Periodic solutions to second-order nonlinear differential equations in Banach spaces
In this paper, the authors study the following second-order dynamic differential equation: \[\begin{cases} u''(t) = f(t, u(t)) + h(t),\\ u(a)-u(b)=u'(a)-u'(b)=0, \end{cases} \] where \(f : [a,b]\times {\mathbb B} \to {\mathbb B}\) is a sequentially weak-strong continuous mapping, \(h : [a,b] \to {\mathbb B}\) is a continuous function on \({\mathbb B}\) and \({\mathbb B}\) is a reflexive real Banach space. The existence of periodic solutions is proved via a generalization of Bolzano-Poincaré-Miranda theorem and Leray-Schauder alternative theorem, under the boundedness and growth conditions. Moreover, three examples are presented, two for systems of second-order differential equations and one for a partial differential equation, illustrating the obtained result.
scientific article; zbMATH DE number 7488607
scientific article