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Lower and upper solutions for damped elastic systems with delay in ordered Banach space

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Abstract

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, a class of nonlocal problem of structural damped elastic systems with delay are studied in the case of noncompact semigroups in ordered Banach space. Firstly, we introduce the concept of lower S-asymptotically \(\omega \)-periodic solution and upper S-asymptotically \(\omega \)-periodic solution, on the premise of the existence of upper and lower S-asymptotically \(\omega \)-periodic solutions, the existence of maximal and minimal S-asymptotically \(\omega \)-periodic mild solutions for the elastic system are obtained. Then, an existence theorem of positive mild solutions for elastic system is obtained without assuming lower and upper S-asymptotically \(\omega \)-periodic solutions. Finally, as the application of abstract results, the existence and uniqueness of S-asymptotically \(\omega \)-periodic mild solutions and positive mild solutions for a classes of nonlocal damped elastic systems with delay are discussed.

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References

  1. Amann, H.: Periodic solutions of semilinear parabolic equations. In: Cesari, L., Kannan, R., Weinberger, R. (eds.) Nonlinear Anal, A Collection of Papers in Honor of Erich H. Rothe, pp. 1–29. Academic Press, New York (1978)

    Google Scholar 

  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)

    MathSciNet  Google Scholar 

  3. Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., New York: Marcel Dekker, (60) (1980)

  4. Cuevas, C., Souza, J.: Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 72, 1683–1689 (2010)

    MathSciNet  Google Scholar 

  5. Cuevas, C., Henriquez, H.R., Soto, H.: Asymptotically periodic solutions of fractional differential equations. Appl. Math. Comput. 236, 524–545 (2014)

    MathSciNet  Google Scholar 

  6. Cheng, P., Li, Y.: Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces. Appl. Math. 59, 99–120 (2014)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Cheng, P., Zhang, X., Li, Y.: Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10, 955–973 (2019)

    MathSciNet  Google Scholar 

  9. Chen, P., Li, Y.: Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces. Nonlinear Anal. 74, 3578–3588 (2011)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. Chen, P., Li, Y., Yang, H.: Perturbation method for nonlocal impulsive evolution equations. Nonlinear Anal. Hybrid Syst 8, 22–30 (2013)

    MathSciNet  Google Scholar 

  12. Chen, P., Zhang, X., Li, Y.: Iterative method for a new class of evolution equations with noninstantaneous impulses. Taiwanese J. Math. 21, 913–942 (2017)

    MathSciNet  Google Scholar 

  13. Chen, X., Cheng, L.: On countable determination of the Kuratowski measure of noncompactness. J. Math. Anal. Appl. 504, 125370 (2021)

    MathSciNet  Google Scholar 

  14. Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39, 433–454 (1982)

    MathSciNet  Google Scholar 

  15. Diagana, T.: Well-posedness for some damped elastic systems in Banach spaces. Appl. Math. Lett. 71, 74–80 (2017)

    MathSciNet  Google Scholar 

  16. Du, S., Lakshmikantham, V.: Monotone iterative technique for differential equations in Banach spaces. J. Math. Anal. Appl. 87, 454–459 (1982)

    MathSciNet  Google Scholar 

  17. Du, Y.: Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 38, 1–20 (1990)

    MathSciNet  Google Scholar 

  18. Deimling, K.: Nonlinear Functional Analysis. Springer-Verlag, New York (1985)

    Google Scholar 

  19. Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630–637 (1993)

    MathSciNet  Google Scholar 

  20. Ezzinbi, K., Liu, J.H.: Nondensely defined evolution equations with nonlocal conditions. Math. Comput. Modelling 36, 1027–1038 (2002)

    MathSciNet  Google Scholar 

  21. Ezzinbi, K., Liu, J.H.: Periodic solutions of some evolution equations with infinite delay. Int. J. Evol. Equ. 2, 19–27 (2007)

    MathSciNet  Google Scholar 

  22. Fan, H., Li, Y.: Monotone iterative technique for the elastic systems with structural damping in Banach spaces. Comput. Math. Appl. 68, 384–391 (2014)

    MathSciNet  Google Scholar 

  23. Fan, H., Li, Y.: Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces. J. Math. Anal. Appl. 410, 316–322 (2014)

    MathSciNet  Google Scholar 

  24. Fan, H., Gao, F.: Asymptotic stability of solutions to elastis systems with structural damping. Electron. J. Differ. Eq. 245, 1–9 (2014)

    Google Scholar 

  25. Guo, D.: Nonlinear Functional Analysis. Shandong Science and Technology, Jinan, (Chinese) (1985)

  26. Guo, D., Sun, J.: Ordinary differential equations in abstract spaces. Shandong Science and Technology, Jinan, (1989) (Chinese)

  27. Gou, H., Li, Y.: Mixed monotone iterative technique for damped elastic systems in Banach spaces. J. Pseudo-Differ. Oper. Appl. 11, 917–933 (2020)

    MathSciNet  Google Scholar 

  28. Gou, H., Li, Y.: A Study on Damped Elastic Systems in Banach Spaces. Numer. Func. Anal. Opt. 41, 542–570 (2020)

    MathSciNet  Google Scholar 

  29. Henríquez, H.R., Pierri, M., Táboas, P.: On \(S\)-asymptotically \(\omega \)-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343, 1119–1130 (2008)

    MathSciNet  Google Scholar 

  30. Huang, F.: On the holomorphic property of the semigroup associated with linear elastic systems with structural damping. Acta Math. Sci. 5, 271–277 (1985)

    MathSciNet  Google Scholar 

  31. Huang, F., Liu, K.: Holomiphic property and exponential stability of the semigroup associated with linear elastic systems with damping. Ann. Diff. Eqs. 4(4), 411–424 (1988)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  33. Li, Y.: The positive solutions of abstract semilinear evolution equations and their applications. Acta Math. Sin. 39, 666–672 (1996)

    MathSciNet  Google Scholar 

  34. Li, Y.: Periodic solutions of semilinear evolution equations in Banach spaces. Acta Math. Sin. 41, 629–636 (1998). ((in Chinese))

    MathSciNet  Google Scholar 

  35. Li, Y.: The global solutions of initial value problems for abstract semilinear evolution equations. Acta Anal. Funct. Appl. 3, 339–347 (2001)

    MathSciNet  Google Scholar 

  36. Li, Y., Liu, Z.: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 66, 83–92 (2007)

    MathSciNet  Google Scholar 

  37. Li, Y.: Existence and uniqueness of periodic solution for a class of semilinear evolution equations. J. Math. Anal. Appl. 349, 226–234 (2009)

    MathSciNet  Google Scholar 

  38. Li, Y.: Existence and asymptotic stability of periodic solution for evolution equations with delays. J. Funct. Anal. 261, 1309–1324 (2011)

    MathSciNet  Google Scholar 

  39. Li, Q.: Monotone iterative technique for delayed evolution equation periodic problems in Banach spaces. Pure Appl. Math. Q. 14, 393–417 (2018)

    MathSciNet  Google Scholar 

  40. Li, Q., Wei, M.: Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evol. Equ. Control Theory 9, 753–772 (2020)

    MathSciNet  Google Scholar 

  41. Li, Q., Wang, G., Wei, M.: Monotone iterative technique for time-space fractional diffusion equations involving delay. Nonlinear Anal. Model. 26, 241–258 (2021)

    MathSciNet  Google Scholar 

  42. Li, Q., Wei, M.: Monotone iterative technique for S-asymptotically periodic problem of fractional evolution equation with finite delay in ordered Banach space. J. Math. Inequal. 15, 521–546 (2021)

    MathSciNet  Google Scholar 

  43. Li, F., Liang, J., Wang, H.: \(S\)-asymptotically \(\omega \)-periodic solution for fractional differential equations of order \(q\in (0,1)\) with finite delay, Adv. Difference Equ., (2017), Paper No. 83, 14 pp

  44. Li, F., Wang, H.: \(S\)-asymptotically \(\omega \)-periodic mild solutions of neutral fractional differential equations with finite delay in Banach space. Mediterr. J. Math. 14, 57 (2017)

    MathSciNet  Google Scholar 

  45. Li, B., Gou, H.: Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces. Chaos, Solitons Fractals 110, 209–215 (2018)

    MathSciNet  Google Scholar 

  46. Luong, V.T., Tung, N.T.: Decay mild solutions for elastic systems with structural damping involving nonlocal conditions. Vestnik St. Petersburg Univ. Math. 50, 55–67 (2017)

    MathSciNet  Google Scholar 

  47. Luong, V.T., Tung, N.T.: Exponential decay for elastic systems with structural damping and infinite delay. Appl. Anal. 99, 13–28 (2020)

    MathSciNet  Google Scholar 

  48. Liu, J.H.: Periodic solutions of infinite delay evolution equations. J. Math. Anal. Appl. 247, 627–644 (2000)

    MathSciNet  Google Scholar 

  49. Liu, K., Liu, Z.: Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces. J. Differ. Eq. 141, 340–355 (1997)

    MathSciNet  Google Scholar 

  50. Pierri, M.: On \(S\)-asymptotically \(\omega \)-periodic functions and applications. Nonlinear Anal. 75, 651–661 (2012)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  52. Ren, L., Wang, J., Fečkan, M.: Asymptotically periodic solutions for Caputo type fractional evolution equations. Fract. Calc. Appl. Anal. 21, 1294–1312 (2018)

    MathSciNet  Google Scholar 

  53. Sun, J., Zhao, Z.: Extremal solutions of initial value problem for integro-differential equations of mixed type in Banach spaces. Ann. Differ. Eq. 8, 469–475 (1992)

    MathSciNet  Google Scholar 

  54. Wei, S.: Global existence of mild solutions for the elastic system with structural damping. Ann. Appl. Math. 35, 180–188 (2019)

    MathSciNet  Google Scholar 

  55. Wei, M., Li, Y.: Existence and global asymptotic behavior of mild solutions for damped elastic systems with delay and nonlocal conditions. J. Anal. Appl. Comput. 13(2), 874–892 (2023)

    MathSciNet  Google Scholar 

  56. Wei, M., Li, Y., Li, Q.: Positive mild solutions for damped elastic systems with delay and nonlocal conditions in ordered Banach space. Qual. Theory Dyn. Syst. 21, 128 (2022)

    MathSciNet  Google Scholar 

  57. Xiao, T., Liang, J.: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Anal. Theory Methods Appl. 63, e225–e227 (2005)

    Google Scholar 

  58. Xue, X.: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach space. Nonlinear Anal. Theory Methods Appl. 70, 2593–2601 (2009)

    MathSciNet  Google Scholar 

  59. Zhang, X., Cheng, P., Li, Y.: Monotone iterative method for retarded evolution equations involving nonlocal and impulsive conditions. Electron. J. Differ. Eq. 68, 1–25 (2020)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and referees for their careful reading of this paper and valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Nos.11661071, 12061062). Science Research Project for Colleges and Universities of Gansu Province (No.2022A-010) and Project of NWNU-LKQN2023-02.

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Supported by the National Natural Science Foundation of China (12061062, 11661071). Science Research Project for Colleges and Universities of Gansu Province (No.2022A-010) Project of NWNU-LKQN2023-02.

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

    Haide Gou & Mei Wei

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  1. Haide Gou
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Correspondence to Haide Gou.

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Gou, H., Wei, M. Lower and upper solutions for damped elastic systems with delay in ordered Banach space. Japan J. Indust. Appl. Math. 41, 475–501 (2024). https://doi.org/10.1007/s13160-023-00615-5

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  • DOI: https://doi.org/10.1007/s13160-023-00615-5

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