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Zbl 1099.34026
Li, Wan-Tong; Liu, Xi-Lan
Eigenvalue problems for second-order nonlinear dynamic equations on time scales.
(English)
[J] J. Math. Anal. Appl. 318, No. 2, 578-592 (2006). ISSN 0022-247X

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. \par 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 $\lambda$, $\lim_{\lambda\rightarrow 0^+}\Vert u_\lambda\Vert =0$ and $\lim_{\lambda\rightarrow +\infty}\Vert u_\lambda\Vert =+\infty$.
[Dingyong Bai (Guangzhou)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
39A10 Difference equations

Keywords: eigenvalue problems; nonlinear dynamic equations; time scale; positive solution; existence; uniqueness

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