Research Article | Open | Published:
A Dual of the Compression-Expansion Fixed Point Theorems
Fixed Point Theory and Applicationsvolume 2007, Article number: 090715 (2007)
This paper presents a dual of the fixed point theorems of compression and expansion of functional type as well as the original Leggett-Williams fixed point theorem. The multi-valued situation is also discussed.
Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana University Mathematics Journal 1979,28(4):673–688. 10.1512/iumj.1979.28.28046
Guo DJ: A new fixed-point theorem. Acta Mathematica Sinica 1981,24(3):444–450.
Anderson DR, Avery RI: Fixed point theorem of cone expansion and compression of functional type. Journal of Difference Equations and Applications 2002,8(11):1073–1083. 10.1080/10236190290015344
Zhang G, Sun J: A generalization of the cone expansion and compression fixed point theorem and applications. Nonlinear Analysis: Theory, Methods & Applications 2007,67(2):579–586. 10.1016/j.na.2006.06.003
Guo DJ: Some fixed point theorems on cone maps. Kexue Tongbao 1984,29(5):575–578.
Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands; 1964:381.
Agarwal RP, Avery RI, Henderson J, O'Regan D: The five functionals fixed point theorem generalized to multivalued maps. Journal of Nonlinear and Convex Analysis 2003,4(3):455–462.
Avery RI: A generalization of the Leggett-Williams fixed point theorem. Mathematical Sciences Research Hot-Line 1999,3(7):9–14.
Avery RI, Henderson J: An extension of the five functionals fixed point theorem. International Journal of Differential Equations and Applications 2000,1(3):275–290.
Avery RI, Henderson J: Two positive fixed points of nonlinear operators on ordered Banach spaces. Communications on Applied Nonlinear Analysis 2001,8(1):27–36.
Agarwal RP, O'Regan D: A generalization of the Petryshyn-Leggett-Williams fixed point theorem with applications to integral inclusions. Applied Mathematics and Computation 2001,123(2):263–274. 10.1016/S0096-3003(00)00077-1
O'Regan D: Integral inclusions of upper semi-continuous or lower semi-continuous type. Proceedings of the American Mathematical Society 1996,124(8):2391–2399. 10.1090/S0002-9939-96-03456-9
Smirnov GV: Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics. Volume 41. American Mathematical Society, Providence, RI, USA; 2002:xvi+226.
Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
Avery RI, Peterson AC: Multiple positive solutions of a discrete second order conjugate problem. PanAmerican Mathematical Journal 1998,8(3):1–12.