López Somoza, LucíaMinhós, Feliz2020-04-022020-04-022019López-Somoza, L., Minhós, F. Existence and multiplicity results for some generalized Hammerstein equations with a parameter. Adv Differ Equ 2019, 423 (2019). https://doi.org/10.1186/s13662-019-2359-yhttp://hdl.handle.net/10347/21100This paper considers the existence and multiplicity of fixed points for the integral operator Tu(t)=λ∫T0k(t,s)f(s,u(s),u′(s),…,u(m)(s))ds,t∈[0,T]≡I, where λ>0 is a positive parameter, k:I×I→R is a kernel function such that k∈Wm,1(I×I), m is a positive integer with m≥1, and f:I×Rm+1→[0,+∞[ is an L1-Carathéodory function. The existence of solutions for these Hammerstein equations is obtained by fixed point index theory on new type of cones. Therefore some assumptions must hold only for, at least, one of the derivatives of the kernel or, even, for the kernel on a subset of the domain. Assuming some asymptotic conditions on the nonlinearity f, we get sufficient conditions for multiplicity of solutions. Two examples will illustrate the potentialities of the main results, namely the fact that the kernel function and/or some derivatives may only be positive on some subintervals, which can degenerate to a point. Moreover, an application of our method to general Lidstone problems improves the existent results in the literature in this fieldeng© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were madehttp://creativecommons.org/licenses/by/4.0/Hammerstein equationsNonlinear boundary value problemsParameter dependenceDegree theoryFixed points in conesjournal article10.1186/s13662-019-2359-y1687-1847open access