Darbo-Type $ \mathcal{Z}_{\rm{m}} $ and $ \mathcal{L}_{\rm{m}} $ contractions and its applications to Caputo fractional integro-differential equations

Abstract

Darbo type $ \mathcal{Z}_{\rm{m}} $-contraction and Darbo type $ \mathcal{L}_{\rm{m}} $-contraction are introduced and some fixed point results are established for such contraction mappings. As an application, we prove the existence of solution of a Caputo fractional Volterra-Fredholm integro-differential equation via integral type boundary conditions and verify the validity of our application by an appropriate example.

Keywords

• measure of noncompactness
• fractional integro-differential equation
• simulation function
• zm-contraction
• lm-contraction
• fixed point