Abstract
Discrete fractional calculus ℱ C 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 ℏ ℤ . 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 n n ∈ 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.
Highlights
fractional calculus (FC) and its concrete utilities have increased a great deal of significance in light of the fact that fractional operators have become a useful asset with more precise and victories in demonstrating a few complex marvels in various apparently differing and broad fields of science and numerous areas, for example, fluid flow, optics, chaos, image processing, virology, and financial economics [1,2,3]
We observe that some variants have been concerned with the qualitative investigation of solutions of discrete fractional difference equations arising in the theory of discrete FC
We demonstrate the proportional fractional sum with memory depending on the proportional difference, which is mainly due to Abdeljawad et al [54]
Summary
FC and its concrete utilities have increased a great deal of significance in light of the fact that fractional operators have become a useful asset with more precise and victories in demonstrating a few complex marvels in various apparently differing and broad fields of science and numerous areas, for example, fluid flow, optics, chaos, image processing, virology, and financial economics [1,2,3]. Several researchers have devoted their concentrations for exploring the novel versions of fractional integral inequalities for a family of n(n ∈ N)-positive increasing functions. In this flow, we observe that some variants have been concerned with the qualitative investigation of solutions of discrete fractional difference equations arising in the theory of discrete FC. Our findings can deliver a prevailing instrument to illustrate the dynamics of discrete complex frameworks all the more profoundly
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