[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5666/KMJ.2019.59.1.65

Positive Solutions of Nonlinear Neumann Boundary Value Problems with Sign-Changing Green's Function

Elsanosi, Mohammed Elnagi M. (Department of Mathematics, Faculty of Educations, University of Khartoum)

Publication Information

Kyungpook Mathematical Journal / v.59, no.1, 2019 , pp. 65-71 More about this Journal

Abstract

This paper is concerned with the existence of positive solutions of the nonlinear Neumann boundary value problems $\{u^{{\prime}{\prime}}+a(t)u={\lambda}b(t)f(u),\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$ , where $$a,b{\in}C[0,1$ $]$ $$$ with $$a(t)$ > $0,\;b(t){\geq}0$$ and the Green's function of the linear problem $\{u^{{\prime}{\prime}}+a(t)u=0,\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$ may change its sign on $$[0,1$ $]$ ${\times}[0,1$ $]$ $$$ . Our analysis relies on the Leray-Schauder fixed point theorem.

Keywords

Neumann boundary value problems; sign-changing Green's function; Leray-Schauder fixed point theorem; positive solutions;

Citations & Related Records

Reference

1	A. Boscaggin, A note on a superlinear indefinite Neumann problem with multiple positive solutions, J. Math. Anal. Appl., 377(2011), 259-268. DOI
2	J. Chu, Y. Sun and H. Chen, Positive solutions of Neumann problems with singularities, J. Math. Anal. Appl., 337(2008), 1267-1272. DOI
3	K. Deimling, Nonlinear functional analysis, New York, Springer-Verlag, 1985.
4	J. R. Graef, L. Kong and H. Wang, A periodic boundary value problem with vanishing Green's function, Appl. Math. Lett., 21(2)(2008), 176-180. DOI
5	D. D. Hai, Positive solutions to a class of elliptic boundary value problems, J. Math. Anal. Appl., 227(1998), 195-199. DOI
6	D. Jiang, Y. Yang, J. Chu and D. O'Regan, The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order, Nonlinear. Anal., 67(2007), 2815-2828. DOI
7	M. Krasnosel'skii, Positive solutions of operator equations, Noordhoff, Groningen, 1964.
8	Z. Li, Existence of positive solutions of superlinear second-order Neumann boundary value problem, Nonlinear Anal., 72(2010), 3216-3221. DOI
9	R. Ma, Nonlinear periodic boundary value problems with sign-changing Green's function, Nonlinear Anal., 74(2011), 1714-1720. DOI
10	Y. Sun, Y. J. Cho and D. O'Regan, Positive solutions for singular second order Neumann boundary value problems via a cone fixed point theorem, Appl. Math. Comput., 210(2009), 80-86. DOI
11	J. Sun and W. Li, Multiple positive solutions to second-order Neumann boundary value problems, Appl. Math. Comput., 146(2003), 187-194. DOI
12	J. Sun, W. Li and S. Cheng, Three positive solutions for second-order Neumann boundary value problems, Appl. Math. Lett., 17(2004), 1079-1084. DOI