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Zbl 1168.34014
Luo, Hua
Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 4, A, 1679-1691 (2009). ISSN 0362-546X

Summary: This paper considers a singular m-point dynamic eigenvalue problem on time scales $\Bbb T$: $$-(p(t)u^\Delta(t))^\nabla= \lambda f(t,u(t)), \quad t\in(0,1]\cap\Bbb T,$$ $$u(0)= \sum_{i=1}^{m-2} a_iu(\xi_i), \quad \gamma u(1)+\delta p(1)u^\Delta(1)= \sum_{i=1}^{m-2} b_ip(\xi_i)u^\Delta(\xi_i).$$ We allow $f(t,w)$ to be singular at $w=0$ and $t=0$. By constructing the Green's function and studying its positivity, eigenvalue intervals in which there exist positive solutions of the above problem are obtained by making use of the fixed point index theory.
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B16 Singular nonlinear boundary value problems
34B10 Multipoint boundary value problems
39A10 Difference equations

Keywords: time scales; singular multi-point boundary-value problems; eigenvalue; existence; Green's function

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