Variation of Parameters for Causal Operator Differential Equations ()
1. 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,
for all
, whenever
for all
.
Example 1.1: All of the delay operators and integral operators are nonanticipating. All compositions or Cartesian products of the nonanticipating operators are nonanticipating.
Example 1.2: Let
be a compact subset of the real line and f a function from the interval
into Y. The knowledge of the state of the system
(1.1)
at a given time
is sufficient to determine its state at any future time. This system has no after-effect or “no memory”.
Example 1.3: In a dynamic system
(1.2)
when T is a nonanticipating operator, to find the state curve
we need to have information about the initial function
for
in order to determine the state of the solution.
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
and
are such that
for
, it follows that
, for
where
and
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.
Example 1.3: Consider
, and let T be a mapping of Z into itself represented by a convolution integral defined of the form
This is a nonanticipating mapping if and only if
for almost all
. This Voltera integral mapping shows that
is independent of x(t) for
.
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 is independent of the past input, meaning that the mapping
is said to be (anticausal) or anticipating if for fixed s in
,
for
, whenever
for
.
Example 1.4: Let T be an operator from the space of a square summable function
into
. We can show that the following mappings are anticausal;
Since, for fixed real number s, the fact that
for
implies
for
and means that the future input
will affect the past. Therefore, this is an anticausal operator.
2. Nonanticipating Operator Differential Equation
Notations. Let S be the interval of all nonpositive numbers. Let I be the compact interval
,
, and define
. Assume Y, Z, and U are Banach spaces. Let
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
the space of all Lipschitzian functions y strongly differentiable almost everywhere from J into Y.
Let
be a fixed initial function from the space
. Denote by
the subset of the lip-space
consisting of all functions y such that
for all t in S.
According to these two definitions,
.
For any Banach space Y and Z, let
denote the space of all functions
from the product
into Z, Lipschitzian in y, and for every fixed y the function
belongs to the space
. 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
into
will be called a nonanticipating operator if for every two functions y and z in
and every point
, the fact that
for almost all
implies that
for almost all
.
An operator P from a subset D of Y into Z is said to be Lipschitzian if there exists a constant b such that
(2.1)
for every
. For
the operator
(2.2)
is called the operator induced by
and the operator F is called Induced Operator generated by the function f.
Lipschitzian Space (or simply the Lip-Space), denoted by
, is the set of all functions
such that
is uniformly bounded, Lipschitzian in y, and is measurable in t. That is,
is Lipschitzian in y and
. The infimum of all Lipschitzian constants L will be denoted by
.
On the space
, we shall introduce a family of norms, called k-norm by the formula
for any fixed real number k. Observe that from this definition follows the inequality
for almost all t in I. Notice that for every k, the k norms
and
are equivalent.
A Lipschitzian operator P from a subset D of
into the space
is called an operator of exponential type if for some constants b and k0,
for all y and z in the domain D and all
.
Example 2.1: The operator
for a constant real number r is an induced operator. Thus for any function
the operator T is nonantipating and Lipschitzian.
Properties of the nonlinear operator F in (2.2) induced by the function
have been studied by Bogdan 1981 and 1982. In particular, it is known that for
and
the function
defined by
belongs to the space of measurable functions
(see Ahangar 1989, [1] - [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
is a well defined, nonanticipating, and Lipschitzian operator.
Definition 2.1 (Direct Sum Operators): Let
be operators from the domain
into the space
. Define the direct sum operator
such that
for every y in
and t in I.
Lemma 2.1: A direct sum operator of two nonanticipating and Lipschitzian operators is nonanticipating and Lipschitzian.
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
and for some point s in the interval I we have
for almost all
. Since T1 and T2 are nonanticipative, then
for almost all
,
for almost all
.
These two equalities will imply that
for almost all
. Thus
for almost all
. This will imply that
and
coincide for almost all
.
Now let us prove that the operator
is Lipschitzian. According to the definition
Since both operators are Lipschitzian, the right hand side will be
Notice that
represents the ess.sup norm in the space measurable functions
. If we let
for
and take the essential supremum norm on the left hand side of the above relation then it will be
(2.3)
for all y and z in the domain D. This proves that the direct sum operator is Lipschitzian. Q.E.D.
Example 2.2: Assume that
for a constant real number r and
. The operator
is nananticipa- ting and Lipschitzian.
Nonanticipating Deterministic Dynamical System: Assume that the operator T is nonanticipating and Lipscitzian. The behavior of a dynamic system
(2.4)
is known as an after effect differential equation with the initial domain
.
Given that
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.
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
has been studied (See 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
with the initial data function
for
. 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
to the Banach space Z is called nonanticipating differential equation
(2.5)
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.
Theorem 2.1: Given a nonanticipating and Lipschitzian operator G from the initial domain
into the space of Bochner measurable functions
, there exists a unique solution
that satisfies the nonlinear operator system (2.5).
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 C0 semigrop of operators.
3) Instead of continuity of the nonlinear functions
, 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.
3. 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
(3.1)
which contain the initial function
for the past time interval
. The solution of the system (3.1) is denoted by x(t) which depends on the initial time
and the initial function
and can described by
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,
3) there exists a function
such that
, for almost all
.
The following proposition 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.
Proposition 3.1: Assume that the operator T is Nant-Lip and functions f and g belong to the Lip-space which is
and
.
1) If g is the perturbation to the Equation (3.1) then there is a unique strong solution y(t) in the initial domain
which satisfies the perturbed system of differential equation
(3.2)
2) Given a solution
of (3.1) then the solution to the pertrubed equation will satisfy the integral equation
(3.3)
Proof: 1) Let us assume that the operator
and
. Define the direct sum operator
.
By Lemma 2.1, the operator G will be Nant-Lip and the differential Equation (3.2) will be in the following form
(3.4)
for 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
to the Equation (3.4).
Proof of 2) The equivalent integral equation of (3.4) will be
(3.5)
Applying the direct sum operators P1 and P2 we get the conclusion which is (3.3). Q.E.D.
Substitute for unperturbed solution
in (3.3) as a solution of (3.1) we will get the following
(3.6)
This completes the proof of part (ii).Q.E.D
4. 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
takes the form
, where
and
are twice differentiable functions. Notice that at the vertices of these solutions,
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
(4.1)
for
,
, where
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
. One way to to get the solution space Y is to work from A and show that it generates a C0-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
such that
(4.2)
We define a weak solution “mild solution” u such that both relations (4.2) and the following are equivalent
(4.3)
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,
is the generator of a C0-semigroup on X,
is open and
be a continuous function such that
is differentiable and
is a continuous in U.
For
, we denote by
the mild solution to the Cauchy problem
(4.4)
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)
(4.5)
Definition 4.1: We define the function x(t) to be a mild solution to the system (4.4) on
if it satisfies (4.5) and
for all t in I.
Lemma 4.1: Every semigroup of operators generated by the operator A is a nonanticipating and Lipschitzian operator.
Proof: Assume that the semigroup
generated by A is given. Thus by the definition of semigroup, for every y in D(A)
Suppose that for y and
in D(A) then
, for every
. Thus the equality
implies that
, for all
. This proves that the semigroup operator Tt 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?
Theorem 4.2: (Existence and Uniqueness of the Solution)
Let the operator A be a semigroup operator and T nonanticipating and Lipschitzian. Assume that
. Then the system (4.4) has a unique solution in the space of initial domain
.
Proof: The homogeneous solution is guaranteed by the semigroup of operators and it will be equal to
. 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.
These types of problems arise in a variety of physical models like heat conduction, population dynamics, and chemical reactions.
5. Variation of Parameters for Perturbed Operator Differential Equations
Suppose X is a Banach space,
is a generator of a C0-semigroup on X,
is open and
be a continuous function such that
is differentiable and
is continuous in U where
and
.
For
, we denote by
the mild solution to the following Cauchy problem
(5.1)
Assume also that y(t) is a solution to the following perturbed system
(5.2)
These solutions in the system (5.1) are then related by the evolutionary property
for all
. The initial function
depends on t, t0, and x0. It is denoted by
. 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 C0-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
is generator of a C0-semigroup on X,
is continuously differentiable with respect to x.
Let us summarize our conditions to present the following hypothesis;
(H1) The operator
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.
(H3) Assume that
is a mild solution to the following unperturbed operator differential Equation (5.1).
(H4) also let
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
as partial derivative on variation with respect to the second parameter
. It satisfies the following nonlinear operator equation
(5.4)
The relation (5.4) shows how fast the unperturbed solution x(t) changes with respect to its initial position x0, and its initial function
. 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
exists and it is denoted by
.
It satisfies the second kind of operator differential equation
(5.6)
Furthermore
(5.7)
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
which will be the initial function
. 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:
for
where the second part of the relation (5.4) can be interpreted as an identity matrix:
(5.8)
2) Notice that, at the starting point
we can re-evluate the rate of change of the solution with respect to the initial moment
A few notes are important: For a vector solution
(5.9)
Similar to (5.8)
Notice that
for all
.
This completes the proof of the first part of (b).
To prove the second part of (b), we can assume that
has a variation on
.
Let us take the derivative of both sides of (5.9) with respect t0:
(5.10)
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:
Substiting (5.10) yields
Theorem (5.1): Alekseev Type Variation of Parameters Theorem for NODE Systems:
Let
and
be solutions of the NODE systems: (5.1) and (5.2) through the initial conditions
respectively. Then for
(5.11)
Notice: As we see in Equations (5.1) and (5.2), the perturbation causes the changes on the initial conditions at
and
and on the initial function
. Up to the initial condition both functions
and
have the past history and they will be identical at
.
Proof: Variations of unperturbed solution x(t) and perturbed solution y(t) when the initial conditions of the moving object change with respect to the independent variable
can be demonstrated by the following chain rule formula
Substitute (5.3) and the perturbed solution y'(s) from (5.2)
Substitute for V(s) by (5.7)
As a result of these substitutions we can integrate the following relation on
:
Now integrate
Now the question is this: what is
? The unperturbed solution x(t) with the perturbed solution as the initial conditions
and
.
Thus
, and the above relation will be concluded as follows:
This is a conclusion of the Alekseev type Theorem for Nonlinear Operator Differential Equations.
6. 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
since
in general is not differentiable with respect to
. We also need the differentiability of the solution
with respect to the parameters
. The variation of parameters was investigated with respect to the parameters
in the previous section and it will be investigated in this section with respect to
.
The relation (5.2) has been generalized in Hale 1992 for infinite dimensional variational operator when
(6.1)
Assume that
exists, then
This argument can lead to the fact that if the operator
, then the solution to the system
(6.2)
has a unique solution. The system (6.2) is called the variational equation. Notice that for all
,
then
Using the chain rule for abstract functions, we get
Thus by integrating the system,
(6.3)
Proposition 6.1 (Alekseev's Theorem for Operator Differential Equations): Suppose
and
are of class
. If
is the solution of Equation (5.1) through the initial state,
and
is the perturbed solution of
(6.4)
through
, then, for any
we have
(6.5)
Proof: For
assume
. Differentiating with respect to the first parameter
,
Using the relation (5.3)
Integrating from t0 to t, we will conclude that
Therefore,
This proves the theorem for
.
Assume that for the initial function
the maximal interval is
for the solution
.
For
let us define
and operators
by the following relations
Since both operators are well defined and continuous on
and coincide on
, they must coincide on
. This proves the theorem.
The next theorem will provide the variation of parameters formula for operator differential equations.
Theorem 6.1 (Variation of Parameters for NODE): The solution of the systems (5.1) and (5.2) satisfy the following
(6.6)
where
and assume the inverse matrix
exists.
Proof: In a variation of parameters, we will determine a function
to satisfy the differential equation for perturbed solution y such that
(6.7)
is a solution process for the system (5.7). From the system (5.7) and differentia- tion of (5.9) we will get
(6.8)
Since
is a solution of (5.5), then
(6.9)
It can be observed that the inverse matrix
exists, then
(6.10)
By integrating we will obtain
(6.11)
Differentiation with respect to the second independent variable s when
implies that
Substituting (6.10) for
we get the following for the right hand side
which implies
Using variation definition (6.7) in the above relation, we will now get the variation of parameters for nonlinear operator differential Equation (6.5)
(6.12)
The operator T in the differential Equation (5.1) and (5.7) could be any delay, integral, composition, or Cartesian product of nonanticipating and Lipschitzian operators which will affect the nonperturbed solution
. The variation formula (5.14) will be effected by the operator T through these changes.
Assuming that the variation of parameters is given, we will investigate some of the properties of this formula through the following conclusions for particular cases.
Corollary 6.1: Suppose that the conditions of Theorem 6.1 satisfy and guarantee the existence and uniqueness of the solution of the system (5.2). Assume also
is the initial state of the system
. Then the relation (5.7) will be
(6.13)
Proof: Assuming that
is a solution to the homogeneous equation
, then by the direct integration of the system (5.1) and applying the variation of parameters formula (5.3) to the nonlinear system (5.2), we will get the formula (6.13).
Corollary 6.2: Suppose that the conditions of H1 through H4 guarantee the existence and uniqueness of the solution of (5.1) and (5.6). Assume also a particular case when
and
, then the Alekseev’s formula (5.7) deduces the variation of parameters formula
(6.16)
for linear differential equation:
.
Proof: Assuming that
is a solution to the homogeneous
, then the fundamental matrix of the homogeneous system will be
By considering the following
and
we conclude that
(6.17)
{It can be verified that
}
Notice that the deterministic function f is identically equal to zero
. This concludes the variation of constants for linear system (6.17).
Corollary 6.3: Suppose that in the differential Equation (5.1)
, then the general solution of (6.16) about the equilibrium solution
will be
(6.18)
Proof: Since the operator
, then the solution
is a constant function. Therefore
. To find the perturbed solution
of the system (5.6), we use the conclusion of the Proposition 5.1 for unperturbed solution of the system (5.1) to obtain the relation (6.18).
We will study the variation of parameters for operator differential equations disturbed force operator functions. These nonlinear operators can involve the following types: delay, integrals, composition, or cartesian products of all nonanticipating and Lipschitzian operators.
7. Conclusions
Assume that
is a matrix function on
into the space
. Suppose that
represents the fundamental matrix solution process of a differential equation
(7.1)
Then
(7.2)
(7.3)
A method of variation of parameters for the systems (7.1) - (7.3) is presented by G. S. Ladde and V. Lakshmikantham, 1980.
Suppose
is Lebesgue summable from I into
and let
be a perturbation in the system (6.1) then the solution process
of the following nonlinear system
(7.4)
will satisfy the following integral equation
(7.5)
for all
. Further study of this general form of the variation of parameters for nonlinear operator differential equations should be very interesting. These nonlinear operators can involve varieties of many types of operators like: delay, integrals, composition, or Cartesian products of all nonanticipating and Lipschitzian operators.
A classical nonlinear system type
for
and
in (1.1) is well known and extensively studied. The variation of parameters discovered by Alekseeve is a great tools to study this kind of nonlinear system and use this conclusion for stability and asymptotic behavior of a nonlinear system. The solutions to a nonlinear operator differential equations of type (1.2) which include all operators T satisfying nonanticipating and lipschitzian conditions also reviewed here, have a huge range of application.
For operator in this paper we proved and demonstrated a general form of Alekseeve Theorem when a non linear system (5.1) includes a C0-semigroup of opeartor A.
All important conditions in (H1) through (H4) are connecting the nonanticipating property of T, semigroup property of At, and Lipschitzian property of f. The variation of parameters helped us to find the solution to the purturbed system. This perturbed solution for nonanticipating dynamic systems will help us in the future to study the stability and asymptotic behavior of the system. Two major issues related to the Variatiion of Parameters can be developed for Nonlinear Operator Differential Equations.
First, is the numerical algorithm and computational program to produce the solution to such a general form of nonlinear variational of parameters method. Second, generalize the stability application to nonlinear system to operator differential equations.