Third order problems with nonlocal conditions of integral type

Abstract

We discuss the existence of solutions of nonlinear third order ordinary differential equations with integral boundary conditions. We provide sufficient conditions on the nonlinearity and the functions appearing in the boundary conditions that guarantee the existence of at least one solution to our problem. We rely on the method of lower and upper solutions to generate an iterative technique, which is not necessarily monotone.

Fitted finite difference technique for singularly perturbed reaction diffusion equations of third order with integral boundary condition

“…Boundary value problems with IBC are a common type of problem that arises in domains such as electrochemistry, 17 thermoelasticity, 18 heat conduction 19 . In References 20‐24 the authors have examined the existence and uniqueness of third order differential equations with IBC, as well as their applications. The existence and applications of systems of second order differential equations with IBC is discussed in References 3,25, and 26.…”

A family of singular ordinary differential equations of the third order with an integral boundary condition

“…Singular boundary value problems arise very frequently in fluid mechanics and in other branches of applied mathematics. There are results on the existence and asymptotic estimates of solutions for third order ordinary differential equations with singularly perturbed boundary value problems, which depend on a small positive parameter see for example [16,19,27], on third order ordinary differential equations with singularly perturbed boundary value problems and with nonlinear coefficients or boundary conditions see for example [3,12,29,50], on third order ordinary differential equations with nonlinear boundary value problems see for example [18,28], on existence results for third order ordinary differential equations see for example [17,24], and particularly third order ordinary differential equations with integral boundary conditions see for example [2,6,7,20,21,39,42,47,49] In the last years there are several papers which consider integral or nonlocal boundary conditions on different branches of applications, e.g. for the heat equations see for example [10,13,14,15,22,26,30,34,35,36,38], for the wave equations [37], for the second order ordinary differential equations see for example [5,31,33,44,52,53,…”

“…There are results on the existence and asymptotic estimates of solutions for third order ordinary differential equations with singularly perturbed boundary value problems, which depend on a small positive parameter, see, for example [1][2][3], on third order ordinary differential equations with singularly perturbed boundary value problems and with nonlinear coefficients or boundary conditions, see for example [4][5][6][7], on third order ordinary differential equations with nonlinear boundary value problems, see for example [8,9], on existence results for third order ordinary differential equations, see for example [10][11][12], and particularly third order ordinary differential equations with integral boundary conditions, see for example [13][14][15][16][17][18][19][20][21].…”