Existence and asymptotic stability of periodic solutions for a class of neutral delayed evolution equation

The purpose of this work is to study the existence and asymptotic stability of ω -periodic mild solutions to a class of neutral delayed evolution equation in Banach space X <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfrac> <m:mrow> <m:mtext>d</m:mtext> </m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mi>c</m:mi> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mi>c</m:mi> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>z</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>τ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>,</m:mo> </m:math> $$d/dt(z(t)-cz(t-δ ))+A(z(t)-cz(t-δ ))=f(t, z(t), z(t-τ )), t∈ R,$$ where ∣ c ∣ < 1, the constants δ , τ > 0 are defined as time lags, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mi mathvariant="script">D</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⊂</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> A : D(A)⊂ X→ X is a closed linear operator, − A generates a strongly continuous semigroup T ( t )( t ⩾ 0), and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>×</m:mo> <m:msup> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> f : R×X²→ X is continuous function which is ω -periodic in t . Firstly, by using theory of operator semigroup and fixed point theorems, the existence results of ω -periodic mild solutions to the above equation are obtained under the assumption that the T ( t )( t ⩾ 0) is compact. Next, in the case that T ( t )( t ⩾ 0) is non-compact, we investigate the asymptotic stability of periodic solution for the equation based on the contract