YEKINI SHEHU, G. C. UGWUNNADI
Abstract. In this paper, we introduce a new iterative process to approximate a common fixed point of an infinite family of multi-valued generalized nonexpansive maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed iterative process. Furthermore, strong convergence theorems for infinite family of multi-valued nonexpansive maps are obtained. Our results extend many known recent results in the literature.
1. Introduction
Let D be a nonempty, closed and convex subset of a real Hilbert space H. The set D is called proximinal if for each x∈H, there exists y ∈D such that||x−y|| =d(x, D), where d(x, D) = inf{||x−z|| :z ∈D}. Let CB(D), K(D) and CK(D) denote the families of nonempty, closed and bounded subsets, nonempty compact subsets and nonempty compact convex subsets ofDrespectively. TheHausdorff metriconCB(D) is defined by
H(A, B) = maxn sup
x∈A
d(x, B),sup
y∈B
d(y, A)o (1.1)
for A, B ∈ CB(D). A single-valued map T : D → D is called nonexpansive if ||T x−T y|| ≤ ||x−y|| for all x, y ∈ D. A multi-valued map T : D → CB(D) is said to be nonexpansive if H(T x, T y) ≤ ||x−y|| for all x, y ∈D. An elementp∈D is called a fixed point ofT :D → D (resp., T : D →CB(D)) if p=T p (resp., p∈T p). The set of fixed points ofT is denoted byF(T). A multi-valued mapT :D →CB(D) is said to be quasi-nonexpansiveifH(T x, T p)≤ ||x−p||for allx∈Dand allp∈F(T). A multi-valued mapT :D→CB(D) is said to begeneralized nonexpansive[9] ifH(T x, T y)≤a||x−y||+b(d(x, T x)+d(y, T y))+c(d(x, T y)+d(y, T x)) for all x, y∈ D, where a+ 2b+ 2c≤1. It can be shown that ifT is a multi-valued generalized nonexpansive map, thenT is a multi-valued quasi-nonexpansive map. Indeed, ifp∈F(T) andT is generalized nonexpansive, we obtain for allx∈D that
H(T x, T p) ≤ a||x−p||+b(d(x, T x) +d(p, T p)) +c(d(x, T p) +d(p, T x))
≤ a||x−p||+b(||x−p||+d(p, T x)) +c(d(x, T p) +d(p, T x))
≤ (a+b+c)||x−p||+ (b+c)d(p, T x)
≤ (a+b+c)||x−p||+ (b+c)H(T x, T p).
Hence,H(T x, T p)≤1−(b+c)a+b+c ||x−p||.Since 1−(b+c)a+b+c ≤1, it follows that H(T x, T p)≤ ||x−p||.
Furthermore, if T : D → CB(D) is a generalized nonexpansive mapping and for some p, T p= {p} then for x∈F(T)− {p}, we have
||x−p|| = H(p, T x) =a||p−x||+b(d(p, T p) +d(x, T x)) +c(d(p, T x) +d(x, T p))
= a||p−x||+c(||p−x||+||x−p||) = (a+ 2c)||x−p||.
Hence a+ 2c = 1 and so b = 0. Therefore if T : D →CB(D) is a generalized nonexpansive mapping and for some p, T p={p} thenF(T) ={p}orb= 0 anda+ 2c= 1.
A mapT :D→CB(D) is said to satisfyCondition (I)if there is a nondecreasing function f : [0,∞)→[0,∞) withf(0) = 0, f(r)>0 forr∈(0,∞) such that
d(x, T x)≥f(d(x, F(T)))
Key words and phrases. Strong convergence; multi-valued generalized nonexpansive maps; multi-valued nonexpansive maps;
uniformly convex Banach spaces.
2000Mathematics Subject Classification: 47H06, 47H09, 47J05, 47J25.
1
for allx∈D.
A family{Ti:D→CB(D), i= 1,2,3, . . .} is said to satisfyCondition (II)if there is a nondecreasing function f : [0,∞)→[0,∞) withf(0) = 0, f(r)>0 for r∈(0,∞) such that
d(x, Tix)≥f(d(x,∩∞i=1F(Ti))) for alli= 1,2,3, . . .and x∈D.
The mapping T :D→CB(D) is calledhemicompactif, for any sequence{xn} in D such thatd(xn, T xn)→0 as n→ ∞, there exists a subsequence {xnk} of {xn} such that xnk → p∈ D. We note that if D is compact, then every multi-valued mappingT :D→CB(D) is hemicompact.
The fixed point theory of multi-valued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multi-valued mappings. The first results in this direction were established by Markin [1] in Hilbert spaces and by Browder [2] for spaces having weakly continuous duality mapping. Lami Dozo [3] generalized these results to a Banach space satisfying Opial’s con- dition.
In 1974, by using Edelstein’s method of asymptotic centers, Lim [4] obtained a fixed point theorem for a multi- valued nonexpansive self-mapping in a uniformly convex Banach space.
Theorem 1.1. (Lim,[4]) LetDbe a nonempty, closed convex and bounded subset of a uniformly convex Banach spaceE andT :D→K(E)a multi-valued nonexpansive mapping. Then T has a fixed point.
In 1990, Kirk-Massa [5] gave an extension of Lim’s theorem proving the existence of a fixed point in a Banach space for which the asymptotic center of a bounded sequence in a closed bounded convex subset is nonempty and compact.
Theorem 1.2. (Kirk-Massa, [5]) LetD be a nonempty, closed convex and bounded subset of a Banach space E andT :D→CK(E)a multi-valued nonexpansive mapping. Suppose that the asymptotic center inE of each bounded sequence ofE is nonempty and compact. Then T has a fixed point.
Banach contraction mapping principle was extended nicely multi-valued mappings by Nadler [6] in 1969. (Below is stated in a Banach space setting).
Theorem 1.3. (Nadler, [6]) Let D be a nonempty closed subset of a Banach space E andT :D →CB(D)a multi-valued contraction. Then T has a fixed point.
In 1953, Mann [7] introduced the following iterative scheme to approximate a fixed point of a nonexpansive mappingT in a Hilbert space H:
xn+1=αnxn+ (1−αn)T xn, ∀n≥1
where the initial point x0 is taken arbitrarily in D and{αn}∞n=1 is a sequence in [0,1]. However, we note that Mann’s iteration yields only weak convergence, see, for example [8].
In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for a multi-valued nonex- pansive mapping T with a fixed point pconverge to a fixed point q of T under certain conditions. They also claimed that the fixed pointpmay be different fromq.
In 2007, Panyanak [10] extended the results of Sastry and Babu to uniformly convex Banach spaces and proved the following theorems.
Theorem 1.4. (Panyanak, [10]) Let E be a uniformly convex Banach space, D a nonempty closed bounded convex subset of E and T : D → P(D) a multi-valued nonexpansive mapping that satisfies condition (I).
Assume that (i) 0 ≤αn < 1 and (ii) Σ∞n=1αn =∞. Suppose that F(T) a nonempty proximinal subset of D.
Then the Mann iterates{xn} defined byx0∈D,
xn+1=αnyn+ (1−αn)xn, αn ∈[a, b], 0< a < b <1, n≥0, (1.2)
whereyn∈T xn such that||yn−un||=d(un, T xn)andun ∈F(T)such that||xn−un||=d(xn, F(T)), converges strongly to a fixed point of T.
Theorem 1.5. (Panyanak, [10]) Let E be a uniformly convex Banach space, D a nonempty compact convex subset of E and T : D → P(D) a multi-valued nonexpansive mapping with a fixed point p. Assume that (i) 0≤αn, βn<1; (ii)βn→0 and(iii) Σ∞n=1αnβn=∞. Then the Ishikawa iterates{xn} defined byx0∈D,
yn=βnzn+ (1−βn)xn, βn∈[0,1], n≥0 zn∈T xn such that ||zn−p||=d(p, T xn), and
xn+1=αnz0n+ (1−αn)xn, αn∈[0,1], n6= 0 zn0 ∈T yn such that ||zn0 −p||=d(p, T yn), converges strongly to a fixed point ofT.
Later, Song and Wang [11] noted there was a gap in the proofs of Theorem 1.5 above and Theorem 5 of [9]. They further solved/revised the gap and also gave the affirmative answer Panyanak [10] question using the Ishikawa iterative scheme. In the main results, the domain ofT is still compact, which is a strong condition (see Theorem 1 of [11]) andT satisfies condition (I) (see Theorem 1 of [11]).
Recently, Abbaset al. [13] introduced the following one-step iterative process to compute common fixed points of two multi-valued nonexpansive mappings.
x1∈D
xn+1=anxn+bnyn+cnzn, ∀n≥1.
(1.3)
Using (1.3), Abbaset al. [13] proved weak and strong convergence theorems for approximation of common fixed point of two multi-valued nonexpansive mappings in Banach spaces.
Motivated by the ongoing research and the above mentioned results, we introduce a new iterative scheme for approximation of common fixed points of infinite family of multi-valued generalized nonexpansive maps in a real Banach space. Furthermore, weprove strong convergence theorems for approximation of common fixed points of infinite family of multi-valued generalized nonexpansive maps in a uniformly convex real Banach space. Next, we prove a necessary and sufficient condition for strong convergence of our new iterative process to a common fixed point of infinite family of multi-valued nonexpansive maps. Our results extend the results of Sastry and Babu [9], Panyanak [10], Song and Wang [11], Abbaset al. [13], Qinet al. [14], and Gang and Shangquan [15].
2. Preliminaries
LetE be Banach space and dimE≥2. Themodulus of convexity ofE is the functionδE: (0,2]→[0,1] defined by
δE() := infn 1−
x+y 2
:||x||=||y||= 1;=||x−y||o .
Eisuniformly convex if for any∈(0,2], there exists aδ=δ()>0 such that ifx, y∈Ewith||x|| ≤1, ||y|| ≤1 and ||x−y|| ≥, then||12(x+y)|| ≤1−δ. Equivalently,E is uniformly convex if and only ifδE()>0 for all ∈(0,2].
Lemma 2.1. (Schu,[16]) SupposeEis a uniformly convex Banach space and0< p≤tn≤q <1for all positive integersn. Also suppose that{xn}and{yn}are two sequences ofE such thatlim sup
n→∞
||xn|| ≤r, lim sup
n→∞
||yn|| ≤r and lim
n→∞||tnxn+ (1−tn)yn||=rhold for somer >0. Then lim
n→∞||xn−yn||= 0.
Lemma 2.2. (Kim et al. [17]) Let E be a uniformly convex real Banach space. For arbitrary r > 0, let Br(0) := {x ∈ E : ||x|| ≤ r}. Then, for any given sequence {xn}∞n=1 ⊂ Br(0) and for any given sequence {λn}∞n=1 of positive numbers such thatP∞
i=1λi= 1, there exists a continuous strictly increasing convex function g: [0,2r]→R, g(0) = 0
such that for any positive integersi, j withi < j, the following inequality holds:
∞
X
n=1
λnxn
2
≤
∞
X
n=1
λn||xn||2−λiλjg(||xi−xj||).
3. Main Results
Lemma 3.1. Let E be a uniformly convex real Banach space and D a nonempty, closed and convex subset of E. Let T1, T2, T3, . . . be multi-valued generalized nonexpansive maps ofD into CB(D) with ∩∞i=1F(Ti)6=∅ for whichTip={p}, ∀p∈ ∩∞i=1F(Ti). Let {αni}∞n=1, i= 0,1,2, . . .be a sequence in [,1−], ∈(0,1) such that P∞
i=0αni= 1 for alln≥1. Let{xn}∞n=1 be a sequence defined iteratively byx1∈D,
xn+1=αn0xn+
∞
X
i=1
αniy(i)n , (3.1)
wherey(i)n ∈Tixn, i= 1,2,3, . . .. Then
n→∞limd(xn, Tixn) = 0, ∀i= 1,2,3, . . . .
Proof. Letx∗∈ ∩∞i=1F(Ti). SinceTi, i= 1,2,3, . . .is generalized nonexpansive, we obtain
H(Tixn, Tix∗) ≤ a||xn−x∗||+b(d(xn, Tixn) +d(x∗, Tix∗)) +c(d(xn, Tix∗) +d(x∗, Tixn))
≤ a||xn−x∗||+b(||xn−x∗||+d(x∗, Tixn)) +c(d(xn, Tix∗) +d(x∗, Tixn))
≤ (a+b+c)||xn−x∗||+ (b+c)d(x∗, Tixn)
≤ (a+b+c)||xn−x∗||+ (b+c)H(Tix∗, Tixn).
Hence,
H(Tixn, Tix∗)≤ a+b+c 1−(b+c)
||xn−x∗||.
Since 1−(b+c)a+b+c ≤1, it follows that
H(Tixn, Tix∗)≤ ||xn−x∗||.
(3.2)
Then from (3.1) and (3.2), we have the following estimates,
||xn+1−x∗|| ≤αn0||xn−x∗||+
∞
X
i=1
αni||y(i)n −x∗||
≤ αn0||xn−x∗||+
∞
X
i=1
αnid(y(i)n , Tix∗)≤αn0||xn−x∗||+
∞
X
i=1
αniH(Tixn, Tix∗)
≤ αn0||xn−x∗||+
∞
X
i=1
αni||xn−x∗||=||xn−x∗||.
(3.3)
Thus, lim
n→∞||xn−x∗||exists. Now using Lemma 2.2, we obtain
||xn+1−x∗||2=
αn0xn+
∞
X
i=1
αniyn(i)−x∗
2
=
αn0(xn−x∗) +
∞
X
i=1
αni(yn(i)−x∗)
2
≤ αn0||xn−x∗||2+
∞
X
i=1
αni||y(i)n −x∗||2−αn0αnig(||xn−yn(i)||)
≤ αn0||xn−x∗||2+
∞
X
i=1
αni(d(y(i)n , Tix∗))2−αn0αnig(||xn−y(i)n ||)
≤ αn0||xn−x∗||2+
∞
X
i=1
αni(H(Tixn, Tix∗))2−αn0αnig(||xn−y(i)n ||)
≤ αn0||xn−x∗||2+
∞
X
i=1
αni||xn−x∗||2−αn0αnig(||xn−y(i)n ||)
= ||xn−x∗||2−αn0αnig(||xn−yn(i)||).
(3.4)
This implies that
0≤2g(||xn−y(i)n ||)≤αn0αnig(||xn−yn(i)||)≤ ||xn−x∗||2− ||xn+1−x∗||2. Hence lim
n→∞g(||xn−y(i)n ||) = 0.By property ofg, we have lim
n→∞||xn−yn(i)||= 0.Then d(xn, Tixn)≤ ||xn−y(i)n || →0, n→ ∞, i= 1,2,3, . . . .
This completes the proof.
Theorem 3.2. Let E be a uniformly convex real Banach space and D a nonempty, closed and convex subset of E. Let T1, T2, T3, . . . be multi-valued generalized nonexpansive maps of D intoCB(D) with ∩∞i=1F(Ti) 6=∅ for which Tip = {p}, ∀p ∈ ∩∞i=1F(Ti) and {Ti}∞i=1 satisfying condition (II). Let {αni}∞n=1, i = 0,1,2, . . . be a sequence in [,1−], ∈ (0,1) such that P∞
i=0αni = 1 for all n ≥1. Let {xn}∞n=1 be a sequence defined iteratively by (3.1). Then,{xn}∞n=1 converges strongly to a common fixed point of{Ti}∞i=1.
Proof. Since {Ti}∞i=1 satisfies condition (II), we have that d(xn,∩∞i=1F(Ti))→ 0 as n→ ∞. Thus, there is a subsequence {xnk} of{xn}and a sequence {pk} ⊂ ∩∞i=1F(Ti) such that
||xnk−pk||< 1 2k for allk. By Lemma 3.1, we obtain that
||xnk+1−pk|| ≤ ||xnk−pk||< 1 2k. We now show that {pk}is a Cauchy sequence in D. Observe that
||pk+1−pk|| ≤ ||pk+1−xnk+1||+||xnk+1−pk||
< 1 2k+1 + 1
2k < 1 2k−1.
This shows that {pk}is a Cauchy sequence in D and thus converges top∈D. Since d(pk, Tip)|| ≤H(Tipk, Tip)≤ ||pk−p||
andpk →pask→ ∞, it follows thatd(p, Tip) = 0 and thusp∈ ∩∞i=1F(Ti) and{xnk}converges strongly to p.
Since lim
n→∞||xn−p|| exists, it follows that{xn} converges strongly top. This completes the proof.
Theorem 3.3. Let E be a uniformly convex real Banach space and D a nonempty, closed and convex subset of E. Let T1, T2, T3, . . . be multi-valued generalized nonexpansive maps of D intoCB(D) with ∩∞i=1F(Ti) 6=∅ for which Tip = {p}, ∀p ∈ ∩∞i=1F(Ti) and Ti is hemicompact for some i ∈ N and Ti is continuous for each i= 1,2,3, . . .. Let{αni}∞n=1, i= 0,1,2, . . . be a sequence in[,1−], ∈(0,1) such that P∞
i=0αni= 1for all
n≥1. Let {xn}∞n=1 be a sequence defined iteratively by (3.1). Then, {xn}∞n=1 converges strongly to a common fixed point of {Ti}∞i=1.
Proof. Since lim
n→∞d(xn, Tixn) = 0, ∀i = 1,2,3, . . . and Ti is hemicompact for each i = 1,2,3, . . ., there is a subsequence {xnk} of {xn} such that xnk → p as k → ∞ for some p ∈ D. Since Ti is continuous for each i= 1,2,3, . . ., we haved(xnk, Tixnk)→d(p, Tip). As a result, we have thatd(p, Tip) = 0, ∀i= 1,2,3, . . . and so p∈ ∩∞i=1F(Ti). Since lim
n→∞||xn−p||exists, it follows that {xn}converges strongly top. This completes the proof.
Theorem 3.4. Let E be a uniformly convex real Banach space and D a nonempty compact convex subset of E. Let T1, T2, T3, . . . be multi-valued nonexpansive maps of D into CB(D) with ∩∞i=1F(Ti) 6= ∅ for which Tip = {p}, ∀p ∈ ∩∞i=1F(Ti). Let {αni}∞n=1, i = 0,1,2, . . . be a sequence in [,1−], ∈ (0,1) such that P∞
i=0αni= 1 for all n≥1. Let{xn}∞n=1 be a sequence defined iteratively by (3.1). Then, {xn}∞n=1 converges strongly to a common fixed point of{Ti}∞i=1.
Proof. From the compactness ofD, there exists a subsequence{xnk}∞n=kof{xn}∞n=1such that lim
k→∞||xnk−q||= 0 for someq∈D. Thus,
d(q, Tiq) ≤ ||xnk−q||+d(xnk, Tixnk) +H(Tixnk, Tiq)
≤ 2||xnk−q||+d(xnk, Tixnk)→0 ask→ ∞.
Hence,q ∈ ∩∞i=1F(Ti). Now, on takingq in place of x∗, we get that lim
n→∞||xn−q|| exists. This completes the
proof.
The following result gives a necessary and sufficient condition for strong convergence of the sequence in (3.1) to a common fixed point of an infinite family of multi-valued nonexpansive maps{Ti}∞i=1.
Theorem 3.5. LetD be a nonempty, closed and convex subset of a real Banach space E. LetT1, T2, T3, . . . ,be multi-valued nonexpansive maps of D intoCB(D) with ∩∞i=1F(Ti)6=∅ for which Tip={p}, ∀p∈ ∩∞i=1F(Ti).
Let {αni}∞n=1, i= 0,1, . . . , m be a sequence in [,1−], ∈ (0,1) such that P∞
i=0αni= 1 for alln ≥1. Let {xn}∞n=1 be a sequence defined iteratively by (3.1). Then, {xn}∞n=1 converges strongly to a common fixed point of {Ti}∞i=1 if and only iflim inf
n→∞d(xn, F) = 0.
Proof. The necessity is obvious. Conversely, suppose that lim inf
n→∞d(xn, F) = 0. By (3.3), we have
||xn+1−x∗|| ≤ ||xn−x∗||.
This gives
d(xn+1, F)≤d(xn, F).
Hence, lim
n→∞d(xn, F) exists. By hypothesis, lim inf
n→∞d(xn, F) = 0 so we must have lim
n→∞d(xn, F) = 0.
Next, we show that{xn}∞n=1 is a Cauchy sequence inD. Let >0 be given and since lim
n→∞d(xn, F) = 0, there existsn0 such that for alln≥n0, we have
d(xn, F)<
4.
In particular, inf{||xn0−p||:p∈F}<4, so that there must exist ap∗∈F such that
||xn0−p∗||<
2. Now, form, n≥n0, we have
||xn+m−xn|| ≤ |||xn+m−p∗||+||xn−p∗||
≤ 2||xn0−p∗||
< 2 2
=.
Hence,{xn}is a Cauchy sequence in a closed subsetD of a Banach spaceE and therefore, it must converge in D. Let lim
n→∞xn =p. Now, for eachi= 1,2,3, . . ., we obtain
d(p, Tip) ≤ d(p, xn) +d(xn, Tixn) +H(Tixn, Tip)
≤ d(p, xn) +d(xn, Tixn) +d(xn, p)→0 asn→ ∞
gives thatd(p, Tip) = 0, i= 1,2,3, . . .which implies thatp∈Tip. Consequently,p∈F =∩∞i=1F(Ti)6=∅.
Corollary 3.6. (Abbas et al.,[13]) LetE be a uniformly convex real Banach space satisfying Opial’s condition.
Let D be a nonempty, closed and convex ofE. LetT, S be multi-valued nonexpansive mappings ofD intoK(D) such thatF(T)∩F(S)6=∅. Let{an}∞n=1, {bn}∞n=1 and{cn}∞n=1 be sequence in(0,1)satisfyingan+bn+cn ≤1.
Let {xn}∞n=1 be a sequence defined iteratively by x1∈D
xn+1=anxn+bnyn+cnzn, ∀n≥1 (3.5)
whereyn ∈T xn, zn∈Sxn such that||yn−p|| ≤d(p, T xn)and||zn−p|| ≤d(p, Sxn)wheneverpis a fixed point of any one of mappings T andS. Then,{xn}∞n=1 converges weakly to a common fixed point ofF(T)∩F(S).
Corollary 3.7. (Abbas et al.,[13]) LetE be a real Banach space andD a nonempty, closed and convex subset of E. Let T, S be multi-valued nonexpansive mappings of D into K(D) such that F(T)∩F(S) 6= ∅. Let {an}∞n=1, {bn}∞n=1 and {cn}∞n=1 be sequence in (0,1) satisfying an+bn+cn ≤ 1. Let {xn}∞n=1 be a sequence defined iteratively by
x1∈D
xn+1=anxn+bnyn+cnzn, ∀n≥1 (3.6)
whereyn ∈T xn, zn∈Sxn such that||yn−p|| ≤d(p, T xn)and||zn−p|| ≤d(p, Sxn)wheneverpis a fixed point of any one of mappingsT andS. Then,{xn}∞n=1 converges strongly to a common fixed point ofF(T)∩F(S)if and only iflim inf
n→∞d(xn, F) = 0.
Remark 3.8. We observe that the iterative scheme (1.3) can be re-written as xn+1=αn0xn+
2
X
i=1
αniyn(i), (3.7)
where αn0 = αn, αn1 = bn, αn2 = cn, y(1)n = yn, yn(2) = zn. Furthermore, this is an iterative scheme for approximation of common fixed point of two multivalued nonexpansive mappings. Motivated by (3.7), we introduced our iterative scheme (3.1) for approximation of common fixed point of an infinite family of multi- valued generalized nonexpansive mappings. This is the proper justification of introducing our iteration scheme (3.1).
Remark 3.9. Our iterative scheme (3.1) reduces to the iterative scheme (1.2) when considering approximation of fixed point of a multivalued nonexpansive mapping. Furthermore, our iterative scheme (3.1) is more general than the Mann iterative scheme considered by Sastry and Babu [9] and Song and Wang [11].
Remark 3.10. Our iterative scheme (3.1) reduces to the iterative schemes (3.5) and (3.6) when considering approximation of common fixed point of two multivalued nonexpansive mappings.
Remark 3.11. Our results extend the results of Sastry and Babu [9], Panyanak [10] and Song and Wang [11]
from approximation of a fixed point of asingle multi-valued nonexpansive mappingto approximation of common fixed point of an infinite family of multi-valued generalized nonexpansive mappingsand Abbas et al. [13] from approximation of a common fixed point oftwo multi-valued nonexpansive mappingsto approximation of common fixed point ofan infinite family of multi-valued generalized nonexpansive mappings.
AcknowledgementsThe authors would like to express their sincere thanks to the anonymous referees for their valuable suggestions and comments which improved the original version of the manuscript greatly.
References
[1] J. T. Markin; A fixed point theorem for set valued mappings,Bull. Amer. Math. Soc.74(1968), 639-640.
[2] F. E. Browder; Nonlinear operators and nonlinear equations of evolution in Banach spaces, in : Nonlinear Functional Analysis (Proc. Sympos. PureMath., vol. 18, Part 2, Chicago, III., 1968), American Mathematical Society, Rhode Island, 1976, 1-308.
[3] E. L. Dozo; multi-valued nonexpansive mappings and Opial’s condition,Bull. Amer. Math. Soc.38(1973), 286-292.
[4] T. C. Lim; A fixed theorem for multi-valued nonexpansive mappings in a uniformly convex Banach space,Bull. Amer. Math.
Soc.80(1974), 1123-1126.
[5] W. A. Kirk, S. Massa; Remarks on asymptotic and Chebyshev centers ,Houston J. Math.16(1990), 357-364.
[6] S. B. Nadler Jr.; multi-valued contraction mappings,Pacific J. Math.30(1969), 475-487.
[7] W. R. Mann; Mean value methods in iteration,Proc. Amer. Math. Soc.4(1953), 506-510.
[8] A. Genel, J. Linderstrauss; An example concerning fixed points,Isreal. J. Math.22(1975), 81-86.
[9] K. P. R. Sastry, G. V. R. Babu; Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point,Czech. Math.
J.55(2005), 817-826.
[10] B. Panyanak; Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces, Comp. Math. Appl. 54 (2007), 872-877.
[11] Y. Song, H. Wang; Erratum to Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces,Comp.
Math. Appl.55(2008), 2999-3002.
[12] N. Shahzad, H. Zegeye; On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,Nonlinear Anal.71 (2009), 838-844.
[13] A. Abbas, S. H. Khan, A. R. Khan, R. P. Agarwal; Common fixed points of two multi-valued nonexpansive mappings by one-step iterative scheme,Appl. Math. Lett.24(2011), 97-102.
[14] X. Qin; Y. J. Cho; S. M. Kang; An iterative method for an infinite family of nonexpansive mappings in Hilbert spaces,Bull.
Malays. Math. Sci. Soc.32(2009), no 2, 161-171.
[15] C. Gang; B. Shangquan; Strong convergence Theorems for Variational Inequality Problems and Fixed Point Problems in Banach Spaces,Bull. Malays. Math. Sci. Soc.Accepted.
[16] J. Schu; Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,Bull. Austral. Math. Soc.43 (1991) 153-159.
[17] S. Chang, J. K. Kim, X. R. Wang; Modified Block iterative algorithm for solving convex feasibility problems in Banach spaces, J. Ineq. Appl.(2010), Article ID 869684,14 pages, doi:10.1155/2010/869684.
Department of Mathematics, University of Nigeria, Nsukka.
E-mail address:[email protected]; [email protected]
