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Existence of positive solutions for singular boundary value problem on time scales. (English) Zbl 1116.34015

This paper deals with the existence of positive solutions to the second-order dynamic equation on a time scale \(\mathbb{T}\) \[ [\phi(t)x^{\triangle}(t)]^{\triangle} + \lambda m(t) f(t, x(\sigma(t))) = 0, \quad t \in [a,b], \] satisfying the boundary conditions \[ \alpha x(a) - \beta x^{\triangle}(b) = 0, \quad \gamma x(\sigma(b)) + \delta x^{\triangle}(\sigma(b)) = 0. \] The nonresonant case is studied under the assumptions \(\alpha, \beta, \gamma, \delta \geq 0\), \(\delta \geq \gamma(\sigma^2(b)-\sigma(b))\), and \[ \frac{\gamma\beta}{\phi(a)} + \frac{\alpha\delta}{\phi(\sigma(b))} + \alpha\gamma\int_a^{\sigma(b)}\frac{\triangle \tau}{\phi(\tau)} > 0. \] The main result is as follows: Assume that \((H_1)\) \(f \in C([a,\sigma(b)] \times [0,\infty), (0,\infty))\) and there exist a constant \(L\) and a function \(F\), which is integrable on \([a,\sigma(b)]\), satisfying \(f^2(t,s) \leq F(t)\), \([a,\sigma(b)] \times [L,\infty)\); \((H_2)\) \(m: (a,\sigma(b)) \to [0,\infty)\) is rd-continuous and may be singular at both \(t = a\) and \(t = \sigma(b)\); \((H_3)\) \(0 < \int_{\xi}^{\omega} G(\sigma(s),s) m(s) \triangle s, \int_{a}^{\sigma(b)} G(\sigma(s),s) m(s) \triangle s < \infty\), where \(G(\sigma(t),s)\) and the constants \(\xi\) and \(\omega\) are given in a paper by L. Erbe and A. Peterson [Math. Comput. Modelling 32, No. 5–6, 571–585 (2000; Zbl 0963.34020)]; \((H_4)\) there exist constants \(r,R \in [\xi,\omega]\), satisfying \[ \liminf_{s \to \infty} \min_{t \in [r,R]} \frac{m(t)f(t,s)}{s} = +\infty. \] Then, for every \(\lambda \in (0,\infty)\), the boundary value problem has at least one positive solution.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations

Citations:

Zbl 0963.34020
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Full Text: DOI

References:

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