A global existence theorem for a nonautonomous differential equation in a Banach space

Abstract

Suppose that <italic>X</italic> is a real or complex Banach space and that <italic>A</italic> is a continuous function from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis times upper X"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty ) \times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <italic>X</italic> . Suppose also that there is a continuous real valued function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis t comma dot right-parenthesis minus rho left-parenthesis t right-parenthesis upper I"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A(t, \cdot ) - \rho (t)I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is dissipative for each <italic>t</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this note we show that, for each <italic>z</italic> in <italic>X</italic> , there is a unique differentiable function <italic>u</italic> from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <italic>X</italic> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis 0 right-parenthesis equals z"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">u(0) = z</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="u prime left-parenthesis t right-parenthesis equals upper A left-parenthesis t comma u left-parenthesis t right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>u</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:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</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:mrow> <mml:annotation encoding="application/x-tex">u’(t) = A(t,u(t))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <italic>t</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo>