Partial Hyperbolic Functional Differential Inclusions

  • Saïd Abbas
  • Mouffak Benchohra
  • Gaston M. N’Guérékata
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 27)

Abstract

In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic differential inclusions with fractional order involving the Caputo fractional derivative, when the right-hand side is convex as well as nonconvex valued. Some results rely on the nonlinear alternative of Leray–Schauder type. In other results, we shall use the fixed-point theorem for contraction multivalued maps due to Covitz and Nadler.

References

  1. 1.
    M.I. Abbas, On the existence of locally attractive solutions of a nonlinear quadratic volterra integral equation of fractional order. Adv. Diff. Equ. 2010, 1–11 (2010)Google Scholar
  2. 2.
    S. Abbas, R.P. Agarwal, M. Benchohra, Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay. Nonlinear Anal. Hybrid Syst. 4, 818–829 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    S. Abbas, R.P. Agarwal, M. Benchohra, Impulsive discontinuous partial hyperbolic differential equations of fractional order on Banach Algebras. Electron. J. Differ. Equat. 2010(91), 1–17 (2010)MathSciNetGoogle Scholar
  4. 4.
    S. Abbas, R.P. Agarwal, M. Benchohra, Existence theory for partial hyperbolic differential inclusions with finite delay involving the Caputo fractional derivative, (submitted)Google Scholar
  5. 5.
    S. Abbas, M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7, 62–72 (2009)MathSciNetMATHGoogle Scholar
  6. 6.
    S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Anal. Hybrid Syst. 3, 597–604 (2009)MathSciNetMATHGoogle Scholar
  7. 7.
    S. Abbas, M. Benchohra, Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4, 406–413 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    S. Abbas, M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses. Discuss. Math. Differ. Incl. Control Optim. 30(1), 141–161 (2010)MathSciNetMATHGoogle Scholar
  9. 9.
    S. Abbas, M. Benchohra, Impulsive partial hyperbolic differential inclusions of fractional order. Demonstratio Math. XLIII(4), 775–797 (2010)Google Scholar
  10. 10.
    S. Abbas, M. Benchohra, Darboux problem for partial functional differential equations with infinite delay and Caputo’s fractional derivative, Adv. Dynamical Syst. Appl. 5(1), 1–19 (2010)MathSciNetGoogle Scholar
  11. 11.
    S. Abbas, M. Benchohra, Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. Frac. Calc. Appl. Anal. 13(3), 225–244 (2010)MathSciNetMATHGoogle Scholar
  12. 12.
    S. Abbas, M. Benchohra, Upper and lower solutions method for the darboux problem for fractional order partial differential inclusions. Int. J. Modern Math. 5(3), 327–338 (2010)MathSciNetMATHGoogle Scholar
  13. 13.
    S. Abbas, M. Benchohra, Existence theory for impulsive partial hyperbolic differential equations of fractional order at variable times. Fixed Point Theory. 12(1), 3–16 (2011)MathSciNetMATHGoogle Scholar
  14. 14.
    S. Abbas, M. Benchohra, Upper and lower solutions method for partial hyperbolic functional differential equations with Caputo’s fractional derivative. Libertas Math. 31, 103–110 (2011)MathSciNetMATHGoogle Scholar
  15. 15.
    S. Abbas, M. Benchohra, Existence results for fractional order partial hyperbolic functional differential inclusions, (submitted)Google Scholar
  16. 16.
    S. Abbas, M. Benchohra, A global uniqueness result for fractional order implicit differential equations. Math. Univ. Comen (submitted)Google Scholar
  17. 17.
    S. Abbas, M. Benchohra, Darboux problem for implicit impulsive partial hyperbolic differential equations. Electron. J. Differ. Equat. 2011, 15 (2011)MathSciNetGoogle Scholar
  18. 18.
    S. Abbas, M. Benchohra, On the set of solutions of fractional order Riemann-Liouville integral inclusions. Demonstratio Math. (to appear)Google Scholar
  19. 19.
    S. Abbas, M. Benchohra, On the set of solutions for the Darboux problem for fractional order partial hyperbolic functional differential inclusions. Fixed Point Theory (to appear)Google Scholar
  20. 20.
    S. Abbas, M. Benchohra, Uniqueness results for Fredholm type fractional order Riemann-Liouville integral equations (submitted)Google Scholar
  21. 21.
    S. Abbas, M. Benchohra, Fractional order Riemann-Liouville integral equations with multiple time delay. Appl. Math. E-Notes (to appear)Google Scholar
  22. 22.
    S. Abbas, M. Benchohra, Nonlinear quadratic Volterra Riemann-Liouville integral equations of fractional order. Nonlinear Anal. Forum 17, 1–9 (2012)Google Scholar
  23. 23.
    S. Abbas, M. Benchohra, On the set of solutions of nonlinear fractional order Riemann-Liouville functional integral equations in Banach algebras (submitted)Google Scholar
  24. 24.
    S. Abbas, M. Benchohra, Fractional order Riemann-Liouville integral inclusions with two independent variables and multiple time delay. Opuscula Math. (to appear)Google Scholar
  25. 25.
    S. Abbas, M. Benchohra, L. Gorniewicz, Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn.. online e- 2010, 271–282Google Scholar
  26. 26.
    S. Abbas, M. Benchohra, L. Gorniewicz, Fractional order impulsive partial hyperbolic differential inclusions with variable times. Discussions Mathe. Differ. Inclu. Contr. Optimiz. 31(1), 91–114 (2011)MathSciNetMATHGoogle Scholar
  27. 27.
    S. Abbas, M. Benchohra, L. Gorniewicz, Fractional order impulsive partial hyperbolic functional differential equations with variable times and state-dependent delay. Math. Bulletin 7, 317–350 (2010)Google Scholar
  28. 28.
    S. Abbas, M. Benchohra, J. Henderson, Global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order. Comm. Appl. Nonlinear Anal. 19, 79–89 (2012)MathSciNetMATHGoogle Scholar
  29. 29.
    S. Abbas, M. Benchohra, J. Henderson, Attractivity results for nonlinear fractional order Riemann-Liouville integral equations in Banach algebras, (submitted)Google Scholar
  30. 30.
    S. Abbas, M. Benchohra, J.J. Nieto, Global uniqueness results for fractional order partial hyperbolic functional differential equations. Adv. in Difference Equ. 2011, Art. ID 379876, 25 ppGoogle Scholar
  31. 31.
    S. Abbas, M. Benchohra, J.J. Nieto, Functional implicit hyperbolic fractional order differential equations with delay, (submitted)Google Scholar
  32. 32.
    S. Abbas, M. Benchohra, G.M. N’Guérékata, B.A. Slimani, Darboux problem for fractional order discontinuous hyperbolic partial differential equations in Banach algebras. Complex Variables and Elliptic Equations 57(2–4), 337–350 (2012)MathSciNetMATHGoogle Scholar
  33. 33.
    S. Abbas, M. Benchohra, J.J. Trujillo, Fractional order impulsive hyperbolic implicit differential equations with state-dependent delay (submitted)Google Scholar
  34. 34.
    S. Abbas, M. Benchohra, A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations. Frac. Calc. Appl. Anal. 15(2), 168–182 (2012)MathSciNetGoogle Scholar
  35. 35.
    S. Abbas, M. Benchohra, Y. Zhou, Darboux problem for fractional order neutral functional partial hyperbolic differential equations, Int. J. Dynamical Systems Differential Equations. 2(3&4), 301–312 (2009)MathSciNetMATHGoogle Scholar
  36. 36.
    S. Abbas, M. Benchohra, Y. Zhou, Fractional order partial functional differential inclusions with infinite delay. Proc. A. Razmadze Math. Inst. 154, 1–19 (2010)MathSciNetMATHGoogle Scholar
  37. 37.
    S. Abbas, M. Benchohra, Y. Zhou, Fractional order partial hyperbolic functional differential equations with state-dependent delay. Int. J. Dyn. Syst. Differ. Equat. 3(4), 459–490 (2011)MathSciNetMATHGoogle Scholar
  38. 38.
    N.H. Abel, Solutions de quelques problèmes à l’aide d’intégrales définies (1823). Œuvres complètes de Niels Henrik Abel, 1, Grondahl, Christiania, 1881, 11–18Google Scholar
  39. 39.
    R.P Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations. Georgian. Math. J. 16, 401–411 (2009)Google Scholar
  40. 40.
    R.P Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109, 973–1033 (2010)Google Scholar
  41. 41.
    R.P. Agarwal, M. Benchohra, B.A. Slimani, Existence results for differential equations with fractional order and impulses. Mem. Differ. Equat. Math. Phys. 44, 1–21 (2008)MathSciNetMATHGoogle Scholar
  42. 42.
    R.P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, in Cambridge Tracts in Mathematics, vol. 141 (Cambridge University Press, Cambridge, 2001)Google Scholar
  43. 43.
    R.P. Agarwal, D. ORegan, S. Stanek, Positive solutions for Dirichlet problem of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)Google Scholar
  44. 44.
    R.P Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59(3), 1095–1100 (2010)Google Scholar
  45. 45.
    R.P. Agarwal, Y. Zhou, J. Wang, X. Luo, Fractional functional differential equations with causal operators in Banach spaces. Math. Comput. Model. 54(5–6), 1440–1452 (2011)MathSciNetMATHGoogle Scholar
  46. 46.
    O.P. Agrawal, O. Defterli, D. Baleanu, Fractional optimal control problems with several state and control variables. J. Vib. Contr. 16(13), 1967–1976 (2010)MathSciNetGoogle Scholar
  47. 47.
    B. Ahmad, J.J. Nieto, Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwanese J. Math. 15(3), 981–993 (2011)MathSciNetMATHGoogle Scholar
  48. 48.
    B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251–258 (2009)MathSciNetMATHGoogle Scholar
  49. 49.
    E. Ait Dads, M. Benchohra, S. Hamani, Impulsive fractional differential inclusions involving the Caputo fractional derivative. Fract. Calc. Appl. Anal. 12(1), 15–38 (2009)MathSciNetMATHGoogle Scholar
  50. 50.
    R. Almeida, D.F.M. Torres, Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 61, 3097–3104 (2011)MathSciNetMATHGoogle Scholar
  51. 51.
    R. Almeida, D.F.M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, 1490–1500 (2011)MathSciNetMATHGoogle Scholar
  52. 52.
    G.A. Anastassiou, in Advances on Fractional Inequalities (Springer, New York, 2011)Google Scholar
  53. 53.
    D. Araya, C. Lizama, Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69, 3692–3705 (2008)MathSciNetMATHGoogle Scholar
  54. 54.
    S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty. Nonlinear Anal. 74, 3685–3693 (2011)MathSciNetMATHGoogle Scholar
  55. 55.
    J.P. Aubin, Impulse differential inclusions and hybrid systems: a viability ap- proach, Lecture Notes, Universit Paris-Dauphine (2002)Google Scholar
  56. 56.
    J.P. Aubin, A. Cellina, in Differential Inclusions (Springer, Berlin, 1984)Google Scholar
  57. 57.
    J.P. Aubin, H. Frankowska, in Set-Valued Analysis (Birkhauser, Boston, 1990)Google Scholar
  58. 58.
    I. Bajo, E. Liz, Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204, 65–73 (1996)MathSciNetMATHGoogle Scholar
  59. 59.
    K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence results for fractional impulsive integrodifferetial equations in Banach spaces. Comm. Nonlinear Sci. Numer. Simul. 16, 1970–1977 (2011)MathSciNetMATHGoogle Scholar
  60. 60.
    K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 72, 4587-4593 (2010)MathSciNetMATHGoogle Scholar
  61. 61.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, in Fractional Calculus Models and Numerical Methods (World Scientific Publishing, New York, 2012)Google Scholar
  62. 62.
    D. Baleanu, S.I. Vacaru, Fractional curve flows and solitonic hierarchies in gravity and geometric mechanics. J. Math. Phys. 52(5), 053514, 15 (2011)Google Scholar
  63. 63.
    J. Banaś, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equation. Nonlinear Anal. 69(7), 1945–1952 (2008)MathSciNetMATHGoogle Scholar
  64. 64.
    E. Bazhlekova, in Fractional Evolution Equations in Banach Spaces (University Press Facilities, Eindhoven University of Technology, 2001)Google Scholar
  65. 65.
    A. Belarbi, M. Benchohra, Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times. J. Math. Anal. Appl. 327, 1116–1129 (2007)MathSciNetMATHGoogle Scholar
  66. 66.
    A. Belarbi, M. Benchohra, A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Appl. Anal. 85, 1459–1470 (2006)MathSciNetMATHGoogle Scholar
  67. 67.
    M. Benchohra, J.R. Graef, S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions. Appl. Anal. 87(7), 851–863 (2008)MathSciNetMATHGoogle Scholar
  68. 68.
    M. Benchohra, J.R. Graef, F-Z. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces. Electron. J. Qual. Theory Differ. Equat. 2010(54), 10 ppGoogle Scholar
  69. 69.
    M. Benchohra, S. Hamani, S.K. Ntouyas, boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)MathSciNetMATHGoogle Scholar
  70. 70.
    M. Benchohra, J. Henderson, S.K. Ntouyas, in Impulsive Differential Equations and Inclusions, vol. 2 (Hindawi Publishing Corporation, New York, 2006)Google Scholar
  71. 71.
    M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for functional differential equations of fractional order. J. Math. Anal. Appl. 338, 1340–1350 (2008)MathSciNetMATHGoogle Scholar
  72. 72.
    M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, On first order impulsive dynamic equations on time scales. J. Difference Equ. Appl. 10, 541–548 (2004)MathSciNetMATHGoogle Scholar
  73. 73.
    M. Benchohra, J.J. Nieto, D. Seba, Measure of noncompactness and hyperbolic partial fractional differential equations in Banach spaces. Panamer. Math. J. 20(3), 27–37 (2010)MathSciNetMATHGoogle Scholar
  74. 74.
    M. Benchohra, S.K. Ntouyas, An existence theorem for an hyperbolic differential inclusion in Banach spaces. Discuss. Math. Differ. Incl. Contr. Optim. 22, 5–16 (2002)MathSciNetMATHGoogle Scholar
  75. 75.
    M. Benchohra, S.K. Ntouyas, On an hyperbolic functional differential inclusion in Banach spaces. Fasc. Math. 33, 27–35 (2002)MathSciNetMATHGoogle Scholar
  76. 76.
    M. Benchohra, S.K. Ntouyas, An existence result for hyperbolic functional differential inclusions. Comment. Math. Prace Mat. 42, 1–16 (2002)MathSciNetMATHGoogle Scholar
  77. 77.
    M. Benchohra, B.A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equat. 2009(10), 11 (2009)Google Scholar
  78. 78.
    F. Berhoun, A contribution of some classes of impulsive differential equations with integer and non integer order, Doctorate thesis, University of Sidi Bel Abbes, 2010Google Scholar
  79. 79.
    A. Bica, V.A. Caus, S. Muresan, Application of a trapezoid inequality to neutral Fredholm integro-differential equations in Banach spaces. J. Inequal. Pure Appl. Math. 7, 5 (2006), Art. 173Google Scholar
  80. 80.
    F.S. De Blasi, G. Pianigiani, V. Staicu: On the solution sets of some nonconvex hyperbolic differential inclusions. Czechoslovak Math. J. 45, 107–116 (1995)MathSciNetMATHGoogle Scholar
  81. 81.
    H.F. Bohnenblust, S. Karlin, On a theorem of ville. Contribution to the theory of games, in Annals of Mathematics Studies, vol. 24 (Priceton University Press, Princeton. N. G., 1950), pp. 155–160Google Scholar
  82. 82.
    A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values. Studia Math. 90, 69–86 (1988)MathSciNetMATHGoogle Scholar
  83. 83.
    T.A. Burton, Fractional differential equations and Lyapunov functionals. Nonlinear Anal. 74, 5648–5662 (2011)MathSciNetMATHGoogle Scholar
  84. 84.
    T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189, 23–31 (1998)MathSciNetMATHGoogle Scholar
  85. 85.
    L. Byszewski, Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation u xt = F(x, t, u, u x). J. Appl. Math. Stochastic Anal. 3, 163–168 (1990)MathSciNetMATHGoogle Scholar
  86. 86.
    L. Byszewski, Theorem about existence and uniqueness of continuous solutions of nonlocal problem for nonlinear hyperbolic equation. Appl. Anal. 40, 173–180 (1991)MathSciNetMATHGoogle Scholar
  87. 87.
    L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional differential evolution nonlocal Cauchy problem, Selected Problems in Mathematics, Cracow Univ. of Tech. Monographs, Anniversary Issue 6, 25–33 (1995)Google Scholar
  88. 88.
    L. Byszewski, V. Lakshmikantam, Monotone iterative technique for non-local hyperbolic differential problem. J. Math. Phys. Sci 26, 345–359 (1992)MathSciNetMATHGoogle Scholar
  89. 89.
    L. Byszewski, S.N. Papageorgiou, An application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem. J. Appl. Math. Stoch. Anal. 12, 179–180 (1999)MathSciNetMATHGoogle Scholar
  90. 90.
    M. Caputo, Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Roy. Astron. Soc. 13, 529–539 (1967), reprinted in Fract. Calc. Appl. Anal. 11, 4–14 (2008)Google Scholar
  91. 91.
    M. Caputo, Linear models of dissipation whose is almost frequency independent-II. Geophys. J. R. Astr. Soc. 13, 529–539 (1967)Google Scholar
  92. 92.
    M. Caputo, in Elasticità e Dissipazione (Zanichelli, Bologna, 1969)Google Scholar
  93. 93.
    C. Castaing, M. Valadier, in Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580 (Springer, Berlin, 1977)Google Scholar
  94. 94.
    Y.-K. Chang, J.J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009)MathSciNetMATHGoogle Scholar
  95. 95.
    C. Corduneanu, in Integral Equations and Applications (Cambridge University Press, Cambridge, 1991)Google Scholar
  96. 96.
    H. Covitz, S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8, 5–11 (1970)MathSciNetMATHGoogle Scholar
  97. 97.
    T. Czlapinski, On the Darboux problem for partial differential-functional equations with infinite delay at derivatives. Nonlinear Anal. 44, 389–398 (2001)MathSciNetMATHGoogle Scholar
  98. 98.
    T. Czlapinski, Existence of solutions of the Darboux problem for partial differential-functional equations with infinite delay in a Banach space. Comment. Math. Prace Mat. 35, 111–122 (1995)MathSciNetMATHGoogle Scholar
  99. 99.
    M.F. Danca, K. Diethelm, Kai. Fractional-order attractors synthesis via parameter switchings. Commun. Nonlinear Sci. Numer. Simul. 15(12), 3745–3753 (2010)MathSciNetMATHGoogle Scholar
  100. 100.
    M.A. Darwish, J. Henderson, D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. Bull. Korean Math. Soc. 48, 539–553 (2011)MathSciNetMATHGoogle Scholar
  101. 101.
    M. Dawidowski, I. Kubiaczyk, An existence theorem for the generalized hyperbolic equation z′ xy ∈ F(x, y, z) in Banach space. Ann. Soc. Math. Pol. Ser. I Comment. Math. 30(1), 41–49 (1990)MathSciNetMATHGoogle Scholar
  102. 102.
    A. Debbouche, Fractional evolution integro-differential systems with nonlocal conditions. Adv. Dyn. Syst. Appl. 5(1), 49–60 (2010)MathSciNetGoogle Scholar
  103. 103.
    A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)MathSciNetMATHGoogle Scholar
  104. 104.
    K. Deimling, in Multivalued Differential Equations (Walter De Gruyter, Berlin, 1992)MATHGoogle Scholar
  105. 105.
    D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)MathSciNetMATHGoogle Scholar
  106. 106.
    Z. Denton, A.S. Vatsala, Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations. Opuscula Math. 31(3), 327–339 (2011)MathSciNetMATHGoogle Scholar
  107. 107.
    B.C. Dhage, A nonlinear alternative in Banach algebras with applications to functional differential equations. Nonlinear Funct. Anal. Appl. 8, 563–575 (2004)MathSciNetGoogle Scholar
  108. 108.
    B.C. Dhage, Some algebraic fixed point theorems for multi-valued mappings with applications. Diss. Math. Differ. Inclusions Contr. Optim. 26, 5–55 (2006)MathSciNetMATHGoogle Scholar
  109. 109.
    B.C. Dhage, Nonlinear functional boundary value problems in Banach algebras involving Carathéodories. Kyungpook Math. J. 46(4), 527–541 (2006)MathSciNetMATHGoogle Scholar
  110. 110.
    B.C. Dhage, Existence theorems for hyperbolic differential inclusions in Banach algebras. J. Math. Anal. Appl. 335, 225–242 (2007)MathSciNetMATHGoogle Scholar
  111. 111.
    B.C. Dhage, Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness. Diff. Equ. Appl. 2(3), 299–318 (2010)MathSciNetMATHGoogle Scholar
  112. 112.
    T. Diagana, G.M. Mophou, G.M. N’Guérékata, On the existence of mild solutions to some semilinear fractional integro-differential equations. Electron. J. Qual. Theory Differ. Equ. 2010 (58), 17Google Scholar
  113. 113.
    K. Diethelm, in The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics (Springer, Berlin, 2010)Google Scholar
  114. 114.
    K. Diethelm, N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetMATHGoogle Scholar
  115. 115.
    K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, ed. by F. Keil, W. Mackens, H. Voss, J. Werther (Springer, Heidelberg, 1999), pp. 217–224Google Scholar
  116. 116.
    X. Dong, J. Wang, Y. Zhou, Yong. On nonlocal problems for fractional differential equations in Banach spaces. Opuscula Math. 31(3), 341–357 (2011)MathSciNetMATHGoogle Scholar
  117. 117.
    S. Dugowson, L’élaboration par Riemann d’une définition de la dérivation d’ordre non entier. revue d’histoire des Mathématiques 3, 49–97 (1997)Google Scholar
  118. 118.
    M.M. El-Borai, On some fractional evolution equations with nonlocal conditions. Int. J. Pure Appl. Math. 24, 405–413 (2005)MathSciNetMATHGoogle Scholar
  119. 119.
    M.M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type. J. Appl. Math. Stoch. Anal. 2004(3), 197–211MathSciNetGoogle Scholar
  120. 120.
    M.M. El-Borai, K. El-Said El-Nadi, E.G. El-Akabawy On some fractional evolution equations. Comput. Math. Appl. 59(3), 1352–1355 (2010)MathSciNetMATHGoogle Scholar
  121. 121.
    M.M. El-Borai, K. El-Nadi, H.A. Fouad, On some fractional stochastic delay differential equations. Comput. Math. Appl. 59(3), 1165–1170 (2010)MathSciNetMATHGoogle Scholar
  122. 122.
    A.M.A. El-Sayed, Fractional order evolution equations. J. Fract. Calc. 7, 89–100 (1995)MathSciNetMATHGoogle Scholar
  123. 123.
    A.M.A. El-Sayed, Fractional order diffusion-wave equations. Int. J. Theo. Phys. 35, 311–322 (1996)MathSciNetMATHGoogle Scholar
  124. 124.
    A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998)MathSciNetMATHGoogle Scholar
  125. 125.
    J.B.J. Fourier, Théorie Analytique de la Chaleur, Didot, Paris, 499–508 (1822)Google Scholar
  126. 126.
    M. Frigon, Théorèmes d’existence de solutions d’inclusions différentielles, Topological Methods in Differential Equations and Inclusions, NATO ASI Series C, vol. 472, ed. by A. Granas, M. Frigon (Kluwer Academic Publishers, Dordrecht, 1995), pp. 51–87Google Scholar
  127. 127.
    M. Frigon, A. Granas, Théorèmes d’existence pour des inclusions différentielles sans convexité. C. R. Acad. Sci. Paris, Ser. I 310, 819–822 (1990)Google Scholar
  128. 128.
    M. Frigon, D. O’Regan, Impulsive differential equations with variable times. Nonlinear Anal. 26, 1913–1922 (1996)MathSciNetMATHGoogle Scholar
  129. 129.
    M. Frigon, D. O’Regan, First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl. 233, 730–739 (1999)MathSciNetMATHGoogle Scholar
  130. 130.
    M. Frigon, D. O’Regan, Second order Sturm-Liouville BVP’s with impulses at variable moments. Dynam. Contin. Discrete Impuls. Syst. 8 (2), 149–159 (2001)MathSciNetMATHGoogle Scholar
  131. 131.
    K.M. Furati, N.-eddine Tatar, Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Frac. Calc. 28, 23–42 (2005)Google Scholar
  132. 132.
    K.M. Furati, N.-eddine Tatar, Power type estimates for a nonlinear fractional differential equation. Nonlinear Anal. 62, 1025–1036 (2005)Google Scholar
  133. 133.
    L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991)Google Scholar
  134. 134.
    W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)Google Scholar
  135. 135.
    L. Gorniewicz, in Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, vol. 495 (Kluwer Academic Publishers, Dordrecht, 1999)Google Scholar
  136. 136.
    A. Granas, J. Dugundji, in Fixed Point Theory (Springer, New York, 2003)Google Scholar
  137. 137.
    A.K. Grunwald, Dérivationen und deren Anwendung. Zeitschrift für Mathematik und Phisik, 12, 441–480 (1867)Google Scholar
  138. 138.
    J. Hale, J. Kato, Phase space for retarded equationswith infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)MathSciNetMATHGoogle Scholar
  139. 139.
    J.K. Hale, S. Verduyn Lunel, in Introduction to Functional -Differential Equations. Applied Mathematical Sciences, vol. 99 (Springer, New York, 1993)Google Scholar
  140. 140.
    F. Hartung, Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays. J. Math. Anal. Appl. 324(1), 504–524 (2006)MathSciNetMATHGoogle Scholar
  141. 141.
    F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays. J. Comput. Appl. Math. 174(2), 201–211 (2005)MathSciNetMATHGoogle Scholar
  142. 142.
    D. Henry, in Geometric Theory of Semilinear Parabolic Partial Differential Equations (Springer, Berlin, 1989)Google Scholar
  143. 143.
    S. Heikkila, V. Lakshmikantham, in Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations (Marcel Dekker Inc., New York, 1994)Google Scholar
  144. 144.
    J. Henderson, A. Ouahab, Fractional functional differential inclusions with finite delay. Nonlinear Anal. 70 (2009) 2091–2105MathSciNetMATHGoogle Scholar
  145. 145.
    J. Henderson, A. Ouahab, Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)MathSciNetMATHGoogle Scholar
  146. 146.
    J. Henderson, C. Tisdell, Topological transversality and boundary value problems on time scales. J. Math. Anal. Appl. 289, 110–125 (2004)MathSciNetMATHGoogle Scholar
  147. 147.
    E. Hernández, A. Prokopczyk, L. Ladeira, A note on partial functional differential equations with state-dependent delay. Nonlinear Anal. Real World Applications 7, 510–519 (2006)MATHGoogle Scholar
  148. 148.
    E. Hernandez M., R. Sakthivel, S. Tanaka Aki, Existence results for impulsive evolution differential equations with state-dependent delay. Electron. J. Differ. Equat. 2008 (28), 1–11 (2008)Google Scholar
  149. 149.
    M.A.E. Herzallah, D. Baleanu, Fractional-order variational calculus with generalized boundary conditions. Adv. Difference Equ. Article ID 357580, 9 p 2011 Google Scholar
  150. 150.
    M.A.E. Herzallah, A.M.A. El-Sayed, D. Baleanu, Perturbation for fractional-order evolution equation. Nonlinear Dynam. 62(3), 593–600 (2010)MathSciNetMATHGoogle Scholar
  151. 151.
    R. Hilfer, in Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)MATHGoogle Scholar
  152. 152.
    Y. Hino, S. Murakami, T. Naito, in Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473 (Springer, Berlin, 1991)Google Scholar
  153. 153.
    Sh. Hu, N. Papageorgiou, in Handbook of Multivalued Analysis, Theory I (Kluwer, Dordrecht, 1997)MATHGoogle Scholar
  154. 154.
    R.W. Ibrahim, Existence and uniqueness of holomorphic solutions for fractional Cauchy problem. J. Math. Anal. Appl. 380, 232–240 (2011)MathSciNetMATHGoogle Scholar
  155. 155.
    R.W. Ibrahim, H.A. Jalab, Existence of the solution of fractiona integral inclusion with time delay. Misk. Math. Notes 11(2), 139–150 (2010)MathSciNetMATHGoogle Scholar
  156. 156.
    T. Kaczorek, in Selected Problems of Fractional Systems Theory (Springer, London, 2011)Google Scholar
  157. 157.
    A. Kadem, D. Baleanu, Homotopy perturbation method for the coupled fractional Lotka-Volterra equations. Romanian J. Phys. 56(3–4), 332–338 (2011)MathSciNetMATHGoogle Scholar
  158. 158.
    Z. Kamont, in Hyperbolic Functional Differential Inequalities and Applications (Kluwer Academic Publishers, Dordrecht, 1999)Google Scholar
  159. 159.
    Z. Kamont, K. Kropielnicka, Differential difference inequalities related to hyperbolic functional differential systems and applications. Math. Inequal. Appl. 8(4), 655–674 (2005)MathSciNetMATHGoogle Scholar
  160. 160.
    S.K. Kaul, V. Lakshmikantham, S. Leela, Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times. Nonlinear Anal. 22, 1263–1270 (1994)MathSciNetMATHGoogle Scholar
  161. 161.
    S.K. Kaul, X.Z. Liu, Vector Lyapunov functions for impulsive differential systems with variable times. Dynam. Contin. Discrete Impuls. Syst. 6, 25–38 (1999)MathSciNetMATHGoogle Scholar
  162. 162.
    S.K. Kaul, X.Z. Liu, Impulsive integro-differential equations with variable times. Nonlinear Stud. 8, 21–32 (2001)MathSciNetMATHGoogle Scholar
  163. 163.
    E.R. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equat. (3), 11 (2007)Google Scholar
  164. 164.
    A.A. Kilbas, B. Bonilla, J. Trujillo, Nonlinear differential equations of fractional order in a space of integrable functions. Dokl. Ross. Akad. Nauk 374(4), 445–449 (2000)MathSciNetGoogle Scholar
  165. 165.
    A.A. Kilbas, S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differ. Equat. 41, 84–89 (2005)MathSciNetMATHGoogle Scholar
  166. 166.
    A.A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, in Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (Elsevier Science B.V., Amsterdam, 2006)Google Scholar
  167. 167.
    M. Kirane, M. Medved, N. Tatar, Semilinear Volterra integrodifferential problems with fractional derivatives in the nonlinearities. Abstr. Appl. Anal. 2011, Art. ID 510314, 11 ppGoogle Scholar
  168. 168.
    V.S. Kiryakova, Y.F. Luchko, The multi-index Mittag-Leffler functions and their appplications for solving fractional order problems in applied analysis. Application of mathematics in technical and natural sciences, 597–613, AIP Conf. Proc., 1301, Amer. Inst. Phys., Melville, NY, 2010Google Scholar
  169. 169.
    M. Kisielewicz, in Differential Inclusions and Optimal Control (Kluwer, Dordrecht, The Netherlands, 1991)Google Scholar
  170. 170.
    S. Labidi, N. Tatar, Blow-up of solutions for a nonlinear beam equation with fractional feedback. Nonlinear Anal. 74(4), 1402–1409 (2011)MathSciNetMATHGoogle Scholar
  171. 171.
    S.F. Lacroix, Traité du Calcul Différentiel et du Calcul Intégral, Courcier, Paris, t.3 (1819), 409–410Google Scholar
  172. 172.
    G.S. Ladde, V. Lakshmikanthan, A.S. Vatsala, in Monotone Iterative Techniques for Nonliner Differential Equations (Pitman Advanced Publishing Program, London, 1985)Google Scholar
  173. 173.
    V. Lakshmikantham, Theory of fractional differential equations. Nonlinear Anal. 60, 3337–3343 (2008)MathSciNetGoogle Scholar
  174. 174.
    V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, in Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989)Google Scholar
  175. 175.
    V. Lakshmikantham, S. Leela, J. Vasundhara, in Theory of Fractional Dynamic Systems (Cambridge Academic Publishers, Cambridge, 2009)Google Scholar
  176. 176.
    V. Lakshmikantham, S.G. Pandit, The method of upper, lower solutions and hyperbolic partial differential equations. J. Math. Anal. Appl. 105, 466–477 (1985)MathSciNetMATHGoogle Scholar
  177. 177.
    V. Lakshmikantham, N.S. Papageorgiou, J. Vasundhara, The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments. Appl. Anal. 15, 41–58 (1993)MathSciNetGoogle Scholar
  178. 178.
    V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)MathSciNetMATHGoogle Scholar
  179. 179.
    V. Lakshmikantham, L. Wen, B. Zhang, in Theory of Differential Equations with Unbounded Delay. Mathematics and its Applications (Kluwer Academic Publishers, Dordrecht, 1994)Google Scholar
  180. 180.
    A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13, 781–786 (1965)MathSciNetMATHGoogle Scholar
  181. 181.
    G.W. Leibniz, Letter from Hanover, Germany, Deptember 30, 1695 to G.A. L’Hospital, in JLeibnizen Mathematische Schriften, vol. 2 (Olms Verlag, Hildesheim, Germany, 1962), pp. 301–302. First published in 1849Google Scholar
  182. 182.
    F. Li, G.M. N’Guérékata, An existence result for neutral delay integrodifferential equations with fractional order and nonlocal conditions Abst. Appl. Anal. (2011), Article ID 952782, 20 pagesGoogle Scholar
  183. 183.
    Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(5), 1810–1821 (2010)MathSciNetMATHGoogle Scholar
  184. 184.
    T.C. Lim, On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J. Math. Anal. Appl. 110, 436–441 (1985)MathSciNetMATHGoogle Scholar
  185. 185.
    J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques. J. l’Ecole Roy. Polytéchn. 13, 529–539 (1832)Google Scholar
  186. 186.
    Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, 538–548 (2011)MathSciNetMATHGoogle Scholar
  187. 187.
    R. Magin, in Fractional Calculus in Bioengineering (Begell House Publishers, Redding, 2006)Google Scholar
  188. 188.
    R. Magin, M.D. Ortigueira, I. Podlubny, J.J. Trujillo, On the fractional signals and systems. Signal Process. 91, 350–371 (2011)MATHGoogle Scholar
  189. 189.
    F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics ed. by A. Carpinteri, F. Mainardi (Springer-Verlag, Wien, 1997), pp. 291–348Google Scholar
  190. 190.
    S. Marano, V. Staicu, On the set of solutions to a class of nonconvex nonclosed differential inclusions. Acta Math. Hungar. 76, 287–301 (1997)MathSciNetMATHGoogle Scholar
  191. 191.
    F. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)Google Scholar
  192. 192.
    K.S. Miller, B. Ross, in An Introduction to the Fractional Calculus and Differential Equations (Wiley, New York, 1993)Google Scholar
  193. 193.
    V.D. Milman, A.A. Myshkis, On the stability of motion in the presence of impulses. Sib. Math. J. 1, 233–237 (1960), [in Russian]Google Scholar
  194. 194.
    V.D. Milman, A.A. Myshkis, Random impulses in linear dynamical systems, in Approximante Methods for Solving Differential Equations (Publishing House of the Academy of Sciences of Ukainian SSR, Kiev, 1963), pp. 64–81, [in Russian]Google Scholar
  195. 195.
    G.M. Mittag-Leffler, Sur la nouvelle function E α. C. R. Acad. Sci. Paris 137, 554–558 (1903)Google Scholar
  196. 196.
    G.M. Mittag-Leffler, Sopra la funzione E α(x). Rend. Accad. Lincei, ser. 5 13, 3–5 (1904)Google Scholar
  197. 197.
    K. Moaddy, S. Momani, I. Hashim, The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics. Comput. Math. Appl. 61(4), 1209–1216 (2011)MathSciNetMATHGoogle Scholar
  198. 198.
    G.M. Mophou, Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68–78 (2011)MathSciNetMATHGoogle Scholar
  199. 199.
    M. Mophou, O. Nakoulima, G.M. N’Guérékata, Existence results for some fractional differential equations with nonlocal conditions. Nonlinear Stud. 17, 15–22 (2010)MathSciNetMATHGoogle Scholar
  200. 200.
    G.M. Mophou, G.M. N’Guérékata, Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79, 315–322 (2009)MathSciNetMATHGoogle Scholar
  201. 201.
    G.M. Mophou, G.M. N’Guérékata, On some classes of almost automorphic functions and applications to fractional differential equations. Comput. Math. Appl. 59, 1310–1317 (2010)MathSciNetMATHGoogle Scholar
  202. 202.
    G.M. Mophou, G.M. N’Guérékata, On integral solutions of some nonlocal fractional differential equations with nondense domain. Nonlinear Anal. 71, 4668–4675 (2009)MathSciNetMATHGoogle Scholar
  203. 203.
    G.M. Mophou, G.N. N’Guérékata, Controllability of semilinear neutral fractional functional evolution equations with infinite delay. Nonlinear Stud. 18, 195–209 (2011)MathSciNetMATHGoogle Scholar
  204. 204.
    G.M. Mophou, G.M. N’Guérékata, V. Valmorin, Pseudo almost automorphic solutions of a neutral functional fractional differential equations. Intern. J. Evol. Equ. 4, 129–139 (2009)Google Scholar
  205. 205.
    S. Muslih, O.P. Agrawal, Riesz fractional derivatives and fractional dimensional space. Int. J. Theor. Phys. 49(2), 270–275 (2010)MathSciNetMATHGoogle Scholar
  206. 206.
    S. Muslih, O.P. Agrawal, D. Baleanu, A fractional Schrdinger equation and its solution. Int. J. Theor. Phys. 49(8), 1746–1752 (2010)MathSciNetMATHGoogle Scholar
  207. 207.
    J.J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23, 1248–1251 (2010)MathSciNetMATHGoogle Scholar
  208. 208.
    K.B. Oldham, J. Spanier, in The Fractional Calculus (Academic Press, New York, 1974)Google Scholar
  209. 209.
    M.D. Ortigueira, in Fractional Calculus for Scientists and Engineers (Springer, Dordrecht, 2011)Google Scholar
  210. 210.
    B.G. Pachpatte, On Volterra-Fredholm integral equation in two variables. Demonstratio Math. XL(4), 839–852 (2007)Google Scholar
  211. 211.
    B.G. Pachpatte, On Fredholm type integrodifferential equation. Tamkang J. Math. 39(1), 85–94 (2008)MathSciNetMATHGoogle Scholar
  212. 212.
    B.G. Pachpatte, On Fredholm type integral equation in two variables. Diff. Equ. Appl. 1, 27–39 (2009)MATHGoogle Scholar
  213. 213.
    S.G. Pandit, Monotone methods for systems of nonlinear hyperbolic problems in two independent variables. Nonlinear Anal. 30, 235–272 (1997)MathSciNetGoogle Scholar
  214. 214.
    I. Podlubny, in Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198 (Academic Press, San Diego, 1999)Google Scholar
  215. 215.
    I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus Appl. Anal. 5, 367–386 (2002)MathSciNetMATHGoogle Scholar
  216. 216.
    I. Podlubny, I. Petraš, B.M. Vinagre, P. O’Leary, L. Dorčak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dynam. 29, 281–296 (2002)MATHGoogle Scholar
  217. 217.
    J.D. Ramrez, A.S. Vatsala, Monotone method for nonlinear Caputo fractional boundary value problems. Dynam. Systems Appl. 20(1), 73–88 (2011)MathSciNetGoogle Scholar
  218. 218.
    A. Razminia, V.J. Majd, D. Baleanu, Chaotic incommensurate fractional order Rssler system: Active control and synchronization. Adv. Difference Equat. 2011(15), 12 (2011)Google Scholar
  219. 219.
    M. Rivero, J.J. Trujillo, L. Vzquez, M.P. Velasco, Fractional dynamics of populations. Appl. Math. Comput. 218, 1089–1095 (2011)MathSciNetMATHGoogle Scholar
  220. 220.
    J. Sabatier, O. Agrawal, J. Machado (eds.), in Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007)Google Scholar
  221. 221.
    J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simul. 15(5), 1318–1326 (2010)MathSciNetMATHGoogle Scholar
  222. 222.
    H.A.H. Salem, On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies. Comput. Math. Appl. 224, 565–572 (2009)MathSciNetMATHGoogle Scholar
  223. 223.
    H.A.H. Salem, On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order. Comput. Math. Appl. 59(3), 1278–1293 (2010)MathSciNetMATHGoogle Scholar
  224. 224.
    H.A.H. Salem Global monotonic solutions of multi term fractional differential equations. Appl. Math. Comput. 217(14), 6597–6603 (2011)MathSciNetGoogle Scholar
  225. 225.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, in Fractional Integrals and Derivatives. Theory and Applications (Gordon and Breach, Yverdon, 1993)Google Scholar
  226. 226.
    N. Samko, S. Samko, B. Vakulov, Fractional integrals and hypersingular integrals in variable order Hlder spaces on homogeneous spaces. J. Funct. Spaces Appl. 8(3), 215–244 (2010)MathSciNetMATHGoogle Scholar
  227. 227.
    A.M. Samoilenko, N.A. Perestyuk, in Impulsive Differential Equations (World Scientific, Singapore, 1995)Google Scholar
  228. 228.
    N.P. Semenchuk, On one class of differential equations of noninteger order. Differents. Uravn. 10, 1831–1833 (1982)MathSciNetGoogle Scholar
  229. 229.
    H. Sheng, Y. Chen, T. Qiu, in Fractional Processes and Fractional-order Signal Processing; Techniques and Applications (Springer-Verlag, London, 2011)Google Scholar
  230. 230.
    B.A. Slimani, A contribution to fractional order differential equations and inclusions with impulses, Doctorate thesis, University of Sidi Bel Abbes, 2009Google Scholar
  231. 231.
    V.E. Tarasov, in Fractional dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Heidelberg, 2010)Google Scholar
  232. 232.
    V.E. Tarasov, Fractional dynamics of relativistic particle. Int. J. Theor. Phys. 49(2), 293–303 (2010)MathSciNetMATHGoogle Scholar
  233. 233.
    V.E. Tarasov, M. Edelman, Fractional dissipative standard map. Chaos 20(2), 023127, 7 (2010)Google Scholar
  234. 234.
    J.A. Tenreiro Machado, Time-delay and fractional derivatives. Adv. Difference Equ. 2011, Art. ID 934094, 12 ppGoogle Scholar
  235. 235.
    J.A. Tenreiro Machado. Entropy analysis of integer and fractional dynamical systems. Nonlinear Dynam. 62(1–2), 371–378 (2010)MathSciNetMATHGoogle Scholar
  236. 236.
    J.A. Tenreiro Machado. Time-delay and fractional derivatives. Adv. Difference Equ. (2011), Art. ID 934094, 12 ppGoogle Scholar
  237. 237.
    J.A. Tenreiro Machado, V. Kiryakova, F. Mainardi, A poster about the old history of fractional calculus. Fract. Calc. Appl. Anal. 13(4), 447–454 (2010)MathSciNetMATHGoogle Scholar
  238. 238.
    J.A. Tenreiro Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)MathSciNetMATHGoogle Scholar
  239. 239.
    J.A. Tenreiro Machado, V. Kiryakova, F. Mainardi, A poster about the old history of fractional calculus. Fract. Calc. Appl. Anal. 13(4), 447–454 (2010)MathSciNetMATHGoogle Scholar
  240. 240.
    J.C. Trigeassou, N. Maamri, J. Sabatier, A.A. Oustaloup, Lyapunov approach to the stability of fractional differential equations. Signal Process. 91, 437–445 (2011)MATHGoogle Scholar
  241. 241.
    L. Vzquez. From Newton’s equation to fractional diffusion and wave equations. Adv. Difference Equ. 2011, Art. ID 169421, 13 ppGoogle Scholar
  242. 242.
    A.N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side (Russian) Ukran. Mat. Zh. 47(4), 531–534 (1995); translation in Ukrainian Math. J. 47 (1995), no. 4, 617–621 (1996)Google Scholar
  243. 243.
    A.N. Vityuk, Existence of Solutions of partial differential inclusions of fractional order. Izv. Vyssh. Uchebn. Ser. Mat. 8, 13–19 (1997)MathSciNetGoogle Scholar
  244. 244.
    A.N. Vityuk, A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7(3), 318–325 (2004)MathSciNetGoogle Scholar
  245. 245.
    A.N. Vityuk, A.V. Golushkov, The Darboux problem for a differential equation containing a fractional derivative. Nonlinear Oscil. 8, 450–462 (2005)MathSciNetGoogle Scholar
  246. 246.
    A.N. Vityuk, A.V. Mykhailenko, On one class of differential quations of fractional order. Nonlinear Oscil. 11(3) (2008), 307–319MathSciNetGoogle Scholar
  247. 247.
    A.N. Vityuk, A.V. Mykhailenko, The Darboux problem for an implicit fractional-order differential equation. J. Math. Sci. 175(4), 391–401 (2011)Google Scholar
  248. 248.
    J. Wang, Y. Zhou, W. Wei, A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16(10), 4049–4059 (2011)MathSciNetMATHGoogle Scholar
  249. 249.
    C. Yu, G. Gao, Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26–29 (2005)MathSciNetMATHGoogle Scholar
  250. 250.
    G. Zaslavsky, in Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, New York, 2005)MATHGoogle Scholar
  251. 251.
    S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations. Electron. J. Differ. Equat. (36), 1–12 (2006)Google Scholar
  252. 252.
    S. Zhang, Existence of positive solutions of a singular partial differential equation. Math. Bohemica 133(1), 29–40 (2008)MATHGoogle Scholar
  253. 253.
    Y. Zhou, Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Syst. Differ. Equat. 1(4), 239–244 (2008)MATHGoogle Scholar
  254. 254.
    Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal. 71, 2724–2733 (2009)MathSciNetMATHGoogle Scholar
  255. 255.
    Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. 71, 3249–3256 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Saïd Abbas
    • 1
  • Mouffak Benchohra
    • 2
  • Gaston M. N’Guérékata
    • 3
  1. 1.Laboratoire de MathématiquesUniversité de SaïdaSaïdaAlgeria
  2. 2.Laboratoire de MathématiquesUniversité de Sidi Bel-AbbèsSidi Bel-AbbèsAlgeria
  3. 3.Department of MathematicsMorgan State UniversityBaltimoreUSA

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