In this article, we deal with the existence of solutions for the following second-order differential equation: u′′(t)=f(t,u(t))+h(t)u(a)-u(b)=u′(a)-u′(b)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned}&u''(t)=f(t,u(t))+h(t)\\\\&u(a)-u(b)= u'(a)-u'(b)=0, \\end{aligned}\\right. \\end{aligned}$$\\end{document}where {mathbb {B}} is a reflexive real Banach space, f:[a,b]times {mathbb {B}}rightarrow {mathbb {B}} is a sequentially weak–strong continuous mapping, and h:[a,b]rightarrow {mathbb {B}} is a continuous function on {mathbb {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.