Landesman–Lazer conditions for difference equations involving sublinear perturbations. Issue 11 (1st November 2016)

• Record Type: Journal Article
• Authors: Volek, Jonáš
• Is Part Of: Journal of difference equations and applications. Volume 22:Issue 11(2016)
• Page Start: 1698
• Page End: 1719
• Publication Date: 2016-11-01
• Subjects: Difference equations -- Neumann problem -- periodic problem -- Landesman–Lazer condition -- existence -- uniqueness; 39A05 -- 39A14 -- 39A23 -- 39A70; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2016.1234617
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.