Set-valued contractions with an application to Fredholm integral inclusions in $m_v^b-$metric spaces

This study investigates set-valued contractions within the framework of $m_v^b-$metric spaces, extending classical contraction principles. By introducing and examining the Hausdorff $m_v^b-$metric, we establish a foundation for set-valued fixed point theorems, thereby contributing significantly to this area of research. Our findings generalize several well-known contraction concepts, including those of Banach, Sehgal, Wardowski, Altun, Bianchini, and Nadler, within the context of $m_v^b-$metric spaces. These advancements have practical implications, particularly in the study of nonlinear systems and the mathematical model of Fredholm integral inclusions. The results presented here emphasize the growing importance of set-valued fixed points and pave the way for further exploration and application across various scientific and engineering domains.