Existence and extremal solutions in Banach algebras for a fractional order differential equation

Abstract

The existence of a solution in Banach algebras for a fractional order nonlinear quadratic differential equation with initial value condition is investigated in this paper. Furthermore, we demonstrate that the solutions to this equation are appealing locally. The primary conclusion is established using basic hybrid fixed point theory methods for three operators. Under certain monotonicity criteria, existence theorems for extremal solutions can also be established. Finally, we provide a specific example to demonstrate our findings.