# Linear overdetermined boundary value problems in Hilbert space

- Vladimir P Maksimov
^{1}Email author

**2014**:140

https://doi.org/10.1186/s13661-014-0140-4

© Maksimov; licensee Springer. 2014

**Received: **13 December 2013

**Accepted: **26 May 2014

**Published: **11 July 2014

## 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 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.

## Introduction

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 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) [[5]–[7]]. For this case sufficient conditions are derived in an explicit form.

## Preliminaries

*D*and

*B*are Banach spaces such that

*D*is isomorphic to the direct product $B\times {R}^{n}$. Let us denote by $\mathcal{J}=\{\mathrm{\Lambda},Y\}:B\times {R}^{n}\to D$ an isomorphism and let ${\mathcal{J}}^{-1}=[\delta ,r]$.

*B*and ${R}^{n}$ into a Banach space

*D*is defined by a pair of linear operators $\mathrm{\Lambda}:B\to D$ and $Y:{R}^{n}\to D$ in such a way that

*D*into a direct product $B\times {R}^{n}$ is defined by a pair of linear operators $\delta :D\to B$ and $r:D\to {R}^{n}$ so that

*I*is the identity operator. We will identify the finite-dimensional operator $Y:{R}^{n}\to D$ with a vector $({y}_{1},\dots ,{y}_{n})$, ${y}_{i}\in D$, such that $Y\beta ={\sum}_{i=1}^{n}{y}_{i}{\beta}^{i}$, $\beta =col\{{\beta}^{1},\dots ,{\beta}^{n}\}$.

Denote the components of the vector functional *r* by ${r}^{1},\dots ,{r}^{n}$. If $\ell =[{\ell}^{1},\dots ,{\ell}^{m}]:D\to {R}^{m}$ is a linear vector functional, and $X=({x}_{1},\dots ,{x}_{n})$ is a vector with components ${x}_{i}\in D$, then *lX* denotes the $m\times n$-matrix, whose columns are the values of the vector functional *l* on the components of $X:lX=({l}^{i}{x}_{j})$, $i=1,\dots ,m$; $j=1,\dots ,n$.

*ℓ*to the two parts of (2.4), we get

*linear boundary value problem*.

*X*called the fundamental vector and

*G*called the Green operator.

*j*th element of

*X*, is nonsingular,

*i.e.*$det\ell X\ne 0$.

In the case that $m>n$ BVP (2.7) lacks the everywhere and unique solvability, namely, it is solvable if and only if the right-hand side $\{f,\gamma \}\in B\times {R}^{m}$ is orthogonal to all the solutions $\{\omega ,\beta \}$ of the homogeneous adjoint equation (2.9), *i.e.*$\omega f+\beta \gamma =0$ [[8]] (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 $\u3008\cdot ,\cdot \u3009$.

## A case of AFDE

*n*components of

*ℓ*and the elements of ${\gamma}_{1}$ in (3.1) are the corresponding components of

*γ*. Thus ${\ell}_{2}$ will stand for the final $(N-n)$ components of

*ℓ*, and elements of ${\gamma}_{2}\in {R}^{N-n}$ are defined as the final $(N-n)$ components of

*γ*. For $\alpha \in {R}^{q}$, $\alpha =col({\alpha}^{1},\dots ,{\alpha}^{q})$, we put ${\lfloor \alpha \rfloor}^{j}={\alpha}^{j}$. Thus in the cases that a vector

*V*is expressed by a complicated formula we will use ${\lfloor V\rfloor}^{j}$ instead of ${V}^{j}$ to indicate the

*j*th component of

*V*.

*H*that generates the functional ${\lambda}^{j}$: for any $f\in H$

### Theorem 1

*Let*

*W*

*be nonsingular*.

*Then BVP*(2.7)

*is solvable for any*$f\in H$

*of the form*

*where*

*and*$\phi \in H$

*is arbitrary element that is orthogonal to each*${\lambda}^{k}$, $k=1,\dots ,N-n$.

### Proof

*α*:

*x*of the form (3.4) satisfies the equality ${\ell}_{2}x={\gamma}_{2}$. For this purpose, apply ${\ell}_{2}$ to both parts of (3.4):

*λ*over the space

*H*:

*φ*is orthogonal to ${\lambda}^{j}$: $\u3008{\lambda}^{j},\phi \u3009=0$ for any $j=1,\dots ,N-n$. Let us use the substitution (3.7) as applied to (3.5):

*c*into (3.7). □

## 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) [[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 $[0,T]\subset R$. By ${L}_{2}^{n}={L}_{2}^{n}[0,T]$ we denote the Hilbert space of square summable functions $v:[0,T]\to {R}^{n}$ endowed with the inner product $(u,v)={\int}_{0}^{T}{u}^{\prime}(t)v(t)\phantom{\rule{0.2em}{0ex}}dt$ (⋅^{′} is the symbol of transposition). The space $A{C}_{2}^{n}=A{C}_{2}^{n}[0,T]$ is the space of absolutely continuous functions $x:[0,T]\to {R}^{n}$ such that $\dot{x}\in {L}_{2}^{n}$ with the norm ${\parallel x\parallel}_{A{C}_{2}^{n}}=|x(0)|+\sqrt{(\dot{x},\dot{x})}$, where $|\cdot |$ stands for the norm of ${R}^{n}$. Thus we have here $D=A{C}_{2}^{n}$, $H={L}_{2}^{n}$, $A{C}_{2}^{n}\cong {L}_{2}^{n}\times {R}^{n}$, and $x(t)={\int}_{0}^{t}z(s)\phantom{\rule{0.2em}{0ex}}ds+x(0)$, $(\mathrm{\Lambda}z)(t)={\int}_{0}^{t}z(s)\phantom{\rule{0.2em}{0ex}}ds$, $Y=I$, $\delta x=\dot{x}$, $rx=x(0)$ (see (2.2)-(2.4)).

*A*has elements that are square summable on $[0,T]$. Therefore, we have here $Q=I-\mathcal{K}$, $Arx=A(\cdot )x(0)$ (see (2.5)).

Recall that, under some natural assumptions, the following equations can be rewritten in the form (4.1):

*i*th column of

*A*.

*ℓ*:

*ℓ*are linearly independent.

*j*th column of

*X*, is nonsingular,

*i.e.*$det\ell X\ne 0$. It should be noted that this condition cannot be verified immediately because

*X*cannot be (as a rule) evaluated explicitly. In addition, even if

*X*were known, then the elements of

*ℓX*, generally speaking, could not be evaluated explicitly. By the theorem about inverse operators, the matrix

*ℓX*is invertible if one can find an invertible matrix Γ such that $\parallel \ell X-\mathrm{\Gamma}\parallel <1/\parallel {\mathrm{\Gamma}}^{-1}\parallel $. As has been shown in [[13]], such a matrix Γ for the invertible matrix

*ℓX*always can be found among the matrices $\mathrm{\Gamma}=\overline{\ell}\overline{X}$, where $\overline{\ell}:A{C}_{2}^{n}\to {R}^{n}$ is a vector functional near

*ℓ*, and $\overline{X}$ 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) [[2], [10], [13]] which opens a way to the computer-assisted study of BVPs.

*n*components of

*ℓ*and the elements of ${\gamma}_{1}$ in (4.6) are the corresponding components of

*γ*. Thus ${\ell}_{2}$ will stand for the final $(N-n)$ components of

*ℓ*, and elements of ${\gamma}_{2}\in {R}^{N-n}$ are defined as the final $(N-n)$ components of

*γ*. Let us write ${\ell}_{1}$ in the form

### Theorem 2

*Let the matrix*$W={\int}_{0}^{T}F(s){F}^{\prime}(s)\phantom{\rule{0.2em}{0ex}}ds$,

*where*

*F*

*is defined by*(4.10),

*be nonsingular*.

*Then BVP*(4.1), (4.4)

*is solvable for all*$f\in {L}_{2}^{n}[0,T]$

*of the form*

*where*

*and*$\phi (\cdot )\in {L}_{2}^{n}$

*is an arbitrary function that is orthogonal to each column of*${F}^{\prime}(\cdot )$:

### Proof

*α*:

*x*of the form (4.11) satisfies the equality ${\ell}_{2}x={\gamma}_{2}$. For this purpose, apply ${\ell}_{2}$ to both parts of (4.11):

*F*whose columns belong to ${L}_{2}^{N-n}[0,T]$.

*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

*f*(see (4.14)) that solves (4.13). This completes the proof. □

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 $N>n$ is proposed in [[14]].

## Declarations

### 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.

## Authors’ Affiliations

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