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Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives. (English) Zbl 1168.34014
Summary: This paper considers a singular m-point dynamic eigenvalue problem on time scales \(\mathbb T\):
\[ -(p(t)u^\Delta(t))^\nabla= \lambda f(t,u(t)), \quad t\in(0,1]\cap\mathbb T, \]
\[ u(0)= \sum_{i=1}^{m-2} a_iu(\xi_i), \quad \gamma u(1)+\delta p(1)u^\Delta(1)= \sum_{i=1}^{m-2} b_ip(\xi_i)u^\Delta(\xi_i). \]
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:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
39A10 Additive difference equations
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