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On best proximity pair theorems for relatively u -continuous mappings
Nonlinear Analysis 74 (2011) 3870–3875
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
On best proximity pair theorems for relatively u-continuous mappings A. Anthony Eldred a , V. Sankar Raj b,∗ , P. Veeramani b a
Department of Mathematics, St.Joseph’s College, Tiruchirappalli 620 002, India
b
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
article
info
Article history: Received 11 April 2010 Accepted 14 February 2011 Keywords: Best proximity points Relatively u-continuous mappings Metric projection Affine mappings Common best proximity points
abstract A new class of mappings, called relatively u-continuous, is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we obtain a generalized version of the Markov–Kakutani theorem for best proximity points in the setting of a strictly convex Banach space. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Let (X , d) be a metric space and consider a mapping T : A → X , where A is a nonempty subset of X . The mapping T is said to have a fixed point in A if the fixed point equation Tx = x has at least one solution. In metric terminology, we say that x ∈ A is a fixed point of T if d(x, Tx) = 0. It is clear that the necessary condition for the existence of a fixed point for T is T (A) ∩ A ̸= ∅ (but not sufficient). If the fixed point equation Tx = x does not possess a solution, then d(x, Tx) > 0 for all x ∈ A. In such a situation, it is our aim to find an element x ∈ A such that d(x, Tx) is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Consider the following well-known best approximation theorem due to Ky Fan [1]. Theorem 1.1 ([1]). Let A be a nonempty compact convex subset of a normed linear space X and T : A → X be a continuous function. Then there exists x ∈ A such that ‖x − Tx‖ = dist (Tx, A) := inf{‖Tx − a‖ : a ∈ A}. Such an element x ∈ A in Theorem 1.1 is called a best approximant of T in A. Note that if x ∈ A is a best approximant, then ‖x − Tx‖ need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem min ‖x − Tx‖ x∈A
(1)
has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A, B of a metric space X and a mapping T : A → B. The natural question is whether one can find an element x0 ∈ A such that d(x0 , Tx0 ) = min{d(x, Tx) : x ∈ A}. Since d(x, Tx) ≥ dist (A, B), the optimal solution to the problem of minimizing the real valued function x → d(x, Tx) over the domain A of the mapping T will be the one for which the valued dist (A, B) is attained. A point x0 ∈ A is called a best proximity point of T if d(x0 , Tx0 ) = dist (A, B). Note that if dist (A, B) = 0, then the best proximity point is nothing but a fixed point of T .
∗
Corresponding address: Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, India. Tel.: +91 94440 40583. E-mail addresses: [email protected] (A.A. Eldred), [email protected] (V.S. Raj), [email protected] (P. Veeramani).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.021
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On other hand, Eldred et al. [2] introduced a new class of single valued mappings called relatively nonexpansive mappings to study the existence of best proximity points for such mappings. Their results generalize the celebrated fixed point theorem for nonexpansive mappings due to Browder [3]–Göhde [4]–Kirk [5] (see [6] also). The notion of relatively nonexpansive mapping is defined as follows. Definition 1.1 ([2]). Let A, B be nonempty subsets of a metric space (X , d). A mapping T : A ∪ B → A ∪ B is said to be a relatively nonexpansive mapping if 1. T (A) ⊆ B, T (B) ⊆ A; 2. d(Tx, Ty) ≤ d(x, y), for all x ∈ A, y ∈ B. Note that a relatively nonexpansive mapping need not be a continuous mapping. Also every nonexpansive self-map can be considered as a relatively nonexpansive mapping. For proving the existence of a best proximity point for relatively nonexpansive mappings, in [2], the authors introduced and used a geometric notion called proximal normal structure. Using the proximal normal structure, the following proximity pair theorem is proved in [2]. Theorem 1.2 ([2]). Let (A, B) be a nonempty, weakly compact convex pair in a Banach space X . Let T : A ∪ B → A ∪ B be a relatively nonexpansive mapping and suppose (A, B) has a proximal normal structure. Then there exists (x0 , y0 ) ∈ A × B such that ‖x0 − Tx0 ‖ = ‖Ty0 − y0 ‖ = dist (A, B). In [7], the authors discussed some topological properties of the set consisting of all best proximity points of a relatively nonexpansive mapping and proved the existence of common best proximity points for a family of relatively nonexpansive mappings. In [8], Eldred and Veeramani investigated the existence of best proximity points for a class of mappings called cyclic contraction. The notion of cyclic contractions is defined as follows. Definition 1.2 ([8]). Let A, B be nonempty subsets of a metric space X . A mapping T : A ∪ B → A ∪ B is said to be a cyclic contraction if there exists k ∈ [0, 1) such that 1. T (A) ⊆ B, T (B) ⊆ A; 2. d(Tx, Ty) ≤ k d(x, y) + (1 − k) dist (A, B), for all x ∈ A, y ∈ B. Note that the class of cyclic contraction mappings defined on A ∪ B, where A, B are nonempty subsets of a metric space, is strictly contained in the class of relatively nonexpansive mappings on A ∪ B. But, the conclusion of the best proximity point theorem in [8], ensures the existence of a unique best proximity point for a cyclic contraction mapping. Using the notion of cyclic contractions, in [9] the authors proved the existence of best proximity points for a relatively nonexpansive mapping without invoking proximal normal structure. In [10], the authors introduced another subclass, called a class of proximal pointwise contractions, contained in the class of relatively nonexpansive mapping and studied the existence of best proximity points for such a class. More related results in this direction can be found in [11,12]. In an application point of view, one can refer [13–16]. In this article, we introduce a new class of mappings called relatively u-continuous mappings which properly contains the class of relatively nonexpansive mappings and we attempt to study sufficient conditions for the existence of a best proximity point for such mappings. As an application of the existence theorem, we obtain a generalized version of the Markov–Kakutani theorem for best proximity points in the setting of a strictly convex Banach space. 2. Preliminaries In this section, we discuss some notations and known results which will be used in subsequent sections. At the end of this section, we bring in the notion of relatively u-continuous mappings and discuss the relation between relatively nonexpansive mappings and relatively u-continuous mappings. Let A, B be nonempty subsets of a normed linear space X . Then the proximity pair associated with the pair (A, B) is denoted by (A0 , B0 ), where A0 = {x ∈ A : ‖x − y‖ = dist (A, B), for some y ∈ B}, B0 = {y ∈ B : ‖x − y‖ = dist (A, B), for some x ∈ A}. In [17], Kirk et al. gave sufficient conditions which guarantee the nonemptiness of A0 and B0 . Let X be a normed linear space and C be a nonempty subset of X . Then the metric projection operator PC : X → 2C is defined as PC (x) = {y ∈ C : ‖x − y‖ = dist (x, C )}, for each x ∈ X , where 2C denotes the set of all subsets of C (including empty set). It is well known that if C is assumed to be a nonempty weakly compact convex subset of a strictly convex Banach space X then, the metric projection operator PC is a single valued mapping from X to C . Let A, B be nonempty weakly compact convex subsets of a Banach space X . Consider the mapping P : A ∪ B → A ∪ B defined as P (x) =
PB (x), if x ∈ A PA (x), if x ∈ B.
(2)
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If X is a strictly convex Banach space, then P is a single valued mapping and satisfies P (A) ⊆ B, P (B) ⊆ A. We use the following convergence theorem, which was proved in [9], in our main results. Theorem 2.1 ([9]). Let X be a strictly convex Banach space and A be a nonempty compact convex subset of X and B be a nonempty closed subset of X . Let {xn } be a sequence in A and y ∈ B. Suppose ‖xn − y‖ → dist (A, B). Then xn → PA (y). Let A, B be nonempty subsets of a normed linear space X . Now, let us define a new class of mappings on A ∪ B, called relatively u-continuous mappings, which properly contains the class of relatively nonexpansive mappings. Definition 2.1. Let A, B be nonempty subsets of a Banach space X . A mapping T : A ∪ B → A ∪ B is said to be a relatively u-continuous mapping if it satisfies 1. T (A) ⊆ B, T (B) ⊆ A; 2. for each ε>0, there exists a δ>0 such that ‖Tx − Ty‖<ε+ dist (A, B), whenever ‖x − y‖<δ+ dist (A, B), for all x ∈ A, y ∈ B. Note that every relatively nonexpansive mapping is a relatively u-continuous mapping. The following example shows that the converse is not true. Example 2.1. Let us consider (X = R2 , ‖ · ‖2 ). Let A = { (0, t ) : 0 ≤ t ≤ 1} and B = { (1, s) : 0 ≤ s ≤ 1}. Define T : A ∪ B → A ∪ B by T (x, y) =
√ (1, √y) (0, y)
if x = 0 if x = 1.
Then T is a relatively u-continuous mapping but not a relatively nonexpansive mapping. Proof. Since
1+
1 = T (0, 0) − T 1, 1 = ( 1 , 0 ) − 0 , 4 4 16 1 = 1+ 1 .
(0, 0) − 1,
1
16
16
T is not a relatively nonexpansive mapping. But we can see that T is a relatively u-continuous mapping. Note that √ dist (A, B) = 1. Let ε > 0 be given. Since the mapping g : [0, 1] → [0, 1], given by y → y, is uniformly continuous, √ √ there exists δ1 > 0 such that whenever |y − t | < δ1 implies | y − t | < ε. Take δ =
1 + δ12 − 1. Hence, whenever ‖(0, y) − (1, t )‖ < δ + dist (A, B), that is,
1 + |y − t |2 = ‖(0, y) − (1, t )‖ <
√
√ δ + dist (A, B) = δ + 1 = 1 + δ12 implies that |y − t | < δ1 . Therefore, | y − t | < ε . Then, ‖T (0, y) − T (1, t )‖ = √ √ √ 1 + | y − t |2 < 1 + ε 2 < 1 + ε. That is, for each ε > 0, there exists δ > 0 such that ‖T (0, y) − T (1, t )‖ < ε+ dist (A, B), whenever ‖(0, y)−(1, t )‖ < δ+ dist (A, B), for all(0, y) ∈ A, (1, t ) ∈ B. Hence T is a relatively u-continuous
mapping.
3. Main results First let us prove the following propositions which we use in our main results. Proposition 3.1. Let A, B be nonempty subsets of a normed linear space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Then T (A0 ) ⊆ B0 , T (B0 ) ⊆ A0 . Proof. If A0 = ∅, then B0 = ∅. Otherwise, for x ∈ A0 , there exists y ∈ B such that ‖x − y‖ = dist (A, B). Since T is a relatively u-continuous mapping, for each ε > 0 there exists δ > 0 such that ‖Ta − Tb‖ < ε + dist (A, B), whenever ‖a − b‖ < δ + dist (A, B), for all a ∈ A, b ∈ B. Since ‖x − y‖ = dist (A, B), for any δ > 0, ‖x − y‖ < δ + dist (A, B). Therefore, dist (A, B) ≤ ‖Tx − Ty‖ < ε + dist (A, B), for each ε > 0. This implies that ‖Tx − Ty‖ = dist (A, B), and hence, T (x) ∈ B0 . That is, T (A0 ) ⊆ B0 . Similarly, we can show that T (B0 ) ⊆ A0 . Using the above proposition, the following example shows that a continuous mapping T : A ∪ B → A ∪ B satisfying T (A) ⊆ B, T (B) ⊆ A, need not be relatively u-continuous. Example 3.1. Let A = [0, 1], and B = [3, 4]. Define a mapping 4, if x ∈ A, 0, if x ∈ B.
Tx =
Then T is a continuous mapping on A ∪ B and satisfies T (A) ⊆ B, T (B) ⊆ A, but it is not a relatively u-continuous mapping.
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Proposition 3.2. Let A, B be nonempty weakly compact convex subsets of a strictly convex Banach space X . Let T : A ∪ B → A ∪ B be a relatively u-continuous mapping and P : A ∪ B → A ∪ B be a mapping defined as in (2). Then TP (x) = P (Tx), for all x ∈ A0 ∪ B0 , i.e. PA (Tx) = T (PB (x)), x ∈ A0 . Proof. Let x ∈ A0 . Then there exists a unique y ∈ B0 such that ‖x − y‖ = dist (A, B). Therefore y = PB (x) and x = PA (y). Since T is a relatively u-continuous mapping, by Proposition 3.1, T (x) ∈ B0 and T (y) ∈ A0 with ‖T (x) − T (y)‖ = dist (A, B). Since X is a strictly convex Banach space, the uniqueness of metric projection operator yields that PA (T (x)) = T (y) = T (PB (x)). Hence for any x ∈ A0 , PA (T (x)) = T (y) = T (PB (x)). Similarly we can show for any y ∈ B0 , T (PA (y)) = PB (T (y)). Hence TP (x) = P (Tx), for all x ∈ A0 ∪ B0 . Now, let us state the following theorem which ensures the existence of best proximity points for a relatively u-continuous mapping. Theorem 3.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Then there exists (x0 , y0 ) ∈ A × B such that ‖x0 − Tx0 ‖ = ‖y0 − Ty0 ‖ = dist (A, B). Proof. Since A and B are compact convex subsets, (A0 , B0 ) is a nonempty compact convex pair. The relatively u-continuity of T implies that T (A0 ) ⊆ B0 , T (B0 ) ⊆ A0 (by Proposition 3.1). Consider the metric projection operator PA : X → A on A. Since T (A0 ) ⊆ B0 and PA (B0 ) ⊆ A0 , the composite mapping PA ◦ T restricted to A0 is a self-map. That is, PA ◦ T : A0 → A0 . Suppose x0 ∈ A0 is a fixed point of the mapping PA ◦ T , i.e., PA (Tx0 ) = x0 , then ‖x0 − Tx0 ‖ = dist (Tx0 , A). But Tx0 ∈ B0 implies that there exists x ∈ A0 such that ‖ x − Tx0 ‖ = dist (A, B). We can conclude dist (Tx0 , A) = dist (A, B) by the following inequality, dist (A, B) ≤ dist (Tx0 , A) ≤ ‖Tx0 − x‖ = dist (A, B). Thus ‖x0 − Tx0 ‖ = dist (Tx0 , A) = dist (A, B). Hence to complete the theorem, it is enough to prove the existence of fixed point for the mapping PA ◦ T : A0 → A0 . Since A is a compact convex subset of X , A0 is also a nonempty compact convex subset of A. Suppose PA ◦ T : A0 → A0 is a continuous function, then by the Schauder fixed point theorem, PA ◦ T has a fixed point in A0 . Since X is a strictly convex Banach space, we know that PA : X → A is a continuous mapping. Hence the theorem follows by showing that T is continuous on A0 . Let {xn } be a sequence in A0 such that xn → x0 for some x0 ∈ A0 . Let us prove that T (xn ) → T (x0 ). By Theorem 2.1, it is enough to show that ‖Txn − PA (Tx0 )‖ → dist (A, B). Since X is a strictly convex Banach space, we can conclude the theorem by the fact Txn −→ PB (PA (Tx0 )) = PB (T (PB x0 )) = T (PA (PB x0 )) = Tx0 . Claim: ‖Txn − PA (Tx0 )‖ → dist (A, B). Consider the following inequality.
‖xn − PB (x0 )‖ ≤ ‖xn − x0 ‖ + ‖x0 − PB (x0 )‖ = ‖xn − x0 ‖ + dist (A, B) → dist (A, B).
(3)
Let ε > 0 be given. Then there exists a δ > 0 such that ‖Tx − Ty‖ < ε dist (A, B), whenever ‖x − y‖ < δ + dist (A, B), for all x ∈ A, y ∈ B. Since ‖xn − PB (x0 )‖ → dist (A, B), for this δ > 0, there exist N0 ∈ N such that ‖xn − PB (x0 )‖ < δ + dist (A, B), for all n ≥ N0 . Then ‖Txn − T (PB x0 )‖ < ε + dist (A, B), for all n ≥ N0 , i.e., ‖Txn − PA (Tx0 )‖ = ‖Txn − T (PB x0 )‖ → dist (A, B). This completes the proof. 4. Common best proximity points In this section, we discuss sufficient conditions for the existence of common best proximity points for a family of relatively u-continuous mappings. Definition 4.1. Let A, B be nonempty convex subsets of a normed linear space. A relatively u-continuous mapping T : A ∪ B → A ∪ B is said to be affine if T (λx + (1 − λ)y) = λTx + (1 − λ)Ty, for all x, y ∈ A or x, y ∈ B and λ ∈ (0, 1). Before proving the existence of common best proximity points for a family of relatively u-continuous mappings, let us have some observations on the structure of the set consisting of all best proximity points of a relatively u-continuous mapping. Define FA (T ) = {x ∈ A : ‖x − Tx‖ = dist (A, B)}, FB (T ) = {y ∈ B : ‖y − Ty‖ = dist (A, B)}. Lemma 4.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Suppose T is affine, then (FA (T ), FB (T )) is a nonempty compact convex pair in (A0 , B0 ) with dist (FA (T ), FB (T )) = dist (A, B). Proof. By Theorem 3.1, FA (T ), FB (T ) are nonempty subsets of A0 , B0 respectively. Also, we have proved that a relatively u-continuous mapping T is continuous on A0 ∪ B0 . Hence, for any xn ∈ FA (T ) such that xn → x, for some x ∈ A0 then, x ∈ FA (T ). This implies that FA (T ) is a closed subset of A0 and hence compact. Similar argument shows that FB (T ) is a
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nonempty compact subset of B0 . Consider x1 , x2 ∈ FA (T ) and λ ∈ [0, 1]. That is, ‖xi − Txi ‖ = dist(A, B), for i = 1, 2. Then,
‖λx1 + (1 − λ)x2 − T (λx1 + (1 − λ)x2 )‖ = ‖λ(x1 − Tx1 ) + (1 − λ)(x2 − Tx2 )‖ ≤ λ‖x1 − Tx1 ‖ + (1 − λ)‖x2 − Tx2 ‖ = dist (A, B) which shows that λx1 + (1 − λ)x2 ∈ FA (T ). Since x1 , x2 ∈ FA (T ) are arbitrary, we conclude that FA (T ) is a convex subset of A0 . Similarly, we can show that FB (T ) is also a convex subset of B0 . Since FA (T ) is nonempty, let x0 ∈ FA (T ) and by considering the following inequality dist (A, B) ≤ dist (FA (T ), FB (T )) ≤ ‖x0 − Tx0 ‖ = dist (A, B), we have dist (FA (T ), FB (T )) = dist (A, B).
Lemma 4.2. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X . Let T , S : A ∪ B → A ∪ B be relatively u-continuous mappings on A ∪ B such that S ◦ T (x) = T ◦ S (x), for all x ∈ A ∪ B. Then, S (FA (T )) ⊆ FB (T ) and S (FB (T )) ⊆ FA (T ). Proof. Consider x ∈ FA (T ). That is, ‖x − Tx‖ = dist (A, B). The relatively u-continuity of S shows that ‖S (x) − S (Tx)‖ = dist (A, B). Since S and T are commuting, we have ‖Sx − T (Sx)‖ = dist (A, B) which implies that, Sx ∈ FB (T ). Hence, S (FA (T )) ⊆ FB (T ). Similarly, S (FB (T )) ⊆ FA (T ). Theorem 4.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T , S : A ∪ B → A ∪ B be commuting, affine, relatively u-continuous mappings on A ∪ B. Then there exists x0 ∈ A such that ‖x0 − Sx0 ‖ = ‖x0 − Tx0 ‖ = dist (A, B). That is, FA (T ) ∩ FA (S ) ̸= ∅. Proof. Consider the pair (FA (T ), FB (T )) ⊆ (A0 , B0 ). By Lemma 4.1, we conclude that (FA (T ), FB (T )) is a nonempty compact convex pair with dist (FA (T ), FB (T )) = dist (A, B). Now, consider the mapping S : FA (T ) ∪ FB (T ) → FA (T ) ∪ FB (T ). By Lemma 4.2, S is a relatively u-continuous mapping on FA (T ) ∪ FB (T ). Hence by Theorem 3.1, there exists x0 ∈ FA (T ) such that
‖x0 − Sx0 ‖ = dist (FA (T ), FB (T )) = dist (A, B) which implies that x0 ∈ FA (S ). Hence x0 ∈ FA (T ) ∩ FA (S ).
The following theorem guarantees the existence of a common best proximity point for a finite commuting family F = {T1 , T2 , . . . , Tn } of affine, relatively u-continuous mappings on A ∪ B. Theorem 4.2. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and F = {T1 , T2 , . . . , Tn } be a family of commuting, affine, relatively u-continuous mappings on A ∪ B. Then there exists x0 ∈ A such that ‖x0 − Ti x0 ‖ = dist (A, B), for all i = 1, 2, . . . , n. Proof. For each i = 1, . . . , n, define FA (Ti ) = {x ∈ A : ‖x − Ti (x)‖ = dist(A, B)} and FB (Ti ) = {y ∈ B : ‖y − Ti (y)‖ = dist (A, B)}. By Lemma 4.1, for each i = 1, . . . , n; (FA (Ti ), FB (Ti )) is a nonempty compact convex pair in (A, B) with dist (FA (Ti ), FB (Ti )) = dist (A, B). Theorem 4.1 implies that (FA (T1 ) ∩ FA (T2 ), FB (T1 ) ∩ FB (T2 )) is a nonempty compact convex pair with dist (FA (T1 ) ∩ FA (T2 ), FB (T 1 ) ∩ FB (T2 )) = dist (A, B). As in Lemma 4.2, we can prove that T3 is a relatively u-continuous mapping on (FA (T1 )∩ FA (T2 )) (FB (T1 )∩ FB (T2 )) and there exists a best proximity point z0 ∈ FA (T1 )∩ FA (T2 ) for T3 , i.e. z0 ∈ ∩3i=1 FA (Ti ). By repeating the argument, we can prove that there exists x0 ∈ ∩ni=1 FA (Ti ) such that ‖x0 − Ti x0 ‖ = dist (A, B), for all i = 1, 2, . . . , n. Next, let us prove the existence of a best proximity point for an arbitrary family of commuting, affine, relatively u-continuous mappings. Theorem 4.3. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and F = {Tγ : γ ∈ J } be a family of commuting, affine, relatively u-continuous mappings on A ∪ B with some index set J. Then there exists x0 ∈ A such that ‖x0 − Tγ x0 ‖ = dist (A, B), for all γ ∈ J . Proof. Consider a collection {FA (Tγ )}γ ∈J of nonempty closed convex subsets of A. By Theorem 4.2, it is clear that the collection {FA (Tγ )}γ ∈J has a finite intersection property. Since A is compact, we conclude that γ ∈J FA (Tγ ) is nonempty. That is, there exists an element x0 ∈ γ ∈J FA (Tγ ) such that ‖x0 − Tγ x0 ‖ = dist (A, B), for all γ ∈ J . As a corollary to Theorem 4.3, we obtain a particular case of the Markov–Kakutani fixed point theorem, in a strictly convex Banach space setting. Corollary 4.1. Let A be a nonempty compact convex subset of a strictly convex Banach space X and A be a commuting family of affine continuous self-mappings on A. Then there exists a common fixed point for the mappings in A .
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Acknowledgements The authors gratefully acknowledge many helpful comments and suggestions of the referee for the improvement of this manuscript. The second author V. Sankar Raj acknowledges the Council of Scientific and Industrial Research (India) for the financial support provided in the form of a Senior Research Fellowship to carry out this research work at Indian Institute of Technology Madras, Chennai. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
On best proximity pair theorems for relatively u-continuous mappings A. Anthony Eldred a , V. Sankar Raj b,∗ , P. Veeramani b a
Department of Mathematics, St.Joseph’s College, Tiruchirappalli 620 002, India
b
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
article
info
Article history: Received 11 April 2010 Accepted 14 February 2011 Keywords: Best proximity points Relatively u-continuous mappings Metric projection Affine mappings Common best proximity points
abstract A new class of mappings, called relatively u-continuous, is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we obtain a generalized version of the Markov–Kakutani theorem for best proximity points in the setting of a strictly convex Banach space. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Let (X , d) be a metric space and consider a mapping T : A → X , where A is a nonempty subset of X . The mapping T is said to have a fixed point in A if the fixed point equation Tx = x has at least one solution. In metric terminology, we say that x ∈ A is a fixed point of T if d(x, Tx) = 0. It is clear that the necessary condition for the existence of a fixed point for T is T (A) ∩ A ̸= ∅ (but not sufficient). If the fixed point equation Tx = x does not possess a solution, then d(x, Tx) > 0 for all x ∈ A. In such a situation, it is our aim to find an element x ∈ A such that d(x, Tx) is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Consider the following well-known best approximation theorem due to Ky Fan [1]. Theorem 1.1 ([1]). Let A be a nonempty compact convex subset of a normed linear space X and T : A → X be a continuous function. Then there exists x ∈ A such that ‖x − Tx‖ = dist (Tx, A) := inf{‖Tx − a‖ : a ∈ A}. Such an element x ∈ A in Theorem 1.1 is called a best approximant of T in A. Note that if x ∈ A is a best approximant, then ‖x − Tx‖ need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem min ‖x − Tx‖ x∈A
(1)
has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A, B of a metric space X and a mapping T : A → B. The natural question is whether one can find an element x0 ∈ A such that d(x0 , Tx0 ) = min{d(x, Tx) : x ∈ A}. Since d(x, Tx) ≥ dist (A, B), the optimal solution to the problem of minimizing the real valued function x → d(x, Tx) over the domain A of the mapping T will be the one for which the valued dist (A, B) is attained. A point x0 ∈ A is called a best proximity point of T if d(x0 , Tx0 ) = dist (A, B). Note that if dist (A, B) = 0, then the best proximity point is nothing but a fixed point of T .
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Corresponding address: Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, India. Tel.: +91 94440 40583. E-mail addresses: [email protected] (A.A. Eldred), [email protected] (V.S. Raj), [email protected] (P. Veeramani).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.021
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On other hand, Eldred et al. [2] introduced a new class of single valued mappings called relatively nonexpansive mappings to study the existence of best proximity points for such mappings. Their results generalize the celebrated fixed point theorem for nonexpansive mappings due to Browder [3]–Göhde [4]–Kirk [5] (see [6] also). The notion of relatively nonexpansive mapping is defined as follows. Definition 1.1 ([2]). Let A, B be nonempty subsets of a metric space (X , d). A mapping T : A ∪ B → A ∪ B is said to be a relatively nonexpansive mapping if 1. T (A) ⊆ B, T (B) ⊆ A; 2. d(Tx, Ty) ≤ d(x, y), for all x ∈ A, y ∈ B. Note that a relatively nonexpansive mapping need not be a continuous mapping. Also every nonexpansive self-map can be considered as a relatively nonexpansive mapping. For proving the existence of a best proximity point for relatively nonexpansive mappings, in [2], the authors introduced and used a geometric notion called proximal normal structure. Using the proximal normal structure, the following proximity pair theorem is proved in [2]. Theorem 1.2 ([2]). Let (A, B) be a nonempty, weakly compact convex pair in a Banach space X . Let T : A ∪ B → A ∪ B be a relatively nonexpansive mapping and suppose (A, B) has a proximal normal structure. Then there exists (x0 , y0 ) ∈ A × B such that ‖x0 − Tx0 ‖ = ‖Ty0 − y0 ‖ = dist (A, B). In [7], the authors discussed some topological properties of the set consisting of all best proximity points of a relatively nonexpansive mapping and proved the existence of common best proximity points for a family of relatively nonexpansive mappings. In [8], Eldred and Veeramani investigated the existence of best proximity points for a class of mappings called cyclic contraction. The notion of cyclic contractions is defined as follows. Definition 1.2 ([8]). Let A, B be nonempty subsets of a metric space X . A mapping T : A ∪ B → A ∪ B is said to be a cyclic contraction if there exists k ∈ [0, 1) such that 1. T (A) ⊆ B, T (B) ⊆ A; 2. d(Tx, Ty) ≤ k d(x, y) + (1 − k) dist (A, B), for all x ∈ A, y ∈ B. Note that the class of cyclic contraction mappings defined on A ∪ B, where A, B are nonempty subsets of a metric space, is strictly contained in the class of relatively nonexpansive mappings on A ∪ B. But, the conclusion of the best proximity point theorem in [8], ensures the existence of a unique best proximity point for a cyclic contraction mapping. Using the notion of cyclic contractions, in [9] the authors proved the existence of best proximity points for a relatively nonexpansive mapping without invoking proximal normal structure. In [10], the authors introduced another subclass, called a class of proximal pointwise contractions, contained in the class of relatively nonexpansive mapping and studied the existence of best proximity points for such a class. More related results in this direction can be found in [11,12]. In an application point of view, one can refer [13–16]. In this article, we introduce a new class of mappings called relatively u-continuous mappings which properly contains the class of relatively nonexpansive mappings and we attempt to study sufficient conditions for the existence of a best proximity point for such mappings. As an application of the existence theorem, we obtain a generalized version of the Markov–Kakutani theorem for best proximity points in the setting of a strictly convex Banach space. 2. Preliminaries In this section, we discuss some notations and known results which will be used in subsequent sections. At the end of this section, we bring in the notion of relatively u-continuous mappings and discuss the relation between relatively nonexpansive mappings and relatively u-continuous mappings. Let A, B be nonempty subsets of a normed linear space X . Then the proximity pair associated with the pair (A, B) is denoted by (A0 , B0 ), where A0 = {x ∈ A : ‖x − y‖ = dist (A, B), for some y ∈ B}, B0 = {y ∈ B : ‖x − y‖ = dist (A, B), for some x ∈ A}. In [17], Kirk et al. gave sufficient conditions which guarantee the nonemptiness of A0 and B0 . Let X be a normed linear space and C be a nonempty subset of X . Then the metric projection operator PC : X → 2C is defined as PC (x) = {y ∈ C : ‖x − y‖ = dist (x, C )}, for each x ∈ X , where 2C denotes the set of all subsets of C (including empty set). It is well known that if C is assumed to be a nonempty weakly compact convex subset of a strictly convex Banach space X then, the metric projection operator PC is a single valued mapping from X to C . Let A, B be nonempty weakly compact convex subsets of a Banach space X . Consider the mapping P : A ∪ B → A ∪ B defined as P (x) =
PB (x), if x ∈ A PA (x), if x ∈ B.
(2)
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If X is a strictly convex Banach space, then P is a single valued mapping and satisfies P (A) ⊆ B, P (B) ⊆ A. We use the following convergence theorem, which was proved in [9], in our main results. Theorem 2.1 ([9]). Let X be a strictly convex Banach space and A be a nonempty compact convex subset of X and B be a nonempty closed subset of X . Let {xn } be a sequence in A and y ∈ B. Suppose ‖xn − y‖ → dist (A, B). Then xn → PA (y). Let A, B be nonempty subsets of a normed linear space X . Now, let us define a new class of mappings on A ∪ B, called relatively u-continuous mappings, which properly contains the class of relatively nonexpansive mappings. Definition 2.1. Let A, B be nonempty subsets of a Banach space X . A mapping T : A ∪ B → A ∪ B is said to be a relatively u-continuous mapping if it satisfies 1. T (A) ⊆ B, T (B) ⊆ A; 2. for each ε>0, there exists a δ>0 such that ‖Tx − Ty‖<ε+ dist (A, B), whenever ‖x − y‖<δ+ dist (A, B), for all x ∈ A, y ∈ B. Note that every relatively nonexpansive mapping is a relatively u-continuous mapping. The following example shows that the converse is not true. Example 2.1. Let us consider (X = R2 , ‖ · ‖2 ). Let A = { (0, t ) : 0 ≤ t ≤ 1} and B = { (1, s) : 0 ≤ s ≤ 1}. Define T : A ∪ B → A ∪ B by T (x, y) =
√ (1, √y) (0, y)
if x = 0 if x = 1.
Then T is a relatively u-continuous mapping but not a relatively nonexpansive mapping. Proof. Since
1+
1 = T (0, 0) − T 1, 1 = ( 1 , 0 ) − 0 , 4 4 16 1 = 1+ 1 .
(0, 0) − 1,
1
16
16
T is not a relatively nonexpansive mapping. But we can see that T is a relatively u-continuous mapping. Note that √ dist (A, B) = 1. Let ε > 0 be given. Since the mapping g : [0, 1] → [0, 1], given by y → y, is uniformly continuous, √ √ there exists δ1 > 0 such that whenever |y − t | < δ1 implies | y − t | < ε. Take δ =
1 + δ12 − 1. Hence, whenever ‖(0, y) − (1, t )‖ < δ + dist (A, B), that is,
1 + |y − t |2 = ‖(0, y) − (1, t )‖ <
√
√ δ + dist (A, B) = δ + 1 = 1 + δ12 implies that |y − t | < δ1 . Therefore, | y − t | < ε . Then, ‖T (0, y) − T (1, t )‖ = √ √ √ 1 + | y − t |2 < 1 + ε 2 < 1 + ε. That is, for each ε > 0, there exists δ > 0 such that ‖T (0, y) − T (1, t )‖ < ε+ dist (A, B), whenever ‖(0, y)−(1, t )‖ < δ+ dist (A, B), for all(0, y) ∈ A, (1, t ) ∈ B. Hence T is a relatively u-continuous
mapping.
3. Main results First let us prove the following propositions which we use in our main results. Proposition 3.1. Let A, B be nonempty subsets of a normed linear space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Then T (A0 ) ⊆ B0 , T (B0 ) ⊆ A0 . Proof. If A0 = ∅, then B0 = ∅. Otherwise, for x ∈ A0 , there exists y ∈ B such that ‖x − y‖ = dist (A, B). Since T is a relatively u-continuous mapping, for each ε > 0 there exists δ > 0 such that ‖Ta − Tb‖ < ε + dist (A, B), whenever ‖a − b‖ < δ + dist (A, B), for all a ∈ A, b ∈ B. Since ‖x − y‖ = dist (A, B), for any δ > 0, ‖x − y‖ < δ + dist (A, B). Therefore, dist (A, B) ≤ ‖Tx − Ty‖ < ε + dist (A, B), for each ε > 0. This implies that ‖Tx − Ty‖ = dist (A, B), and hence, T (x) ∈ B0 . That is, T (A0 ) ⊆ B0 . Similarly, we can show that T (B0 ) ⊆ A0 . Using the above proposition, the following example shows that a continuous mapping T : A ∪ B → A ∪ B satisfying T (A) ⊆ B, T (B) ⊆ A, need not be relatively u-continuous. Example 3.1. Let A = [0, 1], and B = [3, 4]. Define a mapping 4, if x ∈ A, 0, if x ∈ B.
Tx =
Then T is a continuous mapping on A ∪ B and satisfies T (A) ⊆ B, T (B) ⊆ A, but it is not a relatively u-continuous mapping.
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Proposition 3.2. Let A, B be nonempty weakly compact convex subsets of a strictly convex Banach space X . Let T : A ∪ B → A ∪ B be a relatively u-continuous mapping and P : A ∪ B → A ∪ B be a mapping defined as in (2). Then TP (x) = P (Tx), for all x ∈ A0 ∪ B0 , i.e. PA (Tx) = T (PB (x)), x ∈ A0 . Proof. Let x ∈ A0 . Then there exists a unique y ∈ B0 such that ‖x − y‖ = dist (A, B). Therefore y = PB (x) and x = PA (y). Since T is a relatively u-continuous mapping, by Proposition 3.1, T (x) ∈ B0 and T (y) ∈ A0 with ‖T (x) − T (y)‖ = dist (A, B). Since X is a strictly convex Banach space, the uniqueness of metric projection operator yields that PA (T (x)) = T (y) = T (PB (x)). Hence for any x ∈ A0 , PA (T (x)) = T (y) = T (PB (x)). Similarly we can show for any y ∈ B0 , T (PA (y)) = PB (T (y)). Hence TP (x) = P (Tx), for all x ∈ A0 ∪ B0 . Now, let us state the following theorem which ensures the existence of best proximity points for a relatively u-continuous mapping. Theorem 3.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Then there exists (x0 , y0 ) ∈ A × B such that ‖x0 − Tx0 ‖ = ‖y0 − Ty0 ‖ = dist (A, B). Proof. Since A and B are compact convex subsets, (A0 , B0 ) is a nonempty compact convex pair. The relatively u-continuity of T implies that T (A0 ) ⊆ B0 , T (B0 ) ⊆ A0 (by Proposition 3.1). Consider the metric projection operator PA : X → A on A. Since T (A0 ) ⊆ B0 and PA (B0 ) ⊆ A0 , the composite mapping PA ◦ T restricted to A0 is a self-map. That is, PA ◦ T : A0 → A0 . Suppose x0 ∈ A0 is a fixed point of the mapping PA ◦ T , i.e., PA (Tx0 ) = x0 , then ‖x0 − Tx0 ‖ = dist (Tx0 , A). But Tx0 ∈ B0 implies that there exists x ∈ A0 such that ‖ x − Tx0 ‖ = dist (A, B). We can conclude dist (Tx0 , A) = dist (A, B) by the following inequality, dist (A, B) ≤ dist (Tx0 , A) ≤ ‖Tx0 − x‖ = dist (A, B). Thus ‖x0 − Tx0 ‖ = dist (Tx0 , A) = dist (A, B). Hence to complete the theorem, it is enough to prove the existence of fixed point for the mapping PA ◦ T : A0 → A0 . Since A is a compact convex subset of X , A0 is also a nonempty compact convex subset of A. Suppose PA ◦ T : A0 → A0 is a continuous function, then by the Schauder fixed point theorem, PA ◦ T has a fixed point in A0 . Since X is a strictly convex Banach space, we know that PA : X → A is a continuous mapping. Hence the theorem follows by showing that T is continuous on A0 . Let {xn } be a sequence in A0 such that xn → x0 for some x0 ∈ A0 . Let us prove that T (xn ) → T (x0 ). By Theorem 2.1, it is enough to show that ‖Txn − PA (Tx0 )‖ → dist (A, B). Since X is a strictly convex Banach space, we can conclude the theorem by the fact Txn −→ PB (PA (Tx0 )) = PB (T (PB x0 )) = T (PA (PB x0 )) = Tx0 . Claim: ‖Txn − PA (Tx0 )‖ → dist (A, B). Consider the following inequality.
‖xn − PB (x0 )‖ ≤ ‖xn − x0 ‖ + ‖x0 − PB (x0 )‖ = ‖xn − x0 ‖ + dist (A, B) → dist (A, B).
(3)
Let ε > 0 be given. Then there exists a δ > 0 such that ‖Tx − Ty‖ < ε dist (A, B), whenever ‖x − y‖ < δ + dist (A, B), for all x ∈ A, y ∈ B. Since ‖xn − PB (x0 )‖ → dist (A, B), for this δ > 0, there exist N0 ∈ N such that ‖xn − PB (x0 )‖ < δ + dist (A, B), for all n ≥ N0 . Then ‖Txn − T (PB x0 )‖ < ε + dist (A, B), for all n ≥ N0 , i.e., ‖Txn − PA (Tx0 )‖ = ‖Txn − T (PB x0 )‖ → dist (A, B). This completes the proof. 4. Common best proximity points In this section, we discuss sufficient conditions for the existence of common best proximity points for a family of relatively u-continuous mappings. Definition 4.1. Let A, B be nonempty convex subsets of a normed linear space. A relatively u-continuous mapping T : A ∪ B → A ∪ B is said to be affine if T (λx + (1 − λ)y) = λTx + (1 − λ)Ty, for all x, y ∈ A or x, y ∈ B and λ ∈ (0, 1). Before proving the existence of common best proximity points for a family of relatively u-continuous mappings, let us have some observations on the structure of the set consisting of all best proximity points of a relatively u-continuous mapping. Define FA (T ) = {x ∈ A : ‖x − Tx‖ = dist (A, B)}, FB (T ) = {y ∈ B : ‖y − Ty‖ = dist (A, B)}. Lemma 4.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T : A ∪ B → A ∪ B be a relatively u-continuous mapping. Suppose T is affine, then (FA (T ), FB (T )) is a nonempty compact convex pair in (A0 , B0 ) with dist (FA (T ), FB (T )) = dist (A, B). Proof. By Theorem 3.1, FA (T ), FB (T ) are nonempty subsets of A0 , B0 respectively. Also, we have proved that a relatively u-continuous mapping T is continuous on A0 ∪ B0 . Hence, for any xn ∈ FA (T ) such that xn → x, for some x ∈ A0 then, x ∈ FA (T ). This implies that FA (T ) is a closed subset of A0 and hence compact. Similar argument shows that FB (T ) is a
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nonempty compact subset of B0 . Consider x1 , x2 ∈ FA (T ) and λ ∈ [0, 1]. That is, ‖xi − Txi ‖ = dist(A, B), for i = 1, 2. Then,
‖λx1 + (1 − λ)x2 − T (λx1 + (1 − λ)x2 )‖ = ‖λ(x1 − Tx1 ) + (1 − λ)(x2 − Tx2 )‖ ≤ λ‖x1 − Tx1 ‖ + (1 − λ)‖x2 − Tx2 ‖ = dist (A, B) which shows that λx1 + (1 − λ)x2 ∈ FA (T ). Since x1 , x2 ∈ FA (T ) are arbitrary, we conclude that FA (T ) is a convex subset of A0 . Similarly, we can show that FB (T ) is also a convex subset of B0 . Since FA (T ) is nonempty, let x0 ∈ FA (T ) and by considering the following inequality dist (A, B) ≤ dist (FA (T ), FB (T )) ≤ ‖x0 − Tx0 ‖ = dist (A, B), we have dist (FA (T ), FB (T )) = dist (A, B).
Lemma 4.2. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X . Let T , S : A ∪ B → A ∪ B be relatively u-continuous mappings on A ∪ B such that S ◦ T (x) = T ◦ S (x), for all x ∈ A ∪ B. Then, S (FA (T )) ⊆ FB (T ) and S (FB (T )) ⊆ FA (T ). Proof. Consider x ∈ FA (T ). That is, ‖x − Tx‖ = dist (A, B). The relatively u-continuity of S shows that ‖S (x) − S (Tx)‖ = dist (A, B). Since S and T are commuting, we have ‖Sx − T (Sx)‖ = dist (A, B) which implies that, Sx ∈ FB (T ). Hence, S (FA (T )) ⊆ FB (T ). Similarly, S (FB (T )) ⊆ FA (T ). Theorem 4.1. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and T , S : A ∪ B → A ∪ B be commuting, affine, relatively u-continuous mappings on A ∪ B. Then there exists x0 ∈ A such that ‖x0 − Sx0 ‖ = ‖x0 − Tx0 ‖ = dist (A, B). That is, FA (T ) ∩ FA (S ) ̸= ∅. Proof. Consider the pair (FA (T ), FB (T )) ⊆ (A0 , B0 ). By Lemma 4.1, we conclude that (FA (T ), FB (T )) is a nonempty compact convex pair with dist (FA (T ), FB (T )) = dist (A, B). Now, consider the mapping S : FA (T ) ∪ FB (T ) → FA (T ) ∪ FB (T ). By Lemma 4.2, S is a relatively u-continuous mapping on FA (T ) ∪ FB (T ). Hence by Theorem 3.1, there exists x0 ∈ FA (T ) such that
‖x0 − Sx0 ‖ = dist (FA (T ), FB (T )) = dist (A, B) which implies that x0 ∈ FA (S ). Hence x0 ∈ FA (T ) ∩ FA (S ).
The following theorem guarantees the existence of a common best proximity point for a finite commuting family F = {T1 , T2 , . . . , Tn } of affine, relatively u-continuous mappings on A ∪ B. Theorem 4.2. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and F = {T1 , T2 , . . . , Tn } be a family of commuting, affine, relatively u-continuous mappings on A ∪ B. Then there exists x0 ∈ A such that ‖x0 − Ti x0 ‖ = dist (A, B), for all i = 1, 2, . . . , n. Proof. For each i = 1, . . . , n, define FA (Ti ) = {x ∈ A : ‖x − Ti (x)‖ = dist(A, B)} and FB (Ti ) = {y ∈ B : ‖y − Ti (y)‖ = dist (A, B)}. By Lemma 4.1, for each i = 1, . . . , n; (FA (Ti ), FB (Ti )) is a nonempty compact convex pair in (A, B) with dist (FA (Ti ), FB (Ti )) = dist (A, B). Theorem 4.1 implies that (FA (T1 ) ∩ FA (T2 ), FB (T1 ) ∩ FB (T2 )) is a nonempty compact convex pair with dist (FA (T1 ) ∩ FA (T2 ), FB (T 1 ) ∩ FB (T2 )) = dist (A, B). As in Lemma 4.2, we can prove that T3 is a relatively u-continuous mapping on (FA (T1 )∩ FA (T2 )) (FB (T1 )∩ FB (T2 )) and there exists a best proximity point z0 ∈ FA (T1 )∩ FA (T2 ) for T3 , i.e. z0 ∈ ∩3i=1 FA (Ti ). By repeating the argument, we can prove that there exists x0 ∈ ∩ni=1 FA (Ti ) such that ‖x0 − Ti x0 ‖ = dist (A, B), for all i = 1, 2, . . . , n. Next, let us prove the existence of a best proximity point for an arbitrary family of commuting, affine, relatively u-continuous mappings. Theorem 4.3. Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and F = {Tγ : γ ∈ J } be a family of commuting, affine, relatively u-continuous mappings on A ∪ B with some index set J. Then there exists x0 ∈ A such that ‖x0 − Tγ x0 ‖ = dist (A, B), for all γ ∈ J . Proof. Consider a collection {FA (Tγ )}γ ∈J of nonempty closed convex subsets of A. By Theorem 4.2, it is clear that the collection {FA (Tγ )}γ ∈J has a finite intersection property. Since A is compact, we conclude that γ ∈J FA (Tγ ) is nonempty. That is, there exists an element x0 ∈ γ ∈J FA (Tγ ) such that ‖x0 − Tγ x0 ‖ = dist (A, B), for all γ ∈ J . As a corollary to Theorem 4.3, we obtain a particular case of the Markov–Kakutani fixed point theorem, in a strictly convex Banach space setting. Corollary 4.1. Let A be a nonempty compact convex subset of a strictly convex Banach space X and A be a commuting family of affine continuous self-mappings on A. Then there exists a common fixed point for the mappings in A .
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Acknowledgements The authors gratefully acknowledge many helpful comments and suggestions of the referee for the improvement of this manuscript. The second author V. Sankar Raj acknowledges the Council of Scientific and Industrial Research (India) for the financial support provided in the form of a Senior Research Fellowship to carry out this research work at Indian Institute of Technology Madras, Chennai. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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