Advertisement

Acta Applicandae Mathematicae

, Volume 109, Issue 3, pp 973–1033 | Cite as

A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

  • Ravi P. AgarwalEmail author
  • Mouffak Benchohra
  • Samira Hamani
Article

Abstract

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.

Keywords

Boundary value problem Initial value problems Differential equations and inclusions Caputo fractional derivative Fractional integral Existence Uniqueness Impulses Fixed point Selection Nonlocal conditions Integral conditions 

Mathematics Subject Classification (2000)

26A33 34A37 34A60 34B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adomian, G., Adomian, G.E.: Cellular systems and aging models. Comput. Math. Appl. 11, 283–291 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Agarwal, R.P., Belmekki, M., Benchohra, M.: Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative. Dyn. Continuos Discrete Impuls. Syst. (2008, to appear) Google Scholar
  3. 3.
    Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative (2008, submitted) Google Scholar
  4. 4.
    Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order. Adv. Stud. Contemp. Math. 16(2), 181–196 (2008) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for fractional differential equations. Georgian Math. J. (2008, to appear) Google Scholar
  6. 6.
    Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Ahmad, B., Nieto, J.J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. (2008). doi: 10.1016/j.na.2007.09.018 MathSciNetGoogle Scholar
  8. 8.
    Ait Dads, E., Benchohra, M., Hamani, S.: Impulsive fractional differential inclusions involving the Caputo Factional derivative. Fract. Calc. Appl. Anal. (2008, to appear) Google Scholar
  9. 9.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984) zbMATHGoogle Scholar
  10. 10.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990) zbMATHGoogle Scholar
  11. 11.
    Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions for N-term non-autonomous fractional differential equations. Positivity 9(2), 193–206 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients. Electron. J. Differ. Equ. 129, 12 (2006) MathSciNetGoogle Scholar
  13. 13.
    Bainov, D.D., Simeonov, P.S.: Systems with Impulsive Effect. Horwood, Chichester (1989) Google Scholar
  14. 14.
    Belarbi, A., Benchohra, M., Dhage, B.C.: Existence theory for perturbed boundary value problems with integral boundary conditions. Georgian Math. J. 13(2), 215–228 (2006) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Belarbi, A., Benchohra, M., Ouahab, A.: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Appl. Anal. 85, 1459–1470 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Belarbi, A., Benchohra, M., Hamani, S., Ntouyas, S.K.: Perturbed functional differential equations with fractional order. Commun. Appl. Anal. 11(3–4), 429–440 (2007) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Belmekki, M., Benchohra, M.: Existence results for Fractional order semilinear functional differential equations. Proc. A. Razmadze Math. Inst. 146, 9–20 (2008) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Belmekki, M., Benchohra, M., Gorniewicz, L.: Semilinear functional differential equations with fractional order and infinite delay. Fixed Point Theory 9(2), 423–439 (2008) zbMATHMathSciNetGoogle Scholar
  19. 19.
    Benchohra, M., Hamani, S.: Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative. Topol. Methods Nonlinear Anal. (2008, to appear) Google Scholar
  20. 20.
    Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order. Discuss. Math. Differ. Incl. Control Optim. (2008, to appear) Google Scholar
  21. 21.
    Benchohra, M., Slimani, B.A.: Impulsive fractional differential equations (2008, submitted) Google Scholar
  22. 22.
    Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi, New York (2006) zbMATHGoogle Scholar
  23. 23.
    Benchohra, M., Hamani, S., Henderson, J.: Functional differential inclusions with integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 15, 1–13 (2007) MathSciNetGoogle Scholar
  24. 24.
    Benchohra, M., Hamani, S., Nieto, J.J.: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. Rocky Mountain J. Math. (2008, to appear) Google Scholar
  25. 25.
    Benchohra, M., Hamani, S., Nieto, J.J., Slimani, B.A.: Existence results for differential inclusions with fractional order and impulses (2008, submitted) Google Scholar
  26. 26.
    Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87(7), 851–863 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Benchohra, M., Hamani, S., Ntouyas, S.K.: boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008) zbMATHMathSciNetGoogle Scholar
  28. 28.
    Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338(2), 1340–1350 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. Calc. Appl. Anal. 11(1), 35–56 (2008) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Blayneh, K.W.: Analysis of age structured host-parasitoid model. Far East J. Dyn. Syst. 4, 125–145 (2002) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Braverman, E., Zhukovskiy, S.: The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited. Nonlinear Anal.: Hybrid Syst. 2, 971–979 (2008) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values. Stud. Math. 90, 70–85 (1988) MathSciNetGoogle Scholar
  33. 33.
    Byszewski, L.: Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Byszewski, L.: Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem. In: Selected Problems of Mathematics. 50th Anniv. Cracow Univ. Technol. Anniv. Issue, vol. 6, pp. 25–33. Cracow Univ. Technol., Krakow (1995) Google Scholar
  35. 35.
    Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189, 23–31 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977) zbMATHGoogle Scholar
  38. 38.
    Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. (2008, in press) Google Scholar
  39. 39.
    Covitz, H., Nadler, S.B. Jr.: Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8, 5–11 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Daftardar-Gejji, V., Jafari, H.: Boundary value problems for fractional diffusion-wave equation. Aust. J. Math. Anal. Appl. 3(1), 8 (2006). Art. 16 (electronic) MathSciNetGoogle Scholar
  41. 41.
    Daftardar-Gejji, V., Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J. Math. Anal. Appl. 328(2), 1026–1033 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Dai, B., Su, H., Hu, D.: Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse. Nonlinear Anal. (2008, in press) Google Scholar
  43. 43.
    Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin (1992) zbMATHGoogle Scholar
  44. 44.
    Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Dhage, B.C.: Multivalued mappings and fixed points II. Tamkang J. Math. 37(1), 27–46 (2006) zbMATHMathSciNetGoogle Scholar
  46. 46.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999) Google Scholar
  48. 48.
    Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    El-Sayed, A.M.A.: Fractional order evolution equations. J. Fract. Calc. 7, 89–100 (1995) zbMATHMathSciNetGoogle Scholar
  50. 50.
    El-Sayed, A.M.A.: Fractional order diffusion-wave equations. Int. J. Theor. Phys. 35, 311–322 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    El-Sayed, A.M.A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Frigon, M., Granas, A.: Théorèmes d’existence pour des inclusions différentielles sans convexité. C. R. Acad. Sci. Paris, Ser. I 310, 819–822 (1990) zbMATHMathSciNetGoogle Scholar
  53. 53.
    Fryszkowski, A.: Fixed Point Theory for Decomposable Sets. Topological Fixed Point Theory and Its Applications, vol. 2. Kluwer Academic, Dordrecht (2004) zbMATHGoogle Scholar
  54. 54.
    Furati, K.M., Tatar, N.-E.: An existence result for a nonlocal fractional differential problem. J. Fract. Calc. 26, 43–51 (2004) zbMATHMathSciNetGoogle Scholar
  55. 55.
    Furati, K.M., Tatar, N.-E.: Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Fract. Calc. 28, 23–42 (2005) zbMATHMathSciNetGoogle Scholar
  56. 56.
    Gaul, L., Klein, P., Kempfle, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991) CrossRefGoogle Scholar
  57. 57.
    Glockle, W.G., Nonnenmacher, T.F.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995) CrossRefGoogle Scholar
  58. 58.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003) zbMATHGoogle Scholar
  59. 59.
    Guo, H., Chen, L.: Time-limited pest control of a Lotka-Volterra model with impulsive harvest. Nonlinear Anal.: Real World Appl. (2008). doi: 10.1016/j.nonrwa.2007.11.007 MathSciNetGoogle Scholar
  60. 60.
    Hernandez, E., Henriquez, H.R., McKibben, M.A.: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. (2008). doi: 10.1016/j.na.2008.03.062 Google Scholar
  61. 61.
    Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–772 (2006) CrossRefGoogle Scholar
  62. 62.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) zbMATHGoogle Scholar
  63. 63.
    Hu, Sh., Papageorgiou, N.: Handbook of Multivalued Analysis, Theory, vol. I. Kluwer, Dordrecht (1997) Google Scholar
  64. 64.
    Jiang, G., Lu, Q., Qian, L.: Chaos and its control in an impulsive differential system. Chaos, Solitons Fractals 34, 1135–1147 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Jiang, G., Lu, Q., Qian, L.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos, Solitons Fractals 31, 448–461 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Kaufmann, E.R., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 3, 11 (2007) Google Scholar
  67. 67.
    Kilbas, A.A., Marzan, S.A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differ. Equ. 41, 84–89 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) zbMATHCrossRefGoogle Scholar
  69. 69.
    Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, vol. 301. Longman, Harlow (1994). Copublished in the United States with Wiley, New York (1994) zbMATHGoogle Scholar
  70. 70.
    Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991) Google Scholar
  71. 71.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) zbMATHGoogle Scholar
  72. 72.
    Luo, Z., Nieto, J.J.: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. (2008). doi: 10.1016/j.na.2008.03.004 Google Scholar
  73. 73.
    Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997) Google Scholar
  74. 74.
    Metzler, F., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995) CrossRefGoogle Scholar
  75. 75.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) zbMATHGoogle Scholar
  76. 76.
    Mohamad, S., Gopalsamy, K., Akca, H.: Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear Anal.: Real World Appl. 9, 872–888 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  77. 77.
    Momani, S.M., Hadid, S.B.: Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24, 37–44 (2003) zbMATHMathSciNetGoogle Scholar
  78. 78.
    Momani, S.M., Hadid, S.B., Alawenh, Z.M.: Some analytical properties of solutions of differential equations of noninteger order. Int. J. Math. Math. Sci. 2004, 697–701 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) zbMATHGoogle Scholar
  80. 80.
    Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69(11), 3877–3896 (2007). MathSciNetGoogle Scholar
  81. 81.
    Pei, Y., Li, C., Chen, L., Wang, C.: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators. Adv. Complex Syst. 8, 483–495 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999) Google Scholar
  83. 83.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus Appl. Anal. 5, 367–386 (2002) zbMATHMathSciNetGoogle Scholar
  84. 84.
    Podlubny, I., Petraš, I., Vinagre, B.M., O’Leary, P., Dorčak, L.: Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dyn. 29, 281–296 (2002) zbMATHCrossRefGoogle Scholar
  85. 85.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon (1993) zbMATHGoogle Scholar
  86. 86.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995) zbMATHGoogle Scholar
  87. 87.
    Shen, J.H., Li, J.L.: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal.: Real World (2007). doi: 110.1016/j.nonrwa.2007.08.026 Google Scholar
  88. 88.
    Smart, D.R.: Fixed Point Theorems, vol. 66. Cambridge University Press, Cambridge (1980) Google Scholar
  89. 89.
    Wang, W.B., Shen, J.H., Nieto, J.J.: Permanence periodic solution of predator prey system with Holling type functional response and impulses. Discrete Dyn. Nat. Soc. (2007). doi: 10.1155/2007/81756 MathSciNetGoogle Scholar
  90. 90.
    Wei, C., Chen, L.: A delayed epidemic model with pulse vaccination. Discrete Dyn. Nat. Soc. (2008). Article ID 746951, 13 pages Google Scholar
  91. 91.
    Xia, Y.: Positive periodic solutions for a neutral impulsive delayed Lotka-Volterra competition system with the effect of toxic substance. Nonlinear Anal.: Real World Appl. 8, 204–221 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  92. 92.
    Yan, J., Zhao, A., Nieto, J.J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Math. Comput. Model. 40, 509–518 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  93. 93.
    Yu, C., Gao, G.: Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26–29 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  94. 94.
    Zeng, G.Z., Wang, F.Y., Nieto, J.J.: Complexity of delayed predator-prey model with impulsive harvest and Holling type-II functional response. Adv. Complex Syst. 11, 77–97 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  95. 95.
    Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1–12 (2006) CrossRefGoogle Scholar
  96. 96.
    Zhang, H., Chen, L.S., Nieto, J.J.: A delayed epidemic model with stage structure and pulses for management strategy. Nonlinear Anal.: Real World (2007). doi: 10.1016/j.nonrwa.2007.05.004 Google Scholar
  97. 97.
    Zhang, H., Xu, W., Chen, L.: A impulsive infective transmission SI model for pest control. Math. Methods Appl. Sci. 30, 1169–1184 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  98. 98.
    Zhou, J., Xiang, L., Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Phys. A: Stat. Mech. Appl. 384, 684–692 (2007) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
    Email author
  • Mouffak Benchohra
    • 2
  • Samira Hamani
    • 2
  1. 1.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  2. 2.Laboratoire de MathématiquesUniversité de Sidi Bel-AbbèsSidi Bel-AbbèsAlgérie

Personalised recommendations