Before revision:  We prove that if \psi is suitably smooth, the function t\mapsto P\{\zeta>t\} is twice continuously differentiable, and the condition 0<- \frac{d \log \mathbb{P\{\zeta > t\}}{dt} } \{\zeta>t\}}{dt}} <\lambda holds for the hazard rate of \zeta, then there exists a unique continuously differentiable function b solving the smoothed problem.
After revision:  We prove that if \psi is suitably smooth, the function t\mapsto P\{\zeta>t\} is twice continuously differentiable, and the condition 0<- \frac{d \log \mathbb{P\{\zeta > t\}}{dt} } \frac{d\log\mathbb{P\{\zeta>t\}}{dt}} <\lambda holds for the hazard rate of \zeta, then there exists a unique continuously differentiable function b solving the smoothed problem.