Sign-changing solutions to second-order integral boundary value problems

• Published in: Nonlinear analysis Vol. 69; no. 4; pp. 1179 - 1187
• Main Authors: Li, Yuhua, Li, Fuyi
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 15.08.2008; Elsevier
• Subjects: Exact sciences and technology; Fixed point index; Global analysis, analysis on manifolds; Integral boundary value problem; Leray–Schauder degree; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Sciences and techniques of general use; Sign-changing solutions; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Fixed point index; Leray–Schauder degree; Integral boundary value problem; Sign-changing solutions; Second order equation; Fix point; Boundary value problem; Nonlinear analysis; Leray-Schauder degree; Mathematical model
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2007.06.024

In this paper, by using the fixed point index theory and Leray–Schauder degree theory, we consider the existence and multiplicity of sign-changing solutions to nonlinear second-order integral boundary value problem − u ″ ( t ) = f ( u ( t ) ) for all t ∈ [ 0 , 1 ] subject to u ( 0 ) = 0 and u ( 1 ) = g ( ∫ 0 1 u ( s ) d s ) , where f , g ∈ C ( R , R ) . We obtain some new existence results concerning sign-changing solutions by computing hardly eigenvalues and the algebraic multiplicities of the associated linear problem. If f and g satisfy certain conditions, then this problem has at least six different nontrivial solutions: two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if f and g are also odd, then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.