Existence Theorem for Abstract Measure Delay Integro-Differential Equations
S. S. Bellale^{1}, S. B. Birajdar^{2}, D. S. Palimkar^{3}
^{1}Mathematics Research Centre, Dayanand Science College, Maharashtra, India
^{2}Department of Mathematics, Bidve Engineering College, Latur, Maharashtra, India
^{3}Department of Mathematics, Vasantrao Naik College, Nanded, Maharashtra, India
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To cite this article:
S. S. Bellale, S. B. Birajdar, D. S. Palimkar. Existence Theorem for Abstract Measure Delay Integro-Differential Equations. Applied and Computational Mathematics. Vol. 4, No. 4, 2015, pp. 225-231. doi: 10.11648/j.acm.20150404.11
Abstract: In this paper, we have proved the existence and uniqueness results for an abstract measure delay integro-differential equation by using Leray-Schauder nonlinear alternative under certain Caratheodory conditions. The various aspects of the solutions of the abstract measure integro-differential equations have been studied in the literature using the various fixed point techniques such as Schauder,s fixed point principle and Banach contraction mapping principal etc. In this paper we have proved existence and uniqueness condition for Abstract Measure delay integro-differential equations.
Keywords: Time Scale, Abstract Measure Integro-Differential Equation, Abstract Measure Delay Integro-Differential Equation, Existence Theorem and Extermal Solutions
1. Introduction
The concept of stability has been widely used by many scientists under various model formulations. Absolute stability was originally formulated by Lur’e and Postnikov and is connected between with engineering and mathematical considerations. From a mathematical point of view, one arrives at this concept from of continuity. In our study we shall be concerned with a view of such physical phenomena.
Functional integro-differential equations with delay is a hereditary system in which the rate of charge or the derivative of the unknown function or set function depends upon the past history. The functional integro–differential equations of neutral type is a hereditary system in which the derivative of the unknown function is determined by the values of a state variable as well as the derivative of the state variable over some past interval in the phase space. Although the general theory and the basic result for integro-differential equations have now been thoroughly investigated, the study of functional integro-differential equations has not been complete yet. In recent years, this has been an increasing interest for such equations among the mathematicians all over the word.
The study of abstract measure integro-differential equations is initiated and developed at length in a series of papers by Dhage [1,2,3]. The study of abstract measure delay differential equations was initiated by Joshi [9,10], Shendge and Joshi [14] and Bellale [4,5,6].
Using the approach of above mentioned papers, in this paper, we prove the existence and stability results for a abstract measure delay integro-differential equations.
2. Preliminaries
Let denote the real line, an Euclidean space with respect to the norm defined by
(2.1)
For .
Let X be a real Banach space with any convenient norm . For any two points in X, the segments in X is defined by
.
Let and be two fixed points in X, such that , where 0 is the zero vector of X. Let z be a point of X, such that . For this z and , define the sets and as follows.
For , we write if . Let the positive number be denoted by . For each let denote that element of which
.
Note that, and are not the same points unless and.
Let M denote the -algebra of all subsets of X so that (X, M) becomes a measurable space. Let by
(2.2)
Where is a total variation measure of p and is given by
(2.3)
For all with for . It is known that is a Banach space with respect to the norm defined by (2.2). Let be a finite measure on X and let . We say p is absolutely continuous with respect to the measure if implies for all . In this case, we write.
For a fixed let be the smallest -algebra on, containing and the sets. Let be such that and let denote the -algebra of all sets containing and the sets of the form for. We define the set and by
and
Where is given (large enough) and .
Finally let, denote the space of all integrable nonnegative real valued functions h on with the norm , defined by
3. Statement of the Problem
Let be a finite real measure on X. Given a with consider the following abstract measure delay integro differentiate equation involving the delay,
(3.1)
on
,
Where q is a given known vector measure, is Radon – Nikodym derivative of p with respect to and the function is such that is integrable for each .
Definition 3.1 :-
Given an initial real measure q on , a vector is said to be a solution of delay (1.1) if
on
Remark 3.1 :- The delay integro-Differential equation (3.1) is equivalent to the abstract measure delay integro-differential equation.
A solution p of delay integro-differential equation (3.1) on will be denoted by
We apply the Schauder’s fixed point theorem foe formulating the main existence result for the delay integro-differential equation (3.1). Before stating this result, we recall definition.
Definition 3.2. :-
An operator Q on a Banach space X into it self is called compact if for any bounded subset of S of X. Q (X) is a relatively compact subset of X. Q is called totally bounded if Q (S) is a totally bounded subset of X for each bounded subset S of X. If Q is continuous and totally bounded, then it is called completely continuous on X.
Every compact operator is totally bounded, but the converse may not be true however both notions coincide on bounded subset of X.
Theorem 3.1 :-
Let S be a non-empty, closed convex and bounded subset of the Banach space X and let be a continuous and compact operator. Then the operator equation has a solutions.
Now we shall prove the main existence theorem for the delay (3.1) under suitable conditions of the function R.
4. Existence and Uniqueness Theorem
Definition 4.1:-
A function is said to satisfy conditions of Caratheodory or simply it is Caratheodory if
is measurable for each , and is continuous for almost everywhere on .
Definition 4.2:-
A Caratheodory function f is called Caratheodory if
For each given real number there exists a function such that
,
for all with .
Definition 4.3 :-
A function is called Caratheodory if there exists a function such that.
for all
Consider the following set of assumptions.
, For any ,
the-algebra M_{z} is compact with respect to topology generated by the pseudo-metric defined by
is continuous on with respect to the pseudo-metric d defined in
The function is Caratheodory.
The function is continuous and there exist functions such
Theorem 4.1:- Suppose the assumption hold. Then for a given initial measure , the delay (3.1) admits a solutions on for some .
Proof:-Let be a decreasing sequence of real numbers such that as
Then we have
There fore, these exists a real number r and a point such that
Where . This is possible by virtue of and the positiveness of .
How in the Banach space we define a subset S by
(4.2)
if and
Where, the constant k is given by
(4.3)
From (4.2) and (4.3) it follows that for all
Define an operator T from S into by
(4.4)
We shall show that the operator T satisfies all the conditions of theorem (3.1) on S.
Step I:- We show that + continuously maps S into itself. First show that the operator T maps S into itself. Let p t S be arbitrary and let . Then there are sets such that
Then
for all . From the definition of the norm in one has .
This show that T maps S into itself. Now we show that T is continuous on S. Let be a sequence of vector measures in S converging to vector measure p, that is, . Then for by hypothesis , these exists a such that
Therefore, fore any
This shows that T is a continuous operator on S.
Step II:- Next, we show that T(S) is a uniformly bounded and set in . Now as in step I, it is proved that T (S) is a subset of S and hence it is uniformly bounded set in . Now by the definition of the map T are has
Further we show that T (S) is an equi - continuous set in .
Let then there are setsand with and
We know that the set – identities
Therefore, we have
Since is - Caratheodory, we have that
Assume that .
Then we have and consequently and From the continuity of on it follows that
This show that T (S) is an equi-continuous set on, thus T (s) is uniformly bounded and equi- continuous set inso it is compact in the norm topology on . Now an application of Arzela Ascoli Theorem yields that T (S) is a compact subset of . As a result, T is a continuous and compact operator on essences, an application of theorem 3.1 yields that the operator equation has a solution is S. As a result, the delay (3.1) has a solution on . This completes the proof.
5. Extension of Stability
A solution of the delay (3.1) so obtained can be extended to the larger segment, whenever . An existence result in this condition is.
Theorem 5.1. Under the hypothesis of theorem (4.1) let be solution of the delay (4.3.1) on . Then the larger segment if
Proof :-
Definition 5.1:-
We consider the following assumptions.
The function is -integrable and for all for some .
Given there exists such that
For all with
Theorem 5.2:-
Let the assumptions hold. Then for each and a fixed number there exists a unique solution of the delay (3.1) satisfying whenever .
Proof:- Let , where .
Then corresponding this there is a number such that
(5.1)
For all
With
Define a subset
of by
(5.2)
Let Using and we obtain
(5.3)
Define an operator T from into by
(5.4)
Now if , then there are two disjoint sets F and G in such that
.
Hence for , from (5.2) and (5.3) it follows that
For all where and
Therefore for all . This shows that T maps S (e) into itself. Next, we show that T is a contraction operator on . Let . Then by ,
(5.5)
For all . This further implies that.
Where, . This shows that T is a contraction operator on with the contraction constants. Therefore, by an application of contraction mapping principle, there is a unique solution of the delay (3.1) satisfying whenever . This completes the proof.
Example 5.1:- Let the Lebesgue measure on and . Consider the delay AMIGDE
(5.6)
and
(5.7)
Here for , we observe that
If then we have
In this way the solution p for the linear delay AMIGDE (5.6) can be found recursively on .
Acknowledgments
The authors are thankful to the referee for their valuable suggestions. The author S. S. Bellale is also thankful to UGC, New Delhi (MRP-Major, R.N.41-750, 2012) for their financial support.
References