coincidence points and invariant approximation results for multimaps

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’REGANDepartment of Mathematics, National University of Ireland, Galway, Ireland

E-mail: [email protected]

Naseer SHAHZADDepartment 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 subsetsof X. Let H be the generalised Hausdorff distance on CD(X), i.e., for any A, B ∈ CD(X),

H(A, B) := inf{ε > 0 : A ⊆ B(B, ε), B ⊆ B(A, ε)} ∈ [0,∞],

where B(S, r) := ∪x∈SB(x, r) (r > 0). Let T : X −→ CD(X) be a multimap. Then T is saidto be a contraction if there exists 0 ≤ λ < 1 such that H(Tx, Ty) ≤ 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 fixedpoint of T (resp. f) if x∗ ∈ Tx∗ (resp. x∗ = fx∗). The set of fixed points of T (resp. f) isdenoted by F (T ) (resp. F (f)). A point x∗ ∈ X is a coincidence point of f and T if fx∗ ∈ Tx∗.The set of coincidence points of f and T is denoted by C(f, T ). The pair {f, T} is called (1)commuting if fTx = Tfx for all x ∈ X; (2) weakly compatible [1] if f and T commute atpoints in C(f, T ); (3) (IT )-commuting [2] at x ∈ X if fTx ⊆ Tfx; (4) T -weakly commuting [3]at x ∈ X if ffx ∈ Tfx. The mappings f and T are said to satisfy property (EA) [3] if thereexists a sequence {xn} in X, some t ∈ X and A ∈ CD(X), such that

limn→∞ fxn = t ∈ A = lim

n→∞Txn.

Let S be a nonempty subset of a normed space E. Then the set S is called p-starshaped withp ∈ 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) = λfx + (1 − λ)fp for all ∈ S and

Received March 4, 2005, Revised July 15, 2005, Accepted July 28, 2005

1602 O’Regan D. and Shahzad N.

all real λ with 0 ≤ λ ≤ 1. We define Tλx := λTx + (1− λ)p and [Tx, p] := {Tλx : λ ∈ [0, 1]} forT : 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 ∈ Txn are sequences such that {xn} converges weakly tox0 and {yn} converges to y0 in X, then y0 ∈ Tx0. The mapping f is weakly continuous iffxn → fx weakly whenever xn → x weakly.

The set PS(x̂) := {y ∈ S : d(y, x̂) = d(x̂, S)} is called the set of best approximants to x̂ ∈ X

out of S, where d(x̂, S) := inf{d(z, x̂) : 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 noncom-muting maps. Sessa [4] defined the notion of weakly commuting single-valued maps. ThenJungck [5, 6] introduced the class of compatible single-valued maps which includes properly theclass of weakly commuting maps. The concept of weak compatibility was given by Jungck andRhoades [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, intro-duced initially by Itoh and Takahashi [2]. They further noted that the (IT )-commutativity off 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 conceptto 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 ofmaps. Exploiting these concepts, he obtained some coincidence and fixed point theorems. Inthis paper we prove some coincidence point theorems for maps satisfying a new general con-tractive 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 ofthe 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 nondecreasingfunction 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(Tx, Ty) ≤ φ(max{φ1(d(fx, fy)), φ2(d(fx, Tx)), φ3(d(fy, Ty)),

φ4(d(fy, Tx)), φ5(d(fx, Ty)),

φ6(d(fx, fy) + d(fx, Tx) + d(fy, Tx) + d(fy, Ty)),

φ7(d(fx, fy) + d(fx, Tx) + d(fy, Tx) + d(fx, Ty))}),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 1603

A ∈ CD(X) such that limn→∞ fxn = t ∈ A = limn→∞ Txn. Since f(X) is closed, it followsthat t = limn→∞ fxn ∈ f(X). Thus there exists a ∈ X such that limn→∞ fxn = t = fa. Weclaim that fa ∈ Ta. Now suppose this is not so. Then d(fa, Ta) > 0. Now

H(Txn, Ta) ≤ φ(max{φ1(d(fxn, fa)), φ2(d(fxn, Txn)), φ3(d(fa, Ta)),

φ4(d(fa, Txn)), φ5(d(fxn, Ta)),

φ6(d(fxn, fa) + d(fxn, Txn) + d(fa, Txn) + d(fa, Ta)),

φ7(d(fxn, fa) + d(fxn, Txn) + d(fa, Txn) + d(fxn, Ta))}),which, on taking the limit as n → ∞, gives

H(A, Ta) ≤ φ(max{φ1(d(fa, fa)), φ2(d(fa, A)), φ3(d(fa, Ta)),

φ4(d(fa, A)), φ5(d(fa, Ta)),

φ6(d(fa, fa) + d(fa, A) + d(fa, A) + d(fa, Ta)),

φ7(d(fa, fa) + d(fa, A) + d(fa, A) + d(fa, Ta))}).It further implies that (note that fa ∈ A)

d(fa, Ta) ≤ φ(max{φ1(0), φ2(0), φ3(d(fa, Ta)), φ4(0), φ5(d(fa, Ta)),

φ6(0 + 0 + 0 + d(fa, Ta)), φ7(0 + 0 + 0 + d(fa, Ta))}).Since φi(0) = 0 for i = 1, 2, 4, it follows that

d(fa, Ta) ≤ φ(max{φ3(d(fa, Ta)), φ5(d(fa, Ta)), φ6(d(fa, Ta)), φ7(d(fa, Ta))}).If

φ3(d(fa, Ta)) = max{φ3(d(fa, Ta)), φ5(d(fa, Ta)), φ6(d(fa, Ta)), φ7(d(fa, Ta))},then, since d(fa, Ta) > 0, we have (using φ(φ3(z)) < z for z > 0)

d(fa, Ta) ≤ φ(φ3(d(fa, Ta))) < d(fa, Ta),

a contradiction. One can obtain contradictions in the other cases in a similar fashion. Henced(fa, Ta) = 0, which implies fa ∈ Ta = Ta.

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 nondecreasingfunction φ : [0,∞) → [0,∞) and a continuous function φ1 : [0,∞) → [0,∞) satisfying φ(z) < 2z

for z > 0 and φ1(0) = 0, and

H(Tx, Ty) ≤ φ

(max

{φ1(d(fx, fy)),

12[d(fx, Tx) + d(fy, Ty)],

12[d(fy, Tx) + d(fx, Ty)]

})

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

H(Tx, Ty) ≤ max{

d(fx, fy),12[d(fx, Tx) + d(fy, Ty)],


},

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 fx = x2 and Tx = [1, 3x + 1]for all x ∈ X. Then f and T satisfy property (EA). To see this choose the sequence xn = 1+ 1

n .


Also

H(Tx, Ty) = 3|x − y| ≤ 32d(fx, fy)

≤ φ

(max

{d(fx, fy),

12[d(fx, Tx) + d(fy, Ty)],


}),

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 f1 = 1 ∈ [1, 4] = T1.

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) suchthat f and T satisfy property (EA) and f(X) is closed. Suppose that there exists a continuousnondecreasing 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(Tx, Ty) ≤ φ(max{φ1(d(fx, fy)), φ2(d(fx, Tx)), φ3(d(fy, Ty)),

φ4(d(fy, Tx)), φ5(d(fx, Ty)),

φ6(d(fx, fy) + d(fx, Tx) + d(fy, Tx) + d(fy, Ty)),

φ7(d(fx, fy) + d(fx, Tx) + d(fy, Tx) + d(fx, Ty))}),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 pointsin C(f, T ), and limn→∞ fna exists for a ∈ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa 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 thatfa ∈ Ta. If (i) holds, then, since f is (IT )-commuting at points in C(f, T ), we have, fora ∈ C(f, T ),

fn−1Ta = fn−2fTa ⊆ fn−2Tfa = fn−3fTfa ⊆ fn−3Tf2a = . . . ⊆ Tfn−1a

(here we use the fact that fa ∈ C(f, T ) etc.; to see this notice fa ∈ Ta so ffa ∈ fTa ⊆ Tfa

since a ∈ C(f, T )). Thus fna = fn−1fa ∈ fn−1Ta ⊆ Tfn−1a for a ∈ C(f, T ). Let a0 =limn→∞ fna. Then, taking the limit as n → ∞, we get a0 ∈ F (T ). Also since f is continuouswe have a0 ∈ F (f). Thus F (T )

⋂F (f) = ∅.

If (ii) holds, then since f is T -weakly commuting, we have ffa ∈ Tfa. Consequently, wehave fa = ffa ∈ Tfa. Let b0 = fa. Then b0 = fb0 ∈ Tb0. 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. Supposethat

H(Tx, Ty) ≤ max{

d(fx, fy),12[d(fx, Tx) + d(fy, Ty)],


},

for all x, y ∈ X. If f is T -weakly commuting at a and ffa = fa, for any a ∈ C(f, T ), thenF (f) ∩ F (T ) = ∅.Example 2.7 Let X = [0, 1] with the usual metric. Consider fx = x2 and Tx = [0, 3x4

2(x2+1) ],


for all x ∈ X. Then f and T satisfy property (EA); to see this choose the sequence xn = 1n .

Also

H(Tx, Ty) ≤ 32d(fx, fy)

≤ φ

(max

{d(fx, fy),

12[d(fx, Tx) + d(fy, Ty)],


}),

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 f0 = 0 ∈ {0} = T0 and so0 ∈ C(f, T ). Suppose x ∈ (0, 1] and fx ∈ Tx. Note that x2 > 3x4

2(x2+1) since 2x2(x2 + 1) > 3x4

(i.e. x2 < 2). Thus C(f, T ) = {0}. Let a ∈ C(f, T ) so a = 0. Now ffa = ff0 = 0 ∈ {0} =Tf0 = Tfa, and so f is T -weakly commuting at a and ffa = ff0 = f0 = fa for a ∈ C(f, T ).Thus all of the hypotheses of Theorem 2.5 are satisfied. Notice also that Corollary 2.6 does notapply. In fact, here note 0 = f0 ∈ T0 = {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 nondecreasingfunctions φi : [0,∞) → [0,∞) (i = 2, . . . , 7) satisfying φi(0) = 0 for i = 1, 2, 4 and, there existssome α ∈ (0, 2] with φi(z) < z

α for z > 0 and i = 3, 5, 6, 7, and

H(Tx, Ty) ≤ α(max{φ1(||fx − fy||), φ2(d(fx, [Tx, p])), φ3(d(fy, [Ty, p])),

φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),

φ6(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fy, [Ty, p])),

φ7(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fx, [Ty, p]))}),for all x, y ∈ S; and,

(b) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλx := λTx + (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 atpoints in C(f, T ), and limn→∞ fna exists for a ∈ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa 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, defineTn : S −→ CD(S) by Tn(x) := λnTx + (1 − λn)p for each x ∈ S. Note that, for each n,Tn(S) ⊆ S (since S is p-starshaped). Let φ(n)(t) := αλnt. Then

H(Tnx, Tny) = λnH(Tx, Ty)

≤ αλn max{φ1(||fx − fy||), φ2(d(fx, [Tx, p])), φ3(d(fy, [Ty, p])),

φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),



φ7(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fx, [Ty, p]))})≤ αλn max{φ1(||fx − fy||), φ2(d(fx, Tnx)), φ3(d(fy, Tny)),

φ4(d(fy, Tnx)), φ5(d(fx, Tny)),

φ6(||fx − fy|| + d(fx, Tnx) + d(fy, Tnx) + d(fy, Tny)),

φ7(||fx − fy|| + d(fx, Tnx) + d(fy, Tnx) + d(fx, Tny))})= φ(n)(max{φ1(||fx − fy||), φ2(d(fx, Tnx)), φ3(d(fy, Tny)),

φ4(d(fy, Tnx)), φ5(d(fx, Tny)),

φ6(||fx − fy|| + d(fx, Tnx) + d(fy, Tnx) + d(fy, Tny)),

φ7(||fx − fy|| + d(fx, Tnx) + d(fy, Tnx) + d(fx, Tny))})),for all x, y ∈ S. Now Theorem 2.1 guarantees that C(f, Tn) = ∅, that is, fxn ∈ Tnxn for somexn ∈ S. This implies that there is a yn ∈ Txn such that fxn − yn = (1 − λn)(p − yn). SinceT (S) is bounded, it follows that fxn − yn −→ 0 as n −→ ∞. The closedness of (f − T )(S)further implies that 0 ∈ (f − T )(S). Hence C(f, T ) = ∅ and so fa ∈ Ta, for some a ∈ S. Nowthe argument in Theorem 2.5 immediately guarantees that F (f) and F (T ) have a commonelement.

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


α for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),


φ7(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fx, [Ty, 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 Tλx := λTx + (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→∞ fna exists for a ∈ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa, 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 ofTheorem 2.8, there is a yn ∈ Txn for some xn ∈ S such that fxn −yn = (1−λn)(p−yn). SinceT (S) ⊆ S is bounded, it follows that fxn−yn −→ 0 as n −→ ∞. The compactness of S furtherimplies that there exists a subsequence {xm} of {xn} such that xm → a ∈ S. Thus since f iscontinuous and T is closed we have fa ∈ Ta. Now the argument in Theorem 2.5 immediatelyguarantees that F (f) and F (T ) have a common element.


Theorem 2.10 Let S be a weakly compact and p-starshaped subset of a Banach space E withp ∈ S, f : S −→ S, and T : S −→ CD(S). Suppose that


α for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),


φ7(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fx, [Ty, p]))})for all x, y ∈ S;

(b) f is weakly continuous on S; and,


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→∞ fna exists for a ∈ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa 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 ∈ Txn forsome xn ∈ S such that fxn − yn = (1 − λn)(p − yn). Since T (S) ⊆ S is bounded (note weaklycompact sets are bounded; see [17, p. 130]), it follows that fxn − yn −→ 0 as n −→ ∞. Theweak compactness of S further implies that there exists a subsequence {xm} of {xn} such thatxm → a ∈ S weakly. Thus, since f − T is demiclosed at zero, we have fa ∈ Ta.

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) andx̂ ∈ E. Suppose that


α for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),


φ7(||fx − fy|| + d(fx, [Tx, p]) + d(fy, [Tx, p]) + d(fx, [Ty, p]))})for all x, y ∈ PS(x̂);

(b) f is continuous on PS(x̂) and T : PS(x̂) → CD(E) is closed; and,


If PS(x̂) is nonempty, compact, p-starshaped with p ∈ S, and if it is both f-invariant andT -invariant, then PS(x̂) ∩ C(f, T ) = ∅. In addition, if one of the following conditions holds :


(i) f is (IT )-commuting at points in PS(x̂) ∩ C(f, T ), and limn→∞ fna exists for a ∈PS(x̂) ∩ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa=fa for a ∈ PS(x̂) ∩ C(f, T ),then PS(x̂) ∩ F (f) ∩ F (T ) = ∅.Proof Since PS(x̂) is both f -invariant and T -invariant, it follows that f : PS(x̂) → PS(x̂) andT : PS(x̂) → CD(PS(x̂)). 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(x̂) ∈ T (x̂) = {x̂} for some x̂ ∈ E. Suppose that


α for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),



(b) H(Tx, T x̂) ≤ ||fx − fx̂|| for all x ∈ PS(x̂);(c) f is continuous on PS(x̂) and T : PS(x̂) → CD(E) is close; and,(d) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1 here Tλx := λTx + (1 − λ)p.If PS(x̂) is nonempty, compact, p-starshaped with p ∈ S, and is f-invariant, then PS(x̂) ∩

C(f, T ) = ∅. In addition, if one of the following conditions holds:(i) f is (IT )-commuting at points in PS(x̂) ∩ C(f, T ), and limn→∞ fna exists for a ∈

PS(x̂) ∩ C(f, T ); or,(ii) f is T -weakly commuting at a and ffa=fa for a ∈ PS(x̂) ∩ C(f, T ),

then PS(x̂) ∩ F (f) ∩ F (T ) = ∅.Proof Let z ∈ PS(x̂). Then ‖(1 − k)z + kx̂ − x̂‖ = (1 − k)‖z − x̂‖ < d(x̂, S) for k ∈ (0, 1).Therefore, {(1 − k)z + kx̂ : k ∈ (0, 1)}⋂

S = ∅ and so z is not in the interior of S. Thusz ∈ ∂S

⋂S and so Tz ⊆ S since T (∂S

⋂S) ⊂ S. We claim that Tz ⊆ PS(x̂) for all z ∈ P (x̂).

Let y ∈ Tz ⊆ S. Then, since f(x̂) ∈ T (x̂) = {x̂} and fz ∈ f(PS(x̂)) ⊆ PS(x̂), we have

d(y, T x̂) ≤ H(Tz, T x̂) ≤ ||fz − fx̂|| = ||fz − x̂|| = d(x̂, S).

This implies that ||y − x̂|| = d(x̂, S), since T (x̂) = {x̂} and d(x̂, S) ≤ ||y − x̂||. Thus y ∈ PS(x̂)and so Tz ⊆ PS(x̂) if z ∈ P (x̂). Hence PS(x̂) is T -invariant. The result now follows fromTheorem 2.11.

The following extends and complements recent results due to Jungck and Sessa [14], Latifand 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) andx̂ ∈ E. Suppose that

(a) There exists a continuous function φ1 : [0,∞) → [0,∞) and continuous nondecreasingfunctions 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) < zα for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),



(b) f is weakly continuous on PS(x̂); and,


If PS(x̂) is nonempty, weakly compact, p-starshaped with p ∈ S, both f-invariant and T -invariant, and if f − T is demiclosed at zero, then PS(x̂) ∩ C(f, T ) = ∅. In addition, if one ofthe following conditions holds:

(i) f is (IT )-commuting at points in PS(x̂)∩C(f, T ), and limn→∞ fna exists for a ∈ PS(x̂)∩C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa for a ∈ PS(x̂) ∩ C(f, T ),

then PS(x̂) ∩ F (f) ∩ F (T ) = ∅.Proof Since PS(x̂) is both f -invariant and T -invariant, it follows that f : PS(x̂) → PS(x̂) andT : PS(x̂) → CD(PS(x̂)). 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(x̂) ∈ T (x̂) = {x̂} for some x̂ ∈ E. Suppose that


α for z > 0 and i = 3, 5, 6, 7, and


φ4(d(fy, [Tx, p])), φ5(d(fx, [Ty, p])),



(b) H(Tx, T x̂) ≤ ||fx − fx̂|| for all x ∈ PS(x̂);

(c) f is weakly continuous on PS(x̂); and,

(d) f and Tλ satisfy property (EA) for each 0 ≤ λ ≤ 1; here Tλx := λTx + (1 − λ)p.

If PS(x̂) is nonempty, weakly compact, p-starshaped with p ∈ S, f-invariant, and if f − T

is demiclosed at zero, then PS(x̂) ∩ C(f, T ) = ∅. In addition, if one of the following conditionsholds :

(i) f is (IT )-commuting at points in PS(x̂) ∩ C(f, T ), and limn→∞ fna exists for a ∈PS(x̂) ∩ C(f, T ); or,

(ii) f is T -weakly commuting at a and ffa = fa for a ∈ PS(x̂) ∩ C(f, T ),

then PS(x̂) ∩ F (f) ∩ F (T ) = ∅.Proof As in Corollary 2.12, PS(x̂) is T -invariant. The result now follows from Theorem 2.13.


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

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