I have the following bilinear form $B:L^{2}(I) \times L^{2}(I) \rightarrow \mathbb{R}$, where \begin{align} B(u,v)= \int_{I} \alpha u v dx - \int_{I} \left( \int_{x}^{x_{max}} \beta(x,y) \alpha(y) u(y) dy \right) v(x) dx , \end{align} where $I=[0,x_{max}]$, and $\alpha$ and $\beta$ are positive functions.
I want to show that $B$ is coercive (to show well posedness of a related problem), which means that $\exists c>0$ such that \begin{align} B(v,v) \geq c \| v \|_{L^{2}(I)}. \end{align} Any hint?
This is not true, unless more conditions on $\alpha$ and $\beta$ are specified. Take $\alpha$, $\beta$ constant, $x_\max=1$, and take $u=1$: $$ B(u,u) = \alpha - \int_I \beta\alpha (1-x) \ dx = \alpha (1-\beta/2). $$ Hence $B$ is not coercive if $\beta=2$ and $\alpha>0$ constant.