A CLASS OF THIRD-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITION AT RESONANCE

Abstract

In this paper, we consider third-order boundary value problem with, Dirichlet, Neumann and integral conditions at resonance case, where the kernel’s dimension of the ordinary differential operator is equal to one and the ordinary differential equation which can be written as the abstract equation Lu = Nu, called semilinear form, where L is a linear Fredholm operator of index zero, and N is a nonlinear operator. First, we prove a priori estimates, and then we use Mawhin’s coincidence degree theory to deduce the existence of solutions. One important ingredient to be able to apply this abstract results (Mawhin’s coincidence degree theory) is proving the Fredholm property of the operator L. An example is also presented to illustrate the effectiveness of the main results with integral condition at resonance case The existence of solution is etablished via Mawhin's coincidence degree theory. The results are illustrated with an example.