Rockafellar’s Proximal Point Algorithm for a Finite Family of Monotone Operators

is not properly contained in the graph of any other monotone mapping. It is known that T is maximal iff R(I+rT ) = H for every r > 0, where R(I+rT ) = ∪ {z+rTz : z ∈ H,Tz ̸= ∅}. Monotone operators have proven to be a key class of objects in modern Optimization and Analysis; see, e.g., the books [1-7] and the references therein. Let us consider the zero point problem for a monotone operator T on a real Hilbert space H, that is, finding a point z ∈ H, such that 0 ∈ Tz. This problem is closely related to many kinds of important problems, such as minimization problems, saddle point problems, equilibrium problems and others. In order to approximate the solution to this problem, various types of iterative schemes have been proposed. One of the most important methods is Rockafellar proximal point algorithm [8], which generates a sequence {xn} according to the relation: xn+1 = J T rn(xn + en), (1.1)