An efficient iterative method for solving split variational inclusion problem with applications

<p style='text-indent:20px;'>A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.</p>

Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

Modified Proximal Algorithms for Finding Solutions of the Split Variational Inclusions

We investigate the split variational inclusion problem in Hilbert spaces. We propose efficient algorithms in which, in each iteration, the stepsize is chosen self-adaptive, and proves weak and strong convergence theorems. We provide numerical experiments to validate the theoretical results for solving the split variational inclusion problem as well as the comparison to algorithms defined by Byrne et al. and Chuang, respectively. It is shown that the proposed algorithms outrun other algorithms via numerical experiments. As applications, we apply our method to compressed sensing in signal recovery. The proposed methods have as a main advantage that the computation of the Lipschitz constants for the gradient of functions is dropped in generating the sequences.