Studies of Extension of Fixed Point Theorem of Rhoades Using Ishikawa Iteration Process

INTRODUCTION

If c be a non empty subset of x, where x be a Banach space. And let T a mapping from C to itself. The iteration scheme called Ishikawa Scheme is defined as follows: In above Ishikawa scheme, nn, satisfy There are following two contractive conditions to be used. There exists a constant K, 0 < k < 1 such that for all x, y in x.

(A)

(B) At least one of the following conditions holds: In this paper it is shown that, for mapping T which satisfy conditions (A) or (B) above, if the sequence of Ishikawa iterates converges, it converges to the fixed point of T. These results extend the corresponding results of Rhoades [1] and Hicks and Kubicek [2]. Definition 1: A mapping T : XX is called a quasicontraction if there exists a constant K, 0 < k < 1 such that for each x, y ││ X, where X be a Banach space. Theorem 1: Suppose T: CC be a mapping satisfying (A), {Xn} the sequence of the Ishikawa scheme associated with T are such that {││n} is bounded away from zero. If {xn} converges to p, then p is a fixed point of T, where X be a normed linear space and C be a closed convex subset of X. Proof: We have from {I1} that

Since T satisfies (A) we have Thus We have by taking the limit as n, It follows that Using the definition (1) of T and the triangle inequality, we have Thus we obtain, by taking the limit as n, At least This means

Definition 2: A mapping T : CC is called strictly pseuiocontractive if for some k, 0 < k < 1, and all x, y, c, where X be a normed linear space and C be an non-empty subset of X. Definition 2.1: T is called pseuiocontractive if for all x,y ││C, Definition 2.2: T is said to satisfy the condition (T) if for all x ││ C and y ││ F (T) It is clear that any strictly pseuiocontractive mapping is hemicontractive, any mapping satisfying condition (T) is demicontractive and a demicontractive mapping is hemicontractive but not conversely. Theorem 3: Let a mapping T : C C satisfies condition (T). Suppose F (T) is non-empty and diverges and Then for each xO ││ ││C ││ ││where 1nX is defined as in the Ishikawa scheme. Proof: By using the condition (T), the mapping T is demicontractive for any constant K. We get for any x, y and z in H (Hillbert Space) and a real number. Therefore for and each integar By using condition (T) we get By using demi contractiveness of T and definition of yn we get Hence, By induction, we obtain Therefore We note that Therefore Hence which diverges. Therefore diverges. And from (2.4) we get

REFERENCES:

1. Chidume, C.E. : J. Nigerian Math. Soc. 4 (1985) P. 1-11 2 Rhoades, B.E. : Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976). 741-750 3. Chidume, C.E. : J. Nigerian Math. Soc. 7 (1988) P. 1-9 4. Ciric, L.B.: Proc. Amer. Math. Soc. V. 45 (1974), P. 267-273. 5. Ishikawa, S.: Fixed points by a new iterative method. Proc. Amer. Math. Soc. 149 (1974) P. 147-150 6. Rhoades, B.E. : Souchow J. Math. 19 (1993) P. 377-80 8. Rhoades, B.E.: Trans. Amer. Math. Soc. 196 (1974) P. 161-76. 9. Reich, S. : Nonlinear Analysis, 2 (1978) P. 85-92.

PhD, Department of Mathematics, Magadh University, Bodh Gaya