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Positivity

, Volume 22, Issue 5, pp 1387–1402 | Cite as

Positive solutions of nonlinear multi-point boundary value problems

  • Abdulkadir Dogan
Article
  • 54 Downloads

Abstract

This paper deals with the existence of positive solutions of nonlinear differential equation
$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$
subject to the boundary conditions
$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$
where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality
$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

Keywords

Differential equation Nonlinear boundary value problems Positive solutions Fixed point theorem 

Mathematics Subject Classification

34B15 34B18 

Notes

Acknowledgements

The author would like to thank the anonymous referees and editor for their helpful comments and suggestions.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Computer SciencesAbdullah Gul UniversityKayseriTurkey

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