Differential identities involving three generalized derivations on prime rings and Banach algebras

Abstract Let L be a non-central Lie ideal of a prime ring R . The primary objective of this paper is to characterize generalized derivations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℋ</m:mi> </m:math> {\mathcal{H}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℱ</m:mi> </m:math> {\mathcal{F}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒢</m:mi> </m:math> {\mathcal{G}} of R satisfying the following algebraic identity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>⊥</m:mo> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="script">ℱ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">𝒢</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mtext>for all </m:mtext> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>L</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> x\bot\mathcal{H}(y)=\mathcal{F}(x)y+y\mathcal{G}(x)\quad\text{for all }x,y\in L, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>⊥</m:mo> </m:math> {\bot} represents either the Lie product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">[</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[\,\cdot\,,\cdot\,]} , or the Jordan product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∘</m:mo> </m:math> {\circ} . Furthermore, we extend our results to Banach algebras.