If we consider a functional $J(y)=\int_a^b F(x,y,y^{\prime})dx$ then the variation is defined as the linear functional $$\delta J[h]=\frac{d}{d\alpha}F(x,y+\alpha h,(y+\alpha h)^{\prime})\Big|_{\alpha=0}.$$ I believe this is the same as the definition in Gelfand-Fomin.
But it is possible to consider a more general variation, $y(x,\alpha)$ and then to define $$\delta J[\delta y]=\frac{d}{d\alpha}F(x,y(x,\alpha),y^{\prime}(x,\alpha)\Big|_{\alpha=0}$$.
With $\delta y=\frac{dy}{d\alpha}\Big|_{\alpha=0}$.
Are these definitions correct ? Or should one write $\delta y=\frac{dy}{d\alpha}\Big|_{\alpha=0}\alpha$ ?
Now Elsgolc wites on page 22 that the two derivatives do not coincide. Is this true ? What is an example ?
(Some thoughts: If one writes with Gelfand-Fomin: $$J[y+h]-J[y]=\delta J[h]+\epsilon(h)||h||$$ where $\epsilon(h)\to 0$ as $||h||\to 0$, then letting $h=y(x,\alpha)-y(x,0)$ we have
$$J[y(x,\alpha)]-J[y(x,0)]=\delta J[y(x,\alpha)-y(x,0)]+\epsilon(h)||h||$$ and dividing by $\alpha$ taking limit we get $$\frac{d}{d\alpha}F(x,y(x,\alpha),y^{\prime}(x,\alpha)\Big|_{\alpha=0}=\delta J[\delta y]$$ assuming of course that the limit $\lim\limits_{\alpha\to 0} \frac{|h(x,\alpha)|}{\alpha}$ exists.)