Let \(y\) be a continuous positive function vanishing at 0 : \(y(0)=0\). Prove that there exists a unique pair of functions \((z, k)\) such that

(i) \(k(0)=0\), where \(k\) is an increasing continuous function

(ii) \(z(t)+k(t)=y(t), z(t) \geq 0\)

(iii) \(\int_{0}^{t} \mathbb{1}_{\{z(s)>0\}} d k(s)=0\)

(iv) \(\forall t, \exists d(t) \geq t, z(d(t))=0\)

\(k^{*}(t)=\inf _{s \geq t}(y(s))\).