If $\| x-y \|\leq \| x-y +t((I-T)x-(I-T)y)\|$ holds for all $x,y\in D(T)$ and $t>0,$ then $T$ is pseudo-contractive.

Question

I am currently reviewing Chidume and Zegeye where a claim was made that $T$ is pseudo-contractive if $$\| x-y \|\leq \| x-y +t((I-T)x-(I-T)y)\|$$ holds for all $x,y\in D(T)$ and $t>0.$ According to them, result from Kato enables us to use the above inequality to show that $T$ is pseudo-contractive if and only if, $\exists\;j(x-y)\in J(x-y)$ such that $$\langle Tx-Ty,j(x-y) \rangle\leq \| x-y \|^2,\forall\;x,y\in D(T) $$ where $T:D(T)\to R(T)$ but I can't see how. Can anyone show me some hints or how? Thanks for your time and efforts!

Answer 1

They're defining pseudo-contractivity of $T$ as $I-T$ being monotone (definition (M) on the first page of Kato), and applying Kato's lemma (denoted (M') on the second page) to the operator $I-T$ yields the inequality: $$\langle (I-T)x - (I-T)y, j(x-y) \rangle \geq 0$$ which can be rearranged: $$\langle x-y, j(x-y) \rangle \geq \langle Tx-Ty, j(x-y)\rangle$$ and $\langle x-y, j(x-y)\rangle = ||x-y||^2$ since $j$ was chosen with exactly this property in mind. Thus, $T$ is pseudo-contractive if and only if $I-T$ is monotone if and only if $\langle Tx-Ty, j(x-y) \rangle \leq ||x-y||^2$ for all $x,y \in D(T)$ and some $j \in F(x-y)$ (I'm using Kato's notation $F$ here). I hope I haven't missed anything, and if I haven't, I hope that helps.