[tex]g(x)=h(2x)-x\implies h(2x)=g(x)+x[/tex]
[tex]f(x)=g(x+1)\implies g(x)=f(x-1)[/tex]
Taken together, we get
[tex]h(2x)=f(x-1)+x[/tex]
Since [tex]2018=2\cdot1009[/tex], the above gives
[tex]a^3=f(1008)+1009[/tex]
We have
[tex]f(y)=\displaystyle\sum_{i=0}^{1008}(2018-2i)y=1019090y[/tex]
so that
[tex]a^3=1019090\cdot1008+1009[/tex]
[tex]a^3=1027243729=1009^3[/tex]
[tex]\implies a=1009[/tex]
