Positive solutions to boundary value problems with nonlinear boundary conditions

Abstract 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.

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