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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.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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##### References:
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