Abstract
This paper is concerned with new fixed point theorems of operator type that are proved by fixed point index theory and are generalizations of the Leggett–Williams fixed point theorems. With additional structures on the inward and outward boundaries of a conical region, the theorems can be used to prove the existence of solutions of boundary value problems for nonlinear differential equations with dependence on higher-order derivatives, and they can provide nonlocal upper and lower bounds on the solutions. An application is given to demonstrate these advantages.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Agarwal and D. O’Regan, Fixed points of cone compression and expansion multimaps defined on Fréchet spaces: The projective limit approach. J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID 92375, 1–13.
Amann H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)
D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Fixed point theorem utilizing operators and functionals. Electron. J. Qual. Theory Differ. Equ. 2012 (2012), No. 12, 1–16.
D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Operator type expansioncompression fixed point theorem. Electron. J. Differential Equations 2011 (2011), No. 42, 1–11.
R. I. Avery: A generalization of the Leggett-Williams fixed point theorem. Math. Sci. Res. J. 3, 9–14 (1999)
R. I. Avery, D. R. Anderson and J. Henderson, Some fixed point theorems of Leggett-Williams type. Rocky Mountain J. Math. 41 (2011), 371–386.
R. I. Avery, J. Henderson and X. Liu, Omitted ray fixed point theorem. J. Fixed Point Theory Appl., to appear.
R. Avery, J. Henderson and D. O’Regan, Functional compression-expansion fixed point theorem. Electron. J. Differential Equations 2008 (2008), No. 22, 1–12.
R. Avery, J. Henderson and D. O’Regan, Three functionals fixed point theorem. Arab. J. Sci. Eng. ASJE. Math. 34 (2009), 25–37.
R. Avery and A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42 (2001), 313–322.
Z. Bai and W. Ge, Existence of three positive solutions for some second-order boundary value problems. Comput. Math. Appl. 48 (2004), 699–707.
S. Budişan, Generalizations of Krasnosel’skii’s fixed point theorem in cones. Stud. Univ. Babeş-Bolyai Math. 56 (2011), 165–171.
K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, New York, 1985.
A. Granas and J. Dugundji, Fixed Point Theory. Springer Monogr. Math., Springer-Verlag, New York, 2003.
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press, Boston, MA, 1988.
M. A. Krasnosel’skii, Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964.
M. K. Kwong, On Krasnosel’skii’s cone fixed point theorem. Fixed Point Theory Appl. 2008 (2008), Art. ID 164537, 1–18.
R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28 (1979), 673–688.
K. G. Mavridis, Two modifications of the Leggett-Williams fixed point theorem and their applications. Electron. J. Differential Equations 2010 (2010), No. 53, 1–11.
O’Regan D., Precup R.: Compression expansion fixed point theorem in two norms and applications. J. Math. Anal. Appl. 309, 383–391 (2005)
F. Wang and F. Zhang, An extension of fixed point theorems concerning cone expansion and compression and its application. Commun. Korean Math. Soc. 24 (2009), 281–290.
E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Professor Andrzej Granas
Rights and permissions
About this article
Cite this article
Avery, R.I., Graef, J.R. & Liu, X. Compression fixed point theorems of operator type. J. Fixed Point Theory Appl. 17, 83–97 (2015). https://doi.org/10.1007/s11784-015-0228-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0228-1