## Topological Methods in Nonlinear Analysis

### Global structure of positive solutions for superlinear second order $m$-point boundary value problems

#### Abstract

In this paper, we consider the nonlinear eigenvalue problems \begin{gather*} u''+\lambda h(t)f(u)=0, \quad 0< t< 1, \\ u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_i u(\eta_i), \end{gather*} where $m\geq 3$, $\eta_i\in (0,1)$ and $\alpha_i> 0$ for $i=1,\ldots,m-2$, with $\sum_{i=1}^{m-2}\alpha_i\eta_i< 1$; $h\in C([0,1], [0,\infty))$ and $h(t)\ge 0$ for $t\in [0,1]$ and $h(t_0)> 0$ for $t_0\in [0,1]$; $f\in C([0,\infty),[0,\infty))$ and $f(s)> 0$ for $s> 0$, and $f_0=\infty$, where $f_0=\lim_{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using the nonlinear Krein-Rutman Theorem.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 34, Number 2 (2009), 279-290.

Dates
First available in Project Euclid: 27 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461786799

Mathematical Reviews number (MathSciNet)
MR2604448

Zentralblatt MATH identifier
1200.34017

#### Citation

Ma, Ruyun; An, Yulian. Global structure of positive solutions for superlinear second order $m$-point boundary value problems. Topol. Methods Nonlinear Anal. 34 (2009), no. 2, 279--290. https://projecteuclid.org/euclid.tmna/1461786799

#### References

• E. Dancer, Global solutions branches for positive maps , Arch. Rational Mech. Anal., 55 (1974), 207–213 \ref\key 2
• K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin (1985) \ref\key 3
• Z. Deng, Introduction to BVPs and Sturmian Comparison Theory for Ordinary Differential Equations, Central China Normal University Press (1987), (in Chinese) \ref\key 4
• M. K. Kwong and J. S. W. Wong, The shooting method and nonhomogeneous multi-point BVPs of second-order ODE , Boundary Value Problems (2007), pp. 16 \ref\key 5
• R. Ma, Existence of positive solutions for superlinear semipositone $m$-point boundary value problems , Proc. Edinburgh Math. Soc., 46(2003), 279–292 \ref\key 6
• R. Ma and D. O'Regan, Nodal solutions for second-order $m$-point boundary value problems with nonlinearities across several eigenvalues , Nonlinear Anal., 64 (2006), 1562–1577 \ref\key 7
• B. P. Rynne, Global bifurcation for $2m$th-order boundary value problems and infinitely many solutions of superlinear problems , J. Differential Equations, 188 (2003), 461–472 \ref\key 8
• J. R. L. Webb, Positive solutions of some three point boundary value problems via fixed point index theory , Nonlinear Anal., 47 (2001), 4319–4332 \ref\key 9
• G. T. Whyburn, Topological Analysis, Princeton Math. Ser., 23 , Princeton University Press, Princeton, N. J. (1958) \ref\key 10
• E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer–Verlag, New York (1986)