# Show that the variational formulation has at most one solution

We have the problem: $$-u''(x) + u(x) = f(x) ,\quad \quad x \in [0,L]$$ $$u(0) = 0$$ $$u'(L) + u(L) = 4$$ I then put it into variational form (hopefully correctly done) with introduction of $$v \in T(I) = \{v \in C^1 : v(0) =0 \}$$. $$u(L)v(L)+\int_0^L u'v\,' \, dx + \int_0^L uv\,dx = \int_0^L fv\,dx+ 4v(L)$$ We have bilinear operator as $$\therefore a(u,v) = u(L)v(L)+\int_0^L u'v\,' \, dx + \int_0^L uv\,dx$$ as well as $$f(v) = \int_0^L fv\,dx+ 4v(L)$$ I am not sure on how to go about proving/showing that the variational form has at most one solution. The Lax-Milgram Theorem gets thrown around a lot in proving uniqueness. I do not know how to use it in application though. Please guide me or recommend a path/resource that I must follow in order to show this.