We give a new unified method of establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of \emph{arbitrary} order with any allowed number of \emph{nonlocal} boundary conditions (BCs). In particular, we are able to determine the Green's function for these problems with very little explicit calculation, which shows that studying a more general version of a problem with appropriate notation can lead to a simplification in approach. We obtain existence and nonexistence results, some of which are sharp, and give new results for both nonlocal and local BCs. We illustrate the theory with a detailed account of a fourth order problem that models an elastic beam and also determine optimal values of constants that appear in the theory.

Nonlocal boundary value problems of arbitrary order

INFANTE, GENNARO
2009-01-01

Abstract

We give a new unified method of establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of \emph{arbitrary} order with any allowed number of \emph{nonlocal} boundary conditions (BCs). In particular, we are able to determine the Green's function for these problems with very little explicit calculation, which shows that studying a more general version of a problem with appropriate notation can lead to a simplification in approach. We obtain existence and nonexistence results, some of which are sharp, and give new results for both nonlocal and local BCs. We illustrate the theory with a detailed account of a fourth order problem that models an elastic beam and also determine optimal values of constants that appear in the theory.
2009
Boundary value problem; Fixed point index; Positive solution
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/130011
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