Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

Abstract

Abstract. We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

“…This type of problem has been tackled in the past in the case of second and higher order BVPs in [7,24,25,42,52,53,57]. Since in [0, 1] × [0, 1] the kernel k 1 is non-positive only for…”

Nonlinear perturbed integral equations related to nonlocal boundary value problems

“…This type of heat-flow problem was motivated by earlier work by Guidotti and Merino [22] and has been investigated by a number of authors -we refer the reader to the recent papers [30,55] and references therein. The approach of [33] was modified by Cabada and co-authors [7] in order to deal with the case of integral equations with a deviated argument, namely with a signed measure, in the spirit of the paper by Webb and Infante [57]. The results of [7] cover the interesting case of differential equations with reflections and, in particular were applied to the study of the BVP (1.4) u ′′ (t) + g(t)f (t, u(t), u(σ(t))) = 0, t ∈ (0, 1),…”

“…The approach of [33] was modified by Cabada and co-authors [7] in order to deal with the case of integral equations with a deviated argument, namely with a signed measure, in the spirit of the paper by Webb and Infante [57]. The results of [7] cover the interesting case of differential equations with reflections and, in particular were applied to the study of the BVP (1.4) u ′′ (t) + g(t)f (t, u(t), u(σ(t))) = 0, t ∈ (0, 1),…”

“…The BVP (1.4)-(1.5) arises when studying the steady states of a model of a light bulb with a temperature regulating system that includes a feedback controller. One assumption made in this thermostat model is that the feedback controller has a linear response; for more details see Section 4 of [7]. The formulation of nonlocal boundary conditions (BCs) in terms of Stieltjes integrals is fairly general and includes, as special cases, multi-point and integral conditions, namely…”

“…Nevertheless, in a number of applications, the response of the feedback controller can be nonlinear ; for example the nonlocal BVP (1. 7) u (4) (t) − g(t)f (t, u(t)) = 0, u(0) = u ′ (0) = u ′′ (1) = 0, u ′′′ (1) +B(u(η)) = 0, describes a cantilever equation with a feedback mechanism, where a spring reacts (in a nonlinear manner) to the displacement registered in a point η of the beam. Positive solutions of the BVP (1.7) were investigated by Infante and Pietramala in [29] by means of the perturbed integral equation…”