RACSAM (2011) 105:261–272DOI 10.1007/s13398-011-0036-4

ORIGINAL PAPER

Coincidence point theorems for multivalued f -weakcontraction mappings and applications

Mujahid Abbas · Nawab Hussain · Billy E. Rhoades

Received: 20 December 2010 / Accepted: 26 April 2011 / Published online: 14 May 2011© Springer-Verlag 2011

Abstract We prove the existence of coincidence points and common fixed points formultivalued f -weak contraction mappings which assume closed values only. As an appli-cation, related common fixed point, invariant approximation, random coincidence point andrandom invariant approximation results are also obtained. Our results provide extensions aswell as substantial improvements of several well known results in the existing literature.

Keywords Coincidence point · Random fixed point · Multivalued f -weak contractionmap · Random approximation · Measurable space

Mathematics Subject Classification (2000) 47H09 · 47H10 · 47H40 · 54H25 · 60H25

1 Introduction and preliminaries

Let (X, d) be a metric space. We denote by C B(X) and C L(X), the families of all nonemptyclosed bounded and nonempty closed subsets of X, respectively. For A, B ∈ C L(X). Set,E A,B = {ε > 0 : A ⊆ Nε(B), B ⊆ Nε(A)}. We define a generalized Hausdorff metric Hon C L(X) by

M. Abbas (B)Department of Mathematics, Lahore University of Management Sciences,Lahore 54792, Pakistane-mail: mujahid@lums.edu.pk

N. HussainDepartment of Mathematics, King Abdul Aziz University,P.O. Box 80203, Jeddah 21589, Saudi Arabiae-mail: nhusain@kau.edu.sa; hussianjam@hotmail.com

B. E. RhoadesDepartment of Mathematics, Indiana University,Bloomington, IN 47405-7106, USAe-mail: rhoades@indiana.edu

262 M. Abbas et al.

H(A, B) ={

inf E A,B if E A,B �= ∅∞ if E A,B = ∅

where Nε(A) = {x ∈ X : d(x, A) < ε}. For A, B ∈ C L(X), let d(A, B) = inf{d(a, b) :b ∈ B}. A mapping T : X −→ C L(X) is said to be a multivalued weak contraction iff thereexist two constants θ ∈ (0, 1) and L ≥ 0 such that

H(T x, T y) ≤ θd(x, y) + Ld(y, T x)

for every x, y ∈ X . Let f be a self map on X . A mapping T : X −→ C L(X) is said to bea multivalued f -weak contraction or multivalued ( f, θ, L)-weak contraction iff there existtwo constants θ ∈ (0, 1) and L ≥ 0 such that

H(T x, T y) ≤ θd( f x, f y) + Ld( f y, T x) (1.1)

for every x, y ∈ X .

Definition 1.1 Let f : X −→ X and T : X −→ C L(X). A point x in X is said to be (1)

fixed point of f if f (x) = x; (2) a fixed point of T if x ∈ T (x); (3) a coincidence point of apair ( f, T ) if f x ∈ T x; (4) a common fixed point of a pair ( f, T ) if x = f x ∈ T x .

The symbols F( f ), C( f, T ) and F( f, T ) denote the set of all fixed points of f, the setof all coincidence points of the pair ( f, T ) and the set of all common fixed points of the pair( f, T ), respectively.

Definition 1.2 A subset M of a normed space X is called (5) q-starshaped or starshapedwith respect to q if λx + (1 − λ)q ∈ M for all x ∈ M and λ ∈ [0, 1]; (6) convex ifλx + (1 − λ)y ∈ M for all x, y ∈ M and λ ∈ [0, 1].Definition 1.3 Let f be a self map on a normed space X and M ⊆ X, The map f is called(7) affine on M if M is convex and f (λx +(1−λ)y) = λ f x +(1−λ) f y for all x, y ∈ M andλ ∈ [0, 1]; (8) q-affine on M if M is q-starshaped and f (λx + (1 − λ)q) = λ f x + (1 − λ)qfor all x ∈ M and λ ∈ [0, 1].Definition 1.4 Let f : M −→ M and T : M −→ C L(M). The pair ( f, T ) is called (9)commuting if T f x = f T x for all x ∈ M; (10) R-weaklycommuting if f T x ∈ C L(M) andH( f T x, T f x) ≤ Rd( f x, T x) for all x ∈ M and for some R > 0; (11) weakly compatible[17] if they commute at their coincidence points, that is, f T x = T f x whenever x ∈ C( f, T ).If M is q-starshaped with q ∈ F( f ), then the pair ( f, T ) is called (12) Cq -commuting iff T x = T f x for all x ∈ Cq( f, T ) = ∪{C( f, Tλ) : λ ∈ [0, 1]}, where Tλx = λT x+(1−λ)q;(13) R-subweakly commuting on F (with respect to q) if f T x ∈ C L(M) and

H( f T x, T f x) ≤ R inf{d( f x, Tλx) : 0 ≤ λ ≤ 1}for each x ∈ M and for some R > 0;(14) R- subcommuting on M (with respect to q [27])if f T x ∈ C L(M) and H( f T x, T f x) ≤ R

λd( f x, Tλx) for all x ∈ M, λ ∈ (0, 1] and some

R > 0.

Definition 1.5 Let f : X −→ X, T : X −→ C L(X) and M ⊆ X . Then f − T is called(15) demiclosed at 0 if, whenever a sequence {xn} in M converges weakly to x0 in M andyn ∈ ( f − T )xn such that {yn} converges to 0 strongly, then 0 ∈ ( f − T )x0. The map f iscalled (16) T -weaklycommuting at x ∈ X if f 2x ∈ T f x .

Coincidence point theorems 263

It is well known that R-subweakly commuting maps are R-weakly commuting andR-weakly commuting maps are weakly compatible. It is also known that, if the pair ( f, T )

is weakly compatible at x ∈ C( f, T ), then f is T -weakly commuting at x and hencef n(x) ∈ C( f, T ). However the converse is not true in general (for a detailed discussion onthe above mentioned notions and their implications, we refer the reader to [3,15–17,31]).

Definition 1.6 Let M be a q-starshaped subset of a normed space X, f : M −→ M andT : M −→ C L(M). Then f and T are said to satisfy:

(a) the condition (P) ([8]), if for every closed ball B of M with radius r ≥ 0 and anysequence {xn} in M, for which dist (xn, B) → 0 and dist (xn, T xn) → 0 as n → ∞,

there exists a y ∈ B with y ∈ T y.(b) the condition (P0), if for every closed ball B of M with radius r ≥ 0 and any sequence

{xn} in M, for which dist (xn, B) → 0 and dist ( f xn, T xn) → 0 as n → ∞, thereexists a y ∈ B with f y ∈ T y.

Clearly if f = id then condition (P0) becomes condition (P).Let X be a metric space and M be any subset of X . If there exists a y0 ∈ M such that

d(x, y0) = d(x, M), then y0 is called a best approximation to x out of M . We denote byPM (x), the set of all best approximations to x out of M, P f

M (x) = {y ∈ M : f y ∈ PM (x)},and D f

M (x) = PM (x) ∩ P fM (x)

Recently M. Berinde and V. Berinde [7] extended the notion of a weak contraction from sin-gle valued mappings to multivalued mappings. Kamran [20] further extended this notion fora hybrid pair of f : X −→ X and T : X −→ C B(X) and obtained results regarding coinci-dence points of the pair ( f, T ). For practical purposes, a relaxation of the boundedness condi-tion is always desired. The purpose of this paper is to improve the aforementioned coincidencepoint results restricting the range of multivalued mappings to C L(X). As applications, invari-ant approximation, random coincidence point and random best approximation results areestablished, which in turn extend and strengthen various known results in the literature.

2 Main results

We begin with the following result which extends and improves Theorem 2.1 of Al-Thagafi[1], Theorem 2.1 of Al-Thagafi and Shahzad [2], the main result of Jungck [14], Theorem2.9 of Kamran [20], and Lemma 2.2 of Shahzad [29].

Theorem 2.1 Let X be a metric space, f : X −→ X and T : X −→ C L(X) be a ( f, θ, L)-weak contraction with T (X) ⊆ f (X). Suppose that T (X) is complete. Then C( f, T ) �= ∅.Moreover, F( f, T ) �= ∅ if one of the following conditions holds:

(a) f 2x = f x for some x ∈ C( f, T ).(b) f and T are weakly compatible on C( f, T ), f is continuous, and lim

n→∞ f n x exists for

some x ∈ C( f, T ).(c) For some z ∈ C( f, T ), f is continuous at z, and lim

n→∞ f n y = z for some y ∈ X.

(d) f (C( f, T )) is a singleton subset of C( f, T ).

Proof Let x0 be in X . As T (X) ⊆ T (X) ⊆ f (X), we can construct a sequence {xn} inX such that f xn ∈ T xn−1 ⊆ T X, for all n ≥ 1. We conclude as in [20] (see also [21])that { f xn} is a Cauchy sequence in T (X). It follows from the completeness of T (X) thatf xn −→ y ∈ T (X) ⊆ f (X), where y = f x for some x in X . Note that for every n ≥ 1,

we have

264 M. Abbas et al.

d( f x, T x) ≤ d( f x, f xn+1) + d( f xn+1, T x)

≤ d( f x, f xn+1) + H(T xn, T x)

≤ d( f x, f xn+1) + θd( f xn, f x) + Ld( f x, T xn)

≤ d( f x, f xn+1) + θd( f xn, f x) + Ld( f x, f xn+1).

Since f x = limn→∞ f xn = lim

n→∞ f xn+1 and f xn+1 ∈ T xn, taking the limit as n → ∞, we

obtain d( f x, T x) = 0, and hence f x ∈ T x . Thus C( f, T ) �=∅.

(a) By the given hypothesis f 2x = f x for some x ∈ C( f, T ). Suppose that y = f x =f 2x = f y ∈ T x . Now

d(y, T y) ≤ H(T x, T y)

≤ θd( f x, f y) + Ld( f y, T x) = 0,

and we obtain that d(y, T y) = 0 and hence y ∈ T y.(b) Suppose that y = lim

n→∞ f n x for some x ∈ C( f, T ). Since f is continuous, y is fixed

point of f . Also since f and T are weakly compatible on C( f, T ), f n x ∈ C( f, T ) forall n ≥ 1, and hence f n x ∈ T f n−1x . Consider,

d(y, T y) ≤ d(y, f n x) + d( f n x, T y)

≤ d(y, f n x) + H(T f n−1x, T y)

≤ d(y, f n x) + θd( f n x, f y) + Ld( f y, T f n−1x)

≤ d(y, f n x) + θd( f n x, y) + Ld(y, f n x).

Taking the limit as n → ∞, we obtain d(y, T y) = 0 and hence y ∈ T y. ThusF( f, T ) �= ∅.

(c) Suppose that for some z ∈ C( f, T ), f is continuous at z, and limn→∞ f n x = z for some

x ∈ X . Then z = f z ∈ T z, and F( f, T ) �= ∅.(d) Since f (C( f, T )) = x (say) and x ∈ C( f, T ), this implies that x = f x ∈ T x . Thus

F( f, T ) �= ∅.

Corollary 2.2 Let X be a metric space, T : X −→ C L(X) be a (θ, L)-weak contractionwith T (X) ⊆ X. Suppose that T (X) is complete. Then T has a fixed point.

Corollary 2.2 generalizes the Banach contraction principle, and the Nadlar contraction Prin-ciple [26]. Theorem 3 of [7] becomes a special case of our Corollary 2.2.

We also define, δ( f y, T x) = inf{d( f y, Tλx) : 0 ≤ λ ≤ 1}.Theorem 2.3 Let M be a subset of a normed space X, f : M −→ M and T : M −→C L(M). Suppose that f (M) = M, T (M) is bounded, T (M) is complete, T (M) ⊆ f (M),

the pair { f, T } satisfies condition (P0), and

H(T x, T y) ≤ ‖ f x − f y‖ + δ( f y, T x) (2.1)

for all x, y ∈ M. Then C( f, T ) �= ∅. Moreover, F( f, T ) �= ∅ if one of the conditions(a) − (d) of Theorem (2.1) holds.

Proof Let B be a closed ball in M with center at q . Let {λn} be a sequence in (0, 1) suchthat λn → 1. For n ≥ 1, let

Tn(x) = Tλn (x) = λnT x + (1 − λn)q

Coincidence point theorems 265

for all x in B. Since T (M) is complete, (T (B) being a closed subset of complete space T (M)

is complete ), Since T (M) ⊆ f (M) (obviously T (B) ⊆ f (M)) and f (M) = M, we haveTn(B) ⊆ f (M) and Tn(B) is complete for each n ≥ 1. Now consider

H(Tn x, Tn y) = λn H(T x, T y)

≤ λnd( f x, f y) + λnδ( f y, T x)

≤ λnd( f x, f y) + λnd( f y, Tn x)

for all x, y ∈ B, which implies that each Tn is an ( f, λn, λn)-weak contraction on B.Hence, from Theorem 2.1 we conclude that f xn ∈ T xn = λn T xn + (1 − λn)q for somexn ∈ B. As f xn = λn yn + (1 − λn)q for some yn ∈ T xn ⊆ T (M), T (M) is bounded,λn → 1 and

‖ f xn − yn‖ = (1 − λn) ‖q − yn‖≤ (1 − λn)(‖q‖ + ‖yn‖),

so ‖ f xn − yn‖ → 0 and hence d( f xn, T xn) ≤ ‖ f xn − yn‖ → 0. Since the pair { f, T }satisfies the condition (P0), there exists a u ∈ B such that f u ∈ T u. Thus C( f, T ) �= ∅.Using arguments similar to those given in the proof of Theorem 2.1, it can be shown thatF( f, T ) �= ∅ if one of the conditions (a) − (d) of Theorem 2.1 holds.

Clearly an f -nonexpansive multivalued map T satisfies inequality (2.1), so Theorem 2.3improves and generalizes Corollary 2.5 of Hussain and Jungck [11], Corollaries 3.2, 3.4 ofJungck [15], Theorem 6 due to Jungck and Sessa [18], Theorems 2.2–2.5 of Latif and Tweddle[25], Theorem 3 due to Rhoades [27] and Theorems 2.1, 2.2, 2.4, 2.6–2.8 of Shahzad andHussain [31].

Theorem 2.4 Let M be a subset of a normed space X, f : M −→ M and T : M −→C L(M). Suppose that M is q-starshaped, f (M) = M [resp. f is q-affine on M], T (M) iscomplete, T (M) ⊆ f (M), and f and T satisfy (2.1) for all x, y ∈ M. Then C( f, T ) �= ∅ ifone of the following conditions holds.

(e) T (M) is bounded and ( f − T )(M) is closed.(f) M is weakly compact and f − T is demiclosed at 0.(g) T (M) is bounded and T and f satisfy for all x, y ∈ M,

Hr (T x, T y) ≤ θ1(d( f x, T x))dr ( f x, T x) + θ2(d( f y, T y))dr ( f y, T y)

where θi : R → [0, 1) (i = 1, 2) and r ≥ 1 is some fixed positive real number. MoreoverF( f, T ) �= ∅ if one of the conditions (a) − (d) of Theorem 2.1 holds.

Proof (e) Let B be a closed ball in M with radius r ≥ 0. Suppose that {xn} is a sequencein M such that d(xn, B) → 0 and and dist ( f xn, T xn) → 0 as n → ∞. Now for each nin N , there exist yn in T xn such that

d( f xn, yn) ≤ dist ( f xn, T xn) + 1

n

which, from hypothesis, implies that d( f xn, yn) → 0 as n → ∞. As ( f − T )(M) is closed,0 ∈ ( f − T )(M). Hence the pair { f, T } satisfies the condition (P0) and the result followsfrom Theorem 2.3.

( f ) As proved earlier, f xn − yn → 0 as n → ∞ where yn ∈ T xn . By the weak com-pactness of M, there is a subsequence {xm} of the sequence {xn} such that {xm} converges

266 M. Abbas et al.

weakly to y ∈ M as m → ∞. Since f − T is demiclosed at 0, we obtain 0 ∈ ( f − T )y.Hence the pair { f, T } satisfies the condition (P0) and the result follows from Theorem 2.3.

(g) Using Theorem 2.11 [22], it can be shown that the pair { f, T } satisfies the condition(P0) and the result follows from Theorem 2.3.

Corollary 2.5 Let M be a subset of a normed space X, and T : M −→ C L(M). Supposethat M is q-starshaped, T (M) is bounded, T (M) is complete, T (M) ⊆ M, T satisfies thecondition (P) and H(T x, T y) ≤ d(x, y) + δ(y, T x) for all x, y ∈ M. Then T has a fixedpoint.

3 Approximation theory

As an application of Theorem 2.3, we obtain the following result which improves and gen-eralizes Theorems 3.2, 3.3 of Al-Thagafi [1] for D = PM (p), Theorem 3.1, 3.3 due toAl-Thagafi and Shahzad [2], Theorem 7 of Jungck and Sessa [18], Theorem 3.14 of Kamran[20], Theorem 3 of Latif and Bano [24], Theorem 3 of Sahab, Khan and Sessa [28], Theorems2.12, 2.13 of Shahzad and Hussain [31], and many others.

Theorem 3.1 Let M be a subset of a normed space X, f : X −→ X and T : X −→C L(X), T (M ∩ ∂ M) ⊆ M and p ∈ X. Suppose that PM (p) is closed and q-starshaped,

f (PM (p)) = PM (p), T (PM (p)) is complete, the pair { f, T } satisfies the condition (P0) onPM (p), sup

y∈T x‖y − p‖ ≤ ‖ f x − p‖ for all x ∈ PM (p), and the pair { f, T } satisfies (2.1) for

all x, y ∈ PM (p). Then, C( f, T ) ∩ PM (p) �= ∅. Moreover F( f, T ) ∩ PM (p) �= ∅ providedone of the following conditions holds:

(a) f 2x = f x for some x ∈ C( f, T ) ∩ PM (p)

(b) f and T are weakly compatible on C( f, T ) ∩ PM (p), f is continuous, and limn→∞ f n x

exists for some x ∈ C( f, T ) ∩ PM (p).(c) For some x ∈ C( f, T ) ∩ PM (p), f is continuous at x, and lim

n→∞ f n y = x for some

y ∈ PM (p).(d) f (C( f, T )) ∩ PM (p) is a singleton subset of C( f, T ) ∩ PM (p).

Proof Let x ∈ PM (p), ‖(1 − λ)x + λp − p‖ < ‖x − p‖ = d(p, M) for all λ ∈ (0, 1).Thus, {(1 − λ)x + λp : λ ∈ (0, 1)} ∩ M = ∅ and so x ∈ M ∩ ∂ M . Since T (M ∩ ∂ M) ⊆ M,

it follows that T x ⊆ M . Now let z ∈ T x . As f x ∈ PM (p),

‖z − p‖ ≤ supy∈T x

‖y − p‖ ≤ ‖ f x − p‖ = d(p, M).

Thus z ∈ PM (p) and hence T x ⊆ PM (p). Moreover, T (PM (p)) ⊆ PM (p) = f (PM (p)).The result now follows from Theorem 2.3.

Corollary 3.2 Let X be a normed space, and T : X −→ C L(X), M a subset of X withT (M ∩ ∂ M) ⊆ M and, p ∈ X. Suppose that PM (p) is closed and q-starshaped, T (PM (p))

is complete, the map T satisfies condition (P) on PM (p), supy∈T x

‖y − p‖ ≤ ‖x − p‖ for all

x ∈ PM (p), and

H(T x, T y) ≤ d(x, y) + δ(y, T x)

for all x, y ∈ PM (p). Then F(T ) ∩ PM (p) �= ∅.

Proof The result follows by taking f (x) = x in Theorem 3.1.

Coincidence point theorems 267

Theorem 3.3 Let X be a normed space, f : X −→ X and T : X −→ C L(X), M asubset of X with T (M ∩ ∂ M) ⊆ f (M) ∩ M and p ∈ X. Suppose that P f

M (p) is closed

and q-starshaped, f (P fM (p)) = P f

M (p), T (P fM (p)) is complete, the pair { f, T } satisfies

the condition (P0) on P fM (p), sup

y∈T x‖y − p‖ ≤ ‖ f x − p‖ for all x ∈ P f

M (p), and the

pair { f, T } satisfies (2.1) for all x, y ∈ P fM (p). Then, C( f, T ) ∩ PM (p) �= ∅. Moreover,

F( f, T ) ∩ PM (p) �= ∅ if one of the conditions (a) − (d) of Theorem 3.1 holds.

Proof Since, P fM (p) = f (P f

M (p)) ⊆ PM (p), let z ∈ T (P fM (p)). Then there exists an

x ∈ P fM (p) such that z ∈ T x, and

‖z − p‖ ≤ supy∈T x

‖y − p‖ ≤ ‖ f x − p‖ = d(p, M),

which implies that z ∈ PM (p) and hence T (P fM (p)) ⊆ PM (p). Also, since x ∈ P f

M (p),x ∈ M ∩ ∂ M and T x ⊆ f (M). Thus, we have a u ∈ M such that z = f u ∈ T x ⊆ PM (p).

Thus u ∈ P fM (p), and T (P f

M (p)) ⊆ P fM (p) = f (P f

M (p)). The result now follows fromTheorem 2.3.

Theorem 3.4 Let X be a normed space, f : X −→ X and T : X −→ C L(X). Supposethat M is a subset of X with T (M ∩ ∂ M) ⊆ f (M) ∩ M and p ∈ X. Suppose that D f

M (p)

is closed and q-starshaped, f (D fM (p)) = D f

M (p), T (D fM (p)) is complete, the pair { f, T }

satisfies the condition (P0) on D fM (p), sup

y∈T x‖y − p‖ ≤ ‖ f x − p‖ for all x ∈ D f

M (p),

and the pair { f, T } satisfies (2.1) for all x, y ∈ D fM (p) and ‖ f x − p‖ = ‖x − p‖ for all

x ∈ T (D fM (p)). Then C( f, T ) ∩ PM (p) �= φ. Moreover F( f, T ) ∩ PM (p) �= ∅ provided

one of the conditions (a) − (d) of Theorem 3.1 holds.

Proof Since, D fM (p) = f (D f

M (p)) ⊆ PM (p), following arguments similar to those in

the proof of Theorem 3.1, we obtain T (D fM (p)) ⊆ PM (p). Let, x ∈ T (D f

M (p)). Since

‖ f x − p‖ = ‖x − p‖ and x ∈ T (D fM (p)) ⊆ CM (p), it follows that x ∈ C f

M (p), which

further gives x ∈ D fM (p). Hence, T (D f

M (p)) ⊆ D fM (p) = f (D f

M (p)) and the result followsfrom Theorem 2.3.

Theorems 3.3 and 3.4 extend Al-Thagafi ([1], Theorems 3.2, 3.3) for D = D fM (p) and

Al-Thagafi and Shahzad ([2], Theorem 3.2).Let 0 be the class of closed convex subsets M of a normed space X containing 0. For

M ∈ 0 , we define Mu = {x ∈ M : ‖x‖ ≤ 2‖u‖}. Then PM (u) is closed, convex andcontained in Mu ∈ 0 (see [2,11]).

Theorem 3.5 Let X be a normed space, f : X −→ X and T : X −→ C L(X), and M ∈ 0

with T (Mp) ⊂ f (M) = M for some p ∈ X. Suppose that T (Mp) is compact, the pair{ f, T } satisfies the condition (P0) on every A ∈ C L(Mp), ‖ f x − p‖ = ‖x − p‖ for allx ∈ M, sup

y∈T x‖y − p‖ ≤ ‖x − p‖ for all x ∈ Mp, and the pair { f, T } satisfies (2.1) for

all x, y ∈ Mp. Then PM (p) is nonempty, closed and convex, T (PM (p)) ⊂ f (PM (p)) =PM (p), and C( f, T )∩ PM (p) �= ∅. Moreover, F( f, T )∩ PM (p) �= ∅ if one of the conditions(a) − (d) of Theorem 3.1 holds.

268 M. Abbas et al.

Proof We may assume that p /∈ M . If x ∈ M \ Mp , then ‖x‖ > 2‖p‖. Note that

‖x − p‖ ≥ ‖x‖ − ‖p‖ > ‖p‖ ≥ dist (u, M).

Thus, dist (p, Mp) = dist (p, M) ≤ ‖p‖. As T (Mp) is compact, so ‖z − p‖ =dist (p, T (Mp)) for some z ∈ T (Mp). This implies that

dist (p, Mp) ≤ dist (p, T (Mp)) ≤ dist (p, T (Mp)) ≤ dist (p, T (z)) ≤≤ sup

y∈T z‖y − p‖ ≤ ‖z − p‖ .

Hence ‖z − p‖ = dist (p, M) and so PM (u) is nonempty. Moreover it is closed and convex.Let z ∈ PM (p). Then ‖ f z − p‖ = ‖z − p‖ = dist (p, M), which implies that f z ∈ PM (p)

and so f (PM (p)) ⊂ PM (p). For the converse assume that y ∈ PM (p). Then y ∈ M = f (M)

and there is some x ∈ M such that y = f x . Now

‖x − p‖ = ‖ f x − p‖ = ‖y − p‖ = dist (p, M).

This implies that x ∈ PM (p) and hence y = f x ∈ f (PM (p)). Thus f (PM (p)) = PM (p).Further, let y ∈ T (PM (p)) ⊂ f (M). Then there exists a z ∈ PM (p) and x0 ∈ M such thaty = f x0 ∈ T z. Further, we have

‖x0 − p‖ = ‖ f x0 − p‖ = ‖y − p‖ ≤ supx∈T z

‖x − p‖ ≤ ‖z − p‖ = dist (p, M).

Thus, x0, y ∈ PM (p) and hence T (PM (p)) ⊂ f (PM (p)) = PM (p). The conclusion followsfrom Theorem 2.3 by replacing M by PM (p).

Theorem 3.5 extends ([1], Theorems 4.1, 4.2(b)), ([2], Theorems 4.1, 4.2) and ([29],Theorem 2.4).

4 Random coincidence point theorems

Throughout this section, (�,) denotes a measurable space. A multivalued mappingT : � −→ C L(X) is measurable if T −1(U ) ∈ for each open subset U of X, whereT −1(U ) = {ω ∈ � : T (ω) ∩ U �= φ}. A mapping ξ : � → X is said to be a selector of ameasurable mapping T : � → C L(X) if ξ is measurable and, for any ω ∈ �, ξ(ω) ∈ T (ω).A mapping T : � × X −→ C L(X)(T : � × X −→ X) is a random operator if andonly if for each fixed x ∈ X, the mapping T (., x) : � −→ C L(X)(T (., x) : � −→ X)

is measurable. A measurable mapping ξ : � −→ X is a random fixed point of a randomoperator T : � × X −→ C L(X) if and only if ξ(ω) ∈ T (ω, ξ(ω)) for each ω ∈ �. Wedenote the set of all random fixed points of T by RF(T ). Let ξ : � −→ X be a measurablemapping and T : � × X −→ C L(X) and f : � × X −→ X be two random operators; (17)

ξ is a random coincidence point of f and T if and only if f (ω, ξ(ω)) ∈ T (ω, ξ(ω)) for eachω ∈ �. We denote the set of all random coincidence points of f and T by RC( f, T ); (18)ξ

is a random common fixed point of f and T if and only if f (ω, ξ(ω)) = T (ω, ξ(ω)) foreach ω ∈ �. We denote the set of all random common fixed points of f and T by RF( f, T ).For further details we refer to [5,4,13,30] and the references therein.

Random coincidence point theorems are stochastic generalizations of classical coinci-dence point theorems. Recently Shahzad and Hussain [31] have obtained a random coinci-dence point theorem for a pair of noncommuting random operators. In this direction, usingthe ideas of Thagafi and Shahzad [3] and Shahzad and Hussain [31], we obtain random coin-cidence points of multivalued f -weak contraction random operators with closed values. We

Coincidence point theorems 269

also prove results regarding random fixed points of random multivalued weak contractionmappings which in turn are the stochastic generalization of results of [7] and [20]. As anapplication, random invariant approximation results are also derived.

Theorem 4.1 Let (�,) be a measurable space and M be a subset of a Banach spaceX, f : � × M −→ M and T : � × M −→ C L(M) be two continuous random operators.Suppose that M is separable, closed, and for every ω ∈ �, f (ω, M) = M [resp. f (ω, .) isq-affine on M], T (ω, M) ⊆ f (ω, M).T (ω, M) is bounded, the pair { f (ω, .), T (ω, .)} sat-isfies condition P0 and H(T (ω, x), T (ω, y)) ≤ d( f (ω, x), f (ω, y) + δ( f (ω, y), T (ω, x))

for all x, y ∈ M and ω ∈ �. Then RC( f, T ) �= ∅. Moreover f and T have a commonrandom fixed point if one of following conditions holds for every ω ∈ �:

(a) f (ω, f (ω, x)) = f (ω, x) for every x ∈ C( f (ω, .), T (ω, .)).(b) f (ω, .) and T (ω, .) are weakly compatible on C( f (ω, .), T (ω, .)), and

limn→∞ f n(ω, x) exists for every x ∈ C( f (ω, .), T (ω, .)).

(c) For every x ∈ C( f (ω, .), T (ω, .)), there exists a y ∈ M such thatlim

n→∞ f n(ω, y) = x.

(d) There exists a measurable map ξ : � −→ M such thatf (ω, C( f (ω, .), T (ω, .))) = {ξ(ω)} is a subset of C( f (ω, .), T (ω, .)).

Proof Fix ω ∈ � and set, S(ω) = {x ∈ B0 : f (ω, x) ∈ T (ω, x)}; i.e., S : � → 2M .Since M is complete and the pair ( f (ω, .), T (ω, .)) satisfies the conditions of Theorem 2.3,C( f (ω, .), T (ω, .)) �= ∅, and hence S(ω) is nonempty and complete. Now we show thatS : � → C L(M) is a measurable map. Let B0 be an arbitrary closed ball in M . Define

L(B0) =∞⋂

k=1

∞⋃z∈Zk

{ω ∈ � : d( f (ω, xi ), T (ω, xi )) <

1

k

},

where Z = {xi } is a countable dense subset of M and Zk = Z ∩ {x ∈ M : d(x, B0) < 1k }.

Clearly S−1(B0) ⊆ L(B0). For this, let ω ∈ S−1(B0). Then S(ω)∩ B0 �= φ (by the definitionof S−1(B0)). There exists an x in B0 such that f (ω, x) ∈ T (ω, x). Since Z is a countabledense subset of M therefore Z ∩ B(x, 1

k ) �= φ. It follows that for each k there exists an integernk such that

d(x, xnk ) <1

k

Since x ∈ B0, d(B0, xnk ) < 1k . So we have a subsequence {xnk } in Zk . Also, since g :

� × M −→ R, defined by g(ω, x) = d( f (ω, x), T (ω, x)) is upper semi continuous,lim sup

k→∞d( f (ω, xnk ), T (ω, xnk )) ≤ d( f (ω, x), T (ω, x)) = 0. Thus

lim supk→∞

d( f (ω, xnk ), T (ω, xnk )) = 0.

Hence

limk→∞ d( f (ω, xnk ), T (ω, xnk )) = 0.

Therefore, for each k in N , there exists an xnk in Zk such that

d( f (ω, znk ), T (ω, znk )) <1

k,

270 M. Abbas et al.

which implies that ω ∈ L(B0). Fix ω0 in L(B0). Then, for every k, there is an integernk such that d( f (ω0, xnk ), T (ω0, xnk )) < 1

k , and d(xnk , B0) < 1k which further implies

that d( f (ω0, xnk ), T (ω0, xnk )) → 0 and d(xnk , B0) → 0 as k → ∞. Since the pair( f (ω, .), T (ω, .)) satisfies condition (P0), there exists a v ∈ B0 such that f (ω0, v) ∈T (ω0, v). Therefore ω0 ∈ S−1(A) and hence S−1(A) = L(A). By [9]{

ω ∈ � : d( f (ω, x), T (ω, x)) <2

n

}∈

for every x ∈ M, because the map g : � × M −→ R, defined by g(ω, x)= d( f (ω, x),

T (ω, x)), is continuous in x ∈ M and measurable in ω ∈ �. Thus S is measurable. Apply-ing the measurable selection theorem [23], we obtain a measurable selector of the multivaluedrandom operator S, which in turns serve as a random coincidence point of random operatorsf and T . The rest follows as in the proof of our Theorem 2.1 and Theorem 3.7 of [31].

We obtain from Theorem 4.1 the conclusions of Theorem 3.17 and 3.18 of Shahzad [30],without assuming the commutativity of the maps. It also extends Theorems 3.4 and 3.7 dueto Shahzad and Hussain [31].

Corollary 4.2 Let (�,) be a measurable space and M be a subset of a Banach space X,

T : � × M −→ C L(M) be a continuous random operator. Suppose that M is separable,closed and q-starshaped, and for every ω ∈ �, T (ω, M) is bounded, T (ω, .) satisfies con-dition (P) on every set in C L(M) and H(T (ω, x), T (ω, y)) ≤ d(x, y) + δ(y, T (ω, x)) forall x, y ∈ M and ω ∈ �. Then RF(T ) �= ∅.

Theorem 4.3 Let (�,) be a measurable space, X be a Banach space p∈ X, and M ⊆ X.

f : �×X −→ X and T : �×X −→ C L(X) be two continuous random operators. Supposethat PM (p) is separable, closed and q-starshaped, and for every ω ∈ �, T (ω, ∂ M ∩ M) ⊆M, f (ω, PM (p)) = PM (p), supy∈T (ω,x) ‖y − p‖ ≤ ‖ f (ω, x) − p‖ for all x ∈ PM (p), the

pair { f (ω, .), T (ω, .)} satisfies condition (P0) on every set A in C L(PM (p)) and

H(T (ω, x), T (ω, y)) ≤ d( f (ω, x), f (ω, y)) + δ( f (ω, y), T (ω, x))

for all x, y ∈ PM (p) and ω ∈ �. Then there exists a measurable map ξ : � −→ PM (p)

such that ξ ∈ RC( f, T ). Moreover ξ ∈ RF( f, T ) if one of the following conditions holdsfor every ω ∈ �:

(e) f (ω, f (ω, x)) = f (ω, x) for every x ∈ C( f (ω, .), T (ω, .)) ∩ PM (p).(f) (b) f (ω, .) and T (ω, .) are weakly compatible on C( f (ω, .), T (ω, .)) ∩ PM (p), and

limn→∞ f n(ω, x) exists for every x ∈ C( f (ω, .), T (ω, .)) ∩ PM (p).

(g) (c) For every x ∈ C( f (ω, .), T (ω, .)) ∩ PM (p), there exists a y ∈ PM (p) such thatlim

n→∞ f n(ω, y) = x.

(h) There exists a measurable function ξ : � −→ PM (p) such that

f (ω, C( f (ω, .), T (ω, .))) ∩ PM (p) = {ξ(ω)}is a subset of C( f (ω, .), T (ω, .)) ∩ PM (p).

Proof Fix ω ∈ �. As in the proof of Theorem 3.1, T (ω, PM (p)) ⊆ PM (p) and T (ω, .) :PM (p) −→ C L(PM (p)). The rest follows from Theorem 4.1.

Corollary 4.4 Let (�,) be a measurable space, X be a Banach space p ∈ X, andM ⊆ X. T : � × X −→ C L(X) be a continuous random operator. Suppose that

Coincidence point theorems 271

PM (p) is separable, closed and q-starshaped, and for every ω ∈ �, T (ω, ∂ M ∩ M) ⊆M, sup

y∈T (ω,x)

‖y − p‖ ≤ ‖x − p‖ for all x ∈ PM (p), the pair T (ω, .) satisfies condition (P)

on every set in C L(CM (p)) and

H(T (ω, x), T (ω, y)) ≤ d(x, y) + δ(y, T (ω, x))

for all x, y ∈ PM (p) and ω ∈ �. Then there exists a measurable map ξ : � −→ PM (p)

such that ξ ∈ RF(T ).A subset M of a linear space X is said to have property (N ) with respect to T (see [10,12])

if,

(i) T : M → C L(M).(ii) (1 − kn)q + knT x ⊂ M, for some q ∈ M where a fixed sequence of real numbers

kn(0 < kn < 1) converging to 1 for each x ∈ M.

Remark 4.5 (1) As an application of Theorem 4.1, the analogue of Theorem 4.5 and Corol-lary 4.6 due to Al-Thagafi and Shahzad [3] can be established for multivalued ( f, θ, L)-weak contractions.

(2) All of the results of this paper remain valid, provided the q -starshapedness of the setM [res. PM (p)] is replaced by property (N ). Consequently, recent results due to Hussain[10], and Hussain, O’Regan and Agarwal [12] are extended to the more general class ofmultivalued ( f, θ, L)-weak contractions.

(3) Following the above arguments, it is possible to obtain common fixed point and invariantapproximation results for f, g : M −→ M and T : M −→ C L(M) satisfying:

H(T x, T y) ≤ ‖ f x − gy‖ + δ(gy, T x)

for all x, y ∈ M . Consequently, Theorem 2.3, Corollary 2.9, Theorem 2.11 and Corollary2.15 due to Hussain and Jungck [11] are extended and improved.

Acknowledgments The authors are thankful to the reviewers for their suggestions to improve thepresentation of this manuscript.

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