Abstract
The general linear boundary value problem for an abstract functional differential equation is considered in the case that the number of boundary conditions is greater than the dimension of the nullspace to the corresponding homogeneous equation. Sufficient conditions of the solvability of the problem are obtained. A case of a functional differential system with aftereffect is considered separately.
Keywords:
functional differential equations; boundary value problems; illposed problemsIntroduction
Linear boundary value problems (BVPs) for differential equations with ordinary derivatives that lack the everywhere and unique solvability are met with in various applications. Among these applications are some problems in oscillation theory (see, for examples, [[1]]) and economic dynamics [[2]]. Results on the solvability and solutions representation for these BVPs are widely used as an instrument of investigating weakly nonlinear BVPs [[3]]. General results concerning linear BVPs for an abstract functional differential equation (AFDE) are given in [[4]]. In this paper, we consider a case that the number of linearly independent boundary conditions is greater than the dimension of the nullspace of the corresponding homogeneous equation and obtain sufficient conditions of the solvability without recourse to the adjoint BVP and an extension of the original BVP. Our approach is based in essence on the assumption that the derivative of the solution does belong to a Hilbert space. Then we consider a system of functional differential equations that, formally speaking, is a concrete realization of the AFDE and, on the other hand, covers many kinds of dynamic models with aftereffect (integrodifferential, delayed differential, differential difference) [[5]–[7]]. For this case sufficient conditions are derived in an explicit form.
Preliminaries
In this section, we give some necessary facts from the theory of AFDE [[4], [8], [9]]. The linear abstract functional differential equation is the equation
A linear operator acting from the direct product
A linear operator acting from a space D into a direct product
Under the norm
Hence
Denote the components of the vector functional r by
Applying ℒ to both parts of (2.4), we obtain the decomposition
Let
Taking into account (2.5) and (2.6), we can rewrite BVP (2.7) in the form
In the sequel it is assumed that the socalled principal BVP
Problem (2.7) covers a wide class of BVPs for ordinary differential systems, differential delay
systems, some singular and impulsive systems [[7]]. This problem is wellposed if
In the case that
In what follows we derive conditions of solvability for (2.7) in a more explicit form without recourse to the adjoint BVP. Our approach is based
in essence on the assumption that the space B is a Hilbert space H with an inner product
A case of AFDE
Consider BVP (2.7) under the assumption that
Define the vector functional
Let us define the
Theorem 1
LetWbe nonsingular. Then BVP (2.7) is solvable for any
Proof
The general solution of the equation
A case of systems with aftereffect
In this section, we consider a system of functional differential equations with aftereffect that, formally speaking, is a concrete realization of the AFDE, and, on the other hand, it covers many kinds of dynamic models with aftereffect (integrodifferential, delayed differential, differential difference) [[2], [6], [10]].
Despite the case considered in Sections 2, 3 is more general, we derive here conditions of the solvability in detail since the corresponding transformations are based on the properties of operators and spaces as applied to the case under consideration.
Let us introduce the functional spaces where operators and equations are considered.
Fix a segment
Consider the functional differential equation
Recall that, under some natural assumptions, the following equations can be rewritten in the form (4.1):
the differential equation with concentrated delay
the differential equation with distributed delay
the integrodifferential equation
In what follows we will use some results from [[5], [8], [11], [12]] concerning (4.1). The homogeneous equation (4.1) (
The solution of (4.1) with the initial condition
The matrix
The general solution of (4.1) has the form
The general linear BVP is the system (4.1) supplemented by linear boundary conditions
BVP (4.1), (4.4) is wellposed if
We assume in the sequel that
Put
Theorem 2
Let the matrix
Proof
Let us apply
An explicit form of F is simple to derive by elementary transformations taking into account (4.5) and the properties of the Cauchy matrix. To do this, first note that
In view of Theorem 2, the solvability of BVP (4.1), (4.4) can be investigated on the base of the reliable computing experiment [[2], [10], [13]]. A somewhat different approach to the study of BVP (4.1), (4.4) with
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author thanks the referees for their careful reading of the manuscript and useful comments. The author acknowledges the support by the company Prognoz, Perm.
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