I want to prove that if the constant $k>0$ is sufficiently large, then for every $f\in L^2(0,1)$, there exists a unigue $u\in H^2(0,1) $ satisfying $$-u''+ku=f \; on \;(0,1) $$ $$u'(0)=0,\;\; u'(1)=u(1)$$
for this problem, i want to find appropriate Bilinear form B to satisfy the condition for Lax-Milgram thorem (Similar as Energy estimates so that there is such $k$).
so for $B[u,v]=\int -u''vdx=-u(1)v(1)+\int u'v'dx$ it cannot be bounded by $\alpha ||u||||v||$ for some $\alpha$ since the constant must depend on u,v
and also not sure about the coercivity.
Is there proper bilinear form to satisfy the condition for Lax-Milgram theorem to prove the existence of unique solution u?
In your case $$B(u,v)=-u(1)v(1)+\int\limits_0^1 {u'v' + kuvdx}$$ It is clear that this bilinear form is continuous over $$V = \left\{ {u \in {H^1},u(0) = 0,u'(1) = u(1)} \right\}$$ we can prove that $B$ is continuous by using the embedding of ${H^1}(0,1)$ into ${L^\infty }(0,1)$.