Positive solutions of a boundary value problem for a nonlinear fractional differential equation

Electronic Journal of Qualitative Theory of Differential Equations

Volumn , Issue , 2008, Pages 1-11

  Kaufmann, Eric R ^a   Mboumi, Ebene ^a  

^a University of Arkansas at Little Rock ^*  (United States)

Author keywords

Fractional derivative; Nonlinear dynamic equation; Positive solution

Indexed keywords

EID: 38149118737     PISSN: None     EISSN: 14173875     Source Type: Journal    
DOI: 10.14232/ejqtde.2008.1.3     Document Type: Article

References (39)

1. S. Ahmad and A. Lazer, Positive operators and Sturmian theory on nonselfadjoint second order sysyems, Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978, pp. 25-42.
2. R. P. Agarwal, D. O'Regan and P. J. Y. Wong, "Positive Solutions of Differential, Difference, and Integral Equations," Kluwer Academic Publishers, Boston, 1999.
3. T. S. Aleroev, The Sturm-Liouville problem for a second-order differential equation with fractional derivatives in the lower terms, Differentsial'nye Uravneniya 18 (1982), No. 2, 341-342.
4. T. S. Aleroev, On a boundary value problem for a fractional-order differential operator, Differentsial'nye Uravneniya 34 (1998), No. 1, 126.
5. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18 (1976), 620-709.
6. R. I. Avery, J. M. Davis, and J. Henderson, Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem, Electron. J. Diff. Eqns. 2000 (2000), No. 40, 1-15.
7. Z. Bai and H. Lü, Positive solutions for a boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505.
8. A. Belarbi, M. Benchohra, and A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006), No. 12, 1459-1470.
9. V. Daftardar-Gejji, Positive solutions of a system of non-autonomous nonlinear fractional differential equations, J. Math. Anal. Appl. 302 (2004), 56-64.
10. P. W. Eloe and J. Henderson, Comparison of eigenvalues for a class of two-point boundary value problems, Appl. Anal. 34 (1989), 25-34.
11. P. W. Eloe, J. Henderson, and E. R. Kaufmann Multiple Positive Solutions for Difference Equations, J. Differ. Equations Appl. 3 (1998), 219-229.
12. R. D. Gentry and C. C. Travis, Comparison of eigenvalues associated with linear differential equations of arbitrary order, Trans. Amer. Math. Soc. 223 (1976), 167-179.
13. D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, San Diego, 1988.
14. D. Hankerson and A. Peterson, Comparison of eigenvalues for n-th order difference equations, Diff. and Int. Eqns. 3 (1990), 363-380.
15. J. Henderson and E. R. Kaufmann Multiple Positive Solutions for Focal Boundary Value Problems, Commun. Appl. Anal. 1 (1997), 53-60.
16. H. Jafari and V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput. 180 (2006), No. 2, 700-706.
17. H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math. 196 (2006), No. 2, 644-651.
18. M. Jia, X. Liu, and E. R. Kaufmann, Boundary Value Problems for Second-Order Functional Differential Equations on Infinite Intervals, Nonlinear Studies, in press.
19. B. Karna, E. R. Kaufmann, and J. Nobles, Comparison of Eigenvalues for a Fourth Order Four-Point Boundary Value Problem, Electron. J. Qual. Theory Differ. Equ. 2005 (2005), No. 15, 1-9.
20. E. R. Kaufmann, Positive Solutions of a Three-Point Boundary-Value Problem on a Time Scale, Electron. J. Diff. Eqns. 2003 (2003), No. 82, 1-11.
21. E. R. Kaufmann and N. Kosmatov, A Second-Order Singular Boundary Value Problem, Comput. Math. Appl. 47 (2004), 1317-1326.
22. E. R. Kaufmann and N. Kosmatov, Singular Conjugate Boundary Value Problems on a Time Scale, J. Difference Equ. Appl. 10 (2004), No. 2, 119-127.
23. M. S. Keener and C. C. Travis, Positive cones and focal points for a class of nth order differential equations, Trans. Amer. Math. Soc. 237 (1978), 331-351.
24. A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ: Uravn 41 (2005), No. 1, 82-86;
25. translation in Differ. Equ 41 (2005), No. 1, 84-89.
26. A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results, and problems. II, Appl. Anal. 81 (2002), No. 2, 435-493.
27. M. A. Krasnosel'skiĩ, "Topological Methods in the Theory of Nonlinear Integral Equations,"(English) Translated by A.H. Armstrong; A Pergamon Press Book, MacMillan, New York, 1964.
28. 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.
29. K. S. Miller and B. Ross, "An Introduction to the Fractional Calculus and Fractional Differential Equations," Wiley, New York, 1993.
30. K. Nishimoto, "Fractional Calculus: Integrations and Differentiations of Arbitrary Order,"University of New Haven Press, New Haven, 1989.
31. K. B. Oldham and J. Spanier, "The Fractional Calculus," Academic Press, New York, 1974.
32. I. Podlubny, "Fractional Differential Equations," Mathematics in Sciences and Engineering 198, Academic Press, San Diego 1999.
33. G. Samko, A. A. Kilbas, O. I. Marichev, "Fractional Integrals and Derivatives: Theory and Applications," Gordon and Breach, Yverdon, 1993.
34. E. Tomastik, Comparison theorems for conjugate points of nth order nonselfadjoint differential equations, Proc. Amer. Math. Soc. 96 (1986), 437-442.
35. K. Wiener, On solutions of boundary value problems for differential equations of noninteger order, Math. Nachr. 88 (1979), 181-190.
36. X. Xu, Positive Solutions for Singular Semi-positone Boundary Value Problems, J. Math. Anal. Appl. 237 (2007), 480-491.
37. B. Yang, Positive Solutions for a fourth order boundary value problem, Electron. J. Qualitative Theory Diff. Equ. 2005 (2005), No. 3, 1-17.
38. S. Zhang, The Existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2000), 804-812.
39. S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differentional equations, Electron. J. Diff. Eqns. 2006 (2006), No. 36, 1-12.