Multiplicity of positive solutions for a nonlinear m-point BVP with sign changing nonlinearity on time scales

Abstract

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for a nonlinear m-point boundary value problem for the following second-order dynamic equation on time scales $$\begin{array}{l}(\phi(u^{\Delta}))^{\nabla}+a(t)f(t,u(t))=0,\quad t\in(0,T),\\\noalign{\vspace{2mm}}\displaystyle u(0)=\sum_{i=1}^{m-2}a_{i}u(\xi_{i}),\qquad \phi(u^{\Delta}(T))=\sum_{i=1}^{m-2}b_{i}\phi(u^{\Delta}(\xi_{i})),\end{array}$$ where φ:R⟶R is an increasing homeomorphism and positive homomorphism with φ(0)=0. The nonlinear term f may change sign. We obtain several existence theorems of positive solutions for the above boundary value problems. We should point out that the above equation we studied is same as that in Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), but the methods is different from Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), we generalize and improve the results (Han and Jin, Communications in Nonlinear Science and Numerical Simulation, 2009). As an application, a typical example to demonstrate our results is given.