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 Lipschit;: pseudocontractive self-mapping of a closed convex and bounded subset K of E. Suppose F(T) dcnotes the set of fixed points of T and U denot :s the sunny nonexpansive retraction of K onto F(T) and w any point of K, it is proved that the sequence {xn),"Lo generated from an arbitrary xo E K bv (where I denotes the identity operator on E and {a,),"==, and {Pn),"=o are real sequences in (0,1], satisfying certain conditions) converges strongly to Uw. This result is sirriilar to, and in some sence, is an improvement on the theorems of Chidume (Proc. Amcr. Math. Soc. 129(8) (2001) .. 2245-2251) and Ishikawa (Proc. Amer. Math. Soc. 44(l) (l974), 147-150). Furthermore, ..;u?pose 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 E Ax has a solution x* E E for an arbitrary but fixed f E E. Thcn it is proved that the sequence {x~)~~~ generated from an arbitra.ry xo E E by (where {c~):=~ is a, real sequence iil [O,l) satisfying certain conditions) converges strongly to x*. Moreover, :;uppose 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* E D(T). Then there exists a neighbourhood B of x* such that the sequence {x,)~=~ generated from a3 arbitrary xo E B c D(T) by (where {cn)?==, 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 (Operat,or Theory, Advances and Applications 98 (19%') ,7-22), Bruck (Bull. Amer. Math. Soc. 79(1973),1259-1262), Chidume and Moore (J. Math. Anal. Ap~ll. 245(l) (2000) ,142-160) and OsiIilce (Nonlinear Analysis 36 (1) (1999) ,I-g). Finally, if E is a real Banach space and T : E + E a map with F(T) := {z E E : TX = x) # B and satisfying the accretive-type condition (x - Tx, j(x - x*)) 2 Xllx - '1'x(12, for all x E E, x* E F(T) and X > 0, then a necessary and sufficient condition for the convergence of the sequence {2,}~=, generated from an arbitrary so E E by xn+l = (1 - c,)x, + c,Tz,, V n 2 0 . . (where {&}:=, is a real sequence in [0, I) satisfying certain conditions) to a fixed point of T is established. This resu1.t exte~ds 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 Chidurne (J. Nigeriau Math. Soc. 3(1984),57-62).