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

Chunmei Miao, Junfang Zhao, Weigao 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 , + ∞ ) × ℝ , ( 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 of solution to nonlinear boundary value problem for ordinary differential equation of the second order in Hilbert space

Eva Rovderová (1992)

Mathematica Bohemica

In this paper we deal with the boundary value problem in the Hilbert space. Existence of a solutions is proved by using the method of lower and upper solutions. It is not necessary to suppose that the homogeneous problem has only the trivial solution. We use some results from functional analysis, especially the fixed-point theorem in the Banach space with a cone (Theorem 4.1, [5]).

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