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Fractional non-autonomous evolution equation with nonlocal conditions

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Abstract

The aim of this paper is to discuss the existence of mild solutions for a class of time fractional non-autonomous evolution equations with nonlocal conditions and measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to k-set-contractive operator and a new estimation technique of the measure of noncompactness, we obtain the new existence results of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic by deleting the compactness condition on nonlocal function g and extending the study of fractional autonomous evolution equations in recent years to non-autonomous ones. Finally, as samples of applications, we consider a time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition and nonlocal conditions.

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References

  1. McKibben, M.: Discoving evolution equations with applications, Vol. I. Deterministic Equations, Appl. Math. Nonlinear Sci. Ser. Chapman and Hall/CRC, New York (2011)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Liang, J., Liu, J.H., Xiao, T.J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57, 183–189 (2004)

    Article  MathSciNet  Google Scholar 

  8. Liang, J., Liu, J.H., Xiao, T.J.: Nonlocal impulsive problems for integro-differential equations. Math. Comput. Model. 49, 798–804 (2009)

    Article  Google Scholar 

  9. Zhu, L., Li, G.: Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces. Nonlinear Anal. 74, 5133–5140 (2011)

    Article  MathSciNet  Google Scholar 

  10. Zhu, T., Song, C., Li, G.: Existence of mild solutions for abstract semilinear evolution equations in Banach spaces. Nonlinear Anal. 75, 177–181 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Wang, R., Xiao, T.J., Liang, J.: A note on the fractional Cauchy problems with nonlocal conditions. Appl. Math. Lett. 24, 1435–1442 (2011)

    Article  MathSciNet  Google Scholar 

  15. Bajlekova, E.G.: Fractional Evolution Equations in Banach Spaces, Ph.D. thesis, Department of Mathematics, Eindhoven University of Technology (2001)

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

    Article  MathSciNet  Google Scholar 

  17. El-Borai, M.M., El-Nadi, K.E., El-Akabawy, E.G.: On some fractional evolution equations. Comput. Math. Appl. 59, 1352–1355 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Chen, P., Zhang, X., Li, Y.: A blowup alternative result for fractional non-autonomous evolution equation of Volterra type. Commun. Pure Appl. Anal. 17(5), 1975–1992 (2018)

    Article  MathSciNet  Google Scholar 

  20. Li, M., Chen, C., Li, F.B.: On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, 2702–2726 (2010)

    Article  MathSciNet  Google Scholar 

  21. Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012)

    Article  MathSciNet  Google Scholar 

  22. Mei, Zhan, Peng, J., Zhang, Y.: An operator theoretical approach to Riemann–Liouville fractional Cauchy problem. Math. Nachr. 288, 784–797 (2015)

    Article  MathSciNet  Google Scholar 

  23. Shu, X., Shi, Y.: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465–476 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012)

    Article  MathSciNet  Google Scholar 

  25. Wang, J., Fečkan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Part. Differ. Equ. 8, 345–361 (2011)

    Article  MathSciNet  Google Scholar 

  26. Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12, 262–272 (2011)

    Article  MathSciNet  Google Scholar 

  27. Wang, J., Zhou, Y., Fečkan, M.: Abstract Cauchy problem for fractional differential equations. Nonlinear Dyn. 74, 685–700 (2013)

    Article  MathSciNet  Google Scholar 

  28. Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)

    Article  MathSciNet  Google Scholar 

  29. Ouyang, Z.: Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. Comput. Math. Appl. 61, 860–870 (2011)

    Article  MathSciNet  Google Scholar 

  30. Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay. Appl. Math. Lett. 61, 73–79 (2016)

    Article  MathSciNet  Google Scholar 

  31. Zhu, B., Liu, L., Wu, Y.: Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Comput. Math. Appl. (2016). https://doi.org/10.1016/j.camwa.2016.01.028

    Article  MATH  Google Scholar 

  32. Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fract. Calc. Appl. Anal. 20, 1338–1355 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Tanabe, H.: Functional Analytic Methods for Partial Differential Equations. Marcel Dekker, New York (1997)

    MATH  Google Scholar 

  34. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math, vol. 840. Springer, New York (1981)

    Book  Google Scholar 

  35. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  Google Scholar 

  36. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: van Mill, J. (ed.) North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam (2006)

  37. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)

    Book  Google Scholar 

  38. Gorenflo, R., Mainardi, F.: Fractional calculus and stable probability distributions. Arch. Mech. 50(3), 377–388 (1998)

    MathSciNet  MATH  Google Scholar 

  39. Banas̀, J., Goebel, K.: Measures of noncompactness in banach spaces. In: Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)

  40. Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, New York, NY (1969)

    MATH  Google Scholar 

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Acknowledgements

This work is supported by National Natural Science Foundations of China (Nos. 11501455, 11661071), Key Project of Gansu Provincial National Science Foundation (No. 1606RJZA015) and Project of NWNU-LKQN-14-6.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Pengyu Chen, Xuping Zhang & Yongxiang Li

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  1. Pengyu Chen
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  2. Xuping Zhang
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  3. Yongxiang Li
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Correspondence to Pengyu Chen.

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Chen, P., Zhang, X. & Li, Y. Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10, 955–973 (2019). https://doi.org/10.1007/s11868-018-0257-9

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  • DOI: https://doi.org/10.1007/s11868-018-0257-9

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