Fixed Point Theory in Sets with Graph

The goal of this sabbatical leave is to develop and provide new mathematical tools to be used and applied to the study of various problems arising in different areas such as artificial intelligence, economics, physics and engineering. I will mainly focus on both algebraic and combinatorial properties of graphs that can guarantee the existence and uniqueness of fixed points for single and multivalued mappings on metric spaces or discrete sets endowed with a graph or a partial order. We hope to extend the obtained results to more general metric spaces like modular function spaces. In addition, we will use various nonlinear examples which come from fractional differential equations and impulsive differential inclusions to enrich the expected obtained fixed-point theorems via graphs. This sabbatical leave year will provide me an exciting opportunity to develop and make further advances on discrete metric spaces which will benefit King Fahd University of Petroleum & Minerals and the Kingdom of Saudi Arabia. My research enhancement will also help in the strengthening of the newly formed scientific research group in Fixed-Point Theory and Applications at KFUPM. We plan to attract some of our graduate students to do their graduate work in this area. For now, I do have three master graduate students who are ready to gain the benefits of this project in their future career.