Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the p-Laplacian operator
- Serkan Araci^{1}Email author,
- Erdoğan Şen^{2, 3},
- Mehmet Açikgöz^{4} and
- Hari M Srivastava^{5}
https://doi.org/10.1186/s13662-015-0375-0
© Araci et al.; licensee Springer. 2015
Received: 19 November 2014
Accepted: 13 January 2015
Published: 11 February 2015
Abstract
In this article, we investigate the existence of a solution arising from the following fractional q-difference boundary value problem by using the p-Laplacian 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 q-difference boundary value problem in order to show the existence and uniqueness of positive and nondecreasing solutions by means of a familiar fixed point theorem.
Keywords
positive solutions fixed point theorem fractional q-difference equation p-Laplacian operatorMSC
05A30 26A33 34K10 39A13 34A08 34B181 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 q-difference calculus (or the so-called quantum calculus), which is an old subject. New developments in this theory were made. These include (for example) the q-analogs of the fractional integral and the fractional derivative operators, the q-analogs 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 q-calculus).
We now review briefly some concepts of the quantum calculus.
Definition 1
(see [21])
Definition 2
Lemma 1
(see [21])
- (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
Theorem 1
(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.
2 Fractional boundary value problem
We first demonstrate Lemma 3.
Lemma 3
Proof
Lemma 4
- (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
3 Uniqueness of positive solutions
Theorem 2
- (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 \(t\in[0,1]\),$$ \phi_{p}\bigl(\ln(v+2)\bigr)\leqq f(t,v)\leqq f(t,u)\leqq \phi_{p}\bigl(\ln (u+2) (u-v+1)^{\lambda}\bigr). $$
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
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 q-difference 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.
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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