- Open Access
Linear overdetermined boundary value problems in Hilbert space
© Maksimov; licensee Springer. 2014
- Received: 13 December 2013
- Accepted: 26 May 2014
- Published: 11 July 2014
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.
- 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.
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 , ; .
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 .
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)).
Recall that, under some natural assumptions, the following equations can be rewritten in the form (4.1):
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 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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