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http://dx.doi.org/10.4134/JKMS.2014.51.6.1123

UPPER AND LOWER SOLUTION METHOD FOR FRACTIONAL EVOLUTION EQUATIONS WITH ORDER 1 < α < 2  

Shu, Xiao-Bao (Department of Mathematics Hunan University)
Xu, Fei (Department of Mathematics Wilfrid Laurier University)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.6, 2014 , pp. 1123-1139 More about this Journal
Abstract
In this work, we investigate the existence of the extremal solutions for a class of fractional partial differential equations with order 1 < ${\alpha}$ < 2 by upper and lower solution method. Using the theory of Hausdorff measure of noncompactness, a series of results about the solutions to such differential equations is obtained.
Keywords
fractional partial differential equation; mild solution; upper and lower solution method; Hausdorff measure of noncompactness;
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1 S. Abbas and M. Benchohra, Upper and lower solutions method for darboux problem for fractional order implicit impulsive partial hyperbolic differential equations, Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Mathematica 51 (2012), no. 2, 5-18.
2 R. P. Agarwal, M. Benchohra, and B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys. 44 (2008), 1-21.   DOI   ScienceOn
3 B. Ahmad and S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (2009), no. 3, 251-258.   DOI   ScienceOn
4 B. Ahmad, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 1, 134-141.   DOI   ScienceOn
5 K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron. J. Qual. Theory Differ. Equ. 4 (2010), no. 4, 12 pp.
6 J. Bana's and K. Goebel, Measure of Noncompactness in Banach Spaces, Marcel Dekker Inc., New York, 1980.
7 E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.
8 R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943), no. 4, 643-647.   DOI
9 M. Belmekki and M. Benchohra, Existence result for fractional order semilinear functional differential equations with nondense domain, Nonlinear Anal. 72 (2010), no. 2, 925-932.   DOI   ScienceOn
10 M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. Calc. Appl. Anal. 11 (2008), no. 1, 35-56.
11 M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 2009 (2009), no. 10, 11 pp.
12 P. Benilan, Equations d'evolution dans un espace de Banach quelconque et applications, Th'ese de doctorat d'etat, Orsay, 1972.
13 D. Bothe, Multivalued perturbation of m-accretive differential inclusions, lsrael. J. Math. 108 (1998), 109-138.
14 D. J. Guo, V. Lakshmikantham, and X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht, 1996.
15 K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
16 Z. Denton, P. W. Ng, and A. S. Vatsala, Quasilinearization method via lower and upper solutions for Riemann-Liouville fractional differential equations, Nonlinear Dyn. Syst. Theory 11 (2011), no. 3, 239-251.
17 H. Fan and J. Mu, Initial value problem for fractional evolution equations, Adv. Difference Equ. 2012 (2012), no. 49, 10 pp.; doi:10.1186/1687-1847-2012-49.   DOI   ScienceOn
18 H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal 7 (1983), no. 12, 1351-1371.   DOI   ScienceOn
19 L. Hu, Y. Ren, and R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum 79 (2009), no. 3, 507-514.
20 R. W. Ibrahim and S. Momani, Upper and lower bounds of solutions for fractional integral equations, Surv. Math. Appl. 2 (2007), 145-156.
21 O. K. Jaradat, A. Al-Omari, and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal. 69 (2008), no. 9, 3153-3159.   DOI   ScienceOn
22 V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, 1969.
23 L. Lin, X. Liu, and H. Fang, Method of upper and lower solutions for fractional differential equations, Electron. J. Differential Equations 2012 (2012), no. 100, 13 pp.
24 J. Mu, Monotone iterative technique for fractional evolution equations in Banach spaces, J. Appl. Math. 2011 (2011), Art. ID 767186, 13 pp.
25 F. Mainardi, P. Paradisi, and R. Gorenflo, Probability distributions generated by fractional diffusion equations, J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer, Dordrecht, 2000.
26 H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), no. 5, 985-999.   DOI   ScienceOn
27 G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal. 72 (2010), no. 304, 1604-1615.   DOI   ScienceOn
28 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
29 X. Shu, Y. Lai, and Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. 74 (2011), no. 5, 2003-2011.   DOI   ScienceOn
30 X. Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < $\alpha$ < 2, Comput. Math. Appl. 64 (2012), no. 6, 2100-2110.   DOI   ScienceOn
31 Z. Tai and X. Wang, Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces, Appl. Math. Lett. 22 (2009), no. 11, 1760-1765.   DOI   ScienceOn
32 C. Wang, H. Yuan, and S. Wang, On positive solution of nonlinear fractional differential equation, World Appl. Sci. J. 18 (2012), no. 11, 1540-1545.
33 S. Zhang and X. Su, Existence of extreme solutions for fractional order boundary value problem using upper and lower solutions method in reverst order, J. Fract. Calc. Appl. 2 (2012), no. 6, 1-14.
34 A. Yakar, Initial time difference quasilinearization for Caputo fractional differential equations, Adv. Difference Equ. 2012 (2012), no. 92, 9 pp.   DOI   ScienceOn