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In this paper, we first study a hierarchical problem of Baillon's type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon's type. Our result of the ergodic theorem of Baillon's type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results.As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and fixed point set constraints; mathematical programming with a system of mixed type equilibria and fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem. Our results will have many applications in diverse fields of science.

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In this paper, we introduced two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray's type theorem for these nonlinear mappings. Next, we prove weak convergence theorems for Moudafi's iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

In this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of fixed points for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study fixed points of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumption is needed.