On the general solutions of three functional equations
Tóm tắt
The general solutions of the functional equations
$$\begin{aligned} F(pq)= & {} q^\alpha G(p)+p^\alpha H(q)+K(p)L(q),\\ F(pq)= & {} q^\beta G(p)+p^\alpha H(q)+cK(p)K(q) \end{aligned}$$
and
$$\begin{aligned} f(pq)=q^\alpha f(p)+p^\alpha f(q) +cg(p)g(q) \end{aligned}$$
with
$$g(1)=0$$
and c, a given nonzero real constant, are obtained. Here F, G, H, K, L, f and g are real-valued functions each with domain I, the unit closed interval and
$$1\ne \alpha >0$$
,
$$\alpha \in {\mathbb {R}}$$
;
$$1\ne \beta >0$$
,
$$\beta \in {\mathbb {R}}$$
.
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