Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

Abstract

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msubsup> <mrow> <mtext> </mtext> </mrow> <mrow> <mn>0</mn> </mrow> <mrow> <mi>c</mi> </mrow> </msubsup> <msubsup> <mrow> <mi>D</mi> </mrow> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> <mrow> <mi>γ</mi> </mrow> </msubsup> <mi mathvariant="fraktur">x</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> <mo>=</mo> <mi>α</mi> <mi mathvariant="fraktur">x</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> <mo>+</mo> <mi mathvariant="fraktur">P</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> <mo>,</mo> <mi mathvariant="fraktur">x</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> </mrow> </mfenced> <mo>+</mo> <mi mathvariant="fraktur">A</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> <mi mathvariant="fraktur">W</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> <mo>,</mo> <mi mathvariant="fraktur">I</mi> <mo>∈</mo> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>T</mi> </mrow> </mfenced> <mo>,</mo> <mi mathvariant="fraktur">x</mi> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <msub> <mrow> <mi mathvariant="fraktur">x</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> , in which <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>γ</mi> <mo>∈</mo> <mfenced open="(" close=")"> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </mfenced> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </math> is the fuzzy metric space and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>I</mi> <mo>=</mo> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>T</mi> </mrow> </mfenced> </math> is a real line interval. With the help of few conditions on functions <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi mathvariant="fraktur">P</mi> <mo>:</mo> <mi>I</mi> <mo>×</mo> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> <mo>⟶</mo> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi mathvariant="fraktur">W</mi> <mfenced open="(" close=")"> <mrow> <mi mathvariant="fraktur">I</mi> </mrow> </mfenced> </math> is control and it belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi mathvariant="fraktur">A</mi> <mo>∈</mo> <mi>F</mi> <mfenced open="(" close=")"> <mrow> <mi>I</mi> <mo>,</mo> <mi>L</mi> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>E</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </mrow> </mfenced> </mrow> </mfenced> </math> , and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>α</mi> </math> stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.