Nonlinear discrete Sturm–Liouville problems

• Published in: Journal of mathematical analysis and applications Vol. 308; no. 1; pp. 380 - 391
• Main Author: Rodriguez, Jesús
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.08.2005; Elsevier
• Subjects: Boundary value problems; Brower Fixed Point Theorem; Exact sciences and technology; Global analysis, analysis on manifolds; Implicit Function Theorem; Mathematical analysis; Mathematics; Ordinary differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Implicit Function Theorem; Boundary value problems; Brower Fixed Point Theorem; Sturm Liouville problem; Sufficient condition; Boundary value problem; Fixed point theorem; Mathematical analysis; Nonlinearity; Implicit function theorem; Nonlinear problems; Solvability
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.01.032

In this paper we study nonlinear boundary value problems of the form Δ [ p ( t − 1 ) Δ y ( t − 1 ) ] + q ( t ) y ( t ) + λ y ( t ) = f ( y ( t ) ) ; t = a + 1 , … , b + 1 , subject to a 11 y ( a ) + a 12 Δ y ( a ) = 0 and a 21 y ( b + 1 ) + a 22 Δ y ( b + 1 ) = 0 . The parameter λ is an eigenvalue of the associated linear problem; that is, there is a nontrivial function u that satisfies the boundary conditions and also Δ [ p ( t − 1 ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) + λ u ( t ) = 0 for t in { a + 1 , a + 2 , … , b + 1 } . We establish sufficient conditions for the solvability of such problems. In addition, in those cases where the nonlinearity is “small,” we provide a qualitative analysis of the relation between solutions of the nonlinear problem and eigenfunctions of the linear one.