Singular (p,n−p) focal and (n,p) higher order boundary value problems

In this paper we discuss the existence of positive solutions of the fully fourth-order boundary value problem which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where $f:[0,1]\times {{\mathbb{R}}_{+}}^3\times {\mathbb{R}}_{-}\to {\mathbb{R}}_{+}$ is continuous. Some inequality conditions on $f$ guaranteeing the existence of positive solutions are presented. Our conditions allow that $f(t,x_0,x_1,x_2,x_3)$ is superlinear or sublinear growth on $x_0,x_1,x_2,x_3$. For the superlinear case, a Nagumo-type condition is presented to restrict the growth of $f$ on $x_2$ and $x_3$. Our discussion is based on the fixed point index theory in cones.

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo–Krasnosel'skii fixed point theorem of cone expansion–compression type play important parts in this work.

For higher order nonlinear hyperbolic equations nonclassical boundary value problems of Vallée-Poussin type are studied, and unimprovable in a sense sufficient conditions of solvability, unique solvability and well-posedness are established.

For the differential equation where the function $f:(a,b)\times {\mathbb{R}}^2\to \mathbb{R}$ has nonintegrable singularities with respect to the time variable at $t=a$ and $t=b$, new unimprovable sufficient conditions of solvability and unique solvability of the boundary value problems and are established.

The positive solution is studied for a nonlinear fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. By making use of the monotonically iterative technique, a successively iterative sequence is constructed. By applying the successively approximate method, an existence theorem is proved.

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following third-order generalized right-focal boundary value problem with p-Laplacian operator: The main tool is the monotone iterative technique.