Fractional integro-differential inclusions with state-dependent delay

Khalida Aissani; Mouffak Benchohra; Khalil Ezzinbi

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

  • Volume: 34, Issue: 2, page 153-167
  • ISSN: 1509-9407

Abstract

top
In this paper, we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay. The techniques rely on fractional calculus, multivalued mapping on a bounded set and Bohnenblust-Karlin's fixed point theorem. Finally, we present an example to illustrate the theory.

How to cite

top

Khalida Aissani, Mouffak Benchohra, and Khalil Ezzinbi. "Fractional integro-differential inclusions with state-dependent delay." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 34.2 (2014): 153-167. <http://eudml.org/doc/270531>.

@article{KhalidaAissani2014,
abstract = {In this paper, we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay. The techniques rely on fractional calculus, multivalued mapping on a bounded set and Bohnenblust-Karlin's fixed point theorem. Finally, we present an example to illustrate the theory.},
author = {Khalida Aissani, Mouffak Benchohra, Khalil Ezzinbi},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {fractional integro-differential inclusions; Caputo fractional derivative; mild solution; multivalued map; Bohnenblust-Karlin's fixed point; state-dependent delay},
language = {eng},
number = {2},
pages = {153-167},
title = {Fractional integro-differential inclusions with state-dependent delay},
url = {http://eudml.org/doc/270531},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Khalida Aissani
AU - Mouffak Benchohra
AU - Khalil Ezzinbi
TI - Fractional integro-differential inclusions with state-dependent delay
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2014
VL - 34
IS - 2
SP - 153
EP - 167
AB - In this paper, we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay. The techniques rely on fractional calculus, multivalued mapping on a bounded set and Bohnenblust-Karlin's fixed point theorem. Finally, we present an example to illustrate the theory.
LA - eng
KW - fractional integro-differential inclusions; Caputo fractional derivative; mild solution; multivalued map; Bohnenblust-Karlin's fixed point; state-dependent delay
UR - http://eudml.org/doc/270531
ER -

References

top
  1. [1] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations (Springer, New York, 2012). doi: 10.1007/978-1-4614-4036-9 
  2. [2] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations (Nova Science Publishers, New York, 2014). 
  3. [3] R. Agarwal, B. de Andrade, and G. Siracusa, On fractional integro-differential equations with state-dependent delay, Comput. Math. Appl. 62 (2011) 1143-1149. doi: 10.1016/j.camwa.2011.02.033 Zbl1228.35262
  4. [4] R.P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equat. 2009 (2009) Article ID 981728, 1-47. Zbl1182.34103
  5. [5] R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010) 973-1033. doi: 10.1007/s10440-008-9356-6 Zbl1198.26004
  6. [6] W.G. Aiello, H.I. Freedman and J. Wu, Analysis of a model representing stagestructured population growth with state-dependent time delay, SIAM J. Appl. Math. 52 (3) (1992) 855-869. doi: 10.1137/0152048 Zbl0760.92018
  7. [7] K. Aissani and M. Benchohra, Semilinear fractional order integro-differential equations with infinite delay in Banach spaces, Arch. Math. 49 (2013) 105-117. doi: 10.5817/AM2013-2-105 Zbl1299.26008
  8. [8] K. Aissani and M. Benchohra, Existence results for fractional integro-differential equations with state-dependent delay, Adv. Dyn. Syst. Appl. 9 (1) (2014) 17-30. 
  9. [9] K. Aissani and M. Benchohra, Impulsive fractional differential inclusions with infinite delay, Electron. J. Differ. Eq. 2013 (265) 1-13. Zbl1295.34084
  10. [10] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific Publishing, New York, 2012). Zbl1248.26011
  11. [11] M. Benchohra, K. Ezzinbi and S. Litimein, The existence and controllability results for fractional order integro-differential inclusions in Fréchet spaces, Proceedings A. Razm. Math. Inst. 162 (2013) 1-23. Zbl1306.34096
  12. [12] M. Benchohra, J. Henderson, S. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008) 1340-1350. doi: 10.1016/j.jmaa.2007.06.021 Zbl1209.34096
  13. [13] M. Benchohra and S. Litimein, Fractional integro-differential equations with state-dependent delay on an unbounded domain, Afr. Diaspora J. Math. 12 (2) (2011) 13-25. Zbl1244.34098
  14. [14] M. Benchohra, S. Litimein, J.J. Trujillo and M.P. Velasco, Abstract fractional integro-differential equations with state-dependent delay, Int. J. Evol. Equat. 6 (2) (2012) 25-38. Zbl1263.26013
  15. [15] H.F. Bohnenblust and S. Karlin, On a theorem of Ville. Contribution to the theory of games, Ann. Math. Stud. No. 24, Princeton Univ. (1950) 155-160. Zbl0041.25701
  16. [16] A. Cernea, On the existence of mild solutions for nonconvex fractional semilinear differential inclusions, Electron. J. Qual. Theory Differ. Eq. 2012 (64) (2012) 1-15. Zbl06476214
  17. [17] A. Cernea, A note on mild solutions for nonconvex fractional semilinear differential inclusion, Ann. Acad. Rom. Sci. Ser. Math. Appl. 5 (2013) 35-45. Zbl1284.34025
  18. [18] L. Debnath and D. Bhatta, Integral Transforms and Their Applications (Second Edition) (CRC Press, 2007). Zbl1113.44001
  19. [19] K. Deimling, Multivalued Differential Equations (Walter De Gruyter, Berlin-New York, 1992). doi: 10.1515/9783110874228 Zbl0760.34002
  20. [20] K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Berlin, 2010). Zbl1215.34001
  21. [21] J.P.C. dos Santos, C. Cuevas and B. de Andrade, Existence results for a fractional equation with state-dependent delay, Adv. Differ. Eq. 2011 (2011), Article ID 642013, 15 pages. Zbl1216.45003
  22. [22] R.D. Driver, A neutral system with state-dependent delay, J. Differ. Eq. 54 (1) (1984) 73-86. doi: 10.1016/0022-0396(84)90143-8 
  23. [23] M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002) 433-440. doi: 10.1016/S0960-0779(01)00208-9 
  24. [24] A.M.A. El-Sayed and A.G. Ibrahim, Multivalued fractional differential equations of arbitrary orders, Appl. Math. Comput. 68 (1995) 15-25. doi: 10.1016/0096-3003(94)00080-N 
  25. [25] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495 (Kluwer Academic Publishers, Dordrecht, 1999). doi: 10.1007/978-94-015-9195-9 Zbl0937.55001
  26. [26] J.K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funk. Ekvacioj 21 (1978) 11-41. Zbl0383.34055
  27. [27] F. Hartung, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study, Nonlinear Anal. TMA 47 (7) (2001) 4557-4566. doi: 10.1016/S0362-546X(01)00569-7 Zbl1042.34582
  28. [28] F. Hartung, and J. Turi, Identification of parameters in delay equations with state-dependent delays, Nonlinear Anal. TMA 29 (11) (1997) 1303-1318. doi: 10.1016/S0362-546X(96)00100-9 Zbl0894.34071
  29. [29] F. Hartung, T.L. Herdman and J. Turi, Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear Anal. TMA 39 (3) (2000) 305-325. doi: 10.1016/S0362-546X(98)00169-2 Zbl0955.34067
  30. [30] E. Hernández and M.A. McKibben, On state-dependent delay partial neutral functional-differential equations, Appl. Math. Comput. 186 (1) (2007) 294-301. doi: 10.1016/j.amc.2006.07.103 Zbl1119.35106
  31. [31] E. Hernández, M.A. McKibben and H.R. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Mod. 49 (2009) 1260-1267. doi: 10.1016/j.mcm.2008.07.011 Zbl1165.34420
  32. [32] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA 7 (2006) 510-519. doi: 10.1016/j.nonrwa.2005.03.014 Zbl1109.34060
  33. [33] R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000). Zbl0998.26002
  34. [34] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Unbounded Delay (Springer-Verlag, Berlin, 1991). Zbl0732.34051
  35. [35] A.A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Science B.V., Amsterdam, 2006). Zbl1092.45003
  36. [36] A. Lasota and 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 (1965) 781-786. Zbl0151.10703
  37. [37] F. Li and J. Zhang, Existence of mild solutions to fractional integrodifferential equations of neutral type with infinite delay, Adv. Diff. Equat. 2011 (2011), Article ID 963463, 1-15. Zbl1213.45008
  38. [38] F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, in: Econophysics: An Emerging Science, J. Kertesz and I. Kondor, Eds. (Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000). 
  39. [39] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1993). 
  40. [40] A.V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl. 326 (2) (2007) 1031-1045. doi: 10.1016/j.jmaa.2006.03.049 Zbl1178.35370
  41. [41] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Heidelberg; Higher Education Press, Beijing, 2010). Zbl1214.81004

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.