Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces

Abstract The main focus of this study is to discuss the existence of positive <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega -periodic mild solutions for evolution equation with delay in an ordered Banach space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> </m:math> E . Under the ordered conditions of the growth exponent of the nonlinearity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g with respect to the semigroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">T</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mspace width=".25em"/> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{T}}\left(t)\hspace{.25em}\left(t\ge 0) or the first eigenvalue of the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">A</m:mi> </m:math> {\mathcal{A}} , the existence results of mild solutions and positive mild solutions for evolution equation are obtained by applying the Poincaré mapping and monotone iterative method, and adding appropriate conditions to the nonlinearity term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g without assuming the existence of upper and lower solutions. Finally, we give an example to verify the applicability of our abstract results.