices to prove T satisfies the proximal property.
Suppose \(u_{k}\in A_{0}\) such that \(u_{k}\rightharpoonup x^{*}\in A_{0}\) and \(\Vert u_{k}-Tu_{k}\Vert \rightarrow \operatorname {dist}(A,B)\). Since \(A_{0},B_{0}\) are weakly compact, without loss of generality, we may assume that \(u_{k}\rightharpoonup x^{*}\in A_{0}\) and \(Tu_{k}\rightharpoonup y^{*}\in B_{0}\). So we have \(u_{k}-Tu_{k}\rightharpoonup x^{*}-y^{*}\) and \(\Vert u_{k}-Tu_{k}\Vert \rightarrow \operatorname {dist}(A,B)\). This implies that \((I-T)u_{k}=u_{k}-Tu_{k}\rightarrow x^{*}-y^{*}\) by the assumption that \((A_{0}, B_{0})\) satisfies the H-property. Therefore \((I-T)x^{*}=x^{*}-y^{*}\) since \((I-T)\) is demiclosed. It is easy to see that \(\Vert x^{*}-Tx^{*}\Vert =\Vert x^{*}-y^{*}\Vert =\operatorname {dist}(A,B)\). So T satisfies the proximal property. □
Recall that a locally uniformly convex Banach space X must be strictly convex and have the H-property, then each convex subset pair \((A, B)\) is a semi-sharp proximinal pair and \((A_{0}, B_{0})\) satisfies the H-property. So we have the following corollary.
Corollary 3.7
Let \((A,B)\) be a pair of nonempty, closed, and convex subsets of a locally uniformly convex Banach space X and \(A_{0}\neq\emptyset\). Suppose that \(T:A\rightarrow B\) is a proximal nonexpansive and \(T(A_{0})\subseteq B_{0}\). Then T has at least one best proximity point in A provided that \((A_{0}, B_{0})\) is a weakly compact pair and \((I-T)\) is demiclosed.
Recently a new notion of the proximal generalized nonexpansive mapping was introduced in [20], and a few results about the proximity point for this class of mappings are given. For more information, see [20].
Let us end this section by an example.
Example
- (i)
\(p=(0,1)\in B\) and set B is not convex but is a p-starshaped set;
- (ii)
for any \(s\in B_{1}\) and \(O=(0, 0)\in A\), \(d(s, O)=1=\operatorname {dist}(A, B)\) and \(d(s, t)>1\) for any \(t\in A\), \(t\neq O\);
- (iii)
\(A_{0}=A\), \(B_{0}=B\), and \((A, B)\) is a semi-sharp proximinal pair;
- (iv)
\((B, A)\) is not a semi-sharp proximinal pair;
- (v)
T is not a nonexpansive mapping but is a proximal nonexpansive mapping;
- (vi)
T is not continuous or affine;
- (vii)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\) is compact;
- (viii)
\(\Vert (0, 0)-T((0, 0))\Vert =\Vert (0, 0)-(-\frac{1}{2}, \frac {1}{2})\Vert =1=\operatorname {dist}(A, B)\).
4 Proximity point for nonexpansive mapping
In this section, as applications of Theorem 3.3, Lemma 3.5, and Theorem 3.6, three results to ensure the existence of the best proximity for the nonexpansive mapping are given.
In [21], Abkar and Gabeleh obtained two proximity point theorems for the nonexpansive mapping.
Theorem 4.1
[21]
Let \((A,B)\) be a pair of nonempty, closed, and convex subsets of a Banach space X such that A is compact. Suppose that \(A_{0}\) is nonempty and \((A,B)\) has the P-property. Let \(T: A\rightarrow B\) be a nonexpansive non-self-mapping such that \(T(A_{0})\subseteq B_{0}\). Then T has at least one best proximity point in A.
Theorem 4.2
[21]
- (a)
T is weakly continuous.
- (b)
T satisfies the proximal property.
We can find that the ‘P-property’ appears in both of Theorems 4.1 and Theorem 4.2. Next, let us show that this assumption can be replaced by the so-called ‘weak P-property’. Besides, in both of the theorems, the pair \((A, B)\) can be assumed only as a starshaped pair instead of a convex pair.
Definition 4.3
[22]
It is clear that the weak P-property is weaker than the P-property and \((A, B)\) has the P-property if and only if both \((A, B)\) and \((B, A)\) have the weak P-property. See [23] and [24]. Obviously, if a pair \((A,B)\) has the weak P-property it must be a semi-sharp proximinal pair.
Theorem 4.4
Let \((A,B)\) be a pair of nonempty, closed subsets of a Banach space X such that A is a p-starshaped set, B is a q-starshaped set, and \(\Vert p-q\Vert =\operatorname {dist}(A,B)\). Suppose that \(A_{0}\) is compact and \((A,B)\) has the weak P-property. Let \(T: A\rightarrow B\) be a nonexpansive mapping such that \(T(A_{0})\subseteq B_{0}\). Then T has at least one best proximity point in \(A_{0}\).
Proof
It is the direct consequence of Theorem 3.3. □
Theorem 4.5
- (a)
T is weakly continuous;
- (b)
T satisfies the proximal property.
Proof
It is the direct consequence of Lemma 3.5. □
Theorem 4.6
- (a)
\((I-T)\) is demiclosed.
- (b)
X satisfies the Opial property.
Proof
If X satisfies the Opial property, then \((I-T)\) must be demiclosed since T is a nonexpansive mapping. To prove the theorem, it suffices to prove the assertion for the case (a). But this is the direct consequence of Theorem 3.6. □
Declarations
Acknowledgements
The work is supported by Science Foundation of Heilongjiang Province (A201206), Science Foundation of Heilongjiang Province (A201305), Research plan of the National Natural Science Foundation of China (No. 91120303) and National Natural Science Foundation of China (No. 91220301).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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