Some Fractional q-Integrals and q-Derivatives

Abstract

A q-analogue of the integral ∣f(t)dt is defined by means ofwhich is an inverse of the q–derivativeThe present author (2) has recently obtained a q–nalogue of a formula of Cauchy, namely,where, for real or complex α and N a positive integer,

On a class of nonlinear Volterra-Fredholm q-integral equations

“…A q-analogue of the Riemann-Liouville fractional integral operator is introduced in [2] by Al-Salam through…”

A Lyapunov-type inequality for a fractional q-difference boundary value problem

“…Details, basic definitions and properties of q-difference calculus can be found in the book [18]. The fractional q-difference calculus has been introduced by Al-Salam [3] and Agarwal [1]. Due to the intensive works in the field of fractional calculus, several developments in this theory of fractional q-difference were made (see [4,9,11,12,21,23,27,28,29] and references therein).…”

Positive Solutions for Nonlinear Caputo Type Fractional q-Difference Equations with Integral Boundary Conditions

“…Since Al-Salam [1] and Agarwal [2] introduced the fractional q-difference calculus, the theory of fractional q-difference calculus itself and nonlinear fractional q-difference equation boundary value problems have been extensively investigated by many researchers. For some recent developments on fractional q-difference calculus and boundary value problems of fractional q-difference equations, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein.…”

“…where α ∈ (n − 1, n] are a real number and n ≥ 3 is an integer, D α q are the fractional q-derivative of the Riemann-Liouville type, µ > 0 and 0 < q < 1 are two constants, g, h are two given continuous functions, and 1]. By applying monotone iterative method and some inequalities associated with the Green's function, the existence results of positive solutions and two iterative schemes approximating the solutions were established.…”