Multi-point boundary value problems for second-order functional differential equations

AbstractThis paper is concerned with the existence of extreme solutions of multi-point boundary value problem for a class of second-order functional differential equations. We introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solutions

Existence and uniqueness of positive solutions for Neumann problems of second order impulsive differential equations

This work is concerned with the existence and uniqueness of positive solutions for Neumann boundary value problems of second order impulsive differential equations. The result is obtained by using a fixed point theorem of generalized concave operators

Three positive solutions to initial-boundary value problems of nonlinear delay differential equations

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation $$\left\{ \begin{array}{lll} (\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\ x_{0}=0,\\ x(1)=0, \end{array}\right.$$ where $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\mathbb{R})$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $-\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results