Generalized fixed point framework for multivalued mappings with applications to convex optimization and fractional integral equations

In this article, new fixed-point results for coupled α∗-dominated multivalued mappings in ordered orbitally complete double-controlled metric spaces are established. We use a generalized rational-type Nashine–Wardowski Feng–Liu contraction to prove general common fixed-point theorems which unify, improve and extend significantly some well-known results appearing in the literature on nonlinear analysis. The current study includes broader relaxed assumptions like inequality orbital lower semi-continuity and α∗-dominatedadmissibility, enlarging the class of non-linearity on which one can apply these methods to study complex metric space structures. As an application, we consider the proximal point algorithm in convex optimization and show that our framework implies improved convergence characteristics and stability even under generalized metric conditions. Moreover, the results are utilized in proving the existence and uniqueness of solutions for a nonlinear Abel type fractional integral equation of first kind. Numerical comparisons also shows that the proposed iterative methods converges more rapidly and with smaller error than the classical Banach algorithms. In short, the results provide a unified and general method that connects fixed-point theory, convex optimization and fractional calculus that has its applications in various branches of mathematical and applied sciences.