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1 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, aero- dynamics and communications (see [3–5, 13] for example). Originally, the study on frac- tional .-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 .-calculus have lead to a new de- velopment direction of fractional .-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 .-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 .-difference equation

with boundary conditions

where 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 .-difference Schrödinger equation

where. 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

where 0 < q < 1, 2 < α ≤ 3, . may be singular at .. 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 .-Laplacian operator φ.(x(t)) in the turbulent flow model, recently, the fractional differential equations with .-Laplacian operator attracted much attention of scholars; see [8, 10, 18]. In 2016, Li et al. [10] investigated a fractional .-difference equation nonhomogeneous boundary value problem

restricted to

where 0 < γ < 1, 2 < α < 3, is a generalized .-Laplacian operator, which includes two cases: They gave the existence of positive solutions by some fixed point theorems in cones. As a general- ized 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 .-difference equation containing p(t)-Laplacian operator:

where, where Tq denotes the time scale defined by. denote the standard Riemann–Liouville fractional .-derivatives, µ > 0 is a parameter, y[u] denotes a linear functional given by

involving Stieltjes integral with respect to a suitable function A : [0, 1] → R of bounded variation. The measure d. can be a signed measure. Laplacian operator, and it has the following characteristics:

(a) ϕ : R → R is an odd and strictly monotone increasing homeomorphism;

(b) the inverse continuous operator and

Indeed, if . degenerates to constant ., then our research problem turns into (5)–(6); if . then it turns into (1)–(2); on this basis, if y = 0, then it becomes homogeneous boundary value problem (3); if . has two space variables, then it changed into (4), so the boundary condition we studied in this paper is more extensive. The form has not been seen in existing works.

We study problem (7) by using some fixed point theorems of increasing (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.

Throughout this paper, let, we assume that

(H1) is increasing with respect to the second variable, where (there exists at least one point .. such that f (to, 0) /= 0);

(H2) for any y ∈ (0,1), there exists with ln ln . such that ..

Remark 1.

, then assumption (H2) become condition (H1) in [17]. It can be regarded as a particular case of (H2) due to the characteristic of p(t)-Laplacian operator, here 0 + 1- p > 0 is demanded. Besides, this condition covers the superlinear, sublinear and mixed types of superlinear and sublinear functions.

Remark 2.

Condition (H2) implies that, for all Y ≥ 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.

2 Preliminaries and previous results

We present some necessary definitions and lemmas about fractional .-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 . is .-geometric, then it includes all geometric sequences The definition of .-analogue for a ∈ R is

The q-analogue of the Pochhammer symbol is defined by

Let f be a real-valued continuous function defined on a .-geometric set V , q = 1, the .-derivative of . is defined by

And

Furthermore, the .th .-derivativecan be represented by

The q-integral of a function f in the interval [0, b] is defined by

Definition 1.

(See [5].) Let ≥ 0 and . be a function defined on [0, 1]. The fractional .-integral of Riemann–Liouville type is and

Further, when = 1.

Definition 2.

[5] The fractional .-derivative of Riemann–Liouville type of order ≥ 0 is defined by

where [| is the smallest integer greater than or equal to.

Moreover,

Remark 3.

(See [19].) Assume that f (t) is a continuous function on [0, 1] and there exists to ∈ (0, 1) such that f (to) 0. If f (t) ≥ 0, then we have

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

Where

Proof. By Definitions 1, 2 and (8) we can reduce above problem to

It follows from the condition u(0) = Dqu(0) = 0 that c2 = c3 = 0, then

Further, one has

The condition Dqu(1) implies that

By simple calculation we get

and so, substituting it into (11), we deduce that

The proof is complete.

Lemma 2.

The function G₁(t, qs. has the following properties:

Proof. The proof is similar to Lemma 3.0.7. of [7], we omit it.

Remark 4.

From Lemma 1 and (10) we have

And

Next, we consider the following boundary value problem:

Lemma 3.

be a given function with g(0) = 0. Then problem (12) has a unique solution

where G(t, qs) is defined as in (10).

Proof. First, we deduce Then

has the solution From the condition which implies we translate into considering

Lemma 1 implies that problem (12) has a unique solution

The proof is complete.

Moreover, we collect some notations that are already known in literatures [20, 21].Let be a real Banach space and it is partially ordered by a cone for any the nation x-y means that there exist u > 0 and v > 0 such that ux <y < vx for fixed denotes the zero element of E. Define a set we define

Definition 3.

(See [21].) Let T : K_h,e→ E be a given operator. For any x ∈ K_h,e, λ ∈ (0, 1), there exists such that

Then . is called -concave operator.

Now we consider problem (7) in Banach space E = C[0, 1] endowed with the norm ǁUǁ = sup{|u(t)|: t ∈ [0, 1]}. Set the standard cone K = {x ∈ E: x(t) ≥ 0, mint_∈[τ,1] x(t) ≥ τ.⁻¹ǁxǁ, t ∈ [0, 1]}. Obviously, K ⊂ E is normal. Define the operator T : K → E by

Further, let

where

Lemma 4.

Let assumptions (H0)–(H2) hold. In addition, we assume that

(H3) 0 ≤ A < [a − 1]q, ζ(s) ≥ 0, where A, ζ are defined as in assumption (H0). If g(t) ≥ 0 with g(t) = 0, g(0) = 0 for t e [0, 1], then T . K_h,e > E is a ϕ - (h, e).- concave operator.

Proof. For t ∈ [0, 1], we have

Since g(t) ≥ 0, g(t) = 0, thus h(t) = 0. Then we show that 0 ≤ e(t) ≤ .h(t). By Lemma 2 and (H3) we have

And

that is, e ∈ K. Further, from Remark 4 one has

hence, 0 ≤ e(t) ≤ h(t). Moreover, K_h,e = { u e C[0, 1]: u + e e Kh} . In view of Lemma 3, the solution .(.) of problem (7) can be expressed as

For any u ∈ K_h,e, we consider the operator T . , which can also be written as

Evidently, u(t) is the solution of problem (7) if and only if . is the fixed point of T .

Now we show that T : K_h,e→ E is a − (h, e)-concave operator. For y ∈ (0, 1), u ∈ K_h,e, by condition (H2) we can obtain

Let For y ∈ (0, 1), because ln ln in (H2), we have ln and thus

So we have , we get

According to Definition 3, we know that T is a concave operator. The proof is complete.

Remark 5.

If g(t) ≤ 0 with g(0) = 0, (H0)–(H3) hold, it is clear that T is a p - (h, θ)- concave operator.

Remark 6.

Note that the inequalities of condition (H3) are general satisfied provided that dA is positive. Consider the case when the measure d. changes the sign, particularly, take It changes sign and one can see

Let 0 ≤ A < 1, then it requires that

If = 5/2, q = 1/2, then 0 ≤ (b − a)(a − b/2) < (65√2 − 24).4, while if a = 3, q = 1.2, then 0 ≤ (b− a)(a − b/2) < 105/4. Further, we know that if q → 1, dA(t) = (at − b) dt, a, b > 0. Then

Similarly, let 0 ≤ A < 1, it requires that b/α ≤ a − b < α + 1. If a = 5/2, then 2b/5 ≤ a − b < 7.2, while if = 3, then b/4 ≤ a − b < 4.

Lemma 5.

(See [21].) Let K be normal and T be an increasing ϕ − (h, e.-concave operator with Th ∈ K_h,e. Then T has a unique fixed point x. in K_h,e. Moreover, for any w0. ∈ K_h,e, constructing the sequence wn =Twn₋₁, n = 1, 2, . . . , then ||wn − x*ǁ → 0 as n → ∞.

Remark 7.

If e = 0, i.e., T is an increasing concave operator, the above result is still holds.

3 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 prob- lem (7)has a unique solution u. in K_h,e. Further, for any given v. K_h,e, constructing a sequence

one has

Proof. By means of Lemma 4 we know that . :concave operator. Now we prove that T : K_h,e→ E is increasing. For u ∈ K_h,e, we have u + e ∈ Kh, 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 Th E K_h,e, so we have to prove Th + e E Kh. From Lemma 2 and (H1) we get

And

Let

Since from (H1) we can easily get

hence we have r1 ≥ r2 > 0 and r2h ≤ Th + e ≤ r1h, which implies that Th + e K_he.

In view of Lemma 5, the operator . has a unique fixed point u* in K_h,e, and

Namely, u*(t) is the solution of problem (7). In addition, for any vo ∈ K_h,e, the sequence v_n = Tv_n−1, n = 1, 2, . . ., satisfies vn → u* as n → ∞, that is,

and

Corollary 1.

If the conditions of Theorem 1 hold and

then problem (7)has a unique nontrivial positive solution in K_h,e. In addition, we can also construct an iterative scheme

approximating the unique nontrivial positive solution u*(t).

Further, similar to the proof of Theorem 1, by using Remark 7 we have the following result:

Theorem 2.

Assume that (H0), (H3) hold, g(t) ≤ 0, g(0) = 0 and

(H4) f : Ro × R+ → R+ is increasing with respect to the second variable with f (t, 0) /≡ 0;

(H5) for any λ _E(0, 1), there exists ψ() (0, 1) with ln .

Then problem (7)has a unique positive solution u*in K_h, where h(t) = t.^a−1, t _E[0, 1]. Further, making a monotone iterative sequence

for any v_o ∈ K_h, we have vn(t) → u*(t) as n → ∞.

Next, we consider a special case of problem (7) with homogeneous boundary condi- tion

Corollary 2.

Assume (H0)–(H3) hold and g(t) ≥ 0, g(t) 0, g(0) = 0. Then prob- lem (13)has a unique solution u. in K_h,e. Further, for any v_oK_h,e, making a monotone iterative sequence

Corollary 3.

In Corollary ., if only requires ρ(s) - g(s) = 0, s [0, 1], then problem (13)has a unique nontrivial solution in K_h,e. In addition, we can also construct an iterative scheme shown as (14)approximating the unique nontrivial solution u*(t).

Corollary 4.

If g(t) ≤ 0, g(0) = 0 and assumptions (H3)–(H5) hold, then problem (13)has a unique nontrivial positive solution in K.. In addition, we can make an iterative scheme shown as (14)approximating the unique nontrivial solution u*(t).

Remark 8.

(i) From Theorem 1 and Lemma 5 we can see that the unique solution u* of prob- lem (1) is in a special set K_h,e. That is, there exist µ, ν > 0 such that u* [µh e, νh+.]. 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 .-difference equations, our main results has not been seen in previ- ous 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.

4 Examples

Example 1. Consider the following boundary value problem:

where, for . ∈ (0, 1],

for Then we obtain that ,

It is clear that is continuous and increasing with respect to the second variable,

Further, for one has

here .

we claim that condition (H2) holds. Therefore, Theorem 1 implies that problem (15) has a unique solution we construct a sequence

and we have .

Example 2. Consider the following boundary value problem:

where

and with it can be seen that is con- tinuous and increasing with respect to u, and . with . then condition (H4) holds. For ∈ (0, 1), we get

for all t_E [0, 1], u _E R+, where Hence condition (H5) is satisfied. Considering Theorem 2, problem (16) has a unique positive solution . making a sequence

for n = 1, 2, . . . , we have [0, 1].

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