Can I Find an Expression Relating x to x1 and x2 while Decomposing Operator A?

[shaiguy6]

So I am trying to decompose a linear operator A, in the following manner. I am trying to solve Ax=y for x, and I also have that A=(L-G), so I am trying to solve (L-G)x=y. y is given, and so are L, A, and G. Now, I also know the solutions to Lx₁=y and Gx₂=0. I'd like to somehow find an expression relating x to x₁ and x₂. Any help would be appreciated.

 

[shaiguy6]

In case this wasn't clear, I'll write it over this way:

(L-G)v = y
Lv₁=y
Gv₂=0

or equivalently:

(L-G)^-1y=v
L^-1y=v₁
G^-10=v₂

v₁ and v₂ and v are not equal, they are different. I'd like to have an equation for v in terms of v₁ and v₂.