Applications of new contraction mappings on existence and uniqueness results for implicit ϕ-Hilfer fractional pantograph differential equations

Abstract

Abstract In this paper, we consider initial value problems for two different classes of implicit ϕ-Hilfer fractional pantograph differential equations. We use different approach that is based on α − ψ $\alpha -\psi $ -contraction mappings to demonstrate the existence and uniqueness of solutions for the proposed problems. The mappings are defined in appropriate cones of positive functions. The presented examples demonstrate the efficiency of the used method and the consistency of the proposed results.

Keywords

• α − ψ $\alpha -\psi $ -Contraction mapping
• ϕ-Hilfer fractional derivative
• Fractional differential equation
• Pantograph differential equation