On positive periodic solutions of second-order semipositone differential equations

Using the Krasnosel’skii’s fixed point theorem, we establish the existence and multiplicity of positive Tperiodicsolutions of second-order semipositone system $x''(t)+a(t)x(t)=lambda f(t,x(t)),x(0)=x(T),x'(0)=x'(T)$ where $x=(x_1,;x_2,......,;x_n),;f(t,x)=(f_1(t,x),;f_2(t,x),.....,;f_n(t,x))$ is bounded below.

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[1] Agarwal RP, O’Regan D, Wong PJY. Existence of constant-sign solutions to a system of difference equations: the semipositone and singular case. Journal of Difference Equations and Applications 2005; 11: 151-171.
[2] Bahri A, Rabinowitz PH. A minimax method for a class of Hamiltonian systems with singular potentials. Journal of Functional Analysis 1989; 82: 412-428.
[3] Bonheure D, De Coster C. Forced singular oscillators and the method of lower and upper solutions. Topological Methods in Nonlinear Analysis 22 (2003), 297-317.
[4] De Coster C, Habets P. An overview of the method of lower and upper solutions for ODEs. In: Grossinho MR, Ramos M, Rebelo C, Sanchez L, editors. Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and their Applications, vol 43. Boston, MA, USA: Birkhuser, 2001.
[5] Chu J, Torres PJ, Zhang M. Periodic solutions of second order non-autonomous singular dynamical systems. Journal of Differential Equations 2007; 239: 196-212.
[6] Dogan A. The existence of positive solutions for a semipositone second-order m-point boundary value problem. Dynamic Systems and Applications 2015; 24: 419-428.
[7] Dogan A. On the existence of positive solutions for the second-order boundary value problem, Applied Mathematics Letters 2015; 49: 107-112.
[8] Dogan A. Positive solutions of nonlinear multi-point boundary value problems. Positivity 2018; 22: 1387-1402.
[9] Franco D, Torres PJ. Periodic solutions of singular systems without the strong force condition. Proceedings of the American Mathematical Society 2008; 136: 1229-1236.
[10] Franco D, Webb JRL. Collisionless orbits of singular and nonsingular dynamical systems. Discrete & Continuous Dynamical Systems 2006; 15: 747-757.
[11] Guo D, Lakshmikantham V. Nonlinear Problems in Abstract Cones. New York, NY, USA: Academic Press, 1988.
[12] Graef JR, Kong L, Wang H. Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. Journal of Differential Equations 2008; 245: 1185-1197.
[13] Hai DD, Shivaji R. Positive solutions for semipositone systems in an annulus. Rocky Mountain Journal of Mathematics 1999; 29: 1285-1299.
[14] Lee EK, Sankar L, Shivaji R. Positive solutions for infinite semipositone problems on exterior domains. Differential Integral Equations 2011; 24: 861-875.
[15] Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Berlin, Germany: Springer, 1989.
[16] Ma R, Chen R, He Z. Positive periodic solutions of second-order differential equations with weak singularities. Applied Mathematics and Computation 2014; 232: 97-103.
[17] O’Regan D, Wang H. Positive periodic solutions of systems of second order ordinary differential equations. Positivity 2006; 10: 285-298.
[18] Rachunková I, Tvrdý M, Vrkoc̆ I. Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. Journal of Differential Equations 2001; 176: 445-469.
[19] Torres PJ. Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. Journal of Differential Equations 2003; 190: 643-662.
[20] Wang H. Positive periodic solutions of functional differential equations. Journal of Differential Equations 2004; 202: 354-366.
[21] Wang H. Positive periodic solutions of singular systems with a parameter. Journal of Differential Equations 2010; 249: 2986-3002.
[22] Wang H. Periodic solutions to non-autonomous second-order systems. Nonlinear Analysis 2009; 71: 1271-1275.
[23] Wang F, Chu J, Siegmund S. Periodic solutions for second order singular differential systems with parameters. Topological Methods in Nonlinear Analysis 2015; 46: 549-562.
[24] Xu X. Postive solutions for singular semi-positone three-point systems. Nonlinear Analysis 2007; 66: 791-805.
[25] Yao Q. An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem. Applied Mathematics Letters 2010; 23: 1401-1406.
[26] Zhao F, Wu X. Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Nonlinear Analysis 2005; 60: 325-335.