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In this article, the Dirichlet problem for a class of quasilinear differential equation systems is studied. Existence results are given by combining the dual least action principle and mountain pass theorem.

In this article, we consider the solvability for the m-Point boundary value problem: where φ is continuous and strictly increasing function with φ(R) = R. We provide sufficient and necessary conditions for the solvability to the above boundary value problem.

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0 1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.

Consider the existence of positive periodic solutions of the non-autonomous state-dependent delay differential equation with the impulses as follows It is shown that under reasonable conditions, the equation has positive periodic solutions and under certain conditions the equation has no positive T-periodic solution.

We consider the following boundary-value problems: . Using a generalization of Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary-value problems.

By means of continuation theorem of coincidence degree theory, existence, nonexistence and unique existence criteria are established for the periodic solution of a kind of second order neutral functional differential equation.

We consider the ( p , n m p ) right focal boundary value problem: $${\matrix{{(- 1)^{n - p} u^{(n)} \! = \lambda \;f(t, u), } \hfill & \ {{\rm for }\ 0 \lt t \lt 1, } \hfill \cr \quad \quad \,{u^{(i)} (0) = 0, } \hfill & {0 \le i \le p - 1, } \hfill \cr \quad \quad \,{u^{(i)} (1) = 0, } \hfill & {p \le i \le n - 1, } \hfill \cr}} $$ where 1 h p h n m 1 is fixed and u > 0. Using a fixed point theorem for operators on a cone, we develop criteria for the existence of positive solutions of the boundary value problem for u on a suitable interval.

In this paper we are concerned with the nonlinear eigenvalue problem consisting of the functional differential equation with p -Laplacian operator, ( { p ( u '))' + u a ( t ) f ( t , u t ) = 0, { p ( s ) = | s | p m 2 s , p > 1, the initial condition, u ( s ) = } ( s ), m ‰ h s h 0, and the boundary condition, u (0) = 0 = u (1), where a ( t ) is allowed to be singular at the end points of (0, 1). Our results apply to more than just the sublinear and superlinear cases discussed in other references.