1. Introduction

The dynamics of various evolutionary processes often undergo abrupt changes of state within intervals of continuous evolution. Over the years, differential equations had been used to model problems arising from physical phenomena and thereby bringing about solutions to such transformations. As at then, not much attention was given to physical, biological and economical processes, such issues like thresholds, bursting rhythms, optimal control models and pharmacokinetics which are processes known to exhibit abrupt changes at a given time-lag.

In certain phenomenon, these changes are regarded as shocks, perturbations and natural disasters [1]. These perturbations because of its short term durations are rather better handled as having acted instantaneously in the form of impulses. Associated with this development, a theory of impulsive differential equations had been recently given attention [2-5].

Researchers are now exploiting this idea of impulsive differential equations to handle other certain processes that involve hereditary issues such as population dynamics, ecology, chemical technology, biotechnology etc having greater functional analysis concept thus giving rise to functional differential equations [6, 7].

As an application of this theory, models for thresholds of malaria control when spraying occurs had been presented [8, 9].

Despite the rapid attention to impulsive differential and partial differential equations and inclusions with fixed moments or fractional orders [4, 6, 2] cases for which the part governing the derivatives are not completely resolved.

A dynamic process involving the derivative x’(t) of a state x(t) may be known only within a set S(t, x (t)) ⊂ 𝑹 formulated by

Differential inclusions arise more especially in models for control systems, game theory and biological systems.

In this paper, the existence result for solution of dynamic evolutionary processes modeled using second order impulsive differential inclusions of the form

(1.1)

(1.2)

(1.3)

(1.4)

Where is multivalued map with compact values, is the family of all subsets of with representing the right and left limits of u(t) at fixed moment respectively, is established by applying the Schauder’s fixed point theorem.

This paper is organized in three sections. In section two, some preliminaries and theorems are given. The main existence result is formulated and proved in section three. This paper further initiates the study of impulsive differential inclusions using Galerkin’s approximations.

2. Preliminaries

In this section, notations, some basic definitions and some auxiliary results from multivalued analysis which are used in the sequel are presented and with some certain necessary assumptions.

Let with

Consider the space of piece wise continuous functions defined by

and the space of first order differentiable functions given by

hold except for some at which and exist such that and These sets of functions are Banach spaces with the norm:

(2.1)

The space of all absolutely continuous functions are denoted by

Definition 2.1: A function is said to be

L’-caratheodory function if

(i) is measurable for each

(ii) is continous for almost all

Definition 2.2: A multivalued map is said to be L’- caratheodory if

(i) is measurable for each

(ii) upper semicontinous on R for almost all

(iii) for each such that

(2.2)

and t almost everywhere in Hypothesis

Let ‘F’ be an function, then

H1: there exist constants and such that

for each

H2: and are Lipschitz continuous in that for

and

H3: there exist a constant such that

for each and

H4: there exist a continuous non-decreasing function Such that for

With and

Theorem (2.1) (Schauder fixed point) [10, p 367]. Let Ω be a closed bounded and convex subset of the Banach space X and let be continous and compact. Then f possesses at least one fixed point in such that

Lemma 2.1 [11] if and

is non-decreasing with

then the integral equation

has for each a unique solution z.

If satisfies the integral in equality

then

3. Main Result

Considering now the initial value problem of equations 1.1 – 1.4, our existence result concerns the a priori estimates on its possible solution.

Definition 3.1:

A given function

is called a solution of equation 1.1-1.4 if it satisfies the differential inclusion

The Solution representation is given as

(3.1)

Lemma 3.1 Assume that the hypothesis H1-H4 are satisfied. Then the equation 1.1-1.4 has at least one solution.

Proof: A solution to problem 1.1-1.4 is often assumed to be a fixed point of an operator of the form.

defined by

(3.2)

We show that L is a compact operator that is closed bounded and convex.

Step 1: L is continuous

Let be a sequence such that in then

Since and are continous and f is L’-Caratheodory, then

Hence

(3.4)

Step 2: L maps bounded sets into bounded set in

Let be a bounded set f for each it is enough to show that there exist such that since are continous and in particular Lipschitz’s continuous, we have that

(3.5)

(3.6)

Step 3: L maps set into equicontinuous sets of the space let and such that and be a bounded set as defined above. Then

(3.7)

As the right hand side of the inequality tends to zero

By applying the Arzela-Ascoli theorem it is clearly seen from the consequences of step 1- step 3 that L is compact and completely continuous. We state the result thus:

The Existence Theorem

Suppose that the lemma 3.1 and hypothesis H1-H4 are satisfied for such that with

Then the impulsive differential inclusion 1.1-1.4 has at least a solution.

Proof: since the operator L is compact, closed bounded and convex, by applying the Schauder’s fixed point theorem we consider the set

(3.8)

And we show that is bounded. Let by definition we mean

(3.9)

(3.10)

(3.11)

Let represent the right hand side of the inequality, then

With

(3.12)

i.e

Since Ψ is non-decreasing function, we have that

By theorem 1.4.2, p35 [1], we have that

(3.13)

This equality indicates the existence of a constant depending only on the functions such that

Hence

(3.14)

Thus is bounded. We deduce therefore that has a fixed point which is the solution.

4. Conclusions

The Solution is considered at a point t_k which is known and can be estimated. Thus, the solution to the problem exists by establishing or locating the point t_k, where the solution representation is as given in 3.1. For further research, if the problem can be formulated in the finite dimensional space thus

then applying Galerkin’s approximations of the solution and subsequent extension to the entire space, we assumed that the problem can be solved.