Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations

Saïd Abbas; Mouffak Benchohra

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2012)

  • Volume: 51, Issue: 2, page 5-18
  • ISSN: 0231-9721

Abstract

top
In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.

How to cite

top

Abbas, Saïd, and Benchohra, Mouffak. "Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 51.2 (2012): 5-18. <http://eudml.org/doc/246875>.

@article{Abbas2012,
abstract = {In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.},
author = {Abbas, Saïd, Benchohra, Mouffak},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {partial hyperbolic differential equation; fractional order; left-sided mixed; Riemann–Liouville integral; mixed regularized derivative; impulse; upper solution; lower solution; fixed point; partial hyperbolic differential equation; fractional order; left-sided mixed; Riemann-Liouville integral; mixed regularized derivative; impulse; upper solution; lower solution; fixed point},
language = {eng},
number = {2},
pages = {5-18},
publisher = {Palacký University Olomouc},
title = {Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations},
url = {http://eudml.org/doc/246875},
volume = {51},
year = {2012},
}

TY - JOUR
AU - Abbas, Saïd
AU - Benchohra, Mouffak
TI - Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2012
PB - Palacký University Olomouc
VL - 51
IS - 2
SP - 5
EP - 18
AB - In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.
LA - eng
KW - partial hyperbolic differential equation; fractional order; left-sided mixed; Riemann–Liouville integral; mixed regularized derivative; impulse; upper solution; lower solution; fixed point; partial hyperbolic differential equation; fractional order; left-sided mixed; Riemann-Liouville integral; mixed regularized derivative; impulse; upper solution; lower solution; fixed point
UR - http://eudml.org/doc/246875
ER -

References

top
  1. Abbas, S., Agarwal, R. P., Benchohra, M., Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010), 818–829. (2010) Zbl1204.35171MR2680249
  2. Abbas, S., Benchohra, M., Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62–72. (2009) Zbl1178.35371MR2535015
  3. Abbas, S., Benchohra, M., Upper and lower solutions method for the Darboux problem for fractional order partial differential inclusions, Int. J. Modern Math. 5, 3 (2010), 327–338. (2010) Zbl1244.35157MR2779058
  4. Abbas, S., Benchohra, M., Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), 406–413. (2010) Zbl1202.35340MR2645856
  5. Abbas, S., Benchohra, M., 10.7151/dmdico.1116, Discuss. Math. Differ. Incl. Control Optim. 30, 1 (2010), 141–161. (2010) Zbl1203.26005MR2682404DOI10.7151/dmdico.1116
  6. Abbas, S., Benchohra, M., Górniewicz, L., Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative, Sci. Math. Jpn. e-2010 (2010), 271–282, online. (2010) Zbl1200.26004MR2666846
  7. Abbas, S., Benchohra, M., N’Guérékata, G. M., Topics in Fractional Differential Equations, Developments in Mathematics 27, Springer, New York, 2012. (2012) Zbl1273.35001MR2962045
  8. Abbas, S., Benchohra, M., Nieto, J. J., Global uniqueness results for fractional order partial hyperbolic functional differential equations, Adv. Differ. Equations ID 379876, doi:10.1155/2011/379876 (2011), 1–25, online. (2011) Zbl1217.35005MR2774251
  9. Abbas, S., Benchohra, M., Vityuk, A. N., On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal. 15, 2 (2012), 168–182. (2012) MR2897771
  10. Agarwal, R. P., O’Regan, D., Staněk, S., 10.1016/j.jmaa.2010.04.034, J. Math. Anal. Appl. 371 (2010), 57–68. (2010) Zbl1206.34009MR2660986DOI10.1016/j.jmaa.2010.04.034
  11. Benchohra, M., Ouahab, A., 10.1080/00036810601066350, Appl. Anal. 85 (2006), 1459–1470. (2006) Zbl1175.34080MR2282996DOI10.1080/00036810601066350
  12. Benchohra, M., Graef, J. R., Hamani, S., 10.1080/00036810802307579, Appl. Anal. 87, 7 (2008), 851–863. (2008) MR2458962DOI10.1080/00036810802307579
  13. Benchohra, M., Hamani, S., Ntouyas, S. K., Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1–12. (2008) Zbl1157.26301MR2390179
  14. Benchohra, M., Henderson, J., Ntouyas, S. K., Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol 2, New York, 2006. (2006) Zbl1130.34003MR2322133
  15. Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A., 10.1016/j.jmaa.2007.06.021, J. Math. Anal. Appl. 338 (2008), 1340–1350. (2008) MR2386501DOI10.1016/j.jmaa.2007.06.021
  16. Diethelm, K., Ford, N. J., 10.1006/jmaa.2000.7194, J. Math. Anal. Appl. 265 (2002), 229–248. (2002) Zbl1014.34003MR1876137DOI10.1006/jmaa.2000.7194
  17. Glockle, W. G., Nonnenmacher, T. F., 10.1016/S0006-3495(95)80157-8, Biophys. J. 68 (1995), 46–53. (1995) DOI10.1016/S0006-3495(95)80157-8
  18. Granas, A., Dugundji, J., Fixed Point Theory, Springer-Verlag, New York, 2003. (2003) Zbl1025.47002MR1987179
  19. Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. (2000) Zbl0998.26002MR1890104
  20. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006. (2006) Zbl1092.45003MR2218073
  21. Kilbas, A. A., Marzan, S. A., 10.1007/s10625-005-0137-y, Differential Equations 41 (2005), 84–89. (2005) Zbl1160.34301MR2213269DOI10.1007/s10625-005-0137-y
  22. Lakshmikantham, V., Bainov D. D., Simeonov, P. S., Theory of Impulsive Differntial Equations, World Scientific, Singapore, 1989. (1989) MR1082551
  23. Lakshmikantham, V., Pandit, S. G., 10.1016/0022-247X(85)90062-9, J. Math. Anal. Appl. 105 (1985), 466–477. (1985) Zbl0569.35056MR0778480DOI10.1016/0022-247X(85)90062-9
  24. Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, In: Carpinteri, A., Mainardi, F. (eds) Fractional Calculus in Continuum Mechanics Springer-Verlag, Wien, 1997, 291–348. (1997) MR1611587
  25. Metzler, F., Schick, W., Kilian, H. G., Nonnenmacher, T. F., 10.1063/1.470346, J. Chem. Phys. 103 (1995), 7180–7186. (1995) DOI10.1063/1.470346
  26. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. (1993) MR1219954
  27. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, London, 1974. (1974) Zbl0292.26011MR0361633
  28. Pandit, S. G., 10.1016/S0362-546X(96)00265-9, Nonlinear Anal. 30 (1997), 235–272. (1997) Zbl0892.35095MR1602940DOI10.1016/S0362-546X(96)00265-9
  29. 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 Dynam. 29 (2002), 281–296. (2002) MR1926477
  30. Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. (1993) Zbl0818.26003MR1347689
  31. Staněk, S., 10.1016/j.camwa.2011.04.048, Comput. Math. Appl. 62 (2011), 1379–1388. (2011) Zbl1228.34020MR2824725DOI10.1016/j.camwa.2011.04.048
  32. Vityuk, A. N., Existence of solutions of partial differential inclusions of fractional order, Izv. Vyssh. Uchebn., Ser. Mat. 8 (1997), 13–19. (1997) MR1485123
  33. Vityuk, A. N., Golushkov, A. V., 10.1007/s11072-005-0015-9, Nonlinear Oscil. 7, 3 (2004), 318–325. (2004) MR2151816DOI10.1007/s11072-005-0015-9
  34. Vityuk, A. N., Mykhailenko, A. V., 10.1007/s11072-009-0032-1, Nonlinear Oscil. 11, 3 (2008), 307–319. (2008) MR2512745DOI10.1007/s11072-009-0032-1
  35. Vityuk, A. N., Mykhailenko, A. V., 10.1007/s10958-011-0353-3, J. Math. Sci. 175, 4 (2011), 391–401. (2011) Zbl1279.35092MR2977138DOI10.1007/s10958-011-0353-3

Citations in EuDML Documents

top
  1. Saïd Abbas, Eman Alaidarous, Wafaa Albarakati, Mouffak Benchohra, Upper and lower solutions method for partial Hadamard fractional integral equations and inclusions
  2. Saïd Abbas, Mouffak Benchohra, Mohamed Abdalla Darwish, Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses
  3. Saïd Abbas, Mouffak Benchohra, Juan J. Nieto, Ulam Stabilities for Partial Impulsive Fractional Differential Equations

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.