Boundary Value Problem for Fractional q-Difference Equations with Integral Conditions in Banach Spaces

Abstract

The authors investigate the existence of solutions to a class of boundary value problems for fractional q-difference equations in a Banach space that involves a q-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch’s fixed point theorem and the technique of measures of noncompactness. This approach has proved to be an interesting and useful approach to studying such problems. Some basic concepts from the fractional q-calculus are introduced, including q-derivatives and q-integrals. An example of the main result is included as well as some suggestions for future research.

Advances in Boundary Value Problems for Fractional Differential Equations

“…T denote the left Caputo fractional derivative and the right Riemann-Liouville fractional derivative of order γ j , respectively. By using the critical point theory, they prove the existence of infinitely many solutions of problem (15), (16).…”

“…Paper [15] deals with the fractional q-difference equation in a Banach space E, with nonlinear integral conditions…”

Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions

“…3 (1, qs) d q s ≈ 0.13008493,L 4 = 1 1/4 G 4 (1, qs) d q s ≈ 0.02446965, M 1 ≈ 0.26216363, M 2 ≈ 0.02789647, M 1 ≈ 0.00332425, M 2 ≈ 0.00013782. We choose r 1 = 1 1000 , r 2 = 10, σ 1 = 3 < 1 M 1 ≈ 3.8144, σ 2 = 35 < 1 M 2 ≈ 35.8468, σ 3 = 301 > 1 M 1 ≈ 300.8201, and σ 4 = 7256 > 1 t+4 = 0.75 ≥ σ 3 r 1 2 = 0.1505, ∀ t ∈ 1 4 , 1 , u, v ≥ 0, u + v ≤ 1 1000 , x ∈ 0, (37/6) , g(t, u, v, x, y) ≥ e −1/1000 + min t∈[1/4,1] 9(t+1) t 3 +2 ≈ 6.5804 ≥ σ 4 r 1 2 ≈ 3.628, ∀ t ∈ 1 4 , 1 , u, v ≥ 0, u + v ≤ 1 1000 , x ∈ 0, t+4 ≈ 5.3502 ≤ σ 1 r 2 2 = 15, ∀ t ∈ [0, 1], u, v ≥ 0, u + v ≤ 10, x ∈ 0, 10 Γ 1/2 (51/5) , y ∈ 0, 10 Γ 1/2 (37/6) , g(t, u, v, x, y) ≤ 1 + +2 ≈ 15.5719 ≤ σ 2 r 2 2 = 175, ∀ t ∈ [0, 1], u, v ≥ 0, u + v ≤ 10, x ∈ 0, is verified.By Theorem 1, we deduce that problem (81),(82) with the functions in (86) has at least one positive solution (u(t), v(t)), t ∈ [0, 1], such that 1 1000 ≤ ∥u∥ + ∥v∥ ≤ 10, andmin t∈[1/4,1] u(t) ≥ 1 + x 1/3 + 1 6 e −y , ∀ t ∈ [0, 1], u, v ≥ 0, u + v ≤ 1, + 1 6 e −y , ∀ t ∈ [0, 1], u, v ≥ 0, 1 < u + v ≤ 4, + 1 6 e −y , ∀ t ∈ [0, 1], u, v ≥ 0, u + v > 4, g(t, u, v, x, y) = +1 + y 1/4 , ∀ t ∈ [0, 1], u, v ≥ 0, u + v ≤ 1, +1 + y 1/4 , ∀ t ∈ [0, 1], u, v ≥ 0, 1 < u + v ≤ 4, +1 + y 1/4 , ∀ t ∈ [0, 1], u, v ≥ 0, u + v > 4.…”

“…By using varied fixed-point theorems, they obtained the existence and uniqueness results for the solutions of problem (3), (4). In [3], the authors examined the existence of solutions for the fractional q-difference equation with nonlinear integral conditions    ( C D α q y)(t) = f(t, y(t)), for a.e. t ∈ [0, T], y(0) − y ′ (0) = T 0 g(s, y(s) ds, y(T) + y ′ (T) = T 0 h(s, y(s)) ds, (5) where T > 0, q ∈ (0, 1), α ∈ (1, 2], and C D α q is the Caputo fractional q-derivative of order α.…”

Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives

“…They employed fixed point theory, specifically the nonlinear alternative of Schaefer's type proven by Dhage, as well as Dhage's random fixed point theorem in Banach algebras. Another study conducted by Allouch et al [17] focused on the existence of solutions for a class of boundary value problems involving fractional q-difference equations in a Banach space. They utilized Mönch's fixed point theorem and the technique of measures of non-compactness.…”