COINCIDENCE AND INVARIANT APPROXIMATION THEOREMS FOR GENERALIZED f-NONEXPANSIVE MULTIVALUED MAPPINGS

The main purpose of this paper is to prove some new coincidence and common fixed point theorems for noncommuting generalized f -nonexpansive multivalued mappings on non-starshaped domain in the framework of a Banach space. As applications, related common fixed point, invariant approximation, and random coincidence point results are established. This work provides extension as well as substantial improvement of several results in the existing literature.


Introduction and preliminaries
Let M be a subset of a normed space (X, • ).We denote by 2 X ,C(X),CB(X), and K(X), the families of all nonempty, nonempty closed, nonempty closed bounded, and nonempty compact subsets of X, respectively.On C(X), we define the Hausdorff metric H [22], by setting for A,B ∈ C(X), H(A,B) = max sup a∈A d(a,B), sup b∈B d(b,A) , (1.1) where d(a,B) = inf{d(a,x) : x ∈ B}.
The set P M (u) = {x ∈ M : x − u = dist(u,M)} is called the set of best approximants to u ∈ X from M. The diameter of M is denoted and defined by δ(M) = sup{ x − y : x, y ∈ M}.A mapping f : X → X has diminishing orbital diameters (d.o.d.) [12] if for each x ∈ X, δ(O(x)) < ∞ and whenever δ(O(x)) > 0, there exists We denote the boundary of M by ∂M.
Let f : M → M be a mapping.A mapping T : M → C(M) is called f -Lipschitz if, for any x, y ∈ M, there exists k ≥ 0 such that H(Tx,T y) ≤ kd( f x, f y).If k < 1 (resp., k = 1), then T is called f -contraction (resp., f -nonexpansive).The map T is said to be an f -nonexpansive-type mapping [6,19] if given x ∈ M and u x ∈ Tx, there is a u y ∈ T y for each y ∈ M such that d(u x ,u y ) ≤ d( f x, f y).T is said to be * -nonexpansive (cf.[3,6,34] if for all x, y in M and u x ∈ Tx with d(x,u x ) = d(x,Tx), there exists u y ∈ T y with d(y,u y ) = d(y,T y) such that d(u x ,u y ) ≤ d(x, y).The set of fixed points of T (resp., f ) is denoted by F(T) (resp., F( f )).A point x ∈ M is a coincidence point (common fixed point) of f and T if f x ∈ Tx(x = f x ∈ Tx).The set of coincidence points of f and T is denoted by C( f ,T).The pair { f ,T} is called commuting if T f x = f Tx for all x ∈ M. The pair { f ,T} is called R-weakly commuting [27] if for all x ∈ M, f Tx ∈ C(M) and there exists R > 0 such that H( f Tx,T f x) ≤ Rdist( f x,Tx).The pair { f ,T} is called compatible [14] if lim n H(T f x n , f Tx n ) = 0 when {x n } is a sequence such that lim n f x n = t ∈ lim n Tx n for some t in M. The pair { f ,T} is called nontrivially compatible [12] if f and T are compatible and do have a coincidence point.The pair { f ,T} is called weakly compatible if they commute at their coincidence points, that is, if However, the converse is not true, in general.If T is single-valued, then T-weak commutativity at the coincidence points is equivalent to the weak compatibility (see [14]).The mappings f and T are said to satisfy property (E.A) [14], if there exist a sequence and is both Tand f -invariant.Then T and f are called R-subweakly commuting on M (see [27]) if for all x ∈ M, f Tx ∈ CB(M) and there exists a real number R > 0 such that H( f Tx,T f x) ≤ Rdist( f x,T λ x) for each λ ∈ [0,1], where T λ x = (1 − λ)q + λTx.It is well known that R-subweakly commuting maps are R-weakly commuting and R-weakly commuting maps are compatible and compatible maps are weakly compatible.Howevere, the converse is not true, in general (see [11,12,14,27]).
A set M is said to have property (N) [21], if for some q ∈ M and a fixed sequence of real numbers k n (0 < k n < 1) converging to 1 and for each x ∈ M. Each q-starshaped set has the property (N) with respect to any map T : M → C(M) but the converse does not hold, in general (see [8,10]).
A Banach space X satisfies Opial's condition if for every sequence {x n } in X weakly convergent to x ∈ X, the inequality holds for all y = x.Every Hilbert space and the space l p (1 < p < ∞) satisfy Opial's condition.The map T : Meinardus [20] employed the Schauder fixed point theorem to prove a result regarding invariant approximation.Singh [32] proved the following extension of the result of Meinardus.
Theorem 1.1.Let T : X → X be a nonexpansive operator, M a T-invariant subset of X, and u ∈ F(T).If P M (u) is nonempty compact and starshaped, then Sahab et al. [23] established the following result which contains Theorem 1.1.
Theorem 1.2.Let I and T be self-maps of X with u ∈ F(I) ∩ F(T) and M ⊂ X with T(∂M) ⊂ M, and q ∈ F(I).If D = P M (u) is compact and q-starshaped, I(D) = D, I is continuous and linear on D, Iand T are commuting on D, and Jungck and Sessa [13] proved the following result in best approximation theory, which extends Theorems 1.1 and 1.2 and many others.

I is affine and continuous in the weak and strong topology on D. If I and T are commuting on
Recently, Al-Thagafi [2] extended Theorem 1.2 and proved some results on invariant approximations for commuting maps.More recently, Shahzad [24][25][26][27][28], Hussain and Khan [9], and Hussain et al. [10] have further extended and improved the above-mentioned results for noncommuting maps.Latif and Tweddle [18] have proved some coincidence and common fixed point theorems for a commuting pair of f -nonexpansive multivalued mappings defined on starshaped subsets of a Banach space.In [27], Shahzad has obtained extension of the results of Latif and Tweddle [18] for R-subweakly commuting f -nonexpansive maps.Recently, Hussain [8] proved the above-mentioned coincidence point results without any type of commutativity of the maps defined on a non-starshaped domain.
The aim of this paper is to first improve and extend the above-mentioned coincidence point results; in particular, we replace the f -nonexpansiveness of T by generalized f -nonexpansiveness and starshapedness of the set M by a weaker set of conditions.Afterwards, we study the existence of common fixed points of a general class of noncommuting generalized f -nonexpansive mappings.As applications, results regarding * - nonexpansive-and f -nonexpansive-type maps are derived and invariant approximation and random coincidence results are proved.These results improve and extend the recent results of Husain and Latif [6,7], Liu et al. [19], Xu [33,34], Jungck and Sessa [13], Kamran [14], Latif and Bano [17], Sahab et al. [23], Shahzad [24][25][26][27][28][29][30], and many others.Several examples are presented which show that certain hypotheses of our results cannot be relaxed.
The following coincidence point result is a consequence of [22, Theorem 3] of Pathak and Khan, which will be needed for the main results.
Theorem 1.4.Let (X,d) be a metric space and let f : X → X and T : X → C(X) be such that T(X) ⊂ f (X).Assume that T(X) or f (X) is complete and T and f satisfy for all x, y ∈ X and 0 ≤ h < 1, (1.3) Let X be a Banach space which satisfies Opial's condition and M a nonempty weakly compact subset of X.Let f : M → X be a weakly continuous mapping and T : The following general common fixed point result is a consequence of [12, Corollary 3.13].
Theorem 1.6.Let (X,d) be a metric space and g a continuous self-map of X.If g has relatively compact orbits with d.o.d., then g has a fixed point.Moreover, if f is continuous and the pair { f ,g} is nontrivially compatible, then there exists a point z in X such that f z = gz = z.Theorem 1.7 [29,Theorem 3.1].Let M be a nonempty separable weakly compact subset of a Banach space X and f : Ω × M → M a random operator which is both continuous and weakly continuous.Assume that T : If f and T have a deterministic coincidence point, then f and T have a random coincidence point.
Theorem 1.8 [29,Theorem 3.12].Let M be a nonempty separable complete subset of a metric space X and let T : Ω × M → C(X) and f : Ω × M → X be continuous random operators satisfying condition (A 0 ).If f and T have a deterministic coincidence point, then f and T have a random coincidence point.

Coincidence and common fixed point results
The following result extends and improves [18, Theorem 2.1] and [27, Theorem 2.1], in the sense that the maps f and T need not be commuting or R-subweakly commuting, T is not necessarily f -nonexpansive, f is not affine and continuous, and M is not necessarily q-starshaped.
Theorem 2.1.Let f be a selfmap on a nonempty complete subset M of a normed space X such that f (M) = M. Assume that T : M → C(M) satisfies, for all x, y ∈ M and λ ∈ [0,1], (2.1) Proof.Take q ∈ M and define for all x ∈ M and fixed sequence of real numbers k n (0 < k n < 1) converging to 1.Then, for each n, T n (M) ⊂ M = f (M) and for each x, y ∈ M and 0 < k n < 1.By Theorem 1.4, for each n ≥ 1, there exists x n ∈ M such that f x n ∈ T n x n .This implies that there is a Clearly, each T-invariant q-starshaped set satisfies the property (N), if f is affine, then T n (M) ⊂ f (M) provided T(M) ⊂ f (M) and q ∈ F( f ); consequently, we obtain the following result which improves substantially [27, Theorem 2.1].
6 Coincidences and approximations of generalized f -nonexpansive maps Corollary 2.3.Let M be a nonempty q-starshaped subset of a normed space X and f : M → M an affine mapping with q ∈ F( f ).Assume that T : The conclusion of [18, Theorem 2.2(a)] holds without any type of commutativity of f and T, T need not be f -nonexpansive and compact-valued, as follows.
Theorem 2.4.Let f be a selfmap on a nonempty weakly compact subset M of a Banach space X.Assume that T : Proof.For each x ∈ M, Tx ⊂ M, therefore T(M) ⊂ M. Now M is bounded (see [18,27]), so T(M) is bounded.As in the proof of Theorem 2.1, f x n − y n → 0 as n → ∞, where y n ∈ Tx n .By the weak compactness of M, there is a subsequence {x m } of the sequence The following result extends and improves [27, Theorem 2.
Proof.By Theorem 2.1, for each n, there are x n ∈ M and y n ∈ Tx n such that f x n − y n → 0 as n → ∞.It further implies that d( f x n ,Tx n ) → 0 as n → ∞.By the condition (A 0 ) there exists an x 0 ∈ M such that f x 0 ∈ Tx 0 .Hence C( f ,T) = ∅.As in the proof of Theorem 2.1, F( f Corollary 2.7.Let M be a nonempty q-starshaped subset of a normed space X and f : M → M an affine mapping with q ∈ F( f ).Assume that T : , and f and T satisfy condition If we take f = I, the identity map in the above Corollary, then we get the following corollaries which extend and generalize the results of Dotson [4], Habiniak [5], and Lami Dozo [16].Theorem 2.10.Let M be a nonempty compact subset of a normed space X and f : Proof.By Theorem 2.1, f x n − y n → 0 as n → ∞.It further implies that d( f x n ,Tx n ) → 0 as n → ∞.Since M is compact, without loss of generality, we may assume that {x n } converges to some x 0 ∈ M. The continuity of both f and T implies that f x 0 ∈ Tx 0 .Hence C( f ,T) = ∅.As in the proof of Theorem 2.1, F( f The following result improves and extends a recent result due to Beg et al. [3]. Theorem 2.11.Let M be a nonempty complete subset of a normed space X and f : M → M a mapping such that M = f (M).Assume that T : M → C(M) satisfies (2.1), for all x, y ∈ M and λ ∈ [0,1] (or assume that T is f -nonexpansive mapping).If M has property (N), T(M) is bounded, and T and f satisfy for all x, y ∈ M, where θ i : R → [0,1)(i = 1,2) and r is some fixed positive real number, then Thus {x n } is an asymptotically Tregular sequence with respect to f in M. By (2.4), 8 Coincidences and approximations of generalized f -nonexpansive maps where the right-hand side tends to 0 as n, m → ∞.This implies that {Tx n } is a Cauchy sequence in TM ⊂ M. Hence there exists which yields (1 Example 2.12.Let X = R and M = {0, 1} be endowed with the usual metric.Define T : M → K(M) by Tx = {0} for each x ∈ M. Clearly, M is not starshaped but M has the property (N) for q = 0, k n = 1 − 1/(n + 1).Let f : M → M be defined by f (x) = 1 − x for each x ∈ M. All of the conditions of Theorem 2.10 are satisfied; consequently T and f have a coincidence point.Here [27, Theorems 2.1 and 2.2] and [18,Theorem 2.3] cannot be applied because f and T are not R-weakly commuting (and hence not commuting) and M is not q-starshaped.
Example 2.13.Let X = R and M = {0, 1,1 − 1/(n + 1) : n ∈ N} be endowed with the usual metric.Define T(1) = {0} and T(0) = T(1 − 1/(n + 1)) = {1} for all n ∈ N. Clearly, M is not starshaped but M has the property (N) for q = 0 and k n = 1 − 1/(n + 1), n ∈ N. Let f x = x for all x in M. Now f and T satisfy (2.1) together with all other conditions of Theorem 2.1 except the condition that ( Here also note that all of the conditions of Theorem 2.10 are satisfied except the condition that T is continuous.Note that C( f ,T) = ∅.
Example 2.14.Let X = R 2 be endowed with the norm • defined by (a,b) and f (x) = x, for all x ∈ M. All of the conditions of Theorem 2.10 are satisfied except that M has property (N), that is, (1 − k n )q + k n T(M) not contained in M for any choice of q ∈ M and k n .Note that C( f ,T) = ∅. ( and f (x) = x, for all x ∈ M. All of the conditions of Theorem 2.10 are satisfied except that M is compact.Note that C( f ,T) = ∅.
A. R. Khan et al. 9 Example 2.15.Let X = R and M = [0,1] be endowed with the usual metric.Define f : M → M and T : M → K(M) as follows: All of the conditions of Theorem 2.10 are satisfied except that f (M) = M.Note that C( f ,T) = ∅.
The following example reveals that the condition f f v= f v for v ∈ C( f ,T), in Theorem 2.10, is necessary for the result.
Example 2.16.Let X = R and M = [0,1] be endowed with the usual metric.Define T(x) = {0, 1} and f (x) = 1 − x for each x ∈ M. All of the conditions of Theorem 2.10 are satisfied except the condition  Theorem 3.1.Let M be a nonempty subset of a normed space X and let T : M → K(M) be a * -nonexpansive mapping.If M has property (N), then T has a fixed point provided one of the following conditions is satisfied: (i) M is weakly closed, X is complete space satisfying Opial's condition, and T(M) ⊂ B, for some weakly compact set B in X; (ii) M is weakly compact, X is complete, and (I − T) is demiclosed at 0; (iii) M is weakly compact and X is complete space satisfying Opial's condition; where θ i : R → [0,1) (i = 1,2) and r is some fixed positive real number.
Proof.The operator P T : M → K(M) is compact-valued and nonexpansive (see, [3,34]).Further, (1 − k n )q + k n P T x ⊆ (1 − k n )q + k n Tx ⊆ M, for some q ∈ M and a fixed sequence of real numbers k n (0 ≤ k n ≤ 1) converging to 1 and for each x ∈ M. Thus M has property (N) with respect to P T as M has property (N) with respect to T. Also, for each x ∈ M, we Theorem 3.16.Let (Ω, ) be any measurable space, M a nonempty separable closed subset of a Banach space X, and f : Ω × M → M a continuous random operator such that f (ω,M) = M for each ω ∈ Ω. Assume that the random operator T : Ω × M → CB(M) is continuous and satisfies (3.12) (or is f -nonexpansive).If M is q-starshaped, f and T satisfy the condition (A 0 ) and T(ω,M) is bounded for each ω ∈ Ω, then f and T have a random coincidence point.Moreover, if for each ω ∈ Ω and any x ∈ M, f (ω,x) ∈ T(ω,x) implies f (ω, f (ω,x)) = f (ω,x), and f is T-weakly commuting random operator, then f and T have a common random fixed point.
Remark 3.17.(a) If we combine Theorems 2.10 and 1.8, we obtain the conclusion of [29, Theorem 3.17],without the commutativity of maps.Notice that f need not be affine and f (ω,•) need not fix q for each ω ∈ Ω.
(b) Using Theorems 3.13-3.16,we can obtain the random invariant approximation results.In particular, when T is single-valued, we obtain the following result which provides stochastic analogue (for more general maps) of the recent invariant approximation result due to Shahzad [28,Theorem 5].
2 and Corollary 2.3], [4, Theorem 2], [11, Corollary 3.4], and [18, Theorem 2.2].Corollary 2.5 [8, Theorems 2.3 and 2.4].Let f be a selfmap on a nonempty weakly compact subset M of a Banach space X.Assume that T : M → C(M) is f -nonexpansive map such that M = f (M) and M has the property (N).Then C( f ,T) = ∅ provided one of the following two conditions is satisfied: (a) ( f − T) is demiclosed at 0; (b) f is weakly continuous, T is compact-valued, and X satisfies Opial's condition.If, in addition, f is T-weakly commuting at v and f

Theorem 2 . 6 .
Proof.(a) It follows from Theorem 2.4.(b) By Lemma 1.5, ( f − T) is demiclosed at 0. Hence the result from part (a).Let M be a nonempty complete subset of a normed space X and f : M → M a mapping such that M = f (M).Assume that T : M → C(M) satisfies (2.1), for all x, y ∈ M and λ ∈ [0,1].If M has the property (N), f and T satisfy the condition (A 0 ), and

Corollary 2 . 8 .
Let M be a nonempty complete subset of a normed space X and T : M → C(M) a nonexpansive mapping such that T(M) is bounded.If M has property (N) and T satisfies the condition (A), then T has a fixed point.Corollary 2.9.Let M be a nonempty complete subset of a normed space X and T : M → C(M) a nonexpansive hemicompact (or condensing) mapping.If M has property (N) and T(M) is bounded, then T has a fixed point.The following theorem extends and improves [18, Theorem 2.3] of Latif and Tweddle and [4, Theorem 1] of Dotson.

1 .
Deterministic fixed point theory.We obtain the following improvements and generalizations of [3, Theorem 2.4], [6, Theorem 3.2], and [34, Theorem 2] and many other results in the current literature.
is bounded, and T satisfies for all x, y ∈ M, H r (Tx,T y) ≤ θ 1 d(x,Tx) d r (x,Tx) + θ 2 d(y,T y) d r (y,T y),(3.1) [22]aid to be demiclosed at 0 if for every sequence {x n } in M and {y n } in X with y n ∈ Tx n such that {x n } converges weakly to x and {y n } converges to 0 ∈ X, then 0 ∈ Tx.A mapping T :M → C(X) is called upper (resp., lower) semicontinuous if for any closed (resp., open) subset B of X,T −1 (B) = {x ∈ M : T(x) ∩ B = ∅} is closed (resp., open).If T is both upper and lower semicontinuous, then T is called a continuous map.If Tx ∈ K(X) for all x ∈ M, then T is continuous if and only if T ,Tx n ) → 0 as n → ∞, there exists y ∈ D with f y ∈ T y.The sequence {x n } in M is said to be an asymptotically T-regular sequence with respect to f provided d( f x n ,Tx n ) → 0 as n → ∞.If f = I (the identity map on M), then the sequence {x n } is called asymptotically T-regular[22].