On Discrete Fractional Integral Inequalities for a Class of Functions

Abstract

Discrete fractional calculus ℱC is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. ℏ-discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale ℏℤ. The nabla ℏ-fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions nn∈ℕ concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.