Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras

Abstract

In this article, we prove an existence result for partial hyperbolic differential equations of fractional order in Banach algebras under Lipschitz and Carathéodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.

Keywords:

• Darboux problem
• fractional order
• solution
• upper solution
• lower solution
• extremal solutions
• left-sided mixed Riemann–Liouville integral
• Caputo fractional-order derivative
• fixed point

AMS Subject Classification:

• 26A33

Acknowledgement

We are grateful to the referee for pointing out some errors in the first version of this article.