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Existence of solutions for first-order multi-point problems with changing-sign nonlinearity. (English) Zbl 1158.34006

The study of dynamic equations on time scales unifies the theory of differential equations and difference equations. It presents new problems and reveals interesting connections between those traditionally separate areas. Among them are dynamic equations on time scales – introduced by [S. Hilger, Result. Math. 18, No. 1–2, 18–56 (1990; Zbl 0722.39001)] and nicely presented in the monograph by [M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications. Basel: Birkhäuser. (2001; Zbl 0978.39001)].
In this paper the author present the existence of positive solutions for the following class of dynamic equations on time scale with boundary conditions:
\[ y^{\delta}(t)+p(t)y^{\sigma}(t)=\lambda f(t,y^{\sigma}(t)), t\in(a,b)_{\mathbb T}, \]
\[ y(a)=y(b)+\sum_{i=2}^{n-1}\gamma_iy(t_i), \]
where \(n\geq 3\) and \(p: [a,b]_{\mathbb T}\to(0,\infty),\) \(p\in C_{rd}[a,b]_{\mathbb T},\) the points \(t_i\in\mathbb T^k\) for \(i\in\{1,2,\ldots,n\},\) \(\lambda\in(0,\infty),\) \(\gamma_i\in(0,\infty)\) for \(i\in\{2,\ldots,n-1\}\) and \(f:(a,b)_{\mathbb T}\times[0,\infty)\to(-\infty,\infty)\) is an continuous function. Hypotheses imposed on \(f\) are:
\[ \lim_{y\to+\infty}\frac{f(t,y)}{y}=+\infty,\;\;\quad t\in [t_2,t_3]_{\mathbb T};\;-u(t)\leq f(t,y)\leq z(t)h(y) \]
for right-dense continuous functions \(u,z: (a,b)_{\mathbb T}\to(0,\infty)\) and continuous function \(h: [0,\infty)\to(0,\infty).\) The proofs make use of the Guo-Krasonsel’skii fixed point theorem.
This nice paper will be useful to any researcher interested in the theories of difference and differential equations.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
39A10 Additive difference equations
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References:

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