Positive solutions to boundary value problems with nonlinear boundary conditions

• Published in: Nonlinear analysis Vol. 75; no. 1; pp. 417 - 432
• Main Author: Goodrich, Christopher S.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 2012; Elsevier
• Subjects: Boundary conditions; Boundary value problems; Cone; Exact sciences and technology; Mathematical analysis; Mathematics; Nonlinearity; Nonlocal boundary condition; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Positive solution; Sciences and techniques of general use; Second-order boundary value problem; Theorems; Time scales; secondary; Cone; Time scales; Positive solution; Second-order boundary value problem; Nonlocal boundary condition; primary; Second order equation; 34B15; primary 34B09; Nonlinear analysis; 34B10; 39A05; Boundary condition; 34B18; 47H10; Boundary value problem; 34N05; Mathematical model; secondary 26E70
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.08.044

In this paper, we consider the boundary value problem y Δ Δ ( t ) = − λ f ( t , y σ ( t ) ) subject to the boundary conditions y ( a ) = ϕ ( y ) and y ( σ 2 ( b ) ) = 0 . In this setting, ϕ : C rd ( [ a , σ 2 ( b ) ] T , R ) → R is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on ϕ and the nonlinearity f , we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that ϕ may be strictly nonpositive for some y > 0 . Our results are achieved by appealing to the Krasnosel’skiĭ fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford.