## Periodic boundary value problem for second order integro-ordinary differential equations with general kernel and Carathéodory nonlinearities.(English)Zbl 0837.45006

The author considers systems of linear and nonlinear second-order integro-ordinary differential equations $- u''(t) = f \bigl( t,u (t), Ku(t) \bigr),\;u(0) = u(2 \pi),\;u'(0) = u'(2 \pi), \tag{1}$ where $$f : I \times\mathbb{R}^2 \to\mathbb{R}$$ is a Carathéodory function, $$K$$ is an integral operator in $$L^2 (I)$$ with kernel $$k \in L^2 (J)$$, $$J = I \times I$$. Related to (1) consider the linear problem $- u''(t) + Mu (t) + N[Ku] (t) = h(t),\;u(0) = u(2 \pi),\;u'(0) = u'(2 \pi) \tag{2}$ where $$M,N \in\mathbb{R}$$ and $$h \in L^2 (I)$$. The maximum principle is applied and existence conditions are established for periodic solutions. For the problem under consideration the monotone iterative technique is applied as well.
Reviewer: A.Martynyuk (Kiev)

### MSC:

 45J05 Integro-ordinary differential equations 45M15 Periodic solutions of integral equations 45G15 Systems of nonlinear integral equations
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