Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the <inline-formula><math display="inline"><semantics><mrow><msup><mo mathvariant="double-struck">L</mo><mo mathvariant="bold">p</mo></msup></mrow></semantics></math></inline-formula> Space with the Framework of the Ψ-Caputo Derivative

Abstract

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in Lp space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ-Caputo derivative. We generalize the two situations of p=2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end.

Keywords

• fractional calculus
• Ψ-Caputo derivative
• neutral stochastic differential equations
• existence and uniqueness
• averaging principle