A fractional q-difference equation eigenvalue problem with p(t)-Laplacian operator

Abstract. This article is devoted to studying a nonhomogeneous boundary value problem involving Stieltjes integral for a more general form of the fractional q-difference equation with p(t)-Laplacian operator. Here p(t)-Laplacian operator is nonstandard growth, which has been used more widely than the constant growth operator. By using fixed point theorems of φ − (h, e)-concave operators some conditions, which guarantee the existence of a unique positive solution, are derived. Moreover, we can construct an iterative scheme to approximate the unique solution. At last, two examples are given to illustrate the validity of our theoretical results.


Introduction
Fractional calculus, appeared at the beginning of twentieth century, has provided many hot topics of research in many disciplines such as biological sciences, engineering, aerodynamics and communications (see [3][4][5]13] for example). Originally, the study on fractional q-difference calculus can be traced back to Agarwal [1] and Al-Salam [4], then it inspired much interest in theoretical research, many remarkable results have been arisen, which can be found in [2,6,7,14].
Naturally, the widespread applications of fractional q-calculus have lead to a new development direction of fractional q-difference equations, which has exhibited adamantine incorporation to application in fluid mechanics and quantum calculus. After that, kinds of fixed point theorems have been used to deal with various fractional q-difference equation boundary value problems; see [1, 3, 6, 7, 10-12, 15, 16, 18, 19] for instance. As we know, the study of existence, uniqueness and multiplicity of solutions are abundant. In 2011, Ferreira [7] studied a fractional q-difference equation D α q y(x) = −f x, y(x) , x ∈ (0, 1), with boundary conditions where 0 < q < 1, 2 < α 3, f : [0, 1] × R → R is a nonnegative continuous function. By employing Krasnosel'skii fixed point theorem the existence of positive solutions was enunciated.
In 2017, Wang [17] studied twin iterative positive solutions for a fractional q-difference Schrödinger equation where 0 < q < 1, 2 < α < 3, f ∈ C([0, ∞), (0, ∞)), h ∈ C((0, 1), (0, ∞)). The author obtained the existence of twin iterative positive solutions by using a fixed point theorem in cones associated with monotone iterative method. In 2020, Mao et al. [12] generalized the results in [17], the general research problem is D α q u(t) + f t, u(t), v(t)) = 0, 0 < t < 1, where 0 < q < 1, 2 < α 3, f may be singular at v = 0, t = 0, 1. By the iterative algorithm the author obtained a unique positive solution, where the nonlinear term has two space variables. In 2017, we have studied this boundary value problem in [19] by using the monotone iterative technique and lower-upper solution method, the existence of positive or negative solutions are obtained under the nonlinear term is local continuity and local monotonicity.
Since Leibenson [9] presented the p-Laplacian operator φ p (x(t)) in the turbulent flow model, recently, the fractional differential equations with p-Laplacian operator attracted much attention of scholars; see [8,10,18]. In 2016, Li et al. [10] investigated a fractional q-difference equation nonhomogeneous boundary value problem restricted to where 0 < γ < 1, 2 < α < 3, φ : R → R is a generalized p-Laplacian operator, which includes two cases: φ(u) = u and φ(u) = |u| p−2 u, p 1. They gave the existence of positive solutions by some fixed point theorems in cones. As a generalized form of p-Laplacian operator, p(t)-Laplacian operator arises from image restoration, elastic mechanics, nonlinear electro-rheological fluids, which has been widely used in different fields such as physics, image processing, bioengineering, etc, with respect to some valuable results that we can see [6,22]. Different from the above-mentioned works, in this article, we discuss the following nonhomogeneous two-point boundary value problem of a fractional q-difference equation containing p(t)-Laplacian operator: where where T q denotes the time scale defined by T q = {q n : n ∈ N} ∪ {0}. D α q , D β q denote the standard Riemann-Liouville fractional q-derivatives, µ > 0 is a parameter, λ[u] denotes a linear functional given by involving Stieltjes integral with respect to a suitable function Λ : [0, 1] → R of bounded variation. The measure dA can be a signed measure.
We study problem (7) by using some fixed point theorems of increasing ϕ − (h, e)concave operators. Several new existence-uniqueness criteria of nontrivial solutions for problem (7) are obtained. In addition, we can construct a convergent monotone iterative scheme for approximating the unique solution, and the existence of lower-upper solutions is not required, thus our result weakened the restrictions in [19]. It should be pointed out that the compactness condition is not required, when g(t) ≡ 0, our unique results are also new.
Remark 2. Condition (H2) implies that, for all λ 1, we have The paper is organized as follows. Section 2 contains some definitions and lemmas that will be used later. In Section 3, the local unique positive solution of problem (7) is obtained by using fixed point theorems in cones. Two examples are added to illustrate the main results in Section 4.

Preliminaries and previous results
We present some necessary definitions and lemmas about fractional q-calculus; for details, we can see [1,4].
For fixed point q ∈ R, V is a sunset of complex set C, V is called q-geometric if qt ∈ V whenever t ∈ V , that is to say, if V is q-geometric, then it includes all geometric sequences {tq n } ∞ n=0 . The definition of q-analogue for α ∈ R is The q-analogue of the Pochhammer symbol is defined by Let f be a real-valued continuous function defined on a q-geometric set V , |q| = 1, the q-derivative of f is defined by and Furthermore, the nth q-derivative D n q can be represented by The q-integral of a function f in the interval [0, b] is defined by Definition 1. (See [5].) Let α 0 and f be a function defined and where α is the smallest integer greater than or equal to α. Moreover, Remark 3. (See [19].) Assume that f (t) is a continuous function on [0, 1] and there exists . First, we consider the following boundary value problem: We require the following assumption: Lemma 1. Assume (H0) holds and y ∈ C[0, 1], then problem (9) has a unique solution http://www.journals.vu.lt/nonlinear-analysis Proof. By Definitions 1, 2 and (8) we can reduce above problem to It follows from the condition u(0) = D q u(0) = 0 that c 2 = c 3 = 0, then Further, one has By simple calculation we get and so, substituting it into (11), we deduce that The proof is complete.
Lemma 2. The function G 1 (t, qs) has the following properties: Proof. The proof is similar to Lemma 3.0.7. of [7], we omit it.
Moreover, we collect some notations that are already known in literatures [20,21].
Let (E, · ) be a real Banach space and it is partially ordered by a cone K ⊂ E. For any x, y ∈ E, the notation x ∼ y means that there exist µ > 0 and ν > 0 such that µx y νx. For fixed h > θ(i.e., h θ and h = θ), θ denotes the zero element of E.
Remark 6. Note that the inequalities A 0, ζ(s) 0 of condition (H3) are general satisfied provided that dΛ is positive. Consider the case when the measure dΛ changes the sign, particularly, take dΛ(t) = (at − b) d q t, a, b > 0. It changes sign and one can see Let 0 A < 1, then it requires that Similarly, let 0 Lemma 5. (See [21].) Let K be normal and T be an increasing ϕ − (h, e)-concave operator with T h ∈ K h,e . Then T has a unique fixed point x * in K h,e . Moreover, for any w 0 ∈ K h,e , constructing the sequence w n = T w n−1 , n = 1, 2, . . . , then w n − x * → 0 as n → ∞.

Local unique solutions
In this section, we can formulate some results giving sufficient conditions for the existence and uniqueness of solution to problem (7). Theorem 1. Assume that (H0)-(H3) hold, g(t) 0 with g(t) ≡ 0, g(0) = 0. Then problem (7) has a unique solution u * in K h,e . Further, for any given v 0 ∈ K h,e , constructing a sequence v n (t) = Proof. By means of Lemma 4 we know that T : K h,e → E is a ϕ − (h, e)-concave operator. Now we prove that T : K h,e → E is increasing. For u ∈ K h,e , we have u + e ∈ K h , and then there exists m > 0 such that u(t) + e(t) mh(t). We obtain By using the condition (H1) we know T : K h,e → E is increasing. As follows, we prove that T h ∈ K h,e , so we have to prove T h + e ∈ K h . From Lemma 2 and (H1) we get T h(t) + e(t) Let , .

Remark 8.
(i) From Theorem 1 and Lemma 5 we can see that the unique solution u * of problem (1) is in a special set K h,e . That is, there exist µ, ν > 0 such that u * ∈ [µh−e, νh+e]. So we say u * is a local solution. (ii) From Theorem 2 and Remark 7 the unique solution u * of problem (1) is in a special set K h . That is, there exist µ, ν > 0 such that u * ∈ [µh, νh], and thus u * is a positive solution. (iii) For fractional q-difference equations, our main results has not been seen in previous works. The method used here is relatively new, which cannot only guarantee the existence of unique solution, but also can approximate to the unique solution by making an iterative scheme.