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

Abstract In this article, we deal with the existence of solutions for 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 , + h a - b 0 where $${\mathbb {B}}$$ B is a reflexive real Banach space, $$f:[a,b]\times {\mathbb {B}}\rightarrow : [ ] × ? sequentially weak–strong continuous mapping, and $$h:[a,b]\rightarrow function on {B}}.$$ . The proofs are obtained using recent generalization well-known Bolzano–Poincaré–Miranda theorem to infinite-dimensional spaces. last section, present three examples application general result.