In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method and the regularized iteration method. As applications, we consider the algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature. MSC: 47H10; 49J40; 54H25

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The split equality problem has board applications in many areas of applied mathematics. Many researchers studied this problem and proposed various algorithms to solve it. From the literature we know that most algorithms for the split equality problems came from the idea of the projected Landweber algorithm proposed by Byrne and Moudafi (Working paper UAG, 2013), and few algorithms came from the idea of the alternating CQ-algorithm given by Moudafi (Nonlinear Anal. 79:117-121, 2013). Hence, it is important and necessary to give new algorithms from the idea of the alternating CQ-algorithm. In this paper, we first present a hybrid projected Landweber algorithm to study the split equality problem. Next, we propose a hybrid alternating CQ-algorithm to study the split equality problem. As applications, we consider the split feasibility problem and linear inverse problem. Finally, we give numerical results for the split feasibility problem to demonstrate the efficiency of the proposed algorithms. MSC: 49J53; 49M37; 90C25