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SynopsisIn this paper we consider differential-boundary operators T over a finite interval depending on a complex parameter. A differential-boundary operator admits boundary conditions in the differential part. The boundary part contains multipoint boundary conditions and integral conditions. For Birkhoff-regular boundary conditions we prove that every Lp -function is expansible into a series with respect to the eigenfunctions and the associated functions of the differential-boundary operator. Here the Birkhoff-regularity only depends on the boundary conditions at the endpoints of the interval, i.e. T is Birkhoff-regular if and only if T0 is Birkhoff-regular where T0 arises from T by omitting the boundary part in the differential equations, the interior point boundary conditions and the integral condition.

SynopsisIn the recent past many results have been established on non-negative solutions to boundary value problems of the formwhere λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.

SynopsisThis paper considers semilinear elliptic boundary value problems of the formwhere the partial derivative ∂f/∂u is bounded above by the least eigenvalue of the linear elliptic operator L. Existence and uniqueness of solutions is proved by using monotone operator theory and sub and supersolution techniques.

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

SynopsisThe paper considers a linear non-homogeneous boundary value problem for a class of neutral type functional differential equations. A necessary and sufficient condition for the existence of a unique solution of that problem is obtained.

SynopsisThis paper deals with two-point boundary-value problems for ordinary differential equations and the operators which they induce in the appropriate Hilbert space. The problems arenot required to be self-adjoint. No auxiliary condition such as Birkhoff-regularity is imposed. If T is such an operator, it may well have no meaningful spectral structure. It is shown, however, that when T is composed with its adjoint, the result is a non-negative self-adjoint differential operator. The eigenvalues and eigenfunctions of this composite operator are used to delineate the domain, action, range, and generalised inverse of T.

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

SynopsisThis paper is concerned with the existence of solutions of a two point boundary value problem for neutral functional differential equations. We consider the problemwhere M and N are n × n matrices. This is examined by using the “shooting method”. Also, an example is given to illustrate how our result can be applied to yield the existence of solutions of a periodic boundary value problem.

SynopsisThe paper deals with boundary value problems for second-order vector differential equations x″ = f (t, x, x′). Given a region Ω in (t, x)-space we ask whether there exists a solution x(t) of the problem satisfying (t, x(t)) ∊Ω. We arrive at a rather general type of conditions which are sufficient in order that Ω has the desired property. One of these conditions is geometric in nature and depends upon the boundary data only. The second condition can be expressed in terms of inequalities and depends upon the values of f on ∂Ω. These inequalities turn out to be the common background of a variety of conditions which can be found in the literature on boundary value problems and which in the case of a scalar equation reduce to the well-known properties of upper and lower solutions.

SynopsisWe consider a class of convex non-linear boundary value problems of the formwhere L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.

SynopsisThe present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.

SynopsisIn this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

SynopsisSuppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the formare considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).

SynopsisSelf-adjoint operators in L2(0, 1) associated with a formally symmetric differential operator regular in [0, 1] can be determined by boundary conditions or as extensions of the minimal operator. These extensions are determined by extensions of the Cayley transform of the minimal operator. This paper establishes an explicit expression for the extension of the Cayley transform in terms of the boundary conditions and vice versa.

SynopsisIn this paper we study the solutions of the boundary value problemwhere t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).

SynopsisThe analysis developed by the author in a previous paper is used to discuss two magnetohydrodynamic duct flows for which the boundary conditions are not of Dirichlet type. To the accuracy stated, the results obtained confirm those obtained by previous authors who used different approaches to the problems. A feature of the present analysis is that it yields the magnitude of the error entailed by the use of approximate forms for flow quantities. This has been lacking in previous analyses.

SynopsisWe consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion. Based on the results in another paper by the same author, a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation (the Semenov approximation).