Explicit iteration and unbounded solutions for fractional q–difference equations with boundary conditions on an infinite interval

In this work, a proposed system of fractional boundary value problems is investigated concerning its unbounded solutions’ existence for a class of nonlinear fractional q-difference equations in the context of the Riemann–Liouville fractional q-derivative on an infinite interval. The system’s solution is formulated with the help of Green’s function. A compactness criterion is established in a special space. All the obtained results of uniqueness and existence are investigated with the help of fixed-point theorems. Some essential examples are illustrated to support our main outcomes.


Introduction
Fractional differential equations are much better than integer ones with respect to their nature in the descriptions of phenomena and processes of several scientific and engineering phenomena. Various studies of fractional differential equations have been recently conducted in the context of fractional-order derivatives such as the Riemann-Liouville fractional derivative and the Caputo fractional derivative.
A variable's range is studied in this research work on the unbounded domain. A modified compactness criterion is utilized to investigate the studied compact operator H due to the failure of the Arzelá-Ascoli theorem in space C ∞ . With the help of the Banach fixed-point theorem, the Leray-Schauder nonlinear alternative theorem, and the Schauder fixedpoint theorem, the investigated system's uniqueness and existence are proven.
The remainder of this article is organized as follows: the main definitions and lemmas are discussed in Sect. 2 to offer a guide for proving our results by providing a necessary background with some properties to formulate a Green's function that is suitable for the investigated problem on an unbounded domain. In Sect. 3, the solution's existence and uniqueness are investigated for the boundary value problem (4). In Sect. 4, two examples are presented to apply our outcomes.

Definition 2.2 ([40])
The fractional q-derivative of the Riemann-Liouville type of order

Lemma 2.3 ([12])
Assume that ς > 0 and ı ∈ N. Then, an equality holds as follows: Obviously, we have: Lemma 2.4 ([11]) Assume that j 0 and j are intervals involving 0 ∈ j 0 ⊆ j . Assume that ℘ ı , (ı ∈ N) and ℘ are functions defined in j lim ı→∞ ℘ ı (t) = ℘(t), (∀t ∈ j ), and ℘ ı tends uniformly to ℘ on j 0 . Then, The following results are important in the remainder of the paper, where we give the exact expression of the Green's function associated with the system (4). Lemma 2.5 A given function: ∈ C(R + ) is assumed. Then, the boundary value problem: via boundary conditions has a unique solution with defined by ∀t, s ∈ j is the Green's function of boundary value problem (6) and (7).
Proof By integrating both sides of (6) The Definition 2.2 and Lemma 2.3, imply that The boundary conditions implies that ℘(0) = ℘ (0) = ℘ (0) = 0, thus, hence, We obtain: By differentiating both sides of (10) along with (5), we obtain By the second condition of (7) Moreover, equation (11) implies that where is given by (9). Consequently, the unique solution of problem (6) and (7) is given by the following formula The proof is completed.
Let us now mention some Green's function's properties that are essential to our study.

Lemma 2.6
Suppose that 1 G q (t, s) and 2 G q (t, s) are the Green's functions of the linear system (6) and (7) provided in Lemma 2.5. Then, we obtain: .

Main results
First, we define the norm space The space C ∞ is a Banach space [7]. A map ℘(t) ∈ C(j , R) with its q-derivative of Riemann-Liouville of order ς that exists on j is named as a solution of (6) if (7) is satisfied. For ℘ ∈ C ∞ , let an operator H be expressed as: Clearly, we have a continuous, The operator H compactness is shown by proving that H has a fixed point on C ∞ (j , R).
Remark 3.1 Note that to apply the Arzelá-Ascoli theorem in basic space C ∞ , we need to establish the following modified compactness criterion to show that H is compact.
Let us introduce the following hypotheses for convenience: and |w * (t, ℘)| ≤ ψ(t)p(|℘|), on j with nondecreasing function p ∈ C(j , R) and IfΥ is equicontinuous on any compact intervals of j and equiconvergent at ∞, then ϒ is relatively compact on C ∞ Remark 3.2Υ is termed equiconvergent at ∞ iff, for any given positive number , ∃ N = N > 0,

Lemma 3.2 A cone P ⊂ C ∞ is defined by
If (A4) and (A5) hold, then H : P → P is completely continuous.
Proof This proof consists of 3 essential steps. STEP A: Let us prove that H : P → P is continuous. From the continuity and nonnegativity of 1 G q and w, we obtain H℘(t) ≥ 0 for t ∈ j . For each ℘ ∈ P, by (A4), we obtain: Thus, Then by (A4), Taking ℘ n → ℘ as n → +∞ in C ∞ , by (A5), we obtain: So w(t, ℘ n (t)) → w(t, ℘(t)) uniformly on j . By Lemma 2.4, we obtain Hence, combining (15), we obtain H℘ n (t) 1+ ς -1 - Hence, as n → ∞. So, H is continuous. STEP B: We show that H is uniformly bounded. For this case, let Q be any bounded subset of P, i.e., ∃ μ > 0, ℘ ≤ μ for any ℘ ∈ Q. It is enough to prove that H(℘) is bounded in P. For ℘ ∈ Q, we obtain: Hence, H(Q) is uniformly bounded. Now, we show that H(Q) is equicontinuous on any compact interval. First, for each given s > 0, t 1 , t 2 ∈ [0, s], and ℘ ∈ P, and t 2 > t 1 , we deduce: On the other hand, for all ℘ ∈ Q, t 1 → t 2 , we obtain Similar to (16), for all ℘ ∈ Q, t 1 → t 2 , we obtain Hence, H(Q) is equicontinuous on any compact intervals of j .
Hence, H : P → P is equiconvergent at infinity. Thus, the operator H is completely continuous. The proof is completed.
In the next subsections, the boundary value problem's (4) existence and uniqueness are shown via the Banach fixed-point theorem, the Schauder fixed-point theorem, and the Leray-Schauder nonlinear alternative theorem.

Existence via the Leray-Schauder nonlinear alternative theorem
and conditions (A5) and (A6) hold. Then, the boundary value problem (4) has an unbounded solution ℘(t) such that we have ℘ = λH℘ for ℘ ∈ ∂ϒ and λ ∈ (0, 1). If ∃ ℘ ∈ ∂ϒ with ℘ = λH℘, then for λ ∈ (0, 1), we obtain Hence, for ℘ ∈ ∂ϒ, we find which contradicts (18). By the fixed-point theorem of Schauder, the problem (4) has an unbounded solution ℘ = ℘(t), The proof is completed. Proof Assume that an operator H is defined in (14). By testing to see that all Schauder fixed-point theorem's axioms on C ∞ are satisfied. w and 1 G q , 2 G q are given continuous functions indicating that the operator H is continuous. The remainder of the Theorem 3.4 proof is divided as follows:

Existence via the Schauder FpThm
STEP A: Let closed ball. Then, we prove that H : ϒ → ϒ. We select For any ℘ ∈ C ∞ , we prove that Hϒ ⊂ ϒ. Hence, for t ∈ j , we obtain: As a result, H℘ C ∞ ≤ μ, which indicates that Hϒ ⊂ ϒ, i.e., the operator H maps ϒ into ϒ.
STEP B: From Lemma (3.2), we need to prove that H is continuous and completely continuous on C ∞ . Thus, by the Schauder fixed-point theorem, the operator H has a fixed point ℘ in C ∞ which is a solution of problem (4). Then, the boundary value problem (4) has a unique solution:

Existence and uniqueness results via the Banach FpThm
Furthermore, ∃ is an error estimate for the following: and θ = 1 Proof By considering an operator H defined by the equation (14). Assume that ℘ 1 , ℘ 2 ∈ C ∞ . For t ∈ j , we have Consequently, Thus, we collect that As θ < 1, then the Banach fixed-point theorem ensures that H has a unique fixed point ℘ in C ∞ . Hence, the problem (4) has a unique solution ℘ ∈ C ∞ . Furthermore, for each ℘ 0 ∈ C ∞ , ℘ n -℘ ∞ → 0 as n → ∞, where ℘ n = H℘ n-1 (n = 1, 2 . . .). From (21), we find By supposing that n → ∞ on both sides of (22), we can estimate Hence equation (19) holds, and this theorem's proof is completed.  10 13 }, let us have the following boundary value problem for nonlinear fractional q-difference equations on an unbounded domain:

Conclusion
A fractional boundary value problem system has been proposed and studied in terms of its unbounded solutions' existence for a class of nonlinear fractional q-difference equations via the Riemann-Liouville fractional q-derivative on an infinite interval. The system's solution has been formulated using Green's function. In addition, a compactness criterion