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

Abstract

Motivated by the study of systems of higher order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results.2010 Mathematics Subject Classification. Primary 45G15, secondary 34B10, 34B18, 47H30.

Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence

“…Here we discuss in detail the existence and non-existence of positive solutions of the system (1.7), illustrating how the constants that occur in our theory can be computed or estimated. Our results are new and complement the ones in [1,4,15,16,[19][20][21].…”

“…where the functionals h j act on the space C 1 [0, 1], has been studied recently by Infante [16], by means of the classical fixed-point index. Here we develop further this approach and we extend the results of [16] to the case of systems and higher-order dependence in the nonlinearities and the functionals. We also improve the case n = 1 and m 1 = 1, by allowing more freedom in the growth of the nonlinearities near the origin; this is achieved by means of an eigenvalue comparison.…”

“…where the functionals h j act on the space C 1 [0, 1], has been studied recently by Infante [16], by means of the classical fixed-point index. Here we develop further this approach and we extend the results of [16] to the case of systems and higher-order dependence in the nonlinearities and the functionals.…”

Positive solutions of a second-order nonlinear Robin problem involving the first-order derivative

“…For the cases of superlinear nonlinearity at ∞, the Bernstein-Nagumo type condition [7,9,25] is introduced to enable us to obtain the a priori estimate of the first-order derivative for associated boundary value problems. Our results generalize and extend the ones in [20,23], are strikingly different from the ones in [1-3, 8, 12, 14, 15, 17, 21] and also complement the main ones in [16,24].…”

Non-local second-order boundary value problems with derivative-dependent nonlinearity

“…This does not affect fixed points but could influence intermediate calculations. Using that set-up has been done in a number of papers including [17,18] where an integral equation of this type but corresponding to a BVP with nonlinear BCs is studied. It is important for us that v does satisfy the BCs.…”