On the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="script">E</a:mi> </a:math>-Hyperstability of the Inhomogeneous <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:mi>σ</d:mi> </d:math>-Jensen’s Functional Equation on Semigroups

Abstract

In this paper, we study the hyperstability problem for the well-known <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M3"> <a:mi>σ</a:mi> </a:math> -Jensen’s functional equation <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M4"> <c:mi>f</c:mi> <c:mfenced open="(" close=")"> <c:mrow> <c:mi>x</c:mi> <c:mi>y</c:mi> </c:mrow> </c:mfenced> <c:mo>+</c:mo> <c:mi>f</c:mi> <c:mfenced open="(" close=")"> <c:mrow> <c:mi>x</c:mi> <c:mi>σ</c:mi> <c:mfenced open="(" close=")"> <c:mrow> <c:mi>y</c:mi> </c:mrow> </c:mfenced> </c:mrow> </c:mfenced> <c:mo>=</c:mo> <c:mn>2</c:mn> <c:mi>f</c:mi> <c:mfenced open="(" close=")"> <c:mrow> <c:mi>x</c:mi> </c:mrow> </c:mfenced> </c:math> for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>S</m:mi> </m:math> , where <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M6"> <o:mi>S</o:mi> </o:math> is a semigroup and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M7"> <q:mi>σ</q:mi> </q:math> is an involution of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M8"> <s:mi>S</s:mi> </s:math> . We present sufficient conditions on <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M9"> <u:mi mathvariant="script">E</u:mi> <u:mo>⊂</u:mo> <u:msup> <u:mrow> <u:mfenced open="(" close=")"> <u:mrow> <u:msub> <u:mrow> <u:mi>ℝ</u:mi> </u:mrow> <u:mrow> <u:mo>+</u:mo> </u:mrow> </u:msub> </u:mrow> </u:mfenced> </u:mrow> <u:mrow> <u:msup> <u:mrow> <u:mi>S</u:mi> </u:mrow> <u:mrow> <u:mn>2</u:mn> </u:mrow> </u:msup> </u:mrow> </u:msup> </u:math> so that the inhomogeneous form of <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M10"> <z:mi>σ</z:mi> </z:math> -Jensen’s functional equation <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M11"> <bb:mi>f</bb:mi> <bb:mfenced open="(" close=")"> <bb:mrow> <bb:mi>x</bb:mi> <bb:mi>y</bb:mi> </bb:mrow> </bb:mfenced> <bb:mo>+</bb:mo> <bb:mi>f</bb:mi> <bb:mfenced open="(" close=")"> <bb:mrow> <bb:mi>x</bb:mi> <bb:mi>σ</bb:mi> <bb:mfenced open="(" close=")"> <bb:mrow> <bb:mi>y</bb:mi> </bb:mrow> </bb:mfenced> </bb:mrow> </bb:mfenced> <bb:mo>=</bb:mo> <bb:mn>2</bb:mn> <bb:mi>f</bb:mi> <bb:mfenced open="(" close=")"> <bb:mrow> <bb:mi>x</bb:mi> </bb:mrow> </bb:mfenced> <bb:mo>+</bb:mo> <bb:mi>φ</bb:mi> <bb:mfenced open="(" close=")"> <bb:mrow> <bb:mi>x</bb:mi> <bb:mo>,</bb:mo> <bb:mi>y</bb:mi> </bb:mrow> </bb:mfenced> </bb:math> for all <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M12"> <nb:mi>x</nb:mi> <nb:mo>,</nb:mo> <nb:mi>y</nb:mi> <nb:mo>∈</nb:mo> <nb:mi>S</nb:mi> </nb:math> , where the inhomogeneity <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M13"> <pb:mi>φ</pb:mi> </pb:math> is given, can be <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M14"> <rb:mi mathvariant="script">E</rb:mi> </rb:math> -hyperstable on <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" id="M15"> <ub:mi>S</ub:mi> </ub:math> .