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Periodic solutions of second order singular coupled systems. (English) Zbl 1204.34026

Consider the coupled system
\[ \begin{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),\end{aligned}\tag{\(*\)} \]
where \(a_i\in C([0, T],\mathbb{R})\), \(f_i\in C([0,T]\times (0,+\infty),\mathbb{R})\) 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 \[ 0\leq{\widehat b_i(t)\over x^{\alpha_i}}\leq f_i(t,x)\leq {b_i(t)\over x^{\alpha_i}},\quad i= 1,2, \] 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 ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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