Abstract
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.
Highlights
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
Summary
The abstract measure differential equations involve the derivative of the unknown set-function with respect to the σ-finite complete measure. 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. 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
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