Nonlinear Non-Local Boundary-Value Problems and Perturbed Hammerstein Integral Equations

Abstract

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the formwhere G is a compact set in R n . Here α[u] is a positive functional and f is positive, while k and γ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

A Note on a Beam Equation with Nonlinear Boundary Conditions

“…The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case of positive measures, in [21][22][23][24] . Signed measures were used in 12, 25 ; here, as in 21, 22 , due to some inequalities involved in our theory, the functionals α i · are assumed to be given by positive measures.…”

Nodal Solutions of Nonlocal Boundary Value Problems

“…The existence of positive solutions of BVPs with nonlocal BCs, including three-point, multi-point, and integral BCs, have been studied extensively, see, for example, [1,4,5,6,7,8,9,15,16,24,26,28,29,31] and the references therein. In recent years, progress has also been made to the study of nodal solutions, i.e., solutions with a specific zero-counting property in (a, b), for nonlinear BVPs consisting of Eq.…”

Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

“…Subsequent to that paper, Ahmad and Ntouyas have put forth a couple of additional papers devoted to solutions of boundary value problems involving multi-strip integral boundary conditions for both fractional differential equations and fractional differential inclusions; see [11,12]. It can also be pointed out that, under suitable measures, the boundary conditions can be considered in the form of Stieltjes integrals; readers can find of interest the papers, [13][14][15] and [16][17][18].…”