Fractional Retarded Evolution Equations with Measure of Noncompactness Subjected to Mixed Nonlocal Plus Local Initial Conditions

Xuping Zhang 1  and Yongxiang Li 2
  • 1 Department of Mathematics, Northwest Normal University, Lanzhou, People’s Republic of China
  • 2 Department of Mathematics, Northwest Normal University, Lanzhou, People’s Republic of China
Xuping Zhang
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  • Department of Mathematics, Northwest Normal University, Lanzhou, People’s Republic of China
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and Yongxiang Li

Abstract

We consider the fractional retarded evolution equations

CDtqu(t)+Au(t)=f(t,ut,0tw(t,s,us)ds),t[0,a],

where CDtq, q(0,1], is the fractional derivative in the Caputo sense, A is the infinitesimal generator of a C0-semigroup of uniformly bounded linear operators T(t)(t0) on a Banach space X and the nonlinear operators f and w are given functions satisfying some assumptions, subjected to a general mixed nonlocal plus local initial condition of the form u(t)=g(u)(t)+ϕ(t), t[h,0]. Under more general conditions, the existence of mild solutions and positive mild solutions are obtained by means of fractional calculus and fixed point theory for condensing maps. Moreover, we present an example to illustrate the application of abstract results.

  • [1]

    V. Kolmanovskii and A. Myshkis, Introduction to the theory and applications of functional-differential equations, Mathematics and its applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.

  • [2]

    J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations, Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.

  • [3]

    J. Wu, Theory and applications of partial functional differential equations, Appl. Math. Sciences, 119, Springer-Verlag, New York, 1996.

  • [4]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

  • [5]

    V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.

  • [6]

    K. S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.

  • [7]

    I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.

  • [8]

    V. E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer, HEP, 2010.

  • [9]

    R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (2010), 973–1033.

    • Crossref
    • Export Citation
  • [10]

    M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Sol. Frac. 14 (2002), 433-440.

    • Crossref
    • Export Citation
  • [11]

    M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal. 3 (2004), 197-211.

  • [12]

    J. Wang, M. Fečkan and Yong Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), 345-361.

    • Crossref
    • Export Citation
  • [13]

    J. Wang and X. Li, A uniform method to ulam-hyers stability for some linear fractional equations, Mediterr. J. Math. 13 (2016), 625-635.

    • Crossref
    • Export Citation
  • [14]

    J. Wang and Y. Zhou, Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal.: TMA 74 (2011), 5929-5942.

    • Crossref
    • Export Citation
  • [15]

    J. Wang and Y. Zhou, Existence of mild solutions for fractional delay evolution systems, Appl. Math. Comput. 218 (2011), 357-367.

  • [16]

    J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal.: RWA 12 (2011), 262-272.

    • Crossref
    • Export Citation
  • [17]

    J. Wang, Y. Zhou and W. Wei, A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 4049-4059.

    • Crossref
    • Export Citation
  • [18]

    L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.

    • Crossref
    • Export Citation
  • [19]

    L. Byszewski, Application of properties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal. 33 (1998), 413-426.

    • Crossref
    • Export Citation
  • [20]

    P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63 (2013), 731-744.

    • Crossref
    • Export Citation
  • [21]

    P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys. 65 (2014), 711-728.

    • Crossref
    • Export Citation
  • [22]

    Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Functional Anal. 258 (2010), 1709-1727.

    • Crossref
    • Export Citation
  • [23]

    K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the ##InlineEquation:IEq579##$$\alpha$$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (2007), 1613-1622.

    • Crossref
    • Export Citation
  • [24]

    J. Liang, J. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families,Nonlinear Anal. 57 (2004), 183-189.

    • Crossref
    • Export Citation
  • [25]

    J. Liang, J. Liu and T. J. Xiao, Nonlocal impulsive problems for integrodifferential equations, Math. Comput. Modelling 49 (2009), 798-804.

    • Crossref
    • Export Citation
  • [26]

    I. I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012), 1363-1391.

    • Crossref
    • Export Citation
  • [27]

    Vrabie I. I., Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math. 17 (2015), 1350035, 22 pp.

  • [28]

    J. Wang, A. G. Ibrahim and M. Kan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput. 257 (2015), 103-118.

  • [29]

    R. N. Wang, T. J. Xiao and J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lette. 24 (2011), 1435-1442.

    • Crossref
    • Export Citation
  • [30]

    J. Wang, Y. Zhou, W. Wei and H. Xu, Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl. 62 (2011), 1427-1441.

    • Crossref
    • Export Citation
  • [31]

    T.J. Xiao and J. Liang, Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal. 63 (2005), 225-232.

    • Crossref
    • Export Citation
  • [32]

    Kamaljeet, D. Bahuguna, Monotone iterative technique for nonlocal fractional differential equations with finite delay in a Banach space, Electron. J. Qual. Theory Differ. Equ. 9 (2015), 1-16.

  • [33]

    L. Hu, Y. Ren and R. Sakthivel. Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum 79 (2009), 507-514.

    • Crossref
    • Export Citation
  • [34]

    Q. Dong, Z. Fan and G. Li, Existence of solutions to nonlocal neutral functional differential and integrodifferential equations, Int. J. Nonlinear Sci. 5 (2008), 140-151.

  • [35]

    Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal.: RWA 11 (2010), 4465–4475.

  • [36]

    J. Bana&‘{s} and K. Goebel, Measures of noncompactness in Banach spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980.

  • [37]

    K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985.

  • [38]

    H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued Functions, Nonlinear Anal.: TMA 7 (1983), 1351-1371.

    • Crossref
    • Export Citation
  • [39]

    B. N. Sadovskii, A fixed-point principle, Funct. Anal. Appl. 1 (1967), 151-153.

  • [40]

    A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

  • [41]

    Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sin. 37 (1996), 666–672. (in Chinese)

  • [42]

    R. Nagel, One-parameter semigroups of positive operators, Lecture Notes in Math. vol. 1184, Springer-Verlag, Berlin, 1986.

  • [43]

    I. I. Vrabie, C0-Semigroups and Applications. North-Holland Mathematics Studies 191, Elsevier, Amsterdam, 2003.

  • [44]

    K. Balachandran and J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71 (2009), 4471-4475.

    • Crossref
    • Export Citation
  • [45]

    P. Chen, X. Zhang and Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math. 14(226) (2017), 1-16.

  • [46]

    P. Chen, X. Zhang and Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl. 73 (2017), 794-803.

    • Crossref
    • Export Citation
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