Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales

Abstract

We study a system of second-order dynamic equations on time scales $(p_1{u}_1^{\nabla }{)}^{\mathrm{\Delta }}(t)-q_1(t)u_1(t)+\lambda f_1(t,u_1(t),u_2(t))=0,t\in (t_1,t_n),(p_2{u}_2^{\nabla }{)}^{\mathrm{\Delta }}(t)-q_2(t)u_2(t)+\lambda f_2(t,u_1(t)$, $u_2(t))=0$, satisfying four kinds of different multipoint boundary value conditions, $f_i$ is continuous and semipositone. We derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.