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Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods. (English) Zbl 1126.34018

The authors have employed variational techniques and critical point theory to obtain sufficient conditions for the existence of multiple positive solutions of second order dynamic equations with homogeneous Dirichlet boundary conditions of the form
\[ -u^{\Delta\Delta}(t)=f(t,u^\sigma(t)),\quad t\in J\cap T^{k^2},\quad u(a)=0=u(b), \]
with
\[ J=\begin{cases} [a,b)\cap T &\text{if }a<\sigma(a)\\ (a,b)\cap T &\text{if }a=\sigma(a), \end{cases} \] \(T\subset \mathbb R\) an arbitrary bounded time scale such that \(\min T= a\) and \(\max T= b\) and \(f:J\times\mathbb R\to\mathbb R\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
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[1] Agarwal, R.P.; Otero-Espinar, V.; Perera, K.; Vivero, D.R., Basic properties of Sobolev’s spaces on bounded time scales, Adv. difference equ., (2006), 14 pp., Art. ID 38121 · Zbl 1139.39022
[2] R.P. Agarwal, V. Otero-Espinar, K. Perera, D.R. Vivero, Wirtinger’s inequalities on time scales, Canad. Math. Bull., in press · Zbl 1148.26020
[3] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear anal., 58, 69-73, (2004) · Zbl 1070.39005
[4] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular problems by variational methods, Proc. amer. math. soc., 134, 3, 817-824, (2006) · Zbl 1094.34013
[5] Bohner, M.; Peterson, A., Dynamic equations on time scales. an introduction with applications, (2001), Birkhäuser Boston Boston, MA · Zbl 0978.39001
[6] Cabada, A.; Vivero, D.R., Criterions for absolutely continuity on time scales, J. difference equ. appl., 11, 11, 1013-1028, (2005) · Zbl 1081.39011
[7] Cerami, G., An existence criterion for the critical points on unbounded manifolds, Istit. lombardo accad. sci. lett. rend. A, 112, 2, 332-336, (1979), 1978 · Zbl 0436.58006
[8] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS reg. conf. ser. math., vol. 65, (1986), Conference Board of the Mathematical Sciences Washington, DC · Zbl 0609.58002
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