Nadler’s type principle with high order of convergence

Recently Proinov [P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.09.008] generalized Banach contraction principle with high order of convergence. We extend some results of Proinov to the case of multi-valued maps from a complete metric space XX into the space of all nonempty proximinal closed subsets of XX. Our results not only generalize Nadler’s fixed-point theorem (in the case when TT is a mapping from a complete metric space XX into the space of all nonempty proximinal closed subsets of XX) but also gives high order of convergence. As an application, we obtain an existence theorem for first-order initial value problem.