Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Abstract

Let H be a real Hilbert space and let T: H Ä 2 H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v # T &1 0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration. Academic Press

Modified Proximal point algorithms for finding a zero point of maximal monotone operators, generalized mixed equilibrium problems and variational inequalities

“…In 2000, Kamimura and Takahashi [19] proved the following strong convergence theorem in Hilbert spaces, by the following algorithm 12) where J r = (I + rA) -1 J, then the sequence {x n } converges strongly to P A −1 0 (x), where P A −1 0 is the projection from H onto A -1 (0). These results were extended to more general Banach spaces see [20,21].…”

Proximal point algorithms involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces

“…However, as shown by Güler [8], the PPA does not necessarily converge strongly in general. In 2000, Kamimura-Takahashi [11] combined the PPA with Halpern's algorithm [9] so that the strong convergence is guaranteed.…”

On the multilevel nonlinear problem and its convergence algorithms

“…And the problem studied by Theorem 4.13 is one to find a common solution to some variational inclusion problems. Theorem 4.7 and Theorem 4.12 generalize many known results in the literature, for example, [3,4,9,11,12,16,17,19,22,24,25] and references therein.…”

“…For more detail, see the References [9,11,19,25]. Besides of the proximal point algorithm, some other iterative algorithms are also introduced in [4,16,17], which are used to find the approximation solution of the (CVIP).…”