With the help of the Meir-Keeler contraction method and the Mann-type method, two adaptive inertial iterative schemes are introduced for finding solutions of the split variational inclusion problem in Hilbert spaces. The strong convergence of the suggested algorithms are guaranteed by a new stepsize criterion that does not require calculation of the bounded linear operator norm. Some numerical experiments and applications in signal recovery problems are given to demonstrate the efficiency of the proposed algorithms.

In this paper, we discuss the split monotone variational inclusion problem and propose two new inertial algorithms in infinite-dimensional Hilbert spaces. As well as, the iterative sequence by the proposed algorithms converges strongly to the solution of a certain variational inequality with the help of the hybrid steepest descent method. Furthermore, an adaptive step size criterion is considered in suggested algorithms to avoid the difficulty of calculating the operator norm. Finally, some numerical experiments show that our algorithms are realistic and summarize the known results.

This paper attempts to focus on the split common fixed point problem for demicontractive mappings. We give an accelerated hybrid projection algorithm which combines the hybrid projection method and the inertial technique. The strong convergence theorems of this algorithm are obtained under mild conditions by a self-adaptive step-size sequence, which does not need prior knowledge of operator norms. Some numerical experiments in infinite Hilbert space are provided to illustrate the reliability and robustness of the algorithm and also to compare it with existing ones.