On the Existence of Positive Solutions of Ordinary Differential Equations

“…Clearly, y m is the unique solution of the initial value problem (3.15), y«»(0) = 0, 0 < i < n -2 , , " ~ ~ (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) y,?-|> (i) = «. …”

Section: And (36) For the Definitions Of U M (T) And V M (T))mentioning

confidence: 99%

Eigenvalue characterization for (n · p) boundary-value problems

Wong

Agarwal

1998

J. Aust. Math. Soc. Series B, Appl. Math

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“…Existence theorems of positive solutions have been obtained in [18] for the superlinear case and in [7,24,26,28] for "crossing the first eigenvalue" type conditions. In this direction, an existence result has been produced in [20] for (1.1) with f (x, s) as in (1.2).…”

Section: Introductionmentioning

confidence: 99%

Multiple positive solutions for a superlinear problem: A topological approach

Feltrin

Zanolin

2015

Journal of Differential Equations

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We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u + f (x, u) = 0. We allow x → f (x, s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f (x, s)/s is below λ1 as s → 0 + and above λ1 as s → +∞. In particular, we can deal with the situation in which f (x, s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u + a(x)g(u) = 0, where we prove the existence of 2 n − 1 positive solutions when a(x) has n positive humps and a − (x) is sufficiently large.

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“…During the last decades existence of eigenvalues yielding positive solutions for nonlinear second order multi-point boundary value problems is in the focus of interest of many researchers. See, for example [1,5,6,7,9,12,13,14,27]. In particular, existence of positive solutions for systems of second order multi-point boundary value problems was studied in [10,11,18,26,28].…”

Section: Discussionmentioning

confidence: 99%

“…In addition, we put the following assumptions on the functions f and g: The importance of positive solutions for boundary value problems, both theoretically as well as from the perspective of their applications in physical and engineering sciences, has been well documented in the literature; see, for example, [1,5,6,7,9,12,13,14,16,20,27]. While many of these referenced papers have been devoted to scalar problems, there is much emerging interest in boundary value problems for systems of differential equations [10,11,18,22,26,28], and a good deal of research has also involved positive solutions for multipoint nonlinear eigenvalue problems in both scalar and systems contexts [2,8,18,23].…”

Section: Introductionmentioning

confidence: 99%