Research article

Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

  • Received: 10 October 2019 Accepted: 06 December 2019 Published: 12 December 2019
  • MSC : 34N05, 34A12

  • In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Carathéodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

    Citation: Yige Zhao, Yibing Sun, Zhi Liu, Yilin Wang. Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type[J]. AIMS Mathematics, 2020, 5(1): 557-567. doi: 10.3934/math.2020037

    Related Papers:

  • In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Carathéodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.


    加载中


    [1] I. Podlubny, Fractional differential equations, mathematics in science and engineering, Academic Press, New York, 1999.
    [2] D. Kumar, J. Singh, K. Tanwar, et al. A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws, Int. J. Heat Mass Tran., 138 (2019), 1222-1227. doi: 10.1016/j.ijheatmasstransfer.2019.04.094
    [3] D. Kumar, J. Singh, M. Al Qurashi, et al. A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equa., 2019 (2019), 278.
    [4] D. Kumar, J. Singh, S. D. Purohit, et al. A hybrid analytical algorithm for nonlinear fractional wave-like equations, Math. Model. Nat. Pheno., 14 (2019), 304.
    [5] J. Singh, D. Kumar, D. Baleanu, New aspects of fractional Biswas-Milovic model with MittagLeffler law, Math. Model. Nat. Pheno., 14 (2019), 303.
    [6] D. Peng, K. Sun, S. He, et al. Numerical analysis of a simplest fractional-order hyperchaotic system, Theoretical and Applied Mechanics Letters, 9 (2019), 220-228. doi: 10.1016/j.taml.2019.03.006
    [7] S. He, K. Sun, Y. Peng, Detecting chaos in fractional-order nonlinear systems using the smaller alignment index, Phys. Lett. A, 383 (2019), 2267-2271. doi: 10.1016/j.physleta.2019.04.041
    [8] Y. Peng, K. Sun, D. Peng, et al. Dynamics of a higher dimensional fractional-order chaotic map, Physica A: Statistical Mechanics and its Applications, 525 (2019), 96-107. doi: 10.1016/j.physa.2019.03.058
    [9] D. Chergui, T. E. Oussaeif, M. Ahcene, Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions, AIMS Mathematics, 4 (2019), 112-133. doi: 10.3934/Math.2019.1.112
    [10] M. Asaduzzaman, M. Z. Ali, Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations, AIMS Mathematics, 4 (2019), 880-895. doi: 10.3934/math.2019.3.880
    [11] Y. Zhao, X. Hou, Y. Sun, et al. Solvability for some class of multi-order nonlinear fractional systems, Adv. Differ. Equa., 2019 (2019), 23.
    [12] Q. Song, Z. Bai, Positive solutions of fractional differential equations involving the RiemannStieltjes integral boundary condition, Adv. Differ. Equ., 2018 (2018), 183.
    [13] K. Sheng, W. Zhang, Z. Bai, Positive solutions to fractional boundary value problems with pLaplacian on time scales, Bound. Value Probl., 2018 (2018), 70.
    [14] Z. Bai, Y. Chen, H. Lian, et al. On the existence of blow up solutions for a class of fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 1175-1187.
    [15] Z. Bai, Y. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218 (2011), 1719-1725.
    [16] Y. Zhao, S. Sun, Z. Han, et al. Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput., 217 (2011), 6950-6958.
    [17] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surveys in Mathematics & its Applications, 3 (2008), 1-12.
    [18] H. Lu, S. Sun, D. Yang, et al. Theory of fractional hybrid differential equations with linear perturbations of second type, Bound. Value Probl., 2013 (2013), 23.
    [19] S. Sun, Y. Zhao, Z. Han, et al. The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4961-4967. doi: 10.1016/j.cnsns.2012.06.001
    [20] Y. Zhao, S. Sun, Z. Han, et al. Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), 1312-1324. doi: 10.1016/j.camwa.2011.03.041
    [21] Y. Zhao, Y. Sun, Z. Liu, et al. Basic theory of differential equations with mixed perturbations of the second type on time scales, Adv. Differ. Equa., 2019 (2019), 268.
    [22] B. C. Dhage, Basic results in the theory of hybrid differential equations with mixed perturbations of second type, Funct. Differ. Equ., 19 (2012), 87-106.
    [23] B. C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J., 44 (2004), 145-155.
    [24] A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B. V., Amsterdam, 2006.
    [25] C. C. Tisdell, On the solvability of nonlinear first-order boundary-value problems, Electron. J. Differ. Equa., 2006 (2006), 1-8.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3571) PDF downloads(337) Cited by(9)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog