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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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