Resolvent Positive Operators and Positive Fractional Resolvent Families

Media type: E-Article
Title: Resolvent Positive Operators and Positive Fractional Resolvent Families
Contributor: Li, Nan-Ding; Liu, Ru; Li, Miao
imprint: Hindawi Limited, 2021
Published in: Journal of Function Spaces
Language: English
DOI: 10.1155/2021/6418846
ISSN: 2314-8888; 2314-8896
Keywords: Analysis
Origination:
Footnote:
Description: <jats:p>This paper is concerned with positive <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>α</mi> </math> </jats:inline-formula>-times resolvent families on an ordered Banach space <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>E</mi> </math> </jats:inline-formula> (with normal and generating cone), where <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </math> </jats:inline-formula>. We show that a closed and densely defined operator <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>A</mi> </math> </jats:inline-formula> on <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>E</mi> </math> </jats:inline-formula> generates a positive exponentially bounded <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>α</mi> </math> </jats:inline-formula>-times resolvent family for some <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </math> </jats:inline-formula> if and only if, for some <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi>ω</mi> <mo>∈</mo> <mi>ℝ</mi> </math> </jats:inline-formula>, when <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>λ</mi> <mo>></mo> <mi>ω</mi> <mo>,</mo> <mi>λ</mi> <mo>∈</mo> <mi>ρ</mi> <mfenced open="(" close=")"> <mrow> <mi>A</mi> </mrow> </mfenced> </math> </jats:inline-formula>, <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>R</mi> <mfenced open="(" close=")"> <mrow> <mi>λ</mi> <mo>,</mo> <mi>A</mi> </mrow> </mfenced> <mo>≥</mo> <mn>0</mn> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi mathvariant="normal">sup</mi> <mfenced open="{" close="}"> <mrow> <mfenced open="‖" close="‖"> <mrow> <mi>λ</mi> <mi>R</mi> <mfenced open="(" close=")"> <mrow> <mi>λ</mi> <mo>,</mo> <mi>A</mi> </mrow> </mfenced> </mrow> </mfenced> <mtext>: </mtext> <mi>λ</mi> <mo>≥</mo> <mi>ω</mi> </mrow> </mfenced> <mo><</mo> <mo>∞</mo> </math> </jats:inline-formula>. Moreover, we obtain that when <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </math> </jats:inline-formula>, a positive exponentially bounded <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <mi>α</mi> </math> </jats:inline-formula>-times resolvent family is always analytic. While <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14"> <mi>A</mi> </math> </jats:inline-formula> generates a positive <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15"> <mi>α</mi> </math> </jats:inline-formula>-times resolvent family for some <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16"> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </math> </jats:inline-formula> if and only if the operator <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17"> <msup> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>α</mi> </mrow> </msup> <mo>−</mo> <mi>A</mi> </mrow> </mfenced> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math> </jats:inline-formula> is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.</jats:p>
Access State: Open Access

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