MathDB

Let

\theta

and

p(p<1)

) be nonnegative real numbers.
Suppose that

f:X\to Y

is mapping with

f(0)=0

and

\left |\left| 2f\left (\frac{x+y}{2}\right )-f(x)-f(y) \right |\right|_Y \leq \theta\left (\left |\left |x\right |\right |_X^p +\left |\left |y\right |\right |_X^p \right )

for all

x

y\in \mathbb{Z}

with

x\perp y

where

X

is an orthogonality space and

Y

is a real Banach space.
Prove that there exists a unique orthogonally Jensen additive mapping

T:X\to Y

, namely a mapping

T

that satisfies the so-called orthogonally Jensen additive functional equation

2f\left (\frac{x+y}{2}\right )=f(x)+f(y)

for all

x

y\in \mathbb{X}

with

x\perp y

, satisfying the property

\left |\left|f(x)-T(x) \right |\right|_Y \leq \frac{2^p\theta}{2-2^p}\left |\left |x\right |\right |_X^p

for all

x\in X

Problem Statement