## Eigenvalue problems for second-order nonlinear dynamic equations on time scales.(English)Zbl 1099.34026

The authors are concerned with the second-order nonlinear dynamic equation on time scales $u^{\Delta\Delta}(t)+\lambda a(t)f(u(\sigma(t)))=0, t\in [0,1],$
satisfying either the conjugate boundary conditions $$u(0)=u(\sigma(1))=0$$ or the right focal boundary conditions $$u(0)=u^\Delta(\sigma(1))=0$$, where $$a$$ and $$f$$ are positive. The number of positive solutions of the above boundary value problem for $$\lambda$$ belonging to the half-line $$(0,\infty)$$ and the dependence of positive solutions of the problem on the parameter $$\lambda$$ are discussed. It is proved that there exists a $$\lambda^*>0$$ such that the problem has at least two, one and no positive solution(s) for $$0<\lambda<\lambda^*, \lambda=\lambda^*$$ and $$\lambda>\lambda^*$$, respectively.
The main tool is a fixed-point index theorem on cones due to Guo-Lakshmikantham. Furthermore, by using the semi-order method on cones of Banach space, an existence and uniqueness criterion for a positive solution of the problem is established. In particular, such a positive solution $$u_\lambda(t)$$ of the problem depends continuously on the parameter $$\lambda$$, i.e., $$u_\lambda(t)$$ is nondecreasing in $$\lambda$$, $$\lim_{\lambda\rightarrow 0^+}\| u_\lambda\| =0$$ and $$\lim_{\lambda\rightarrow +\infty}\| u_\lambda\| =+\infty$$.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 39A10 Additive difference equations
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