Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation.

1. Introduction

Some researchers have paid close attention to the research of q-difference equation since the q-difference calculus and quantum calculus were discovered by Jackson [1, 2]. After the fractional q-difference calculus was developed by Al-Salam et al. [3-6], many papers on the fractional q-difference equation kept emerging, such as the papers [7-21] and their references. Among them, Li and Yang  established the existence of positive solutions for a class of nonlinear fractional q-difference equations with integral boundary conditions by applying monotone iterative method. Koca  provided an analytical method that can be used to solve analytically the Caputo fractional q-differential equations with initial condition x(0) = [x.sub.0]. The advantage of the method is that it can be applied to the integer order q-difference equations. By applying the monotone iterative technique combined with the method of lower and upper solutions, Wang et al.  obtained the existence of extremal solutions for fractional q-difference equation with initial value problem.

There are also many papers about boundary value problems of fractional q-difference equations; see [10-19] and the references therein. These experts did researches about the existence of a positive solution and multiple positive solutions to this problem by applying some well-known fixed-point theories such as Krasnosel'skii and Schauder fixed-point theorems. Thereinto, in [7, 15-20], the authors focused on the fractional q-difference equation with integral boundary value conditions.

Motivated by the methods of [21-23] and the above works, we study the criteria of three positive solutions for a Caputo fractional q-difference equation with integral boundary value conditions by employing properties of Green's function and the Leggett-Williams fixed-point theorem in this paper. We mainly consider the following problem:

([sup.C][D.sup.v.sub.q]x) (t) + f (t, x (t)) = 0, 0 < q < 1, 0 < t < 1, 2 < v < 3, (1)

x (0) = [D.sup.2.sub.q]x (0) = 0, x (1) = [lambda][[integral].sup.1.sub.0]x (s) [d.sub.q]s, (2)

where C[D.sup.v.sub.q] denotes the Caputo fractional q-derivative of order ] and f [member of] C([0, 1]x [0, + [infinity])) [right arrow] [0, +[infinity]) is continuous function.

In Section 2, basic definitions and some lemmas that will be used in the latter part are presented. In Section 3, some results for the existence of three positive solutions to problem (1)-(2) are established. And some examples to corroborate our results are given in Section 4.

2. Background Materials and Preliminaries

In this piece, we show some basic definitions and some lemmas that will be used to demonstrate our main results in the latter section.

Setting 0 < q < 1, we define

[[v].sub.q] = [1 - [q.sup.v]/1 - q], v [member of] R. (3)

The q-analogue of the power function [(t - s).sup.n] with n [member of] N is

[mathematical expression not reproducible]. (4)

If [alpha] [member of] R, then

[mathematical expression not reproducible]. (5)

We define the q-gamma function as follows:

[[GAMMA].sub.q] (s) = [(1 - q).sup.(s-1)] [(1 - q).sup.1-s], s [member of] R \ {0, -1, -2, ...}, (6)

which satisfies [[GAMMA].sub.q](s + 1) = [[s].sub.q][[GAMMA].sub.q](s).

For 0 < q < 1, the q-derivative of a function f is defined by

[mathematical expression not reproducible]. (7)

The higher order q-derivatives are defined by

([D.sup.0.sub.q]f) (t) = f (t), ([D.sup.n.sub.q]f) (t) = [D.sub.q] ([D.sup.n-1.sub.q]f) (t), n[member of] N. (8)

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

([I.sup.q.sub.f]) (t) = [[integral].sup.t.sub.0]f (s) [d.sub.q]s = t (1 - q) [[infinity].summation over (n=0)]f (t[q.sup.n]) [q.sup.n], t [member of] [0, b] (9)

provided that the series converges.

If f is defined on the interval [0, b] and a [member of] [0, b], its q-integral from a to b is defined by

[[integral].sup.b.sub.a] f (s) [d.sub.q]s = [[integral].sup.b.sub.0] f (s) [d.sub.q]s - [[integral].sup.a.sub.0]f (s) [d.sub.q]s. (10)

The higher order q-integrals are defined by

([I.sup.0.sub.q]f) (t) = f (t), ([I.sup.n.sub.q]f) (t) = [I.sub.q] ([I.sup.n-1.sub.q]f) (t), n[member of] N. (11)

We note that ([D.sub.q][I.sub.q]f)(t) = f(t) and if f is continuous at x=0, we get ([I.sub.q][D.sub.q]f)(t) = f(t) - f(0). For more details on the basic material of q-calculus, the readers can refer to [1-6].

Now let us give definitions of fractional q-integral and q-derivative.

Definition 1 (see [4, 6]). Let v [greater than or equal to] 0 and let f be a function defined on [L.sup.1]([0, 1]).The fractional q-integral of the Riemann-Liouville type is ([I.sup.0.sub.q]f)(t) = f(t) and

([I.sup.v.sub.q]f) (t) = [1/[GAMMA]q (v)] [[integral].sup.t.sub.0][(t - qs).sup.(v-1)] f (s) [d.sub.q]s, v > 0, t [member of] [0, 1], (12)

where [L.sup.1]([0, 1]) denotes the classical Banach space consisting of measurable functions on [0, 1] that are integrable.

Definition 2 (see ). The fractional q-derivative of the Riemann-Liouville type of order v [greater than or equal to] 0 is given by [D.sup.0.sub.q]f(t) = f(t) and

([D.sup.v.sub.q]f) (t) = ([D.sup.n.sub.q][I.sup.n-v.sub.q] f) (t), (13)

where n is the smallest integer greater than or equal to v.

Definition 3 (see ). The fractional q-derivative of Caputo type of order v [greater than or equal to] 0 for a function f is defined by

([sup.C][D.sup.v.sub.q]f) (t) = ([I.sup.n-v.sub.q] [D.sup.n.sub.q]f) (t). (14)

Lemma 4 (see ). Let v > 0 and let n be the smallest integer greater than or equal to v. Then, for t [member of] [0, 1], the following equality holds:

([I.sup.v.sub.q][sup.C][D.sup.v.sub.q]f) (t) = f (t) + [n-1.summation over (k=0)] [[t.sup.k]/[[GAMMA].sub.q] (k+1)][D.sup.k.sub.q]f (0). (15)

Lemma 5 (see ). Let [alpha] [member of] [R.sub.+], [beta] [member of] (-1, +[infinity]), and the following is valid:

[I.sup.[alpha].sub.q] [(t-s).sup.([beta])] = [[[GAMMA].sub.q] ([beta] + 1)/[[GAMMA].sub.q] ([beta] + [alpha] + 1)] [(t- s).sup.([beta]+[alpha])]. (16)

Next, Green's function for integral boundary value problem (1)-(2) is derived and the properties of Green's function are concluded. These properties will be used to demonstrate the main results in Section 3.

Lemma 6. Given g(t) [member of] C[0, 1], the unique solution of the following problem

([sup.C][D.sup.v.sub.q]x) (t) + g (t) = 0, 0 < t < 1, 2 < v < 3, (17)

x (0) = [D.sup.2.sub.q]x (0) = 0, x (1) = [lambda][[integral].sup.1.sub.0] x (s) [d.sub.q]s (18)

is

x (t) = [[integral].sup.1.sub.0] G(t, qs) g (s) [d.sub.q]s, (19)

where

[mathematical expression not reproducible], (20)

and [tau] = ([.sub.q] - [lambda])[[v].sub.q]. Here G(t, qs) is called Green's function of boundary value problem (17)-(18).

Proof. By Lemma 4, it is clear that (17) is equivalent to

x (t) = - [1/[[GAMMA].sub.q] (v)][[integral].sup.t.sub.0][(t - qs).sup.(v-1)] g (s) [d.sub.q]s+[c.sub.1] + [c.sub.2]t+ [c.sub.3][t.sup.2] (21)

for some [c.sub.i] [member of] R, i = 1, 2, 3. Applying the boundary condition x(0) = [D.sup.2.sub.q]x(0) = 0, there is [c.sub.1] = [c.sub.3] = 0, and then

x (t) = - [1/[[GAMMA].sub.q] (v)][[integral].sup.t.sub.0][(t - qs).sup.(v-1)] g (s) [d.sub.q]s+[c.sub.2]t. (22)

Using the condition x(1) = [lambda] [[integral].sup.1.sub.0] x(s)[d.sub.q]s, there is

[c.sub.2] = [1/[[GAMMA].sub.q] (v)][[integral].sup.1.sub.0][(1 - qs).sup.(v-1)] g (s) [d.sub.q]s+[lambda][[integral].sup.1.sub.0]x (s) [d.sub.q]s. (23)

Substituting [c.sub.2] into (22), we get

[mathematical expression not reproducible]. (24)

Let [xi]=[[integral].sup.1.sub.0] x(s)[d.sub.q]s; integrating equality (24) with respect to t from t=0 to t=1 and then exchanging integral order, we obtain that

[mathematical expression not reproducible]. (25)

Solving the above equation, then

[mathematical expression not reproducible]. (26)

Substituting [xi] into (24), we get

[mathematical expression not reproducible]. (27)

The proof is complete.

Remark 7. It is obvious that G(0, qs) = G(t, 1) = 0 for all t, qs [member of] [0, 1] and [lambda][not equal to][.sub.q].

Lemma 8. Suppose 2 < v < 3 and 0 < [lambda] < [.sub.q]. Then the function G(t, qs) defined by (20) satisfies the following inequalities:

tG(1, qs) [less than or equal to] G(t, qs) [less than or equal to][[.sub.q] [[v].sub.q]/[lambda] ([[v].sub.q] - [.sub.q])] G(1, qs), (28)

(t, qs) [member of] [0, 1] x [0, 1].

Proof. According to the expression of G(t, qs), we get

[mathematical expression not reproducible]. (29)

For the case 0 [less than or equal to] t [less than or equal to] qs [less than or equal to] 1, it is clear that

[mathematical expression not reproducible]. (30)

For the case 0 [less than or equal to] qs [less than or equal to] t [less than or equal to] 1, it is easy to see that

[mathematical expression not reproducible]. (31)

On the other hand,

[mathematical expression not reproducible]. (32)

Therefore

tG(1, qs) [less than or equal to] G(t, qs) [less than or equal to] [[.sub.q] [[v].sub.q]/[lambda] ([[v].sub.q] - [.sub.q])] G(1, qs), 0 [less than or equal to] qs < t [less than or equal to] 1. (33)

When t=1, inequalities (28) are obvious. In conclusion, inequalities (28) are fulfilled. The proof is complete.

Corollary 9. If [[v].sub.q] - [.sub.q] > 0, then G(t, qs) > 0, for t, qs [member of] (0, 1).

Definition 10 (see ). If P is a cone of the real Banach space E, a mapping [theta] : P [right arrow] [0, [infinity]) is continuous and with

[theta] (tx + (1-t) y) [greater than or equal to] t[theta] (x) + (1-t) [theta] (y), x, y [member of] P, t [member of] [0, 1], (34)

it is called a nonnegative concave continuous functional [theta] on P.

Assuming that r, a, b are positive constants, we will employ the following notations:

Pr = {x[member of]P: [parallel]x[parallel] < r}, [bar.[P.sub.r]] = {x[member of]P: [parallel]x[parallel] [less than or equal to] r}, P ([theta], a, b) = {x[member of]P:[theta] (x) [greater than or equal to] a, [parallel]x[parallel] [less than or equal to] b}. (35)

Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.

Lemma 11 (see ). Let E = (E, [parallel]*[parallel]) be a Banach space, P[subset]E be a cone of E, and c>0 be a constant. Suppose there exists a concave nonnegative continuous functional [theta] on P with [theta](x) [less than or equal to] [parallel]x[parallel] for x [member of] [bar.[P.sub.c]]. Let T : [bar.[P.sub.c]] [right arrow] [bar.[P.sub.]c] be a completely continuous operator. Assume there are numbers, a, and b with 0<d<a<b[less than or equal to]c such that

(i) the set {x [member of] P([theta], a, b) : [theta](x) > a} is nonempty and [theta](Tx) > a for all x [member of] P([theta], a, b);

(ii) [parallel]Tx[parallel] < d for x [member of] [bar.[P.sub.d]];

(iii) [theta](Tx) > a for all x [member of] P([theta], a, c) with [parallel]Tx[parallel] > b.

Then T has at least three fixed points [x.sub.1], [x.sub.2], and [x.sub.3] [member of] [bar.[P.sub.c]]. Furthermore, we have

[mathematical expression not reproducible]. (36)

3. Existence of Three Positive Solutions

In this section, the above lemmas will be applied to obtain the main results of this paper.

Let C[0, 1] be the space of all continuous real functions defined on [0, 1] with the maximum norm [parallel]x[parallel] = [max.sub.t[member of][0,1]][absolute value of x(t)]. We can know it is a Banach space. Define the cone P [subset] C[0, 1] as follows:

P = {x[member of]C[0, 1] : x (t) [greater than or equal to] 0, t [member of] [0, 1]}. (37)

From Lemma 6, we know that x(t) is a solution of boundary value problem (1)-(2) if and only if it satisfies

x (t) = [[integral].sup.1.sub.0]G(t, qs) f (s, x (s)) [d.sub.q]s, t [member of] [0, 1]. (38)

That is to say, the positive solutions of problem (1)-(2) are equivalent to the fixed points of T in C[0, 1] defined by

(Tx) (t) = [[integral].sup.1.sub.0]G(t, qs) f (s, x (s)) [d.sub.q]s, t [member of] [0, 1]. (39)

Then T(P) [subset] P and using the Ascoli-Arzela theorem, we are able to confirm that T is completely continuous.

We shall use Lemma 11 to discuss the existence of three fixed points to T. We then obtain sufficient conditions for the existence of three positive solutions to problem (1)-(2). To establish our main results, we take a positive number [mu] [member of] (0, 1), letting the nonnegative concave continuous function [theta] on P be defined by

[mathematical expression not reproducible]. (40)

Denote

[mathematical expression not reproducible]. (41)

And suppose that the function f(t, x) satisfies the following condition:

(C) f(t, x) is a nonnegative continuous function on [0, 1] x [0, +[infinity]) and there exists [t.sub.n] [right arrow] 0 such that f([t.sub.n], x([t.sub.n])) > 0, n = 1, 2, ....

Theorem 12. Assume that condition (C) holds and there exist constants 0<d<a such that

(C1) f(t, x) < Ad for (t, x) [member of] [0, 1] x [0, d];

(C2) f(t, x) [greater than or equal to] (B/[mu])a for (t, x) [member of] [[mu], 1] x [a, c], where c > [rho]a;

(C3) f(t, x) [less than or equal to] [kappa]x + [beta] for (t, x) [member of] [0, 1] x [0, +[infinity]), where [kappa], [beta] are positive numbers.

Then the boundary value problem (1)-(2) has at least three positive solutions [x.sub.1], [x.sub.2], and [x.sub.3].

Proof. Set c > max{[beta]/(A - [kappa]), [rho]a}, and then, for x [member of] [bar.[P.sub.c]], we have from (28)

[mathematical expression not reproducible]. (42)

That is, Tx [member of] [P.sub.c]. Therefore T : [bar.[P.sub.c]] [right arrow] [bar.[P.sub.c]] is a completely continuous operator. By (C1), we can get

[mathematical expression not reproducible]. (43)

Hence condition (ii) of Lemma 11 is satisfied.

We choose [x.sub.0] = ([rho] + 1)a/2 for t [member of] [[mu], 1]; then [x.sub.0] [member of] {x [member of] P([theta], a, [rho]a) : [theta](x) > a}, which implies {x [member of] P([theta], a, [rho]a) : [theta](x) > a} [not equal to] 0. Hence, if x [member of] P([theta], a, [rho]a), then a [less than or equal to] x(t) [less than or equal to] [rho]a for [mu] [less than or equal to] t [less than or equal to] 1. Thus

[mathematical expression not reproducible]. (44)

From the above inequality, we see that [theta](Tx) > a for all x [member of] P([theta], a, [rho]a). This affirms that condition (i) of Lemma 11 is satisfied.

Finally, for x [member of] P([theta], a, c) with [parallel]Tx[parallel] > [rho]a, we get

[mathematical expression not reproducible]. (45)

This confirms that condition (iii) of Lemma 11 is fulfilled. By virtue of Lemma 11, the boundary value problem (1)-(2) has at least three solutions [x.sub.1], [x.sub.2], and [x.sub.3]. Taking into account the fact that condition (C) holds, we have [x.sub.i](t) > 0, 0 < t < 1, i = 1, 2, 3. The proof is complete.

Theorem 13. Let condition (C) hold. Assume that there exist constants 0 < d < a < c (c > [rho]a) such that (C1), (C2), and (C4) are satisfied, where

(C4) f(t, x) [less than or equal to] Ac for (t, x) [member of] [0, 1] x [0, c].

Then the boundary value problem (1)-(2) has at least three positive solutions [x.sub.1], [x.sub.2], and [x.sub.3] such that

[mathematical expression not reproducible]. (46)

Proof. From (C4), we get

[mathematical expression not reproducible]. (47)

Therefore, T : [[bar.P].sub.c] [right arrow] [[bar.P].sub.c]. The remainder of the proof is similar to the proof of Theorem 12 and is therefore omitted. By Lemma 11, the boundary value problem (1)-(2) has at least three positive solutions [x.sub.1], [x.sub.2], and [x.sub.3] satisfying

[mathematical expression not reproducible]. (48)

The proof is complete.

Theorem 14. Let condition (C) hold. Assume that there exist constants 0 < d < a such that (C1) and (C2) are satisfied, and function f(t, s) satisfies

(C5) [f.sup.[infinity]] < A.

Then the boundary vale problem (1)-(2) has at least three positive solutions.

Proof. From hypothesis (C5), there exist 0 < [sigma] < A and R > 0; when x[greater than or equal to]R, we have

f (t, x) [less than or equal to] [sigma]u. (49)

Set M = [max.sub.(t,x)[member of][0,1]x[0,R]]f(t, x); consequently we get

0[less than or equal to]f (t, x) [less than or equal to] [sigma]x + M, 0 [less than or equal to] x < +[infinity]. (50)

This shows that condition (C3) of Theorem 12 is satisfied. By Theorem 12, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.

Theorem 15. Assume that there exist two positive constants a, c (c > [rho]a) such that conditions (C), (C2), and (C4) hold. And function f(t, x) satisfies

(C6) [f.sup.0] < A.

Then the boundary value problem (1)-(2) has at least three positive solutions.

Proof. In line with (C6), it is easy to see that there exists a positive constant d<a such that, for [parallel]x[parallel] < d, we have

f (t, x (t)) < Ax. (51)

That is to say,

f (t, x (t)) < Ad, [parallel]x[parallel] < d. (52)

This implies that conditions of Theorem 13 are satisfied. By Theorem 13, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.

In light of the proof of Theorems 14 and 15, we obtain one theorem and four corollaries as follows.

Theorem 16. Assume that the function f(t, x) satisfies conditions (C), (C2), (C5), and (C6). Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 17. Assume that conditions (C), (C2), and (C3) hold. The function f(t, x) satisfies [f.sup.0] = 0. Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 18. Assume that conditions (C), (C1), and (C2) hold. The function f(t, x) satisfies [f.sup.[infinity]] = 0. Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 19. Assume that conditions (C), (C2), and (C4) hold. The function f(t, x) satisfies [f.sup.[infinity]] = 0. Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 20. Assume that conditions (C) and (C2) hold. The function f(t, x) satisfies [f.sup.0] = 0 and [f.sup.[infinity]] = 0. Then the boundary value problem (1)-(2) has at least three positive solutions.

4. Examples

In this section, we present three examples to illustrate our results. We take v = 5/2, q = 1/2, [mu] = 1/2, [lambda] = 1/2, and by estimation, we then have A > 0.65, B < 17.785.

Consider the Caputo fractional q-difference

([sup.C][D.sup.v.sub.q]x) (t) + f (t, x) = 0, 0 < t < 1, (53)

with the boundary conditions

x (0) = [D.sup.2.sub.q]x (0) = 0, x (1) = [1/2] [[integral].sup.1.sub.0]x (s) [d.sub.q]s. (54)

Example 1. We take

[mathematical expression not reproducible]. (55)

There exist constants d = 1/36 and a = 253/250 such that

[mathematical expression not reproducible]. (56)

All the conditions of Theorem 12 hold. Thus, at this moment, by virtue of Theorem 12 we know that the boundary value problem (53)-(54) has three positive solutions.

Example 2. We take

[mathematical expression not reproducible]. (57)

There exist constants d = 0.25 and a = 1.06 such that

[mathematical expression not reproducible]. (58)

All the conditions of Theorem 14 hold. Thus, in this case, by Theorem 14 we know that the boundary value problem (53)-(54) has three positive solutions.

Example 3. We seek

[mathematical expression not reproducible]. (59)

There exists constant a = 1.5 such that

[mathematical expression not reproducible]. (60)

All the conditions of Theorem 16 hold. Thus, in this case, by using Theorem 16 we know that the boundary value problem (53)-(54) has three positive solutions.

5. Conclusions

The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractional q-difference equation with integral boundary value conditions are discussed. The study in the paper was to provide an analytical method: The Leggett-Williams fixed-point theorem can be used to solve fractional q-difference equation. In order to use the Leggett-Williams fixed-point theorem, Green's function and its properties were derived. By applying these properties and the Leggett-Williams fixed-point theorem, we presented the existence of three positive solutions of this class of fractional q-difference equations with integral boundary value conditions. An important advantage of this method is that it can be used to study three positive solutions for integer order q-differential equations and fractional differential equation, and so forth.

https://doi.org/10.1155/2018/2475284

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant no. 11271235), the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111117, and 20111020), and Shanxi Datong University Institute (2016K9 and 2017K4).

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Huiqin Chen, Shugui Kang (iD), Lili Kong, and Ying Gao

School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China

Correspondence should be addressed to Shugui Kang; dtkangshugui@126.com

Received 28 November 2017; Revised 9 January 2018; Accepted 28 January 2018; Published 26 February 2018

Academic Editor: Douglas R. Anderson
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