SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

SOME HARDY-TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA DELTA FRACTIONAL INTEGRALS

In this paper, some Jensen- and Hardy-type inequalities for convex functions are extended by using Riemann–Liouville delta fractional integrals. Further, some Pólya–Knopp-type inequalities and Hardy–Hilbert-type inequality for convex functions are also proved. Moreover, some related inequalities are proved by using special kernels. Particular cases of resulting inequalities provide the results on fractional calculus, time scales calculus, quantum fractional calculus and discrete fractional calculus.

Discrete Fractional Inequalities Pertaining a Fractional Sum Operator with Some Applications on Time Scales

This content replicates some discrete nonlinear fractional inequalities by virtue of the fractional sum operator Ψ ¯ on time scales. Through the recognition of the principle of discrete fractional calculus, we are able to recover the precise estimates for unknown functions of inequalities of the Gronwall type. The resultant inequalities are of unique structure comparative with the latest reviewing disclosures and can be described as a complementary tool for numerically testing the solutions of discrete partial differential equations. The foremost consequences are probably confirmed via handling of assessment procedure and technique of mean value speculation. We display few examples of the proposed inequalities to represent the incentives of our effort.

On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.