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In this paper, we investigate the positive solutions of periodic boundary value problem. By using critical point theory the existence of multiple positive solutions is obtained.

In this paper, we generalize the fixed point theorem of cone expansion and compression of norm type to the theorem of functional type. As an application, the existence of positive solutions for some fourth–order beam equation boundary value problems is obtained. The emphasis is put on that the nonlinear term is dependent on all lower order derivatives.

By the use of continuation theorems and the duality principle it’s proved that for both homogeneous and nonhomogeneous Sturm–Liouville boundary value problems with Laplacian type operators, relative superlinearity implies the existence of infinitely many solutions under a few weakly added conditions.

The main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows: $$ {\left( {x{\left( t \right)} - {\sum\limits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} }} \right)}^{{\prime \prime }} = f{\left( {x{\left( t \right)}} \right)}{x}\ifmmode{'}\else$'$\fi{\left( t \right)} + g{\left( {x{\left( {t - \tau } \right)}} \right)} + p{\left( t \right)}. $$ In order to do so, we analyze the structure of the linear difference operator A : C 2π →C 2π, \( {\left[ {Ax} \right]}{\left( t \right)} = x{\left( t \right)} - {\sum\nolimits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} } \) to determine some fundamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodic solutions.

In this paper, the authors consider the existence of periodic solutions for a kind of second neutral functional differential equation as follows: $$ {\left( {x{\left( t \right)} - cx{\left( {t - \tau } \right)}} \right)}^{{\prime \prime }} = g{\left( {t,x{\left( {t - \mu {\left( t \right)}} \right)}} \right)} + e{\left( t \right)}, $$ in the critical case |c| = 1. By employing Mawhin's continuation theorem and some analysis techniques, some new results are obtained.

By means of the continuation theorem of coincidence degree theory, some new results on the non–existence, existence and unique existence of periodic solutions for a kind of second order neutral functional differential equation are obtained.

This paper uses the method of upper and lower solutions and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive functional differential equations.

In this paper, we consider the multipoint boundary value problem for the one-dimensional $p$-Laplacian $$\left(\phi_p(u'(t))\right)' + q(t) f\left(t,u(t),u'(t)\right) = 0,~~~~\ \ \ \ \ \ t \in (0,1), $$ subject to the boundary value conditions: $$ u(0) = \sum_{i=1}^{m-2} a_{i} u(\xi_{i}),\ \ \ \ \ \ u(1) = \sum_{i=1}^{m-2} b_{i} u(\xi_{i}), $$ where $\phi_{p}(s) = |s|^{p-2}s$, $p \gt 1$, $\xi_{i} \in (0,1)$ with $0 \lt \xi_{1} \lt \xi_{2} \lt \cdots \lt \xi_{m-2} \lt 1$ and $a_{i}, b_{i} \in [0,1)$, $0 \leq \sum\limits_{i=1}^{m-2} a_{i} \lt 1$, $0 \leq \sum\limits_{i=1}^{m-2} b_{i} \lt 1$. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term $f$ is involved with the first-order derivative explicitly.

In this paper, the following fourth-order three-point boundary value problem with $p$-Laplacian operator is studied: $$ \left\{ \begin{array}{ll} (\phi_p(u''(t)))'' = a(t) f(u(t)), \quad t \in (0,1), \\ u(0) = \xi u(1), \; u'(1) = \eta u'(0), \\ u''(0) = \alpha_1 u''(\delta), \; u''(1) = \beta_1 u''(\delta), \end{array} \right. $$ where $\alpha_1, \; \beta_1 \ge 0$, $\xi \neq 1$, $\eta \neq 1$, $0 \lt \delta \lt 1$ and $\phi_p(z) = |z|^{p-2} z$ for $p \gt 1$. We impose growth conditions on $f$ which guarantee the existence of at least three positive solutions for the problem.