Abstract
A sharp theorem by Kiguradze and Partsvania ensures the existence of extremal solutions between given lower and upper solutions for singular Dirichlet problems. This paper has a twofold purpose: first, we present a new sufficient condition for one of Kiguradze and Partsvania’s assumptions, and we illustrate its applicability in the study of a new family of examples; second, we combine Kiguradze and Partsvania’s theorem with Heikkila’s iterative technique to obtain a new result on the existence of extremal solutions for a more general class of discontinuous and singular functional boundary value problems. In particular, our framework includes classical equations with delay (or advance), singularities with respect to the independent variable, and implicit functional boundary conditions.
Highlights
1 Introduction and first results We are going to review the results in a paper by Kiguradze and Partsvania [ ] and we are going to use them in the proof of a new more general existence result of extremal solutions for functional and singular second-order problems
Proposition . provides us with a new sufficient condition for a technical assumption in [ ], and we will use it in the analysis of a new family of examples
Let φi : [ , ] → R, i = , , be nonincreasing functions (not necessarily continuous), and consider the following functional problem, which includes both a past and a future dependence:
Summary
Introduction and first resultsWe are going to review the results in a paper by Kiguradze and Partsvania [ ] and we are going to use them in the proof of a new more general existence result of extremal solutions for functional and singular second-order problems.Our revision of the results in [ ] is not merely a reproduction, as it includes some contributions of our own. [ , Theorem ] Let f : (a, b) × R → R be a function satisfying (i), (ii), and (iii) and assume that σ and σ are lower and upper solutions of
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
