Upper and lower solutions to boundary value problems for functional-differential equations and theorems on functional-differential inequalities (Q2702981)

In the first part of the paper, the authors investigate the existence of positive solutions to the linear boundary value problem for functional-differential equations NEWLINE\[NEWLINE x'(t)=p(x)(t)+q(t), t\in I;\quad l(x)=c, NEWLINE\]NEWLINE with \(I=[a,b]\), \(c\in {\mathbb R}^n\), \(q\in L(I,{\mathbb R}^n)\), and \(p:C(I,{\mathbb R}^n)\to L(I,{\mathbb R}^n)\), \(l:C(I,{\mathbb R}^n)\to {\mathbb R}^n\) are linear bounded operators. For it, some related functional-differential inequalities are studied. NEWLINENEWLINENEWLINEIn the second part, sufficient conditions are found to assure the existence of the extremal (minimal and maximal) solutions to the nonlinear problem NEWLINE\[NEWLINE x'(t)=p(x)(t)+f(x)(t), t\in I;\quad l(x)=h(x), NEWLINE\]NEWLINE where \(f:C(I,{\mathbb R}^n)\to L(I,{\mathbb R}^n)\), \(h:C(I,{\mathbb R}^n)\to {\mathbb R}^n\) are continuous operators. NEWLINENEWLINENEWLINERemark: In the paper, the extremal solutions are called upper and lower solutions, while in the literature the names of upper and lower solutions refer in general to other type of functions. See, for example, the survey by \textit{C. de Coster} and \textit{P. Habets} [in: Zanolin, F. (ed.), Nonlinear analysis and boundary value problems for ordinary differential equations. Wien: Springer. CISM Courses Lect. 371, 1-78 (1996; Zbl 0889.34018)].