### A multiplicity fixed point theorem in Fréchet spaces.

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Existence of positive solution to certain classes of singular and nonsingular third order nonlinear two point boundary value problems is examined using the idea of Topological Transversality.

In this paper, we prove existence and controllability results for first and second order semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces, with nonlocal conditions. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.

New fixed point results are presented for multivalued maps defined on subsets of a Fréchet space E. The proof relies on the notion of a pseudo open set, degree and index theory, and on viewing E as the projective limit of a sequence of Banach spaces.

In this paper we study the existence of nontrivial solutions for a nonlinear boundary value problem posed on the half-line. Our approach is based on Ekeland’s variational principle.

This paper discusses the existence and multiplicity of solutions for a class of $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $$\left\}\begin{array}{cc}-\left(a+b{\int}_{\Omega}\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}\phantom{\rule{0.277778em}{0ex}}dx\right){\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)=f(x,u)\phantom{\rule{0.166667em}{0ex}},\hfill & in\phantom{\rule{1.0em}{0ex}}\Omega \hfill \\ u=0\hfill & on\phantom{\rule{1.0em}{0ex}}\partial \Omega \phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\right.$$ where $\Omega $ is a smooth open subset of ${\mathbb{R}}^{N}$ and $p\in C\left(\overline{\Omega}\right)$ with $N<{p}^{-}={inf}_{x\in \Omega}p\left(x\right)\le {p}^{+}={sup}_{x\in \Omega}p\left(x\right)<+\infty $, $a$, $b$ are positive constants and $f:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

This paper investigates the existence of positive solutions for a fourth-order differential system using a fixed point theorem of cone expansion and compression type.

We prove controllability results for first and second order semilinear differential inclusions in Banach spaces with nonlocal conditions.

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