Transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations

- We use a new method to deal with impulse term of second impulsive differential equations.
- The dependence of positive solution xλ(t) on the parameter λ is also studied.
- We extend and improve some known results of non-impulsive equations and are compared with some recent results.

Using a new method for dealing with the impulse term of second impulsive differential equations, we are concerned with determining values of λ, for which there exist positive solutions of the following boundary value problems in the form of {λx″(t)+f(t,x(t))=0,t∈J,t≠tk,x(tk+)−x(tk)=ckx(tk),k=1,2,…,n,ax(0)−bx′(0)=ax(1)−bx′(1)=∫01h(s)x(t)dt, where λ>0 is a positive parameter, {ck} is a real sequence with ck>−1,k=1,2,…,n. The dependence of positive solution xλ(t) on the parameter λ is also studied, i.e.,  limλ→+∞‖xλ‖=+∞orlimλ→+∞‖xλ‖=0. The proof of our main result is based upon transformation techniques and fixed point theories. This is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique.