## Topological Methods in Nonlinear Analysis

### Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type

#### Abstract

New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))\,ds \equiv Au(t)$$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$.

These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)\,ds.$$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],$$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a unified approach applicable to many BVPs.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 27, Number 1 (2006), 91-115.

Dates
First available in Project Euclid: 12 May 2016

https://projecteuclid.org/euclid.tmna/1463081848

Mathematical Reviews number (MathSciNet)
MR2236412

Zentralblatt MATH identifier
1146.34020

#### Citation

Webb, J. R. L.; Lan, K. Q. Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 91--115. https://projecteuclid.org/euclid.tmna/1463081848

#### References

• H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces , SIAM Rev., 18(1976), 620–709 \ref\key 2
• Ch. Bai and J. Fang, Existence of multiple positive solutions for nonlinear $m$-point boundary value problems , J. Math. Anal. Appl., 281(2003), 76–85 \ref\key 3 ––––, Existence of multiple positive solutions for nonlinear $m$-point boundary value problems , Appl. Math. Comp., 140(2003), 297–305 \ref\key 4
• L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems , Math. Comput. Modelling, 32(2000), 529–539 \ref\key 5
• L. H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems , J. Math. Anal. Appl., 184(1994), 640–648 \ref\key 6
• D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press (1988) \ref\key 7
• C. P. Gupta, S. K. Ntouyas and P. CH. Tsamatos, On an $m$-point boundary-value problem for second-order ordinary differential equations , Nonlinear Anal., 23(1994), 1427–1436 \ref\key 8 ––––, Solvability of an $m$-point boundary-value problem for second-order ordinary differential equations , J. Math. Anal. Appl., 189(1995), 575–584 \ref\key 9
• G. Infante and J. R. L. Webb, Nonzero solutions of Hammerstein integral equations with discontinuous kernels , J. Math. Anal. Appl., 272(2002), 30–42 \ref\key 10
• G. L. Karakostas and P. Ch. Tsamatos, Multiple positive solutions for a nonlocal boundary-value problem with response function quiet at zero , Electron. J. Differential Equations (2001, 13 ), 10 pp \ref\key 11 ––––, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems , Electron. J. Differential Equations (2002, 30 ), 17 pp \ref\key 12
• M. A. Krasnosel'skiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Springer–Verlag, Berlin (1984) \ref\key 13
• K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities , Dynam. Systems Differential Equations, 8 (2000), 175–192 \ref\key 14 ––––, Multiple positive solutions of semilinear differential equations with singularities , J. London Math. Soc. (2), 63(2001), 690–704 \ref\key 15
• K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities , J. Differential Equations, 148(1998), 407–421 \ref\key 16
• Y. Li, Abstract existence theorems of positive solutions for nonlinear boundary value problems , Nonlinear Anal., 57(2004), 211–227 \ref\key 17
• F. Li and G. Han, Generalization for Amann's and Leggett–Williams's three-solution theorems and applications , J. Math. Anal. Appl., 298(2004), 638–654 \ref\key 18
• Z. Liu and F. Li, Multiple positive solutions of nonlinear two-point boundary value problems , J. Math. Anal. Appl., 203(1996), 610–625 \ref\key 19
• R. Ma and N. Castaneda, Existence of solutions of nonlinear $m$-point boundary-value problems , J. Math. Anal. Appl., 256(2001), 556–567 \ref\key 21
• R. H. Martin, Nonlinear operators and differential equations in Banach spaces, Wiley, New York (1976) \ref\key 22
• R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein–Rutman theorem , Fixed Point Theory (E. Fadell and G. Fournier, eds.), Springer (1981, Lecture Notes in Math., 886 ), 309–330 \ref\key 23
• 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 24 ––––, Remarks on positive solutions of three point boundary value problems , Dynamical Systems and Differential Equations (Wilmington, NC, 2002) Discrete Contin. Dynam. Systems (2003). Added Volume, 905–915 \ref\key 25
• G. Zhang and J. Sun, Positive solutions of $m$-point boundary value problems , J. Math. Anal. Appl., 291 (2004), 406–418