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AIMS Mathematics

2020, Volume 5, Issue 6: 7510-7530. doi: 10.3934/math.2020481
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Research article

Periodic boundary value problem involving sequential fractional derivatives in Banach space

  • 1.
    Department of Mathematics, China University of Mining and Technology-Beijing, 100083, Ding No.11 Xueyuan Road, Haidian District, Beijing, China
  • 2.
    Department of Basic Courses, Beijing Union University, 100101, No.97 Beisihuan East Road, Chaoyang District, Beijing, China
  • Received: 10 July 2020 Accepted: 17 September 2020 Published: 23 September 2020

  • MSC : 26A33, 34B15

  • In this paper, by the method of upper and lower solutions coupled with the monotone iterative technique, we investigate the existence and uniqueness results of solutions for a periodic boundary value problem of nonlinear fractional differential equation involving conformable sequential fractional derivatives in Banach space. An example is given to illustrate our main result.

    Citation: Xinwei Su, Shuqin Zhang, Lixin Zhang. Periodic boundary value problem involving sequential fractional derivatives in Banach space[J]. AIMS Mathematics, 2020, 5(6): 7510-7530. doi: 10.3934/math.2020481

    Related Papers:

  • In this paper, by the method of upper and lower solutions coupled with the monotone iterative technique, we investigate the existence and uniqueness results of solutions for a periodic boundary value problem of nonlinear fractional differential equation involving conformable sequential fractional derivatives in Banach space. An example is given to illustrate our main result.


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    [1] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
    [2] Z. L. Wei, Q. D. Li, J. L. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367 (2010), 260-272.
    [3] B. Ahmad, R. Luca, Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions, Chaos Soliton. Fract., 104 (2017), 378-388.
    [4] B. Ahmad, J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 64 (2012), 3046-3052.
    [5] B. Ahmad, J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Space. Appl., 2013 (2013), 1-8.
    [6] B. Ahmad, S. K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput., 266 (2015), 615-622.
    [7] Y. Chen, X. H. Tang, Solvability of sequential fractional order multi-point boundary value problems at resonance, Appl. Math. Comput., 218 (2012), 7638-7648.
    [8] Q. D. Li, H. Su, Z. L. Wei, Existence and uniqueness result for a class of sequential fractional differential equations, J. Appl. Math. Comput., 38 (2012), 641-652.
    [9] C. Promsakon, N. Phuangthong, S. K. Ntouyas, et al. Nonlinear sequential Riemann-Liouville and Caputo fractional differential equations with generalized fractional integral conditions, Adv. Differ. Equ., 2018 (2018), 1-17.
    [10] H. Ye, R. Huang, Initial value problem for nonlinear fractional differential equations with sequential fractional derivative, Adv. Differ. Equ., 2015 (2015), 1-13.
    [11] H. Ye, R. Huang, On the nonlinear fractional differential equations with Caputo sequential fractional derivative, Adv. Math. Phys., 2015 (2015), 1-9.
    [12] O. Abu Arqub, Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space, Fundam. Inf., 166 (2019), 87-110.
    [13] O. Abu Arqub, Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis, Fundam. Inf., 166 (2019), 111-137.
    [14] O. Abu Arqub, N. Shawagfeh, Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media, J. Porous Med., 22 (2019), 411-434.
    [15] A. M. A. El-Sayed, H. H. G. Hashem, E. A. A. Ziada, Picard and Adomian decomposition methods for a quadratic integral equation of fractional order, Comput. Appl. Math., 33 (2014), 95-109.
    [16] A. Lotfi, S. A. Yousefi, A numerical technique for solving a class of fractional variational problems, J. Comput. Appl. Math., 237 (2013), 633-643.
    [17] P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.
    [18] H. D. Gou, Y. X. Li, Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions, Bound. Value Probl., 2019 (2019), 1-25.
    [19] C. Hu, B. Liu, S. Xie, Monotone iterative solutions for nonlinear boundary value problems of fractional differential equations with deviating arguments, Appl. Math. Comput., 222 (2013), 72-81.
    [20] F. A. Mcrae, Monotone iterative technique and existence results for fractional differential equations, Nonlinear Anal., 71 (2009), 6093-6096.
    [21] J. Mu, Y. Li, Monotone interactive technique for impulsive fractional evolution equations, J. Inequal. Appl., 2011 (2011), 1-12.
    [22] G. Wang, Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments, J. Comput. Appl. Math., 236 (2012), 2425-2430.
    [23] L. Zhang, B. Ahmad, G. Wang, Explicit iterations and extremal solutions for fractional differential equation with nonlinear integral boundary conditions, Appl. Math. Comput., 268 (2015), 338-392.
    [24] C. Z. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 384 (2011), 211-231.
    [25] R. Khalil, M. Al-Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
    [26] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
    [27] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903-917.
    [28] D. R. Anderson, D. J. Ulness, Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56 (2015), 1-15.
    [29] D. Zhao, X. Pan, M. Luo, A new framework for multivariate general conformable fractional calculus and potential applications, Phys. A, 15 (2018), 271-280.
    [30] S. Yang, L. Wang, S. Q. Zhang, Conformable derivative: application to non-Darcian flow in lowpermeability porous media, Appl. Math. Lett., 79 (2018), 105-110.
    [31] H. W. Zhou, S. Yang, S. Q. Zhang, Conformable derivative approach to anomalous diffusion, Phys. A, 491 (2018), 1001-1013.
    [32] M. Al-Refai, T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, 2017 (2017), 1-7.
    [33] S. Asawasamrit, S. K. Ntouyas, P. Thiramanus, et al. Periodic boundary value problems for impulsive conformable fractional integro-differential equations, Bound. Value Probl., 2016 (2016), 1-18.
    [34] H. Batarfi, J. Losada, J. J. Nieto, et al. Three-point boundary value problems for conformable fractional differential equations, J. Funct. Space., 2015 (2015), 1-6.
    [35] B. Bayour, D. F. M. Torres, Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math., 312 (2017), 127-133.
    [36] M. Bouaouid, M. Hannabou, K. Hilal, Nonlocal conformable fractional differential equations with a measure of noncompactness in Banach spaces, J. Math., 2020 (2020), 1-6.
    [37] L. M. Feng, S. R. Sun, Oscillation theorems for three classes of conformable fractional differential equations, Adv. Differ. Equ., 2019 (2019), 1-30.
    [38] A. Jaiswal, D. Bahuguna, Semilinear conformable fractional differential equations in Banach spaces, Differ. Equ. Dyn. Syst., 27 (2019), 313-325.
    [39] M. M. Li, J. R. Wang, D. O'Regan, Existence and Ulam's stability for conformable fractional differential equations with constant coefficients, Bull. Malays. Math. Sci. Soc., 42 (2019), 1791-1812.
    [40] S. M. Meng, Y. J. Cui, Multiplicity results to a conformable fractional differential equations involving integral boundary condition, Complexity, 2019 (2019), 1-8.
    [41] E. Ünal, A. Gökdoğan, Solution of conformable fractional ordinary differential equations via differential transform method, Optik, 128 (2017), 264-273.
    [42] J. F. Wang, C. Z. Bai, Antiperiodic boundary value problems for impulsive fractional functional differential equations via conformable derivative, J. Funct. Space., 2018 (2018), 1-11.
    [43] W. Y. Zhong, L. F. Wang, Positive solutions of conformable fractional differential equations with integral boundary conditions, Bound. Value Probl., 2018 (2018), 1-12.
    [44] W. Y. Zhong, L. F. Wang, Basic theory of initial value problems of conformable fractional differential equations, Adv. Differ. Equ., 2018 (2018), 1-14.
    [45] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
    [46] T. Cardinali, R. Precup, P. Rubbioni, A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math. Anal. Appl., 432(2015), 1039-1057.
    [47] H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371.
    [48] D. O'Regan, R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl., 6 (2001), 77-97.
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