Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type

  • 1 Department of Mathematics, University of Peshawar, 25000, Peshawar, Pakistan
  • 2 Department of Mathematics and General Sciences, Prince Sultan University, 11586, Riyadh, Saudi Arabia

Abstract

In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.

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  • [1] Agarwal R. P., Zhou Y., He Y., Existence of fractional neutral functional differential equations, Comput.Math. Appl., 2010, 59, 1095–1100

  • [2] Ahmad N., Ali Z., Shah K., Zada A., Rahman G., Analysis of implicit type nonlinear dynamical problem of impulsive fractional differentail equations, Complexity, 2018, Article ID 6423974

  • [3] Ali Z., Zada A., Shah K., On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations, Bull. Malays. Math. Sci. Soc., 2019, 42(5), 2681–2699

  • [4] Khan A., Syam M. I., Zada A., Khan H., Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives, Eur. Phys. J. Plus, 2018, 133:264

  • [5] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-HollandMathematics Studies, Elsevier Science B.V., Amsterdam, 2006, 204

  • [6] Magin R., Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, 2004, 32, 1–104

  • [7] Oldham K. B., Fractional differential equations in electrochemistry, Adv. Eng. Software, 2010, 41, 9–12

  • [8] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999

  • [9] Rizwan R., Zada A., Wang X., Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses, Adv. Difference Equ., 2019, 2019:85

  • [10] Zada A., Ali S., Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul., 2018, 19(7), 763–774

  • [11] Zada A., Ali S., Stability of integral Caputo-type boundary value problem with noninstantaneous impulses, Int. J. Appl. Comput. Math., 2019, 5:55

  • [12] Jarad F., Abdeljawad T., Alzabut J., Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Special Topics, 2017, 226(16-18), 3457–3471

  • [13] Zada A., Ali S., Li Y., Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Difference Equ., 2017, 2017:317

  • [14] Zada A., Yar M., Li T., Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math., 2018, 17, 103–125

  • [15] Zhou H., Alzabut J., Yang L., On fractional Langevin differential equations with anti-periodic boundary conditions, Eur. Phys. J. Special Topics, 2017, 226(16-18), 3577–3590

  • [16] Abdeljawad T., Alzabut J., On Riemann-Liouville fractional q-difference equations and their application to retarded logistic type model, Math. Meth. Appl. Sci., 2018, 41(18), 8953–8962

  • [17] Alzabut J., Abdeljawad T., Baleanu D., Nonlinear delay fractional difference equationswith applications on discrete fractional Lotka-Volterra competition model, J. Comput. Anal. Appl., 2018, 25(5), 889–898

  • [18] Liu S., Wang J., Zhou Y., Feckan M., Iterative learning control with pulse compensation for fractional differential equations, Math. Solv., 2018, 68, 563–574

  • [19] Luo D., Wang J., Shen D., Learning formation control for fractional-order multi-agent systems, Math. Meth. Appl. Sci., 2018, 41, 5003–5014

  • [20] Wang J., Ibrahim A. G., O’Regan D., Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions, J. Fixed Point Theory Appl., 2018, 20(59), 1–25

  • [21] Wang Y., Liu L.,Wu Y., Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 2011, 74, 3599–3605

  • [22] Zhang X., Liu L., Wu Y., Wiwatanapataphee B., Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion, Appl. Math. Letters, 2017, 66, 1–8

  • [23] Zhu B., Liu L., Wu Y., Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equation with delay, Appl. Math. Lett., 2016, 61, 73–79

  • [24] Zhang J., Wang J., Numerical analysis for a class of Navier-Stokes equations with time fractional derivatives, Appl. Math. Comput., 2018, 336, 481–489

  • [25] Berhail A., Tabouche N., Matar M. M., Alzabut J., On nonlocal integral and derivative boundary value problem of nonlinear Hadamard Langevin equation with three different fractional orders, Bol. Soc. Mat. Mex., 2019, https://doi.org/10.1007/s40590-019-00257-z

  • [26] Hyers D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 1941, 27, 222–224

  • [27] Ali Z., Zada A., Shah K., Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem, Bound. Value Prob., 2018, 2018:175

  • [28] Li T., Zada A., Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ., 2016, 2016:153

  • [29] Obloza M., Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat., 1993, 13, 259–270

  • [30] Shah R., Zada A., A fixed point approach to the stability of a nonlinear Volterra integrodiferential equation with delay, Hacettepe J. Math. Stat., 2018, 47(3), 615–623

  • [31] Shah S. O., Zada A., Hamza A. E., Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales, Qual. Theory Dyn. Syst., DOI: 10.1007/s12346-019-00315-x

  • [32] Ulam S. M., A Collection of Mathematical Problems, Interscience Publ. New York, 1960

  • [33] Wang J., Lv L., Zhou Y., Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011, 63, 1–10

  • [34] Wang J., Zada A., Ali W., Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. Num., 2018, 19(5), 553–560

  • [35] Wang X., Arif M., Zada A., β-Hyers-Ulam-Rassias stability of semilinear nonautonomous impulsive system, Symmetry, 2019, 11(2), 231

  • [36] Xu B., Brzdek J., Zhang W., Fixed point results and the Hyers-Ulam stability of linear equations of higher orders, Pacific J. Math., 2015, 273, 483–498

  • [37] Zada A., AliW., Farina S., Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Meth. App. Sci., 2017, 40(15), 5502–5514

  • [38] Zada A., Ali A., Park C., Ulam type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type, Appl. Math. Comput., 2019, 350, 60–65

  • [39] Zada A., Wang P., Lassoued D., Li T., Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems, Adv. Difference Equ., 2017, 2017:192

  • [40] Zada A., Riaz U., Khan F. U., Hyers-Ulamstability of impulsive integral equations, Boll. UnioneMat. Ital., 2019, 12(3), 453–467

  • [41] Zada A., Shah S. O., Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacettepe J. Math. Stat., 2018, 47(5), 1196–1205

  • [42] Zada A., Shah O., Shah R., Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., 2015, 271, 512–518

  • [43] Zada A., Shaleena S., Li T., Stability analysis of higher order nonlinear differential equations in β-normed spaces, Math. Meth. App. Sci., 2019, 42(4), 1151–1166

  • [44] Abbas S., Benchohra M., Lagreg J. E., Alsaedi A., Zhou Y., Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equ., 2017, 2017:180

  • [45] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000

  • [46] Wang J., Shah K., Ali A., Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Meth. Appl. Sci., 2018, 41, 1–11

  • [47] Furati K. M, Kassim M. D., Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differ. Equ., 2013, 235

  • [48] Furati K. M., Kassim M. D., Tatar N. E., Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 2012, 64, 1616–1626

  • [49] Hilfer R., Threefold introduction to fractional derivatives, In: Anomalous Transport, Foundations and Applications, 2008, 17–73

  • [50] Kamocki R., Obczynski C., On fractional Cauchy–type problems containing Hilfer’s derivative, Electron. J. Qual. Theory Differ. Equ., 2016, 50, 1–12

  • [51] Rassias T. M., On the stability of the linear mapping in Banach spaces, In: Proc. Amer. Math. Soc., 1978, 72, 297–300

  • [52] Rus I. A., Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 2010, 26, 103–107

  • [53] Tomovski Z., Hilfer R., Srivastava H. M., Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct., 2010, 21(11), 797–814

  • [54] Wang J., Zhang Y., Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 2015, 266, 850–859

  • [55] Shen Y., Li Y., A general method for the Ulam stability of linear differential equations, Bull. Malays. Math. Sci. Soc., 2019, 42(6), 3187–3211

  • [56] Guo Y., Shu X., Li Y., Xu F., The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β< 2, Bound. Value Prob., 2019, 2019:59

  • [57] Urs C., Coupled fixed point theorem and application to periodic boundary value problem, Miskolic Math Notes, 2013, 14, 323–333

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