×

Existence results for second-order system with impulse effects via variational methods. (English) Zbl 1208.34032

This paper is devoted to the investigation of the existence of positive solutions to the second-order system with a \(p\)-Laplacian and impulses
\[ \frac{d}{dt}(\Phi_p(\dot{x}(t)))+\nabla F(t,x(t))=0,\qquad t \in [0,T], \]
\[ -\Delta \Phi_p(\dot{x}(t_i))=\nabla I_i(x(t_i)), \qquad i=1,2,\dots,l, \]
\[ x(0)=x(T)=0 \]
by using variational methods, where \(T>0\), \(x \in \mathbb{R}^N\), \(p>1\), \(0= t_0<t_1<\ldots<t_l<t_{l+1}=T\), \(\Delta \Phi_p(\dot{x}(t_i))=\Phi_p(\dot{x}(t_i^+))-\Phi_p(\dot{x}(t_i^-))\), \(\dot{x}(t_i^+)\) and \(\dot{x}(t_i^-)\) denote, respectively, the right and left limits of \(\dot{x}(t)\) at \(t =t_i\), \(\Phi_p(x)=|x|^{p-2} x\), \(\nabla F(t,x)=\frac{\partial}{\partial x}F(t,x)\), \(\nabla I_i(x)=\left( \frac{\partial I_i}{\partial x_1},\dots, \frac{\partial I_i}{\partial x_N} \right)\), \(\nabla F(t,0) \not \equiv 0\), and \(\nabla I_i \in C((\mathbb{R^+})^N, (\mathbb{R^+})^N)\).
The solutions of the problem are transferred, by considering an auxiliary problem, into the critical points of some functional, and the mountain pass theorem [see D. Guo, Nonlinear functional analysis, Shandong Science and Technology Press, Jinan (1985)] allows to prove the existence of at least one positive solution. Some results of [J. Simon, Lect. Notes Math. 665, 205–227 (1978; Zbl 0402.35017)], are also useful for the procedure. A particular case of this system has been studied in [X. Lin and D. Jiang, J. Math. Anal. Appl. 321, 501–514 (2006; Zbl 1103.34015)] by using the fixed point index in cones.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R.P., O’Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114, 51–59 (2000) · Zbl 1047.34008 · doi:10.1016/S0096-3003(99)00074-0
[2] Averna, D., Bonanno, G.: A three critical points theorem and its applications to the ordinary Dirichlet problem. Topol. Methods Nonlinear Anal. 22, 93–104 (2003) · Zbl 1048.58005
[3] Guo, D.: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan (1985)
[4] Hristova, S.G., Bainov, D.D.: Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential-difference equations. J. Math. Anal. Appl. 1997, 1–13 (1996) · Zbl 0849.34051 · doi:10.1006/jmaa.1996.0001
[5] Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002
[6] Lee, E.K., Lee, Y.H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation. Appl. Math. Comput. 158, 745–759 (2004) · Zbl 1069.34035 · doi:10.1016/j.amc.2003.10.013
[7] Lin, X., Jiang, D.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501–514 (2006) · Zbl 1103.34015 · doi:10.1016/j.jmaa.2005.07.076
[8] Liu, X., Guo, D.: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Appl. Math. Comput. 216, 284–302 (1997) · Zbl 0889.45016
[9] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989) · Zbl 0676.58017
[10] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applicatins to Differential Equations. CBMS Regional Conf. Ser. in Math., vol. 65. Am. Math. Soc., Providence (1986) · Zbl 0609.58002
[11] Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000) · Zbl 0979.35040
[12] Ricceri, B.: A general multiplicity theorem for certain nonlinear equations in Hilbert spaces. Proc. Am. Math. Soc. 133, 3255–3261 (2005) · Zbl 1069.47068 · doi:10.1090/S0002-9939-05-08218-3
[13] Simon, J.: Regularité de la Solution d’une Equation Non Lineaire dans R n . Lecture Notes in Mathematics, vol. 665. Springer, Berlin (1978). P. Benilan, J. Robert (eds.)
[14] Tian, Y., Ge, W.: Periodic solutions of non-autonomous second-order systems with a p-Laplacian. Nonlinear Anal. 66, 192–203 (2007) · Zbl 1116.34034 · doi:10.1016/j.na.2005.11.020
[15] Tian, Y., Ge, W.: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509–527 (2008) · Zbl 1163.34015 · doi:10.1017/S0013091506001532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.