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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.