Existence Theorems for Strongly Accretive Operators in Banach Spaces

Let X be a (real) Banach space and let https://www.w3.org/1998/Math/MathML"> α : [ 0 , ∞ ) → [ 0 , ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10017.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be a function for which https://www.w3.org/1998/Math/MathML"> α ( 0 ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10018.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the https://www.w3.org/1998/Math/MathML"> l i m   i n f r → r 0   α ( r ) > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10019.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for every https://www.w3.org/1998/Math/MathML"> r 0 > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10020.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . An operator https://www.w3.org/1998/Math/MathML"> A : D ( A ) ⊂ X → 2 X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10021.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called α-strongly accretive if for each https://www.w3.org/1998/Math/MathML"> x , y ∈ D ( A ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10022.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> there exists https://www.w3.org/1998/Math/MathML"> j ∈ J ( x - y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10023.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> such that 1 https://www.w3.org/1998/Math/MathML"> 〈 u − v , j 〉 ≥ α ( ‖ x − y ‖ ) ‖ x − y ‖ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq10024.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>