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Several existence theorems of nonlinear \(m\)-point BVP for an increasing homeomorphism and homomorphism on time scales. (English) Zbl 1157.39010

The authors consider the problem
\[ (\varphi(u^\Delta))^\nabla+a(t)f(t,u(t))=0, \quad t\in(0,T), \]
\[ \varphi(u^\Delta(0))=\sum_{i=1}^{m-2}a_i\varphi(u^\Delta(\xi_i)), \quad u(T)=\sum_{i=1}^{m-2}b_iu(\xi_i), \]
on a time scale, where \(0<\xi_1<\dots<\xi_{m-2}<\varrho(T)\), and the coefficient and the nonlinearities satisfy quite standard assumptions. By means of a cone fixed point theorem they establish conditions guaranteeing the existence of at least one positive solutions of the above problem. An example is given. According to the reviewer the assumptions posed on \(\varphi\) are such that the only possibility is \(\varphi\) being a power function.
Reviewer: Pavel Rehak (Brno)

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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