Periodic Solutions to Second-Order Nonlinear Differential Equations in Banach Spaces

Abstract

In this article, we deal with the existence of solutions for the following second-order differential equation: $$\begin{aligned} \left\{ \begin{aligned}&u''(t)=f(t,u(t))+h(t)\\&u(a)-u(b)= u'(a)-u'(b)=0, \end{aligned}\right. \end{aligned}$$ u ′ ′ ( t ) = f ( t , u ( t ) ) + h ( t ) u ( a ) - u ( b ) = u ′ ( a ) - u ′ ( b ) = 0 , where $${\mathbb {B}}$$ B is a reflexive real Banach space, $$f:[a,b]\times {\mathbb {B}}\rightarrow {\mathbb {B}}$$ f : [ a , b ] × B → B is a sequentially weak–strong continuous mapping, and $$h:[a,b]\rightarrow {\mathbb {B}}$$ h : [ a , b ] → B is a continuous function on $${\mathbb {B}}.$$ B . The proofs are obtained using a recent generalization of the well-known Bolzano–Poincaré–Miranda theorem to infinite-dimensional Banach spaces. In the last section, we present three examples of application of the general result.