Variation of Parameters for Causal Operator Differential Equations

The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters ( ) 0 0 , t x . The initial information is described ( ) ( ) x t t φ = for almost all 0 t t ≤ and ( ) 0 0 t φ φ = . We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP.

t φ φ = .We will study the nonlinear variation of parameters (NVP) for this

Definitions and Example of Nonanticipative Operators
An important feature of ordinary differential equations is that the future behavior of solutions depends only upon the present (initial) values of the solution.There are many physical and social phenomena which have hereditary dependence.That means the future state of the system depends not only upon the present state, but also upon past information (see [1]- [6]).
Twins before the time of conception share all of their genetic history and may go to a different path in their future life.We are going to study the phenomenon which can be formulated in principle that the "present" events are independent of the "future".These kinds of events are called nonanticipation or causal events.

Definition 1.1: Continuous Nonanticipating System
A mapping of T from the space of functions Y into itself is said to be a nonanticipating mapping if for every fixed s in the real line R, ( )( ) ( )( ) The following are examples of continuous anticipating operators (see Naylor and Sell 1982 [7]).Definition 1.2: A mapping T of Y into itself is said to be causal if for each integer N, whenever two inputs In other words, if the inputs x and y agree up to some time N, then the outputs T(x) and T(y) agree up to time N.In particular, T(x) and T(y) agree up to time N no matter what the inputs x and y are in the future beyond N.The events in the past and present are independent from the future.
∫ This is a nonanticipating mapping if and only if ( ) 0 h t s − = for almost all 0 t s − < .This Voltera integral mapping shows that ( )( ) Notice that when a mapping is not nonanticipating it will be an anticipating mapping, meaning that the past and the present depend on the future.Anticipating (anticausal) Mapping: This is a mapping that the future output ( ) ( ) ( ) Since, for fixed real number s, the fact that ( ) ( )

Nonanticipating Operator Differential Equation
Notations.Let S be the interval of all nonpositive numbers.Let I be the compact for almost all t s < implies that ( )( ) ( )( ) An operator P from a subset D of Y into Z is said to be Lipschitzian if there exists a constant b such that ( ) ( )  [6]).
When an operator T is nonanticipating, the future values of the input will have no effect on the present state.One can prove that the composition and the Cartesian product of nonanticipating and Lipschitzian operators are Nonanticipating and Lipschitzian.Furthermore, the operator F induced by the function f is a well defined, nonanticipating, and Lipschitzian operator.

T T y t T y t T y t ⊕ = +
for every y in

( )
, M J Y and t in I. Proof: First let us prove that the direct sum operator is a nonanticipating operator.Assume that two functions y and z are in the space of ( ) , M J Y and for some point s in the interval I we have ( ) ( ) for almost all t s < .
Since T 1 and T 2 are nonanticipative, then ( )( ) ( )( ) T y t T z t = for almost all t s < .
These two equalities will imply that

T y T y t T z T z t + = +
for almost all t s

T T y t T T z t ⊕ = ⊕
for almost all t s < .
This will imply that ( )( ) T y t and ( )( ) Now let us prove that the operator

T y t T z t T T y t T T z t T y T y t T z T z t T y t T z t T y t T z t
Since both operators are Lipschitzian, the right hand side will be represents the ess.sup norm in the space measurable functions ( ) and take the essential supremum norm on the left hand side of the above relation then it will be ( ) ( ) for all y and z in the domain D. This proves that the direct sum operator is .The operator ( ) ( )( )

T y T T y =
⊕ is nananticipating and Lipschitzian.
Nonanticipating Deterministic Dynamical System: Assume that the operator T is nonanticipating and Lipscitzian.The behavior of a dynamic system is known as an after effect differential equation with the initial domain there exists a unique solution y to the system (2.4).
Equations of this type arise in many mathematical modeling problems.In a simplest case, T as a constant delay operator can be applied (see Hale 77 [8] and Driver 77 [9]).The following is a single species growth model with time delay.
Example 2.3: A single species model with delay can be described by ( ) ( ) ( ) where r is the growth rate of the species y, and K is called the environment capacity for y.
Our goal is to investigate the conditions which guarantee the solution of the system (2.4) when there is a random perturbation in the system.Solution to the Nonanticipating Operator differential Equations: The following operator differential equation when G is a nonanticipating operator from the initial domain ( ) for almost all t in the interval I.We define that a function y from the space M(I,Y) is a solution to the nonanticipating operator differential equation if it is strongly differentiable and satisfies the system (2.5) (see Bogdan 1981 [11], Bogdan 1982 [12], Ahangar 1989, [1], and Ahangar 1986 [2]).We accept the following theorem without proof.Note: The purpose of this paper is to develop a generalized nonlinear variation of parameters formula, analogous to Alekseev's result (see Alekseev 1961 [13]).The generalization is listed below: 1) The classical existence and uniqueness theorem for the solution of abstract Cauchy problems no longer holds if the underlying space is an infinite dimensional Banach space (See Lakshmikantham 1972, [14] [15] and [16]).
2) The nonlinear system in this paper includes all evolutionary equations of C 0 semigrop of operators.
3) Instead of continuity of the nonlinear functions , , f t y t T y t , we will replace the more general form of these functions in Banach spaces to be Bochner measurable in t and Liptschitzian in y.For regulatory conditions, we will assume the nonlinear operator involved in the nonlinear system is nonanticipating and liptchitzian.
4) The solution functions either x or y are assumed strongly differential.

Strong Solution to the Perturbed Nonanticipating
Operator Differential Equations Definition 3.1: By Nant-Lip we mean nonanticipating and Lipschitzian operators.
The operator G in the system (2.5) is nonanticipating and Lipschitzian.We need to clarify the meaning of the solution to the nonlinear system of operator differential Equation (2.5).The important part is when we accept some other principles indirectly hidden in the proof of Theorem (2.1).In fact we use the equivalent relationship between (2.5) and the integral Notice that this equivalent relation requires the absolute continuity of function y and the summability of the operator G which implies the differentiability of y.The above nonlinear operator system similarly could be presented by the following operator differential equation which contain the initial function φ for the past time interval The solution of the system (3.1) is denoted by x(t) which depends on the initial time 0 t and the initial function 0 φ and can described by ( ) 0 , , x t t φ which is called the strong solution to the system.Definition 3.2: A function x(t) is said to be a strong solution to the system (3.1) if it satisfies the following conditions: 1) x is strongly differentiable, 2) x satisfies the system (3.1)almost everywhere in the interval I, 2) Given a solution ( ) 0 , , x t t φ of (3.1) then the solution to the pertrubed equation will satisfy the integral equation This completes the proof of part (ii).Q.E.D

Generalized Operator Differential Equations
Introduction to the mild (Weak) solutions: For the definition of strong solution in the previous section, it was assumed equivalent relations between the differential and integral forms.This assumption required the differentiability of the solution.This condition may not be true in a large class partial differential equations.We are going to review the difficulties of applying the concepts of strong solution to the operator differential equations.The following are some examples.
The collection of solutions of the problem of free oscillations of an infinite string expressible in the form , where φ and ψ are twice differentiable functions.Notice that at the vertices of these solutions, ( ) , u x t will not be differentiable.Notice also the Lipschitzian condition for the nonlinear operator G which is required for the unique solution to the system (2.5) may not hold for unbounded operators in evolutionary equations.Thus, we need to have a new concept which includes the nondifferentiable solutions for unbodied operators.We are going to demonstrate this study by a linear system of abstract Cauchy problem ( ) α − may be an unbounded operator in the space X.Assume that the domain of this operator is denoted by We are looking for a solution space Y X ⊂ .One way to to get the solution space Y is to work from A and show that it generates a C 0 -semigroup.
When the operator is PDE, it may be unbounded, thus the solution in (4.1) may not be well defined.
We use a test function We define a weak solution "mild solution" u such that both relations (4.2) and the following are equivalent Most of the physical models can be described by a PDE system with evolution equations.One can interpret the solution as an ODE solution in an appropriate infinite dimensional space.
Nonlinear Operator Differential Equations(NODE): Suppose X is a Banach space, ( ) x t t φ the mild solution to the Cauchy problem x Ax f t x t T x t t t which has not been defined yet.We can define it by employing a similar argument and using the integral form of the system (4.4) , for all t ξ ≤ .This proves that the semigroup operator T t is nonanticipating.
Remarks: 1) The converse is not true.There may be a nonanticipating operator which may not be a semigroup.
2) It would be interesting to find out what conditions we may impose on the nonanticipating operators to generate a semigroup?These types of problems arise in a variety of physical models like heat conduction, population dynamics, and chemical reactions.

Variation of Parameters for Perturbed Operator Differential Equations
Suppose X is a Banach space, ( ) ( ) For ( ) Assume also that y(t) is a solution to the following perturbed system  .The solution to the system says that the future is determined completely by the present, with the past being involved only in that it determines the present.This is a deterministic version of the Markov property.
We make use of the following theorem in developing the variation formula for nonlinear differential equations.The Alekseev's formula for C 0 -semigroups was generalized by Hale 1992 [17].In addition, F. Bruaer 1966 [18] and 1967 [19] studied the perturbation of Nonlinear Systems of Differential Equations [10], [11].
We will use the same approach to develop the Nonlinear Variation of Parameter (NVP) for operator differential equations.
Let X be a Banach space, operator ( ) Let us summarize our conditions to present the following hypothesis; (H1) The operator t A in (5.1) and (5.2) is a Semigroup.
(H2) Assume that functions f and g belong to the following Lip spaces.That is they are Bochner measurable on the first variable and Lipschitzian on the other variables.
x t t φ is a mild solution to the following unperturbed operator differential Equation (5.1).
(H4) also let ( ) 0 , , y t t φ be a solution to the following perturbed nonlinear operator differential Equation (5.2).Lemma (5.1): Assume that all conditions for the existence of the solution to the nonlinear operator system of the unpeturbed equation hold.Then 1) The derivative ( ) ( ) exists and it is denoted by ( ) x t t x φ ∂ as partial derivative on variation with respect to the second parameter 0 x .It satisfies the following nonlinear operator equation ( ) ( ) ( ) The relation shows how fast the unperturbed solution x(t) changes with respect to its initial position x 0 , and its initial function φ .This is a partial derivative with respect the variable x(s) for new initial value ( ) 2) Also assume that the function x(t) is Frechet differentiable with respect the first parameter variable 0 t ( ) ( ) exists and it is denoted by ( ) , , , It satisfies the second kind of operator differential equation Proof: Part 1): We are assuming that the transformation T will be applied on the solution function x(t) and will produce a function at ( ) 0 0 , t x which will be the initial function 0 φ .Though the unperturbed solution can be described by ( ) ( ) Let us take the derivative of both sides of (5.3) w.r.t variable t: ( , , , , Substitute its equivalent from (5.3) then we can conclude: where the second part of the relation (5.4) can be interpreted as an identity matrix: ( ) ( ) ( ) This completes the proof of the first part of (b).
To prove the second part of (b), we can assume that ( ) Let us take the derivative of both sides of (5.9) with respect t 0 : This completes the second part of the result in (2).
Proof of the last part of (2): using the definition of operators U and V: , , , y t t x φ be solutions of the NODE systems: (5.1)   and (5.2) through the initial conditions ( ) 0 0 0 , , t x φ respectively.Then for 3) and the perturbed solution y'(s) from (

s y s x t s y s f s y s g s y s v
x t t y t x t t x x t s y s t u g s y s s

Generalized Alekseev's VOP of NODE with Initial Functions
When the operator A is unbounded, one cannot expect to derive the same result for any 0 x X ∈ since ( ) 0 0 0 , , , x t t x φ in general is not differentiable with respect to 0 t .We also need the differentiability of the solution ( ) 0 0 0 , , , x t t x φ with respect to the parameters ( ) x t t φ is the solution of Equation (5.1) through the initial state, ( ) T be a mapping of Z into itself represented by a convolution integral defined of the form and define J S I =  .Assume Y, Z, and U are Banach spaces.Let ( ) , M I Y be the space of all essentially bounded Bochner measurable functions with respect to classical Lebesgue measure from the interval I into the Banach space Y. Denote by ( ) , L J Y the space of all Lipschitzian functions y strongly differentiable almost everywhere from J into Y.Let φ be a fixed initial function from the space ( ) Z denote the space of all functions ( ) , f t y from the product I Y × into Z, Lipschitzian in y, and for every fixed y the function ( ) ., f y belongs to the space ( ) , M I Z .This space is called Lip-space.We apply the definition of nonanticipative operators in Section 1.1 to the initial domain.An operator T from the initial domain ( ) will be called a nonanticipating operator if for every two functions y and z in

Lemma 2 . 1 :
A direct sum operator of two nonanticipating and Lipschitzian operators is nonanticipating and Lipschitzian.

Theorem 2 . 1 :
Given a nonanticipating and Lipschitzian operator G from the initial domain

Proposition 3 . 1 : 1 )
will show the existence and uniqueness of the solution to the perturbed operator differential Equation (3.1).For introductory perturbation theory see Brauer 66 and Brauer 67.Assume that the operator T is Nant-Lip and functions f and g belong to the Lip-space which is If g is the perturbation to the Equation (3.1) then there is a unique strong solution y(t) in the initial domain

Theorem 4 . 2 :
(Existence and Uniqueness of the Solution)Let the operator A be a semigroup operator and T nonanticipating and Lipschitzian.Assume that homogeneous solution is guaranteed by the semigroup of operators and it will be equal to ( )e At tφ .The unique solution of the entire system (4.4) will be obtained by the nonanticipating and Lipschitzian properties of T and the Theorem 2.1. φ section and it will be investigated in this section with respect to 0 φ .The relation (5.2) has been generalized inHale 1992 for infinite dimensional [10]Kuang 1996, p.173,[10]).Example 2.4: Let T be the operator defined in example 2.2.One can verify the existence and uniqueness of the solution of the system The chaotic behavior induced by time delays was presented by Yang Kuang 1996.The global existence of the general single species with stage structured model described by a system ( ) ( )( ) y' t T y t = with the initial data function ( ) ( ) Applying the direct sum operators P 1 and P 2 we get the conclusion which is(3.3).Q.E.D.
[12]almost all t in I.According to Bogdan's theorm (see Bogdan 1981 and 1982,[11],[12]), there exists a unique solution y(t) in( ) d This argument can lead to the fact that if the operator