Convergence of solutions to set optimization problems with variable ordering structures

In this article, we consider set optimization problems with variable ordering structures. Within the framework of the set less order relation with variable ordering structures, we investigate the existence, the upper convergence, and the lower convergence of solutions to such problems in the image spaces. For both the existence and the upper convergence of solutions, we employ new techniques to obtain various results without assuming the compactness of the constraint sets. Additionally, we utilize the domination property concerning the variable ordering cones to address the lower convergence of solutions. The obtained results are presented in several versions from different aspects for convenient comparison with existing results. Many examples are provided to illustrate the novelty of our results or to compare them with existing ones in the literature.