Almost automorphic solutions of semilinear evolution equations

Abstract

We are concerned with the semilinear differential equation in a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals upper A x left-parenthesis t right-parenthesis plus upper F left-parenthesis t comma x left-parenthesis t right-parenthesis right-parenthesis comma t element-of double-struck upper R comma"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mtext> </mml:mtext> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x’(t)=Ax(t)+F(t,x(t)),\;\ t\in \mathbb {R} \, ,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generates an exponentially stable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -semigroup and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis t comma x right-parenthesis colon double-struck upper R times double-struck upper X right-arrow double-struck upper X"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">F(t,x): \mathbb {R} \times \mathbb {X} \to \mathbb {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a function of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis t comma x right-parenthesis equals upper P left-parenthesis t right-parenthesis upper Q left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F(t,x)=P(t)Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Under appropriate conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.