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

Yuming Wei, Beijing Institute of Technology, Beijing, P. R. China
Patricia J. Y. Wong, Nanyang Technological University, Jurong, Singapore

E. J. Qualitative Theory of Diff. Equ., No. 14. (2009), pp. 1-11.

Abstract: 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 $\phi: \R \rightarrow \R$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\R)$ 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.

You can download the full text of this paper in DVI, PostScript or PDF format.