Studying Maximal Monotone Operators and the Minty-Browder Theory, I have stumbled accross the following expression which I cannot justify.
It can be found at Page 40, of the book "Nonlinear Differential Equations of Monotone types in Banach Spaces, Viorel Barbu, Springer". It is a step in proving that for the norm of Yosida estimation $A_\lambda$, it is : $\left\|A_\lambda\right\|\leq \left\|Ax\right\|= \inf\left\{\|y\|: y \in Ax\right\}$. Note that $A$ is a monotone operator.
For those not familiar, $A_\lambda$ is defined as $A_\lambda = \lambda^{-1}J\left(x-x_\lambda\right)$ where $J_\lambda x = x_\lambda$ and $x_\lambda$ is the unique solution to the equation: $0 \in J(x_\lambda-x) + \lambda Ax_\lambda$. Note that $J$ is the duality mapping $J:X \to 2^{X^*}$ or simply to $X^*$ and also $\left\langle x,x^*\right\rangle \in A$ where $x \in X$ and $x^* \in X^*$.
The line states:
$$0 \leq \left\langle x - J_\lambda x, x^* - A_\lambda x \right\rangle \leq \|x^*\|\|x-x_\lambda\| - \lambda^{-1}\|x-x_\lambda\|^2$$
Now, decomposing it, I understand how the first inequality is derived, as $A_\lambda x \in AJ_\lambda x$ and $A$ is monotone. Thus: $$\left\langle x - J_\lambda x, x^* - A_\lambda x \right\rangle \geq 0$$ Now, it seems really weird to me that straight from the duality brackets $\left\langle \cdot, \cdot\right\rangle$ there is an inequality involving norms derived. All I can see, is: \begin{align*} \left| \left\langle x - J_\lambda x, x^* - A_\lambda x\right\rangle \right| &\leq \left\| x - J_\lambda x\right\| \left\|x^* - A_\lambda x\right\| \\ &\leq \left\|x-J_\lambda x\right\| \left( \left\|x^*\right\| + \left\|Ax_\lambda\right\|\right) \\ &= \left\| x - x_\lambda\right\| \left( \|x^*\| + \lambda^{-1}\left\|x-x_\lambda\right\|\right) \\ &= \left\|x^*\right\| \left\|x-x_\lambda\right\| +\lambda^{-1} \left\|x-x_\lambda\right\|^2 \end{align*} Any help will be greatly appreciated.