Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces

Nonlinear Analysis 66 (2007) 2345–2354 www.elsevier.com/locate/na

Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces Jong Soo Jung ∗ Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea Received 1 April 2003; accepted 14 March 2006

Abstract Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty closed convex subset of E, and T : C → K(E) a nonexpansive mapping. For u ∈ C and t ∈ (0, 1), let xt be a fixed point of a contraction G t : C → K(E), defined by G t x := t T x + (1 − t)u, x ∈ C. It is proved that if C is a nonexpansive retract of E, {xt } is bounded and T z = {z} for any fixed point z of T , then the strong limt →1 xt exists and belongs to the fixed point set of T . Furthermore, we study the strong convergence of {xt } with the weak inwardness condition on T in a reflexive Banach space with a uniformly Gˆateaux differentiable norm. c 2006 Elsevier Ltd. All rights reserved.  MSC: 47H10; 47H09 Keywords: Multivalued nonexpansive mapping; Fixed points; Inwardness; Weak inwardness; Nonexpansive retract; Banach limits; Uniformly convex; Uniformly Gˆateaux differentiable norm

1. Introduction Let E be a Banach space and C a nonempty closed subset of E. We shall denote by F (E) the family of nonempty closed subsets of E, by CB(E) the family of nonempty closed bounded subsets of E, by K(E) the family of nonempty compact subsets of E, and by KC(E) the family of nonempty compact convex subsets of E. Let H (·, ·) be the Hausdorff distance on CB(E), that is,   H (A, B) = max sup d(a, B), sup d(b, A) a∈A

b∈B

∗ Tel.: +82 51 200 7213; fax: +82 51 200 7217.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2006.03.023

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for all A, B ∈ CB(E), where d(a, B) = inf{a − b : b ∈ B} is the distance from the point a to the subset B. A multivalued mapping T : C → F (E) is said to be a contraction if there exists a constant k ∈ [0, 1) such that H (T x, T y) ≤ kx − y

(1)

for all x, y ∈ C. If (1) is valid when k = 1, the T is called nonexpansive. A point x is a fixed point for a multivalued mapping T if x ∈ T x. Banach’s Contraction Principle was extended to a multivalued contraction by Nadler [13] in 1969. Given a u ∈ C and a t ∈ (0, 1), we can define a contraction G t : C → K(C) by G t x := t T x + (1 − t)u,

x ∈ C.

(2)

Then G t is multivalued and hence it has a (non-unique, in general) fixed point x t ∈ C (see [13]): that is x t ∈ t T x t + (1 − t)u.

(3)

If T is a single valued, we have x t = t T x t + (1 − t)u.

(4)

(Such a sequence {x t } is said to be an approximating fixed point of T since it possesses the property that if {x t } is bounded, then limt →1 T x t − x t  = 0.) The strong convergence of {x t } as t → 1 for a single-valued nonexpansive self- or nonself-mapping T was studied in Hilbert space or certain Banach spaces by many authors (see for instance, Browder [2], Halpern [7], Jung and Kim [8], Jung and Kim [9], Kim and Takahashi [10], Reich [17], Singh and Watson [20], Takahashi and Kim [23], Xu [25], and Xu and Yin [28]). Let yt ∈ T x t be such that x t = t yt + (1 − t)u.

(5)

Now a natural question arises of whether Browder’s theorem can be extended to the multivalued case. A simple example given by Pietramala [14] shows that the answer is negative even if E is Euclidean. Example 1. Let C = [0, 1]×[0, 1] be the square in the real plane and T : C → K(C) be defined by T (a, b) = the triangle with vertices (0, 0), (a, 0), (0, b),

(a, b) ∈ C.

Then it is easy to see that for any (ai , bi ) ∈ C, i = 1, 2, H (T (a1 , b1 ), T (a2 , b2 )) = max{|a1 − a2 |, |b1 − b2 |} ≤ (a1 , b1 ) − (a2 , b2 ), showing that T is nonexpansive. It is also easy to see that the fixed point set of T is F(T ) = {(a, 0) : 0 ≤ a ≤ 1} ∪ {(0, b) : 0 ≤ b ≤ 1}. Let u = (1, 0). Then the mapping G t defined by (2) has the fixed point set F(G t ) = {(a, 0) : 1 − t ≤ a ≤ 1}. Let

⎧  ⎨ 1 ,0 , xt = n ⎩ (1, 0)

if t = 1 − otherwise.

1 n

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Then {x t } satisfies (3) but does not converge. This example also shows that the sequence {F(G t )} of fixed point sets of G t ’s does not converge as t → 1 to the fixed point set F(T ) of T under the Hausdorff metric. However, L´opez Acedo and Xu [12] gave the strong convergence of {x t } under the restriction F(T ) = {z} in Hilbert space. Kim and Jung [11] extended the result of L´opez Acedo and Xu [12] to a Banach space with a sequentially continuous duality mapping. Recently, Sahu [19] also studied the multivalued case in a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm. In this paper, we establish the strong convergence of {x t } defined by (3) for the multivalued nonexpansive nonself-mapping T in a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm. We also study the strong convergence of {x t } for the multivalued nonexpansive nonself-mapping T satisfying the inwardness condition in a reflexive Banach space with a uniformly Gˆateaux differentiable norm. Our results improve and extend the results in [8, 9,25,28] to the multivalued case. We also point out that the condition F(T ) = {z} should be included in the main results of Sahu [19]. 2. Preliminaries Let E be a real Banach space with norm  ·  and let E ∗ be its dual. The value of x ∗ ∈ E ∗ at x ∈ E will be denoted by x, x ∗ . A Banach space E is called uniformly convex if δ(ε) > 0 for every ε > 0, where the modulus δ(ε) of convexity of E is defined by   x + y : x ≤ 1, y ≤ 1, x − y ≥ ε δ(ε) = inf 1 − 2 for every ε with 0 ≤ ε ≤ 2. It is well known that if E is uniformly convex, then E is reflexive and strictly convex (cf. [5]). The norm of E is said to be Gˆateaux differentiable (and E is said to be smooth) if lim

t →0

x + t y − x t

(6)

exists for each x, y in its unit sphere U = {x ∈ E : x = 1}. It is said to be uniformly Gˆateaux differentiable if for each y ∈ U , this limit is attained uniformly for x ∈ U . Finally, the norm is said to be uniformly Fr´echet differentiable (and E is said to be uniformly smooth) if the limit in (6) is attained uniformly for (x, y) ∈ U × U . Since the dual E ∗ of E is uniformly convex if and only if the norm of E is uniformly Fr´echet differentiable, every Banach space with a uniformly convex dual is reflexive and has a uniformly Gˆateaux differentiable norm. The converse implication is false. A discussion of these and related concepts may be found in [3]. The (normalized) duality mapping J from E into the family of nonempty (by Hahn–Banach theorem) weak-star compact subsets of its dual E ∗ is defined by J (x) = {x ∗ ∈ E ∗ : x, x ∗ = x2 = x ∗ 2 } for each x ∈ E. It is single valued if and only if E is smooth. Let μ be a linear continuous functional on ∞ and let a = (a1 , a2 , . . .) ∈ ∞ . We will sometimes write μn (an ) in place of the value μ(a). A linear continuous functional μ such that μ = 1 = μ(1) and μn (an ) = μn (an+1 ) for every a = (a1 , a2 , . . .) ∈ ∞ is called a Banach limit. We know that if μ is a Banach limit, then lim inf an ≤ μn (an ) ≤ lim sup an n→∞

n→∞

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for every a = (a1 , a2 , . . .) ∈ ∞ . Let {x n } be a bounded sequence in E. Then we can define the real valued continuous convex function φ on E by φ(z) = μn x n − z2 for each z ∈ E. The following lemma which was given in [6,21] is, in fact, a variant of Lemma 1.3 in [16]. Lemma 1. Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gˆateaux differentiable norm and let {x n } be a bounded sequence in E. Let μ be a Banach limit and u ∈ C. Then μn x n − u2 = min μn x n − y2 y∈C

if and only if μn x − u, J (x n − u) ≤ 0

(7)

for all x ∈ C. We also need the following result, which was essentially given by Reich [18, pp. 314–315] and was also proved by Takahashi and Jeong [22]. We present the brief proof for the sake of completeness. Lemma 2. Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E, and {x n } a bounded sequence of E. Then the set   M = u ∈ C : μn x n − u2 = min μn x n − z2 z∈C

consists of one point. Proof. Let φ(z) = μn x n − z2 for each z ∈ E and r = inf{φ(z) : z ∈ C}. Then, since the function φ on C is convex and continuous, φ(z) → ∞ as z → ∞, and E is reflexive, there exists u ∈ C with φ(u) = r (cf. [1, p. 79]). Therefore M is nonempty. By Theorem 2 of [24], ·2 is uniformly convex on any bounded subset of E; in particular, we have a continuous increasing function g = gr : [0, ∞) → [0, ∞), with g(0) = 0, such that λx + (1 − λ)y2 ≤ λx2 + (1 − λ)y2 − λ(1 − λ)g(x − y) for 0 ≤ λ ≤ 1 and x, y ∈ Br , where Br is the closed ball centered at 0 and with radius r that is big enough that Br contains {x n }. It follows that φ(λx + (1 − λ)y) ≤ λφ(x) + (1 − λ)φ(y) − λ(1 − λ)g(x − y) for 0 ≤ λ ≤ 1 and x, y ∈ Br . This implies that φ is a strictly convex function on E. Thus the minimum point u of φ is unique, that is, M consists of one point.  A subset C of E is said to be a retract if there exists continuous mapping Q : E → C with C = F(Q), the fixed point set of Q. Any such mapping Q is a retraction of E onto C. If Q is nonexpansive, then C is said to be a nonexpansive retract of E (cf. [5,16]). Finally, we introduce some terminology for boundary conditions for nonself-mappings. The inward set of C at x is defined by IC (x) = {z ∈ E : z = x + λ(y − x) : y ∈ C, λ ≥ 0}.

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Let I¯C (x) = x + TC (x) with

  d(x + λy, C) =0 TC (x) = y ∈ E : lim inf λ λ→0+

for any x ∈ C. Note that for a convex set C, we have I¯C (x) = IC (x), the closure of IC (x). A multivalued mapping T : C → F (E) is said to satisfy the inwardness condition if T x ⊂ IC (x) for all x ∈ C and to satisfy the weak inwardness condition if T x ⊂ I¯C (x) for all x ∈ C. We notice that a fixed point theorem for nonexpansive mappings satisfying the inwardness condition is given in Corollary 3.5 of Reich [15]. Recently, the following lemma was given by Xu [27] (also see Lemma 2.3.2 of Xu [26]). Lemma 3. If C is a compact convex subset of a Banach space E and T : C → KC(E) is a nonexpansive mapping satisfying the boundary condition: T x ∩ I¯C (x) = ∅,

x ∈ C,

then T has a fixed point. 3. Main results In this section, we first prove a strong convergence theorem for multivalued nonexpansive nonself-mappings in a Banach space with a uniformly Gˆateaux differentiable norm. Theorem 1. Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty closed convex subset of E, and T : C → K(E) a nonexpansive nonselfmapping. Suppose that C is a nonexpansive retract of E. Suppose that T (y) = {y} for any fixed point y of T and that for each u ∈ C and t ∈ (0, 1), the contraction G t defined by (2) has a fixed point x t ∈ C. Then T has a fixed point if and only if {x t } remains bounded as t → 1 and in this case, {x t } converges strongly as t → 1 to a fixed point of T . Proof. If T z = {z}, then {x t } is uniformly bounded. In fact, given any x t , we have some yt ∈ T x t such that x t = t yt + (1 − t)u. Since yt − z = d(yt , T z) ≤ H (T x t , T z) ≤ x t − z

(8)

for all t ∈ (0, 1), we have x t −z ≤ tyt −z+(1−t)u−z. This implies that x t −z ≤ u−z and so {x t } is uniformly bounded. Suppose conversely that {x t } remains bounded as t → 1. We now show that T has a fixed point z and that {x t } converges strongly as t → 1 to a fixed point of T . To this end, let tn → 1 and x n = x tn . Define φ : E → [0, ∞) by φ(z) = μn x n − z2 . Since φ is continuous and convex, φ(z) → ∞ as z → ∞, and E is reflexive, φ attains its infimum over C (cf. [1, p. 79]). Let z ∈ C be such that μn x n − z2 = min μn x n − y2 y∈C

and let M = {x ∈ C : μn x n − x2 = min μn x n − y2 }. y∈C

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Then M is a nonempty bounded closed convex subset of C. Since C is a nonexpansive retract of E, the point z is the unique global minimum (over all of E). In fact, let Q be a nonexpansive retraction of E onto C. Then for any y ∈ E, we have μn x n − y2 ≥ μn Qx n − Qy2 = μn x n − Qy2 ≥ μn x n − z2 and hence μn x n − z2 = min μn x n − y2 . y∈E

This global minimum point z is also unique by Lemma 2. On the other hand, since x t = t yt + (1 − t)u for some yt ∈ T x t , it follows that x t − yt  = (1 − t)u − yt  → 0

(9)

as t → 1. Since T is compact valued, we have for each n ≥ 1, some wn ∈ T z for z ∈ M such that yn − wn  = d(yn , T z) ≤ H (T x n , T z) ≤ x n − z.

(10)

Let w = limn→∞ wn ∈ T z. It follows from (9) and (10) that μn x n − w2 ≤ μn yn − wn 2 ≤ μn x n − z2 .

(11)

Since z is the unique global minimum, we have w = z ∈ T z, that is, z is a fixed point of T and so T z = {z} by the assumption. On the another hand, for T z = {z}, we have from (8) x n − yn , J (x n − z) = (x n − z) + (z − yn ), J (x n − z)

≥ x n − z2 − yn − zx n − z ≥ x n − z2 − x n − z2 = 0, and it follows that 0 ≤ x n − yn , J (x n − z) = (1 − tn ) u − yn , J (x n − z) .

(12)

Hence from (9) and (12), we obtain μn x n − u, J (x n − z) ≤ 0

(13)

for T z = {z} = M. But, from (7) in Lemma 1, we have μn x − z, J (x n − z) ≤ 0 for all x ∈ C. In particular, we have μn u − z, J (x n − z) ≤ 0.

(14)

Combining (13) and (14), we get μn x n − z, J (x n − z) = μn x n − z2 ≤ 0. Therefore, there is a subsequence {x n j } of {x n } which converges strongly to z. To complete the proof, suppose that there is another subsequence {x n j } of {x n } which converges strongly to (say) y. Since d(x nk , T x nk ) ≤ x nk − ynk  = (1 − tnk )u − ynk  → 0

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as k → ∞, we have d(y, T y) = 0 and hence y ∈ T y, that is, y is a fixed point of T . By assumption, T y = {y}. It then follows from (13) that z − u, J (z − y) ≤ 0 and y − u, J (y − z) ≤ 0. Adding these two inequalities yields z − y, J (z − y) = z − y2 ≤ 0 and thus z = y. This proves the strong convergence of {x t } to z.



Corollary 1. Let H be a real Hilbert space, C a nonempty closed convex subset of H , and T : C → K(H ) a nonexpansive nonself-mapping. Suppose that T (y) = {y} for any fixed point y of T and that for each u ∈ C and t ∈ (0, 1), the contraction G t defined by (2) has a fixed point x t ∈ C. Then T has a fixed point if and only if {x t } remains bounded as t → 1 and in this case, {x t } converges strongly as t → 1 to a fixed point of T . Proof. Note that a closed convex subset C of Hilbert space H is a nonexpansive retract, where the nearest point projection P of H onto C is a nonexpansive retraction. Thus the result follows from Theorem 1.  Remark 1. (1) Theorem 1 and Corollary 1 are the multivalued versions of Theorem 1 in [8] and Theorem 1 in [28], respectively. (2) Theorem 1 can be considered as an extension of Theorem 1 in [19] to a nonself-mapping case. In particular, we point out that the condition T z = {z} should be added in the assumptions of the results in [19]. (3) In the case where E is a Banach space with a weakly sequentially continuous duality mapping and T : C → K(C) is a nonexpansive mapping, the convergence of {x t } was given in [11]. So Theorem 1 can also be considered as a nonself-mapping version of Theorem 4.1 in [11]. (4) It is still an open question whether the assumption T z = {z} in Theorem 1 can be omitted. We also do not know whether Theorem 1 is valid in a Banach space with a Fr´echet differentiable norm. Remark 2. To guarantee the existence of a fixed point of the contraction G t defined by (2), the weak inwardness condition: T x ⊂ I¯C (x),

x ∈ C,

upon the mapping T : C → K(E) is used. In fact, it is well known that if C is a nonempty closed subset of a Banach space E, T : C → F (E) is a contraction satisfying the weak inwardness condition, and x ∈ E has a nearest point in T x, then T has a fixed point (Theorem 11.4 of Deimling [4]). A fixed point theorem for multivalued strict contractions was given in Theorem 3.4 of Reich [15], too. It has recently also been shown that if C is a closed bounded convex subset of a uniformly convex Banach space E and T : C → K(E) is a nonexpansive mapping satisfying the weak inwardness condition, then T has a fixed point (Theorem 3.4 of Xu [27]). Using Remark 2, we have the following.

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Corollary 2. Let X, C, T be as in Theorem 1. Suppose in addition that C is bounded and T satisfies the weak inwardness condition. If T z = {z} for any fixed point z of T , then for each u ∈ C, the sequence {x t } defined by (3) converges strongly as t → 1− to a fixed point. Proof. Fix u ∈ C and define for each t ∈ (0, 1), the contraction G t : C → K(E) by G t x := t T x + (1 − t)u,

x ∈ C.

As it is easily seen that G t also satisfies the weak inwardness condition: G t x ⊂ I¯C (x) for all x ∈ C, we have by Remark 2 that G t has a fixed point denoted by x t . Also, by Remark 2, the fixed point set of T is nonempty. Thus the result follows from Theorem 1.  Corollary 3. Let H, C, T be as in Corollary 1. Suppose in addition that C is bounded and T satisfies the weak inwardness condition. If T z = {z} for any fixed point z of T , then for each u ∈ C, the sequence {x t } defined by (3) converges strongly as t → 1− to a fixed point. Remark 3. (1) Corollary 2 extends Corollary 2 in [8], Corollary 2 in [9] and Theorem 2 in [25] to the multivalued case. (2) Corollary 3 is also a multivalued version of Corollary 1 in [28]. (3) Theorem 1 and Corollary 2 apply to all L p spaces or  p spaces for 1 < p < ∞. Theorem 2. Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty closed convex subset of E, and T : C → KC(E) a nonexpansive nonself-mapping satisfying the inwardness condition. Assume that every closed bounded convex subset of C is compact. If the fixed point set F(T ) of T is nonempty and T y = {y} for any y ∈ F(T ), then the sequence {x t } defined by (3) converges strongly as t → 1 to a fixed point of T . Proof. Let w ∈ F(T ). Then by assumption, T w = {w}. As in proof of Theorem 1, we have x t − w ≤ u − w for all t ∈ (0, 1) and hence {x t } is uniformly bounded. We now show that {x t } converges strongly as t → 1− to a fixed point of T . To this end, let tn → 1 and x n = x tn . As in the proof of Theorem 1, we define the same function φ : E → [0, ∞) by φ(z) = μn x n − z2 and let   2 2 M = x ∈ C : μn x n − x = min μn x n − y . y∈C

Then M is a nonempty closed bounded convex subset of C and by assumption, M is compact convex. Now we prove that the inwardness condition of T on C implies a weaker inwardness of T on M, that is, T z ∩ I M (z) = ∅,

z ∈ M.

(15)

In fact, if z ∈ M, by the compactness of T z, we have for each n ≥ 1, some wn ∈ T z such that yn − wn  = d(yn , T z) ≤ H (T x n , T z) ≤ x n − z, where x n = tn yn + (1 − tn )u for some yn ∈ T x n . Let w = limn→∞ wn ∈ T z. It follows from (9) that φ(w) = μn x n − w2 ≤ μn yn − wn 2 ≤ μn x n − z2 = φ(z).

(16)

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It remains to show that w ∈ I M (z). Since T z ∈ IC (z), we have for some λ ≥ 0 and v ∈ C that w = z + λ(v − z). If λ ≤ 1, then by the convexity of C, w ∈ C and hence by (16), w ∈ M ⊂ I M (z) and we have done. So assume that λ > 1. Then we can write v in the form v = r w + (1 − r )z, where r = λ−1 ∈ (0, 1). By the convexity of φ and (16), we obtain φ(v) ≤ r φ(w) + (1 − r )φ(z) ≤ φ(z) for any z ∈ M. This implies that v ∈ M and therefore w = z + λ(v − z) belongs to I M (z) for z ∈ M. Thus we have T z ∩ I M (z) = ∅,

z ∈ M.

Then it follows from Lemma 3 that T has a fixed point z ∈ M and by the assumption, T z = {z}. The strong convergence of {x t } to z is the same as is given in the proof of Theorem 1.  Corollary 4. Let E be a uniformly smooth Banach space, C a nonempty closed convex subset of E, and T : C → KC(E) a nonexpansive nonself-mapping satisfying the inwardness condition. Assume that every closed bounded convex subset of C is compact. If the fixed point set F(T ) of T is nonempty and T y = {y} for any y ∈ F(T ), then the sequence {x t } defined by (3) converges strongly as t → 1 to a fixed point of T . Corollary 5. Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty compact convex subset of E, and T : C → KC(E) a nonexpansive nonself-mapping satisfying the inwardness condition. If the fixed point set F(T ) of T is nonempty and T y = {y} for any y ∈ F(T ), then the sequence {x t } defined by (3) converges strongly as t → 1 to a fixed point of T . Corollary 6. Let E be a uniformly smooth Banach space, C a nonempty compact convex subset of E, and T : C → KC(E) a nonexpansive nonself-mapping satisfying the inwardness condition. If the fixed point set F(T ) of T is nonempty and T y = {y} for any y ∈ F(T ), then the sequence {x t } defined by (3) converges strongly as t → 1 to a fixed point of T . Remark 4. (1) Theorem 2 is also a multivalued version of Theorem 1 in [9]. (2) Corollary 4 improves Corollary 1 in [9] and Theorem 1 in [25] to a multivalued case. Acknowledgement The author thanks the anonymous referee for his/her careful reading and helpful comments and suggestions, which improved the presentation of this manuscript. References [1] V. Barbu, Th. Precupanu, Convexity and Optimization in Banach spaces, Editura Academiei R. S. R., Bucharest, 1978. [2] F.E. Browder, Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967) 82–90. [3] M.M. Day, Normed Linear Spaces, 3rd ed., Springer-Verlag, Berlin, New York, 1973. [4] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.

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