Positive solutions of three-point boundary-value problems for p-Laplacian singular differential equations
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In this paper we prove the existence of positive solutions for the three-point singular boundary-value problem -[ϕp (u')]' = q(t) ƒ(t, u(t)), 0 < t < 1 subject to u(0) - g(u'(0)) = 0, u(1) - βu(η) = 0 or to u(0) - αu(η) = 0, u(1) + g(u'(1)) = 0, where ϕp is the p-Laplacian operator, 0 < η < 1; 0 < α, β < 1 are fixed points and g is a monotone continuous function defined on the real line ℝ with g(0) = 0 and ug(u) ≥ 0. Our approach is a combination of Nonlinear Alternative of Leray-Schauder with the properties of the associated vector field at the (u, u') plane. More precisely, we show that the solutions of the above boundary-value problem remains away from the origin for the case where the nonlinearity is sublinear and so we avoid its singularity at u = 0.
CitationGalanis, G. N., & Palamides, A. P. (2005). Positive solutions of three-point boundary-value problems for p-Laplacian singular differential equations. Electronic Journal of Differential Equations, 2005(106), pp. 1-18.
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