- Research
- Open Access
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 operator
MSC
- 05A30
- 26A33
- 34K10
- 39A13
- 34A08
- 34B18
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 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
References
- Jackson, FH: On q-definite integrals. Pure Appl. Math. Q. 41, 193-203 (1910) MATHGoogle Scholar
- Agarwal, RP: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365-370 (1969) View ArticleMATHGoogle Scholar
- Atici, FM, Eloe, PW: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14, 333-344 (2007) View ArticleMathSciNetGoogle Scholar
- Ernst, T: The history of q-Calculus and a new method. U.U.D.M. Report 2000:16, Department of Mathematics, Uppsala University (2000) Google Scholar
- Han, Z-H, Lu, H-L, Sun, S-R, Yang, D-W: Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary. Electron. J. Differ. Equ. 2012, 213 (2012) View ArticleMathSciNetGoogle Scholar
- Hilfer, R (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) MATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
- Lakshmikantham, V: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337-3343 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677-2682 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Miao, F-G, Liang, S-H: Uniqueness of positive solutions for fractional q-difference boundary value problems with p-Laplacian operator. Electron. J. Differ. Equ. 2013, 174 (2013) View ArticleMathSciNetGoogle Scholar
- Petráš, I: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer Series on Nonlinear Physical Science. Springer, Berlin (2011) View ArticleGoogle Scholar
- Podlubny, I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999) MATHGoogle Scholar
- Rajković, PM, Marinković, SD, Stanković, MS: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1, 311-323 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993) MATHGoogle Scholar
- Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390-444 (2011) MathSciNetGoogle Scholar
- Al-Salam, WA: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15, 135-140 (1966/1967) Google Scholar
- Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340-1350 (2008) View ArticleMATHMathSciNetGoogle Scholar
- El-Sayed, AMA, El-Mesiry, AEM, El-Saka, HAA: On the fractional-order logistic equation. Appl. Math. Lett. 20, 817-823 (2007) View ArticleMATHMathSciNetGoogle Scholar
- El-Shahed, M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007, Article ID 10368 (2007) View ArticleMathSciNetGoogle Scholar
- Ferreira, RAC: Nontrivial solutions for fractional q-difference boundary-value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010) Google Scholar
- Ferreira, RAC: Positive solutions for a class of boundary value problems with fractional q-differences. Comput. Math. Appl. 61, 367-373 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Agarwal, RP, O’Regan, D, Wong, PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (1999) View ArticleMATHGoogle Scholar
- Chai, G: Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator. Bound. Value Probl. 2012, Article ID 18 (2012) View ArticleGoogle Scholar
- Lakshmikantham, V, Vatsala, AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828-834 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Liang, S-H, Zhang, J-H: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71, 5545-5550 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363-1375 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Mena, JC, Harjani, J, Sadarangani, K: Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems. Bound. Value Probl. 2009, Article ID 421310 (2009) Google Scholar
- Şen, E, Açikgöz, M, Seo, JJ, Araci, S, Oruçoğlu, K: Existence and uniqueness of positive solutions of boundary value problems for fractional differential equations with p-Laplacian operator and identities on the some special polynomials. J. Funct. Spaces Appl. 2013, Article ID 753171 (2013) Google Scholar
- Yang, W-G: Positive solution for fractional q-difference boundary value problems with ϕ-Laplacian operator. Bull. Malays. Math. Soc. 36, 1195-1203 (2013) MATHGoogle Scholar
- Zhang, S-Q: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36 (2006) Google Scholar
- Zhang, S-Q: Existence of solution for a boundary value problem of fractional order. Acta Math. Sci. 26, 220-228 (2006) View ArticleMATHGoogle Scholar
- Zhou, Y: Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Syst. Differ. Equ. 1, 239-244 (2008) MATHMathSciNetGoogle Scholar
- Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403-3410 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223-239 (2005) View ArticleMATHMathSciNetGoogle Scholar
- O’Regan, D, Petrusel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241-1252 (2008) View ArticleMATHMathSciNetGoogle Scholar