Hybrid fixed point theorem for abstract measure Integro-differential equation

Introduction

The study of abstract measure differential equations is iniated by Sharma [8,9] and subsequently developed by Joshi [7] , Shendge and Joshi [10] , Similarly, the study of abstract measure Integrodifferential equation is studied by Bellale and Dhage [4] for various aspect of solutions. In such models of differential equations, the ordinary derivative is replaced by the derivative of the set functions which there by gives the generalization of the ordinary derivative and measure differential equations. The various aspects of the solution of the abstract measure differential equations have been studied in the literature using the fixed point techniques such as Schauder's fixed point principle and Banach contraction mapping principle etc. In such situation one needs to show that the operator under consideration maps a certain set into itself. This is a serve restriction, which motivated us to persue the study of abstract measure Integrodifferential equations using the nonlinear alternative of Leray-Schauder. In the present chapter we shall prove the existence and uniqueness result for an abstract measure-Integro differential equation in question is also proved under certain monotonicity conditions of the nonlinearity involved in the equation. In the following section we give some preliminaries in the sequel.

Definitions and Notations

Let X be real Banach space with a norm . . Let X y x ∈ , . Then the line segment y x in X is defined by

. In this case we also write

Let M denote the -σ algebra of all subsets of X so that ( )

be the space of all vector measures (signed measures) and define a norm . on ( )

Where p is a total variation measure of p and is given by

Where supremum is taken over all partition { }

is a Banach space with respect to the norm . defined by (2.4). Let µ be a σ -finite measure on X and let ( )

. We say p is absolutely continuous with respect to the measure µ if ( ) .

Statement of the problem:

Let µ be a real or σ -finite positive measure on X . Given a ( )

Where q is a given known vector measure, µ d dp is a Radon-Nikodym derivative of p with respect to µ and

A solution p of AMIDE (3.1) on z x 0 will be denoted by ( )

In the following section we shall prove the main existence theorem for AMIDE (3.1) under suitable conditions on f . We shall use the following form of the Leray-Schauder's nonlinear alternative. See Dugundji and Granas [6] .

denote respectively the open and closed balls in a Banach space X centered at the origin 0 of radius r , for some 0

be a completely continuous operator. Then either ( )

Existence theorems:

We need the following definition in the sequel.

Definition 4.1: A function

ii ψ is non decreasing, and

The class of all D -function on We consider the following set of assumptions.

the σ -algebra z M is compact with respect to the topology generated by the Pseudo-metric d defined by ( ) ( )

A q is continuous on z M with respect to the Pseudo-metric d defined in ( )

We shall show that the operator T satisfies all the conditions of Theorem (3.1) on [ ] r B , 0 .

Step I: First we show that T is a continuous on [ ]

And, so T is a continuous operator on [ ] r B , 0 .

For all z M E∈ , By definition of the norm . we have ( ) ( )

This further implies that ( ) ( )

Substituting r u = in the above inequality yields that ( )

Which is a contradiction to the inequality (4.1).

Hence the operator equation p Tp = has a solution in v with r v ≤ . Consequently the AMIDE(3.1) has a solution ( ) Below we shall show that the functions f and g satisfy all the conditions of theorem (4.1).

Obviously f is continuous on the domain of its definition. Let IR y y ∈

References

1. Banas J, Goebel K. Measures of non compactness in Banach space, Lecture Notes in Pure And Applied Mathematics, Dekker, New York, 1980, 60.
2. Bellale SS, Dapke GB. Existence theorem and extremal solutions for perturbed Measure Differential equations with maxima, Inter. jour. math.arc. 2016; 7(10):1-11.
3. Bellale SS, Dapke GB. Hybrid fixed point theorem for nonlinear differential Equation, Inter. jour. eng. sci. and res. tech. 2017.
4. Bellale SS, Dhage BC. Existence theorem for perturbed abstract measure differential Equations, Nonlinear Analysis, theory; methods and application Nonlinear Analysis, theory; methods and application, 2008. doi:1016j.na.2008.11.057
5. Das PC, Sharma RR. Existence and stability of measure differential equations, Zech, Math J. 1972; 22:145-158.
6. Dugundji J, Granas A. Fixed point theory, Monograhie Math. PNW, Warsaw, 1982.
7. Joshi SR. A system of abstract measure delay differential equations, J. Math. Phy. Soc. 1979; 48:497-506.
8. Sharma RR. An abstract measure differential equation, Proc. Amer. Math. soc. 1972; 32:503-510.
9. Sharma RR. A measure differential inequality with applications, Proc.Amer. Math. Soc. 1975; 48:87-97.
10. Shendge GR, Joshi SR. Abstract measure differential inequalities and application, Acta, Math. Hung. 1983; 41:53-54.