On the existence of solutions to a new k-dimensional p-Laplacian fractional inclusion system

Abstract This study investigates the presence of positive solutions for a new category of coupled systems characterized by k -dimensional fractional differential inclusions that utilize the p -Laplacian operator. The analysis is conducted under specific initial and boundary conditions that ensure the mathematical soundness of the model. Due to these systems’ intricate nature and nonlinearity, conventional analytical techniques may not be readily applicable. To address these difficulties, we apply a fixed-point theorem on a cone, which is instrumental in demonstrating the existence of positive solutions within our framework. To enhance our findings, we develop two iterative sequences that help us converge towards an approximate solution. These sequences are meticulously crafted to refine the solution space while upholding the required mathematical rigor. This iterative methodology ensures that our approach remains computationally viable while yielding significant approximations for practical use. Moreover, we establish appropriate conditions on the nonlinear components to ensure the efficacy of our method in producing solutions that meet both theoretical and computational standards. A specific numerical example underscores the importance of our results and showcases the practical relevance of our theoretical conclusions.