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Abstrakty
An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.
Czasopismo
Rocznik
Tom
Strony
447--456
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
autor
autor
autor
autor
- Baylor University Department of Mathematics Waco, Texas 76798-7328 USA, Johnny_Henderson@baylor.edu
Bibliografia
- [1] D.R. Anderson, R.I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equ. Appl. 8 (2002), 1073–1083.
- [2] D.R. Anderson, R.I. Avery, J. Henderson, X.Y. Liu, J.W. Lyons, Existence of a positive solution for a right focal discrete boundary value problem, J. Difference Equ. Appl., in press.
- [3] R.I. Avery, A generalization of the Leggett-Williams fixed point theorem, MSR Hotline 2 (1998), 9–14.
- [4] R.I. Avery, D.R. Anderson, J. Henderson, Functional expansion-compression fixed point theorem of Leggett-Williams type, submitted.
- [5] R.I. Avery, C.J. Chyan, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 42 (2001), 695–704.
- [6] R.I. Avery, J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8 (2001), 27–36.
- [7] R.I. Avery, J. Henderson, D.R. Anderson, A topological proof and extension of the Leggett-Williams fixed point theorem, Comm. Appl. Nonlinear Anal. 16 (2009) 4, 39–44.
- [8] R.I. Avery, J. Henderson, D. O’Regan, A dual of the compression- expansion fixed point theorems, Fixed Point Theory Appl. 2007, Art. ID 90715, 11 pp.
- [9] R.I. Avery, A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (2001), 313–322.
- [10] X. Cai, J. Yu, Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2006), 649–661.
- [11] P.W. Eloe, J. Henderson, E. Kaufmann, Multiple positive solutions for difference equations, J. Difference Equ. Appl. 3 (1998), 219–229.
- [12] W. Ge, Boundary Value Problems of Nonlinear Differential Equations, Science Publications, Beijing, 2007.
- [13] J.R. Graef, J. Henderson, Double solutions of boundary value problems for 2mth-order differential equations and difference equations, Comput. Math. Anal. 45 (2003), 873–885.
- [14] D. Guo, Some fixed point theorems on cone maps, Kexeu Tongbao 29 (1984), 575–578.
- [15] Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), 193–201.
- [16] I.Y. Karaca, Discrete third-order three-point boundary value problems, J. Comput. Appl. Math. 205 (2007), 458–468.
- [17] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964.
- [18] M.K. Kwong, The topological nature of Krasnosel’skii’s cone fixed point theorem, Nonlinear Anal. 69 (2008), 891–897.
- [19] R.W. Leggett, L.R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688.
- [20] Y. Li, L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006), 1019–1023.
- [21] Y. Liu, W. Ge, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl. 278 (2003), 551–561.
- [22] F. Merdivenci, Two positive solutions for a boundary value problem for difference equations, J. Difference Equ. Appl. 1 (1995), 262–270.
- [23] H. Pang, H. Feng, W. Ge, Multiple positive solutions of quasi-linear boundary value problems for finite difference equations, Appl. Math. Comput. 197 (2008), 451–456.
- [24] J. Sun, G. Zhang, A generalization of the cone expansion and compression fixed point theorem and applications, Nonlinear Anal. 67 (2007), 579–586.
- [25] D. Wang, W. Guan, Three positive solutions of boundary value problems for p-Laplacian difference equations, Comput. Math. Anal. 55 (2008), 1943–1949.
- [26] P.J.Y. Wong, R.P. Agarwal, Eigenvalue intervals and double positive solutions for certain discrete boundary value problems, Commun. Appl. Anal. 3 (1999), 189–217.
- [27] P.J.Y. Wong, R.P. Agarwal, Existence of multiple solutions of discrete two-point right focal boundary value problems, J. Difference Equ. Appl. 5 (1999), 517–540.
- [28] C. Yang, P. Weng, Green’s functions and positive solutions for boundary value problems of third-order difference equations, Comput. Math. Appl. 54 (2007), 567–578.
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Bibliografia
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