Generalized Functional Differential Equations: Existence and Uniqueness of Solutions

We study generalized nonlinear functional differential equations arising in various applications, for instance in the control theory, or if there is a need to incorporate impulsive and/or delay effects into the underlying system. The main result of the paper provides a general existence and uniqueness theorem for such equations, and we also give many illustrative examples. The proofs are based on the theory of generalized Volterra operators in the spaces of continuous and discontinuous functions.


Introduction
We introduce a broad class of functional differential equations driven by a general measure (in the paper we call these equations generalized for brevity).The class includes ordinary, delay, impulsive, difference equations and their combinations as well as important types of equations with distributed control and equations with discontinuous noise (e.g. of Poisson type).We illustrate the general theory with several examples.However, we do not intend to present an exhaustive theory of the equations included in the examples treating them rather as auxiliary to the main framework.That is why the list of references related to the particular classes of equations considered below [1-3, 10, 12, 13] is by far not complete.We cite therefore only very few papers and refer the reader to the references in these and other papers for more information.
The analysis framework is organized in a way that has become customary in the contemporary theory of functional differential equations (see e.g.[4] and the references therein).An essential feature of this construction is to consider the initial (prehistory) function as a part of the equation itself, which in particular, gives an opportunity to include equations with unbounded delays and avoid "nasty" functional spaces.
To be able to establish the well-posedness of the initial value problem, we formulate and prove a fixed point theorem for generalized Volterra operators in L p -spaces with respect to an arbitrary measure, thus extending similar results proved in the series of papers [5,15,16].The fixed point theorem of the present paper can also be used in many other applications.
We start with a brief description of the functional spaces which we need to define a generalized functional differential equation.
Let μ be a σ-additive, finite measure defined on the family B of all Borel subsets of the interval [0, T], and let µ be its standard extension, i.e. a σ-additive, finite and complete measure which is defined on the minimal σ-algebra L containing B and all subsets of measure zero and which satisfies µ(E) = μ(E) for any E ∈ B. Any set from L will alternatively be called µ-measurable.
As usual, we say that a function y : [0, T] → R is µ-measurable if it satisfies the following condition: for any Borel subset B ⊂ R the set y −1 (B) ∈ L. The restriction of µ to the familiy of all µ-measurable subsets of an arbitrary set S ⊂ [0, T], S ∈ L, will again be denoted by µ.The Lebesgue integral of a µ-measurable function y defined on S will be denoted by S y(s) µ(ds).If µ = mes is the Lebesgue measure, then we will write S y(s) ds.The measure µ ⊗ ν stands below for the product of two measures µ and ν.The indicator (the characteristic function) 1 S of a set S is given by The space L p (S, R n , µ), 1 ≤ p < ∞ consists of all functions y : S → R n (more exactly, of equivalence classes), which are p-integrable with respect to (w.r.t.) the measure µ; the standard norm in this space is given by y L p = S |y(s)| p µ(ds) 1/p .The space L ∞ (S, R n , µ) contains all µ-bounded (i.e.bounded up to a set of measure zero), µ-measurable functions y : S → R n , the norm being defined y L ∞ = ess sup t∈ S |y(t)|.In the case S = [0, T] we will use the shorter notation Let us now define the space It contains all µmeasurable functions y : [0, T] → R n which are absolutely continuous w.r.t. the measure µ and whose "derivative", w.r.t.µ belongs to L p : y(s)µ(ds). (1.1) For notational convenience, we will assume that the functions from W p 1 have an auxiliary value at −0, which we will treat as the left-hand limit at 0. That is why we introduced the "interval" [−0, T] in the notation of W p 1 .From (1.1) we conclude that the functions x ∈ W p 1 are cadlag, (see e.g.[7]) i.e. they are right-continuous and have left-hand limits at any point t ∈ [0, T] including t = 0.The definition (1.1) also implies that the value of the jump of a function x ∈ W p 1 is equal to = 0 and for continuity of x at a point t of positive measure we have to require that y(t) = 0.
The definition (1.1) determines a one-to-one mapping x → (α, y) between the spaces W p 1 and R n × L p , and in our notation we may also write α = x(−0).The mapping x → y from (1.1) produces "differentiation" operator δ µ which can be used to indroduce a norm in W p 1 : With this definition, the spaces W p 1 and R n × L p become isometric.In the particular case of the Lebesgue measure µ = mes, we obtain the usual differentiation of an absolutely continuous function x: δ mes x = ẋ.In this case we also may write x(−0) = x(0) arriving at the standard space W p 1 ([0, T] , R n , mes) of absolutely continuous functions [4].The main target of the paper is the following generalized nonlinear differential equation: where F : W p 1 → L p is a given (nonlinear) operator and x ∈ W p 1 is an unknown function (solution) that should satisfy the initial condition (1.4) The central result of the paper describes the conditions providing existence and uniqueness of solutions of the initial value problem (1.3)- (1.4).
Using the introduced notation of the "derivative" of a function w.r.t. the measure µ we can rewrite the equation (1.3) as Applying the isomorphism between the spaces W p 1 and R n × L p described in (1.1) yields the following integral equation in the space W p 1 : Equivalently, we can rewrite (1.5) in the form of an integral equation w.r.t.y = δ µ x in the space L p : Both representations of the main equation (1.5) will be used below.Normally, the continuity assumption is required in existence and uniqueness theorems: (1.1) The operator F : However, we will in many cases only assume that the operator F has the following Volterratype property adjusted to arbitrary measures: for any t ∈ [0, T] such that µ([0, t]) > 0, the equality x(s) = x(s), s ∈ [−0, t) implies the equality (Fx)(s) = (F x)(s) s ∈ [0, t].In particular, if µ({0}) > 0, then the Volterra operator F produces the same value (Fx)(0) for any x ∈ W p 1 ([−0, T] , R n , µ) with the same auxiliary value (1.4).
Remark 1.1.At the points, where µ({t}) = 0, we can assume, without loss of generality, that in the definition of the Volterra property the intervals are equal, i.e. both are either [0, t] or [0, t).However, in the case µ({t}) > 0, it is essential that the intervals differ, i.e. that the image of a function completely depends on the values of the function at strictly preceding times.
A method of studying existence and uniqueness we propose in this paper goes back to the theory of generalized Volterra operators originally suggested by the second author, see e.g.[15]).We will apply this theory either to equation (1.6) or to equation (1.7).We stress that these results do not require continuity of the operator F. We also remark that some examples described in Section 2 are non-Volterra.These examples are only meant to illustrate the general algorithm of how to represent various equations with deviated argument in the standard form (1.3) (or (1.5)).This algorithm is an essential part of the theory of functional differential equations known as Azbelev's theory, see e.g.[4] and the references therein.
We also note that the existence and uniqueness in the case when F in (1.3) is an affine operator (more precisely, (Fx)(t) = [0,t) Q(t, s)dx(s) + f (t)) was studied in [10].
To be able to proceed with further analysis, we need some auxiliary results about the introduced functional spaces and mappings in these spaces.
First of all, we will often use the following "integration by parts formula": which holds for arbitrary functions u, v : [0, T] → R n of bounded variation and any points 0 ≤ t 1 < t 2 ≤ T.
Without loss of generality, we may assume that all functions of finite variation (in particular, functions belonging to W p 1 ) are cadlag.Therefore, we can always replace v(s + 0) with v(s) in formula (1.8).
Proposition 1.2.Let S be a µ-measurable subset of the interval [0, T].The linear integral operator (Qy)(t) ≡ S Q(t, s) y(s) µ(ds) is bounded as an operator from L p (S, R n , µ) to L q (S, R n , µ) (1 ≤ p, q < ∞) if the kernel Q : S × S → R n×n is a µ µ-measurable and satisfies the following condition: and the function ϑ, given as ϑ(t) ≡ Q(t, •) L p , belongs to the space L q (S, R, µ).
for almost all t ∈ S and all u ∈ R l , where b ≥ 0 and a ∈ L p (S, R, µ).

Some examples of the equation (1.3)
In this section we review the notions of a difference equation and its solutions as finite collections of vectors and describe the concept of a functional differential equation and its absolutely continuous solutions, which was suggested and developed by the participants of the Perm Seminar in Russia led by Prof. N. V. Azbelev [4].Let us also remark that a constantly growing interest to hybrid systems has initiated analysis of objects combining functional differential and difference equations [11].

Functional differential equations
The example below is a functional differential equation (see e.g.[4]) where ) is a (nonlinear) operator, mes is the Lebesgue measure, 1 ≤ p < ∞.In the results presented in the monograph [4] equation (2.1) is assumed to satisfy the following condition.
(2.1) The operator F : In Section 3 we describe more specific examples of equation (2.1).All of them include the Volterra property on F, which is not necessarily fulfilled in (2.1).

Nonlinear difference equations
By this we mean the following system of equations: where x 0 , . . ., x m ∈ R n are unknown vectors and x −1 = α is the initial condition.It is assumed that the functions f i : R (m+2)n → R n are continuous.In this case, the measure µ of a set S ⊂ [0, m] is equal to the number of integers contained in S. Now we put Then "the derivative" of x at t = 0, 1, . . ., m is given as while its values (δ µ x)(t) where t ∈ (i − 1, i), i = 1, . . ., m may be defined arbitrarily or may remain undefined, as µ((i − 1, i)) = 0. Indeed, for any t ∈ [i − 1, i) we have and similarly for t = m: and equation (2.2) becomes the functional differential equation (1.5).
Note that for the measure just defined we have

Examples with Volterra operators
In this section we assume that the operator F in (1.3) is Volterra.

Linear nonhomogeneous equation with the unknown function in the differential
This equation, which was studied in [10], is given by or, equivalently, by 2) The assumptions we put on the equation (1.3) are as follows.
In some cases we find it convenient to extend the function Q(t, s) to the set [0, T] × [0, T] assuming that Q(t, s) = 0 for the corresponding (t, s).
In [10] it is shown that under the assumptions (3.1a)-(3.1d)equation (3.1) with the initial condition (1.4) has a unique solution x ∈ W p 1 ([−0, T] , R n , µ) for any α ∈ R n .The proof suggested in [10] is based on the standard iteration procedure.
Specific examples of the equation (3.1) can be found in [10].Below we generalize these examples to the nonlinear case.

Nonlinear differential equations with delay
In this subsection we demonstrate how delay equations can be written in the standard form (2.1).Note that we consider only the case of distributed delays.Some more involved examples can be found in [4]. Let It is assumed that this equation is supplied with the "prehistory" condition: Following [4] we will now include this condition into the equation (3.3) in such a way that the initial condition (1.4) remains unchanged.We separate conditions for s < 0 and s = 0, in particular, for the following reason: since ψ is often assumed to belong to a space consisting of measurable functions, the functional ψ(•) −→ ψ(0) may have no sense.On the other hand, if ψ is continuous and the solutions of (3.3) are supposed to be continuous for all t ∈ (−∞, T], as well, then we can assume that ψ(0) = α.Let us however stress that even in this continuous case separating the conditions for s < 0 and s = 0 may be technically useful (see e.g.[4]).
We now list the assumptions on R(t, s) and f (t, u), which we need to be able to rewrite (3.3) in the form (2.1).Let us choose two real numbers p, q ∈ [1, ∞) and a natural number m.
For instance, the equation with the delay condition h(t) ≤ t, t ∈ [0, T], can be rewritten in the form (3.3) if we put , and using the integration by parts formula (1.8) we obtain we see that ).This operator is Volterra.

Linear difference equations with delay
We describe a particular case of the difference equation (2.2) which can also be represented in the form (3.1) or (3.2).Let where we assume that A ij are n × n−matrices and g 0 , g i are n−vectors, i = 1, . . ., m, j = 0, . . ., m − 1.Using the equality x j = x −1 + ∑ j p=0 ∆x p we rewrite equation (3.6) as follows: Then we define Q ij = ∑ i−1 p=j A ip , and represent equation (3.6) as As in Subsection 2.2, the measure µ of a set S ⊂ [0, m] is now equal to the number of integers contained in S. The µ ⊗ µ-measurable function is defined as integers, while the values of Q(t, s) at the points (t, s), where at least one component is not an integer, are not needed.Then we define the µ-measurable function g : [0, m] → R n by setting g(i) = g i and observing that for t ∈ (i − 1, i), i = 1, . . ., m the values g(t) may be disregarded.Finally, we choose an arbitrary 1 ≤ p ≤ ∞ and define the function of this function can be defined arbitrarily (or remain undefined) for any t ∈ (i − 1, i), i = 1, . . ., m.

Impulsive differential equations with delay
We return to the functional differential equation (2.1), but in this subsection we assume that a countable (in particularly, finite) set T ⊂ (0, T] is given and at any time τ ∈ T the solution can make a jump ∆x(τ) ≡ x(τ) − x(τ − 0).To formalize the notion of such an impulsive functional differential equation we suppose that to any τ ∈ T a positive number M(τ) is assigned in such a way that the series ∑ τ∈T M(τ) converges.Then we are able to define a finite measure µ on [0, T] by putting where ν τ is the Dirac measure at τ.In other words, the measure µ(S) of a set S ∈ L is equal to the sum of its Lebesgue measure mes(S) and ∑ τ∈T ∩S M(τ).
Below we consider an impulsive functional differential equation under the following assumptions.
The behavior of the solution x(•) outside T is governed by equation (2.1) with the nonlinear operator , satisfying the following condition.
Further, we assume that the value of the jump ∆x(τ) at time τ ∈ T may only depend on the values of the solution x(t) for t ∈ [0, τ).More precisely, we impose the following requirement on the jumps: where the vector functional (possibly nonlinear) Υ : T × W p 1 ([−0, T], R n , µ) → R n satisfies the following assumptions.
To see it, we put We claim further that the operator F : W p 1 → L p (defined by (3.9)) becomes continuous if the following conditions are fulfilled.
To prove it, we choose any convergent sequence Let us now look closer at the affine case.In this case, the equation (2.1) converts into We observe as well that defining the measure µ by the formula (3.7) yields the following "derivative" of a function x which is absolutely continuous w.r.t.this measure: Therefore, an arbitrary affine and bounded vector functional which is affine and bounded w.r.t. the second variable, becomes if the following three assumptions are fulfilled.
(3.4f)For any τ ∈ T the n × n-matrix function W (τ, •) belongs to L p (S(τ), R n×n , mes), where and the function ϑ, defined by (3.4g)For any τ ∈ T the n × n-matrix function W(τ, •) ∈ L p (S(τ), R n×n , µ T ), where p and S(τ) are defined above, and v ∈ L p ([0, T], R n , µ T ), where the function v is defined by

Volterra operators in the space L p ([0, T], R n , µ)
In the sequel we will always assume that the (nonlinear) operator The operators considered in Section 3 have this property, including the operator defined by (3.9) if the assumptions (3.4a), (3.4b) are fulfilled.We just remark that, unlike the usual derivative, "the differentiation" of which belongs to the space W Let us make use of the representation (1.7) of the functional differential equation (1.3) and rewrite the initial value problem for this equation with the initial condition (1.4) as an equation in the space L p ([0, t] , R n , µ) This is an equation w.r.t.y = δ µ x.Given x ∈ W p 1 ([−0, T] , R n , µ), the norm of the restriction of the image y = Fx : [0, T] → R n to the subinterval [0, t), calculated in the space L p ([0, t) , R n , µ), is, in general, a discontinuous function of t.This is due to the fact that the measure µ is not assumed to be absolutely continuous w.r.t. the Lebesgue measure.This fact explains why a straightforward application of the classical Volterra theory and its known generalizations to equation (1.7) is impossible.Below we apply an idea of a generalized Volterra property which was suggested in the paper [15].
Let B be a normed space.Suppose that to any γ ∈ [0, 1] we assign an equivalence relation υ(γ) for the elements of the space B. Assume further that the family Finally, we assume that the relations υ(γ) ∈ V are closed under addition and multiplication by scalars, i.e. that for every γ ∈ (0, 1) and any x, x, y, y ∈ B, λ we have Definition 4.2.We say that an operator Φ : B → B is Volterra w.r.t. the family V of equivalence relations (satisfying the above conditions) if for every γ ∈ (0, 1) and any x, y ∈ B the equality (x, y) ∈ υ(γ) implies the equality (Φx, Φy) ∈ υ(γ).
Proof.The proof consists of verifying the conditions of Corollary from Theorem 4 proved in the paper [15, p. 448] for the operator (4.2).These conditions describe the property which in this paper is called "local contraction" and which guarantee unique solvability of the equation (4.1) and hence of the initial value problem (1.3), (1.4).
Remark 4.5.The operators F and (4.2) in the above theorem do not need to be continuous.An example of a local contraction in the space L p ([0, T] , R n , mes) which is nowhere continuous, can be found in [16].
Further steps are performed similarly.Let us show that the final result of the described procedure consisting of countably or finitely many (if ξ n = T for some n) steps yields a global or an unextendable solution.Indeed, we would otherwise obtain numbers , T, for which ξ i ≤ T < T and z ξ i L p ([0,ξ 2 ),R n ,µ) ≤ .after any step.But in this case we would obtain the estimate ∆(r i , ε) ≥ ∆( , ε) for any i.Therefore µ (ξ i−1 , ξ i ] ≥ ∆( , ε), which contradicts the assumption that µ is a finite measure on [0, T].

Local solutions of the initial value problem for impulsive equations
Theorem 4.7 gives us an opportunity to prove one more result on solvability of the impulsive system (2.1), (3.8).
The functional which determines the size of the jumps, can be considered acting from T × W 1 1 ([0, 1], R, µ) to R, and it also satisfies the assumptions (3.4b), (3.4c), (3.4e) for p = 1.This gives us opportunity to search for solutions in the space W 1  1 ([0, 1], R, µ).Moreover, the functional Υ satisfies the estimate x W 1 1 ≤ r.Thus, all the assumptions of Theorem 4.8 are fulfilled, so that any initial value x(0) = α gives rise to a solution for any real α.

Outlook
The central results of the present paper can be used to a further development of the theory of functional differential equations with an arbitrary driven measure, in the spirit of the monograph [4].This development might include boundary value problems, stability analysis, control theory, difference equations, stochastic functional differential and difference equations driven by Poisson-type noises etc.Some preliminary results can e.g.be found in the papers [2,3,8,10,11,13].