Displaying similar documents to “A symmetric solution of a multipoint boundary value problem at resonance.”

Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)

Yuji Liu, Weigao Ge (2005)

Archivum Mathematicum

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In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation x ( n ) ( t ) = f ( t , x ( t ) , x ' ( t ) , , x ( n - 1 ) ( t ) ) + e ( t ) , 0 < t < 1 , ( * ) and the following multi-point boundary value conditions 1 * - 1 x ( i ) ( 0 ) = 0 f o r i = 0 , 1 , , n - 3 , x ( n - 1 ) ( 0 ) = α x ( n - 1 ) ( ξ ) , x ( n - 2 ) ( 1 ) = i = 1 m β i x ( n - 2 ) ( η i ) . * * Sufficient conditions for the existence of at least one solution of the BVP ( * ) and ( * * ) at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002),...

Solvability of a generalized third-order left focal problem at resonance in Banach spaces

Youwei Zhang (2013)

Mathematica Bohemica

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This paper deals with the generalized nonlinear third-order left focal problem at resonance ( p ( t ) u ' ' ( t ) ) ' - q ( t ) u ( t ) = f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , t ] t 0 , T [ , m ( u ( t 0 ) , u ' ' ( t 0 ) ) = 0 , n ( u ( T ) , u ' ( T ) ) = 0 , l ( u ( ξ ) , u ' ( ξ ) , u ' ' ( ξ ) ) = 0 , where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions...