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

Yuji LiuWeigao Ge — 2005

Archivum Mathematicum

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), 475–494...

Existence of positive solutions for singular four-point boundary value problem with a p -Laplacian

Chunmei MiaoJunfang ZhaoWeigao Ge — 2009

Czechoslovak Mathematical Journal

In this paper we deal with the four-point singular boundary value problem ( φ p ( u ' ( t ) ) ) ' + q ( t ) f ( t , u ( t ) , u ' ( t ) ) = 0 , t ( 0 , 1 ) , u ' ( 0 ) - α u ( ξ ) = 0 , u ' ( 1 ) + β u ( η ) = 0 , where φ p ( s ) = | s | p - 2 s , p > 1 , 0 < ξ < η < 1 , α , β > 0 , q C [ 0 , 1 ] , q ( t ) > 0 , t ( 0 , 1 ) , and f C ( [ 0 , 1 ] × ( 0 , + ) × , ( 0 , + ) ) may be singular at u = 0 . By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p -Laplacian operator

De-xiang MaWeigao GeZhan-Ji Gui — 2007

Czechoslovak Mathematical Journal

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ( φ p ( u ' ) ) ' + q ( t ) f ( u ) = 0 , 0 < t < 1 , where φ p ( s ) : = | s | p - 2 s , p > 1 , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q ( t ) may be singular at t = 0 , 1 .

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