A SYSTEM OF ABSTRACT MEASURE DELAY DIFFERENTIAL EQUATIONS

In this paper existence and uniqueness results for an abstract measure delay differential equation are proved, by using Leray-Schauder nonlinear alternative, under Carathéodory conditions.


Introduction
Functional differential equations with delay is a hereditary system in which the rate of change or the derivative of the unknown function or set-function depends upon the past history.The functional differential equations of neutral type is a hereditary system in which the derivative of the set-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 results for differential equations have now been thoroughly investigated, the study of functional differential equations has not been complete yet.In recent years, there has been an increasing interest for such equations among the mathematicians of the world.The study of functional abstract measure differential equations is very rare.
The study of abstract measure delay differential equations was initiated by Joshi [6], Joshi and Deo [7] and Shendge and Joshi [11] and subsequently developed by Dhage [1]- [3].Recently, the authors in [4] proved existence and uniqueness results for abstract measure differential equations, by using Leray-Schauder alternative [5], under Carathéodory conditions.In this paper, by using the same method, we extend the results of [4] to a system of abstract measure delay differential equations.In that our approach is different from that of Joshi [6].The results of this paper complement and generalize the results of Joshi [6] on abstract measure delay differential equations under weaker conditions.

Preliminaries
Let IR denote the real line, IR n an Euclidean space with repect to the norm |•| n defined by Let X be a real Banach space with any convenient norm • .For any two points x, y in X, the segment xy in X is defined by Let x 0 and y 0 be two fixed points in X, such that 0y 0 ⊂ 0x 0 , where 0 is the zero vector of X.Let z be a point of X, such that 0x 0 ⊂ 0z.For this z and x ∈ y 0 z, define the sets S x and S x as follows Let the positive number x 0 − y 0 be denoted by w.For each x ∈ x 0 z, z > x 0 , let x w denote that element of y 0 z which Note that, x w and wx are not the same points, unless w = 0 and x = 0. Let M denote the σ-algebra of all subsets of X so that (X, M ) becomes a measurable space.Let ca(X, M ) be the space of all vector measures (signed measures) and define a norm where |p| n is a total variation measure of p and is given by It is known that ca(X, M ) is a Banach space with respect to the norm • defined by (2).Let µ be a σ-finite measure on X and let p ∈ ca(X, M ).We say p is absolutely continuous with respect to the measure µ if µ(E) = 0 implies p(E) = 0 for some E ∈ M. In this case we write p << µ.
For a fixed x 0 ∈ X, let M 0 be the smallest σ-algebra on S x 0 , containing {x 0 } and the sets S x , x ∈ y 0 x 0 .Let z ∈ X be such that z > x 0 and let M z denote the σ-algebra of all sets containing M 0 and the sets of the form S x for x ∈ x 0 z.Finally let L 1 µ (S z , R) denote the space of all µ -integrable nonnegative real-valued functions h on S z with the norm

Statement of the problem
Let µ be a σ-finite real measure on X.Given a p ∈ ca(X, M ) with p << µ, consider the abstract measure delay differential equation (in short AMDDE), involving the delay w, where q is a given known vector measure, dp/dµ is a Radon-Nikodym derivative of p with respect to µ and f : Definition 3.1 Given an initial real measure q on M 0 , a vector p ∈ ca(S z , M z ) (z > x) is said to be a solution of AMDDE (4) if A solution p of AMDDE (4) on x 0 z will be denoted by p(S x 0 , q).
In the following section we shall prove the main existence theorem for AMDDE (4) under suitable conditions on f.We shall use the following form of the Leray-Schauder's nonlinear alternative.See Dugundji and Granas [5].
Theorem 3.1 Let B(0, r) and B[0, r] denote respectively the open and closed balls in a Banach space X centered at the origin 0 of radius r, for some r > 0. Let T : B[0, r] → X be a completely continuous operator.Then either (i) the operator equation T x = x has a solution in B[0, r], or (ii) there exists an u ∈ X with u = r such that u = λT u for some 0 < λ < 1.

Existence and Uniqueness Theorems
We need the following definition in the sequel.
(ii) (y, z) → β(x, y, z) is continuous for almost everywhere µ on x ∈ x 0 z, and (iii) for each given real number ρ > 0 there exists a function We consider the following set of assumptions.
(A1) For any z > x 0 , the σ-algebra M z is compact with respect to the topology generated by the pseudo-metric d defined by (A3) q is continuous on M z with respect to the Pseudo-metric d defined in (A1).
Proof.Let X = ca(S z , M z ) and consider an open ball B(0, r) in ca(S z , M z ) centered at the origin and of radius r, where the real number r > 0 satisfies (5).Define an operator T from B[0, r] into ca(S z , M z ) by We shall show that the operator T satisfies all the conditions of Theorem 3.1 on B[0, r].
Step I: First we show that T is continuous on B[0, r].Let {p n } be a sequence of vector measures in B[0, r] converging to a vector measure p. Then by Dominated Convergence Theorem, and, so T is a continuous operator on B[0, r].
Step II: Next we show that T (B[0, r]) is a uniformly bounded and equi-continuous set in ca(S z , M z ).Let p ∈ B[0, r] be arbitrary.Then we have p ≤ r.Now by the definition of the map T one has Therefore for any We know the set-identities Therefore we have From the continuity of q on M 0 it follows that This shows that T (B[0, r]) is an equi-continuous set in ca(S z , M z ).Thus T (B[0, r]) is uniformly bounded and equi-continuous set in ca(S z , M z ), so it is compact in the norm topology on ca(S z , M z ).Now an application of Arzelá-Ascoli Theorem yields that T (B[0, r]) is a compact subset of ca(S z , M z ).As a result T is a continuous and totally bounded operator on B[0, r].Hence an application of Theorem 3.1 yields that either x = T x has a solution or the operator equation x = λT x has a solution u with u = r for some 0 < λ < 1.We shall show that this later assertion is not possible.We assume the contrary.Then there is an u ∈ X with u = r satisfying u = λT u for some 0 < λ < 1.Now for any E ∈ M z , we have µ ψ( u ).Substituting u = r in the above inequality, this yields r ≤ q + φ L 1 µ ψ(r), which is a contradiction to the inequality (5).
Hence the operator equation p = T p has a solution v with v ≤ r.Consequently the AMDDE (4) has a solution p = p(S x 0 , q) in B[0, r].This completes the proof.
To prove the uniqueness theorem, we consider the following AMDDE dr dµ = g(x, r(S x ), r(S xw )) a.e.[µ] on where g : S z × IR + × IR + → IR + and g(x, r(S x ), r(S xw )) is µ-integrable for each r ∈ ca(S z , M z ) with r ≥ 0, and g(x, y, z) is nondecreasing in y, z almost everywhere [µ] on x 0 z.Theorem 4.2 Assume that the function g satisfies all the conditions of theorem 4.1 with the function f replaced by g.Suppose further that [µ] on x 0 z and the identically zero measure is the only solution of AMDDE ( 7) on M z .Then AMDDE (4) has at most one solution on M z .
Proof.Suppose that AMDDE (4) has two solutions, namely p 1 and p 2 on M z .Then we have Therefore, Since AMDDE (7) has a identically zero function on M z , one has Therefore AMDDE has at most one solution on M z .This completes the proof.

Special case
In this section it is shown that, in a certain situation, the AMDDE (4) reduces to an ordinary differential-difference equation where g is continuous real function on [x 0 − w, x 0 ], and f satisfies Carathéodory conditions.Let X = IR, µ = m, the Lebesgue measure on IR, S xw = (−∞, x], x ∈ IR, and q a given real Borel measure on M 0 .Then equation ( 4) takes the form It will now be shown that, the equations ( 8) and ( 9) are equivalent in the sense of the following theorem.Theorem 5.1 Let q({x}) = 0, x ∈ [x 0 − w, x 0 ].Then (a) to each solution p = p(S x 0 , q) of ( 9) existing on [x 0 , x 1 ), there corresponds a solution y of (8) satisfying (b) Conversely, if g is a continuous function of bounded variation on [x 0 − w, x 0 ], then to every solution y(x) of ( 8), there corresponds a solution p(S x 0 , q), of (9) existing on [x 0 , x 1 ) with a suitable initial measure q.
Proof.(a) Let p = p(S x 0 , q) be a solution of ( 9), existing on [x 0 , x 1 ).Define a real Borel measure p 1 on IR as follows. and Define the functions y 1 (x), y(x) and g(x) by The condition q({x}) = 0, x ∈ [x 0 − w, x 0 ], the definition of the solution p, and the definitions of y(x), g(x) imply that Hence by [8] (Theorem 8.14, p. 163) g is continuous on [x 0 − w, x 0 ].Now for each x ∈ [x 0 − w, x 1 ) we obtain from (10) and the definition of y(x) Since p is a solution of (9) we have p << m on [x 0 , x 1 ).Hence y(x) is absolutely continuous on [x 0 , x 1 ).This shows that y (x) exists a.e. on [x 0 , x 1 ).Now for each x ∈ [x 0 , x 1 ), we have, by virtue of ( 11) and ( 9) This proves that y(x) is a solution of ( 8) on [x 0 , x 1 ) satisfying (b) Let y(x) be a solution of (8) existing on [x 0 , x 1 ], where g is continuous and of bounded variation on [x 0 − w, x 0 ].Define the function g 1 on IR as follows.
Clearly g 1 ∈ N BV (where NBV is the class of left continuous functions φ of bounded variation such that φ(x) → 0 as x → ∞).Hence by [8] [Theorem 8.14, p. 163] there exists a real Borel measure q 1 on IR, such that, Let us now define the initial measure q on M 0 as follows.
From (12), (13) and the definition of q we have Similarly corresponding to the function y(x) which is a solution of (8) on [x 0 , x 1 ), we can construct a real Borel measure p on M x 1 , such that, Since y(x) is a solution of (8) we have for x ∈ [x 0 , x 1 ) Hence by ( 14) it follows that This shows that p is a solution of ( 9) on [x 0 , x 1 ) and the proof of (b) is complete.
Remark 5.1 In proving (b) part of the above theorem we required g ∈ BV.That is not surprising, since g 1 is constructed from g, such that, g 1 ∈ N BV.
Remark 5.2 Theorem 5.1 shows that our results for the equation ( 4) are general in the sense that they include the corresponding results for the equation (8).
Remark 5.3 If we allow w to be zero then S xw = S x 0 for each x ≥ x 0 .Hence if we define the initial measure q by q(S x 0 ) = α, q(E) = 0 if E = S x 0 , the equation ( 4) takes the form which is the AMDDE studied in [9], [10].Thus our results include as particular cases, the results in [9], [10].

Examples
Example 1.Let X = IR, S x = (−∞, x], x 0 = 0, w = 2 and M 0 be the σ-algebra defined on (−∞, 0].Define an initial measure q on M 0 as follows Define a real measure µ by where N is the set of natural numbers.Consider the AMDDE The above AMDDE is equivalent to It is not difficult to show that the operator T defined by the right hand side of ( 17) is a contraction on ca(R, M ) with the usual total variation norm.Hence AMDDE (15)-( 16) has a unique solution on [0, ∞).

Remark 6.1
The above examples suggest a method to compute the solution of an AMDDE, in the particular case when f (x, y, z) is linear in y and z.