### A cantilever equation with nonlinear boundary conditions.

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The generalized periodic boundary value problem -[g(u’)]’ = f(t,u,u’), a < t < b, with u(a) = ξu(b) + c and u’(b) = ηu’(a) is studied by using the generalized method of upper and lower solutions, where ξ,η ≥ 0, a, b, c are given real numbers, $g\left(s\right)={\left|s\right|}^{p-2}s$, p > 1, and f is a Carathéodory function satisfying a Nagumo condition. The problem has a solution if and only if there exists a lower solution α and an upper solution β with α(t) ≤ β(t) for a ≤ t ≤ b.

We describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem u''(t) = f(t,u(t),u'(t)), u(0) = u(2π), u'(0) = u'(2π) in the presence of a lower solution α(t) and an upper solution β(t) with β(t) ≤ α(t).

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.

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.