Convergence of the proximal point method for metrically regular mappings

In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) ∋ 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions g_n : X → Y with g_n(0) = 0 that are Lipschitz continuous in a neighborhood of the origin. Then pick an initial guess x₀ and find a sequence x_n by applying the iteration g_n(x_n1^-x_n)+T(x_n+1) ∋ 0 for n = 0,1,... We prove that if the Lipschitz constants of g_n are bounded by half the reciprocal of the modulus of regularity of T, then there exists a neighborhood O of x̅ (x̅ being a solution to T(x) ∋ 0) such that for each initial point x₀ ∈ O one can find a sequence x_n generated by the algorithm which is linearly convergent to x̅. Moreover, if the functions g_n have their Lipschitz constants convergent to zero, then there exists a sequence starting from x₀ ∈ O which is superlinearly convergent to x̅. Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular.