CLASSICAL RESULTS VIA MANN–ISHIKAWA ITERATION

. New proofs of existence and uniqueness results for the solution of the Cauchy problem with delay are obtained by use of Mann–Ishikawa iteration. MSC 2000. 47H10.


INTRODUCTION
Consider the following delay differential equation (1) x The existence of an approximative solution for equation ( 1) is given by theorem 1 from [1] (see also [2], [6], [5]).The proof of this theorem is based on the contraction principle.We shall prove it here by applying Mann iteration.
In the last decades, numerous papers were published on the iterative approximation of fixed points of contractive type operators in metric spaces, see for example [7], [8].The Mann iteration [4] and the Ishikawa iteration [3] are certainly the most studied of these fixed point iteration procedures.
Let X be a real Banach space and T : X → X a given operator, let u 0 , x 0 ∈ X.

FIXED POINT THEOREMS
We consider the delay differential equation ( 4) with initial condition ( 5) We suppose that the following conditions are fulfilled (H By a solution of the problem ( 4)-( 5) we mean the function The problem (4)-( 5) is equivalent with the integral equation ( 6) Applying contraction principle we have Theorem 1. [1] We suppose conditions (H 1 )-(H 5 ) are satisfied.Then the problem (4)-( 5) The following lemma is well-known.For sake of completeness, we shall give a proof here.

Proof. Consider Mann iteration
for the operator Denote by x * := T x * the fixed point of T.
For t ∈ [t 0 − τ, t 0 ] we get Consider Lemma 2 to obtain Assumption (H 5 ) leads us to We take Set τ = 0, in (4), to obtain the classical existence and uniqueness result, i.e.Theorem 6, for the Cauchy problem.This problem, see [1], [6], is proved by use of contraction principle.
We suppose that the following conditions are fulfilled (H Note that we have supplied here a new proof for the following result using Mann-Ishikawa iteration.

Theorem 4 . [ 7 ]
and use Lemma 2 to obtain lim n→∞ u n − x * = 0. Let X be a normed space, D a nonempty, convex, closed subset of X and T : D → D a contraction.If u 0 , x 0 ∈ D, then the following are equivalent: (i) the Mann iteration (2) converges to x * ; (ii) the Ishikawa iteration (3) converges to x * .Remark 5.Because Mann iteration and Ishikawa iteration are equivalent, it is possible to consider Ishikawa iteration in order to prove Theorem 3.