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FIXED POINT APPROXIMATION OF GENERALIZED NONEXPANSIVE MULTI-VALUED MAPPINGS IN BANACH SPACES VIA NEW ITERATIVE ALGORITHMS.

1. INTRODUCTION

The fixed point theory of multi-valued nonexpansive mappings is relatively more involved and cumbersome than the corresponding theory of single-valued nonexpansive mappings. Fixed point theory for multi-valued mappings has many fruitful applications in diverse fields, e.g. game theory, mathematical economics and several others. Therefore, it is natural to extend the known fixed point results for single-valued mappings to multi-valued mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multi-valued mappings. The earliest results in this direction were respectively established by Markin [10] in Hilbert spaces while by Browder [4] for spaces admitting weakly continuous duality mapping. Dozo [5] generalized these results in a Banach space satisfying Opial's condition. Though nonexpansive mappings are most extensively studied class of mappings in metric fixed point theory, yet there also exists considerable literature on the classes of mappings enlarging the class of nonexpansive mappings.

Throughout the paper, E stands for a real Banach space with the norm [parallel]*[parallel] and K a nonempty subset of E. Let N denotes the set of all positive integers. Let CB(K), C(K) and P(K) denote the families of nonempty closed and bounded subsets, nonempty compact subsets and nonempty proximinal bounded subsets of K respectively. Recall that the set K is said to be proximinal if for any x [member of] E, there exists an element y [member of] K such that d(x,y) = dist(x,K), where dist(x,K) = inf{[parallel]x - y[parallel]; y [member of] K}.

Let H be the Hausdorff metric on CB(E) defined by

[mathematical expression not reproducible].

A multi-valued mapping T : K [right arrow] CB (E) is said to be nonexpansive if

H(Tx, Ty) [less than or equal to] [parallel]x - y[parallel] for all x, y [member of] K.

A point z [member of] K is called a fixed point of T if z [member of] Tz. As usual, F(T) stands for the set of fixed points of a multi-valued mapping T. A multi-valued mapping T : K [right arrow] CB(K) is said to be quasi-nonexpansive ([20]) if F(T) [not equal to] 0 and

H(Tx, Tz) [less than or equal to] [parallel]x - z[parallel] for all x [member of] K and z [member of] F(T).

As mentioned earlier, the study of fixed points for multi-valued nonexpansive mappings using the Hausdorff metric was initiated by Markin [10] while the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces can be found in Lim [9].

In 2008, Suzuki [23] defined a generalization of nonexpansive mapping and called it a mapping satisfying condition (C). Further, Garcla-Falset et al. [7] proposed two new generalizations of condition (C) and term them as condition (E) and condition ([C.sub.[lambda]]) and studied the existence of fixed points for these classes of mappings whose set-valued versions were studied in [1, 2, 8] whose relevant details can be described as follows:

Definition 1.1 ([8]). Let T : K [right arrow] CB(E) be a multi-valued mapping. Then T is said to satisfy condition ([C.sub.[lambda]]) if for some [lambda] [member of] (0,1) and for each x,y [member of] K

[lambda] dist(x,Tx) [less than or equal to] [parallel]x - y[parallel] [??] H(Tx, Ty) [less than or equal to] [parallel]x - y[parallel].

For [lambda] = 1/2 we recapture the class of mappings satisfying condition (C). It is easy to see that for 0 < [[lambda].sub.1] < [[lambda].sub.2] < 1, condition [mathematical expression not reproducible] implies condition [mathematical expression not reproducible].

Lemma 1.2 ([8]). Let T : K [right arrow] CB(E) be a multi-valued mapping.

(i) If T is nonexpansive, then T satisfies condition ([C.sub.[lambda]]).

(ii) If T satisfies condition ([C.sub.[lambda]]) and F(T) [not equal to] 0, then T is quasi-nonexpansive.

Lemma 1.3 ([6]). Let K be a nonempty subset of a Banach space E and T : K [right arrow] P(E) a multi-valued map satisfying condition (C). Then

H(Tx, Ty) [less than or equal to] 2 dist(x, Tx) + [parallel]x - y[parallel] for all x, y [member of] K.

Very recently, Abkar and Eslamian [2] used a modified Suzuki condition for multivalued mappings which runs as follows:

Definition 1.4 ([2]). A multi-valued mapping T : K [right arrow] CB(E) is said to satisfy condition ([E.sub.[mu]]) if for some [mu] [greater than or equal to] 1, for all x,y [member of] K

dist(x, Ty) [less than or equal to] [mu] dist(x, Tx) + [parallel]x - y[parallel].

We say that T satisfies condition (E) on K whenever T satisfies condition ([E.sub.[mu]]) for some [mu] [greater than or equal to] 1.

Lemma 1.5 ([2]). Let T : K [right arrow] CB(E) be a multi-valued nonexpansive mapping. Then T satisfies condition ([E.sub.1]).

In the sequel we need the following definitions:

Definition 1.6 ([11]). A Banach space E is said to satisfy Opial's condition if for any sequence {[x.sub.n]} in E with [x.sub.n] [??] x implies that

[mathematical expression not reproducible].

Examples of Banach spaces satisfying Opial's condition are Hilbert spaces and all [l.sub.p] spaces (1 < p < [infinity]). On the other hand, [L.sub.p][0, 2] with 1 < p [not equal to] 2 fail to satisfy Opial's condition.

Definition 1.7 ([21]). A multi-valued mapping T : K [right arrow] CB(K) is said to satisfy condition (I) if there exists a nondecreasing function h : [0, [infinity]) [right arrow] [0, [infinity]) with h(0) = 0 and h(r) > 0 for all r [member of] (0, to) such that dist(x, Tx) [greater than or equal to] h(dist(x, F(T)) for all x [member of] K.

Definition 1.8. Multi-valued mappings T,S : K [right arrow] CB(K) are said to satisfy condition (I') if there exists a nondecreasing function g : [0, [infinity]) [right arrow] [0, [infinity]) with g(0) = 0 and g(r) > 0 for all r [member of] (0, [infinity]) such that either dist(x, Tx) [greater than or equal to] g(dist(x, F)) or dist(x, Sx) [greater than or equal to] g(dist(x, F)) for all x [member of] K where F = F(T) [integral] F(S).

Remark 1.9. With S = T in Definition 1.8, condition (I') reduces to condition (I).

Definition 1.10 ([5], [7]). A multi-valued mapping T : K [right arrow] P(E) is said to be demiclosed at y [member of] K if for any sequence {[x.sub.n]} in K weakly convergent to an element x and [y.sub.n] [member of] [Tx.sub.n] strongly convergent to y, we have y [member of] Tx.

Definition 1.11 ([3]). Let [{[a.sub.n]}.sub.n [member of] N] and [{[b.sub.n]}.sub.n[member of]N] be two sequences of real numbers that converge to a and b, respectively. Assume that there exists

[mathematical expression not reproducible].

If l = 0, then we say that [{[a.sub.n]}.sub.n[member of]N] converges to a faster than [{[b.sub.n]}.sub.n [member of] N] to b.

An important property for the class of uniformly convex Banach spaces is contained in following lemma due to Schu [15].

Lemma 1.12 ([15]). Let E be a uniformly convex Banach space and 0 < p [less than or equal to] [t.sub.n] [less than or equal to] q < 1 for all n [member of] N. If {[x.sub.n]} and {[y.sub.n]} are two sequences of E such that [lim.sub.n[right arrow][infinity]] sup [parallel][x.sub.n][parallel] [less than or equal to] [mathematical expression not reproducible] hold for some r [greater than or equal to] 0, then [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - [y.sub.n][parallel] = 0.

The purpose of this paper is to approximate the fixed points of generalized nonexpansive multi-valued mappings in Banach spaces via new iterative algorithms and establish weak and strong convergence theorems of these iterative algorithms under suitable conditions. For further details one can be referred to [16]-[19].

2. PRELIMINARIES

Different iterative schemes have been utilized to approximate the fixed points of multi-valued nonexpansive mappings. Sastry and Babu [14] studied the Mann and Ishikawa iterative schemes for multi-valued mappings and proved that these schemes for a multi-valued map T with a fixed point z converges to a fixed point q of T under certain conditions. They also claimed that the fixed point q may be different from z. To describe some relevant iterative processes, let K be a nonempty convex subset of E and T : K [right arrow] P(K) a multi-valued mapping with z [member of] Tz. Then, the sequence of Mann iterates is defined by with [u.sub.1] [member of] K,

(2.1) [u.sub.n+1] (1 - [a.sub.n])[u.sub.n] + [a.sub.n][t.sub.n], n [member of] N)

where [t.sub.n] [member of] T[u.sub.n] is such that [parallel][t.sub.n] - z[parallel] = dist(z, T[u.sub.n]) and {[a.sub.n]} is a sequence of numbers in (0,1) satisfying [lim.sub.n[right arrow][infinity]] [a.sub.n] = 0 and [summation][a.sub.n] = [infinity].

The sequence of Ishikawa iterates is defined by [v.sub.1] [member of] K,

[mathematical expression not reproducible],

where [u.sub.n] [member of] T[q.sub.n], [t'.sub.n] [member of] T[v.sub.n] are such that [parallel][u.sub.n] - z[parallel] = dist(z, T[q.sub.n]) and [parallel][t'.sub.n] - z[parallel] = dist(z, T[v.sub.n]) and {[a.sub.n]}, {[b.sub.n]} are real sequences of numbers in (0,1) satisfying [lim.sub.n[right arrow][infinity]] [b.sub.n] = 0 and [summation][a.sub.n][b.sub.n] = [infinity].

Panyanak [12] extended the result of Sastry and Babu [14] by modifying the iteration schemes of Sastry and Babu [14] in the setting of uniformly convex Banach spaces but the domain of T remains compact while Song and Wang [21] employed the condition Tz = {z} to prove their results.

Recently, Sahu [13] introduced an iterative scheme, which has been studied extensively in connection with fixed points of single-valued nonexpansive mappings as follows: Let K be a nonempty convex subset of E and f : K [right arrow] K a single-valued mapping. Then, for arbitrary [w.sub.1] [member of] K, the iterative process is defined by

[mathematical expression not reproducible],

where {[a.sub.n]} [member of] (0,1).

In the following, we extend the above iterative scheme to the case of multi- valued nonexpansive mappings on convex subset of E modifying the above ones. Let K be a nonempty convex subset of E and T : K [right arrow] P(K) a multi-valued mapping with z [member of] Tz. Then, the sequence of iterates is defined by

(2.2) [mathematical expression not reproducible]

where [v.sub.n] [member of] T[s.sub.n], [z.sub.n] [less than or equal to] T[w.sub.n] are such that [parallel][v.sub.n] - z[parallel] = dist(z, [Tu.sub.n]) and [parallel][z.sub.n] - z[parallel] = dist(z, T[w.sub.n]) and {[a.sub.n]} is a sequence of numbers in (0,1) satisfying [lim.sub.n[right arrow][infinity]] [a.sub.n] = 0 and [summation] [a.sub.n] < [infinity].

Motivated and inspired by the work of Sahu [13], we introduced a new iterative scheme in the context of multi-valued mappings as follows:

(2.3) [mathematical expression not reproducible],

where [u.sub.n] [member of] T[y.sub.n], [v.sub.n] [member of] T[x.sub.n] and [w.sub.n] [member of] S[x.sub.n] are such that [parallel][v.sub.n] - z[parallel] = dist(z,T[x.sub.n]), [mathematical expression not reproducible] is a sequence of numbers in (0,1) satisfying [lim.sub.n[right arrow][infinity]] [a.sub.n] = 0 and [summation] [a.sub.n] < [infinity].

3. CONVERGENCE THEOREMS VIA ALGORITHM (2.2)

In this section we prove some weak and strong convergence theorems by approximating the fixed points of a multi-valued quasi-nonexpansive mapping and generalized nonexpansive multi-valued mappings by using iterative scheme (2.2). In the sequel, F(T) denotes the set of fixed point of mapping T.

Theorem 3.1. Let E be a uniformly convex Banach space satisfying Opial's condition and K a nonempty closed and convex subset of E. Let T : K [right arrow] P(K) be a multi-valued quasi-nonexpansive mapping and {[w.sub.n]} a sequence as defined by (2.2). If F(T) [not equal to] 0 and (I - T) is demiclosed at zero, then {[w.sub.n]} converges weakly to a fixed point of T.

Proof. Let z [member of] F (T). Hence from (2.2) we have

(3.1) [mathematical expression not reproducible],

(3.2) [less than or equal to] [parallel][w.sub.n] - z[parallel].

Hence from (3.1) and (3.2), we have

(3.3) [parallel][w.sub.n+1] - z[parallel] [less than or equal to] [parallel][w.sub.n] - z[parallel].

Therefore, [lim.sub.n[right arrow][infinity]] [parallel][w.sub.n] - z[parallel] exists for each z [member of] F(T). Let [lim.sub.n[right arrow][infinity]] [parallel][w.sub.n] - z[parallel] = a for some a [greater than or equal to] 0. Then if a = 0, we are done. Suppose that a > 0. Next, we show that [lim.sub.n[right arrow][infinity]] dist(T[w.sub.n], [w.sub.n]) = 0. Taking lim sup on both sides of (3.2), we have

(3.4) [mathematical expression not reproducible].

As,

(3.5) [mathematical expression not reproducible].

Moreover, [lim.sub.n[right arrow][infinity]] [parallel][w.sub.n+1] - z[parallel] = a means that

(3.6) [mathematical expression not reproducible].

From (3.4) and (3.6), we have

[mathematical expression not reproducible].

As,

(3.7) [mathematical expression not reproducible].

Therefore from (3.5), (3.7), and Lemma 1.12, we have

(3.8) [mathematical expression not reproducible].

As dist(T[w.sub.n], [w.sub.n]) [less than or equal to] [parallel][z.sub.n] - [w.sub.n][parallel], we have, [lim.sub.n[right arrow][infinity]] dist(T[w.sub.n], [w.sub.n]) = 0. Now, we prove that {[w.sub.n]} has a unique weak subsequential limit in F(T). Let [mathematical expression not reproducible] be the subsequences of {[w.sub.n]} while [z.sub.1] and [z.sub.2] be the weak limits of [mathematical expression not reproducible] respectively. Since (I - T) is demiclosed at zero, therefore using the fact [z.sub.n] [member of] T[w.sub.n] and equation (3.8), we obtain that [z.sub.1] [member of] F(T). Similarly we can show that [z.sub.2] [member of] F(T). Now, we show the uniqueness of weak limit. Let us suppose that [z.sub.1] [not equal to] [z.sub.2]. Since [mathematical expression not reproducible], by Opial's condition, we have

[mathematical expression not reproducible],

which is a contradiction. Hence {[w.sub.n]} converges weakly to a fixed point of T.

Now, we prove some strong convergence theorems involving generalized nonexpansive multi-valued mapping.

Theorem 3.2. Let E be a Banach space and K a nonempty closed convex subset of E. Let T : K [right arrow] P(K) be a multi-valued quasi-nonexpansive mapping and satisfies condition (E). Let {[w.sub.n]} be a sequence as defined by (2.2). If F(T) [not equal to] 0, then {[w.sub.n]} converges strongly to a fixed point of T if and only if [lim.sub.n[right arrow][infinity]] inf dist([w.sub.n], F(T)) = 0.

Proof. The necessary part is evident. For the reverse part, let us suppose that

[mathematical expression not reproducible].

Then by (3.3), we have

[mathematical expression not reproducible],

which implies that [lim.sub.n[right arrow][infinity]] dist([w.sub.n], F(T)) exists. Therefore by hypothesis, we have [lim.sub.n[right arrow][infinity]] dist([w.sub.n], F(T)) = 0. Now, we show that {[w.sub.n]} is a Cauchy sequence in K. Let [epsilon] > 0. As [lim.sub.n[right arrow][infinity]] dist([w.sub.n], F(T)) = 0, there exists a positive integer m such that for all n [greater than or equal to] m, we have dist([w.sub.n], F(T)) < [epsilon]/4. In particular,

inf{[parallel][w.sub.m] - z[parallel]: z [member of] F(T)} < [epsilon]/4.

Therefore there exists l [member of] F(T) such that [parallel][w.sub.m] - l[parallel] < [epsilon]/2. Now for n,p [greater than or equal to] m, we have

[mathematical expression not reproducible].

Hence {[w.sub.n]} is a Cauchy sequence in a closed subset K of a Banach space E. Therefore it converges in K. Let [lim.sub.n[right arrow][infinity]] [w.sub.n] = w. Then by using condition (E), we have

[mathematical expression not reproducible].

Thus dist(w, Tw) = 0, which in turn implies that w [member of] F(T).

Theorem 3.3. Theorem 3.2 also holds if condition (E) is replaced by condition (C).

Proof. From Theorem 3.2, we conclude that the sequence {[w.sub.n]} converges to w [member of] K. Hence, by using condition (C) and Lemma 1.3, we have

[mathematical expression not reproducible] (by using (3.8)).

Therefore dist(w, Tw) = 0, which in turn implies that w [member of] F(T).

We now apply Theorem 3.2 to obtain our next result in a uniformly convex Banach space wherein T satisfies condition (I).

Theorem 3.4. Let E be a uniformly convex Banach space and K, T, F(T) and {[w.sub.n]} be as in Theorem 3.2. If T satisfies condition (I) and F(T) [not equal to] 0, then {[w.sub.n]} converges strongly to a fixed point of T.

Proof. By Theorem 3.1, [lim.sub.n[right arrow][infinity]] [parallel][w.sub.n] - z[parallel] exists for all z [member of] F(T). Let [lim.sub.n[right arrow][infinity]] [parallel][w.sub.n] - z[parallel] = a, for some a [greater than or equal to] 0. If a = 0, then there is nothing to prove. Suppose a > 0. Then again from Theorem 3.1, [parallel][w.sub.n+1] - z[parallel] [less than or equal to] [parallel][w.sub.n] - z[parallel], which implies that, [mathematical expression not reproducible], so that dist([w.sub.n+1], F(T)) [less than or equal to] dist([w.sub.n], F(T)) and [lim.sub.n[right arrow][infinity]] dist([w.sub.n], F(T)) exists. On using condition (I) and Theorem 3.1, we have,

[mathematical expression not reproducible],

that is, [lim.sub.n[right arrow][infinity]] h(dist([w.sub.n], F(T))) = 0. Since h is a nondecreasing function and h(0) = 0, it follows that [lim.sub.n[right arrow][infinity]] dist([w.sub.n], F(T)) = 0. Now applying Theorem 3.2, we obtain the result.

Theorem 3.5. Theorem 3.4 also holds if condition (E) is replaced by condition (C). Proof. The proof of this theorem is same as that of Theorem 3.4.

4. CONVERGENCE THEOREMS VIA ALGORITHM (2.3)

We start this section to approximate the common fixed points of generalized nonexpansive multi-valued mappings to prove some weak and strong convergence theorems by using iterative algorithm (2.3). In the sequel, F = F(T) [intersection] F(S) denotes the set of common fixed points of mappings T and S.

Lemma 4.1. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and T, S : K [right arrow] P(K) two multi-valued quasi- nonexpansive mappings. Let {[x.sub.n]} be a sequence as defined by (2.3). If F [not equal to] 0 and

dist(x, Tx) [less than or equal to] H(Tx,Sx) [for all]x [member of] K,

then [lim.sub.n[right arrow][infinity]] , dist(T[x.sub.n], [x.sub.n]) = 0 = [lim.sub.n[right arrow][infinity]] dist(S[x.sub.n], [X.sub.n]).

Proof. Let z [member of] F. Then from (2.3) we have

(3.9) [mathematical expression not reproducible],

and

[mathematical expression not reproducible]

(3.10) = [parallel][x.sub.n] - z[parallel].

Hence from (3.9) and (3.10), we have

(3.11) [parallel][x.sub.n+1] - z[parallel] [less than or equal to] [parallel][x.sub.n] - z[parallel].

Therefore, [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - z[parallel] exists for each z [member of] F(T). Assume that [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - z[parallel] = b for some b [greater than or equal to] 0. Then if b = 0, we are done. Suppose that b > 0. Next, we show that [lim.sub.n[right arrow][infinity]] dist(T[x.sub.n], [x.sub.n]) = 0. Taking limit as n [right arrow] [infinity] on both sides of (3.10), we have

(3.12) [mathematical expression not reproducible].

Moreover, as [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n+1] - z[parallel] = b, from (3.9) we have

(3.13) [mathematical expression not reproducible].

From (3.12) and (3.13), we have

[mathematical expression not reproducible].

As,

(3.14) [mathematical expression not reproducible].

In the same way, we get

(3.15) [mathematical expression not reproducible].

As,

[mathematical expression not reproducible],

hence by using (3.14), (3.15) and by applying Lemma 1.12, we get,

(3.16) [mathematical expression not reproducible].

As H(T[x.sub.n], S[x.sub.n]) [less than or equal to] [parallel][v.sub.n] - [w.sub.n][parallel], we have [lim.sub.n[right arrow][infinity]] H(T[x.sub.n], S[x.sub.n]) = 0. Now, we have

dist (T[x.sub.n], [x.sub.n]) [less than or equal to] H (T[x.sub.n], S[x.sub.n]).

Taking limit as n [right arrow] [infinity] on both sides, we get [lim.sub.n[right arrow][infinity]] dist(T[x.sub.n], [x.sub.n]) = 0. Again, we have for each n [member of] N,

dist(S[x.sub.n], [x.sub.n]) < H(S[x.sub.n], T[x.sub.n]) + dist(T[x.sub.n], [x.sub.n]),

which on taking limit as n [right arrow] [infinity] follows, [lim.sub.n[right arrow][infinity]] dist(S[x.sub.n], [x.sub.n]) =0.

Remark 4.2. It is well known that every nonexpansive mapping satisfies conditions (C) and (E). Hence for the sake of simplicity, we present the following example of nonexpansive mappings with the common non empty fixed point set and satisfies the inequality

dist(x,Tx) [less than or equal to] H(Tx, Sx) [for all]x [member of] K.

Example 4.3. Let E = R and K = [1, to). Let us define the mappings T and S by T, S : K [right arrow] CB(K) by

Tx = [0, 1+x/2], Sx = [0, 5-2x/3] [for all]x [member of] K.

Then obviously, S and T are nonexpansive mappings with the common fixed points 0 and 1 as follows: for x, y [member of] K,

[mathematical expression not reproducible].

In a similar way,

[mathematical expression not reproducible].

Now, for any x, y [member of] K,

[mathematical expression not reproducible],

and

[mathematical expression not reproducible],

that is, for all x [member of] K,

dist(x, Tx) [less than or equal to] H(Tx, Sx).

Theorem 4.4. Let E be a uniformly convex Banach space satisfying Opial's condition and K,T,S and {[x.sub.n]} be same as in Lemma 4.1. If F [not equal to] 0, (I - T) and (I - S) are demiclosed at zero, then {[x.sub.n]} converges weakly to a common fixed point of T and S.

Proof. Let z [member of] F. Then as proved in Lemma 4.1, [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - z[parallel] exists. Since E is a uniformly convex Banach space. Thus there exists a subsequence [mathematical expression not reproducible] such that [mathematical expression not reproducible] converges weakly to [z.sub.1] [member of] K. From Lemma 4.1, we have [mathematical expression not reproducible]. Since (I - T) and (I - S) are demiclosed at zero, therefore S[z.sub.1] = [z.sub.1]. Similarly T[z.sub.1] = [z.sub.1]. Finally, we prove that {[x.sub.n]} converges weakly to [z.sub.1]. Let on contrary that there exists a subsequence [mathematical expression not reproducible] such that [mathematical expression not reproducible] converges weakly to [z.sub.2] [member of] K and [z.sub.1] [not equal to] [z.sub.2]. Again in the same way, we can prove that [z.sub.2] [member of] F. Again from Lemma 4.1, [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - [z.sub.1][parallel] and [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - [z.sub.2][parallel] exist. Let [z.sub.1] [not equal to] [z.sub.2]. Then by Opial's condition, we have

[mathematical expression not reproducible],

which is a contradiction. Hence {[x.sub.n]} converges weakly to a common fixed point of T and S.

Theorem 4.5. Let E be a Banach space and K a nonempty closed and convex subset of E. Let T, S : K [right arrow] P(K) be two multi-valued quasi-nonexpansive mappings satisfying condition (E). Let {[x.sub.n]} be a sequence as defined by (2.3). If F [not equal to] 0, then {[x.sub.n]} converges strongly to a common fixed point of T and S if and only if [lim.sub.n[right arrow][infinity]] inf dist([x.sub.n], F) = 0.

Proof. The first part is obvious. Let us suppose that [lim.sub.n[right arrow][infinity]] inf dist([x.sub.n], F) = 0. Then from (3.11), we have

[mathematical expression not reproducible],

which implies that [lim.sub.n[right arrow][infinity]] dist([x.sub.n], F) exists. Then on the similar lines of proof of Theorem 3.2, we can say that {[x.sub.n]} converges in K. Let [lim.sub.n[right arrow][infinity]] [x.sub.n] = x. Then by using condition (E) and Lemma 4.1, we have

[mathematical expression not reproducible],

that is, dist(x, Tx) = 0 that is, x G F(T). Similarly, by using condition (E), we have

dist(x, Sx) [less than or equal to] 2[parallel][x.sub.n] - x[parallel] + [mu] dist([x.sub.n], S[x.sub.n]).

On taking limit n [right arrow] [infinity], we have dist(x, Sx) = 0, as from Lemma 4.1, [lim.sub.n[right arrow][infinity]] dist([x.sub.n], S[x.sub.n]) = 0. Therefore we have x [member of] F(S), which implies that x [member of] F(T) [intersection] F(S) = F.

Theorem 4.6. Theorem 4.5 also holds if condition (E) is replaced by condition (C).

Proof. From Theorem 4.5, we conclude that the sequence {[x.sub.n]} converges to x [member of] K. Hence, by using condition (C) and Lemma 1.3, we have

[mathematical expression not reproducible] (by Lemma 4.1).

Therefore dist(x, Tx) = 0, that is, x [member of] F(T). Similarly, we have dist(x, Sx) = 0, that is, x [member of] F(S), which implies that x [member of] F(T) [intersection] F(S) = F.

Now by using Theorem 4.5, we obtain a strong convergence theorem of the iterative scheme (2.2) under condition (I').

Theorem 4.7. Let E be a uniformly convex Banach space and K,T,S,F and {[x.sub.n]} be as in Theorem 4.5. If mappings T and S satisfy condition (I') and F [not equal to] 0, then {[x.sub.n]} converges strongly to a common fixed point of T and S.

Proof. By Lemma 4.1, [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - z[parallel] exists for all z [member of] F. Let [lim.sub.n[right arrow][infinity]] [parallel][x.sub.n] - z[parallel] = b, for some b [greater than or equal to] 0. If b = 0, then there is nothing to prove. Suppose that b > 0. Then again from Lemma 4.1, [parallel][x.sub.n+1] - z[parallel] [less than or equal to] [parallel][x.sub.n] - z[parallel], which implies that, [inf.sub.z [member of] F] [parallel][x.sub.n+1] - z[parallel] [less than or equal to] [inf.sub.z[member of]F] [parallel][x.sub.n] - z[parallel], so that dist([x.sub.n+1], F) [less than or equal to] dist([x.sub.n], F) and [lim.sub.n[right arrow][infinity]] dist([x.sub.n], F) exists. On using condition (I') and Lemma 4.1, we have,

[mathematical expression not reproducible],

that is,

[mathematical expression not reproducible].

Since g is a nondecreasing function and g(0) = 0, it follows that [lim.sub.n[right arrow][infinity]] dist([x.sub.n], F) = 0. Now applying Theorem 4.5, we obtain the result.

Remark 4.8. The rate of convergence of iterative algorithm (2.3) is faster than iterative algorithms (2.1) and (2.2) for contraction mappings as shown by the given proposition.

Proposition 4.9. Let K be a nonempty closed and convex subset of a Banach space E. Let T, S : K [right arrow] P(K) be multi-valued contraction mappings with Lipschitz constants [k.sub.1] and [k.sup.2] respectively where [k.sub.1], [k.sub.2] < k < 1, and a unique fixed point z. Define sequences {[u.sub.n]}, {[w.sub.n]} and {[x.sub.n]} in K by (2.1), (2.2) and (2.3) respectively. Then we have the following:

1. [parallel][u.sub.n+1] - z[parallel] [less than or equal to] [[1 - [a.sub.n](1 - k)].sup.n] [parallel][u.sub.1] - z[parallel] for all n [member of] N,

2. [mathematical expression not reproducible] for all n [member of] N,

3. [parallel][x.sub.n+1] - z[parallel] [less than or equal to] [k.sup.2n] [parallel][x.sub.1] - z[parallel] for all n [member of] N.

Proof. Suppose that z is a common fixed point of mappings T and S. Then from iterative algorithm (2.1), we have

[mathematical expression not reproducible].

Let [A.sub.n] = [[1 - (1 - k)[a.sub.n]].sup.n] [parallel][u.sub.1] - z[parallel].

Now, from iterative algorithm (2.2), we have

[mathematical expression not reproducible].

Assume that [B.sub.n] = [k.sup.n] [[1 - (1 - k)[a.sub.n]].sup.n] [parallel][w.sub.1] - z[parallel].

By iterative algorithm (2.3), we have

[mathematical expression not reproducible].

Assume that [C.sub.n] = [k.sup.2n] [parallel][x.sub.1] - z[parallel].

Now,

[mathematical expression not reproducible].

Since k < 1, [lim.sub.n[right arrow][infinity]] [k.sup.2n] = 0 and as [a.sub.n] < 1 with [lim.sub.n[right arrow][infinity]] [a.sub.n] = 0, we have [lim.sub.n[right arrow][infinity]] [C.sub.n]/[A.sub.n] = 0. Thus {[x.sub.n]} converges faster than {[u.sub.n]} to z. Similarly,

[mathematical expression not reproducible],

so that [lim.sub.n[right arrow][infinity]] [C.sub.n]/[B.sub.n] = 0. Therefore {[x.sub.n]} converges faster than {[w.sub.n]} to z.

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ANUPAM SHARMA (a) * AND MOHAMMAD IMDAD (b)

(a) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208 016, India anupam@iitk.ac.in

(b) Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India mhimdad@gmail.com
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