# SEMILINEAR PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH THE NONLINEAR OPERATOR IS LIPSHITZ

### Abstract

On a Banach space X, we consider the semilinear partial functional differential equations of the form

(1)

where the mapping t → B(t) is a possibly unbounded operator such that the family (B(t))t≥0 generates an evolution family (U(t, s))t≥s≥0 have an exponential dichotomy; the operator h : BC(R_{+} , X ) → BC(R_{+} , X ) is Lipshitz; f (t)∈L(M, X ) ; xt is the history function defined by x_{t} (s) := x(t + s) for s ∈[-θ , 0] . By using the Lipschitz of the nonlinear operator and the inequalities of evolution family in Banach spaces, we prove the equivalence of the two forms solutions to equation (1) as a mild solution and a Lyapunov-perron solution.

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*UTEHY Journal of Science and Technology*,

*35*, 43-47. Retrieved from https://tapchi.utehy.edu.vn/index.php/jst/article/view/554