Let be a metric space and be a multi-valued operator. is said to be a generalized -weak contraction if there exist a constant and a function , with for every , such that
for all . When for all , we say that is a -weak contraction. The authors prove that a generalized -weak contraction has a fixed point whenever is complete and has nonempty bounded and closed values. Moreover, if is a -weak contraction, then for any there exists an orbit converging to a fixed point of for which the following estimate holds:
for a certain constant .