Estimation of Unknown Function of a Class of Integral Inequalities with pth Power

Abstract

Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities are important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of Fractional differential equations and fractional integral equations. In this paper, we discuss a class of integral inequalities with pth power, which includes a nonconstant term outside the integrals. Using the definitions and properties of modified Riemann-Liouville fractional derivative and Riemann-Liouville fractional integral, the upper bounds of the unknown function is estimated explicitly. The derived result can be applied in the study of qualitative properties of solutions of fractional integral equations. Introduction Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities [1, 2] are important tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of fractional differential equations and fractional integral equations. In 2011, Abdeldaim et al. [3] studied a new iterated integral inequality with pth power ds d d g u h s u s u s f u t u s t ] ] ) ( ) ( [ ) ( ) ( )[ ( ) ( ) ( 0 0 0 0              . (1) In 2014, El-Owaidy, Abdeldaim, and El-Deeb [4] discussed a new nonlinear integral inequality with a nonconstant term outside the integrals ds s u s h ds s u s g t f t u t