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

Mouffak Benchohra 1 , Soufyane Bouriah 2 ,  and Juan J. Nieto 3
  • 1 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, Department of Mathematics, College of Science, King Saud University, 22000, Sidi Bel Abbes, Algeria
  • 2 Department of Mathematics Faculty of Exact Sciences, Informatics Hassiba Benbouali University, 02000, Algeria
  • 3 Departamento de Estatistica, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

Abstract

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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 integrodifferential 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, 295–307

  • [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

OPEN ACCESS

Journal + Issues

Demonstratio Mathematica, founded in 1969, is a fully peer-reviewed, open access journal that publishes original and significant research works and review articles devoted to functional analysis, approximation theory, and related topics. The journal provides the readers with free, instant, and permanent access to all content worldwide (all 53 volumes are available online!)

Search