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Coincidence points, generalized I-nonexpansive multimaps, and applications
Nonlinear Analysis 67 (2007) 2180–2188 www.elsevier.com/locate/na
Coincidence points, generalized I -nonexpansive multimaps, and applications M.A. Al-Thagafi, Naseer Shahzad ∗ Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Received 7 May 2006; accepted 28 August 2006
Abstract We establish coincidence point results for multimaps satisfying generalized I -nonexpansive type conditions. Several invariant approximation results are obtained as applications. A random coincidence point result is also proved and, as applications, random invariant approximation results are obtained. Our results unify, extend and complement several well-known results. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Coincidence point; Invariant approximation; I -nonexpansive map; Random coincidence point; Random invariant approximation; Random operator; Normed space
1. Preliminaries and introduction Let D := (D, d) be a metric space. We denote by C L(D) the set of all nonempty closed subsets of D and by H the generalized Hausdorff distance function on C L(D) defined by H (A, B) := inf{ε > 0 : A ⊆ N (ε, B), B ⊆ N (ε, A)} ∈ [0, ∞] for all A, B ∈ C L(D) where N (ε, A) := {x ∈ X : d(x, a) < ε for some a ∈ A}. For A, B ∈ C L(D), let δ(A, B) := inf{d(a, b) : a ∈ A, b ∈ B} and δ(a, B) := δ({a}, B). Let I : D → D and T : D → C L(D). A fixed point of I [resp. T ] is a point x ∈ D for which x = I x [resp. x ∈ T x]. A coincidence point of the pair (I, T ) is a point x ∈ D for which I x ∈ T x. The set of coincidence [resp. common fixed] points of the pair (I, T ) is denoted by C(I, T ) [resp. F(I, T )]. The set of fixed points of T is denoted by F(T ). A subset D of a normed space X := (X, k.k) is called (a) q-starshaped if kx + (1 − k)q ∈ D for all x ∈ D and all k ∈ [0, 1]; and (b) convex if kx + (1 − k)y ∈ D for all x, y ∈ D and all k ∈ [0, 1]. The selfmap I is called (c) affine if D is convex and I (kx + (1 − k)y) = k I x + (1 − k)I y for all x, y ∈ D and all k ∈ [0, 1]; and (d) q-affine [2] if D is q-starshaped and I (kx + (1 − k)q) = k I x + (1 − k)q for all x ∈ D and all k ∈ [0, 1]. Note that I q = q whenever I is a q-affine selfmap of a q-starshaped set D. The map T is called (e) an I -contraction if H (T x, T y) ≤ kkI x − I yk for all x, y ∈ D and some k ∈ [0, 1); and (f) I -nonexpansive if H (T x, T y) ≤ kI x − I yk for all x, y ∈ D. The map ∗ Corresponding author.
E-mail addresses: [email protected] (M.A. Al-Thagafi), [email protected] (N. Shahzad). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.08.042
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T is called closed if it has a closed graph. The map I − T is called demiclosed at zero if whenever {xn } is a sequence in D such that xn → x0 ∈ D weakly and yn ∈ (I − T )xn such that yn → 0 strongly, then 0 ∈ (I − T )x0 . The pair (I, T ) is called (g) commuting if T I x = I T x for all x ∈ D; (h) R-weakly commuting [28] if I T x ∈ C L(D) and H (I T x, T I x) ≤ Rδ(I x, T x) for all x ∈ D and some R > 0; and (i) weakly compatible [11] if I T x = T I x for all x ∈ C(I, T ). If D is q-starshaped with q ∈ F(I ), the pair (I, T ) is called (j) Cq -commuting [2] if I T x = T I x for all x ∈ Cq (I, T ) := ∪{C(I, Tk ) : 0 ≤ k ≤ 1} where Tk x := kT x + (1 − k)q; (k) R-subweakly commuting (w.r.t. q) [25] if I T x ∈ C L(D) and H (T I x, I T y) ≤ R inf{δ(I x, Tk x) : 0 ≤ k ≤ 1} for all x ∈ D and some R > 0; and (l) R-subcommuting (w.r.t. q) [21] if I T x ∈ C L(D) and H (T I x, I T y) ≤ Rk δ(I x, Tk x) for all x ∈ D, all k ∈ (0, 1], and some R > 0. The map I is called (m) T -weakly commuting [13] at x ∈ D if I 2 x ∈ T I x. We note that the T -weak commutativity of I at x ∈ D is equivalent to I x ∈ C(I, T ). If the pair (I, T ) is weakly compatible at x ∈ C(I, T ), then I is T -weakly commuting at x and, hence, I n x ∈ C(I, T ) for every n ≥ 1. However, the converse is not true in general (see [13]). Note that on C(I, T ), (g) [resp. (h), (j), (k), or (l)] H⇒ (i) H⇒ (m). Note that if T : D → D, then the T -weak commutativity of I at x ∈ C(I, T ) is equivalent to the weak compatibility of the pair (I, T ) at x (see [13]). During the last four decades, several invariant approximation results for selfmaps were obtained as applications of fixed and common fixed point results (see [2, p.2]). Recently, Al-Thagafi and Shahzad [2] extended most of the well-known invariant approximation results to the new class of Cq -commuting selfmaps. Moreover, Kamran [13], Latif and Bano [17], and O’Regan and Shahzad [19,20] obtained invariant approximation results for multimaps. More recently, Shahzad and Hussain [27] obtained ordinary and random invariant approximation theorems for multimaps using results of Shahzad [24]. For details on the subject, we refer the reader to Singh et al. [29]. In this paper, we obtain coincidence point results for multimaps and, as applications, we derive several invariant approximation results. A random coincidence point result is also proved and random invariant approximation results are obtained as applications. Our results complement, extend, and unify all the above mentioned results as well as several well-known results. 2. Coincidence points results Let D be a metric space, I : D → D, T : D → C L(D), and T (D) := ∪x∈D T x. For every x, y ∈ D, define 1 ω I,T (x, y) := max d(I x, I y), δ(I x, T x), δ(I y, T y), [δ(I x, T y) + δ(I y, T x)] . 2 If I is the identity selfmap of D, ω I,T (x, y) will be denoted by ωT (x, y). Theorem 2.1. Let D be a metric space, I : D → D, T : D → C L(D), and T (D) ⊆ I (D). Suppose that T (D) is complete and H (T x, T y) ≤ kω I,T (x, y) for all x, y ∈ D and some k ∈ [0, 1). Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the following conditions holds: (a) For some v ∈ C(I, T ), I is T -weakly commuting at v and I 2 v = I v. (b) I and T are weakly compatible on C(I, T ), I is continuous, T is closed, and limn→∞ I n v exists for some v ∈ C(I, T ). (c) For some v ∈ C(I, T ), I is continuous at v and limn→∞ I n u = v for some u ∈ D. (d) I (C(I, T )) is a singleton subset of C(I, T ). Proof. Let x0 ∈ D. As T (D) ⊆ T (D) ⊆ I (D), we construct a sequence {xn } in D such that I xn ∈ T xn−1 ⊆ T (D) for all n ≥ 1. We conclude, as in [14], that {I xn } is a Cauchy sequence in T (D). It follows from the completeness of T (D) that I xn → z ∈ T (D) ⊆ I (D) where z = I u for some u ∈ D. Note that, for every n ≥ 1, we have δ(I xn , T u) ≤ H (T xn−1 , T u) ≤ kω I,T (xn−1 , u) 1 ≤ k max d(I xn−1 , I u), δ(I xn−1 , T xn−1 ), δ(I u, T u), [δ(I xn−1 , T u) + δ(I u, T xn−1 )] 2 1 ≤ k max d(I xn−1 , I u), d(I xn−1 , I xn ), δ(I u, T u), [δ(I xn−1 , T u) + d(I u, I xn )] . 2
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Letting n → ∞, we obtain δ(I u, T u) ≤ kδ(I u, T u). Therefore, I u ∈ T u and, hence, C(I, T ) 6= φ. (a) Suppose that I is T -weakly commuting at some v ∈ C(I, T ) and I 2 v = I v := s. Then s = I s ∈ T s and so F(I, T ) 6= φ. (b) Suppose that t := limn→∞ I n v exists for some v ∈ C(I, T ). Since I and T are weakly compatible on C(I, T ), we have, for all n ≥ 1, I n v ∈ C(I, T ) and, hence, I n v ∈ T I n−1 v. Then, as T is closed, we get t ∈ F(T ). Also, since I is continuous, we have t ∈ F(I ). Thus F(I, T ) 6= φ. (c) Suppose that limn→∞ I n u = v for some v ∈ C(I, T ) and some u ∈ D. Since I is continuous at v, it follows that v = I v ∈ T v. Thus F(I, T ) 6= φ. (d) Suppose that I (C(I, T )) = {w} ⊆ C(I, T ). This implies that w = I w ∈ T w. Thus F(I, T ) 6= φ. Theorem 2.1 extends Al-Thagafi [1, Theorem 2.1], Al-Thagafi and Shahzad [2, Theorem 2.1], the main result of Jungck [9], and Shahzad [23, Theorem 2.1]. Example. Let D = [0, 1) with the usual metric. Define I x = x 2 and T x = [0, 23 x 2 ] for all x ∈ D. Then all hypotheses of Theorem 2.1 are satisfied. Note that 0 ∈ C(I, T ). Note also that Nadler’s theorem cannot be used. Corollary 2.2. Let D be a metric space, T : D → C L(D), and T (D) ⊆ D. Suppose that T (D) is complete and H (T x, T y) ≤ kωT (x, y) for all x, y ∈ D and some k ∈ [0, 1). Then F(T ) 6= φ. Corollary 2.2 generalizes the Banach Contraction Principle, Nadler’s Contraction Principle, and Daffer and Kaneko [3, Theorem 2.4]. Let D be a q-starshaped subset of a normed space X , I : D → D, and T : D → C L(D). The pair (I, T ) satisfies the coincidence point condition (in short, CPC) on A ∈ C L(D) if whenever {xn } is a sequence in A such that δ(I xn , T xn ) → 0, then I z ∈ T z for some z ∈ A. The map T satisfies the fixed point condition (for short, FPC) on A ∈ C L(D) if whenever {xn } is a sequence in A such that δ(xn , T xn ) → 0, then z ∈ T z for some z ∈ A. For every x, y ∈ D, we define 1 σ I,T (x, y) := max kI x − I yk, γ (I x, T x), γ (I y, T y), [γ (I x, T y) + γ (I y, T x)] . 2 where γ (I x, T y) := inf{δ(I x, Tk y) : 0 ≤ k ≤ 1}. If I is the identity selfmap of D, ω I,T (x, y) and σ I,T (x, y) will be denoted by ωT (x, y) and σT (x, y), respectively. Note that σ I,T (x, y) ≤ ω I,Tk (x, y) and σT (x, y) ≤ ωTk (x, y) for all x, y ∈ D and all k ∈ [0, 1]. Theorem 2.3. Let D be a subset of a normed space X , I : D → D, and T : D → C L(D). Suppose that D is q-starshaped, I (D) = D [resp. I is q-affine], T (D) is bounded, T (D) is complete, T (D) ⊆ I (D), the pair (I, T ) satisfies the CPC on D, and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ D. Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the conditions (a)–(d) of Theorem 2.1 holds. Proof. Let {kn } be a sequence in (0, 1) such that kn → 1. For n ≥ 1, let Tn x := Tkn x = kn T x + (1 − kn )q for all x ∈ D. As D is q-starshaped, T (D) ⊆ I (D), T (D) is complete, and I (D) = D [resp. I is q-affine], we have Tn (D) ⊆ I (D), and each Tn (D) is complete. Moreover, for every n ≥ 1, we have H (Tn x, Tn y) = kn H (T x, T y) ≤ kn σ I,T (x, y) ≤ kn ω I,Tn (x, y) for all x, y ∈ D. It follows from Theorem 2.1 that I xn ∈ Tn xn = kn T xn + (1 − kn )q for some xn ∈ D. Since I xn = kn yn + (1 − kn )q for some yn ∈ T xn ⊆ T (D), T (D) is bounded, kn → 1, and kI xn − yn k = (1 − kn )kq − yn k ≤ (1 − kn )(kqk + kyn k), then I xn − yn → 0 strongly and so δ(I xn , T xn ) ≤ kI xn − yn k → 0. Since the pair (I, T ) satisfies the CPC on D, then there exists v ∈ D such that I v ∈ T v. Therefore, C(I, T ) 6= φ. The rest follows as in the proof of Theorem 2.1. Theorem 2.3 extends and improves well-known results such as Habiniak [5, Theorem 4], Jungck [10, Corollaries 3.2, 3.4], Jungck and Sessa [12, Theorem 6], Latif and Tweddle [18, Theorems 2.1, 2.2, 2.3], Rhoades [21, Theorem 3], Shahzad [25, Theorems 2.1, 2.2], and Shahzad and Hussain [27, Theorems 2.1, 2.2, 2.4, 2.6, 2.7, 2.8, 2.9, 2.11].
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Corollary 2.4. Let D be a subset of a normed space X , I : D → D, and T : D → C L(D). Suppose that D is q-starshaped, I (D) = D [resp. I is q-affine], T (D) is complete, T (D) ⊆ I (D), H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ D, and one of the following conditions holds: (C1) T (D) is bounded and (I − T )(D) is closed; (C2) D is weakly compact and I − T is demiclosed at zero. Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the conditions (a)–(d) of Theorem 2.1 holds. Proof. Note that the following are equivalent for a sequence {xn } in D: (i) δ(I xn , T xn ) → 0 and (ii) there exists a sequence {yn } in D with yn ∈ T xn , such that I xn − yn → 0 strongly. Note also that the pair (I, T ) satisfies the CPC on D means, by definition, that if (i) holds, then C(I, T ) 6= φ. Since “(ii) and (I − T )(D) being closed” imply that C(I, T ) 6= φ, then (C1) implies that the pair (I, T ) satisfies the CPC on D. Next, consider (C2). It is evident, by the Eberlein–Smulian theorem and the definition of “demiclosed” (see [4, Theorem 2]), that the pair (I, T ) satisfies the CPC on D. Now the result follows from Theorem 2.3. Corollary 2.5. Let D be a subset of a normed space X and T : D → C L(D). Suppose that D is q-starshaped, T (D) is bounded, T (D) is complete, T (D) ⊆ D, and T satisfies the FPC on D, and H (T x, T y) ≤ σT (x, y) for all x, y ∈ D. Then F(T ) 6= φ. Corollary 2.5 extends and improves Dotson [4, Theorems 1 and 2] and Lami Dozo [16, Theorem 3.2]. 3. Invariant approximation results I
For M ⊆ X and p ∈ X , let B M ( p) := {x ∈ M : kx − pk = δ( p, M)}, C M ( p) := {x ∈ M : I x ∈ B M ( p)}, I D M ( p) := B M ( p) ∩ C M ( p), and M p := {x ∈ M : kxk ≤ 2k pk}. The set B M ( p) is called the set of best M-approximants to p. Let C0 denote the class of closed convex subsets M of X containing 0. Then B M ( p) is closed, convex and contained in M p ∈ C0 . I
Theorem 3.1. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ M, and p ∈ X . Suppose that B M ( p) is closed and q-starshaped, I (B M ( p)) = B M ( p), T (B M ( p)) is complete, the pair (I, T ) satisfies the CPC on B M ( p), sup y∈T x ky − pk ≤ kI x − pk for all x ∈ B M ( p), and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ B M ( p). Then C(I, T )∩ B M ( p) 6= φ. Moreover, F(I, T )∩ B M ( p) 6= φ if one of the following conditions holds: (a) For some v ∈ C(I, T ) ∩ B M ( p), I is T -weakly commuting at v and I 2 v = I v. (b) I and T are weakly compatible on C(I, T ) ∩ B M ( p), I is continuous, T is closed, and limn→∞ I n v exists for some v ∈ C(I, T ) ∩ B M ( p). (c) For some v ∈ C(I, T ) ∩ B M ( p), I is continuous at v and limn→∞ I n u = v for some u ∈ B M ( p). (d) I (C(I, T ) ∩ B M ( p)) is a singleton subset of C(I, T ) ∩ B M ( p). Proof. Let x ∈ B M ( p). Then k(1 − k)x + kp − pk < δ( p, M) for all k ∈ (0, 1). Thus {(1 − k)x + kp : k ∈ (0, 1)} ∩ M = ∅ and so x ∈ ∂ M ∩ M. Since T (∂ M ∩ M) ⊆ M, T x ⊆ M. Let z ∈ T x. Then kz − pk ≤ sup ky − pk ≤ kI x − pk = δ( p, M). y∈T x
This implies that z ∈ B M ( p) and so T x ⊆ B M ( p). Thus T (B M ( p)) ⊆ B M ( p) = I (B M ( p)). Now the result follows from Theorem 2.3 with D = B M ( p). Theorem 3.1 extends Al-Thagafi [1, Theorems 3.2, 3.3; Remark p.321] for D = B M ( p), Al-Thagafi and Shahzad [2, Theorem 3.1, Theorem 3.3], Sahab, Khan and Sessa [22, Theorem 3], and Shahzad and Hussain [27, Theorems 2.12, 2.13]. It also contains Hicks and Humphries [6, p.221], Jungck and Sessa [12, Theorem 7], and Latif and Bano [17, Theorem 3].
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Corollary 3.2. Let X be a normed space, T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ M, and p ∈ X . Suppose that B M ( p) is closed and q-starshaped, T (B M ( p)) is complete, the map T satisfies the FPC on B M ( p), sup y∈T x ky − pk ≤ kx − pk for all x ∈ B M ( p) and H (T x, T y) ≤ σT (x, y) for all x, y ∈ B M ( p). Then F(T ) ∩ B M ( p) 6= φ. Corollary 3.2 includes [2, Theorem 1.1(a,b,c)] and Hicks and Humphries [6, p. 221] as special cases. It also generalizes [2, Theorem 1.1(d,e)]. Theorem 3.3. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ I (M) ∩ M, I ( p) is closed and q-starshaped, I (C I ( p)) = C I ( p), T (C I ( p)) is complete, the pair and p ∈ X . Suppose that C M M M M I ( p), sup I (I, T ) satisfies the CPC on C M y∈T x ky − pk ≤ kI x − pk for all x ∈ C M ( p), and H (T x, T y) ≤ σ I,T (x, y) I ( p). Then C(I, T )∩ B ( p) 6= φ. Moreover, F(I, T )∩ B ( p) 6= φ if one of the conditions (a)–(d) of for all x, y ∈ C M M M Theorem 3.1 holds. I ( p) = I (C I ( p)) ⊆ B ( p), an argument similar to that in the proof of Theorem 3.1 shows that Proof. As C M M M I I ( p)). Then there exists x ∈ C I ( p) such that z ∈ T x. Since x ∈ ∂ M ∩ M T (C M ( p)) ⊆ B M ( p). Let z ∈ T (C M M I ( p) and, hence, and T x ⊆ I (M), there exists u ∈ M such that z = I u ∈ T x ⊆ B M ( p). Thus u ∈ C M I ( p)) ⊆ C I ( p) = I (C I ( p)). Now the result follows from Theorem 2.3 with D = C I ( p). T (C M M M M
I ( p) and Al-Thagafi and Theorem 3.3 extends Al-Thagafi [1, Theorems 3.2, 3.3; Remark p. 321] for D = C M Shahzad [2, Theorem 3.2].
Theorem 3.4. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ I (M) ∩ M, I ( p) is closed and q-starshaped, I (D I ( p)) = D I ( p), T (D I ( p)) is complete, the pair and p ∈ X . Suppose that D M M M M I ( p), sup I (I, T ) satisfies the CPC on D M y∈T x ky − pk ≤ kI x − pk for all x ∈ D M ( p), H (T x, T y) ≤ σ I,T (x, y) I ( p), and kI x − pk = kx − pk for all x ∈ T (D I ( p)). Then C(I, T ) ∩ B ( p) 6= φ. Moreover, for all x, y ∈ D M M M F(I, T ) ∩ B M ( p) 6= φ if one of the conditions (a)–(d) of Theorem 3.1 holds I ( p) = I (D I ( p)) ⊆ B ( p), an argument similar to that in the proof of Theorem 3.1 shows that Proof. As D M M M I I ( p)). Since kI x − pk = kx − pk and x ∈ T (D I ( p)) ⊆ B ( p), it follows T (D M ( p)) ⊆ B M ( p). Let x ∈ T (D M M M I ( p). Thus x ∈ D I ( p) and, hence, T (D I ( p)) ⊆ D I ( p) = I (D I ( p)). Now the result follows from that x ∈ C M M M M M I ( p). Theorem 2.3 with D = D M
Theorem 3.3 extends Al-Thagafi [1, Theorems 3.2, 3.3]. Theorem 3.5. Let X be a normed space, I : X → X , T : X → C L(X ), and M ∈ C0 with T (M p ) ⊆ I (M) = M for some p ∈ X . Suppose that T (M p ) is compact, the pair (I, T ) satisfies the CPC on every A ∈ C L(M p ), kI x − pk = kx − pk for all x ∈ M, sup y∈T x ky − pk ≤ kx − pk for all x ∈ M p , and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, closed and convex, T (B M ( p)) ⊆ I (B M ( p)) = B M ( p), and C(I, T ) ∩ B M ( p) 6= φ. Moreover, F(I, T ) ∩ B M ( p) 6= φ if one of the conditions (a)–(d) of Theorem 3.1 holds. Proof. Assume that p 6∈ M. If u ∈ M \ M p , then kuk > 2k pk. As 0 ∈ M, we have ku − pk ≥ kuk − k pk > k pk ≥ δ( p, M). Thus α := δ( p, M p ) = δ( p, M). Since T (M p ) is compact and the norm is continuous, there exists v ∈ T (M p ) such that β := δ( p, T (M p )) = kv − pk. Note that α ≤ β ≤ δ( p, T (M p )) ≤ δ( p, T v) ≤ sup kx − pk ≤ kv − pk. x∈T v
Thus α = β and B M ( p) is nonempty closed, and convex. Since B M ( p) ⊆ M = I (M) and kI x − pk = kx − pk for all x ∈ M, then I (B M ( p)) = B M ( p). Now, let z ∈ T (B M ( p)) ⊆ I (M). Then there exist y0 ∈ B M ( p) and w ∈ M such that I w = z ∈ T y0 . Since kw − pk = kI w − pk = kz − pk ≤ sup kx − pk ≤ ky0 − pk = δ( p, M), x∈T y0
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we have z, w ∈ B M ( p). Therefore, T (B M ( p)) ⊆ I (B M ( p)) = B M ( p). The rest follows from Theorem 2.3 with D = B M ( p). Theorem 3.5 extends Al-Thagafi [1, Theorems 4.1, 4.2(b)], Al-Thagafi and Shahzad [2, Theorems 4.1, 4.2], Shahzad [23, Theorem 2.9], and Shahzad [26, Theorem 2.4]. Corollary 3.6. Let X be a normed space, T : X → C L(X ), and M ∈ C0 with T (M p ) ⊆ M for some p ∈ X . Suppose that T (M p ) is compact, the map T satisfies the FPC on every A ∈ C L(M p ), sup y∈T x ky − pk ≤ kx − pk for all x ∈ M p , and H (T x, T y) ≤ σT (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, closed and convex, T (B M ( p)) ⊆ B M ( p) and F(T ) ∩ B M ( p) 6= φ. Corollary 3.6 extends Shahzad [23, Theorem 2.8]. 4. Random coincidence points results Let (Ω , Σ ) be a measurable space where Ω is a set and Σ is a sigma algebra of subsets of Ω , and D a nonempty subset of a normed space X . Let ξ : Ω → D, S : Ω → C L(D), I : Ω × D → D, and T : Ω × D → C L(D). Then ξ is measurable if ξ −1 (V ) ∈ Σ for every open subset V of D. The map S is measurable if S −1 (V ) ∈ Σ for every open subset V of D where S −1 (V ) = {ω ∈ Ω : S(ω) ∩ V 6= φ}. The map ξ is a measurable selector of S if ξ is measurable and ξ(ω) ∈ S(ω) for every ω ∈ Ω . A function f : Ω × D → R is a Caratheodory function if it is continuous in x ∈ D and measurable in ω ∈ Ω . For every x ∈ D, define ξx : Ω → D and Sx : Ω → C L(D) by ξx (ω) := I (ω, x) and Sx (ω) := T (ω, x) for every ω ∈ Ω . For every ω ∈ Ω , define Iω : D → D and Tω : D → C L(D) by Iω (x) := I (ω, x) and Tω (x) := T (ω, x) for every x ∈ D. Then T [resp. I ] is a random operator if Sx [resp. ξx ] is measurable for every x ∈ D. If I and T are random operators and ξ is measurable, then ξ is a random fixed point of T [resp. I ] if ξ(ω) ∈ Tω (ξ(ω)) [resp. Iω (ξ(ω)) = ξ(ω)] for every ω ∈ Ω . The map ξ is a random coincidence [resp. common fixed] point of random operators I and T if Iω (ξ(ω)) ∈ Tω (ξ(ω)) [resp. ξ(ω) = Iω (ξ(ω)) ∈ Tω (ξ(ω))] for every ω ∈ Ω . The set of all random coincidence [resp. common fixed] points of random operators I and T is denoted by RC(I, T ) [resp. RF(I, T )]. The set of all random fixed points of a random operator T is denoted by RF(T ). We use ideas from Itoh [8], Shahzad [24], and Tan and Yuan [30] in the proof of the following result. Theorem 4.1. Let (Ω , Σ ) be a measurable space, D a subset of a Banach space X , I : Ω × D → D, and T : Ω × D → C L(D). Suppose that D is separable, closed and q-starshaped, I and T are continuous random operators and, for every ω ∈ Ω , we have Iω (D) = D [or Iω is q-affine on D and Tω (D) ⊆ Iω (D)], Tω (D) is bounded, the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(D), and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ D. Then RC(I, T ) 6= φ. Moreover, RF(I, T ) 6= φ if one of the following conditions holds for every ω ∈ Ω : (A) (B) (C) (D)
For every v ∈ C(Iω , Tω ), Iω is Tω -weakly commuting at v and Iω2 v = Iω v. Iω and Tω are weakly compatible on C(Iω , Tω ) and limn→∞ Iωn v exists for every v ∈ C(Iω , Tω ). For every v ∈ C(Iω , Tω ), there exists u ∈ D such that limn→∞ Iωn u = v. There exists a measurable function ξ : Ω → D such that Iω (C(Iω , Tω )) = {ξ(ω)} ⊆ C(Iω , Tω ).
Proof. Fix ω ∈ Ω and set S(ω) := {x ∈ D : Iω (x) ∈ Tω (x)}. Note that D is complete and the pair (Iω , Tω ) satisfies the conditions of Theorem 2.3. Thus C(Iω , Tω ) 6= φ and, consequently, S(ω) is nonempty and complete. Now, to show that S : Ω → C L(D) is a measurable map, let A ∈ C L(D). Define ∞ [ ∞ \ 2 L(A) := ω ∈ Ω : δ(Iω (xi ), Tω (xi )) < , n n=1 i=1 −1
−1
where {xi } is a countable dense subset of A. Clearly, S (A) ⊆ L(A). Moreover, to show that L(A) ⊆ S (A), fix ω0 ∈ L(A). Then, for every n, there is an integer i(n) such that δ(Iω0 (xi(n) ), Tω0 (xi(n) )) < n2 . It follows that
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δ(Iω0 (xi(n) ), Tω0 (xi(n) )) → 0 as n → ∞. Since the pair (Iω , Tω ) satisfies the CPN on A, it follows that there exists −1 −1 v ∈ A such that Iω0 (v) ∈ Tω0 (v). Therefore, ω0 ∈ S (A) and, hence, S (A) = L(A). By [7], 2 ∈Σ ω ∈ Ω : δ(Iω (x), Tω (x)) < n for every x ∈ D because the map f : Ω × D → R, defined by f (ω, x) = δ(Iω (x), Tω (x)), is a Caratheodory function. Thus S is measurable. By [15], S has a measurable selector ξ : Ω → D. Thus ξ ∈ RC(I, T ). The rest follows as in the proofs of Theorem 2.1 (above) and Theorem 3.7 of [27]. Theorem 4.1 extends and improves Shahzad [24, Theorems 3.17, 3.18] and Shahzad and Hussain [27, Theorems 3.4, and 3.7]. Corollary 4.2. Let (Ω , Σ ) be a measurable space, D a subset of a Banach space X , and T : Ω × D → C L(D). Suppose that D is separable, closed and q-starshaped, T is a continuous random operator and, for every ω ∈ Ω , we have Tω (D) is bounded, Tω satisfies the FPC on every A ∈ C L(D), and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ D. Then RF(T ) 6= φ. For the rest of the paper, the set of all measurable maps ξ : Ω → B M ( p) will be denoted by M p (Ω ) and the set of all nonempty, closed and bounded subsets of a set D will be denoted by C B(D). Theorem 4.3. Let (Ω , Σ ) be a measurable space, X a Banach space, I : Ω × X → X , T : Ω × X → C L(X ), M ⊆ X and p ∈ X . Suppose that B M ( p) is separable, closed, and q-starshaped, I and T are continuous random operators and, for every ω ∈ Ω , we have Tω (∂ M ∩ M) ⊆ M, Iω (B M ( p)) = B M ( p), the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(B M ( p)), sup y∈Tω x ky − pk ≤ kIω x − pk for all x ∈ B M ( p), and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ B M ( p). Then RC(I, T ) ∩ M p (Ω ) 6= φ. Moreover, RF(I, T ) ∩ M p (Ω ) 6= φ if one of the following conditions holds for every ω ∈ Ω : (a) (b) (c) (d)
For every v ∈ C(Iω , Tω ) ∩ B M ( p), Iω is Tω -weakly commuting at v and Iω2 v = Iω v. Iω and Tω are weakly compatible on C(Iω , Tω )∩ B M ( p) and limn→∞ Iωn v exists for every v ∈ C(Iω , Tω )∩ B M ( p). For every v ∈ C(Iω , Tω ) ∩ B M ( p), there exists u ∈ B M ( p) such that limn→∞ Iωn u = v. There exists a measurable function ξ : Ω → D such that Iω (C(Iω , Tω ) ∩ B M ( p)) = {ξ(ω)} ⊆ C(Iω , Tω ) ∩ B M ( p).
Proof. Fix ω ∈ Ω . As in the proof of Theorem 3.1, Tω (B M ( p)) ⊆ B M ( p) and Tω : B M ( p) → C B(B M ( p)). The rest follows from Theorem 4.1 with D = B M ( p). Corollary 4.4. Let (Ω , Σ ) be a measurable space, X a Banach space, T : Ω × X → C L(X ), M ⊆ X and p ∈ X . Suppose that B M ( p) is separable, closed and q-starshaped, T is a continuous random operator and, for every ω ∈ Ω , we have Tω (∂ M ∩ M) ⊆ M, Tω satisfies the FPC on every A ∈ C L(B M ( p)), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ B M ( p), and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ B M ( p). Then RF(T ) ∩ M p (Ω ) 6= φ. Theorem 4.5. Let (Ω , Σ ) be a measurable space, X a Banach space, I : Ω × X → X , T : Ω × X → C L(X ), M ∈ C0 and p ∈ X . Suppose that I and T are continuous random operators, Tω0 (M p ) is compact for some ω0 ∈ Ω , and, for every ω ∈ Ω , we have Tω (M p ) ⊆ Iω (M) = M, kIω x − pk = kx − pk for all x ∈ M, the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(M p ), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ M p , and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, complete and convex, and, for every ω ∈ Ω , we have Tω (B M ( p)) ⊆ Iω (B M ( p)) = B M ( p) and C(Iω , Tω ) ∩ B M ( p) 6= φ. If, in addition, B M ( p) is separable, then RC(I, T ) ∩ M p (Ω ) 6= φ. Moreover, RF(I, T ) ∩ M p (Ω ) 6= φ if one of the conditions (a)–(d) of Theorem 4.3 holds. Proof. Since Tω0 (M p ) is compact for some ω0 ∈ Ω , then, as in Theorem 3.5, B M ( p) is nonempty, complete and convex, Tω (B M ( p)) ⊆ Iω (B M ( p)) = B M ( p), and Tω : B M ( p) → C B(B M ( p)). The rest follows from Theorem 4.1 with D = B M ( p).
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Corollary 4.6. Let (Ω , Σ ) be a measurable space, X a Banach space, T : Ω × X → C L(X ), M ∈ C0 and p ∈ X . Suppose that T is a continuous random operator, Tω0 (M p ) is compact for some ω0 ∈ Ω , and, for every ω ∈ Ω , we have Tω (M p ) ⊆ M, Tω satisfies the FPC on every A ∈ C L(M p ), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ M p , and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, complete and convex, and, for every ω ∈ Ω , we have Tω (B M ( p)) ⊆ B M ( p) and F(Tω ) ∩ B M ( p) 6= φ. If, in addition, B M ( p) is separable, then RF(T ) ∩ M p (Ω ) 6= φ. 5. Concluding remarks 1. All results can be extended easily to p-normed spaces. 2. The starshapedness of D [resp. B M ( p)], in the related results, can be replaced with the condition: “There exists q ∈ D and a fixed sequence {kn } with 0 < kn < 1 converging to 1 such that kn T x + (1 − kn )q ∈ D for every x ∈ D” 3. Following arguments as above and as in [10], it is possible to obtain common fixed point and invariant approximation results for I, J : D → D and S, T : D → C L(D) satisfying H (Sx, T y) ≤ kI x − J yk for all x, y ∈ D. 4. Following the above arguments, it is possible to obtain common fixed point and invariant approximation results for I, J : D → D and T : D → C L(D) satisfying: 1 H (T x, T y) ≤ max kI x − J yk, δ(I x, T x), δ(J y, T y), [δ(I x, T y) + δ(J y, T x)] 2 for all x, y ∈ D. References [1] M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318–323. [2] M.A. Al-Thagafi, N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006) 2778–2786. [3] P.Z. Daffer, H. Kaneko, Applications of f -contraction mappings to nonlinear integral equations, Bull. Inst. Math. Acad. Sinica 22 (1994) 69–74. [4] W.J. Dotson Jr., Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces, J. London Math. Soc. 4 (1972) 408–410. [5] L. Habiniak, Fixed point theorems and invariant approximations, J. Approx. Theory 56 (1989) 241–244. [6] T.L. Hicks, M.D. Humphries, A note on fixed point theorems, J. Approx. Theory 34 (1982) 221–225. [7] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975) 53–72. [8] S. Itoh, Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979) 261–273. [9] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976) 261–263. [10] G. Jungck, Coincidence and fixed points for compatible and relatively nonexpansive maps, Int. J. Math. Math. Sci. 16 (1993) 95–100. [11] G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (1998) 227–238. [12] G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japonica 42 (1995) 249–252. [13] T. Kamran, Coincidence and fixed points for hybrid strict contractions, J. Math. Anal. Appl. 299 (2004) 235–241. [14] H. Kaneko, S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Int. J. Math. Math. Sci. 12 (1989) 257–262. [15] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 397–403. [16] E. Lami Dozo, Multi-valued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc. 38 (1973) 286–292. [17] A. Latif, A. Bano, A result on invariant approximation, Tamkang J. Math. 33 (2002) 89–92. [18] A. Latif, I. Tweddle, On multi-valued f -nonexpansive maps, Demonstratio Math. 32 (1999) 565–574. [19] D. O’Regan, N. Shahzad, Coincidence points and best proximity pair results for R-subweakly commuting multimaps, Demonstratio Math. 39 (2006) 845–854. [20] D. O’Regan, N. Shahzad, Coincidence points and invariant approximation results for multimaps, Acta Math. Sinica (2006) (in press). [21] R.E. Rhoades, On multi-valued f -nonexpansive maps, Fixed Point Theory Appl. 2 (2001) 89–92. [22] S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349–351. [23] N. Shahzad, Invariant approximations, generalized I -contractions and R-subweakly commuting maps, Fixed Point Theory Appl. 1 (2005) 79–86. [24] N. Shahzad, Some general random coincidence point theorems, New Zealand J. Math. 33 (2004) 95–103. [25] N. Shahzad, Coincidence points and R-subweakly commuting multivalued maps, Demonstratio Math. 36 (2003) 427–431. [26] N. Shahzad, Invariant approximation and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39–45.
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[27] N. Shahzad, N. Hussain, Deterministic and random coincidence point results for f -nonexpansive maps, J. Math. Anal. Appl. 323 (2006) 1038–1046. [28] N. Shahzad, T. Kamran, Coincidence points and R-weakly commuting maps, Arch. Math. (Brno) 37 (2001) 179–183. [29] S.P. Singh, B. Watson, P. Srivastava, Fixed Point Theory and Best Approximation: The KKM-map Principle, Kluwer Academic Publishers, Dordrecht, 1997. [30] K.K. Tan, X.Z. Yuan, Random fixed point theorems and approximation in cones, J. Math. Anal. Appl. 185 (1994) 378–390.
Coincidence points, generalized I -nonexpansive multimaps, and applications M.A. Al-Thagafi, Naseer Shahzad ∗ Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Received 7 May 2006; accepted 28 August 2006
Abstract We establish coincidence point results for multimaps satisfying generalized I -nonexpansive type conditions. Several invariant approximation results are obtained as applications. A random coincidence point result is also proved and, as applications, random invariant approximation results are obtained. Our results unify, extend and complement several well-known results. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Coincidence point; Invariant approximation; I -nonexpansive map; Random coincidence point; Random invariant approximation; Random operator; Normed space
1. Preliminaries and introduction Let D := (D, d) be a metric space. We denote by C L(D) the set of all nonempty closed subsets of D and by H the generalized Hausdorff distance function on C L(D) defined by H (A, B) := inf{ε > 0 : A ⊆ N (ε, B), B ⊆ N (ε, A)} ∈ [0, ∞] for all A, B ∈ C L(D) where N (ε, A) := {x ∈ X : d(x, a) < ε for some a ∈ A}. For A, B ∈ C L(D), let δ(A, B) := inf{d(a, b) : a ∈ A, b ∈ B} and δ(a, B) := δ({a}, B). Let I : D → D and T : D → C L(D). A fixed point of I [resp. T ] is a point x ∈ D for which x = I x [resp. x ∈ T x]. A coincidence point of the pair (I, T ) is a point x ∈ D for which I x ∈ T x. The set of coincidence [resp. common fixed] points of the pair (I, T ) is denoted by C(I, T ) [resp. F(I, T )]. The set of fixed points of T is denoted by F(T ). A subset D of a normed space X := (X, k.k) is called (a) q-starshaped if kx + (1 − k)q ∈ D for all x ∈ D and all k ∈ [0, 1]; and (b) convex if kx + (1 − k)y ∈ D for all x, y ∈ D and all k ∈ [0, 1]. The selfmap I is called (c) affine if D is convex and I (kx + (1 − k)y) = k I x + (1 − k)I y for all x, y ∈ D and all k ∈ [0, 1]; and (d) q-affine [2] if D is q-starshaped and I (kx + (1 − k)q) = k I x + (1 − k)q for all x ∈ D and all k ∈ [0, 1]. Note that I q = q whenever I is a q-affine selfmap of a q-starshaped set D. The map T is called (e) an I -contraction if H (T x, T y) ≤ kkI x − I yk for all x, y ∈ D and some k ∈ [0, 1); and (f) I -nonexpansive if H (T x, T y) ≤ kI x − I yk for all x, y ∈ D. The map ∗ Corresponding author.
E-mail addresses: [email protected] (M.A. Al-Thagafi), [email protected] (N. Shahzad). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.08.042
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T is called closed if it has a closed graph. The map I − T is called demiclosed at zero if whenever {xn } is a sequence in D such that xn → x0 ∈ D weakly and yn ∈ (I − T )xn such that yn → 0 strongly, then 0 ∈ (I − T )x0 . The pair (I, T ) is called (g) commuting if T I x = I T x for all x ∈ D; (h) R-weakly commuting [28] if I T x ∈ C L(D) and H (I T x, T I x) ≤ Rδ(I x, T x) for all x ∈ D and some R > 0; and (i) weakly compatible [11] if I T x = T I x for all x ∈ C(I, T ). If D is q-starshaped with q ∈ F(I ), the pair (I, T ) is called (j) Cq -commuting [2] if I T x = T I x for all x ∈ Cq (I, T ) := ∪{C(I, Tk ) : 0 ≤ k ≤ 1} where Tk x := kT x + (1 − k)q; (k) R-subweakly commuting (w.r.t. q) [25] if I T x ∈ C L(D) and H (T I x, I T y) ≤ R inf{δ(I x, Tk x) : 0 ≤ k ≤ 1} for all x ∈ D and some R > 0; and (l) R-subcommuting (w.r.t. q) [21] if I T x ∈ C L(D) and H (T I x, I T y) ≤ Rk δ(I x, Tk x) for all x ∈ D, all k ∈ (0, 1], and some R > 0. The map I is called (m) T -weakly commuting [13] at x ∈ D if I 2 x ∈ T I x. We note that the T -weak commutativity of I at x ∈ D is equivalent to I x ∈ C(I, T ). If the pair (I, T ) is weakly compatible at x ∈ C(I, T ), then I is T -weakly commuting at x and, hence, I n x ∈ C(I, T ) for every n ≥ 1. However, the converse is not true in general (see [13]). Note that on C(I, T ), (g) [resp. (h), (j), (k), or (l)] H⇒ (i) H⇒ (m). Note that if T : D → D, then the T -weak commutativity of I at x ∈ C(I, T ) is equivalent to the weak compatibility of the pair (I, T ) at x (see [13]). During the last four decades, several invariant approximation results for selfmaps were obtained as applications of fixed and common fixed point results (see [2, p.2]). Recently, Al-Thagafi and Shahzad [2] extended most of the well-known invariant approximation results to the new class of Cq -commuting selfmaps. Moreover, Kamran [13], Latif and Bano [17], and O’Regan and Shahzad [19,20] obtained invariant approximation results for multimaps. More recently, Shahzad and Hussain [27] obtained ordinary and random invariant approximation theorems for multimaps using results of Shahzad [24]. For details on the subject, we refer the reader to Singh et al. [29]. In this paper, we obtain coincidence point results for multimaps and, as applications, we derive several invariant approximation results. A random coincidence point result is also proved and random invariant approximation results are obtained as applications. Our results complement, extend, and unify all the above mentioned results as well as several well-known results. 2. Coincidence points results Let D be a metric space, I : D → D, T : D → C L(D), and T (D) := ∪x∈D T x. For every x, y ∈ D, define 1 ω I,T (x, y) := max d(I x, I y), δ(I x, T x), δ(I y, T y), [δ(I x, T y) + δ(I y, T x)] . 2 If I is the identity selfmap of D, ω I,T (x, y) will be denoted by ωT (x, y). Theorem 2.1. Let D be a metric space, I : D → D, T : D → C L(D), and T (D) ⊆ I (D). Suppose that T (D) is complete and H (T x, T y) ≤ kω I,T (x, y) for all x, y ∈ D and some k ∈ [0, 1). Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the following conditions holds: (a) For some v ∈ C(I, T ), I is T -weakly commuting at v and I 2 v = I v. (b) I and T are weakly compatible on C(I, T ), I is continuous, T is closed, and limn→∞ I n v exists for some v ∈ C(I, T ). (c) For some v ∈ C(I, T ), I is continuous at v and limn→∞ I n u = v for some u ∈ D. (d) I (C(I, T )) is a singleton subset of C(I, T ). Proof. Let x0 ∈ D. As T (D) ⊆ T (D) ⊆ I (D), we construct a sequence {xn } in D such that I xn ∈ T xn−1 ⊆ T (D) for all n ≥ 1. We conclude, as in [14], that {I xn } is a Cauchy sequence in T (D). It follows from the completeness of T (D) that I xn → z ∈ T (D) ⊆ I (D) where z = I u for some u ∈ D. Note that, for every n ≥ 1, we have δ(I xn , T u) ≤ H (T xn−1 , T u) ≤ kω I,T (xn−1 , u) 1 ≤ k max d(I xn−1 , I u), δ(I xn−1 , T xn−1 ), δ(I u, T u), [δ(I xn−1 , T u) + δ(I u, T xn−1 )] 2 1 ≤ k max d(I xn−1 , I u), d(I xn−1 , I xn ), δ(I u, T u), [δ(I xn−1 , T u) + d(I u, I xn )] . 2
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Letting n → ∞, we obtain δ(I u, T u) ≤ kδ(I u, T u). Therefore, I u ∈ T u and, hence, C(I, T ) 6= φ. (a) Suppose that I is T -weakly commuting at some v ∈ C(I, T ) and I 2 v = I v := s. Then s = I s ∈ T s and so F(I, T ) 6= φ. (b) Suppose that t := limn→∞ I n v exists for some v ∈ C(I, T ). Since I and T are weakly compatible on C(I, T ), we have, for all n ≥ 1, I n v ∈ C(I, T ) and, hence, I n v ∈ T I n−1 v. Then, as T is closed, we get t ∈ F(T ). Also, since I is continuous, we have t ∈ F(I ). Thus F(I, T ) 6= φ. (c) Suppose that limn→∞ I n u = v for some v ∈ C(I, T ) and some u ∈ D. Since I is continuous at v, it follows that v = I v ∈ T v. Thus F(I, T ) 6= φ. (d) Suppose that I (C(I, T )) = {w} ⊆ C(I, T ). This implies that w = I w ∈ T w. Thus F(I, T ) 6= φ. Theorem 2.1 extends Al-Thagafi [1, Theorem 2.1], Al-Thagafi and Shahzad [2, Theorem 2.1], the main result of Jungck [9], and Shahzad [23, Theorem 2.1]. Example. Let D = [0, 1) with the usual metric. Define I x = x 2 and T x = [0, 23 x 2 ] for all x ∈ D. Then all hypotheses of Theorem 2.1 are satisfied. Note that 0 ∈ C(I, T ). Note also that Nadler’s theorem cannot be used. Corollary 2.2. Let D be a metric space, T : D → C L(D), and T (D) ⊆ D. Suppose that T (D) is complete and H (T x, T y) ≤ kωT (x, y) for all x, y ∈ D and some k ∈ [0, 1). Then F(T ) 6= φ. Corollary 2.2 generalizes the Banach Contraction Principle, Nadler’s Contraction Principle, and Daffer and Kaneko [3, Theorem 2.4]. Let D be a q-starshaped subset of a normed space X , I : D → D, and T : D → C L(D). The pair (I, T ) satisfies the coincidence point condition (in short, CPC) on A ∈ C L(D) if whenever {xn } is a sequence in A such that δ(I xn , T xn ) → 0, then I z ∈ T z for some z ∈ A. The map T satisfies the fixed point condition (for short, FPC) on A ∈ C L(D) if whenever {xn } is a sequence in A such that δ(xn , T xn ) → 0, then z ∈ T z for some z ∈ A. For every x, y ∈ D, we define 1 σ I,T (x, y) := max kI x − I yk, γ (I x, T x), γ (I y, T y), [γ (I x, T y) + γ (I y, T x)] . 2 where γ (I x, T y) := inf{δ(I x, Tk y) : 0 ≤ k ≤ 1}. If I is the identity selfmap of D, ω I,T (x, y) and σ I,T (x, y) will be denoted by ωT (x, y) and σT (x, y), respectively. Note that σ I,T (x, y) ≤ ω I,Tk (x, y) and σT (x, y) ≤ ωTk (x, y) for all x, y ∈ D and all k ∈ [0, 1]. Theorem 2.3. Let D be a subset of a normed space X , I : D → D, and T : D → C L(D). Suppose that D is q-starshaped, I (D) = D [resp. I is q-affine], T (D) is bounded, T (D) is complete, T (D) ⊆ I (D), the pair (I, T ) satisfies the CPC on D, and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ D. Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the conditions (a)–(d) of Theorem 2.1 holds. Proof. Let {kn } be a sequence in (0, 1) such that kn → 1. For n ≥ 1, let Tn x := Tkn x = kn T x + (1 − kn )q for all x ∈ D. As D is q-starshaped, T (D) ⊆ I (D), T (D) is complete, and I (D) = D [resp. I is q-affine], we have Tn (D) ⊆ I (D), and each Tn (D) is complete. Moreover, for every n ≥ 1, we have H (Tn x, Tn y) = kn H (T x, T y) ≤ kn σ I,T (x, y) ≤ kn ω I,Tn (x, y) for all x, y ∈ D. It follows from Theorem 2.1 that I xn ∈ Tn xn = kn T xn + (1 − kn )q for some xn ∈ D. Since I xn = kn yn + (1 − kn )q for some yn ∈ T xn ⊆ T (D), T (D) is bounded, kn → 1, and kI xn − yn k = (1 − kn )kq − yn k ≤ (1 − kn )(kqk + kyn k), then I xn − yn → 0 strongly and so δ(I xn , T xn ) ≤ kI xn − yn k → 0. Since the pair (I, T ) satisfies the CPC on D, then there exists v ∈ D such that I v ∈ T v. Therefore, C(I, T ) 6= φ. The rest follows as in the proof of Theorem 2.1. Theorem 2.3 extends and improves well-known results such as Habiniak [5, Theorem 4], Jungck [10, Corollaries 3.2, 3.4], Jungck and Sessa [12, Theorem 6], Latif and Tweddle [18, Theorems 2.1, 2.2, 2.3], Rhoades [21, Theorem 3], Shahzad [25, Theorems 2.1, 2.2], and Shahzad and Hussain [27, Theorems 2.1, 2.2, 2.4, 2.6, 2.7, 2.8, 2.9, 2.11].
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Corollary 2.4. Let D be a subset of a normed space X , I : D → D, and T : D → C L(D). Suppose that D is q-starshaped, I (D) = D [resp. I is q-affine], T (D) is complete, T (D) ⊆ I (D), H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ D, and one of the following conditions holds: (C1) T (D) is bounded and (I − T )(D) is closed; (C2) D is weakly compact and I − T is demiclosed at zero. Then C(I, T ) 6= φ. Moreover, F(I, T ) 6= φ if one of the conditions (a)–(d) of Theorem 2.1 holds. Proof. Note that the following are equivalent for a sequence {xn } in D: (i) δ(I xn , T xn ) → 0 and (ii) there exists a sequence {yn } in D with yn ∈ T xn , such that I xn − yn → 0 strongly. Note also that the pair (I, T ) satisfies the CPC on D means, by definition, that if (i) holds, then C(I, T ) 6= φ. Since “(ii) and (I − T )(D) being closed” imply that C(I, T ) 6= φ, then (C1) implies that the pair (I, T ) satisfies the CPC on D. Next, consider (C2). It is evident, by the Eberlein–Smulian theorem and the definition of “demiclosed” (see [4, Theorem 2]), that the pair (I, T ) satisfies the CPC on D. Now the result follows from Theorem 2.3. Corollary 2.5. Let D be a subset of a normed space X and T : D → C L(D). Suppose that D is q-starshaped, T (D) is bounded, T (D) is complete, T (D) ⊆ D, and T satisfies the FPC on D, and H (T x, T y) ≤ σT (x, y) for all x, y ∈ D. Then F(T ) 6= φ. Corollary 2.5 extends and improves Dotson [4, Theorems 1 and 2] and Lami Dozo [16, Theorem 3.2]. 3. Invariant approximation results I
For M ⊆ X and p ∈ X , let B M ( p) := {x ∈ M : kx − pk = δ( p, M)}, C M ( p) := {x ∈ M : I x ∈ B M ( p)}, I D M ( p) := B M ( p) ∩ C M ( p), and M p := {x ∈ M : kxk ≤ 2k pk}. The set B M ( p) is called the set of best M-approximants to p. Let C0 denote the class of closed convex subsets M of X containing 0. Then B M ( p) is closed, convex and contained in M p ∈ C0 . I
Theorem 3.1. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ M, and p ∈ X . Suppose that B M ( p) is closed and q-starshaped, I (B M ( p)) = B M ( p), T (B M ( p)) is complete, the pair (I, T ) satisfies the CPC on B M ( p), sup y∈T x ky − pk ≤ kI x − pk for all x ∈ B M ( p), and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ B M ( p). Then C(I, T )∩ B M ( p) 6= φ. Moreover, F(I, T )∩ B M ( p) 6= φ if one of the following conditions holds: (a) For some v ∈ C(I, T ) ∩ B M ( p), I is T -weakly commuting at v and I 2 v = I v. (b) I and T are weakly compatible on C(I, T ) ∩ B M ( p), I is continuous, T is closed, and limn→∞ I n v exists for some v ∈ C(I, T ) ∩ B M ( p). (c) For some v ∈ C(I, T ) ∩ B M ( p), I is continuous at v and limn→∞ I n u = v for some u ∈ B M ( p). (d) I (C(I, T ) ∩ B M ( p)) is a singleton subset of C(I, T ) ∩ B M ( p). Proof. Let x ∈ B M ( p). Then k(1 − k)x + kp − pk < δ( p, M) for all k ∈ (0, 1). Thus {(1 − k)x + kp : k ∈ (0, 1)} ∩ M = ∅ and so x ∈ ∂ M ∩ M. Since T (∂ M ∩ M) ⊆ M, T x ⊆ M. Let z ∈ T x. Then kz − pk ≤ sup ky − pk ≤ kI x − pk = δ( p, M). y∈T x
This implies that z ∈ B M ( p) and so T x ⊆ B M ( p). Thus T (B M ( p)) ⊆ B M ( p) = I (B M ( p)). Now the result follows from Theorem 2.3 with D = B M ( p). Theorem 3.1 extends Al-Thagafi [1, Theorems 3.2, 3.3; Remark p.321] for D = B M ( p), Al-Thagafi and Shahzad [2, Theorem 3.1, Theorem 3.3], Sahab, Khan and Sessa [22, Theorem 3], and Shahzad and Hussain [27, Theorems 2.12, 2.13]. It also contains Hicks and Humphries [6, p.221], Jungck and Sessa [12, Theorem 7], and Latif and Bano [17, Theorem 3].
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Corollary 3.2. Let X be a normed space, T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ M, and p ∈ X . Suppose that B M ( p) is closed and q-starshaped, T (B M ( p)) is complete, the map T satisfies the FPC on B M ( p), sup y∈T x ky − pk ≤ kx − pk for all x ∈ B M ( p) and H (T x, T y) ≤ σT (x, y) for all x, y ∈ B M ( p). Then F(T ) ∩ B M ( p) 6= φ. Corollary 3.2 includes [2, Theorem 1.1(a,b,c)] and Hicks and Humphries [6, p. 221] as special cases. It also generalizes [2, Theorem 1.1(d,e)]. Theorem 3.3. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ I (M) ∩ M, I ( p) is closed and q-starshaped, I (C I ( p)) = C I ( p), T (C I ( p)) is complete, the pair and p ∈ X . Suppose that C M M M M I ( p), sup I (I, T ) satisfies the CPC on C M y∈T x ky − pk ≤ kI x − pk for all x ∈ C M ( p), and H (T x, T y) ≤ σ I,T (x, y) I ( p). Then C(I, T )∩ B ( p) 6= φ. Moreover, F(I, T )∩ B ( p) 6= φ if one of the conditions (a)–(d) of for all x, y ∈ C M M M Theorem 3.1 holds. I ( p) = I (C I ( p)) ⊆ B ( p), an argument similar to that in the proof of Theorem 3.1 shows that Proof. As C M M M I I ( p)). Then there exists x ∈ C I ( p) such that z ∈ T x. Since x ∈ ∂ M ∩ M T (C M ( p)) ⊆ B M ( p). Let z ∈ T (C M M I ( p) and, hence, and T x ⊆ I (M), there exists u ∈ M such that z = I u ∈ T x ⊆ B M ( p). Thus u ∈ C M I ( p)) ⊆ C I ( p) = I (C I ( p)). Now the result follows from Theorem 2.3 with D = C I ( p). T (C M M M M
I ( p) and Al-Thagafi and Theorem 3.3 extends Al-Thagafi [1, Theorems 3.2, 3.3; Remark p. 321] for D = C M Shahzad [2, Theorem 3.2].
Theorem 3.4. Let X be a normed space, I : X → X , T : X → C L(X ), M ⊆ X with T (∂ M ∩ M) ⊆ I (M) ∩ M, I ( p) is closed and q-starshaped, I (D I ( p)) = D I ( p), T (D I ( p)) is complete, the pair and p ∈ X . Suppose that D M M M M I ( p), sup I (I, T ) satisfies the CPC on D M y∈T x ky − pk ≤ kI x − pk for all x ∈ D M ( p), H (T x, T y) ≤ σ I,T (x, y) I ( p), and kI x − pk = kx − pk for all x ∈ T (D I ( p)). Then C(I, T ) ∩ B ( p) 6= φ. Moreover, for all x, y ∈ D M M M F(I, T ) ∩ B M ( p) 6= φ if one of the conditions (a)–(d) of Theorem 3.1 holds I ( p) = I (D I ( p)) ⊆ B ( p), an argument similar to that in the proof of Theorem 3.1 shows that Proof. As D M M M I I ( p)). Since kI x − pk = kx − pk and x ∈ T (D I ( p)) ⊆ B ( p), it follows T (D M ( p)) ⊆ B M ( p). Let x ∈ T (D M M M I ( p). Thus x ∈ D I ( p) and, hence, T (D I ( p)) ⊆ D I ( p) = I (D I ( p)). Now the result follows from that x ∈ C M M M M M I ( p). Theorem 2.3 with D = D M
Theorem 3.3 extends Al-Thagafi [1, Theorems 3.2, 3.3]. Theorem 3.5. Let X be a normed space, I : X → X , T : X → C L(X ), and M ∈ C0 with T (M p ) ⊆ I (M) = M for some p ∈ X . Suppose that T (M p ) is compact, the pair (I, T ) satisfies the CPC on every A ∈ C L(M p ), kI x − pk = kx − pk for all x ∈ M, sup y∈T x ky − pk ≤ kx − pk for all x ∈ M p , and H (T x, T y) ≤ σ I,T (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, closed and convex, T (B M ( p)) ⊆ I (B M ( p)) = B M ( p), and C(I, T ) ∩ B M ( p) 6= φ. Moreover, F(I, T ) ∩ B M ( p) 6= φ if one of the conditions (a)–(d) of Theorem 3.1 holds. Proof. Assume that p 6∈ M. If u ∈ M \ M p , then kuk > 2k pk. As 0 ∈ M, we have ku − pk ≥ kuk − k pk > k pk ≥ δ( p, M). Thus α := δ( p, M p ) = δ( p, M). Since T (M p ) is compact and the norm is continuous, there exists v ∈ T (M p ) such that β := δ( p, T (M p )) = kv − pk. Note that α ≤ β ≤ δ( p, T (M p )) ≤ δ( p, T v) ≤ sup kx − pk ≤ kv − pk. x∈T v
Thus α = β and B M ( p) is nonempty closed, and convex. Since B M ( p) ⊆ M = I (M) and kI x − pk = kx − pk for all x ∈ M, then I (B M ( p)) = B M ( p). Now, let z ∈ T (B M ( p)) ⊆ I (M). Then there exist y0 ∈ B M ( p) and w ∈ M such that I w = z ∈ T y0 . Since kw − pk = kI w − pk = kz − pk ≤ sup kx − pk ≤ ky0 − pk = δ( p, M), x∈T y0
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we have z, w ∈ B M ( p). Therefore, T (B M ( p)) ⊆ I (B M ( p)) = B M ( p). The rest follows from Theorem 2.3 with D = B M ( p). Theorem 3.5 extends Al-Thagafi [1, Theorems 4.1, 4.2(b)], Al-Thagafi and Shahzad [2, Theorems 4.1, 4.2], Shahzad [23, Theorem 2.9], and Shahzad [26, Theorem 2.4]. Corollary 3.6. Let X be a normed space, T : X → C L(X ), and M ∈ C0 with T (M p ) ⊆ M for some p ∈ X . Suppose that T (M p ) is compact, the map T satisfies the FPC on every A ∈ C L(M p ), sup y∈T x ky − pk ≤ kx − pk for all x ∈ M p , and H (T x, T y) ≤ σT (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, closed and convex, T (B M ( p)) ⊆ B M ( p) and F(T ) ∩ B M ( p) 6= φ. Corollary 3.6 extends Shahzad [23, Theorem 2.8]. 4. Random coincidence points results Let (Ω , Σ ) be a measurable space where Ω is a set and Σ is a sigma algebra of subsets of Ω , and D a nonempty subset of a normed space X . Let ξ : Ω → D, S : Ω → C L(D), I : Ω × D → D, and T : Ω × D → C L(D). Then ξ is measurable if ξ −1 (V ) ∈ Σ for every open subset V of D. The map S is measurable if S −1 (V ) ∈ Σ for every open subset V of D where S −1 (V ) = {ω ∈ Ω : S(ω) ∩ V 6= φ}. The map ξ is a measurable selector of S if ξ is measurable and ξ(ω) ∈ S(ω) for every ω ∈ Ω . A function f : Ω × D → R is a Caratheodory function if it is continuous in x ∈ D and measurable in ω ∈ Ω . For every x ∈ D, define ξx : Ω → D and Sx : Ω → C L(D) by ξx (ω) := I (ω, x) and Sx (ω) := T (ω, x) for every ω ∈ Ω . For every ω ∈ Ω , define Iω : D → D and Tω : D → C L(D) by Iω (x) := I (ω, x) and Tω (x) := T (ω, x) for every x ∈ D. Then T [resp. I ] is a random operator if Sx [resp. ξx ] is measurable for every x ∈ D. If I and T are random operators and ξ is measurable, then ξ is a random fixed point of T [resp. I ] if ξ(ω) ∈ Tω (ξ(ω)) [resp. Iω (ξ(ω)) = ξ(ω)] for every ω ∈ Ω . The map ξ is a random coincidence [resp. common fixed] point of random operators I and T if Iω (ξ(ω)) ∈ Tω (ξ(ω)) [resp. ξ(ω) = Iω (ξ(ω)) ∈ Tω (ξ(ω))] for every ω ∈ Ω . The set of all random coincidence [resp. common fixed] points of random operators I and T is denoted by RC(I, T ) [resp. RF(I, T )]. The set of all random fixed points of a random operator T is denoted by RF(T ). We use ideas from Itoh [8], Shahzad [24], and Tan and Yuan [30] in the proof of the following result. Theorem 4.1. Let (Ω , Σ ) be a measurable space, D a subset of a Banach space X , I : Ω × D → D, and T : Ω × D → C L(D). Suppose that D is separable, closed and q-starshaped, I and T are continuous random operators and, for every ω ∈ Ω , we have Iω (D) = D [or Iω is q-affine on D and Tω (D) ⊆ Iω (D)], Tω (D) is bounded, the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(D), and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ D. Then RC(I, T ) 6= φ. Moreover, RF(I, T ) 6= φ if one of the following conditions holds for every ω ∈ Ω : (A) (B) (C) (D)
For every v ∈ C(Iω , Tω ), Iω is Tω -weakly commuting at v and Iω2 v = Iω v. Iω and Tω are weakly compatible on C(Iω , Tω ) and limn→∞ Iωn v exists for every v ∈ C(Iω , Tω ). For every v ∈ C(Iω , Tω ), there exists u ∈ D such that limn→∞ Iωn u = v. There exists a measurable function ξ : Ω → D such that Iω (C(Iω , Tω )) = {ξ(ω)} ⊆ C(Iω , Tω ).
Proof. Fix ω ∈ Ω and set S(ω) := {x ∈ D : Iω (x) ∈ Tω (x)}. Note that D is complete and the pair (Iω , Tω ) satisfies the conditions of Theorem 2.3. Thus C(Iω , Tω ) 6= φ and, consequently, S(ω) is nonempty and complete. Now, to show that S : Ω → C L(D) is a measurable map, let A ∈ C L(D). Define ∞ [ ∞ \ 2 L(A) := ω ∈ Ω : δ(Iω (xi ), Tω (xi )) < , n n=1 i=1 −1
−1
where {xi } is a countable dense subset of A. Clearly, S (A) ⊆ L(A). Moreover, to show that L(A) ⊆ S (A), fix ω0 ∈ L(A). Then, for every n, there is an integer i(n) such that δ(Iω0 (xi(n) ), Tω0 (xi(n) )) < n2 . It follows that
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δ(Iω0 (xi(n) ), Tω0 (xi(n) )) → 0 as n → ∞. Since the pair (Iω , Tω ) satisfies the CPN on A, it follows that there exists −1 −1 v ∈ A such that Iω0 (v) ∈ Tω0 (v). Therefore, ω0 ∈ S (A) and, hence, S (A) = L(A). By [7], 2 ∈Σ ω ∈ Ω : δ(Iω (x), Tω (x)) < n for every x ∈ D because the map f : Ω × D → R, defined by f (ω, x) = δ(Iω (x), Tω (x)), is a Caratheodory function. Thus S is measurable. By [15], S has a measurable selector ξ : Ω → D. Thus ξ ∈ RC(I, T ). The rest follows as in the proofs of Theorem 2.1 (above) and Theorem 3.7 of [27]. Theorem 4.1 extends and improves Shahzad [24, Theorems 3.17, 3.18] and Shahzad and Hussain [27, Theorems 3.4, and 3.7]. Corollary 4.2. Let (Ω , Σ ) be a measurable space, D a subset of a Banach space X , and T : Ω × D → C L(D). Suppose that D is separable, closed and q-starshaped, T is a continuous random operator and, for every ω ∈ Ω , we have Tω (D) is bounded, Tω satisfies the FPC on every A ∈ C L(D), and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ D. Then RF(T ) 6= φ. For the rest of the paper, the set of all measurable maps ξ : Ω → B M ( p) will be denoted by M p (Ω ) and the set of all nonempty, closed and bounded subsets of a set D will be denoted by C B(D). Theorem 4.3. Let (Ω , Σ ) be a measurable space, X a Banach space, I : Ω × X → X , T : Ω × X → C L(X ), M ⊆ X and p ∈ X . Suppose that B M ( p) is separable, closed, and q-starshaped, I and T are continuous random operators and, for every ω ∈ Ω , we have Tω (∂ M ∩ M) ⊆ M, Iω (B M ( p)) = B M ( p), the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(B M ( p)), sup y∈Tω x ky − pk ≤ kIω x − pk for all x ∈ B M ( p), and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ B M ( p). Then RC(I, T ) ∩ M p (Ω ) 6= φ. Moreover, RF(I, T ) ∩ M p (Ω ) 6= φ if one of the following conditions holds for every ω ∈ Ω : (a) (b) (c) (d)
For every v ∈ C(Iω , Tω ) ∩ B M ( p), Iω is Tω -weakly commuting at v and Iω2 v = Iω v. Iω and Tω are weakly compatible on C(Iω , Tω )∩ B M ( p) and limn→∞ Iωn v exists for every v ∈ C(Iω , Tω )∩ B M ( p). For every v ∈ C(Iω , Tω ) ∩ B M ( p), there exists u ∈ B M ( p) such that limn→∞ Iωn u = v. There exists a measurable function ξ : Ω → D such that Iω (C(Iω , Tω ) ∩ B M ( p)) = {ξ(ω)} ⊆ C(Iω , Tω ) ∩ B M ( p).
Proof. Fix ω ∈ Ω . As in the proof of Theorem 3.1, Tω (B M ( p)) ⊆ B M ( p) and Tω : B M ( p) → C B(B M ( p)). The rest follows from Theorem 4.1 with D = B M ( p). Corollary 4.4. Let (Ω , Σ ) be a measurable space, X a Banach space, T : Ω × X → C L(X ), M ⊆ X and p ∈ X . Suppose that B M ( p) is separable, closed and q-starshaped, T is a continuous random operator and, for every ω ∈ Ω , we have Tω (∂ M ∩ M) ⊆ M, Tω satisfies the FPC on every A ∈ C L(B M ( p)), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ B M ( p), and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ B M ( p). Then RF(T ) ∩ M p (Ω ) 6= φ. Theorem 4.5. Let (Ω , Σ ) be a measurable space, X a Banach space, I : Ω × X → X , T : Ω × X → C L(X ), M ∈ C0 and p ∈ X . Suppose that I and T are continuous random operators, Tω0 (M p ) is compact for some ω0 ∈ Ω , and, for every ω ∈ Ω , we have Tω (M p ) ⊆ Iω (M) = M, kIω x − pk = kx − pk for all x ∈ M, the pair (Iω , Tω ) satisfies the CPC on every A ∈ C L(M p ), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ M p , and H (Tω x, Tω y) ≤ σ I,T (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, complete and convex, and, for every ω ∈ Ω , we have Tω (B M ( p)) ⊆ Iω (B M ( p)) = B M ( p) and C(Iω , Tω ) ∩ B M ( p) 6= φ. If, in addition, B M ( p) is separable, then RC(I, T ) ∩ M p (Ω ) 6= φ. Moreover, RF(I, T ) ∩ M p (Ω ) 6= φ if one of the conditions (a)–(d) of Theorem 4.3 holds. Proof. Since Tω0 (M p ) is compact for some ω0 ∈ Ω , then, as in Theorem 3.5, B M ( p) is nonempty, complete and convex, Tω (B M ( p)) ⊆ Iω (B M ( p)) = B M ( p), and Tω : B M ( p) → C B(B M ( p)). The rest follows from Theorem 4.1 with D = B M ( p).
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Corollary 4.6. Let (Ω , Σ ) be a measurable space, X a Banach space, T : Ω × X → C L(X ), M ∈ C0 and p ∈ X . Suppose that T is a continuous random operator, Tω0 (M p ) is compact for some ω0 ∈ Ω , and, for every ω ∈ Ω , we have Tω (M p ) ⊆ M, Tω satisfies the FPC on every A ∈ C L(M p ), sup y∈Tω x ky − pk ≤ kx − pk for all x ∈ M p , and H (Tω x, Tω y) ≤ σT (x, y) for all x, y ∈ M p . Then B M ( p) is nonempty, complete and convex, and, for every ω ∈ Ω , we have Tω (B M ( p)) ⊆ B M ( p) and F(Tω ) ∩ B M ( p) 6= φ. If, in addition, B M ( p) is separable, then RF(T ) ∩ M p (Ω ) 6= φ. 5. Concluding remarks 1. All results can be extended easily to p-normed spaces. 2. The starshapedness of D [resp. B M ( p)], in the related results, can be replaced with the condition: “There exists q ∈ D and a fixed sequence {kn } with 0 < kn < 1 converging to 1 such that kn T x + (1 − kn )q ∈ D for every x ∈ D” 3. Following arguments as above and as in [10], it is possible to obtain common fixed point and invariant approximation results for I, J : D → D and S, T : D → C L(D) satisfying H (Sx, T y) ≤ kI x − J yk for all x, y ∈ D. 4. Following the above arguments, it is possible to obtain common fixed point and invariant approximation results for I, J : D → D and T : D → C L(D) satisfying: 1 H (T x, T y) ≤ max kI x − J yk, δ(I x, T x), δ(J y, T y), [δ(I x, T y) + δ(J y, T x)] 2 for all x, y ∈ D. References [1] M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318–323. [2] M.A. Al-Thagafi, N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006) 2778–2786. [3] P.Z. Daffer, H. Kaneko, Applications of f -contraction mappings to nonlinear integral equations, Bull. Inst. Math. Acad. Sinica 22 (1994) 69–74. [4] W.J. Dotson Jr., Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces, J. London Math. Soc. 4 (1972) 408–410. [5] L. Habiniak, Fixed point theorems and invariant approximations, J. Approx. Theory 56 (1989) 241–244. [6] T.L. Hicks, M.D. Humphries, A note on fixed point theorems, J. Approx. Theory 34 (1982) 221–225. [7] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975) 53–72. [8] S. Itoh, Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979) 261–273. [9] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976) 261–263. [10] G. Jungck, Coincidence and fixed points for compatible and relatively nonexpansive maps, Int. J. Math. Math. Sci. 16 (1993) 95–100. [11] G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (1998) 227–238. [12] G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japonica 42 (1995) 249–252. [13] T. Kamran, Coincidence and fixed points for hybrid strict contractions, J. Math. Anal. Appl. 299 (2004) 235–241. [14] H. Kaneko, S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Int. J. Math. Math. Sci. 12 (1989) 257–262. [15] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 397–403. [16] E. Lami Dozo, Multi-valued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc. 38 (1973) 286–292. [17] A. Latif, A. Bano, A result on invariant approximation, Tamkang J. Math. 33 (2002) 89–92. [18] A. Latif, I. Tweddle, On multi-valued f -nonexpansive maps, Demonstratio Math. 32 (1999) 565–574. [19] D. O’Regan, N. Shahzad, Coincidence points and best proximity pair results for R-subweakly commuting multimaps, Demonstratio Math. 39 (2006) 845–854. [20] D. O’Regan, N. Shahzad, Coincidence points and invariant approximation results for multimaps, Acta Math. Sinica (2006) (in press). [21] R.E. Rhoades, On multi-valued f -nonexpansive maps, Fixed Point Theory Appl. 2 (2001) 89–92. [22] S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349–351. [23] N. Shahzad, Invariant approximations, generalized I -contractions and R-subweakly commuting maps, Fixed Point Theory Appl. 1 (2005) 79–86. [24] N. Shahzad, Some general random coincidence point theorems, New Zealand J. Math. 33 (2004) 95–103. [25] N. Shahzad, Coincidence points and R-subweakly commuting multivalued maps, Demonstratio Math. 36 (2003) 427–431. [26] N. Shahzad, Invariant approximation and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39–45.
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