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

Abstract

Abstract:In this paper, by applying some well-known fixed point theorems, we investigate the existence of positive solutions for a class of nonlinear Caputo type fractional q-difference equations with integral boundary conditions. Finally, some interesting examples are presented to illustrate the main results.

Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation

“…Suppose 2 < ] < 3 and 0 < < [2] . Then the function ( , ) defined by (20) satisfies the following inequalities:…”

“…Some researchers have paid close attention to the research of -difference equation since the -difference calculus and quantum calculus were discovered by Jackson [1,2]. After the fractional -difference calculus was developed by Al-Salam et al [3][4][5][6], many papers on the fractional -difference equation kept emerging, such as the papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and their references. Among them, Li and Yang [7] established the existence of positive solutions for a class of nonlinear fractionaldifference equations with integral boundary conditions by applying monotone iterative method.…”

“…These experts did researches about the existence of a positive solution and multiple positive solutions to this problem by applying some well-known fixed-point theories such as Krasnosel'skii and Schauder fixed-point theorems. Thereinto, in [7,[15][16][17][18][19][20], the authors focused on the fractional -difference equation with integral boundary value conditions.…”

Unique positive solution for nonlinear Caputo-type fractional q-difference equations with nonlocal and Stieltjes integral boundary conditions

“…Where C D α q is the generalized Caputo fractional q-derivative of order α, with 0 < q < 1, m−2 i=1 γ i < 1, β i ≥ 0, ζ i ∈ (0, 1), i = 1, 2, ..., m − 2, ζ 1 < ζ 2 < ... < ζ m−2 , λ > 0 is a parameter, ν, µ > 0, h : [0, 1] → R + , f : (0, 1) × R → R is a continuous function, andα[x] := problems such as nonlocal, integral, multiple-point, sub-strip boundary problems and some others, see [5][6][7][8][9][10][11][12][13][14][15] and references therein.…”

A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions

“…BVPs with integral BCs arise naturally in semiconductor problems [10], thermal conduction problems [11], hydrodynamic problems [12], population dynamics model [13], and so on (see also [14]). Recently, these BVPs were extensively studied by (among others) Akcan and Çetin [15], Boucherif [16], Benchohra et al [17], Chalishajar and Kumar [18], Dou et al [19], Li and Zhang [20], Liu et al [21], Song et al [22], Tokmagambetov and Torebek [23], Wang et al [24] and Yang and Qin [25] (see also the references to the related earlier works which are cited in each of these investigations).…”