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Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the pLaplacian operator
Advances in Difference Equations volume 2015, Article number: 40 (2015)
Abstract
In this article, we investigate the existence of a solution arising from the following fractional qdifference boundary value problem by using the pLaplacian operator: \(D_{q}^{\gamma}(\phi_{p}(D_{q}^{\delta}y(t)))+f(t,y(t))=0\) (\(0< t<1\); \(0<\gamma<1\); \(3<\delta<4\)), \(y(0)=(D_{q}y)(0)=(D_{q}^{2}y)(0) =0\), \(a_{1}(D_{q}y)(1)+a_{2}(D_{q}^{2}y)(1)=0\), \(a_{1} +\vert a_{2}\vert \neq0\), \(D_{0+}^{\gamma}y(t)_{t=0}=0\). We make use of such a fractional qdifference boundary value problem in order to show the existence and uniqueness of positive and nondecreasing solutions by means of a familiar fixed point theorem.
1 Introduction, definitions, and preliminaries
Recently, many mathematicians, physicists and engineers have extensively studied various families of fractional differential equations and their applications. The development of the theory of fractional calculus stems from the applications in many widespread disciplines such as engineering, economics and other fields. Jackson [1] introduced the qdifference calculus (or the socalled quantum calculus), which is an old subject. New developments in this theory were made. These include (for example) the qanalogs of the fractional integral and the fractional derivative operators, the qanalogs of the Laplace, Fourier, and other integral transforms, and so on (see, for details, [2–13], and [14]; see also a very recent work [15] dealing with qcalculus).
Throughout our present investigation, we make use of the following notations:
Moreover, as usual, ℝ denotes the set of real numbers, \(\mathbb{R}_{+}\) denotes the set of positive real numbers, \(\mathbb{Z}_{}\) denotes the set of negative integers, and ℂ denotes the set of complex numbers.
AlSalam [16] and Agarwal [2] investigated several properties and results for some fractional qintegrals and fractional qderivatives which are based on the qanalog of the ordinary integral:
Atici and Eloe [3] constructed interesting links between the fractional qcalculus in the existing literature and the fractional qcalculus on a time scale given by
They also derived some properties of a qLaplace transform, which are used to solve fractional qdifference equations. Benchohra et al. [17] investigated the existence of solutions for fractionalorder functional equations by means of the Banach fixed point theorem and its nonlinear alternative of LeraySchauder type. ElSayed et al. [18] studied the stability, existence, uniqueness, and numerical solution of the fractionalorder logistic equation. The work of ElShahed [19] was concerned with the existence and nonexistence of positive solutions for some nonlinear fractional boundary value problems. Ferreira (see [20] and [21]) investigated the existence of nontrivial solutions to some nonlinear qfractional boundary value problems by applying a fixed point theorem in cones. For more information on the positive solutions (or nontrivial solutions) for a class of boundary value problems with the fractional differential equations (or qfractional differential equations), we refer the reader to such earlier works as (for example) [5, 10, 22–31], and [32].
We now review briefly some concepts of the quantum calculus.
For \(q\in(0,1)\), the qinteger \([\lambda]_{q}\) is defined by
Clearly, we have
so we say that \([ \lambda ] _{q}\) is a qanalog of the number λ. The qanalog of the binomial formula \((ab)^{n}\) is given by
More generally, we have
Clearly, if we set \(b=0\) in Eq. (1.1), it reduces immediately to
The qgamma function is defined as follows:
and satisfies the formula:
The qderivative of a function \(f(x)\) is given by
For the qderivatives of higher order, we have
Suppose now that \(0< a<b\). Then the definite qintegral is defined as follows:
and
The operator \(I_{q}^{n}\) can be defined by
The Fundamental Theorem of Calculus does indeed apply mutatis mutandis to the operators \(I_{q}\) and \(D_{q}\). We thus have
and if f is continuous at \(x=0\), then
Denoting by \(_{x}D_{q}\) the qderivative with respect to the variable x, we now recall the following three formulas which will be used in the remainder of this paper:
Definition 1
(see [21])
Let \(\delta\geqq0\) and f be a function defined on \([0,1]\). The fractional qintegral of the RiemannLiouville type is given by
and
Definition 2
The fractional qderivative of the RiemannLiouville type of order δ (\(\delta\geqq0 \)) is defined by
and
where m is the smallest integer greater than or equal to δ.
Lemma 1
(see [21])
Let \(\delta \geqq0\), \(\beta\geqq0\), and f be a function defined on \([0,1]\). Then the following two formulas hold true:

(1)
\((I_{q}^{\beta}I_{q}^{\delta }f)(x)=(I_{q}^{\delta +\beta}f)(x)\);

(2)
\((D_{q}^{\delta}I_{q}^{\delta}f)(x)=f(x)\).
Lemma 2
Let \(\delta>0\) and p be a positive integer. Then the following equality holds:
Theorem 1
(a) Let \(( E,\leqq )\) be a partially ordered set and suppose that there exists a metric d in E such that \(( E,d )\) is a complete metric space. Assume that E satisfies the condition that, if \(\{ x_{n} \}\) is a nondecreasing sequence in E such that \(x_{n}\rightarrow x\), then
Let \(T:E\rightarrow E\) be a nondecreasing mapping such that
where
is a continuous and nondecreasing function such that ψ is positive in \(( 0,\infty )\), \(\psi ( 0 ) =0\), and
If there exists \(x_{0}\in E\) with \(x_{0}\leqq T ( x_{0} )\), then T has a fixed point.
(b) If we assume that \(( E,\leqq )\) satisfies the condition that, for \(x\in E\) and \(y\in E\), there exists \(z\in E\) which is comparable to x and y and the hypothesis of (a), then it leads to the uniqueness of the fixed point.
Mena et al. [27] investigated the existence and uniqueness of positive and nondecreasing solutions for the following singular fractional boundary value problem:
Miao and Liang [10], on the other hand, studied the existence and uniqueness of a positive and nondecreasing solution for the following fractional qdifference boundary value problem:
Motivated essentially by the aforementioned work by Miao and Liang [10], we introduce and investigate here the following qdifference boundary value problem by using the pLaplacian operator:
We prove the existence and uniqueness of a positive and nondecreasing solution for the boundary value problem given by Eqs. (1.5) and (1.6) by means of a fixed point theorem involving partially ordered sets.
2 Fractional boundary value problem
Throughout of this paper, we always make use of the usual space \(E=C[0,1]\) which is known as the space of continuous functions on \([ 0,1 ]\). We note that E is a real Banach space with the norm given by
Suppose that \(x\in C[0,1]\) and \(y\in C[0,1]\). Then we have
We know from the recent work [34] that \((C[0,1],\leqq)\) with the familiar metric:
satisfies the hypothesis of Theorem 1(a). Moreover, for \(x\in C[0,1]\) and \(y\in C[0,1]\) such that \(\max\{x,y\}\in C[0,1]\), \((C[0,1],\leqq )\) satisfies the condition of Theorem 1(b).
We first demonstrate Lemma 3.
Lemma 3
If \(h\in C[0,1]\), the following boundary value problem:
has a unique solution given by
where
Proof
By applying Lemma 1, Lemma 2 (with \(p=4\)) and Eq. (2.1), we have
and
From Eq. (2.2), we get \(c_{4}=0\). Thus, upon differentiating both sides of Eq. (2.5), if we make use of Eqs. (1.2) and (1.3), we see that
Using the boundary condition (2.2), we have \(c_{3}=0\). Moreover, by differentiating both sides of Eq. (2.6), and using Eqs. (1.2) and (1.3), we obtain
Similarly, by using the boundary condition (2.2), we have \(c_{2}=0\) and
Consequently, we have the following unique solution of the boundary value problem given by Eqs. (2.1) and (2.2):
We thus arrive at the desired result asserted by Lemma 3. □
By using the method in [10] mutatis mutandis, it can easily be proven that, if \(f\in C([0,1]\times[0,+\infty ),[0,+\infty))\), then the boundary value problem given by Eqs. (1.5) and (1.6) is equivalent to the following integral equation:
where \(G(t,s)\) is defined, as before, by Eq. (2.4).
Lemma 4
The function \(G(t,s)\) given by Eq. (2.4) has the following properties:

(1)
\(G(t,s)\) is a continuous function and \(G(t,qs)\geqq0\);

(2)
\(G(t,s)\) is strictly increasing in the first variable t.
Proof
The continuity of \(G(t,s)\) can easily be checked. We, therefore, omit the details involved. Next, for \(0\leqq s\leqq t\leqq1\), we let
and, for \(0\leqq t\leqq s\leqq1\), we suppose that
Then it is not difficult to see that
Now, for \(g_{1}(0,qs)=0\), \(\delta>0\), and \(a\leqq b\leqq t\), we have
We thus find that
So, clearly, \(G(t,qs)\geqq0\) for all \((t,s)\in[0,1]\times [ 0,1]\). This completes the proof of Lemma 4(1).
Next, for a fixed \(s\in[0,1]\), we see that
This implies that \(g_{1}(t,qs)\) is an increasing function of the first argument t. Furthermore, obviously, \(g_{2}(t,qs)\) is an increasing function of the first argument t. Therefore, \(G(t,qs)\) is an increasing function of t for a fixed \(s\in[0,1]\). This completes the proof of Lemma 4. □
3 Uniqueness of positive solutions
For notational convenience, we write
The main result of this paper is the assertion in Theorem 2.
Theorem 2
The boundary value problem given by Eqs. (1.5) and (1.6) has a unique positive and increasing solution \(u(t)\) if each of the following two conditions is satisfied:

(i)
the function \(f:[0,1]\times[0,\infty )\rightarrow[0,\infty)\) is continuous and nondecreasing with respect to the second variable;

(ii)
there exist λ and M given by Eq. (3.1) (\(0<\lambda+1<M\)) such that, for \(u\in[0,\infty)\) and \(v\in [0,\infty)\) with \(u\geqq v\) and \(t\in[0,1]\),
$$ \phi_{p}\bigl(\ln(v+2)\bigr)\leqq f(t,v)\leqq f(t,u)\leqq \phi_{p}\bigl(\ln (u+2) (uv+1)^{\lambda}\bigr). $$
Furthermore, if \(f(t,0)>0\) for \(t\in[0,1]\), then the solution \(u(t) \) of the boundary value problem given by Eqs. (1.5) and (1.6) is strictly increasing on \([0,\infty)\).
Proof
First of all, we set
We then consider the set K (called a cone) given by
Since K is a closed set, K is a complete metric space in accordance with the usual metric
Let us now consider the operator T as follows:
Then, by applying Lemma 4 and the condition (i) of Theorem 2, we see that \(T(K)\subset K\).
We now show that all conditions of Theorem 1 are satisfied. Firstly, by the condition (i) of Theorem 2, for \(u,v\in K\) and \(u\geqq v\), we have
This shows that T is a nondecreasing operator. On the other hand, for \(u\geqq v\) and by the condition (ii) of Theorem 2, we have
Since the function \(h(x)=\ln(x+1)\) is nondecreasing, from the condition (ii) of Theorem 2, we have
We now let \(\psi(x)=x\ln(x+1)\). Then, obviously, the function ψ given by
is continuous, nondecreasing, and positive in \((0,\infty)\). It is also clearly seen that \(\psi(x)\) satisfies the following conditions:
Thus, for \(u\geqq v\), we have
As \(G(t,qs)\geqq0\) and \(f\geqq0\), we have
Consequently, in view of Theorem 1, the boundary value problem given by Eqs. (1.5) and (1.6) has at least one nonnegative solution. Since \((K,\leqq)\) satisfies the condition (ii) of Theorem 2, Theorem 1 implies the uniqueness of the solution. Thus, clearly, the proof of the last assertion of Theorem 2 follows immediately from the proof of a wellknown result in [10, Theorem 4.2]. Our proof Theorem 2 is thus completed. □
4 Concluding remarks and observations
Our present study was motivated by several aforementioned recent works. Here, we have successfully addressed the problem involving the existence and uniqueness of positive and nondecreasing solutions of a family of fractional qdifference boundary value problems given by Eqs. (1.5) and (1.6). The proof of our main result asserted by Theorem 2 of the preceding section has made use of some familiar fixed point theorems. We have also indicated the relevant connections of the results derived in this investigation with those in earlier works on the subject.
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Araci, S., Şen, E., Açikgöz, M. et al. Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the pLaplacian operator. Adv Differ Equ 2015, 40 (2015). https://doi.org/10.1186/s1366201503750
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DOI: https://doi.org/10.1186/s1366201503750