Topological Methods in Nonlinear Analysis

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

J. R. L. Webb and K. Q. Lan

Full-text: Open access

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

Permanent link to this document
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


Export citation

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