A self-adaptive projection-type method for nonlinear multiple-sets split feasibility problem

In this article, we consider the nonlinear multiple-sets split feasibility problem (NMSFP), which is to find a vector x* such that x* ∈ C and F(x*) ∈ Q, where C and Q are intersections of a family of simple nonempty closed convex sets in ℝⁿ and ℝ^m, respectively, and F: ℝⁿ → ℝ^m is a continuous mapping. When the mapping F is linear, the problem reduces to the multiple-sets split feasibility problem (MSFP), see e.g. Censor et al. [Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl. 21 (2005), pp. 2071-2084]. While MSFP has been considered extensively and many numerical methods have been designed to solve it, there are few results on NMSFP. In this article, after introducing the nonlinear multiple-sets split feasibility model, we propose a projection-type algorithm to solve it. Under some suitable conditions, we prove the global convergence of the proposed algorithm. Finally, we use an example to illustrate the effect of the proposed algorithm.