| Title:
|
Existence of solutions of generalized fractional differential equation with nonlocal initial condition (English) |
| Author:
|
Bhairat, Sandeep P. |
| Author:
|
Dhaigude, Dnyanoba-Bhaurao |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
203-220 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
fractional derivative |
| Keyword:
|
fractional integral |
| Keyword:
|
existence of solution |
| Keyword:
|
fractional differential equation |
| Keyword:
|
fixed point theorem |
| MSC:
|
26A33 |
| MSC:
|
34A08 |
| MSC:
|
34A12 |
| MSC:
|
47H10 |
| idZBL:
|
Zbl 07088846 |
| idMR:
|
MR3974188 |
| DOI:
|
10.21136/MB.2018.0135-17 |
| . |
| Date available:
|
2019-06-21T11:35:30Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147760 |
| . |
| Reference:
|
[1] Abbas, S., Benchohra, M., Lagreg, J.-E., Zhou, Y.: A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability.Chaos Solitons Fractals 102 (2017), 47-71. Zbl 1374.34004, MR 3671994, 10.1016/j.chaos.2017.03.010 |
| Reference:
|
[2] Agarwal, R. P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions.Acta. Appl. Math. 109 (2010), 973-1033. Zbl 1198.26004, MR 2596185, 10.1007/s10440-008-9356-6 |
| Reference:
|
[3] Bagley, R. L., Torvik, P. J.: A different approach to the analysis of viscoelastically damped structures.AIAA J. 21 (1983), 741-748. Zbl 0514.73048, 10.2514/3.8142 |
| Reference:
|
[4] Bagley, R. L., Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity.J. Rheol. 27 (1983), 201-210. Zbl 0515.76012, 10.1122/1.549724 |
| Reference:
|
[5] Bagley, R. L., Torvik, P. J.: On the appearance of the fractional derivative in the behavior of real material.J. Appl. Mech. 51 (1984), 294-298. Zbl 1203.74022, 10.1115/1.3167615 |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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. MR 3820828 |
| Reference:
|
[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. MR 3820828 |
| Reference:
|
[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. MR 3820828 |
| Reference:
|
[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. MR 3820828 |
| Reference:
|
[13] Dhaigude, D. B., Bhairat, S. P.: Ulam stability for system of nonlinear implicit fractional differential equations.Progress in Nonlinear Dynamics and Chaos 6 (2018), 29-38. 10.22457/pindac.v6n1a4 |
| Reference:
|
[14] Furati, K. M., Kassim, M. D., Tatar, N.-E.: Existence and uniqueness for a problem involving Hilfer fractional derivative.Comput. Math. Appl. 64 (2012), 1616-1626. Zbl 1268.34013, MR 2960788, 10.1016/j.camwa.2012.01.009 |
| Reference:
|
[15] Furati, K. M., Tatar, N.-E.: An existence result for a nonlocal fractional differential problem.J. Fractional Calc. 26 (2004), 43-51. Zbl 1101.34001, MR 2096756 |
| Reference:
|
[16] Gaafar, F. M.: Continuous and integrable solutions of a nonlinear Cauchy problem of fractional order with nonlocal conditions.J. Egypt. Math. Soc. 22 (2014), 341-347. Zbl 1306.34007, MR 3260773, 10.1016/j.joems.2013.12.008 |
| Reference:
|
[17] Hilfer, R.: Applications of Fractional Calculus in Physics.World Scientific, London (2000). Zbl 0998.26002, MR 1890104, 10.1142/9789812817747 |
| Reference:
|
[18] Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials.Chemical Physics 284 (2002), 399-408. MR 1890106, 10.1016/S0301-0104(02)00670-5 |
| Reference:
|
[19] Kassim, M. D., Tatar, N.-E.: Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative.Abstr. Appl. Anal. 2013 (2013), Article ID 605029, 12 pages. MR 3139483, 10.1155/2013/605029 |
| Reference:
|
[20] Katugampola, U. N.: New approach to a generalized fractional integral.Appl. Math. Comput. 218 (2011), 860-865. Zbl 1231.26008, MR 2831322, 10.1016/j.amc.2011.03.062 |
| Reference:
|
[21] Katugampola, U. N.: A new approach to generalized fractional derivatives.Bull. Math. Anal. Appl. 6 (2014), 1-15. Zbl 1317.26008, MR 3298307 |
| Reference:
|
[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). |
| Reference:
|
[23] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). Zbl 1092.45003, MR 2218073, 10.1016/s0304-0208(06)x8001-5 |
| Reference:
|
[24] Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelastisity. An Introduction to Mathematical Models.World Scientific, Hackensack (2010). Zbl 1210.26004, MR 2676137, 10.1142/9781848163300 |
| Reference:
|
[25] Oliveira, D. S., Oliveira, E. Capelas de: Hilfer-Katugampola fractional derivative.Available at https://arxiv.org/abs/1705.07733 (2017). MR 3826051 |
| Reference:
|
[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). Zbl 0924.34008, MR 1658022 |
| Reference:
|
[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. |
| Reference:
|
[28] Wang, J., Zhang, Y.: Nonlocal initial value problems for differential equations with Hilfer fractional derivative.Appl. Math. Comput. 266 (2015), 850-859. MR 3377602, 10.1016/j.amc.2015.05.144 |
| . |
