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Existence of positive solutions for a class of arbitrary order boundary value problems involving nonlinear functionals

Kyriakos G. Mavridis (2014)

Annales Polonici Mathematici

We give conditions which guarantee the existence of positive solutions for a variety of arbitrary order boundary value problems for which all boundary conditions involve functionals, using the well-known Krasnosel'skiĭ fixed point theorem. The conditions presented here deal with a variety of problems, which correspond to various functionals, in a uniform way. The applicability of the results obtained is demonstrated by a numerical application.

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

We study the existence of positive solutions of the nonlinear fourth order problem u ( 4 ) ( x ) = λ a ( x ) f ( u ( x ) ) , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

Existence of positive solutions for second order m-point boundary value problems

Ruyun Ma (2002)

Annales Polonici Mathematici

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨ α u ( 0 ) - β u ' ( 0 ) = i = 1 m - 2 a i u ( ξ i ) γ u ( 1 ) + δ u ' ( 1 ) = i = 1 m - 2 b i u ( ξ i ) , where ξ i ( 0 , 1 ) , a i , b i ( 0 , ) (for i ∈ 1,…,m-2) are given constants satisfying ϱ - i = 1 m - 2 a i ϕ ( ξ i ) > 0 , ϱ - i = 1 m - 2 b i ψ ( ξ i ) > 0 and Δ : = - i = 1 m - 2 a i ψ ( ξ i ) ϱ - i = 1 m - 2 a i ϕ ( ξ i ) ϱ - i = 1 m - 2 b i ψ ( ξ i ) - i = 1 m - 2 b i ϕ ( ξ i ) < 0 . We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point...

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 , + ) × , ( 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.

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