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 null-space 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; ill-posed problems
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, []) and economic dynamics []. Results on the solvability and solutions representation for these BVPs are widely used as an instrument of investigating weakly nonlinear BVPs []. General results concerning linear BVPs for an abstract functional differential equation (AFDE) are given in []. In this paper, we consider a case that the number of linearly independent boundary conditions is greater than the dimension of the null-space 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 (integro-differential, delayed differential, differential difference) [–]. For this case sufficient conditions are derived in an explicit form.
A linear operator acting from the direct product of the Banach spaces B and into a Banach space D is defined by a pair of linear operators and in such a way that
A linear operator acting from a space D into a direct product is defined by a pair of linear operators and so that
Under the norm
Denote the components of the vector functional r by . If is a linear vector functional, and is a vector with components , then lX denotes the -matrix, whose columns are the values of the vector functional l on the components of , ; .
Applying ℒ to both parts of (2.4), we obtain the decomposition
Let be a linear bounded vector functional with linearly independent components, . The system
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 so-called principal BVP] (Theorem 1.16, p.11)
Problem (2.7) covers a wide class of BVPs for ordinary differential systems, differential delay systems, some singular and impulsive systems []. This problem is well-posed if . In such a situation, BVP (2.7) is uniquely solvable for any and if and only if the matrix
In the case that BVP (2.7) lacks the everywhere and unique solvability, namely, it is solvable if and only if the right-hand side is orthogonal to all the solutions of the homogeneous adjoint equation (2.9), i.e. [] (Corollary 1.15, p.11).
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 and the system , can be split into two subsystems and such that the BVP
Define the vector functional , by the equality
Let us define the -matrix by the equalities
LetWbe nonsingular. Then BVP (2.7) is solvable for anyof the form
The general solution of the equation has the representation
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 (integro-differential, delayed differential, differential difference) [, , ].
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 . By we denote the Hilbert space of square summable functions endowed with the inner product (⋅′ is the symbol of transposition). The space is the space of absolutely continuous functions such that with the norm , where stands for the norm of . Thus we have here , , , and , , , , (see (2.2)-(2.4)).
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 integro-differential equation
The solution of (4.1) with the initial condition has the representation
The matrix is expressed in terms of the resolvent kernel of the kernel . Namely,
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 well-posed if . In such a situation, the BVP is uniquely solvable for any and if and only if the matrix], such a matrix Γ for the invertible matrix ℓX always can be found among the matrices , where is a vector functional near ℓ, and is an approximation of X. That is why the basis of the so-called constructive study of linear BVPs includes a special technique of approximate constructing the solutions to FDE with guaranteed explicit error bounds as well as the reliable computing experiment (RCE) [, , ] which opens a way to the computer-assisted study of BVPs.
We assume in the sequel that and the system , can be split into two subsystems and such that the BVP
Let the matrix, whereFis defined by (4.10), be nonsingular. Then BVP (4.1), (4.4) is solvable for allof the form
Let us apply to both parts of (4.3):
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 a similar way,
In view of Theorem 2, the solvability of BVP (4.1), (4.4) can be investigated on the base of the reliable computing experiment [, , ]. A somewhat different approach to the study of BVP (4.1), (4.4) with is proposed in [].
The author declares that he has no competing interests.
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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Maksimov, VP, Rumyantsev, AN: Reliable computing experiment in the study of generalized controllability of linear functional differential systems. In: Uvarova L, Latyshev A (eds.) Mathematical Modelling. Problems, Methods, Applications, pp. 91–98. Kluwer Academic, New York (2002)
Maksimov, VP, Chadov, AL: The constructive investigation of boundary-value problems with approximate satisfaction of boundary conditions. Russ. Math.. 54, 71–74 (2010). Publisher Full Text