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Periodic solutions of second order singular coupled systems. (English) Zbl 1204.34026
Consider the coupled system $$\aligned \ddot x+ a_1(t)x &= f_1(t,y)+ e_1(t),\\ \ddot y+ a_2(t) y &= f_2(t,x)+ e_2(t),\endaligned\tag{$*$}$$ where $a_i\in C([0, T],\bbfR)$, $f_i\in C([0,T]\times (0,+\infty),\bbfR)$ for $i= 1,2$, $f_i(t,z)$ can be singular at $z= 0$. The authors assume that the Green’s function $G_i(t,s)$ belonging to the boundary value problem $$\ddot z+ a_i(t)z= e_i(t),\quad z(0)= z(T),\quad \dot z(0)=\dot z(T),\quad i= 1,2,$$ is nonnegative and that the function $f_i$ satisfies $$0\le{\widehat b_i(t)\over x^{\alpha_i}}\le f_i(t,x)\le {b_i(t)\over x^{\alpha_i}},\quad i= 1,2,$$ for $x> 0$ and $t\in(0, T)$, where $0<\alpha_i< 1$. Under additional conditions on the maximum and minimum of the function $$\gamma_i(t)= \int^T_0 G_i(t, s) e_i(s)\,ds,$$ they prove the existence of positive solutions of $(*)$ satisfying periodic boundary conditions. Since there is no assumption on the $T$-periodicity of the functions $a_i$ and $f_i$ with respect to $t$, the solutions are not necessarily $T$-periodic solutions.

34B16Singular nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] L., H.; Yu, H.; Liu, Y.: Positive solutions for singular boundary value problems of a coupled system of differential equations. J. math. Anal. appl. 302, 14-29 (2005) · Zbl 1076.34022
[2] Agarwal, R. P.; O’regan, D.: Multiple solutions for a coupled system of boundary value problems. Dynam. contin. Discrete impuls. Syst. 7, 97-106 (2000) · Zbl 0958.34022
[3] Agarwal, R. P.; O’regan, D.: A coupled system of boundary value problems. Appl. anal. 69, 381-385 (1998) · Zbl 0919.34022
[4] Jiang, D.; Chu, J.; Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. differential equations 211, 282-302 (2005) · Zbl 1074.34048
[5] Torres, P. J.: Weak singularities May help periodic solutions to exist. J. differential equations 232, 277-284 (2007) · Zbl 1116.34036
[6] Chu, J.; Torres, P. J.: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. London math. Soc. 39, 653-660 (2007) · Zbl 1128.34027
[7] Torres, P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. differential equations 190, 643-662 (2003) · Zbl 1032.34040
[8] Chu, J.; Torres, P. J.; Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. differential equations 239, 196-212 (2007) · Zbl 1127.34023
[9] Zhang, M.; Li, W.: A Lyapunov-type stability criterion using $L{\alpha}$ norms. Proc. amer. Math. soc. 130, 3325-3333 (2002) · Zbl 1007.34053
[10] Fonda, A.; Toader, R.: Periodic orbits of radially symmetric Keplerian- like systems, A topological degree approach. J. differential equations 244, 3235-3264 (2008) · Zbl 1168.34031
[11] Franco, D.; Torres, P. J.: Periodic solutions of singular systems without the strong force condition. Proc. amer. Math. soc. 136, No. 4, 1229-1236 (2008) · Zbl 1129.37033