Existence and stability for Volterra integral inclusions in two-metric spaces

Abstract

We establish novel existence, localization, and stability results for multi-valued $(\phi,\psi)$-contractions of Feng--Liu type in a two-metric framework: the contraction condition is imposed with respect to an auxiliary metric $\rho$, while completeness is assumed with respect to another metric $d$ that is topologically stronger than $\rho$ ($d\le R\rho$). Our approach yields a constructive retraction-displacement estimate and a new continuation principle. As a direct application, we prove the existence of solutions for Volterra type integral inclusions in the space of continuous functions equipped with the Bielecki norm, where the required metric comparison is naturally satisfied. Moreover, we establish generalized Ulam--Hyers stability, well-posedness in the sense of Reich and Zaslavski, Ostrowski stability, and data dependence of the fixed point set. The results extend and unify several recent developments in multi-valued fixed point theory and its applications to integral inclusions.