The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies

Abstract

Gronwall's inequalities are important in the study of differential equations and integral inequalities. Gronwall inequalities are a valuable mathematical technique with several applications. They are especially useful in differential equation analysis, stability research, and dynamic systems modeling in domains spanning from science and math to biology and economics. In this paper, we present new generalizations of Gronwall inequalities of integral versions. The proposed results involve $( \rho ,\varphi)-$Riemann-Liouville fractional integral with respect to another function. Some applications on differential equations involving $( \rho ,\varphi)-$Riemann-Liouville fractional integrals and derivatives are established.

Keywords

• fractional calculus
• fractional differential equation
• gronwall inequality