Pseudo-almost periodic $C^{0}$-solution for evolution inclusion with mixed nonlocal plus local initial conditions

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1480 - 1491, 03.11.2023

Li Ye , Yongjian Liu

https://doi.org/10.15672/hujms.1143058

Abstract

This paper is devoted to the study of a class of evolution inclusion in Banach spaces with nonlocal plus local mixed initial conditions. Under some mild assumptions, a unique solvability result to the multivalued evolution problem is obtained via the arguments of fixed point principle and the theory of $C^0$-semigroup.

Keywords

Almost periodic solution, Pseudo-almost periodic solution, Mixed nonlocal plus local initial conditions, Existence and uniqueness, Evolution inclusion

Supporting Institution

National Natural Science Foundation of China

Project Number

11961074

References

• [1] S. Aizicovici and H. Lee, Nonlinear nonlocal cauchy problems in Banach spaces, Applied Mathematics Letters, 18 (4), 401–407, 2005.
• [2] G. Akagi and U. Stefanelli, Periodic solutions for doubly nonlinear evolution equations, Journal of Differential Equations, 251 (7), 1790–1812, 2011.
• [3] E. Alvarez and C. Lizama, Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces, Advances in Difference Equations, 2015 (1), 31, 2015.
• [4] V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Science & Business Media2010.
• [5] S. Bilal, O. Cârjă, T. Donchev, N. Javaid and A. I. Lazu, Nonlocal evolution inclusions under weak conditions, Advances in Difference Equations, 2018 (1), 1–14, 2018.
• [6] M.-D. Burlică and D. Roşu, A class of nonlinear delay evolution equations with non-local initial conditions, Proceedings of the American Mathematical Society, 142 (7), 2445–2458, 2014.
• [7] M.-D. Burlică, D. Roşu and I. I. Vrabie, Abstract reaction–diffusion systems with nonlocal initial conditions, Nonlinear Analysis: Theory, Methods & Applications, 94, 107–119, 2014.
• [8] Y. Chen, J. J. Nieto and D. O’Regan, Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Applied Mathematics Letters, 24 (3), 302–307, 2011.
• [9] Y. Cheng, F. Cong and H. Hua, Anti-periodic solutions for nonlinear evolution equations, Advances in Difference Equations, 2012 (1), 1–15, 2012.
• [10] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, American Journal of Mathematics, 93 (2), 265–298, 1971.
• [11] C. Cuevas and M. Pinto, Existence and uniqueness of pseudo almost periodic solutions of semilinear cauchy problems with nondense domain, Nonlinear Analysis: Theory, Methods & Applications, 45 (1), 73–73, 2001.
• [12] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 179 (2), 630– 637, 1993.
• [13] T. Diagana, Pseudo almost periodic functions in Banach spaces, New York (NY): Nova Science Publishers Inc2007.
• [14] J. K. Hale, Functional differential equations, Springer1971.
• [15] C. Lizama and E. Alvarez-Pardo, Pseudo asymptotic solutions of fractional order semilinear equations, Banach Journal of Mathematical Analysis, 7 (2), 42–52, 2013.
• [16] M. McKibben, Discovering evolution equations with applications: Volume 1- Deterministic equations, Chapman and Hall/CRC2010.
• [17] B. Meknani, The existence and uniqueness of integral solutions to some nonlinear reaction diffusion system with nonlocal retarded initial conditions, Journal of Taibah University for Science, 14 (1), 569–578, 2020.
• [18] B. Meknani, J. Zhang and T. Abdelhamid, Pseudo-almost periodic C0 solutions to the evolution equations with nonlocal initial conditions, Applicable Analysis, (3), 1–11, 2021.
• [19] A. Paicu and I. I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Analysis: Theory, Methods & Applications, 72 (11), 4091–4100, 2010.
• [20] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Periodic solutions for a class of evolution inclusions, Computers and Mathematics with Applications, 75 (8), 3047–3065, 2018, doi:https://doi.org/10.1016/j.camwa.2018.01.031.
• [21] P. Sattayatham, S. Tangmanee and W. Wei, On periodic solutions of nonlinear evolution equations in Banach spaces, Journal of Mathematical Analysis and Applications, 276 (1), 98–108, 2002.
• [22] I. I. Vrabie, Compactness methods for nonlinear evolutions, vol. 75, CRC Press1995.
• [23] I. I. Vrabie, Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions, Journal of Evolution Equations, 13 (3), 693–714, 2013.
• [24] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Communications in Contemporary Mathematics, 17 (02), 1350035, 2015.
• [25] X. Xue and Y. Cheng, Existence of periodic solutions of nonlinear evolution inclusions in Banach spaces, Nonlinear Analysis: Real World Applications, 11 (1), 459–471, 2010.
• [26] Z.-H. Zhao, Y.-K. Chang and W.-S. Li, Asymptotically almost periodic, almost periodic and pseudo-almost periodic mild solutions for neutral differential equations, Nonlinear Analysis: Real World Applications, 11 (4), 3037–3044, 2010.