Two-point boundary value problems for 4th order ordinary differential equations

Abstract

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u ( 4 ) ( t ) = p ( t ) u ( t )+ q ( t ) for t E [ a , b ] , u ( 4 ) ( t ) = p ( t ) u ( t ) + f ( t , u ( t )) for t E [ a , b ] , under the following two -point boundary conditions u ( i ) ( a ) = 0 , u ( i ) ( b ) = 0 ( i = 0 , 1 ) , and u ( i ) ( a ) = 0 ( i = 0 , 1 , 2 ) , u ( b ) = 0 , where p E L ([ a , b ] ; R ) is a nonconstant sign function and f E K ([ a , b ] x R; R ) .The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u ( 4 ) ( t ) = p ( t ) u ( t )+ q ( t ) for t E [ a , b ] , u ( 4 ) ( t ) = p ( t ) u ( t ) + f ( t , u ( t )) for t E [ a , b ] , under the following two -point boundary conditions u ( i ) ( a ) = 0 , u ( i ) ( b ) = 0 ( i = 0 , 1 ) , and u ( i ) ( a ) = 0 ( i = 0 , 1 , 2 ) , u ( b ) = 0 , where p E L ([ a , b ] ; R ) is a nonconstant sign function and f E K ([ a , b ] x R; R ) .