Applied and Computational Mathematics
Volume 4, Issue 4, August 2015, Pages: 225-231

Existence Theorem for Abstract Measure Delay Integro-Differential Equations

S. S. Bellale1, S. B. Birajdar2, D. S. Palimkar3

1Mathematics Research Centre, Dayanand Science College, Maharashtra, India

2Department of Mathematics, Bidve Engineering College, Latur, Maharashtra, India

3Department of Mathematics, Vasantrao Naik College, Nanded, Maharashtra, India

Email address:

(S. S. Bellale)
(S. B. Birajdar)
(D. S. Palimkar)

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 Mz 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

  1. B. C. Dhage "On abstract measure integro-differential equations", J. Math. Phy. Sci.20 (1986), 367 – 380
  2. B. C. Dhage "On system of abstract measure integro – differential inequations and applications", Bull. Inst. Math. Acad. Sinica18 (1989), 65 – 75
  3. B. C. Dhage "Mixed monotonicity theorems for a system of abstract of measure delay integro – differential equations", An. Stint. Univ. "Al. I. Cuza" IasiXLII (1996) 355 – 366
  4. B. C. Dhage and S. S. Bellale, "Abstract measure integro – differential equations", Global Jour Math. Anal.1 (2007), 91 – 108.
  5. B. C. Dhage and S. S. Bellale, "Existence theorem for per- turbd abstract measure differential equations", Nonlinear Analysis, 71(2009),e319-e328
  6. S. S. Bellale, "Hybrid Fixed Point Theorem For Abstract Measure Differential Equation", World Academy Of Science, Engineering and Technology, 73(2013) 782-785 ISSN- e2010-3778
  7. B. C. Dhage, D. N. Chate and S. K. Ntouyas, "Abstract measure differential equations", Dynamic Systems & Appl. 13 (2004) 105 – 108.
  8. J. Dagudji and A. Granas, "Fixed point Theory", Monograhie Math. PNW. Warsaw 1982.
  9. S. R. Joshi, "A system of abstract measure delay differential equations", J. Math. Phy. Sci. 13 (1979), 496 – 506.
  10. S. R. Joshi and S. G. Deo, "On abstract measure delay differential equations", An. Stint. Univ. Al I. CuzaIassiXXVI (1980), 327 – 335.
  11. W. Rudin, "Real and Complex, Analysis, McGraw", – Hill Inc.New York, 1966.
  12. R. R. Sharma, "An abstract measure differential equation", Proc, Amer, Math. Soc. 32 (1972) 503 – 510.
  13. R. R. Sharma, "A measure differential inequality with applications", Proc. Amer. Math. Soc. 48 (1975) 87 – 97.
  14. G. R. Shendge and S. R. Joshi, "Abstract measure differential inequalities and applications", Acta Math Hung. 41 (1983), 53 – 54.
  15. D. R. Smart, "Fixed point Theorems", Cambridge Unive. Press, Cambridge 1974.

Article Tools
  Abstract
  PDF(221K)
Follow on us
ADDRESS
Science Publishing Group
548 FASHION AVENUE
NEW YORK, NY 10018
U.S.A.
Tel: (001)347-688-8931