MASSERA-TYPE THEOREM TO A CLASS  OF SEMI-LINEAR DIFFERENTIAL EQUATIONS

Nguyen Thi Loan; Dang Van Tien

Nguyen Thi Loan Faculty of Fundamental Sciences, PHENIKAA University
Dang Van Tien Department of Science Basic, Hung Yen University of Technology and Education

Abstract

In the present work, by applying a Massera-type theorem, we derive sufficient conditions for the existence and uniqueness of periodic solutions to a class of partial functional differential equations of the form

1794277903692_1924329918512077757_5690375557446009841_14631b9d9a89d3e24aeb71cd73f9956d.jpg

here the operator A(t): E ⊇ D(A) → E is τ-periodic and possibly unbounded; the operator Γ(·, x(·)) takes values in a Banach space E, the function α takes values in the Banach space D(A), and are τ-periodic.

References

J.L. Massera, “The existence of periodic solutions of systems of differential equations,” Duke Math. J., vol.17, pp. 457-475, 1950.

J.K. Hale, O. Lopes, “Fixed point theorems and dissipative processes,” J. Diff. Equ., vol. 13, pp. 391-402, 1966.

A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer, Berlin, 1983.

R. Benkhalti, H. Bouzahir, K. Ezzinbi, “Existence of a periodic solution for some partial functional differential equations with infinite delay,” J. Math. Anal. Appl., vol. 256, pp. 257-280, 2001.

Nguyen Thieu Huy, Ngo Quy Dang, “Existence, uniqueness and conditional stability of periodic solutions to evolution equations,” J. Math. Anal. Appl., vol. 433, pp. 1190-1203, 2016. DOI: 10.1016/j.jmaa.2015.07.059.

N.T. Huy, N.T. Loan, V.T.N. Ha, “Periodic solutions and their conditional stability for partial neutral functional differential equations,” J. Evo. Equ., vol. 19, pp. 1091-1110, 2019. DOI: 10.1007/s00028-019-00511-x