A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

Abstract

Abstract. In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described.

Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence

“…In particular, Goodrich studied the equation The functional formulation (1.3) is well suited for handling, in a unified way, multi-point and integral BCs. For an introduction to nonlocal BCs we refer the reader to the reviews [2,4,17,19,18,21,25] and the manuscripts [14,15,22].…”

Nonzero positive solutions of nonlocal elliptic systems with functional BCs

“…whereα ijk are non-negative coefficients, ω j ∈ Ω,α ik are non-negative continuous functions on Ω. There exists a wide literature on these kind of BCs, we refer the reader to the reviews [4,10,29,33,39,42] and the papers [24,25,34,35,41]. We point that that this setting can also be applied to nonlinear, nonlocal BCs, which have seen recently attention in the framework of (local) elliptic equations, we refer the reader to the papers by Cianciaruso and co-authors [8] and Goodrich [18,19,20,21].…”

Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions

“…Furthermore, a number of applications of nonlocal problems for evolution equations are illustrated in Section 10.2 of [22]. Initial nonlocal conditions have been investigated in a variety of settings, for example in the case of multi-point [10], integral [9,13,23,24,26,27] and nonlinear conditions [2,6,7,8], see also the recent review [28].…”