Existence and stability of the solution of an implicit set-valued functional differential equation with parameters

A set-valued function, also called a correspondence or set-valuedrelation, is a mathematical function that maps elements from one set,the domain of the function, to subsets of another set. Set-valuedfunctions are used in a variety of mathematical fields, including optimization, control theory and game theory.The set-valued functional differential equation has been widely applied in mathematics, physics, optimizations, optimal control, as wellas economics and finance.[2, 3, 5, 6].Here, we study the initial-value problem of a set-valued implicate functional differential equation with parameters.The existence of solution and its continuous dependence on parameters will be provedwe study the existence of solutions of the parametric set-valued implicitfunctional differential equationThe existence of solutions x ∈ C[0, T], C[0, T] is the Banach space of continuous functions defined on [0, T], will be proved. The continuous dependenceof the solutions on the initial value x0 and the parameters γ. µ will be provedalso.