The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included. MSC: 47J25; 47N20; 34G20; 65J15

No abstract

No abstract

We introduce theQ-lasso which generalizes the well-known lasso of Tibshirani (1996) withQa closed convex subset of a Euclideanm-space for some integerm≥1. This setQcan be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the lasso. Solutions of theQ-lasso depend on a tuning parameterγ. In this paper, we obtain basic properties of the solutions as a function ofγ. Because of ill posedness, we also applyl1-l2regularization to theQ-lasso. In addition, we discuss iterative methods for solving theQ-lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for continuous monotone mapping, solution of equilibrium problem and a common fixed point of finite family of asymptotically regular uniformly continuous relatively asymptotically nonexpansive mappings in Banach spaces. Our scheme does not involve computation of C n+1 from C n for each n ≥ 1. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. Mathematics Subject Classification (2000): 47H05, 47H09, 47H10, 47J05, 47J25