Research  Open  Published:
A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued
Fixed Point Theory and Applicationsvolume 2011, Article number: 54 (2011)
Abstract
Bruck [Pac. J. Math. 53, 5971 1974 Theorem 1] proved that for a nonempty closed convex subset E of a Banach space X, if E is weakly compact or bounded and separable and suppose that E has both (FPP) and (CFPP), then for any commuting family S of nonexpansive selfmappings of E, the set F(S) of common fixed points of S is a nonempty nonexpansive retract of E. In this paper, we extend the above result when one of its elements in S is multivalued. The result extends previously known results (on common fixed points of a pair of single valued and multivalued commuting mappings) to infinite number of mappings and to a wider class of spaces.
2000 Mathematics Subject Classification: 47H09; 47H10
1 Introduction
For a pair (t, T) of nonexpansive mappings t : E → E and T : E → 2 ^{X} defined on a bounded closed and convex subset E of a convex metric space or a Banach space X, we are interested in finding a common fixed point of t and T. In [1], Dhompongsa et al. obtained a result for the CAT(0) space setting:
Theorem 1.1. [[1], Theorem 4.1] Let E be a nonempty bounded closed and convex subset of a complete CAT(0) space X, and let t : E → E and T : E → 2 ^{X}be nonexpansive mappings with T(x) a nonempty compact convex subset of X. Assume that for some p ∈ Fix(t),
If t and T are commuting, then Fix(t) ∩ Fix(T) ≠ ∅.
Shahzad and Markin [2] improved Theorem 1.1 by removing the assumption that the nonexpansive multivalued mapping T in that theorem has a convexvalued contractive approximation. They also noted that Theorem 1.1 needs the additional assumption that T(·) ∩ E ≠ ∅ for that result to be valid.
Theorem 1.2. [[2], Theorem 4.2] Let X be a complete CAT(0) space, and E a bounded closed and convex subset of X. Assume t : E → E and T : E → 2 ^{X} are nonexpansive mappings with T(x) a compact convex subset of X and T(x) ∩ E ≠ ∅ for each x ∈ E. If the mappings t and T commute, then Fix(t) ∩ Fix(T) ≠ ∅.
Dhompongsa et al. [3] extended Theorem 1.1 to uniform convex Banach spaces.
Theorem 1.3. [[3], Theorem 4.2] Let E be a nonempty bounded closed and convex subset of a uniform convex Banach space X. Assume t : E → E and T : E → 2 ^{E}are nonexpansive mappings with T(x) a nonempty compact convex subset of E. If t and T are commuting, then Fix(t) ∩ Fix(T) ≠ ∅.
The result has been improved, generalized, and extended under various assumptions. See for examples, [[4], Theorem 3.3], [[5], Theorem 3.4], [[6], Theorem 3.9], [[7], Theorem 4.7], [[8], Theorem 5.3], [[9], Theorem 5.2], [[10], Theorem 3.5], [[11], Theorem 4.2], [[12], Theorem 3.8], [[13], Theorem 3.1].
Recall that a bounded closed and convex subset E of a Banach space X has the fixed point property for nonexpansive mappings (FPP) (respectively, for multivalued nonexpansive mappings (MFPP)) if every nonexpansive mapping of E into E has a fixed point (respectively, every nonexpansive mapping of E into 2 ^{E} with compact convex values has a fixed point). The following concepts and result were introduced and proved by Bruck [14, 15]. For a bounded closed and convex subset E of a Banach space X, a mapping t : E → X is said to satisfy the conditional fixed point property (CFP) if either t has no fixed points, or t has a fixed point in each nonempty bounded closed convex set that leaves t invariant. A set E is said to have the hereditary fixed point property for nonexpansive mappings (HFPP) if every nonempty bounded closed convex subset of E has the fixed point property for nonexpansive mappings; E is said to have the conditional fixed point property for nonexpansive mappings (CFPP) if every nonexpansive t : E → E satisfies (CFP).
Theorem 1.4. [[15], Theorem 1] Let E be a nonempty closed convex subset of a Banach space X. Suppose E is weakly compact or bounded and separable. Suppose E has both (FPP) and (CFPP). Then for any commuting family S of nonexpansive selfmappings of E, the set F(S) of common fixed points of S is a nonempty nonexpansive retract of E.
The object of this paper is to extend Theorems 1.3 and 1.4 for a commuting family S of nonexpansive mappings one of which is multivalued. As consequences,

(i)
Theorem 1.3 is extended to a bigger class of Banach spaces while a class of mappings is no longer finite;

(ii)
Theroem 1.4 is extended so that one of its members in S can be multivalued.
2 Preliminaries
Let E be a nonempty subset of a Banach space X. A mapping t : E → X is said to be nonexpansive if
The set of fixed points of t will be denoted by Fix(t) := {x ∈ E : tx = x}. A subset C of E is said to be tinvariant if t(C) ⊂ C. As usual, B(x, ε) = {y ∈ X : x  y < ε} stands for an open ball. For a subset A and ε > 0, the εneighborhood of A is defined as
Note that if A is convex, then B_{ ε } (A) is also convex. We write $\u0100$ for the closure of A.
We shall denote by 2 ^{E} the family of all subsets of E, CB(E) the family of all nonempty closed bounded subsets of E and denote by KC(E) the family of all nonempty compact convex subsets of E. Let H(·,·) be the Hausdorff distance defined on CB(X), i.e.,
where dist(a, B) := inf{a  b : b ∈ B} is the distance from the point a to the subset B.
A multivalued mapping T : E → CB(X) is said to be nonexpansive if
T is said to be upper semicontinuous if for each x_{0} ∈ E, for each neighborhood U of Tx_{0}, there exists a neighborhood V of x_{0} such that Tx ⊂ U for each x ∈ V. Clearly, every upper semicontinuous mapping T has a closed graph, i.e., for each sequence {x_{ n } } ⊂ E converging to x_{0} ∈ E, for each y_{ n } ∈ Tx_{ n } with y_{ n } → y_{0}, one has y_{0} ∈ Tx_{0}. Fix(T ) is the set of fixed points of T, i.e., Fix(T):= {x ∈ E : x ∈ Tx}. A subset C of E is said to be Tinvariant if Tx ∩ C ≠ ∅ for all x ∈ C. For λ ∈ (0, 1), we say that a multivalued mapping T : E → CB(X) satisfies condition (C_{ λ } ) if λdist(x, Tx) ≤ x  y implies H(Tx, Ty) ≤ x  y for x, y ∈ E. The following example shows that a mapping T satisfying condition (C_{ λ } ) for some λ ∈ (0, 1) can be discontinuous:
Let λ ∈ (0, 1) and $a=\frac{2\left(\lambda +1\right)}{\lambda \left(\lambda +2\right)}$. Define a mapping $T:\left[0,\frac{2}{\lambda}\right]\to KC\left(\left[0,\frac{2}{\lambda}\right]\right)$ by
Clearly, $\frac{1}{\lambda}<a<\frac{2}{\lambda}$ and T is nonexpansive on $\left[0,\frac{2}{\lambda}\right)$. Thus, we only verify that, for $x\in \left[0,\frac{2}{\lambda}\right)$,
and
If $\lambda dist\left(x,Tx\right)\le \phantom{\rule{0.3em}{0ex}}\left\rightx\frac{2}{\lambda}\left\right$, then $x\le \frac{4}{\lambda \left(\lambda +2\right)}$ and
Hence (2.1) holds. If $\lambda dist\left(\frac{2}{\lambda},T\frac{2}{\lambda}\right)\le \phantom{\rule{0.3em}{0ex}}\left\right\frac{2}{\lambda}x\left\right$, then $x\le \frac{4}{\lambda \left(\lambda +2\right)}$ and
Thus (2.2) holds. Therefore, T satisfies condition (C_{ λ } ). Clearly, T is upper semicontinuous but not continuous (and hence T is not nonexpansive).
For a multivalued mapping T : E → CB(X), a sequence {x_{ n } } in E of a Banach space X for which lim_{n→∞}dist(x_{ n } , Tx_{ n } ) = 0 is called an approximate fixed point sequence (afps for short) for T.
Let (M, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X is a map c from a closed interval [0, r] ⊂ ℝ to X such that c(0) = x, c(r) = y and d(c(t), c(s)) = t  s for all s, t ∈ [0, r]. The mapping c is an isometry and d(x, y) = r. The image of c is called a geodesic segment joining x and y which when unique is denoted by seg[x, y]. A metric space (M, d) is said to be of hyperbolic type if it is a metric space that contains a family L of metric segments (isometric images of real line bounded segments) such that (a) each two points x, y in M are endpoints of exactly one member seg[x, y] of L, and (b) if p, x, y ∈ M and m ∈ seg[x, y] satisfies d(x, m) = αd(x, y) for α ∈ [0, 1], then d(p, m) ≤ (1  α)d(p, x) + αd(p, y). M is said to be metrically convex if for any two points x, y ∈ M with x ≠ y there exists z ∈ M, x ≠ z ≠ y, such that d(x, y) = d(x, z) + d(z, y). Obviously, every metric space of hyperbolic type is always metrically convex. The converse is true provided that the space is complete: If (M, d) is a complete metric space and metrically convex, then (M, d) is of hyperbolic type (cf. [[16], Page 24]). Clearly, every nonexpansive retract is of hyperbolic type.
Proposition 2.1. [[17], Proposition 2] Suppose (M, d) is of hyperbolic type, let {α_{ n } } ⊂ [0, 1), if {x_{ n } } and {y_{ n } } are sequences in M which satisfy for all i, n,
(i) x_{n+1}∈ seg[x_{ n } , y_{ n } ] with d(x_{ n } , x_{n+1}) = α_{ n }d(x_{ n } , y_{ n } ),
(ii) d(y_{n+1}, y_{ n } ) ≤ d(x_{n+1}, x_{ n } ),
(iii) d(y_{i+n}, x_{ i }) ≤ d < ∞,
(iv) α_{ n } ≤ b < 1, and
(v)${\sum}_{s=0}^{\infty}\phantom{\rule{0.3em}{0ex}}{\alpha}_{s}=+\infty $.
Then lim_{n→∞}d(y_{ n } , x_{ n } ) = 0.
Let E be a nonempty bounded closed subset of a Banach space X and {x_{ n } } a bounded sequence in X. For x ∈ X, define the asymptotic radius of {x_{ n } } at x as the number
Let
and
The number r(E, {x_{ n } }) and the set A(E, {x_{ n } }) are, respectively, called the asymptotic radius and asymptotic center of {x_{ n } } relative to E. The sequence {x_{ n } } is called regular relative to E if r(E, {x_{ n } }) = r(E, {x_{n′}}) for each subsequence {x_{n′}} of {x_{ n } }. It is well known that: every bounded sequence contains a subsequence that is regular relative to a given set (see [[16], Lemma 15.2] or [[18], Theorem 1]). Further, {x_{ n } } is called asymptotically uniform relative to E if A(E, {x_{ n } }) = A(E, {x_{n′}}) for each subsequence {x_{n′}} of {x_{ n } }. It was noted in [16] that if E is nonempty and weakly compact, then A(E, {x_{ n } }) is nonempty and weakly compact, and if E is convex, then A(E, {x_{ n } }) is convex.
A Banach space X is said to satisfy the KirkMassa condition if the asymptotic center of each bounded sequence of X in each bounded closed and convex subset is nonempty and compact. A more general space than spaces satisfying the KirkMassa condition is a space having property (D). Property (D) introduced in [19] is defined as follows:
Definition 2.2. [[19], Definition 3.1] A Banach space X is said to have property (D) if there exists λ ∈ [0, 1) such that for any nonempty weakly compact convex subset E of X, any sequence {x_{ n } } ⊂ E that is regular and asymptotically uniform relative to E, and any sequence {y_{ n } } ⊂ A(E, {x_{ n } }) that is regular and asymptotically uniform relative to E we have
Theorem 2.3. [[19], Theorem 3.6] Let E be a nonempty weakly compact convex subset of a Banach space X that has property (D). Assume that T : E → KC(E) is a nonexpansive mapping. Then, T has a fixed point.
A direct consequence of Theorem 2.3 is that every weakly compact convex subset of a space having property (D) has both (MFPP) for multivalued nonexpansive mappings and (CFPP). The class of spaces having property (D) contains several wellknown ones including kuniformly rotund, nearly uniformly convex, uniformly convex in every direction spaces, and spaces satisfying Opial condition (see [3, 19–23] and references therein).
The following useful result is due to Bruck:
Theorem 2.4. [[14], Theorem 1] Let E be a nonempty closed convex subset of a Banach space X. Suppose E is locally weakly compact and F is a nonempty subset of E. Let N(F) = {ff} : E → E is nonexpansive and fx = x for all x ∈ F}. Suppose that for each z in E, there exists h in N(F) such that h(z) ∈ F. Then, F is a nonexpansive retract of E.
3 Main results
We first state three main results:
Theorem 3.1. Let E be a weakly compact convex subset of a Banach space X. Suppose E has (MFPP) and (CFPP). Let S be any commuting family of nonexpansive selfmappings of E. If T : E → KC(E) is a multivalued nonexpansive mapping that commutes with every member of S. Then, F(S) ∩ Fix(T) ≠ ∅.
Theorem 3.2. Let X be a Banach space satisfying the KirkMassa condition and let E be a weakly compact convex subset of X. Let S be any commuting family of nonexpansive selfmappings of E. Suppose T : E → KC(E) is a multivalued mapping satisfying condition (C_{ λ } ) for some λ ∈ (0, 1) that commutes with every member of S. If T is upper semicontinuous, then F(S) ∩ Fix(T) ≠ ∅.
Theorem 3.3. Let E be a weakly compact convex subset of a Banach space X. Suppose E has (MFPP) and (CFPP). Let S be any commuting family of nonexpansive selfmappings of E. If T : E → KC(E) is a multivalued nonexpansive mapping that commutes with every member of S. Suppose in addition that T satisfies.
(i) there exists a nonexpansive mapping s : E → E such that sx ∈ Tx for each x ∈ E,
(ii) Fix(T) = {x ∈ E : Tx = {x}} ≠ ∅.
Then, F(S) ∩ Fix(T) is a nonempty nonexpansive retract of E.
Remark 3.4.

(i)
As corollaries, the results in Theorems 3.1 and 3.3 are valid for spaces X having property (D).

(ii)
Theorem 3.3 can be viewed as a generalization of Theorem 1.4 of Bruck for weakly compact convex domains.
Definition 3.5. Let E be a nonempty bounded closed and convex subset of a Banach space X. Let t : E → E be a single valued mapping, T : E → KC(E) a multivalued mapping. Then, t and T are said to be commuting mappings if tTx ⊂ Ttx for all x ∈ E.
If in Theorem 2.4, we put F = Fix(t) where t : E → E is nonexpansive, then it was noted in [[15], Remark 1] that a retraction c ∈ N(F) can be chosen so that cW ⊂ W for all tinvariant closed and convex subsets W of E. With the same proof, we can show that the same result is valid for F = F(S). In this case, we define N(F(S)) = {f  f : E → E is nonexpansive, Fix(f) ⊃ F(S), f(W) ⊂ W whenever W is a closed convex Sinvariant subset of E}. Here, by an "Sinvariant"subset, we mean a subset that is left invariant under every member of S.
Lemma 3.6. Let E be a nonempty weakly compact convex subset of a Banach space X and let S be any commuting family of nonexpansive selfmappings of E. Suppose that E has (FPP) and (CFPP). Then, F(S) is a nonempty nonexpansive retract of E, and a retraction c can be chosen so that every Sinvariant closed and convex subset of E is also cinvariant.
Proof. Note by Theorem 1.4 that F(S) is nonempty. According to Theorem 2.4, it suffices to show that for each z in E, there exists h in N(F(S)) such that h(z) ∈ F(S).
Let z ∈ E and K = {f(z)f ∈ N(F(S))} ⊂ E. Since K is weakly compact convex and invariant under every member in S, we obtain by Theorem 1.4 that F(S)∩K ≠ ∅, i.e., there exists h in N(F(S)) such that h(z) ∈ F(S). Theorem 2.4 then implies that F(S) is a nonexpansive retract of E, where a retraction is chosen from N(F(S)). □
Proof of Theorem 3.1 Let c be a nonexpansive retraction of E onto F(S) obtained in Lemma 3.6. Set Ux := Tcx for x ∈ E. Clearly,
Thus, U is nonexpansive, and since E has (MFPP), there exists p ∈ Up = Tcp. Since Tcp is Sinvariant, by the property of c, Tcp is also cinvariant, i.e., cp ∈ Tcp. Therefore, F(S) ∩ Fix(T) ≠ ∅. □
The following proposition is needed for a proof of Theorem 3.2.
Proposition 3.7. Let A be a compact convex subset of a Banach space X and let a nonempty subset F of A be a nonexpansive retract of A. Suppose a mapping U : A → KC(A) is upper semicontinuous and satisfies:
(i) c(Ux) ⊂ Ux for all x ∈ F where c is a nonexpansive retraction of A onto F, and
(ii) F is U invariant.
Then, U has a fixed point in F.
Proof. Let ε > 0. Since F is compact, there exists a finite εdense subset {z_{1}, z_{2}, ..., z_{ n } } of F , i.e., $F\subset {\bigcup}_{i=1}^{n}\phantom{\rule{2.77695pt}{0ex}}B\left({z}_{i},\frac{\epsilon}{2}\right)$. Put $K:=\overline{co}\left({z}_{1},{z}_{2},\dots ,{z}_{n}\right)$ and define $Vx={\overline{B}}_{\epsilon}\left(Ucx\right)\cap K$ for x ∈ K. Clearly, V : K → KC(K). For x ∈ K, cx ∈ F thus by (ii) there exists y ∈ Ucx ∩ F. Then, choose z_{ i } for some i such that $\left\right{z}_{i}y\left\right\phantom{\rule{0.3em}{0ex}}\le \frac{\epsilon}{2}$. Therefore, ${z}_{i}\in {\stackrel{\u0304}{B}}_{\epsilon}\left(Ucx\right)\cap K$, i.e., V x is nonempty for x ∈ K. We now show that V is upper semicontinuous. Let {x_{ n } } be a sequence in K converging to some x ∈ K and y_{ n } ∈ V x_{ n } with y_{ n } → y. Choose a_{ n } ∈ Ucx_{ n } such that y_{ n }  a_{ n }  ≤ ε. As A is compact, we may assume that a_{ n } → a for some a ∈ A. By upper semicontinuity of U, a ∈ Ucx. Consider
By letting n → ∞, we obtain y  a ≤ ε, i.e., y ∈ V x and the proof that V is upper semicontinuous is complete. By Kakutani fixed point theorem, there exists p_{ ε } ∈ V p_{ ε } , that is, p_{ ε }  b_{ ε }  ≤ ε for some b_{ ε } ∈ Ucp_{ ε } .
By the assumption on U, we see that cb_{ ε } ∈ Ucp_{ ε } and cp_{ ε }  cb_{ ε }  ≤  p_{ ε }  b_{ ε }  ≤ ε. Taking $\epsilon =\frac{1}{n}$ and write q_{ n } for $c{p}_{\frac{1}{n}}$ and b_{ n } for $c{b}_{\frac{1}{n}}$, we obtain a sequence {q_{ n } } ⊂ F and b_{ n } ∈ Uq_{ n } ∩F with q_{ n }  b_{ n }  → 0. By the compactness of F, we assume that q_{ n } → q and b_{ n } → b. It is seen that q = b ∈ Uq. □
Proof of Theorem 3.2 As observed earlier, E has both (FPP) and (CFPP), thus we start with a nonexpansive retraction c of E onto F(S) obtained by Lemma 3.6. For each x ∈ F(S), Tx is invariant under every member of S and Tx is convex, thus Tx is cinvariant. Clearly, c is a nonexpansive retraction of Tx onto Tx ∩ F(S) that is nonempty by Theorem 1.4.
Next, we show that there exists an afps for T in F(S). Let x_{0} ∈ F (S). Since Tx_{0} ∩ F (S) ≠ ∅, we can choose y_{0} ∈ Tx_{0} ∩ F (S). Since F (S) is of hyperbolic type, there exists x_{1} ∈ F (S) such that
Choose y′_{1} ∈ Tx_{1} for which y_{ o }  y′_{1} = dist(y_{0}, Tx_{1}). Set y_{1} = cy′_{1}. Clearly, y_{0}  y_{1} = cy_{0}  cy′_{1} ≤ y_{0}  y′_{1}. Therefore, we can choose y_{1} ∈ Tx_{1} ∩ F (S) so that y_{0}  y_{1} = dist(y_{0}, Tx_{1}). In this way, we will find a sequence {x_{ n } } ⊂ F (S) satisfying
where y_{ n } ∈ Tx_{ n } ∩ F (S) and y_{ n }  y_{n+1} = dist(y_{ n } , Tx_{n+1}).
Since λdist(x_{ n } , Tx_{ n } ) ≤ λx_{ n }  y_{ n }  = x_{ n }  x_{n+1},
From Proposition 2.1, lim_{n→∞}y_{ n }  x_{ n }  = 0 and {x_{ n } } is an afps for T in F(S). Assume that {x_{ n } } is regular relative to E. By the KirkMassa condition, A := A(E, {x_{ n } }) is assumed to be nonempty compact and convex. Define Ux = Tx ∩ A for x ∈ A. We are going to show that Ux is nonempty for each x ∈ A. First, let r := r(E, {x_{ n } }). If r = 0 and if x ∈ A, then x_{ n } → x and y_{ n } → x. Using upper semicontinuity of T , we see that x ∈ Tx, i.e., F(S) ∩ Fix(T) ≠ ∅.
Therefore, we assume for the rest of the proof that r > 0. Let x ∈ A. If for some subsequence $\left\{{x}_{{n}_{k}}\right\}$ of {x_{ n } }, $\lambda dist\left({x}_{{n}_{k}},T{x}_{{n}_{k}}\right)\ge \phantom{\rule{0.3em}{0ex}}\left\right{x}_{{n}_{k}}x\left\right$ for each k, we have
since {x_{ n } } is regular relative to E and this is a contradiction. Therefore,
Now, we show that Ux is nonempty. Choose y_{ n } ∈ Tx_{ n } so that x_{ n }  y_{ n }  = dist(x_{ n } , Tx_{ n } ) and choose z_{ n } ∈ Tx such that y_{ n }  z_{ n }  = dist(y_{ n } , Tx). As Tx is compact, we may assume that {z_{ n } } converges to z ∈ Tx. Using (3.1) and the fact that T satisfies condition (C_{ λ } ), we have
Taking lim sup_{n→∞}in the above inequalities to obtain
that implies that z ∈ Ux proving that Ux is nonempty as claimed.
Now, we show that U is upper semicontinuous. Let {z_{ k } } be a sequence in A converging to some z ∈ A and y_{ k } ∈ Uz_{ k } with y_{ k } → y. Consider the following estimates:
Letting k → ∞, it follows that
Hence y ∈ A. From upper semicontinuity of T, y ∈ Tz. Therefore, y ∈ Uz and thus U is upper semicontinuous. Put F := F(S) ∩ A. Since A is cinvariant, it is clear that F is a nonexpansive retract of A by the retraction c. Now, if x ∈ F, then Ux is Sinvariant which implies Ux is cinvariant. Therefore, condition (i) in Proposition 3.7 is justified. To verify condition (ii), we let x ∈ F. Take y ∈ Ux. It is obvious that cy ∈ Ux ∩ F(S), so U satisfies condition (ii) of Proposition 3.7. Upon applying Proposition 3.7 we obtain a fixed point in F of U and thus of T and we are done. □
Proof of Theorem 3.3 By (i) and (ii), it is seen that Fix(T) = Fix(s). Note by Theorem 3.1 that F(S) ∩ Fix(s) is nonempty. Let c be a retraction from E onto F(S) obtained by Lemma 3.6. Here, c belongs to the set N(F(S)) = {f  f : E → E is nonexpansive, Fix(f) ⊃ F(S), f(W) ∩ W whenever W is a closed convex Sinvariant subset of E}. Put F = F(S) ∩ Fix(s) and let N(F) = {f  f : E → E is nonexpansive, Fix(f) ⊃ F}. Let z ∈ E and consider the weakly compact and convex set K := {f(z)f ∈ N(F)}. It is left to show that h(z) ∈ F for some h ∈ N(F). Since K is Sinvariant, K is therefore cinvariant. It is evident that K is sinvariant. Thus sc : K → K. Therefore, sc has a fixed point, say x, in K, i.e., sc(x) = x. By (i), sc(x) ∈ Tcx. Since Tcx is cinvariant, we have cx ∈ Tcx. That is cx ∈ Fix(T) = Fix(s). Hence scx = x = cx, i.e., cx ∈ F(S) ∩ Fix(s), and the proof is complete. □
When S consists of only the identity mapping of E, we immediately have the following corollary:
Corollary 3.8. Let E be a weakly compact convex subset of a Banach space X. Suppose E has (MFPP). If T : E → KC(E) is a multivalued nonexpansive mapping satisfying.
(i) there exists a nonexpansive mapping s : E → E such that sx ∈ Tx for each x ∈ E,
(ii) Fix(T) = {x ∈ E : Tx = {x}} ≠ ∅.
Then Fix(T) is a nonempty nonexpansive retract of E.
Of course, when T is single valued, condition (i) is satisfied. Even a very simple example shows that condition (ii) in Corollary 3.8 may not be dropped.
Example 3.9. Let X be the Hilbert space ℝ^{2}with the usual norm, and let f : [0, 1] → [0, 1] be a continuous function that is strictly concave,$f\left(0\right)=\frac{1}{2}$and f(1) = 1. Moreover let f′(x) ≤ 1 for x ∈ [0, 1]. Let T : [0, 1]^{2} → KC([0, 1]^{2}) be defined by T(x, y) = [0, x] × [f(x), 1]. We show that T is nonexpansive, but Fix(T)≠ {x : Tx = {x}} and Fix(T) is not metrically convex. If (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ [0, 1]^{2}, then
Hence T is nonexpansive. However,$a=\left(0,\frac{1}{2}\right)$is a fixed point but Ta ≠ {a}. Finally, Fix(T) is not metrically convex since, putting b = (1, 1), we see that b ∈ Tb, but$\frac{a+b}{2}=\left(\frac{1}{2},\frac{3}{4}\right)\notin T\frac{a+b}{2}$since f is strictly concave.
In [[14], Lemma 6] it was stated that: Let E be a nonempty weakly compact convex subset of a Banach space X. Suppose E has (HFPP). Suppose F is a nonempty nonexpansive retract of E and t : E → E is a nonexpansive mapping which leaves F invariant. Then Fix(t) ∩ F is a nonempty nonexpansive retract of E.
Here, we have a multivalued version (with a similar proof) of this result.
Corollary 3.10. Let E and T be as in Corollary 3.8. Suppose F is a nonexpansive retract of E by a retraction c. If Tx is cinvariant for each x ∈ F, then Fix(T) ∩ F is a nonempty nonexpansive retract of E.
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Acknowledgements
The authors are grateful to the referees for their valuable comments. They also wish to thank the National Research University Project under Thailand's Office of the Higher Education Commission for financial support.
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Keywords
 Common fixed point
 Nonexpansive retract
 Property (D)
 KirkMassa condition