Multiple Positive Solutions for a Class of Boundary Value Problem of Fractional (p, q)-Difference Equations under (p, q)-Integral Boundary Conditions

<jats:p>This paper is mainly concerned with a class of fractional <jats:inline-formula>
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                  </jats:inline-formula>-difference equations under <jats:inline-formula>
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                  </jats:inline-formula>-integral boundary conditions. Multiple positive solutions are established by using the topological degree theory and Krein–Rutman theorem. Finally, two examples are worked out to illustrate the main results.</jats:p>


Introduction
e q-difference operator was first systematically studied by Jackson [1]. en, q-calculus has been studied extensively. See [2][3][4] and references therein. q-calculus and q-difference equations have been used by many researchers to solve physical problems such as molecular problems and chemical physics [1,[5][6][7]. For example, in 1967, Floreanini and Vinet [3] studied the behaviors of hydrogen atoms by using Schrödinger equation and q-calculus. Diaz and Osler [8] investigated the q-field theory.
In the last decades, the theory of quantum calculus based on two-parameter (p, q)-integer has been studied since it can be used efficiently in many fields such as difference equations, Lie group, hypergeometric series, and physical sciences. e (p, q)-calculus was first studied by Chakrabarti and Jagannathan [2] in the field of quantum algebra in 1991. Njionou Sadjang [9] systematically established the basic theory of (p, q)-calculus and some (p, q)-Taylor formula. Milovanovic and Gupta [10] developed the concept of (p, q)-beta and (p, q)-gamma functions. ese basic concepts and theories promote the development of (p, q)-calculus. For detailed results on (p, q)-calculus, please see [9][10][11][12][13] and references therein.
On the contrary, the research of fractional calculus in discrete settings was initiated in [8,11,14]. In 2020, Soontharanonl and Sitthiwirattham [15] introduced the fractional (p, q)-calculus, which has been found in a wide range of applications in many fields such as concrete mathematical models of quantum mechanics and fluid mechanics [7,13,15].
As we all know, in recent decades, more and more researchers pay much attention to the fractional differential equations and have obtained substantial achievements, we refer the readers to see [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and references therein. Although the results of discrete fractional calculus are similar to those of continuous fractional calculus, the theory of discrete fractional calculus remains much less developed than that of continuous fractional calculus [35,36]. erefore, it is very important to develop discrete calculus. In particular, the fractional (p, q)-difference equations involving (p, q)-integral boundary conditions have rarely been studied. In order to make up for this gap, the paper mainly studies the following boundary value problem of fractional (p, q)-difference equations under (p, q)-integral boundary conditions: It should be pointed out that the boundary conditions of BVP equation (1) are more extensive. Furthermore, two parameters in the discrete environment makes the boundary value problem more complex. In order to overcome these difficulties, we constructed a special cone. e existence and multiplicity of the positive solution for the BVP equation (1) are obtained by using the topological degree theory, Krein-Rutman theorem.
is paper is structured as follows. In Section 2, we introduce some definitions of (p, q)-fractional integral and differential operator together with some basic properties and lemmas. e main results are given and proved in Section 3. Finally, in Section 4, two examples are given to show the applicability of our main results.

Preliminaries
In this section, we list some basic definitions and lemmas that will be used in this paper. For 0 < q < p ≤ 1, we let e (p, q)-analogue of the power function (a − b) n p,q with n ∈ N 0 \coloneq 0, 1, 2, . . ., is given by For α ∈ R, By [15], we obtain Note that a α q � a α p,q � a α and (0) α q � (0) α p,q � 0 for α > 0. e (p, q)-gamma and (p, q)-beta functions are defined by respectively.
Journal of Mathematics Based on the hypothesis in Lemma 4, we can deduce that us,

Proof
(1) On the one hand, for 0 ≤ s ≤ t ≤ 1, we know G 0 (t, qs) � 1 us, for t ≠ 0, it is easy to see that G 0 (t, qs) � 1 Similarly, for 0 ≤ s ≤ t ≤ 1, we know us, for t ≠ 0, it is also easy to see that

Journal of Mathematics
On the other hand, for 0 ≤ t ≤ s ≤ 1, it is easy to see that, from Lemma 4, the conclusion is obviously established. erefore, G i (t, qs) ≥ 0, for t, s ∈ [0, 1].

Lemma 8 (see [37]). Let Ω be a bounded open set in a
Banach space E with 0 ∈ Ω, and T: Ω ⟶ E is a continuous compact operator. If then the topological degree deg(I − T, Ω, 0) � 1. Let

en, (E, ‖ · ‖) is a real Banach space and P is a cone on E. From Lemma 4, we can define operator T: E ⟶ E as follows:
where G is determined in Lemma 4. Obviously, T is a completely continuous operator.
In addition, from Lemma 4, we can obtain that the solution of BVP equation (12) is equivalent to 6 Journal of Mathematics For our purposes, we need to define the operator L by It is easy to prove that L: E ⟶ E is a linear completely continuous operator and L(P) ⊂ P. Obviously, we know that L has a spectral radius, denoted by r(L), that is not equal to 0. From Krein-Rutman theorem, we know that L has a positive eigenfunction φ 1 corresponding to its first eigenvalue λ 1 � (r(L)) − 1 , i.e., φ 1 � λ 1 Lφ 1 .

Main Results
In this section, we shall establish the existence and multiplicity results of BVP equation (1), which is based on the topological degree theory. For convenience, let λ 1 be the first eigenvalue of the following eigenvalue problem: Now, let us list the following assumptions satisfied throughout the paper: (H6) ere exist r * > 0 and a continuous function ψ r * such that x ∈ 0, r * , Now, we are in a position to give our main results. Proof. First, assumption (H1) implies that there exists r > 0 such that We claim that, for μ ≥ 0, Suppose, on the contrary, that there exist x 1 ∈ zB r ∩ P, μ 1 > 0 such that Without loss of generality, suppose μ 1 > 0. en, Let us, It is a contradiction with the definition of μ * . According to Lemma 7, one obtains deg T, B r ∩ P, P � 0. (41) On the contrary, we can choose ε 0 > 0 such that 0 < (λ 1 − ε 0 )‖L‖ < 1. en, from (H2), there exists R > 0 such that us, one can easily find that Choose R 0 > max R, r, m 1 0 θ 1 (qs)d p,q s/1 − (λ 1 − ε 0 ) ‖L‖}. We claim that, for μ ≥ 1, Suppose, on the contrary, that there exist x 2 ∈ zB R 0 ∩ P and μ 2 ≥ 1 such that
(50) erefore, which means that BVP equation (1)  Proof. On the one hand, assumption (H3) implies that there exist ε ∈ (0, λ 1 ) and r 1 > 0 such that We claim that, for μ ∈ [0, 1], Suppose, on the contrary, that there exist x 1 ∈zB r 1 ∩ P and μ ∈ [0, 1] such that Consequently, we have e nth iteration of this inequality shows that en, It means that which is a contradiction. It follows from Lemma 8 that deg T, B r 1 ∩ P, P � 1.
On the other hand, let L n x(t) � where n > 1. It is easy to see that L n : P ⟶ P is completely continuous operator and spectral radius r(L n ) > 0, denoted by λ n � r − 1 (L n ). We know lim n⟶+∞ λ n � λ 1 .
Up to now, some existence results of BVP equation (1) have been obtained by using the topological degree theory and Krein-Rutman theorem. In the following, the multiple solutions will be considered for BVP equation (1).