On a separation for the Cauchy equation on spheres (Q452432)

Let \(X\) be a real linear space, \(Z\) a nonempty set and let \(\gamma: X\to Z\), \(\alpha: X\to [0,\infty)\), \(\upsilon: X\to X\) be given auxiliary mappings satisfying some (variants of) assumptions. Suppose that \(p,q: X\to \mathbb{R}\), \(q\leq p\), satisfy conditional functional inequalities NEWLINE\[NEWLINE \gamma(x)=\gamma(y)\quad\Longrightarrow \quad p(x+y)\leq p(x)+p(y)\eqno{(1)} NEWLINE\]NEWLINE NEWLINE\[NEWLINE \gamma(x)=\gamma(y)\quad\Longrightarrow \quad q(x+y)\geq q(x)+q(y)\eqno{(2)} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE |q(x)-p(\upsilon(x))|\leq\alpha(x),\qquad x\in X. NEWLINE\]NEWLINE The authors prove that in such a case there exists a unique additive mapping \(A: X\to\mathbb{R}\) separating \(p\) and \(q\), i.e., such that \(q\leq A\leq p\) on \(X\).NEWLINENEWLINEThe main result is formulated in a quite general form which admits, in particular, discontinuous functionals \(\alpha\) satisfying the assumptions. However, a natural realization of the result is with \(\alpha\) being a norm. In particular, in a real inner product space \(X\) with \(\dim X\geq 2\) or in a real normed space \(X\) with \(\dim X\geq 3\), if \(p,q: X\to \mathbb{R}\), \(q\leq p\), satisfy (1) and (2) as well as, with some \(\lambda\geq 0\) and \(r\in (0,1)\cup(1,\infty)\), NEWLINE\[NEWLINE |q(x)-p(x)|\leq \lambda\|x\|^r,\qquad x\in X, NEWLINE\]NEWLINE or NEWLINE\[NEWLINE |q(x)-p(-x)|\leq \lambda\|x\|^r,\qquad x\in X, NEWLINE\]NEWLINE then there exists a unique additive mapping \(A\) separating \(p\) and \(q\).NEWLINENEWLINEThe results improve these of \textit{W. Fechner} [Ann. Math. Sil. 21, 31--40 (2007; Zbl 1155.39014)]. Proofs rely on earlier papers of \textit{R. Ger} and \textit{J. Sikorska} [Pr. Nauk. Uniw. Ślask. Katowicach 1665, Ann. Math. Silesianae 11, 89--99 (1997; Zbl 0894.39009)] and \textit{J. Sikorska} [Nonlinear Anal., Theory Methods Appl. 70, No. 7, A, 2673--2684 (2009; Zbl 1162.39018)].