# Existence and multiplicity results for some generalized Hammerstein equations with a parameter.

This paper considers the existence and multiplicity of fixed points for the integral operator \begin{equation*} {\mathcal{T}}u(t)=\lambda \,\int_{0}^{T}k(t,s)\,f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,\dif s,\quad t\in \lbrack 0,T]\equiv I, \end{equation*} where $\lambda >0$ is a positive parameter, $k:I\times I\rightarrow \mathbb{R% }$ is a kernel function such that $k\in W^{m,1}\left( I\times I\right) $, $m$ is a positive integer with $m\geq 1$, and $f:I\times \mathbb{R}^{m+1}\rightarrow \lbrack 0,+\infty \lbrack $ is a $L^{1}$-Carath\'{e}odory 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 on the literature in this field.

Publisher URL: http://arxiv.org/abs/1811.06118

DOI: arXiv:1811.06118v1

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.