Coincidence Points and Invariant Approximation ... - Springer Link

Acta Mathematica Sinica, English Series Sep., 2007, Vol. 23, No. 9, pp. 1601–1610 Published online: Sep. 18, 2006 DOI: 10.1007/s10114-005-0768-1 Http://www.ActaMath.com

Coincidence Points and Invariant Approximation Results for Multimaps Donal O’REGAN Department of Mathematics, National University of Ireland, Galway, Ireland E-mail: [email protected]

Naseer SHAHZAD Department of Mathematics, King Abdul Aziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia E-mail: [email protected] Abstract Some coincidence point theorems satisfying a general contractive condition are proved. As applications, some invariant approximation results are also obtained and several related results in the literature are either extended or improved. Keywords

coincidence point, weak commutativity, best approximant

MR(2000) Subject Classification 47H10, 54H25

1

Introduction and Preliminaries

Let X := (X, d) be a metric space. We denote by CD(X) the family of nonempty closed subsets of X. Let H be the generalised Hausdorff distance on CD(X), i.e., for any A, B ∈ CD(X), H(A, B) := inf{ > 0 : A ⊆ B(B, ), B ⊆ B(A, )} ∈ [0, ∞], where B(S, r) := ∪x∈S B(x, r) (r > 0). Let T : X −→ CD(X) be a multimap. Then T is said to be a contraction if there exists 0 ≤ λ < 1 such that H(T x, T y) ≤ λd(x, y) for all x, y ∈ X. If λ = 1, then T is called nonexpansive. Let f : X −→ X be a map. A point x∗ ∈ X is a fixed point of T (resp. f ) if x∗ ∈ T x∗ (resp. x∗ = f x∗ ). The set of fixed points of T (resp. f ) is denoted by F (T ) (resp. F (f )). A point x∗ ∈ X is a coincidence point of f and T if f x∗ ∈ T x∗ . The set of coincidence points of f and T is denoted by C(f, T ). The pair {f, T } is called (1) commuting if f T x = T f x for all x ∈ X; (2) weakly compatible [1] if f and T commute at points in C(f, T ); (3) (IT )-commuting [2] at x ∈ X if f T x ⊆ T f x; (4) T -weakly commuting [3] at x ∈ X if f f x ∈ T f x. The mappings f and T are said to satisfy property (EA) [3] if there exists a sequence {xn } in X, some t ∈ X and A ∈ CD(X), such that lim f xn = t ∈ A = lim T xn .

n→∞

n→∞

Let S be a nonempty subset of a normed space E. Then the set S is called p-starshaped with p ∈ S if λx+(1−λ)p ∈ S for all x ∈ S and all real λ with 0 ≤ λ ≤ 1. Suppose S is p-starshaped. Then a map f : S −→ S is called affine if f (λx + (1 − λ)p) = λf x + (1 − λ)f p for all ∈ S and Received March 4, 2005, Revised July 15, 2005, Accepted July 28, 2005

O’Regan D. and Shahzad N.

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all real λ with 0 ≤ λ ≤ 1. We define Tλ x := λT x + (1 − λ)p and [T x, p] := {Tλ x : λ ∈ [0, 1]} for T : S → CD(S). A multimap T : S −→ CD(X) is said to be demiclosed at y0 ∈ X, if whenever {xn } ⊂ S and {yn } ⊂ X with yn ∈ T xn are sequences such that {xn } converges weakly to x0 and {yn } converges to y0 in X, then y0 ∈ T x0 . The mapping f is weakly continuous if f xn → f x weakly whenever xn → x weakly. The set PS (ˆ x) := {y ∈ S : d(y, x ˆ) = d(ˆ x, S)} is called the set of best approximants to x ˆ∈X out of S, where d(ˆ x, S) := inf{d(z, x ˆ) : z ∈ S}. In recent years, various useful notions of noncommuting maps have been introduced and several coincidence point and fixed point results have appeared for these classes of noncommuting maps. Sessa [4] defined the notion of weakly commuting single-valued maps. Then Jungck [5, 6] introduced the class of compatible single-valued maps which includes properly the class of weakly commuting maps. The concept of weak compatibility was given by Jungck and Rhoades [1]. Pant, et al. [7–9] initiated the investigation of noncompatible single-valued maps. Singh and Mishra [10] studied coincidence and fixed points of (IT )-commuting maps, introduced initially by Itoh and Takahashi [2]. They further noted that the (IT )-commutativity of f and T at a coincidence point is more general than their weak compatibility at the same point. Aamri and Moutawakil [11] defined property (EA) for single-valued maps and used this concept to establish some fixed point results for generalized strict contractions. Recently, Kamran [3] extended property (EA) to the setting of multimaps and introduced T -weak commutativity of maps. Exploiting these concepts, he obtained some coincidence and fixed point theorems. In this paper we prove some coincidence point theorems for maps satisfying a new general contractive condition, which contains Ciric maps, Reich maps and other important classes of maps. For a comparison of various definitions of contractive mappings, we refer the reader to Rhoades [12]. As applications, we derive some invariant approximation results. For a brief history of the subject, we refer the reader to [13] and the references cited therein. Our results extend, improve and complement several related results in the literature including those due to Kamran [3], Aamri and Moutawakil [11], Jungck and Sessa [14], Latif and Bano [15], and Shahzad [16]. 2

Main Results

We begin by stating and proving a result for a general contractive condition. Theorem 2.1 Let X := (X, d) be a metric space, f : X −→ X and T : X −→ CD(X) such that f and T satisfy property (EA). Suppose that there exists a continuous nondecreasing function φ : [0, ∞) → [0, ∞) and continuous functions φi : [0, ∞) → [0, ∞) (i = 1, 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and φ(φi (z)) < z for z > 0 and i = 3, 5, 6, 7, and H(T x, T y) ≤ φ(max{φ1 (d(f x, f y)), φ2 (d(f x, T x)), φ3 (d(f y, T y)), φ4 (d(f y, T x)), φ5 (d(f x, T y)), φ6 (d(f x, f y) + d(f x, T x) + d(f y, T x) + d(f y, T y)), φ7 (d(f x, f y) + d(f x, T x) + d(f y, T x) + d(f x, T y))}), for all x, y ∈ X. If f (X) is closed, then C(f, T ) = ∅. Proof Since f and T satisfy property (EA), there exists a sequence {xn } in X, t ∈ X and

Coincidence Points for Multimaps

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A ∈ CD(X) such that limn→∞ f xn = t ∈ A = limn→∞ T xn . Since f (X) is closed, it follows that t = limn→∞ f xn ∈ f (X). Thus there exists a ∈ X such that limn→∞ f xn = t = f a. We claim that f a ∈ T a. Now suppose this is not so. Then d(f a, T a) > 0. Now H(T xn , T a) ≤ φ(max{φ1 (d(f xn , f a)), φ2 (d(f xn , T xn )), φ3 (d(f a, T a)), φ4 (d(f a, T xn )), φ5 (d(f xn , T a)), φ6 (d(f xn , f a) + d(f xn , T xn ) + d(f a, T xn ) + d(f a, T a)), φ7 (d(f xn , f a) + d(f xn , T xn ) + d(f a, T xn ) + d(f xn , T a))}), which, on taking the limit as n → ∞, gives H(A, T a) ≤ φ(max{φ1 (d(f a, f a)), φ2 (d(f a, A)), φ3 (d(f a, T a)), φ4 (d(f a, A)), φ5 (d(f a, T a)), φ6 (d(f a, f a) + d(f a, A) + d(f a, A) + d(f a, T a)), φ7 (d(f a, f a) + d(f a, A) + d(f a, A) + d(f a, T a))}). It further implies that (note that f a ∈ A) d(f a, T a) ≤ φ(max{φ1 (0), φ2 (0), φ3 (d(f a, T a)), φ4 (0), φ5 (d(f a, T a)), φ6 (0 + 0 + 0 + d(f a, T a)), φ7 (0 + 0 + 0 + d(f a, T a))}). Since φi (0) = 0 for i = 1, 2, 4, it follows that d(f a, T a) ≤ φ(max{φ3 (d(f a, T a)), φ5 (d(f a, T a)), φ6 (d(f a, T a)), φ7 (d(f a, T a))}). If φ3 (d(f a, T a)) = max{φ3 (d(f a, T a)), φ5 (d(f a, T a)), φ6 (d(f a, T a)), φ7 (d(f a, T a))}, then, since d(f a, T a) > 0, we have (using φ(φ3 (z)) < z for z > 0) d(f a, T a) ≤ φ(φ3 (d(f a, T a))) < d(f a, T a), a contradiction. One can obtain contradictions in the other cases in a similar fashion. Hence d(f a, T a) = 0, which implies f a ∈ T a = T a. Corollary 2.2 Let X := (X, d) be a metric space, f : X −→ X and T : X −→ CD(X) such that f and T satisfy property (EA). Suppose that there exists a continuous nondecreasing function φ : [0, ∞) → [0, ∞) and a continuous function φ1 : [0, ∞) → [0, ∞) satisfying φ(z) < 2z for z > 0 and φ1 (0) = 0, and 1 1 H(T x, T y) ≤ φ max φ1 (d(f x, f y)), [d(f x, T x) + d(f y, T y)], [d(f y, T x) + d(f x, T y)] 2 2 for all x, y ∈ X. If f (X) is closed, then C(f, T ) = ∅. Corollary 2.3 (Kamran [3, Theorem 3.4]) Let X := (X, d) be a metric space, f : X −→ X and T : X −→ CD(X) such that f and T satisfy property (EA). Suppose that 1 1 H(T x, T y) ≤ max d(f x, f y), [d(f x, T x) + d(f y, T y)], [d(f y, T x) + d(f x, T y)] , 2 2 for all x, y ∈ X. If f (X) is closed, then C(f, T ) = ∅. The following simple example shows the generality of our result: Example 2.4 Let X = [1, ∞) with the usual metric. Consider f x = x2 and T x = [1, 3x + 1] for all x ∈ X. Then f and T satisfy property (EA). To see this choose the sequence xn = 1 + n1 .


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Also 3 H(T x, T y) = 3|x − y| ≤ d(f x, f y) 2 1 1 ≤ φ max d(f x, f y), [d(f x, T x) + d(f y, T y)], [d(f y, T x) + d(f x, T y)] , 2 2 for all x, y ∈ X, where φ(t) := 32 t. All of the hypotheses of Corollary 2.2 are satisfied. Note that Theorem 3.4 of Kamran [3] cannot be used here. In fact here note that f 1 = 1 ∈ [1, 4] = T 1. Next, we prove some common fixed point theorems. Theorem 2.5 Let X := (X, d) be a metric space, f : X −→ X and T : X −→ CD(X) such that f and T satisfy property (EA) and f (X) is closed. Suppose that there exists a continuous nondecreasing function φ : [0, ∞) → [0, ∞) and continuous functions φi : [0, ∞) → [0, ∞) (i = 1, 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and φ(φi (z)) < z for z > 0 and i = 3, 5, 6, 7, and H(T x, T y) ≤ φ(max{φ1 (d(f x, f y)), φ2 (d(f x, T x)), φ3 (d(f y, T y)), φ4 (d(f y, T x)), φ5 (d(f x, T y)), φ6 (d(f x, f y) + d(f x, T x) + d(f y, T x) + d(f y, T y)), φ7 (d(f x, f y) + d(f x, T x) + d(f y, T x) + d(f x, T y))}), for all x, y ∈ X. If one of the following conditions holds : (i) f is continuous, T is closed (that is, has closed graph), f is (IT )-commuting at points in C(f, T ), and limn→∞ f n a exists for a ∈ C(f, T ); or, (ii) f is T -weakly commuting at a and f f a = f a for any a ∈ C(f, T ); then F (f ) ∩ F (T ) = ∅. Proof Theorem 2.1 guarantees that C(f, T ) = ∅. Thus there exists a point a ∈ X such that f a ∈ T a. If (i) holds, then, since f is (IT )-commuting at points in C(f, T ), we have, for a ∈ C(f, T ), f n−1 T a = f n−2 f T a ⊆ f n−2 T f a = f n−3 f T f a ⊆ f n−3 T f 2 a = . . . ⊆ T f n−1 a (here we use the fact that f a ∈ C(f, T ) etc.; to see this notice f a ∈ T a so f f a ∈ f T a ⊆ T f a since a ∈ C(f, T )). Thus f n a = f n−1 f a ∈ f n−1 T a ⊆ T f n−1 a for a ∈ C(f, T ). Let a0 = limn→∞ f n a. Then, taking the limit as n → ∞, we get a0 ∈ F (T ). Also since f is continuous we have a0 ∈ F (f ). Thus F (T ) F (f ) = ∅. If (ii) holds, then since f is T -weakly commuting, we have f f a ∈ T f a. Consequently, we have f a = f f a ∈ T f a. Let b0 = f a. Then b0 = f b0 ∈ T b0 . Thus F (f ) ∩ F (T ) = ∅. Corollary 2.6 (Kamran [3, Theorem 3.10]) Let X := (X, d) be a metric space, f : X −→ X and T : X −→ CD(X) such that f and T satisfy property (EA) and f (X) is closed. Suppose that 1 1 H(T x, T y) ≤ max d(f x, f y), [d(f x, T x) + d(f y, T y)], [d(f y, T x) + d(f x, T y)] , 2 2 for all x, y ∈ X. If f is T -weakly commuting at a and f f a = f a, for any a ∈ C(f, T ), then F (f ) ∩ F (T ) = ∅. Example 2.7

4

Let X = [0, 1] with the usual metric. Consider f x = x2 and T x = [0, 2(x3x 2 +1) ],


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for all x ∈ X. Then f and T satisfy property (EA); to see this choose the sequence xn = Also 3 H(T x, T y) ≤ d(f x, f y) 2 1 1 ≤ φ max d(f x, f y), [d(f x, T x) + d(f y, T y)], [d(f y, T x) + d(f x, T y)] , 2 2

1 n.

for all x, y ∈ X, where φ(t) := 32 t. We now show that condition (ii) of Theorem 2.5 is satisfied. We first find C(f, T ). Let x ∈ X = [0, 1]. If x = 0, then f 0 = 0 ∈ {0} = T 0 and so 4 2 2 4 0 ∈ C(f, T ). Suppose x ∈ (0, 1] and f x ∈ T x. Note that x2 > 2(x3x 2 +1) since 2x (x + 1) > 3x 2 (i.e. x < 2). Thus C(f, T ) = {0}. Let a ∈ C(f, T ) so a = 0. Now f f a = f f 0 = 0 ∈ {0} = T f 0 = T f a, and so f is T -weakly commuting at a and f f a = f f 0 = f 0 = f a for a ∈ C(f, T ). Thus all of the hypotheses of Theorem 2.5 are satisfied. Notice also that Corollary 2.6 does not apply. In fact, here note 0 = f 0 ∈ T 0 = {0}. Theorem 2.8 Let S be a closed and p-starshaped subset of a normed space E with p ∈ S, f : S −→ S, and T : S −→ CD(S). Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)

≤ α(max{φ1 (||f x − f y||), φ2 (d(f x, [T x, p])), φ3 (d(f y, [T y, p])), φ4 (d(f y, [T x, p])), φ5 (d(f x, [T y, p])), φ6 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f y, [T y, p])), φ7 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f x, [T y, p]))}),

for all x, y ∈ S; and, (b) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. If f (S) and (f − T )(S) are closed and T (S) is bounded, then C(f, T ) = ∅. In addition, if one of the following conditions holds : (i) f is continuous and T is closed (that is, has closed graph), f is (IT )-commuting at points in C(f, T ), and limn→∞ f n a exists for a ∈ C(f, T ); or, (ii) f is T -weakly commuting at a and f f a = f a for any a ∈ C(f, T ), then F (f ) ∩ F (T ) = ∅. Proof Choose a sequence {λn } ⊂ (0, 1) such that λn −→ 1 as n → ∞. For each n, define Tn : S −→ CD(S) by Tn (x) := λn T x + (1 − λn )p for each x ∈ S. Note that, for each n, Tn (S) ⊆ S (since S is p-starshaped). Let φ(n) (t) := αλn t. Then H(Tn x, Tn y)

= λn H(T x, T y) ≤ αλn max{φ1 (||f x − f y||), φ2 (d(f x, [T x, p])), φ3 (d(f y, [T y, p])), φ4 (d(f y, [T x, p])), φ5 (d(f x, [T y, p])), φ6 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f y, [T y, p])),


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φ7 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f x, [T y, p]))}) ≤ αλn max{φ1 (||f x − f y||), φ2 (d(f x, Tn x)), φ3 (d(f y, Tn y)), φ4 (d(f y, Tn x)), φ5 (d(f x, Tn y)), φ6 (||f x − f y|| + d(f x, Tn x) + d(f y, Tn x) + d(f y, Tn y)), φ7 (||f x − f y|| + d(f x, Tn x) + d(f y, Tn x) + d(f x, Tn y))}) = φ(n) (max{φ1 (||f x − f y||), φ2 (d(f x, Tn x)), φ3 (d(f y, Tn y)), φ4 (d(f y, Tn x)), φ5 (d(f x, Tn y)), φ6 (||f x − f y|| + d(f x, Tn x) + d(f y, Tn x) + d(f y, Tn y)), φ7 (||f x − f y|| + d(f x, Tn x) + d(f y, Tn x) + d(f x, Tn y))})), for all x, y ∈ S. Now Theorem 2.1 guarantees that C(f, Tn ) = ∅, that is, f xn ∈ Tn xn for some xn ∈ S. This implies that there is a yn ∈ T xn such that f xn − yn = (1 − λn )(p − yn ). Since T (S) is bounded, it follows that f xn − yn −→ 0 as n −→ ∞. The closedness of (f − T )(S) further implies that 0 ∈ (f − T )(S). Hence C(f, T ) = ∅ and so f a ∈ T a, for some a ∈ S. Now the argument in Theorem 2.5 immediately guarantees that F (f ) and F (T ) have a common element. Theorem 2.9 Let S be a compact and p-starshaped subset of a normed space E with p ∈ S, f : S −→ S, and T : S −→ CD(S). Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)

≤ α(max{φ1 (||f x − f y||), φ2 (d(f x, [T x, p])), φ3 (d(f y, [T y, p])), φ4 (d(f y, [T x, p])), φ5 (d(f x, [T y, p])), φ6 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f y, [T y, p])), φ7 (||f x − f y|| + d(f x, [T x, p]) + d(f y, [T x, p]) + d(f x, [T y, p]))})

for all x, y ∈ S; (b) f is continuous on S; and, (c) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. If T is closed, then C(f, T ) = ∅. In addition, if one of the following conditions holds : (i) f is (IT )-commuting at points in C(f, T ), and limn→∞ f n a exists for a ∈ C(f, T ); or, (ii) f is T -weakly commuting at a and f f a = f a, for any a ∈ C(f, T ), then F (f ) ∩ F (T ) = ∅. Proof Since S is compact and f is continuous, we have f (S) is compact. As in the proof of Theorem 2.8, there is a yn ∈ T xn for some xn ∈ S such that f xn − yn = (1 − λn )(p − yn ). Since T (S) ⊆ S is bounded, it follows that f xn − yn −→ 0 as n −→ ∞. The compactness of S further implies that there exists a subsequence {xm } of {xn } such that xm → a ∈ S. Thus since f is continuous and T is closed we have f a ∈ T a. Now the argument in Theorem 2.5 immediately guarantees that F (f ) and F (T ) have a common element.


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Theorem 2.10 Let S be a weakly compact and p-starshaped subset of a Banach space E with p ∈ S, f : S −→ S, and T : S −→ CD(S). Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)


for all x, y ∈ S; (b) f is weakly continuous on S; and, (c) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. If f − T is demiclosed at zero, then C(f, T ) = ∅. In addition, if one of the following conditions holds : (i) f is (IT )-commuting at points in C(f, T ), and limn→∞ f n a exists for a ∈ C(f, T ); or, (ii) f is T -weakly commuting at a and f f a = f a for any a ∈ C(f, T ), then F (f ) ∩ F (T ) = ∅. Proof Since S is weakly compact and f is weakly continuous, we have f (S) is weakly compact, and so f (S) is weakly closed, so closed. As in the proof of Theorem 2.8, there is a yn ∈ T xn for some xn ∈ S such that f xn − yn = (1 − λn )(p − yn ). Since T (S) ⊆ S is bounded (note weakly compact sets are bounded; see [17, p. 130]), it follows that f xn − yn −→ 0 as n −→ ∞. The weak compactness of S further implies that there exists a subsequence {xm } of {xn } such that xm → a ∈ S weakly. Thus, since f − T is demiclosed at zero, we have f a ∈ T a. The following result contains Theorem 3.14 of Kamran [3], as a special case: Theorem 2.11 Let S be a subset of a normed space E, f : E −→ E, T : E −→ CD(E) and x ˆ ∈ E. Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)


x); for all x, y ∈ PS (ˆ (b) f is continuous on PS (ˆ x) and T : PS (ˆ x) → CD(E) is closed; and, (c) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. x) is nonempty, compact, p-starshaped with p ∈ S, and if it is both f -invariant and If PS (ˆ T -invariant, then PS (ˆ x) ∩ C(f, T ) = ∅. In addition, if one of the following conditions holds :


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(i) f is (IT )-commuting at points in PS (ˆ x) ∩ C(f, T ), and limn→∞ f n a exists for a ∈ PS (ˆ x) ∩ C(f, T ); or, (ii) f is T -weakly commuting at a and ffa=fa for a ∈ PS (ˆ x) ∩ C(f, T ), then PS (ˆ x) ∩ F (f ) ∩ F (T ) = ∅. x) is both f -invariant and T -invariant, it follows that f : PS (ˆ x) → PS (ˆ x) and Proof Since PS (ˆ T : PS (ˆ x) → CD(PS (ˆ x)). The results now follow from Theorem 2.9. Corollary 2.12 Let S be a subset of a normed space E, f : E −→ E, T : E −→ CD(E) such that T (∂S S) ⊂ S and f (ˆ x) ∈ T (ˆ x) = {ˆ x} for some x ˆ ∈ E. Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)


x); for all x, y ∈ PS (ˆ (b) H(T x, T x ˆ) ≤ ||f x − f x ˆ|| for all x ∈ PS (ˆ x); (c) f is continuous on PS (ˆ x) and T : PS (ˆ x) → CD(E) is close; and, (d) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1 here Tλ x := λT x + (1 − λ)p. If PS (ˆ x) is nonempty, compact, p-starshaped with p ∈ S, and is f -invariant, then PS (ˆ x) ∩ C(f, T ) = ∅. In addition, if one of the following conditions holds: (i) f is (IT )-commuting at points in PS (ˆ x) ∩ C(f, T ), and limn→∞ f n a exists for a ∈ PS (ˆ x) ∩ C(f, T ); or, (ii) f is T -weakly commuting at a and ffa=fa for a ∈ PS (ˆ x) ∩ C(f, T ), then PS (ˆ x) ∩ F (f ) ∩ F (T ) = ∅. x). Then (1 − k)z + kˆ x−x ˆ = (1 − k) z − x ˆ < d(ˆ x, S) for k ∈ (0, 1). Proof Let z ∈ PS (ˆ Therefore, {(1 − k)z + kˆ x : k ∈ (0, 1)} S = ∅ and so z is not in the interior of S. Thus z ∈ ∂S S and so T z ⊆ S since T (∂S S) ⊂ S. We claim that T z ⊆ PS (ˆ x) for all z ∈ P (ˆ x). Let y ∈ T z ⊆ S. Then, since f (ˆ x) ∈ T (ˆ x) = {ˆ x} and f z ∈ f (PS (ˆ x)) ⊆ PS (ˆ x), we have d(y, T x ˆ) ≤ H(T z, T x ˆ) ≤ ||f z − f x ˆ|| = ||f z − x ˆ|| = d(ˆ x, S). x) This implies that ||y − x ˆ|| = d(ˆ x, S), since T (ˆ x) = {ˆ x} and d(ˆ x, S) ≤ ||y − x ˆ||. Thus y ∈ PS (ˆ and so T z ⊆ PS (ˆ x) if z ∈ P (ˆ x). Hence PS (ˆ x) is T -invariant. The result now follows from Theorem 2.11. The following extends and complements recent results due to Jungck and Sessa [14], Latif and Bano [15], and Shahzad [16]. Theorem 2.13 Let S be a subset of a Banach space E, f : E −→ E, T : E −→ CD(E) and x ˆ ∈ E. Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists


some α ∈ (0, 2] with φi (z) < H(T x, T y)

z α

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for z > 0 and i = 3, 5, 6, 7, and


for all x, y ∈ PS (ˆ x); x); and, (b) f is weakly continuous on PS (ˆ (c) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. x) is nonempty, weakly compact, p-starshaped with p ∈ S, both f -invariant and T If PS (ˆ invariant, and if f − T is demiclosed at zero, then PS (ˆ x) ∩ C(f, T ) = ∅. In addition, if one of the following conditions holds: x)∩C(f, T ), and limn→∞ f n a exists for a ∈ PS (ˆ x)∩ (i) f is (IT )-commuting at points in PS (ˆ C(f, T ); or, x) ∩ C(f, T ), (ii) f is T -weakly commuting at a and f f a = f a for a ∈ PS (ˆ x) ∩ F (f ) ∩ F (T ) = ∅. then PS (ˆ Proof Since PS (ˆ x) is both f -invariant and T -invariant, it follows that f : PS (ˆ x) → PS (ˆ x) and T : PS (ˆ x) → CD(PS (ˆ x)). The results now follow from Theorem 2.10. Corollary 2.14 Let S be a subset of a Banach space E, f : E −→ E, T : E −→ CD(E) such that T (∂S S) ⊂ S and f (ˆ x) ∈ T (ˆ x) = {ˆ x} for some x ˆ ∈ E. Suppose that (a) There exists a continuous function φ1 : [0, ∞) → [0, ∞) and continuous nondecreasing functions φi : [0, ∞) → [0, ∞) (i = 2, . . . , 7) satisfying φi (0) = 0 for i = 1, 2, 4 and, there exists some α ∈ (0, 2] with φi (z) < αz for z > 0 and i = 3, 5, 6, 7, and H(T x, T y)


for all x, y ∈ PS (ˆ x); x); (b) H(T x, T x ˆ) ≤ ||f x − f x ˆ|| for all x ∈ PS (ˆ x); and, (c) f is weakly continuous on PS (ˆ (d) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλ x := λT x + (1 − λ)p. x) is nonempty, weakly compact, p-starshaped with p ∈ S, f -invariant, and if f − T If PS (ˆ is demiclosed at zero, then PS (ˆ x) ∩ C(f, T ) = ∅. In addition, if one of the following conditions holds : x) ∩ C(f, T ), and limn→∞ f n a exists for a ∈ (i) f is (IT )-commuting at points in PS (ˆ PS (ˆ x) ∩ C(f, T ); or, x) ∩ C(f, T ), (ii) f is T -weakly commuting at a and f f a = f a for a ∈ PS (ˆ x) ∩ F (f ) ∩ F (T ) = ∅. then PS (ˆ Proof As in Corollary 2.12, PS (ˆ x) is T -invariant. The result now follows from Theorem 2.13.

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Acknowledgement


The authors thank the referee for his/her suggestions and comments.

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