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Positive solution for qfractional fourpoint boundary value problems with pLaplacian operator
Journal of Inequalities and Applications volume 2014, Article number: 481 (2014)
Abstract
This paper investigates a class of fourpoint boundary value problems of fractional qdifference equations with pLaplacian operator ${D}_{q}^{\beta}({\phi}_{p}({D}_{q}^{\alpha}u(t)))=f(t,u(t))$, $t\in (0,1)$, $u(0)=0$, $u(1)=au(\xi )$, ${D}_{q}^{\alpha}u(0)=0$, and ${D}_{q}^{\alpha}u(1)=b{D}_{q}^{\alpha}u(\eta )$, where ${D}_{q}^{\alpha}$ and ${D}_{q}^{\beta}$ are the fractional qderivative of the RiemannLiouville type, pLaplacian operator is defined as ${\phi}_{p}(s)={s}^{p2}s$, $p>1$, and $f(t,u)$ may be singular at $t=0,1$ or $u=0$. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. Furthermore, two examples are presented to illustrate the main results.
MSC:39A13, 34B18, 34A08.
1 Introduction
Recently, fractional differential equations with pLaplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, nonNewtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science. There have appeared some results for the existence of solutions or positive solutions of boundary value problems for fractional differential equations with pLaplacian operator; see [1–7] and the references therein. For example, under different conditions, Wang et al. [8] and Ren and Chen [9] established the existence of positive solutions to fourpoint boundary value problems for nonlinear fractional differential equations with pLaplacian operator by using the upper and lower solutions method and fixed point theorems, respectively.
Since AlSalam [10] and Agarwal [11] proposed the fractional qdifference calculus, new developments in this theory of fractional qdifference calculus have been made due to the explosion in research within the fractional differential calculus setting. For example, some researcher obtained qanalogs of the integral and differential fractional operators properties such as the qLaplace transform, the qTaylor formula, the MittagLeffler function [12–15], and so on.
Recently, the theory of boundary value problems for nonlinear fractional qdifference equations has been addressed extensively by several researchers. There have been some papers dealing with the existence and multiplicity of solutions or positive solutions for boundary value problems involving nonlinear fractional qdifference equations by the use of some wellknown fixed point theorems and the upper and lower solutions method. For some recent developments on the subject, see [16–25] and the references therein. ElShahed and AlAskar [26] studied the existence of multiple positive solutions to the nonlinear qfractional boundary value problems by using the GuoKrasnoselskii fixed point theorem in a cone. Under different conditions, Graef and Kong [27, 28] investigated the existence of positive solutions for the boundary value problem with fractional qderivatives in terms of different ranges of λ, respectively. Zhao et al. [29] showed some existence results of positive solutions to nonlocal qintegral boundary value problem of nonlinear fractional qderivatives equation using the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii fixed point theorem. Ahmad et al. [30] considered the existence of solutions for the nonlinear fractional qdifference equation with nonlocal boundary conditions by applying some wellknown tools of fixed point theory such as the Banach contraction principle, the Krasnoselskii fixed point theorem, and the LeraySchauder nonlinear alternative. By applying the nonlinear alternative of LeraySchauder type and Krasnoselskii fixed point theorems, the author [31] established sufficient conditions for the existence of positive solutions for nonlinear semipositone fractional qdifference system with coupled integral boundary conditions. Relying on the standard tools of fixed point theory, Agarwal et al. [32] and Ahmad et al. [33] discussed the existence and uniqueness of solutions for a new class of sequential qfractional integrodifferential equations with qantiperiodic boundary conditions and nonlocal fourpoint boundary conditions, respectively.
In [34], Aktuǧlu and Özarslan dealt with the following Caputo qfractional boundary value problem involving the pLaplacian operator:
where ${a}_{0},{a}_{1}\ne 0$, $1<\alpha \in \mathbb{R}$, and $f\in C([0,1]\times \mathbb{R},\mathbb{R})$. Under some conditions, the authors obtained the existence and uniqueness of the solution for the above boundary value problem by using the Banach contraction mapping principle.
In [35], Miao and Liang studied the following threepoint boundary value problem with pLaplacian operator:
where $0<\beta {\eta}^{\alpha 2}<1$, $0<q<1$. The authors proved the existence and uniqueness of a positive and nondecreasing solution for the boundary value problems by using a fixed point theorem in partially ordered sets.
In [36], the author investigated the following fractional qdifference boundary value problem with pLaplacian operator:
where $1<\alpha ,\beta \le 2$. The existence results for the above boundary value problem were obtained by using the upper and lower solutions method associated with the Schauder fixed point theorem.
In this paper, motivated greatly by the above mentioned works, we consider the following fractional qdifference boundary value problem with pLaplacian operator:
where ${D}_{q}^{\alpha}$, ${D}_{q}^{\beta}$ are the fractional qderivative of the RiemannLiouville type with $1<\alpha ,\beta \le 2$, $0\le a,b\le 1$, $0<\xi ,\eta <1$, ${\phi}_{p}(s)={s}^{p2}s$, $p>1$, ${({\varphi}_{p})}^{1}={\varphi}_{r}$, $(1/p)+(1/r)=1$, and $f(t,u):(0,1)\times (0,+\mathrm{\infty})\to [0,\mathrm{\infty})$ is continuous and may be singular at $t=0,1$ or $u=0$. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at least one positive solution for the above fractional qdifference boundary value problem with pLaplacian operator are established. This work improves essentially the results of [36]. At the end of this paper, we will give two examples to show the effectiveness of the main results.
2 Preliminaries
For the convenience of the reader, we present some necessary definitions and lemmas of fractional qcalculus theory to facilitate the analysis of problem (1.1). These details can be found in the recent literature; see [37] and references therein.
Let $q\in (0,1)$ and define
The qanalog of the Pochhammer symbol (the qshifted factorial) is defined by
The qanalog of the power ${(ab)}^{n}$ with $n\in {\mathbb{N}}_{0}=\{0,1,2,\dots \}$ is
The following relation between them holds:
Their natural extensions to the reals are
Clearly, ${(ab)}^{(\gamma )}={a}^{\gamma}{(b/a;q)}_{\gamma}$, $a\ne 0$. Note that, if $b=0$ then ${a}^{(\alpha )}={a}^{\alpha}$. The qgamma function is defined by
and satisfies ${\mathrm{\Gamma}}_{q}(x+1)={[x]}_{q}{\mathrm{\Gamma}}_{q}(x)$.
The qderivative of a function f is here defined by
and qderivatives of higher order by
The qintegral of a function f defined in the interval $[0,b]$ is given by
If $a\in [0,b]$ and f is defined in the interval $[0,b]$, its integral from a to b is defined by
Similarly as done for derivatives, an operator ${I}_{q}^{n}$ can be defined, namely,
The fundamental theorem of calculus applies to these operators ${I}_{q}$ and ${D}_{q}$, i.e.,
and if f is continuous at $x=0$, then
Basic properties of the two operators can be found in the book [37]. We now point out three formulas that will be used later (${}_{i}D_{q}$ denotes the derivative with respect to variable i):
Denote that if $\alpha >0$ and $a\le b\le t$, then ${(ta)}^{(\alpha )}\ge {(tb)}^{(\alpha )}$ [18].
Definition 2.1 ([11])
Let $\alpha \ge 0$ and f be function defined on $[0,1]$. The fractional qintegral of the RiemannLiouville type is ${I}_{q}^{0}f(x)=f(x)$ and
Definition 2.2 ([13])
The fractional qderivative of the RiemannLiouville type of order $\alpha \ge 0$ is defined by ${D}_{q}^{0}f(x)=f(x)$ and
where m is the smallest integer greater than or equal to α.
Lemma 2.3 ([14])
Let $\alpha ,\beta \ge 0$ and f be a function defined on $[0,1]$. Then the next formulas hold:

(1)
$({I}_{q}^{\beta}{I}_{q}^{\alpha}f)(x)={I}_{q}^{\alpha +\beta}f(x)$,

(2)
$({D}_{q}^{\alpha}{I}_{q}^{\alpha}f)(x)=f(x)$.
Lemma 2.4 ([18])
Let $\alpha >0$ and p be a positive integer. Then the following equality holds:
Lemma 2.5 Let $y\in C[0,1]$, $1<\alpha \le 2$, $0<\xi <1$, and $0\le a\le 1$. Then the unique solution of the following linear fractional qdifference boundary value problem:
is given by
where
Proof By applying Lemma 2.4, we may reduce (2.1) to an equivalent integral equation
From $u(0)=0$ and (2.4), we have ${c}_{2}=0$. Consequently the general solution of (2.1) is
By (2.5), one has
And from $u(1)=au(\xi )$, then we have
So, the unique solution of problem (2.1) is
where $G(t,s)$ is defined in (2.3). The proof is completed. □
Lemma 2.6 Let $y\in C[0,1]$, $1<\alpha ,\beta \le 2$, $0<\xi ,\eta <1$, and $0\le a,b\le 1$. Then the following fractional qdifference boundary value problem with pLaplacian operator:
has unique solution given by
where ${b}_{1}={b}^{p1}$, $G(t,s)$ is defined by (2.3) and
Proof By applying Lemma 2.4, we may reduce (2.7) to an equivalent integral equation,
From ${D}_{q}^{\alpha}u(0)=0$ and (2.10), we have ${c}_{4}=0$. Consequently the general solution of (2.7) is
By (2.11), one has
From ${D}_{q}^{\alpha}u(1)=b{D}_{q}^{\alpha}u(\eta )$, we have
where ${b}_{1}={b}^{p1}$. Similar to Lemma 2.6, we have
Consequently, the fractional boundary value problem (2.7) is equivalent to the following problem:
Lemma 2.6 implies that the fractional boundary value problem (2.7) has a unique solution
The proof is completed. □
Lemma 2.7 Let $1<\alpha ,\beta \le 2$, $0<\xi ,\eta <1$, and $0\le a,b\le 1$. Then functions $G(t,s)$ and $H(t,s)$ defined by (2.3) and (2.9), respectively, are continuous on $[0,1]\times [0,1]$ satisfying

(a)
$G(t,qs)\ge 0$, $H(t,qs)\ge 0$, for all $t,s\in [0,1]$;

(b)
for all $t,s\in [0,1]$, ${\sigma}_{1}(qs){t}^{\alpha 1}\le G(t,qs)\le {\sigma}_{2}(qs){t}^{\alpha 1}$, where
$${\sigma}_{1}(s)=\frac{ag(\xi ,s)}{1a{\xi}^{\alpha 1}},\phantom{\rule{2em}{0ex}}{\sigma}_{2}(s)=\frac{{(1s)}^{(\alpha 1)}}{{\mathrm{\Gamma}}_{q}(\alpha )}+\frac{ag(\xi ,s)}{1a{\xi}^{\alpha 1}}.$$
Proof The proof is obvious, so we omit the proof. □
From Lemmas 2.5 and 2.7, it is easy to obtain the following lemma.
Lemma 2.8 Let $u(t)\in C([0,1],\mathbb{R})$ satisfies $u(0)=0$, $u(1)={\phi}_{p}(b)u(\eta )$, and ${D}_{q}^{\beta}u(t)\ge 0$ for any $t\in (0,1)$, then $u(t)\le 0$, for $t\in [0,1]$.
Let $E=\{u:u,{\phi}_{p}({D}_{q}^{\alpha}u)\in {C}^{2}[0,1]\}$. Now we introduce the following definitions about the upper and lower solutions of the fractional qdifference boundary value problem (1.1).
Definition 2.9 A function $\varphi (t)$ is called a lower solution of the fractional qdifference boundary value problem (1.1), if $\varphi (t)\in E$ and $\varphi (t)$ satisfies
Definition 2.10 A function $\psi (t)$ is called an upper solution of the fractional qdifference boundary value problem (1.1), if $\psi (t)\in E$ and $\psi (t)$ satisfies
3 Main results
For the sake of simplicity, we make the following assumptions throughout this paper.
(H_{1}) $f(t,u)\in C[(0,1)\times (0,+\mathrm{\infty}),[0,+\mathrm{\infty})]$ and $f(t,u)$ is decreasing in u.
(H_{2}) Set $e(t)={t}^{\alpha 1}$. For any constant $\rho >0$, $f(t,\rho )\not\equiv 0$, and
We define P ={$u\in C[0,1]$: there exist two positive constants $0<{l}_{u}<{L}_{u}$ such that ${l}_{u}e(t)\le u(t)\le {L}_{u}e(t)$, $t\in [0,1]$}. Obviously, $e(t)\in P$. Therefore, P is not empty. For any $u\in P$, define an operator T by
Theorem 3.1 Suppose that conditions (H_{1})(H_{2}) are satisfied, then the boundary value problem (1.1) has at least one positive solution u, and there exist two positive constants $0<{\lambda}_{1}<1<{\lambda}_{2}$ such that ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$.
Proof We will divide our proof into four steps.
Step 1. We show that T is well defined on P and $T(P)\subset P$, and T is decreasing in u.
In fact, for any $u\in P$, by the definition of P, there exist two positive constants $0<{l}_{u}<1<{L}_{u}$ such that ${l}_{u}e(t)\le u(t)\le {L}_{u}e(t)$ for any $t\in [0,1]$. It follows from Lemma 2.7 and conditions (H_{1})(H_{2}) that
On the other hand, it follows from Lemma 2.7 that
Take
then by (3.1) and (3.2), ${l}_{u}^{\prime}e(t)\le (Tu)(t)\le {L}_{u}^{\prime}e(t)$, which implies that T is well defined and $T(P)\subset P$. It follows from (H_{1}) that the operator T is decreasing in u. By direct computations, we can state that
Step 2. We focus on lower and upper solutions of the fractional qdifference boundary value problem (1.1). Let
then, if $e(t)=(Te)(t)$, the conclusion of Theorem 3.1 holds. If $e(t)\ne (Te)(t)$, clearly, $m(t),n(t)\in P$, and
We will prove that the functions $\varphi (t)=Tn(t)$, $\psi (t)=Tm(t)$ are a couple of lower and upper solutions of the fractional qdifference boundary value problem (1.1), respectively.
From (H_{1}), we know that T is nonincreasing relative to u. Thus it follows from (3.4) and (3.5) that
and $\varphi (t),\psi (t)\in P$. It follows from (3.3)(3.6) that
that is, $\varphi (t)$ and $\psi (t)$ are a couple of lower and upper solutions of the fractional qdifference boundary value problem (1.1), respectively.
Step 3. We will show that the fractional qdifference boundary value problem
has at least one positive solution, where
It follows from (H_{1}) and (3.9) that $g(t,u):[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous. To see this, we consider the operator $A:C[0,1]\to C[0,1]$ defined as follows:
where $G(t,s)$ is defined as (2.3), $H(t,s)$ is defined as (2.9). It is clear that $Au\ge 0$, for all $u\in P$, and a fixed point of the operator A is a solution of the boundary value problem (3.8). Noting that $\varphi (t)\in P$, there exists a positive constant $0<{l}_{\varphi}<1$ such that $\varphi (t)\ge {l}_{\varphi}e(t)$, $t\in [0,1]$. It follows from Lemma 2.7, (3.9), and (H_{2}) that
which implies that the operator A is uniformly bounded.
On the other hand, since $G(t,s)$ is continuous on $[0,1]\times [0,1]$, it is uniformly continuous on $[0,1]\times [0,1]$. So, for fixed $s\in [0,1]$ and for any $\epsilon >0$, there exists a constant $\delta >0$, such that any ${t}_{1},{t}_{2}\in [0,1]$ and ${t}_{1}{t}_{2}<\delta $,
Then, for all $u(t)\in C[0,1]$, we have
that is to say, A is equicontinuous. Thus, from the ArzelaAscoli theorem, we know that A is a compact operator, by using the Schauder fixed point theorem, the operator A has a fixed point u such that $u=Au$; i.e., the fractional qdifference boundary value problem (3.8) has a positive solution.
Step 4. We will prove that the boundary value problem (1.1) has at least one positive solution. Suppose that $u(t)$ is a solution of (3.3), we only need to prove that $\varphi (t)\le u(t)\le \psi (t)$, $t\in [0,1]$. Now we claim that $\varphi (t)\le u(t)\le \psi (t)$, $t\in [0,1]$. In fact, since u is fixed point of A and (3.7), we get
Suppose by contradiction that $u(t)\ge \psi (t)$. According to the definition of g, one verifies that
On the other hand, since ψ is an upper solution to (1.1), we obviously have
Let $z(t)={\phi}_{p}({D}_{q}^{\alpha}\psi (t)){\phi}_{p}({D}_{q}^{\alpha}u(t))$, $0<t<1$. From (3.11) and (3.12), we can get
Thus, by Lemma 2.8, we have $z(t)\le 0$, $t\in [0,1]$, which implies that
Since ${\phi}_{p}$ is monotone increasing, we obtain ${D}_{q}^{\alpha}\psi (t)\le {D}_{q}^{\alpha}u(t)$, i.e., ${D}_{q}^{\alpha}(\psi u)(t)\le 0$. Combining Lemma 2.8, we have $(\psi u)(t)\ge 0$. Therefore, $\psi (t)\ge u(t)$, $t\in [0,1]$, a contradiction to the assumption that $u(t)>\psi (t)$. Hence, $u(t)>\psi (t)$ is impossible.
Similarly, suppose by contradiction that $u(t)\le \varphi (t)$. According to the definition of g, one verifies that
Consequently, we obtain
On the other hand, since ϕ is an upper solution to (1.1), we obviously have
Let $z(t)={\phi}_{p}({D}_{q}^{\alpha}u(t)){\phi}_{p}({D}_{q}^{\alpha}\varphi (t))$, $0<t<1$. From (3.13) and (3.14), we get
Thus, by Lemma 2.5, we have $z(t)\le 0$, $t\in [0,1]$, which implies that
Since ${\phi}_{p}$ is monotone increasing, we obtain ${D}_{q}^{\alpha}u(t)\le {D}_{q}^{\alpha}\varphi (t)$, i.e., ${D}_{q}^{\alpha}(u\varphi )(t)\le 0$. Combining Lemma 2.5, we have $(u\varphi )(t)\ge 0$. Therefore, $u(t)\ge \varphi (t)$, $t\in [0,1]$, a contradiction to the assumption that $u(t)<\varphi (t)$. Hence, $u(t)<\varphi (t)$ is impossible.
Consequently, we have $\varphi (t)\le u(t)\le \psi (t)$, $t\in [0,1]$, that is, $u(t)$ is a positive solution of the boundary value problem (1.1). Furthermore, $\varphi (t),\psi (t)\in P$ implies that there exist two positive constants $0<{\lambda}_{1}<1<{\lambda}_{2}$ such that ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$. Thus, we have finished the proof of Theorem 3.1. □
Theorem 3.2 If $f(t,u)\in C([0,1]\times [0,+\mathrm{\infty}),[0,+\mathrm{\infty}))$ is decreasing in u and $f(t,\rho )\not\equiv 0$ for any $\rho >0$, then the boundary value problem (1.1) has at least one positive solution u, and there exist two positive constants $0<{\lambda}_{1}<1<{\lambda}_{2}$ such that ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$.
Proof The proof is similar to Theorem 3.1, we omit it here. □
4 Two examples
Example 4.1 Consider the pLaplacian fractional qdifference boundary value problem
It is easy to check that (H_{1}) holds. For any $\rho >0$, $f(t,\rho )\not\equiv 0$, we have
which implies that (H_{2}) holds. Theorem 3.1 implies that the boundary value problem (4.1) has at least one positive solution.
Example 4.2 Consider the pLaplacian fractional qdifference boundary value problem
It is not difficult to check that $f(t,u):[0,1]\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is continuous and decreasing in u and $f(t,\rho )\not\equiv 0$ for any $\rho >0$. Theorem 3.2 implies that the boundary value problem (4.2) has at least one positive solution.
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Yuan, Q., Yang, W. Positive solution for qfractional fourpoint boundary value problems with pLaplacian operator. J Inequal Appl 2014, 481 (2014) doi:10.1186/1029242X2014481
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Keywords
 fractional qdifference equations
 fourpoint boundary conditions
 pLaplacian operator
 positive solution
 upper and lower solutions method