Existence of solutions of generalized fractional differential equation with nonlocal initial condition

Sandeep P. Bhairat; Dnyanoba-Bhaurao Dhaigude

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 2, page 203-220
  • ISSN: 0862-7959

Abstract

top
This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.

How to cite

top

Bhairat, Sandeep P., and Dhaigude, Dnyanoba-Bhaurao. "Existence of solutions of generalized fractional differential equation with nonlocal initial condition." Mathematica Bohemica 144.2 (2019): 203-220. <http://eudml.org/doc/294625>.

@article{Bhairat2019,
abstract = {This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.},
author = {Bhairat, Sandeep P., Dhaigude, Dnyanoba-Bhaurao},
journal = {Mathematica Bohemica},
keywords = {fractional derivative; fractional integral; existence of solution; fractional differential equation; fixed point theorem},
language = {eng},
number = {2},
pages = {203-220},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of solutions of generalized fractional differential equation with nonlocal initial condition},
url = {http://eudml.org/doc/294625},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Bhairat, Sandeep P.
AU - Dhaigude, Dnyanoba-Bhaurao
TI - Existence of solutions of generalized fractional differential equation with nonlocal initial condition
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 2
SP - 203
EP - 220
AB - This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.
LA - eng
KW - fractional derivative; fractional integral; existence of solution; fractional differential equation; fixed point theorem
UR - http://eudml.org/doc/294625
ER -

References

top
  1. Abbas, S., Benchohra, M., Lagreg, J.-E., Zhou, Y., 10.1016/j.chaos.2017.03.010, Chaos Solitons Fractals 102 (2017), 47-71. (2017) Zbl1374.34004MR3671994DOI10.1016/j.chaos.2017.03.010
  2. Agarwal, R. P., Benchohra, M., Hamani, S., 10.1007/s10440-008-9356-6, Acta. Appl. Math. 109 (2010), 973-1033. (2010) Zbl1198.26004MR2596185DOI10.1007/s10440-008-9356-6
  3. Bagley, R. L., Torvik, P. J., 10.2514/3.8142, AIAA J. 21 (1983), 741-748. (1983) Zbl0514.73048DOI10.2514/3.8142
  4. Bagley, R. L., Torvik, P. J., 10.1122/1.549724, J. Rheol. 27 (1983), 201-210. (1983) Zbl0515.76012DOI10.1122/1.549724
  5. Bagley, R. L., Torvik, P. J., 10.1115/1.3167615, J. Appl. Mech. 51 (1984), 294-298. (1984) Zbl1203.74022DOI10.1115/1.3167615
  6. Bhairat, S. P., New approach to existence of solution of weighted Cauchy-type problem, Available at http://arxiv.org/abs/1808.03067 (2018), 10 pages. (2018) 
  7. Bhairat, S. P., Dhaigude, D. B., Existence and stability of fractional differential equations involving generalized Katugampola derivative, Available at https://arxiv.org/ abs/1709.08838 (2017), 15 pages. (2017) 
  8. Chitalkar-Dhaigude, C. D., Bhairat, S. P., Dhaigude, D. B., Solution of fractional differential equations involving Hilfer fractional derivative: Method of successive approximations, Bull. Marathwada Math. Soc. 18 (2017), 1-13. (2017) 
  9. Dhaigude, D. B., Bhairat, S. P., Existence and continuation of solutions of Hilfer fractional differential equations, Available at http://arxiv.org/abs/1704.02462v1 (2017), 18 pages. (2017) MR3820828
  10. Dhaigude, D. B., Bhairat, S. P., Existence and uniqueness of solution of Cauchy-type problem for Hilfer fractional differential equations, Commun. Appl. Anal. 22 (2017), 121-134. (2017) MR3820828
  11. Dhaigude, D. B., Bhairat, S. P., On existence and approximation of solution of nonlinear Hilfer fractional differential equations, (to appear) in Int. J. Pure Appl. Math. Available at http://arxiv.org/abs/1704.02464 (2017), 9 pages. (2017) MR3820828
  12. Dhaigude, D. B., Bhairat, S. P., Local existence and uniqueness of solutions for Hilfer-Hadamard fractional differential problem, Nonlinear Dyn. Syst. Theory 18 (2018), 144-153. (2018) MR3820828
  13. Dhaigude, D. B., Bhairat, S. P., 10.22457/pindac.v6n1a4, Progress in Nonlinear Dynamics and Chaos 6 (2018), 29-38. (2018) DOI10.22457/pindac.v6n1a4
  14. Furati, K. M., Kassim, M. D., Tatar, N.-E., 10.1016/j.camwa.2012.01.009, Comput. Math. Appl. 64 (2012), 1616-1626. (2012) Zbl1268.34013MR2960788DOI10.1016/j.camwa.2012.01.009
  15. Furati, K. M., Tatar, N.-E., An existence result for a nonlocal fractional differential problem, J. Fractional Calc. 26 (2004), 43-51. (2004) Zbl1101.34001MR2096756
  16. Gaafar, F. M., 10.1016/j.joems.2013.12.008, J. Egypt. Math. Soc. 22 (2014), 341-347. (2014) Zbl1306.34007MR3260773DOI10.1016/j.joems.2013.12.008
  17. Hilfer, R., 10.1142/9789812817747, World Scientific, London (2000). (2000) Zbl0998.26002MR1890104DOI10.1142/9789812817747
  18. Hilfer, R., 10.1016/S0301-0104(02)00670-5, Chemical Physics 284 (2002), 399-408. (2002) MR1890106DOI10.1016/S0301-0104(02)00670-5
  19. Kassim, M. D., Tatar, N.-E., 10.1155/2013/605029, Abstr. Appl. Anal. 2013 (2013), Article ID 605029, 12 pages. (2013) MR3139483DOI10.1155/2013/605029
  20. Katugampola, U. N., 10.1016/j.amc.2011.03.062, Appl. Math. Comput. 218 (2011), 860-865. (2011) Zbl1231.26008MR2831322DOI10.1016/j.amc.2011.03.062
  21. Katugampola, U. N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1-15. (2014) Zbl1317.26008MR3298307
  22. Katugampola, U. N., Existence and uniqueness results for a class of generalized fractional differenital equations, Available at https://arxiv.org/abs/1411.5229 (2016). (2016) 
  23. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., 10.1016/s0304-0208(06)x8001-5, North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). (2006) Zbl1092.45003MR2218073DOI10.1016/s0304-0208(06)x8001-5
  24. Mainardi, F., 10.1142/9781848163300, World Scientific, Hackensack (2010). (2010) Zbl1210.26004MR2676137DOI10.1142/9781848163300
  25. Oliveira, D. S., Oliveira, E. Capelas de, Hilfer-Katugampola fractional derivative, Available at https://arxiv.org/abs/1705.07733 (2017). (2017) MR3826051
  26. Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198. Academic Press, San Diego (1999). (1999) Zbl0924.34008MR1658022
  27. Salamooni, A. Y. A., Pawar, D. D., Hilfer-Hadamard-type fractional differential equation with Cauchy-type problem, Available at https://arxiv.org/abs/1802.07483 (2018), 18 pages. (2018) 
  28. Wang, J., Zhang, Y., 10.1016/j.amc.2015.05.144, Appl. Math. Comput. 266 (2015), 850-859. (2015) MR3377602DOI10.1016/j.amc.2015.05.144

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.