Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

• Autorzy: Benchohra Mouffak , Bouriah Soufyane , Nieto Juan J.

Identyfikatory

DOI

10.1515/dema-2019-0032

• Języki publikacji: EN

Abstrakty

EN

In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

Słowa kluczowe

EN

Riemann-Liouville fractional derivative; initial value problem; existence; uniqueness; Ulam stability; fixed point theorem

PL

pochodna ułamkowa Riemanna-Liouville'a; zagadnienie początkowe; istnienie; niepowtarzalność; stabilność Ulama; twierdzenie o punkcie stałym

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 437--450
• Opis fizyczny: Bibliogr. 29 poz.

Twórcy

autor

Benchohra Mouffak

• Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89 Sidi Bel Abbes 22000, Algeria
• Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

autor

Bouriah Soufyane

• Department of Mathematics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University, P.O. Box 151 Chlef 02000, Algeria

autor

Nieto Juan J.

• Departamento de Estatistica, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

Bibliografia

• [1] Capelas de Oliveira E., Vanterler da C. Sousa J., Leibniz type rule: ψ-Hilfer fractional operator, Comm. Nonl. Sci. Numer. Simul., 2019, 77, 305-311
• [2] Capelas de Oliveira E., Vanterler da C. Sousa J., Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Let., 2018, 81, 50-56
• [3] Bai Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 2010, 72, 916-924
• [4] Bai Z., Chen Y., Sun S., Lian H., On the existence of blow up solutions for a class of fractional differential equations, Frac. Calc. Anal., 2014, 17, 1175-1187
• [5] Bai Z., Lu H., Positive solutions of boundary value problems of nonlinear fractional differential equation, J. Math. Anal. Appl., 2005, 311, 495-505
• [6] Kilbas A. A., Srivastava Hari M., Trujillo Juan J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006
• [7] Samko S. G., Kilbas A. A., Marichev O.nI., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993
• [8] Abbas S., Benchohra M., Graef J. R., Henderson J., Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018
• [9] Abbas S., Benchohra M., N’Guérékata G. M., Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012
• [10] Abbas S., Benchohra M., N’Guérékata G. M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014
• [11] Ahmad B., Alsaedi A., Ntouyas S. K., Tariboon J., Hadamard-type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017
• [12] Zhou Y., Wang J.-R., Zhang L., Basic Theory of Fractional Differential Equations, Second edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017
• [13] Abbas S., Benchohra M., Bouriah S., Nieto J. J., Periodic solutions for nonlinear fractional differential systems, Differ. Equ. Appl., 2018, 10(3), 299–316
• [14] Ahmad B., Nieto J. J., Existence results for nonlinear boundary value problems of fractional integro-differential equations with integral boundary conditions, Boundary Value Problems, 2009, Article ID 708576
• [15] Benchohra M., Bouriah S., Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure. Appl. Anal., 2015, 1, 22-36
• [16] Benchohra M., Bouriah S., Existence and stability results for neutral functional differential equations of fractional order with delay, Dyn. Contin., Discrete Impul. Syst. Series A: Math. Anal., 2016, 23, 295-30
• [17] Benchohra M., Bouriah S., Darwish M. A., Nonlinear boundary value problem for implicit differential equations of fractional order in Banach spaces, Fixed Point Theory, 2017, 18, 457-470
• [18] Benchohra M., Bouriah S., Graef J. R., Boundary value problems for nonlinear implicit Caputo-Hadamard type fractional differential equations with impulses, Mediterr. J. Math., 2017, 14:206
• [19] Benchohra M., Bouriah S., Graef J. R., Nonlinear implicit differential equations of fractional order at resonance, Electron. J. Differential Equations, 2016, 2016(324), 1-10
• [20] Benchohra M., Bouriah S., Henderson J., Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses, Comm. Appl. Nonlinear Anal., 2015, 22(1), 46-67
• [21] Benchohra M., Lazreg J. E., Nonlinear fractional implicit differential equations, Commun. Appl. Anal., 2013, 17, 471-482
• [22] Benchohra M., Lazreg J. E., Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions, Rom. J. Math. Comput. Sc., 2014, 4, 60-72
• [23] Bai Z., Zhang S., Sun S., Yin C., Monotone iterative method for fractional differential equations, Electron. J. Differential Equations, 2016, 2016(06), 1-8
• [24] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999
• [25] Furati K. M., Kassim M. D., Tahar N.-E., Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 2012, 64, 1616-1626
• [26] Ye H., Gao J., Ding Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 2007, 328, 1075-1081
• [27] Rus I. A., Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 2010, 26, 103-107
• [28] Wei W., Xiang X., Peng Y., Nonlinear impulsive integro-differential equations of mixed type and optimal controls, Optimization, 2006, 55, 141-156
• [29] Granas A., Dugundji J., Fixed Point Theory, Springer-Verlag, New York, 2003

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).