Existence of solutions to first-order periodic boundary value problems

This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution.     

On first-order discrete boundary value problems

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations.     

Symmetric positive solutions of nonlinear boundary value problems

We study the nonlinear boundary value problem

     

Uniqueness and parameter dependence of positive solutions to higher order boundary value problems with fractional $q$-derivatives

The authors study a class of nonlinear higher order boundary value problem with fractional $q$-erivativesand dependence on a positive parameter $\lm$.The existence, uniqueness, and dependence of positive solutions on $\lm$ are discussed.Two sequences are constructed so that they converge uniformly to the unique solution of the problems.Two examples are included in the paper. Numerical computations of the examples confirm their theoretical results.     

Solutions of periodic boundary value problems for second-order nonlinear differential equations via variational methods

By introducing a variational framework for a class of second order nonlinear differential equations with non-separated periodic boundary value conditions, some results on the existence of non-trivial, positive and negative solutions of the problems are obtained. Some results by Atici-Guseinov, Graef-Kong, etc. obtained by topological degree methods are extended. The resonant case of the problems where the nonlinearities are unbounded and satisfy Ahmad-Lazer-Paul type conditions is also considered.     

Discontinuous first-order functional boundary value problems

In this paper we prove a new existence result for functional boundary value problems with first-order functional differential equations under weak conditions on the nonlinear part and monotonicity (but not continuity) with respect to the functional variable. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of unviability or solubility condition.     

Green's function for second-order periodic boundary value problems with piecewise constant arguments

We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function.     

Existence theorems for boundary value problems in difference equations

In this research, we study linear difference equations with constant coefficients subject to boundary conditions. Necessary and/or sufficient conditions for the existence of a unique solution will be established. The proofs of the existence and uniqueness theorems are established by means of special types of determinants called Mosaic Vandermonde determinants.     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

Constructive proof for existence of nonlinear two-point boundary value problems

Owing to the importance of differential equations in physics, the existence of solutions for differential equations has been paid much attention. In this paper, the existence of solution are obtained for the nonlinear second order two-point boundary value problem in the reproducing kernel space. Under certain assumptions on right-hand side, we propose constructive proof for the existence result, and a method is presented to obtain the exact solution expressed by the form of series. This paper is a extension of previous paper [Wei Jiang, Minggen Cui, The exact solution and stability analysis for integral equation of third or first kind with singular kernel, Appl. Math. Comput. 202 (2) (2008) 666-674], which extends a method of solving linear problems to present method for solving nonlinear problems.     

On multiple fixed-sign solutions of a discrete system with Hermite boundary conditions

We consider the discrete system