Upper and lower solutions for an integral boundary problem with two different orders ( p , q ) -fractional difference

In this paper, a ( p , q ) -fractional nonlinear diﬀerence equation of diﬀerent orders is considered and discussed. With the help of ( p , q ) -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.


Introduction
In 1910, F.H. Jackson launched an important study in q-calculus and introduced the qderivative and q-integral, which can be found in [12,13].With these results, many researchers joined this field and applied this calculus to several fields, such as orthogonal polynomials, combinatorics, number theory, fundamental hypergeometric functions, mechanics, quantum theory, relativity theory, equations, and derivatives, see [3, 4, 7-11, 14, 16] and the references mentioned therein.
Subsequently, the concept of (p, q)-calculus underwent generalization and advancement from the q-calculus theory to the two-parameter (p, q)-integer calculus.This particular mathematical framework has proven to be highly effective across various disciplines.
Given the importance of the subject of fractional order derivatives and (p, q)-calculus, many researchers have studied the existence of general or positive solutions, as well as their stability for many problems.So, we present to the reader some references on this matter to enrich the subject of the study [1,5,6,18,24,29].
Xu and Sun [29] investigated the positivity of a class of integral boundary value problems of fractional differential equations with two nonlinear terms of the form where D δ is the standard Riemann-Liouville derivative, 1 < δ ≤ 2.
In 2018, Xu and Han [28] used the method of upper and lower solutions and studied the problem where D δ and D τ are the standard Riemann-Liouville derivatives, 1 < δ ≤ 2, 0 < τ < δ.
The aim of this paper is to investigate an integral boundary value problem of (p, q)fractional difference equations which encompasses two nonlinear terms defined as follows: are continuous with g ξ , y being nondecreasing in y, and where p As far as we are aware, no publication has examined the possibility of finding positive solutions for the (p, q)-fractional difference equations with integral boundary conditions and two nonlinear terms with different fractional order derivatives, by constructing upper and lower control functions and then giving some existence and uniqueness results using Schauder fixed point theorem and Banach contracting mapping principle.On the other hand, this problem is more general than the others in the literature.
This paper is produced as follows.In Sect.2, we give some necessary concepts, tools, and results used in the analysis.Section 3 is devoted to studying the existence of positive solutions.An example is given in Sect. 4. Some conclusions and generalizations are drawn in Sect. 5.

Essential materials
This part presents some essential material, which is required for our study.We begin with some fundamental definitions and results of the q-calculus and (p, q)-calculus, which can be found in [9,[25][26][27].Let 0 < q < p ≤ 1 be constants, The q-analogue of the power function (ab) (n)  q is given by The (p, q)-analogue of the power function (ab) (n) p,q is given by and for δ ∈ R, the general form of the above is given by where p ( δ 2 ) := δ(δ-1) 2 .
Definition 1 ([26]) Let 0 < q < p ≤ 1.The (p, q)-derivative of a function h is defined as and D p,q h (0) = lim ξ →0 D p,q h (ξ ), whenever h is differentiable at 0. Moreover, the high order (p, q)-derivative D n p,q h (ξ ) is defined by ]) Let 0 < q < p ≤ 1 and h be an arbitrary function of a real variable ξ .The (p, q)-integral of the function h is defined as as long as the series on the right-hand side converges.In this case, h is called (p, q)integrable on [0, ξ ].

Main results
In this part, we shall demonstrate the positivity of solutions for boundary value problems (1.4) by employing Schauder fixed point theorem, and then establish sufficient conditions for uniqueness using Banach fixed point theorem.We will utilize the method of ULS in the analysis.

4) if and only if y (ξ ) satisfies the integral equation
where Proof The proof is divided into two cases.
Case 1. τ ≤ 1.From Definition 3 and Lemma 1, applying the operator I δ p,q on both sides of (1.4), one has p,q h pν, y (pν) d p,q ν = I δ-τ p,q I τ p,q D τ p,q g pξ , y (pξ ) p,q g pν, y (pν) for some constants c 1 , c 2 , c 3 ∈ R. Due to the boundary condition y (0) = 0, and the last formula, we get c 2 = 0 and, since As in Case 1, we can write for some constants c 1 , c 2 , c 3 , c 4 ∈ R. Due to the boundary conditions, we get c 2 = and . Therefore, p,q (δ) p,q (δ) The proof is finished, and this process is reversible.

Lemma 3 ([24])
The function G defined by (3.1) satisfies the following properties: Let the Banach space E = C ([0, 1]) be endowed with the norm y = max ξ ∈[0,1] y (ξ ) .Denote Let a and b be positive real numbers, where b is greater than a.For any y belonging to the interval [a, b], we define the lower control function and the upper control function Obviously, V ξ , y and W ξ , y are monotonous and nondecreasing in y, and Now, let the functions α (ξ ) and β (ξ ) be the lower and upper solutions of (1.4), respectively.We need the following hypotheses: and It is evident that y ≤ b.Consequently, K ⊂ E is convex, bounded, and closed.If y ∈ K, there exist positive constants M h and M g such that g pν, y (pν) < M g .
We define the operator φ as The above operator φ is continuous on K due to the continuity of h and g.If y ∈ K, we can obtain where, due to Lemma 3, 0 ≤ G (ξ , qν) ≤ 1, and then we get Thus, p,q is continuous with respect to ξ and ν on [0, 1] × [0, 1], it can be inferred that the function is uniformly continuous on for any ν ∈ [0, 1], we can deduce the following: So we can say that, as ξ 1 → ξ 2 , the right-hand side of the inequality (3.5) converges to zero.
We will now demonstrate that φ (K) ⊂ K. Consider an element y ∈ K.It can be inferred and, in the same way, we also have Using Schauder fixed point theorem, it can be concluded that φ possesses at least one fixed point y within K. Consequently, equation (1.4) will have at least one positive solution y within E, satisfying the conditions α (ξ ) ≤ y (ξ ) ≤ β (ξ ) for all ξ ∈ [0, 1].
Corollary 1 Suppose there are nonnegative bounded continuous functions κ 1 , κ 2 , κ 3 , and κ 4 such that and at least one of κ 1 (ξ ) and κ 3 (ξ ) is not identically equal to 0. Then, equation (1.4) will have at least one positive solution y ∈ E such that for δτ ≥ 1, and Proof Consider the problem which is equivalent to where G is given by (3.1).According to the control function definitions, we have where a, b are minimal and maximal values of y (ξ ) on [0, 1].Hence, we can acquire Since p ( δ-τ 2 ) is positive for δτ ≥ 1, then (3.11) is an upper solution of (1.4).Similarly, we can demonstrate using the same method that is a lower solution of (1.4).Thus the problem (1.4) has at least one positive solution y ∈ E that satisfies (3.8), according to Theorem 1.
We use the same steps for the case δτ ≤ 1 to obtain (3.9), and take into account that p ( δ-τ 2 ) changes the inequality (3.7).
Theorem 2 Assume (3.2) and (3.3) hold and that there exist constants L h , L g > 0 such that Then, the problem (1.4) possesses a unique positive solution on K.
Proof After establishing the validity of Theorem 1, it becomes evident that φ : K → K.Then, for any x and y belonging to K, we get φx (ξ )φy (ξ ) where, due to Lemma 3, 0 ≤ G (ξ , qν) ≤ 1, and then < 1.Thus, there is a fixed point y ∈ K which serves as the sole positive solution to the problem (1.4) on K.

Conclusion
Fractional calculus, which involves studying derivatives of arbitrary order, is an important field of research due to its extensive theoretical advancements and practical applications over the past few decades.So, we investigated in this paper a class of integral boundary value problems of (p, q)-difference equations with two nonlinear terms containing fractional derivatives.The method of ULS, along with Schauder fixed point theorem, was utilized to obtain positive solutions.Additionally, Banach contraction mapping principle was employed to establish uniqueness results.The results obtained in this paper are good and important, as the existence and uniqueness results in [24] can be obtained by the method used in this research when the function g ≡ 0.
A compelling avenue for future research would be to explore fractional (p, q)-difference equations with variable orders, as opposed to the constant order examined in this study.Additionally, investigating boundary conditions of the Riemann-Stieltjes integral type presents another promising direction.Incorporating impulsive effects into the analysis would further broaden the scope of potential applications, see [15] and some related works that can be applied.