## Multiplicity of positive solutions to period boundary value problems for second order impulsive differential equations.(English)Zbl 1193.34058

The authors consider the impulsive boundary value problem
\begin{aligned} &-x''(t) + \rho^2x = f(t,x), \quad t\neq t_k, t \in [0,2\pi],\\ &\Delta x(t_k) = I_k(x(t_k)),\;-\Delta x'(t_k) = J_k(x(t_k)),\;k = 1,\dots,p, \\ &x(0) = x(2\pi),\quad x'(0) = x'(2\pi), \end{aligned}
where $$0 < t_1 < \dots < t_p < 2\pi$$, $$\rho > 0$$; the functions $$f$$, $$I_k$$, $$J_k$$ are continuous. Existence and multiplicity results for positive solutions are obtained. As a main tool the fixed point theorem in cones is used.

### MSC:

 34B37 Boundary value problems with impulses for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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### 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 [2] Cong, F.Z. Periodic solutions for second order differential equations. Appl. Math. Letters, 18: 957–961 (2005) · Zbl 1094.34523 [3] Deimling, K. Nonlinear Functional Analysis, Springer-Verlag, New York, 1985 · Zbl 0559.47040 [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., 197: 1–13 (1996) · Zbl 0849.34051 [5] Jiang, D.Q. On the existence of positive solutions to second order periodic BVPs. Acta Math. Sci., 18:31–35 (1998) [6] Jiang, D.Q., Wei, J.J. Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. Nonlinear Anal., 50: 885–898 (2002) · Zbl 1014.34049 [7] Krasnoselskii, M.A. Positive Solution of Operator Equation. P.imprenta: Groningen, 1985 [8] Ladde, G.S., Lakshmikantham, V., Vatsala, A.S. Monotone Iterative Teachniques for Nonlinear Differential Equations. Pitman Advanced Publishing Program, Pitman, London, 1985 · Zbl 0658.35003 [9] Lakshmikntham, V., Bainov, D.D, Simeonov, P.S. Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989 [10] Lee, E.K., Lee, Y.H. Multiple positive of singular two point boundary value problems for second order impulsive differential equation. Appl. Math. Comput., 158: 745–759 (2004) · Zbl 1069.34035 [11] Lin, X.N., Jiang, D.Q. 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 [12] Wei, Z.L. Periodic boundary value problem for second order impulsive integrodifferential equations of mixed type in banach space. J. Math. Appl. Anal., 195: 214–229 (1995) · Zbl 0849.45006 [13] Zhang, Z.X., Wang, J.Y. On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations. J. Math. Anal. Appl., 281: 99–107 (2003) · Zbl 1030.34024
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