A solution of delay differential equations via Picard–Krasnoselskii hybrid iterative process

The purpose of this paper is to introduce Picard–Krasnoselskii hybrid iterative process which is a hybrid of Picard and Krasnoselskii iterative processes. In case of contractive nonlinear operators, our iterative scheme converges faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative processes in the sense of Berinde (Iterative approximation of fixed points, 2002). We support our analytic proofs with a numerical example. Using this iterative process, we also find the solution of delay differential equation.


Introduction and preliminaries
Throughout this paper, N denotes the set of all positive integers.
Let C be a nonempty convex subset of a normed space E and T : C → C a mapping. The mapping T : C → C is said to be a contraction if T x − T y ≤ δ x − y for each x, y ∈ C and δ ∈ (0, 1). (1.1) The Picard or successive or repeated function iterative process [30] is defined by the sequence {u n } as follows: (1. 2) The Mann iterative process [26] is defined by the sequence {v n }: where {α n } is appropriately chosen sequence in (0, 1). This is a one-step iterative process. The Krasnoselskii iterative process [25] is defined by the sequence {s n } as follows: where λ ∈ (0, 1). This is an averaging process. The sequence {z n } defined by ⎧ ⎨ ⎩ z 1 = z ∈ C, z n+1 = (1 − α n )z n + α n T y n , y n = (1 − β n )z n + β n T z n , n ∈ N (1.5) is known as Ishikawa iterative process [22], where {α n } and {β n } are appropriately chosen sequences in (0, 1). Most of the physical problems of applied sciences and engineering are usually formulated in the form of fixed point equations. The study of iterative processes to approximate the solution of these equations is an active area of research (see e.g., [1,23,24,28,29] and the references therein). The Picard iterative scheme is one of the simplest iteration scheme used to approximate the solution of fixed point equations involving nonlinear contractive operators. Chidume and Olaleru [13] established some interesting fixed points results using the Picard iteration process. Chidume [12] generalized and improved the results in [3]. Chidume et al. [11] established some convergence theorems for multivalued nonexpansive mappings for a Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type iterations (see [11]). Okeke and Abbas [28] proved the convergence and almost sure T -stability of Mann-type and Ishikawa-type random iterative schemes.
Recently Khan [24] introduced the Picard-Mann hybrid iterative process. This new iterative process for one mapping case is given by the sequence {m n } as follows: where {α n } is an appropriately chosen sequence in (0, 1). Motivated by the facts above, we now introduce the Picard-Krasnoselskii hybrid iterative process defined by the sequence {x n }: where λ ∈ (0, 1). Let {u n } and {v n } be two fixed point iteration processes that converge to a certain fixed point p of a given operator T. The sequence {u n } is better than {v n } in the sense of Rhoades [31] if The following definitions are due to Berinde [6]. Several mathematicians have obtained interesting results dealing with the rate of convergence of various iterative processes (see for example, [2,5,[7][8][9]18,20,31,32,39]). Some authors have also investigated the stability of various iterative processes for certain nonlinear operators. See, for example, Dogan and Karakaya [18], Akewe et al. [3] and the references therein.
The following lemma will be needed in the sequel.

Lemma 1.3 [34]
Let {s n } be a sequence of positive real numbers which satisfies: If {μ n } ⊂ (0, 1) and ∞ n=1 μ n = ∞, then lim n→∞ s n = 0. Interest in the study of delay differential equations stems from the fact that several models in real-life problems involves delay differential equations. For instance, delay models are common in many branches of biological modeling (see [19]). They have been used for describing several aspects of infectious disease dynamics: primary infection [14], drug therapy [27] and immune response [16], among others. These models have also appeared in the study of chemostat models [40], circadian rhythms [33], epidemiology [17], the respiratory system [37], tumor growth [38] and neural networks [10]. Statistical analysis of ecological data (see e.g., [35,36]) has shown that there is evidence of delay effects in the population dynamics of many species.
The aim of this paper is to introduce the Picard-Krasnoselskii hybrid iterative process and to show that this new iterative process is faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative processes in the sense of Berinde [6]. Finally, we show that our iterative process can be used to find the solution of delay differential equations.

Rate of convergence
In this section, we prove that the Picard-Krasnoselskii hybrid iterative process (1.7) converges at a rate faster than all of Picard iterative process (1. By (1.1) and the Krasnoselskii iterative process (1.4), we get From (1.1) and the Ishikawa iterative process (1.5), it follows that From (1.5), (1.1) and (2.7), we obtain that Using (1.1) and the Picard-Krasnoselskii hybrid iterative process (1.7), we have (2.10) Set: We now compute the rate of convergence of our iterative process (1.7) as follows: Thus, {x n } converges faster than {u n } to p. That is, the Picard-Krasnoselskii hybrid iterative process (1.7) converges faster than the Picard iterative process (1.2) to p.

Application to delay differential equations
We now employ our iterative process (1.7) to find the solution of delay differential equations.
Let the space C([a, b]) of all continuous real-valued functions on a closed interval [a, b] be endowed with the Chebyshev norm x − y ∞ = max t∈ [a,b] |x(t) − y(t)|. It is known that (C([a, b]), . ∞ ) is a Banach space ( [21]).
In this section, we consider the following delay differential equation. with initial condition

1)
By the solution of above problem, we mean a function

2).
Assume that the following conditions are satisfied.