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OPTIMAL CONSTANTS FOR TWO POINT BOUNDARY VALUE PROBLEMS

By K. Q. Lan and G. C. Yang

Abstract

Dedicated to Professor J. R. L. Webb on the occasion of his sixtieth birthday Abstract. The upper and lower bounds of the smallest positive characteristic value µ1 of a linear differential equation of the form u ′ ′ (t) + µg(t)u(t) = 0 a.e. on [0, 1], subject to the general separated boundary conditions (BCs) are estimated. It is shown that m < µ1 < M(a, b), where m and M(a, b) are computable definite integrals related to the kernels arising from the above boundary value problems. The mimimum values for M(a, b) are discussed when g ≡ 1 and g(s) = 1/sα (α> 0) for some of these BCs. All of these values obtained here are useful in studying the existence of nonzero positive solutions for the nonlinear differential equations of the form subject to the above BCs. u ′ ′ (t) + g(t)f(t, u(t)) = 0 a.e. on [0, 1], 1. Introduction. W

Year: 2013
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