Self-adaptive Forward–Backward Contraction-type Methods for Generalized Split Feasibility Problems

Based on the recent important results of Takahashi–Xu–Yao [Set-Valued and Variational Analysis 23(2015), 205–221] and other related results on split feasibility problems, we study a certain class of generalized split feasibility problems which includes many other split-type problems. We propose some new self-adaptive forward–backward contraction-type methods and prove that they converge strongly to a minimum-norm solution of the generalized split feasibility problems in real Hilbert spaces. As a by-product, we obtain self-adaptive methods for solving other classes of generalized split feasibility problems in real Hilbert spaces. Finally, we apply our results to solve an optimal control problem and an image restoration problem through numerical implementations, and compare our methods with related strongly convergent methods in the literature.