### A nonlinear bounday value problem for a nonlinear ordinary differential operator in weighted Sobolev spaces.

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This paper is devoted to the problem of existence of a solution for a non-resonant, non-linear generalized multi-point boundary value problem on the interval $[0,1]$. The existence of a solution is obtained using topological degree and some a priori estimates for functions satisfying the boundary conditions specified in the problem.

This paper surveys a number of recent results obtained by C. Bereanu and the author in existence results for second order differential equations of the form (ϕ(u'))' = f(t,u,u') submitted to various boundary conditions. In the equation, ϕ: ℝ → ≤ ]-a,a[ is a homeomorphism such that ϕ(0) = 0. An important motivation is the case of the curvature operator, where ϕ(s) = s/√(1+s²). The problems are reduced to fixed point problems in suitable function space, to which Leray-Schauder...

Let $\mathbb{T}$ be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay ${x}^{\u25b5}\left(t\right)=-a\left(t\right)h\left({x}^{\sigma}\left(t\right)\right)+c\left(t\right){x}^{\tilde{\u25b5}}\left(t-r\left(t\right)\right)+G\left(t,x\left(t\right),x\left(t-r\left(t\right)\right)\right)$, $t\in \mathbb{T}$, where ${f}^{\u25b5}$ is the $\u25b5$-derivative on $\mathbb{T}$ and ${f}^{\tilde{\u25b5}}$ is the $\u25b5$-derivative on $(id-r)\left(\mathbb{T}\right)$. We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show...

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where $\overline{{e}^{isp\left(f\right)}}$ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*) has a bounded...

We give an existence result for a periodic boundary value problem involving mean curvature-like operators. Following a recent work of R. Manásevich and J. Mawhin, we use an approach based on the Leray-Schauder degree.

The existence and multiplicity results are shown for certain types of problems with nonlinear boundary value conditions.

Existence results are established for the resonant problem ${y}^{\text{'}\text{'}}+{\lambda}_{m}\phantom{\rule{0.166667em}{0ex}}a\phantom{\rule{0.166667em}{0ex}}y=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in {L}_{\mathrm{l}oc}^{1}(0,1)$ with $a>0$ a.e. on $[0,1]$ and ${\int}_{0}^{1}x(1-x)a\left(x\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x<\infty $.