Abstract
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.
Highlights
The main purpose of this article is to study the existence of solutions of the second-order dynamical equation u (t) = f (t, u(t)) + h(t) u(a) − u(b) = u (a) − u (b) = 0, (1.1)where B is a reflexive real Banach space, f : [a, b] × B → B is a continuous mapping, and h : [a, b] → B is a continuous function on B
Some classical tools have been used in the study of periodic solutions of Eq (1.1) in finite-dimensional Banach spaces, including the method of upper and lower solutions [6,10,13], and degree theory and fixed point theory [8,12]
The existence of periodic solutions of second-order differential equations on a general Banach space has been treated in several papers; for instance, see [14,17,19]
Summary
Some classical tools have been used in the study of periodic solutions of Eq (1.1) in finite-dimensional Banach spaces, including the method of upper and lower solutions [6,10,13], and degree theory and fixed point theory [8,12]. In this sense, in [20], the author gave a generalization of Miranda– Poincare theorem and using this generalization proved theorems about the
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