Approximating Solutions of Non Linear First Order Abstract Measure Differential Equations by Using Dhage Iteration Method

In this paper we have proved the approximating solutions of the nonlinear first order abstract measure differential equation by using Dhage’s iteration method. The main result is based on the iteration method included in the hybrid fixed point theorem in a partially ordered normed linear space. Also we have solved an example for the applicability of given results in the paper. Sharma [2] initiated the study of nonlinear abstract differential equations and some basic results concerning the existence of solutions for such equations. Later, such equations were studied by various authors for different aspects of the solutions under continuous and discontinuous nonlinearities. The study of fixed point theorem for contraction mappings in partial ordered metric space is initiated by different authors. The study of hybrid fixed point theorem in partially ordered metric space is initiated by Dhage with applications to nonlinear differential and integral equations. The iteration method is also embodied in hybrid fixed point theorem in partially ordered spaces by Dhage [12]. The Dhage iteration method is a powerful tool for proving the existence and approximating results for nonlinear measure differential equations. The approximation of the solutions are obtained under weaker mixed partial continuity and partial Lipschitz conditions. In this paper we adopted this iteration method technique for abstract measure differential equations.


Introduction
The abstract measure differential equations involve the derivative of the unknown set-function with respect to the σ-finite complete measure. Some of the abstract measure differential equations have been studied in a series of papers by Joshi [3], Shendge and Joshi [4], Dhage [14,15], Dhage et al. [6] and Dhage and Bellale [9,10] and Suryawanshi and bellale [17] for different aspects of the solutions. The fixed point theorems so far used in the above papers of Dhage [15], Joshi [6], Bellale [13] study the abstract measure integro differential equation and existence theorem. This is a required condition and recently, the authors in Dhage [16], Suryawanshi and Bellale [18] have proved the existence and uniqueness results for abstract measure differential equations. Here our approach is different from that of Sharma [2] and Joshi [3].
The results of this paper complement and generalize the results of the above-mentioned papers on abstract measure differential equations under weaker conditions. The perturbed ordinary differential equations have been treated in Krasnoselskii [1] and it is mentioned that the inverse of such equations gets the sum of two operators in appropriate function spaces. The Krasnoselskii [1] fixed point theorem is useful for proving the existence results for such perturbed differential equations under mixed geometrical and topological conditions on the nonlinearities involved in them.

Preliminaries
A mapping if there exists a continuous and non-decreasing

Statement of the Problem
LetX be a real Banach algebra with a convenient norm || . || . Let , x y X ∈ . Then the line segment xy in X is defined by Let 0 x X ∈ be a fixed point and z X ∈ . Then for any and { | 1} Let 1 2 , x x xy ∈ be arbitrary. We say or equivalently, 0 1 0 2 x x x x ⊂ . In this case we also write 2 1 x x > .
Let M denote the σ-algebra of all subsets of X such that (X, M) is a measurable space. Let ca(X, M) be the space of all vector measures (real signed measures) and define a norm || · || on ca(X, M) by where |p| is a total variation measure of p and is given by Where the supremum is taken over all possible partitions It is known that ca(X,M) is a Banach space with respect to the norm ||.|| given by (4).
Let µ be a σ-finite positive measure on X, and let ( , ) p ca X M ∈ . We say p is absolutely continuous with respect to the measure µ if µ (E) = 0 implies p(E) = 0 for some E M ∈ . In this case we also write p << µ .
The conditions guaranteeing the regularity of E may be found in Heikkiländand Lakshmikantham [8] and the references therein. We need the following definitions (see Dhage [14] and the references therein) in what follows.
T is called monotonic or simply monotone if it is either non decreasing non increasing on E.
For all comparable elements , , x y E ∈ where 0 ( ) r r < ψ < for r > 0. In particular, if ( ) , 0, r kr k T ψ = > is called a partial Lipschitz operator with a Lipschitz constant k and more over, if 0 < k < 1, T is called a partial linear contraction on E with a contraction constant k. The Dhage iteration method or Dhage iteration principle embodied in the following applicable hybrid fixed point theorem of Dhage [12] in a partially ordered normed linear space is used as a key tool for our work contained in this paper. The details of the Dhage iteration method or principle is given in Dhage [14,15], and the references therein.

Main Result
In this section, we prove an existence and approximation result for the AMDE (9) Clearly, ( , ) C J R is a Banach space with respect to above supremum norm and also partially ordered w. r. t. the above partially order relation ≤. It is known that the partially ordered Banach space ( , ) C J R is regular and lattice so that every pair of elements of E has a lower and an upper bound in it. Consider the first order ordinary nonlinear abstract measure differential equation,  (9), where C (X,M) is the space of continuous real-valued functions defined on 0 .
x z The equation (9) has already been discussed for different aspects of the solutions using the usual Picard iteration method. See Bainov and Hristova [11] and the references therein for the details. In this paper we discuss the AMDE (9) for existence and approximation of solutions via a new approach based upon the Dhage iteration method.
We need the following definition as follows.
for all 0 . x x z ∈ We shall show that the operator T satisfies all the conditions of Theorem 3.1. This is achieved in the series of following steps.
Step I: T is a non decreasing operator on E.
Let , x y E ∈ be such that x y x z ∈ . This shows that T is a non decreasing operator on E into E.
Step II: T is a partially continuous operator on E. Let {p n } be a sequence in a chain C in E such that n p p → for all n N ∈ .
Then, by dominated convergence theorem, we have uniformly for all n N ∈ . This shows that the convergence n Tp Tp → is uniformly and hence T is a partially continuous operator on E into itself.
Step III: T is a partially compact operator on E.
Let C be an arbitrary chain in E. We show that T (C) is a uniformly bounded and equicontinuous set in E. First we show that T (C) is uniformly bounded. Let x C ∈ be arbitrary. for all x C ∈ . Hence T is a uniformly bounded subset of E.
Next, we will show that T (C) is an equicontinuous set in E.
Let 1 2 0 , x x x z ∈ be arbitrary with x 1 < x 2 . Then uniformly for all x C ∈ . Hence T (C) is a compact subset of E and consequently T is a partially compact operator on E into itself.
Step IV: u satisfies the operator inequality u Tu ≤ .
By hypothesis (H 2 ), the AMDE (9) has a lower solution u. Then we have 0