Multiple positive solutions for nonlinear three-point boundary value problems on time scales

gðð yðaÞ; yðs2ðbÞÞ; yðeÞ; ð yDðaÞ; yDðsðbÞÞÞÞ 1⁄4 ð0; 0Þ; a , e , s2ðbÞ; e [ T; ð2Þ where f : 1⁄2a; b £ R! R; g : R £ R £ R ! R are continuous, and t is from a so-called “time scale” T. It is assumed that the reader is familiar with the time scale calculus and associated definitions such as delta derivative, jump operators and right-dense continuity. If not, then we refer the reader to Ref. [2]. In Ref. [5], the authors introduced the idea of compatibility of boundary conditions for two point boundary value problems (BVPs) on time scales. In this paper, we extend these ideas to three point BVPs on time scales. The new compatibility conditions are then applied to give some results for the existence of solutions to three point BVPs on time scales. The BVPs treated in this paper include a very wide range of boundary conditions, including nonlinear BVPs. A solution y to Eq. (1) is a function y : 1⁄2a;s2ðbÞ : !R satisfying Eq. (1) with y [ C2rd: