Multi-point boundary value problems on time scales at resonance

We study the nonlinear second order differential equation on a time scale

     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

Semilinear degenerate elliptic boundary value problems at resonance

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems at resonance which include as particular cases the Dirichlet and Robin problems. The approach here is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Lyapunov–Schmidt procedure and the global inversion theorem, we prove existence and uniqueness theorems for our problem. The results here extend an earlier theorem due to Landesman and Lazer to the degenerate case.     

Remarks on nonlocal boundary value problems at resonance

We show that it is important to allow the nonlinear term to change sign when discussing existence of a positive solution for multipoint, or more general nonlocal, boundary value problems in the resonant case. When the nonlinear term has a fixed sign we obtain simple necessary and sufficient conditions for the existence of positive solutions.     

Second order nonlocal boundary value problems at resonance

We present sufficient conditions for the existence of positive solutions for some second order boundary value problems at resonance. The boundary conditions that we study are quite general, involve a Stieltjes integral and include, as particular cases, multi‐point and integral boundary conditions. Our results are based on a Leggett‐Williams norm‐type theorem due to O'Regan and Zima. We employ a general abstract approach which allows us to improve and complement recent results in the literature. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Existence theorems for linear hyperbolic boundary value problems at resonance

Summary Questions of existence, uniqueness, and continuous dependence for weak solutions of linear hyperbolic boundary value problems are considered. The differential equations have the form u_tt + Au=f, where A is elliptic in the spatial variables, and the boundary conditions are homogeneous in both space and time. Resolution of these questions depends on the relationship of the eigenvalues of A and those of an associated scalar problem in time.     

Positive solutions of multi-point boundary value problems at resonance

We establish new results on the existence of positive solutions for some multi-point boundary value problems at resonance. Our results are based on a recent Leggett–Williams norm-type theorem due to O’Regan and Zima. We also derive a new result for a three-point problem, previously studied by several authors.     

Existence of positive solutions for three-point boundary value problems at resonance

With the help of the theorem of a fixed point index for A-proper semilinear operators established by Cremins, we get a existence theorem concerning the existence of positive solution for the second order ordinary differential equation of three-point boundary value problems at resonance.     

Eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance

The abstract boundary value problem Lu + Gu = f, u ϵ dom(L) ⊂ H, is considered. Here H is used to denote a real separable Hilbert space, L a closed symmetric linear operator, and G a nonlinear operator assumed to be Lipschitz continuous and strongly monotone. In addition L is assumed to have a complete set of eigenfunctions in H, and is allowed to have an infinite dimensional null space. The existence of unique solutions, depending continuously on f, is established by a constructive approach. Galerkin approximations are considered and error estimates are given. As an application of the main result, the existence of time periodic weak solutions of the n-dimensional wave equation is shown.     

Solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations at resonance

In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin''s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results.     

Comparison results for first and second order boundary value problems at resonance

Various types of comparison results for first and second order periodic boundary value problems are developed. It is hoped that these comparison results play an important role in the existence theory of boundary value problems at resonance.     

Solvability for second-order three-point boundary value problems at resonance on a half-line

This paper deals with the solvability and uniqueness of the second-order three-point boundary value problems at resonance on a half-line