A discrete fractional logistic map is proposed in the left Caputo discrete delta’s sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.

Discrete fractional logistic map and its chaos

In this paper, we consider a discrete fractional boundary value problem of the form -@D^@ny(t)=f(t+@n-1,y(t+@n-1)), y(@n-2)=g(y), y(@n+b)=0, where f:[@n-1,...,@n+b-1]"N"""@n"""-"""2xR->R is continuous, g:C([@n-2,@n+b]"N"""@n"""-"""2,R) is a given functional, and 1<@n@?2. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel'skii theorem.

Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions

In this manuscript, we define the generalized fractional derivative on ACγ [a,b], the space of functions defined on [a,b] such that γn−1f ∈ AC[a,b], where γ = x1−ρ d dx . We present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version. c ©2017 All rights reserved.

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On the generalized fractional derivatives and their Caputo modification

On Fractional Derivatives with Exponential Kernel and their Discrete Versions

In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.

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Sevgi Şengül

Papers

Modeling with fractional difference equations