Existence of solutions of generalized fractional differential equation with nonlocal initial condition. (English) Zbl 07088846

The authors consider the IVP of the Katugampola fractional differential equation of order \(\alpha\in(0,1)\) and type \(\beta\in[0,1]\) having nonlocal initial conditions. The initial condition involves the Katugampola fractional integral with \(\rho >0\). The equation is given by \[(^\rho D^{\alpha,\beta}_{a+}x)(t)=f(t,x(t)),\] \[(^\rho I^{1-\gamma}_{a+}x)({a+})=\sum_{j=1}^{m}\eta_jx(\xi_j),\] where \((^\rho D^{\alpha,\beta}_{a+})\) is the generalized Katugampola fractional derivative of order \(\alpha\in(0,1)\), and \((^\rho I^{1-\gamma}_{a+})\) is the Katugampola fractional integral with \(\rho >0\), and is denoted by NIVP.
The authors study the existence of solution for the above problem using (i) Krasnosel’skii fixed point theorem and (ii) Schauder fixed point theorem. To achieve their goal first they establish an equivalence between the NIVP and a mixed type nonlinear Voltera integral equation given by \[x(t)=\frac{K}{\Gamma(\alpha)}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\gamma -1}\sum_{j=1}^{m}n_j\int_{a}^{\xi_j}s^{\rho -1}\left(\frac{{\xi}^{\rho}-s^{\rho}}{\rho}\right)^{\alpha -1}f(s,x(s))ds\] \[+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho -1}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{\alpha -1}f(s,x(s))ds.\] They illustrate their results through examples.


34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47H10 Fixed-point theorems
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[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
[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
[3] Bagley, R. L.; Torvik, P. J., A different approach to the analysis of viscoelastically damped structures, AIAA J. 21 (1983), 741-748
[4] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. 27 (1983), 201-210
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[15] Furati, K. M.; Tatar, N.-E., An existence result for a nonlocal fractional differential problem, J. Fractional Calc. 26 (2004), 43-51
[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
[17] Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, London (2000)
[18] Hilfer, R., Experimental evidence for fractional time evolution in glass forming materials, Chemical Physics 284 (2002), 399-408
[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
[20] Katugampola, U. N., New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), 860-865
[21] Katugampola, U. N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1-15
[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)
[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)
[24] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelastisity. An Introduction to Mathematical Models, World Scientific, Hackensack (2010)
[25] Oliveira, D. S.; Oliveira, E. Capelas de, Hilfer-Katugampola fractional derivative, Available at https://arxiv.org/abs/1705.07733 (2017)
[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)
[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
[28] Wang, J.; Zhang, Y., Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850-859
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