### A cantilever equation with nonlinear boundary conditions.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We study the existence of positive solutions to a class of singular nonlinear fourth-order boundary value problems in which the nonlinearity may lack homogeneity. By introducing suitable control functions and applying cone expansion and cone compression, we prove three existence theorems. Our main results improve the existence result in [Z. L. Wei, Appl. Math. Comput. 153 (2004), 865-884] where the nonlinearity has a certain homogeneity.

We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications,...

This paper is concerned with the existence of positive solutions of a multi-point boundary value problem for higher-order differential equation with one-dimensional $p$-Laplacian. Examples are presented to illustrate the main results. The result in this paper generalizes those in existing papers.

Utilizing the theory of fixed point index for compact maps, we establish new results on the existence of positive solutions for a certain third order boundary value problem. The boundary conditions that we study are of nonlocal type, involve Stieltjes integrals and are allowed to be nonlinear.

We consider the singular boundary value problem $${\left({t}^{n}{u}^{\text{'}}\left(t\right)\right)}^{\text{'}}+{t}^{n}f(t,u\left(t\right))=0,\phantom{\rule{1.0em}{0ex}}\underset{t\to 0+}{lim}{t}^{n}{u}^{\text{'}}\left(t\right)=0,\phantom{\rule{1.0em}{0ex}}{a}_{0}u\left(1\right)+{a}_{1}{u}^{\text{'}}(1-)=A,$$ where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \mathbb{N}$, $n\ge 2$, and ${a}_{0}$, ${a}_{1}$, $A$ are real constants such that ${a}_{0}\in (0,\infty )$, whereas ${a}_{1},A\in [0,\infty )$. The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in...

This paper studies the existence of solutions for fractional hybrid differential inclusions of Hadamard type by using a fixed point theorem due to Dhage. The main result is illustrated with the aid of an example.