Iterative Approximation of Equilibrium Points of Evolution Equations

Abstract

Suppose that E is a real Banach space .which is both uniformly convex and q-uniformly smooth and that T is a Lipschitz pseudocontractive self-mapping of a closed convex and bounded subset K of E. Suppose F(T) denotes the set of fixed points of T and U denotes the sunny nonexpansive retraction of K onto F(T) and w any point of K, it is proved that the sequence {Xn} n=0∞generated from an arbitrary x0 € K by Xn+1 = βnw +(1+βn) 1/n+1 ∑3=0n{(1-a3)1+a3T}Xn, (where I denotes the identity operator on E and {an}∞n=0, and {βn}∞n=0 are real sequences in (0,1], satisfying certain conditions) converges strongly to Uw. This result i s similar to, and in some sence, is an improvement on the theorems of Chidume (Proc. Amer. Math. Soc. 129(8) (2001) .2245-2251) and Ishikawa (Proc. Amer. Math. Soc. 44(l) (l974), 147-150). Furthermore, suppose that E is an arbitrary real normed linear space and A : E + 2E is a uniformly continuous and uniformly quasi-accretive multi-valued map with nonempty closed values such that the range of (I - A) is bounded and the inclusion f € Ax has a solution x* € E for an arbitrary but fixed f € E. Then it is proved that the sequence { xn)∞n=0 generated from an arbitrary xo € E by

Xn+1= (1-cn)Xn+cnCn1 Cn € (1-A)xn Ɐ n ≥0

(where {cn)∞n=0 is a, real sequence in [ O , l ) satisfying certain conditions) converges strongly to x*. Moreover, suppose E is an arbitrary real normed linear space and T : D(T) c

E → E is locally Lipschitzian and uniformly hemicontractive map with open domain D(T) and a fixed point x* € D(T). Then there exists a neighbourhood B of x* such that the sequence { xn, ) ∞n=0 generated from a3 arbitrary x0 € B c D(T) by

xn+1=(1-cn)xn+cnTxn,Ɐ n ≥ 0

(where {cn}∞n=0 is a real sequence in [O,1) satisfying certain conditions) remains in B and converges strongly to x*. These results are improvements on the results of Alber and Delabriere (Operator Theory, Advances and Applications 98 (1997) ,7-22), Bruck (Bull. Amer. Math. Soc. 79(1973),1259-1262), Chidume and Moore (J. Math. Anal. Appl 245(l) (2000) ,142-160) and OsiIike (Nonlinear Analysis 36 (1) (1999) ,I-9). Finally, if E is a real Banach space and T : E → E a map with F(T) := {x € E : Tx = x}≠0 and satisfying the accretive-type condition (x - Tx, j(x - x*)) ≥גּ║x- Tx║2 for all x € E, x* € F(T) and גּ > 0, then a necessary and sufficient condition for the convergence of the sequence {xn} ∞n=0=, generated from an arbitrary x0 € E by xn+l = (1 – cn)xn + c,Txn, Ɐ n ≥ 0 . (where {cn}∞n=0 is a real sequence in [0, I) satisfying certain conditions) to a fixed point of T is established. This result extends the results of Maruster (Proc. Amer. Math. Soc.66 (1977), 69-73) and Chidume (J. Nigerian Math. Soc. 3(1984),57-62) and resolves a question raised by Chidume (J. Nigerian Math. Soc. 3(1984),57-62).