A SYSTEM OF ABSTRACT MEASURE DELAY DIFFERENTIAL EQUATIONS

B. C. Dhage^{1}, D. N. Chate^{1} and S. K. Ntouyas^{2}

1Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur Maharashtra, India e-mail: bcd20012001@yahoo.co.in

2 Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece e-mail: sntouyas@cc.uoi.gr

Abstract 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.

Key words and phrases: Abstract measure delay differential equation, existence, fixed point. AMS (MOS) Mathematics Subject Classifications: 34G99

1 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.

2 Preliminaries

Let IR denote the real line, IR^{n} an Euclidean space with repect to the norm |·|_{n} defined by

|x|_{n} = max{|x_{1}|, …, |x_{n}|} for x = (x_{1}, …, x_{n}) ∈ IR^{n}.  (1)

Let X be a real Banach space with any convenient norm ∥ · ∥. For any two points x, y in X, the segment \overline{xy} in X is defined by

\overline{xy} = \{z ∈ X|z = x + r(y − x), 0 ≤ r ≤ 1\}.

Let x_{0} and y_{0} be two fixed points in X, such that \overline{0y_{0}} ⊂ \overline{0x_{0}}, where 0 is the zero vector of X. Let z be a point of X, such that \overline{0x_{0}} ⊂ \overline{0z}. For this z and x ∈ \overline{y_{0}z}, define the sets S_{x} and \overline{S}_{x} as follows

S_{x} = {rx : −∞ < r < 1}

and

\overline{S}_{x} = \{rx : −∞ < r ≤ 1\}.

For x_{1}, x_{2} ∈ \overline{y_{0}z}, we write x_{1} < x_{2} (or x_{2} > x_{1}) if \overline{y_{0}x_{1}} ⊂ \overline{y_{0}x_{2}}. Let the positive number ∥x_{0} − y_{0}∥ be denoted by w. For each x ∈ \overline{x_{0}z}, z > x_{0}, let x_{w} denote that element of \overline{y_{0}z} which

x_{w} < x, ∥x − x_{w}∥ = w.

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 ∥ · ∥ on ca(X, M) by

∥p∥ = |p|_{n}(X)  (2)

where |p|_{n} is a total variation measure of p and is given by

|p|_{n}(X) = ∑_{i=1}^{∞} |p(E_{i})|_{n}, ∀E_{i} ⊂ X.  (3)

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 \overline{S_{x_{0}}}, containing \{x_{0}\} and the sets S_{x}, x ∈ \overline{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 \overline{S}_{x} for x ∈ \overline{x_{0}z}. Finally let L_{μ}^{1}(S_{z}, ℝ) denote the space of all μ -integrable nonnegative real-valued functions h on S_{z} with the norm ∥ · ∥_{L_{μ}^{1}} defined by

∥h∥_{L_{μ}^{1}} = ∫_{S_{z}} |h(x)| dμ.

3 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,

\begin{matrix}\frac{dp}{dμ} = f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}})), a.e. [μ] on \overline{x_{0}z}, \\ p(E) = q(E), E ∈ M_{0}, \end{matrix}  (4)

where q is a given known vector measure, dp/dμ is a Radon-Nikodym derivative of p with respect to μ and f : S_{z} × IR^{n} × IR^{n} → IR^{n} is such that f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}})) is μ-integrable for each p ∈ ca(S_{z}, M_{z}).

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

(i) p(E) = q(E), E ∈ M_{0},

(ii) p << μ on \overline{x_{0}z},

(iii) p satisfies (4) a.e. [μ] on \overline{x_{0}z}.

Remark 3.1 The AMDDE (4) is equivalent to the abstract measure integral equation

p(E) = \{ \begin{matrix}∫_{E} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ, & E ∈ M_{z}, E ⊂ \overline{x_{0}z} \\ q(E), & E ∈ M_{0}. \end{matrix}

A solution p of AMDDE (4) on \overline{x_{0}z} will be denoted by p(\overline{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.

4 Existence and Uniqueness Theorems

We need the following definition in the sequel.

Definition 4.1 A function β : S_{z} × IR^{n} × IR^{n} → IR^{n} is said to satisfy conditions of Carathéodory or simply it is Carathéodory if

(i) x → β(x, y, z) is μ-measurable for each (y, z) ∈ IR^{n} × IR^{n}.

(ii) (y, z) → β(x, y, z) is continuous for almost everywhere μ on x ∈ \overline{x_{0}z}, and

(iii) for each given real number ρ > 0 there exists a function h_{ρ} ∈ L_{μ}^{1}(S_{z}, IR) such that

|β(x, y, z)| ≤ h_{ρ}(x) a.e. [μ] x ∈ \overline{x_{0}z}, for each y, z ∈ IR with |y| ≤ ρ, |z| ≤ ρ.

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

d(E_{1}, E_{2}) = |μ|_{n}(E_{1} △ E_{2}), E_{1}, E_{2} ∈ M_{z}.

(A2) μ({x_{0}}) = 0.

(A3) q is continuous on M_{z} with respect to the Pseudo-metric d defined in (A1).

(A4) The function f(x, y, z) is L_{μ}^{1}-Carathéodory.

(A5) There exists a function ϕ ∈ L_{μ}^{1}(S_{z}, IR^{+}) such that ϕ(x) > 0 a.e. [μ], x ∈ S_{z} and a continuous and nondecreasing function ψ : [0, ∞) → (0, ∞) such that

|f(x, y, z)|_{n} ≤ ϕ(x)ψ(max \{|y|_{n}, |z|_{n}\}) a.e. [μ] on \overline{x_{0}z}, ∀y ∈ IR^{n}, ∀z ∈ IR^{n}.

Theorem 4.1 Suppose that assumptions (A1)–(A5) hold. Further if there exists a real number r > 0 such that

r > ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}ψ(r)  (5)

then AMDDE (4) has a solution on M_{z}.

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

T p(E) = \{ \begin{matrix}∫_{E} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ, & E ∈ M_{z}, E ⊂ \overline{x_{0}z} \\ q(E), & E ∈ M_{0}. \end{matrix}

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,

lim_{n} T p_{n}(E) = lim_{n→∞} ∫_{E} f(x, p_{n}(\overline{S}_{x}), p_{n}(\overline{S}_{x_{w}}))dμ

= ∫_{E} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ
= T p(E)

for all E ∈ M_{z}, E ⊂ \overline{x_{0}z}. Similarly if E ∈ M_{0}, then

lim_{n} T p_{n}(E) = q(E) = T p(E)

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

T p(E) = \{ \begin{matrix}∫_{E} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ, & if E ∈ M_{z}, E ⊂ \overline{x_{0}z} \\ q(E), & if E ∈ M_{0}. \end{matrix}

Therefore for any E = F ∪ G, F ∈ M_{0} and G ∈ M_{z}, G ⊂ \overline{x_{0}z}, we have

|T p(E)|_{n} ≤ |q(E)|_{n} + ∫_{E} |f(x, p(\overline{S}_{x}), p(\overline{S}_{x}))|_{n}dμ
≤ ∥q∥ + ∫_{E} ϕ(x)ψ(max\{|p(\overline{S}_{x})|_{n}, |p(\overline{S}_{x_{w}})|_{n}\})dμ
≤ ∥q∥ + ∫_{E} ϕ(x)ψ(∥p∥)dμ
≤ ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}ψ(r)

for all E ∈ M_{z}. By definition of the norm ∥ · ∥ we have

∥T p∥ = |T p|_{n}(S_{z})
≤ ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}ψ(r).

This shows that the set T (B[0, r]) is uniformly bounded in ca(S_{z}, M_{z}).

Now we show that T (B[0, r]) is an equi-continuous set in ca(S_{z}, M_{z}). Let E_{1}, E_{2} ∈ M_{z}. Then there are sets F_{1}, F_{2} ∈ M_{0} and G_{1}, G_{2} ∈ M_{z} with G_{1}, G_{2} ⊂ \overline{x_{0}z}, and

F_{i} ∩ G_{i} = ∅, i = 1, 2.

We know the set-identities

G_{1} = (G_{1} − G_{2}) ∪ (G_{2} ∩ G_{1}) and G_{2} = (G_{2} − G_{1}) ∪ (G_{2} ∩ G_{1}).  (6)

Therefore we have

T p(E_{1}) − T p(E_{2}) = q(F_{1}) − q(F_{2})
+ ∫_{G_{1}−G_{2}} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ − ∫_{G_{2}−G_{1}} f(x, p(\overline{S}_{x}), p(\overline{S}_{x_{w}}))dμ.

Since f(x, y, z) is L_{μ}^{1}- Carathéodory, we have that

|T p(E_{1}) − T p(E_{2})|_{n} ≤ |q(F_{1}) − q(F_{2})|_{n} + ∫_{G_{1}△G_{2}} |f(x, p(\overline{S}_{x}), p_{n}(\overline{S}_{x_{w}}))|_{n}dμ
≤ |q(F_{1}) − q(F_{2})| + ∫_{G_{1}△G_{2}} h_{r}(x)dμ.

Assume that d(E_{1}, E_{2}) = |μ|_{n}(E_{1} △ E_{2}) → 0. Then we have E_{1} → E_{2} and consequently F_{1} → F_{2} and |μ|_{n}(G_{1} △ G_{2}) → 0. From the continuity of q on M_{0} it follows that

|T p(E_{1}) − T p(E_{2})|_{n} ≤ |q(F_{1}) − q(F_{2})|_{n} + ∫_{G_{1}△G_{2}} h_{r}(x)dμ
→ 0 as E_{1} → E_{2}.

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 E = F ∪ G, where F ∈ M_{0} and G ⊂ \overline{x_{0}z} satisfying F ∩ G = ∅.

Now

u(E) = λT u(E) = \{ \begin{matrix}λq(F), & F ∈ M_{0} \\ λ ∫_{G} f(x, u(\overline{S}_{x}), u(\overline{S}_{x_{w}}))dμ, & G ∈ M_{z}, G ⊂ \overline{x_{0}z}. \end{matrix}

Therefore

|u(E)|_{n} = |λq(F)|_{n} + |λ ∫_{G} f(x, u(\overline{S}_{x}), u(\overline{S}_{x_{w}}))dμ|
≤ ∥q∥ + |∫_{G} |f(x, u(\overline{S}_{x}), u(\overline{S}_{x_{w}}))|_{n}dμ|
≤ ∥q∥ + ∫_{G} ϕ(x)ψ(max \{|p(\overline{S}_{x})|_{n}, |u(\overline{S}_{x_{w}})|_{n}\})dμ
≤ ∥q∥ + ∫_{G} ϕ(x)ψ(∥u∥)dμ
= ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}ψ(∥u∥).

This further implies that

∥u∥ = |u|_{n}(S_{z}) ≤ ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}ψ(∥u∥).

Substituting ∥u∥ = r in the above inequality, this yields

r ≤ ∥q∥ + ∥ϕ∥_{L_{μ}^{1}}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

\begin{matrix}\frac{dr}{dμ} = g(x, r(\overline{S}_{x}), r(\overline{S}_{x_{w}})) a.e. [μ] on \overline{x_{0}z} \\ r(E) = 0, E ∈ M_{0}, \end{matrix} \}  (7)

where g : S_{z} × IR^{+} × IR^{+} → IR^{+} and g(x, r(\overline{S}_{x}), r(\overline{S}_{x_{w}})) 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 \overline{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

|f(x, y, z) − f(x, y_{1}, z_{1})|_{n} ≤ g(x, |y − y_{1}|_{n}, |z − z_{1}|_{n}) a.e. [μ] on \overline{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

p_{1}(E) = q(F) + ∫_{G} f(x, p_{1}(\overline{S}_{x}), p_{1}(\overline{S}_{x_{w}}))dμ

and

p_{2}(E) = q(F) + ∫_{G} f(x, p_{2}(\overline{S}_{x}), p_{2}(\overline{S}_{x_{w}}))dμ,

for all E ∈ M_{z} with E = F ∪ G, F ∈ M_{0}, G ⊂ \overline{x_{0}z} and F ∩ G = ∅. Now

p_{1}(E) − p_{2}(E) = ∫_{G} f(x, p_{1}(\overline{S}_{x}), p_{1}(\overline{S}_{x_{w}}))dμ − ∫_{G} f(x, p_{2}(\overline{S}_{x}), p_{2}(\overline{S}_{x_{w}}))dμ
= ∫_{G}[f(x, p_{1}(\overline{S}_{x}), p_{1}(\overline{S}_{x_{w}})) − f(x, p_{2}(\overline{S}_{x}), p_{2}(\overline{S}_{x_{w}}))]dμ.

Therefore,

|p_{1}(E) − p_{2}(E)|_{n} ≤ ∫_{G} |f(x, p_{1}(\overline{S}_{x}), p_{1}(\overline{S}_{x_{w}})) − f(x, p_{2}(\overline{S}_{x}), p_{2}(\overline{S}_{x_{w}}))|_{n}dμ
≤ ∫_{G} g(x, |p_{1} − p_{2}|_{n}(\overline{S}_{x}), |p_{1} − p_{2}|_{n}(\overline{S}_{x_{w}}))dμ.

Since AMDDE (7) has a identically zero function on M_{z}, one has ∥p_{1} − p_{2}∥ = |p_{1} − p_{2}|_{n}(S_{z}) = 0 ⇒ p_{1} = p_{2}.

Therefore AMDDE has at most one solution on M_{z}. This completes the proof.

5 Special case

In this section it is shown that, in a certain situation, the AMDDE (4) reduces to an ordinary differential-difference equation

\begin{matrix}\frac{dy}{dx} = f(x, y(x), y(x − w)), x ≥ x_{0}, \\ y(x) = g(x), x ∈ [x_{0} − w, x_{0}], \end{matrix} \}  (8)

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, \overline{S}_{x_{w}} = (−∞, x], x ∈ IR, and q a given real Borel measure on M_{0}. Then equation (4) takes the form

\begin{matrix}\frac{dp}{dm} = f(x, p((−∞, x]), p((−∞, x − w])), \\ p(E) = q(E), E ∈ M_{0}. \end{matrix}  (9)

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(\overline{S}_{x_{0}}, q) of (9) existing on [x_{0}, x_{1}), there corresponds a

solution y of (8) satisfying

y(x) = g(x), x ∈ [x_{0} − w, x_{0}].

(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(\overline{S}_{x_{0}}, q), of (9) existing on [x_{0}, x_{1}) with a suitable initial measure q.

Proof. (a) Let p = p(\overline{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.

p_{1}((−∞, x)) = \{ \begin{matrix}0, & if x ≤ x_{0} − w, \\ p((−∞, x]) − p((−∞, x_{0} − w]), & if x_{0} − w < x < x_{1} \\ p((−∞, x_{1})), & if x ≥ x_{1}, \end{matrix}  (10)

and

p_{1}(E) = p(E), if E ⊂ [x_{0} − w, x_{1}).

Define the functions y_{1}(x), y(x) and g(x) by

\begin{aligned}y_{1}(x) & = p_{1}((−∞, x)), & x ∈ IR \\ y(x) & = y_{1}(x) + p((−∞, x_{0} − w]), & x ∈ [x_{0} − w, x_{1}) \end{aligned}

and

g(x) = y(x), x ∈ [x_{0} − w, x_{0}].

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

p_{1}({x}) = p({x}) = 0, x ∈ [x_{0} − w, x_{0}].

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)

\begin{aligned}y(x) & = y_{1}(x) + p((−∞, x_{0} − w]) \\ & = p_{1}((−∞, x)) + p((−∞, x_{0} − w]) \\ & = p(\overline{S}_{x_{w}}). \end{aligned}  (11)

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)

p([x_{0}, x]) = ∫_{[x_{0},x]}(dp/dm)dm,

that is,

p((−∞, x]) − p((−∞, x_{0}]) = ∫_{x_{0}}^{x}(dp/dm)dm.

This further implies that

p(\overline{S}_{x_{w}}) = p(\overline{S}_{x_{0}}) + ∫_{x_{0}}^{x} f(t, p(\overline{S}_{x}), p(\overline{S}_{x−w}))dt.

That is

y(x) = y(x_{0}) + ∫_{x_{0}}^{x} f(t, y(t), y(t − w))dt.

Hence

y′(x) = f(x, y(x), y(x − w)) a.e on [x_{0}, x_{1}).

This proves that y(x) is a solution of (8) on [x_{0}, x_{1}) satisfying

y(x) = g(x), x ∈ [x_{0} − w, x_{0}].

(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.

g_{1}(x) = \{ \begin{matrix}0, & if x < x_{0} − w, \\ g(x) − g(x_{0} − w), & if x_{0} − w ≤ x ≤ x_{0} \\ g(x_{0}) − g(x_{0} − w), & if x > x_{0}. \end{matrix}  (12)

Clearly g_{1} ∈ NBV (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,

q_{1}((−∞, x)) = g_{1}(x).  (13)

Let us now define the initial measure q on M_{0} as follows.

q((−∞, x]) = q_{1}((−∞, x)) + g(x_{0} − w), x ∈ [x_{0} − w, x_{0}],
q(E) = q_{1}(E), E ⊂ [x_{0} − w, x_{0}].

From (12), (13) and the definition of q we have

q(\overline{S}_{x_{w}}) = q((−∞, x]) = g(x), x ∈ [x_{0} − w, x_{0}].

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,

\begin{aligned}p(E) & = q(E), if E ∈ M_{0}, \\ p(\overline{S}_{x_{w}}) & = y(x), x ∈ [x_{0}, x_{1}). \end{aligned}  (14)

Since y(x) is a solution of (8) we have for x ∈ [x_{0}, x_{1})

y(x) = y(x_{0}) + ∫_{x_{0}}^{x} f(t, y(t), y(t − w))dt.

Hence by (14) it follows that

p(\overline{S}_{x_{w}}) − p(\overline{S}_{x_{0}}) = ∫_{[x_{0},x]} f(t, p(\overline{S}_{t}), p(\overline{S}_{t_{w}}))dm.

That is

p([x_{0}, x]) = ∫_{[x_{0},x]} f(t, p(\overline{S}_{t}), p(\overline{S}_{t_{w}}))dm.

In general, if E ∈ M_{x_{1}}, E ⊂ \overline{x_{0}x_{1}}, then

p(E) = ∫_{E} f(t, p((−∞, x], p((−∞, x − w]))dm.

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} ∈ NBV.

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 \overline{S}_{x_{w}} = \overline{S}_{x_{0}} for each x ≥ x_{0}. Hence if we define the initial measure q by

q(\overline{S}_{x_{0}}) = α, q(E) = 0 if E ≠ \overline{S}_{x_{0}},

the equation (4) takes the form

\frac{dp}{dμ} = f(x, p(\overline{S}_{x_{w}})), p(\overline{S}_{x_{0}}) = α

which is the AMDDE studied in [9], [10]. Thus our results include as particular cases, the results in [9], [10].

6 Examples

Example 1. Let X = IR, \overline{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

\begin{aligned}q(E) & = ∑_{n∈E∩\{−1,−2\}} 2^{n}, & if E ∩ \{−1, −2\} ≠ ∅ \\ & = 0, & if E ∩ \{−1, −2\} = ∅. \end{aligned}

Define a real measure μ by

\begin{aligned}μ(E) & = ∑_{n∈N∩(E)} \frac{1}{3^{n}} , & if E ⊂ IR, E ∩ N ≠ ∅ \\ & = 0, & if E ∩ N = ∅. \end{aligned}

where N is the set of natural numbers. Consider the AMDDE

\frac{dp}{dμ} = p(\overline{S}_{x}) + p(\overline{S}_{x−2}),  (15)
p(E) = q(E), E ∈ M_{0}.  (16)

The above AMDDE is equivalent to

p(E) = \{ \begin{matrix}∫_{E} p(\overline{S}_{x})dμ + ∫_{E} p(\overline{S}_{x−2})dμ, E ⊂ [0, ∞), \\ p(E) = q(E), E ∈ M_{0}. \end{matrix}  (17)

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, ∞).

We also observe that

p(\overline{S}_{1}) = p(\overline{S}_{0}) + ∫_{(0,1]} p(\overline{S}_{x})dμ + ∫_{(0,1]} p(\overline{S}_{x−2})dμ
= q(\overline{S}_{0}) + p(\overline{S}_{1})μ(\{1\}) + p(\overline{S}_{−1})μ(\{1\})
= 1 + \frac{1}{3}p(\overline{S}_{1})(1/2)
= 3/2.

Similarly

p(\overline{S}_{2}) = p(\overline{S}_{1}) + ∫_{(1,2]} p(\overline{S}_{x})dμ + ∫_{(1,2]} p(\overline{S}_{x−2})dμ
= p(\overline{S}_{1}) + p(\overline{S}_{2})μ(\{2\}) + p(\overline{S}_{0})μ(\{2\})
= 1 + \frac{3}{2} + \frac{1}{9}p(\overline{S}_{1}) + \frac{1}{12}
= 19/12.

Thus we have

p(\overline{S}_{0}) = \frac{3}{4}, p(\overline{S}_{1}) = \frac{3}{2}, p(\overline{S}_{0}) = \frac{57}{32}, and so on.

It is easy to verify that the sequence \{p(\overline{S}_{n})\}, n = 0, 1, 2, 3, … is convergent, showing thereby that the solution p of the above AMDDE is a finite measure.

Example 2. Let X = IR, μ the Lebesgue measure on IR, \overline{S}_{t} = [0, t], t > 0, and q(E) = μ(E), E ⊂ [0, 1]. Consider the AMDDE

\frac{dp}{dμ} = 6p(\overline{S}_{t−1}),

p(E) = q(E), E ⊂ [0, 1].

Here w = 1. For 0 ≤ t ≤ 1, we observe that

p(\overline{S}_{t}) = p([0, t]) = q([0, t]) = t.

If t ∈ [1, 2], we have

p(\overline{S}_{t}) = q(\overline{S}_{1}) + ∫_{[1,t]} 6p(\overline{S}_{s−1})ds
= 1 + ∫_{1}^{t} 6(s − 1)ds
= 1 + 3(t − 1)^{2}.

Again, if 2 ≤ t ≤ 3, we obtain

p(\overline{S}_{t}) = 6t + 6(t − 2)^{3} − 8,

and so on, the solution p can be found recursively 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.

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 inequalities and applications, Bull. Inst. Math. Acad. Sinica 18 (1989), 65-75.

[3] B. C. Dhage, Mixed monotonicity theorems for a system of abstract measure delay integro-differential equations, An. Stint. Univ. ”Al. I. Cuza” Iasi XLII (1996), 355-366.

[4] B. C. Dhage, D. N. Chate and S. K. Ntouyas, Abstract measure differential equations, (submitted).

[5] J. Dugundji and A. Granas, Fixed Point Theory, Monograhie Math. PNW, Warsaw, 1982.

[6] S. R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979), 497-506.

[7] S. R. Joshi and S. G. Deo, On abstract measure delay differential equations, An. Stiint. Univ. Al I. Cuza Iasi XXVI (1980), 327-335.

[8] W. Rudin, Real and Complex Analysis, McGraw-Hill Inc., New York, 1966.

[9] R. R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972), 503-510.

[10] R. R. Sharma, A measure differential inequality with applications, Proc. Amer. Math. Soc. 48 (1975), 87-97.

[11] G. R. Shendge and S. R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41(1983), 53-54.