# Approximation of fixed points of strongly successively pseudocontractive maps in a Banach space.

1. Introduction

Let X be a real Banach space.

Definition 1.1. [1] A mapping T with domain D(T) and range R(T) in X is said to be strongly successively pseudocontractive if there exists a constant 0 < k < 1 and [n.sub.0] [member of] N such that

[parallel]x - y[parallel] < [parallel]x - y + r[(I - [T.sup.n] - kI)x - (I - [T.sup.n] - kI)y] [parallel] (1.1) for all x, y [member of] D(T) and n [greater than or equal to] n0.

Definition 1.2. [1] A mapping T with domain D(T) and range R(T) in X is said to be uniformly Lipschitzian if there exists L> 0 such that

[parallel][T.sup.n]x - [T.sup.n] y[parallel] [greater than or equal to] L[parallel]x - y[parallel] (1.2)

for all x, y [member of] D(T) and n [greater than or equal to] [n.sub.0].

Remark 1.3. If we set n = [n.sub.0] = 1 in (1.1), we get the definition of a strongly pseudocontractive map. An example of a map which is not strongly pseudocontractive but which is strongly successively pseudocontractive can be found in [1].

One of the effective methods for approximating fixed points of an operator T : D(T)[subset] X [right arrow] X is the Mann iteration process [10], starting with arbitrary [x.sub.0] [member of] D(T) and for n [greater than or equal to] 0 defined by

[x.sub.n+1] = (1 - [[alpha].sub.n])[x.sub.n] + [[alpha].sub.n] T [x.sub.n] (1.3)

where [[alpha].sub.n] [member of][0, 1] satisfy suitable conditions.

In 1998, Xu [8] introduced the Mann iterative process with errors defined by

[x.sub.0] [member of] D(T)

[x.sub.n+1] = [[alpha].sub.n][x.sub.n] + [b.sub.n]T[x.sub.n] + [C.sub.n][U.sub.n], n [greater than or equal to] 0 (1.4)

where {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are sequences in [0,1] such that [a.sub.n] + [b.sub.n] + [c.sub.n] = 1 and {[u.sub.n]} is a bounded sequence in D(T). Clearly, this iteration process with errors has (1.3) as its special case. Furthermore, this Mann iterative process is the same as the one introduced by Liu [12] if [c.sub.n] = 1 for all n. However, due to the defects of the one introduced by Liu [12] as pointed out in [], we are interested in Xu's more general Mann iterative process of (1.4).

Now, we consider the following iteration introduced by Schu [9] for approximation of fixed points of Lipschitz pseudocontractive maps.

[x.sub.n+1] = [a.sub.n][x.sub.n] + [b.sub.n][T.sup.n][x.sub.n] + [c.sub.n],[u.sub.n] [greater than or equal to] 0 (1.5)

This iteration is known as modified Mann iteration with errors, where {un}e X is a bounded sequence and are error terms and {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are real sequences in [0,1] satisfying some conditions. Replacing Tn by T in (1.5) one obtains Mann iteration with errors in (1.4).

It is easy to see that if [c.sub.n] = 0, then (1.5) reduces to

[x.sub.n+1] = (1 - [b.sub.n])[x.sub.n] + [b.sub.n][T.sup.n][x.sub.n] (1.5a)

where {[b.sub.n]} is a real number in [0,1].

Liu [12] proved that the Mann iteration process defined by (1.3) converges strongly to the unique fixed point of a Lipschitzian strictly pseudocontractive mapping. Sastry and Babu [5] proved that any fixed point of a Lipschitzian pseudocontractive self mapping of a non-empty closed convex subset K of a Banach space X is unique and may be norm approximated by Mann iterative procedure (1.3). Their result generalized the result of Liu [12] in the sense that, the assumption that K is bounded was removed and a general convergence rate estimate was provided. Recently, Rafiq [7] extended the above results to the Mann iteration sequence with errors (1.4) in a real Banach space.

In this paper, our purpose is to show that the more general modified Mann iteration sequence with errors converges to the unique fixed point of T if T : X [right arrow] X is a uniformly continuous strongly successively pseudocontractive mapping with a bounded range or T : X [right arrow] X is uniformly Lipschitzian and strongly successively pseudocontractive mapping without necessarily having a bounded range. Our results extend and improve the results of Rafiq [7], Liu [12], Sastry and Babu [5] Mogbademu et al. [11]. Furthermore, we are able to obtain a more general and better convergence rate than those obtained by Liu [12], Sastry and Babu [5], mogbademu et al [11] and the very recent estimate of Ciric et al. [3].

The main results of Liu, Sastry and Babu are the following.

Theorem 1.4. (Liu) Let X be a Banach space, and let K be a nonempty closed convex and bounded subset of X. Let T : K [right arrow] K be a Lipschitzian strictly pseudocontractive mapping. if F(T) [not equal to] [phi] then {[x.sub.n]}[subset] K generated by [x.sub.1] [member of] K

[X.sub.n+1] = (1 - [[alpha].sub.n])[X.sub.n] + [[alpha].sub.n]T[x.sub.n] (1.3a)

with {[a.sub.n]}[subset] (0, 1] satisfying [[infinity].summation over (n = 1) = [infinity],[[alpha].sub.n] [right arrow] 0, strongly converges to q [member of] F(T) and F(T) is a single set.

Theorem 1.5. (Liu) Let K and T be as in Theorem 1.1. If [[alpha].sub.n] = k/2(3 + 3L + [L.sup.2]),

where k = t - 1/t, {q} = F(T), then the sequence {[x.sub.n]} generated by (1.3a) converges strongly to the unique fixed point of T, and we have the estimate

[parallel][x.sub.n+1] - q[parallel] < [[rho].sup.n] [parallel][x.sub.1] - q[parallel]

where [rho] = 1 - [k.sub.2]/4(3 +3L + [L.sub.2]).

Theorem 1.6. (Sastry and Babu) Let (X, [parallel].[parallel]), K, T, L and k be as described above.

Let q [member of] K be a fixed point of T. Suppose that [{[[alpha].sub.n]}.sub.n[member of]N] is a sequence in (0, 1] such that for some [eta] [member of] (0,k), for all n [member of] N, [[alpha].sub.n] [less than or equal to] k - [eta]/(L + 1)(L + 2 - k) while [[infinity].summation over (n = 1) [[alpha].sub.n] = [infinity] Fix [x.sub.1] [member of] K. Define for all n [member of] N

[X.sub.n+1] = (1 - [[alpha].sub.n])[x.sub.n] + [[alpha].sub.n]T[x.sub.n]

then there exists [{[[beta].sub.n]}.sub.n[member of]N], a sequence in (0, 1) with each [[beta].sub.n] [greater than or equal to] (n/1 + k) an such that

for all n [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, [{[x.sub.n]}.sub.n[member of]N] converges strongly to q and q is the unique fixed point of T.

Recently, Ciric et al studied the Mann implicit iteration sequence for strongly accretive and strongly pseudocontractive mappings. They showed that this implicit scheme gives better convergence rate estimate, improving Theorem 1.4, Theorem 1.5 and theorem 1.6.

Theorem 1.7. (Ciric et al) Let E be an arbitrary real Banach space, A : E [right arrow] E be a Lipschitz and strongly accretive map with strong accretive constant k [member of] (0, 1). Let [x.sup.*] denote a solution of the equation Ax = 0. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each x, y [member of] E. For arbitrary [x.sub.0] [member of] E, define the sequence [{[x.sub.n]}.sup.[infinity].sub.n = 1] in E by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {an} is the sequence in (0, 1] satisfying [[alpha].sub.n] [greater than or equal to](2 - k)L/(2 - k)L + k - [eta], [eta] [member of] (0,k) and

[[infinity].summation over (n = 1)](1 - [[alpha].sub.n]), then [{[x.sub.n]}.sup.[infinity].sub.n=0] converges strongly to [x.sup.*] and [x.sup.*] is the unique solution of the equation Ax = 0. Moreover, if [[alpha].sub.n] = (2 k)L/(2 k)L + k - [eta], [eta] [member of] (0,k), then we have the following geometric convergence rate estimate for [{[x.sub.n]}.sup.[infinity].sub.n = 1],

[parallel][X.sub.n] - [X.sup.*] [parallel] [member of] [[rho].sup.n] [parallel][X.sub.0] - [x.sub.*] [parallel]

where

[rho] = 1 - k - [eta]/(2 - k)L + 2(k - [eta]) [eta]

. Thus, the choice [eta] = k/2 yields

[rho] = 1 - [k.sup.2]/4[k + (2 - k)L]

Theorem 1.8. (Ciric et al) Let E be an arbitrary real Banach space, C [subset or equal to] E be nonempty and convex. Let T : C [right arrow] C be Lipschitz (with constant L > 0) and strongly pseudocontractive. Assume that T has a fixed point [x.sup.*] [member of] C.

Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {[[alpha].sub.n]} is a sequence in (0, 1] satisfying [[alpha].sub.n] [greater than or equal to] (2 - k)L /(2 - k)L + (k - [eta]), [eta] [member of] (0,k) and

[[infinity].summation over (n = 1)] (1 - [[alpha].sub.n]) = [infinity]. Then [{[x.sub.n]}.sup.[infinity].sub.n=0] converges strongly to [x.sup.*] and [x.sup.*] is the unique solution

of the equation Ax = 0. Moreover, if

[[alpha].sub.n] = (2 - k)L/(2 - k)L + k - [eta], [eta] = k/2

then we have the following geometric convergence rate estimate for. [{[x.sub.n]}.sup.[infinity].sub.n=1]

[parallel][x.sub.n] - [x.sup.*][parallel] [less than or equal to][[rho].sup.n] [parallel][X.sub.0] - [x.sup.*][parallel]

where

[rho] = 1 - [k.sup.2]/4[k + (2 - k)L].

Lemma 1.9. Suppose that {[[alpha].sub.n]}, {[[beta].sub.n]} and {[[omega].sub.n]} are nonnegative sequences such that

[[alpha].sub.n + 1] [less than or equal to] (1 - [[omega].sub.n])[[alpha].sub.n] + [[beta].sub.n] + [greater than or equal to] 0

with {[[omega].sub.n]} [subset] [0, 1], [summation][[omega].sub.n] = [infinity] and [lim.sub.n[right arrow][infinity]][[beta].sub.n] = 0. Then [lim.sub.n[right arrow][infinity]][[alpha].sub.n] = 0.

2. Main Results

Theorem 2.1. Let X be a real Banach space and T : X [right arrow] X be a uniformly continuous and strongly pseudocontractive mapping with a bounded range. Let p be a fixed point of T and the modified Mann iteration with errors {[x.sub.n]} be defined by (1.5) with {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} [subset] [0, 1] satisfying the following conditions:

(i)[[infinity].summation over (n = 0)] [b.sub.n] = [infinity]

(ii) [c.sub.n] = 0

(iii) [lim.sub.n[right arrow][infinity]] [b.sub.n] = 0

where {[u.sub.n]} is a bounded sequence in X. Then the sequence [{[x.sub.n]}.sup.[infinity].sub.n = 0] converges strongly to the unique fixed point of T.

Proof. Since p is a fixed point of T, then the set of fixed points F(T) of T is nonempty. Set

M = {[parallel][x.sub.0] - p[parallel]+ sup.sub.n[greater than or equal to]0]{ [parallel][T.sup.n][X.sub.n] - p[parallel]} + [sup.sub.n[greater than or equal to]0]{ [parallel][u.sub.n] - p[parallel]}}

It is clear that [parallel][x.sub.0] - p[parallel][less than or equal to] M. Suppose that [parallel][x.sub.n] - p[parallel][less than or equal to] M, a similar argument leads to [parallel][x.sub.n+1] - p[parallel][less than or equal to] M, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we get

[parallel][x.sub.n+1] - p[parallel] [less than or equal to] M for all n [member of] N

We have from (1.5)

[a.sub.n][x.sub.n] + [b.sub.n]T[x.sub.n] + [c.sub.n][u.sub.n] = [X.sub.n+1] (1 - [b.sub.n] - [c.sub.n])[n.sub.n] + [b.sub.n][T.sup.n][x.sub.n] + [c.sub.n][U.sub.n] = [x.sub.n+1] (2.1)

If we denote [d.sub.n] = [b.sub.n] + [c.sub.n], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Using the inequality (1.1) with x = xn+1 and y = p

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

0 [less than or equal to] r(I - [T.sup.n] - kI)[x.sub.n+l] - p[parallel] (2.3)

In view of equality (2.2), (2.3) and the fact that (1 - [d.sub.n])p = (1 - (1 - k)[d.sub.n])p + [d.sub.n] (I - [T.sup.n] - kI)p, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

implying in view of (2.3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

If [T.sup.n] = [parallel][T.sup.n][x.sub.n+1] - [T.sup.n][x.sub.n][parallel], from conditions (ii-iii), (2.6) and the uniform continuity of T, we obtain as n [right arrow] 0,

[T.sup.n] = [parallel][T.sup.n][x.sub.n+1] - [T.sup.n][x.sub.n] [parallel] [right arrow] 0 (2.7)

(2.5) gives the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Put [[alpha].sub.n] = [parallel][x.sub.n] - p[parallel], [[omega].sub.n] = [kd.sub.n], [[beta].sub.n] = ([T.sup.n] + M[c.sub.n]/[d.sub.n]) [k.sub.-2]. Then it follows from our lemma that [[alpha].sub.n] converges to 0, i.e. the sequence {[x.sub.n]} strongly converges to the unique fixed point p.

Corollary 2.2. [7] Let T : K [right arrow] K be a uniformly continuous and strongly pseudocontractive mapping with a bounded range. Let p be a fixed point of T and let the Mann iterative scheme [{[x.sub.n]}.sup.[infinity].sub.n=0] be defined by

[x.sub.n+1] - [a.sub.n][x.sub.n] + [b.sub.n]T[x.sub.n] + [c.sub.n][u.sub.n], n [greater than or equal to] 0

where {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are sequences in [0, 1] such that [a.sub.n] + [b.sub.n] + [c.sub.n] = 1 satisfying the conditions:

a)[[infinity].summation over (n = 0)] [b.sub.n] = O

b) [c.sub.n] = o([b.sub.n])

c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and {[u.sub.n]} is a bounded sequence in X. Then the sequence [{[x.sub.n]}.sup.[infinity].sub.n=0] converges strongly to the unique fixed point p of T

Theorem 2.3. Let T : X [right arrow] X be a strongly successively pseudocontractive (without having a bounded range) and uniformly Lipschitzian with L [less than or equal to] 1. Let p be a fixed point of T and {[x.sub.n]} be defined by (1.5a) with {[b.sub.n]} [subset] [0, 1], satisfying the conditions:

(i) [[infinity].summation over (n = 0)] [b.sub.n] = [infinity]

(ii) [b.sub.n] [less than or equal to] k - n/(1 + L) for some [eta] [member of] (0,k) [for all] n [member of] N and {un} is a bounded sequence in X.

Then the sequence [{[x.sub.n]}.sup.[infinity].sub.n=0] converges strongly to the unique fixed point p of T.

Proof. Since T is uniformly Lipschitzian, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

[parallel][T.sup.n][x.sub.n+1] - [T.sup.n] [X.sub.n] [parallel] [less than or equal to] L [[b.sub.n] (1 + L) + [c.sub.n] [parallel][x.sub.n] - p[parallel] + [c.sub.n]L[parallel][U.sub.n] - p[parallel] (2.9)

Also,

[parallel][u.sub.n] - [T.sup.n] [x.sub.n-][parallel][u.sub.n] - P[parallel] + [parallel][T.sup.n][x.sub.n] - p[parallel] [less than or equal to] [parallel][u.sub.n] - p[parallel] + L [parallel][x.sub.n] - p[parallel] (2.10)

Substituting (2.9) and (2.10) into (2.5) and setting [d.sub.n] = [b.sub.n] + [c.sub.n], for n [member of] N, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

Since in (1.5a) [c.sub.n] = 0 for all n, we have [d.sub.n] = [b.sub.n] for all n e N and thus (2.11) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

Since [b.sub.n] [less than or equal to] k -([eta] (1 + L)L for some [eta] [member of] (0,k) [for all] n [member of] N, then k - (1 + L)L[b.sub.n]/1 - (1 - k)[b.sub.n] < 1 and hence {[x.sub.n]} strongly converges to p.

Remark 2.4. Theorem 2.1 and Theorem 2.3 extend and improve Theorem 2 of Rafiq [7], Theorem 2 of Liu [12], Theorem 2 of Xu [8], Theorem 2 of Osilike [6], Theorem of Sastry and Babu [5] in the following ways:

a) The class of strongly successively pseudocontractive operators is more general than the class of strongly pseudocontractive maps considered by them.

b) The Mann iteration (with errors) are replaced with a more modified Mann iteration with errors.

3. Error Estimate

We observe that

k - (1 + L)L[b.sub.n]/ 1 - (1 - k)[b.sub.n][greater than or equal to] k - (1 + L)L[b.sub.n], [for all]n

if [b.sub.n] = k/2(1 + L)L for each n, then in view of (2.12), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So that

[parallel][x.sub.n+1] - p[parallel] [less than or equal to] [[lambda].sup.n] [parallel][x.sub.1] = p[parallel]

where

[lambda] = (1 - [k.sup.2]/4(1 + L)L) [member of] (0, 1)

In order to have a more detailed analysis of the convergence rate estimate of our A value and the ones obtained by Liu [12], Sastry and Babu [5], Mogbademu et al [11] and Ciric et al [3], we designed a [C.sub.++] program whose input are the specific parameters (k, [eta] and L) and which produces as output a number of iterates, depending on the stopping criterion adopted. The significant results are the following: From Theorem 2.3 of this paper:

[lambda] = (1 - [k.sup.2]/4(1 + L)L) [member of] (0, 1)

In Theorem 1 of Liu [12]:

[lambda] = (1 - [k.sup.2]/4(3 + 3L)[L.sup.2]) [member of] (0, 1)

In Theorem 2 of Sastry and Babu [5]:

[lambda] = (1 - [k.sup.2]/4(1 + L)(L + 2 - k) + 2k) [member of] (0, 1)

In Theorem 2.1 of Mogbademu et al. [11]:

[lambda] = (1 - [k.sup.2]L(L + 1)/4[L.sup.2][(L + 1).sup.2] + [k.sub.2][eta]) for some [eta] [member of] (0, k)

In Theorem 10 of Ciric et al. [3]:

[lambda] = (1 - [k.sup.2]/4(k + (2 - k)L] [member of] (0, 1)

The comparison ofthe convergence rateestimate of our [lambda], [lambda] of Liu, [lambda] of Sastry and Babu and [lambda] of Mogbademu et al is given in Table 3.1 below.

From Table 3.1, we observe that our error estimate improves on the general convergence rate estimate obtained by Liu, Sastry and Babu, and Mogbademu et al.

References

[1] Z. Q. Liu, J. K. Kim and K. H. Kim, Convergence theorems and stability problems of the modified Ishikawa iterative sequences for strictly successively hemi- contractive mappings, Bull. Korean Math. Soc., 39(3):455-469, 2002.

[2] Lj. B. Ciric, J. S. Ume, Ishikawa iterative process for strongly pseudocontractive operators in arbitrary Banach spaces, Math. Comm., 8(1):43-48, 2003.

[3] L. Ciric et al., On Mann implit interations for strongly accretive and strongly pseudocontractive, Appl. Math. Comm., 198:128-137, 2008.

[4] L. Liu, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc., 125:1363-1366, 1997.

[5] K. P. R. Sastry and G. V. R. Babu, Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed convex sets in a Banach space, Proc. Amer. Math. Soc., 128:2907-2909, 2000.

[6] M. O. Osilike, Strong convergence of iterative methods to solutions of certain nonlinear operator equations, ICTP preprint, No. 1c/98/207.

[7] A. Rafiq, Mann iterative scheme for nonlinear equations, Math. Comm., 12:25- 31, 2007.

[8] Y. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., 224:91-101, 1998.

[9] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158:407-413, 1991.

[10] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4:506- 510, 1953.

[11] A. A. Mogbademu, O. A. Akinfenwa, J. O. Olaleru and J. A. Adepoju, On the convergence rate estimate of Mann iteration process for a strictly pseudocontractive operator, Proc. Worldcom (2008), Georgia, USA (accepted, to appear in the journal of supercomputing).

[12] L. S. Liu, Fixed points of local strictly pseudocontractive mappings using Mann and Ishikawa iteration with errors, Indian J. Pure Appl. Math., 26:649-659, 1995.

J. O. Olaleru and A. A. Mogbademu

Mathematics Department, University of Lagos, Lagos, Nigeria.

E-mail: olaleru1@yahoo.co.uk, prinsmo@yahoo.com
```Table 3.1

n [lambda](Liu[4]) [lambda]
(Sastry&Babu[5])

1 0.964286 0.944444
2 0.995192 0.994186
3 0.999669 0.999642
4 0.999986 0.999985
5 1.000000 1.000000

n [lambda] [lambda]

1 0.882353 0.875
2 0.989601 0.989583
3 0.9999421 0.999421
4 0.999978 0.999978
5 0.99999
```