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Solvability of anti-periodic boundary value problem for coupled system of fractional p-Laplacian equation
Advances in Difference Equations volume 2015, Article number: 305 (2015)
Abstract
This paper studies the existence of solutions for anti-periodic boundary value problem for a coupled system of the fractional p-Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, a new existence result is obtained by using the Schaefer fixed point theorem. As an application, an example to illustrate our result is given.
1 Introduction
The subject of fractional calculus has gained considerable popularity and importance due to its frequent appearance in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, electromagnetic, etc. (see [1–4]). Recently, fractional differential equations have been of great interest due to the intensive development of theory of itself and its applications (see [5–10]). Moreover, the existence of solutions to some coupled systems of fractional differential equations have been studied by many authors (see [11–16]). For instance, Ahmad and Nieto (see [11]) considered a three-point boundary value problem for a coupled system of nonlinear fractional differential equations given by
where \(1<\alpha,\beta<2\), \(p,q,\gamma>0\), \(0<\eta<1\), \(\alpha-q,\beta-p\geq1\), \(\gamma\eta^{\alpha-1},\gamma\eta^{\beta-1}<1\), and \(D^{\alpha}\) is the standard Riemann-Liouville fractional derivative. Under certain growth conditions on f and g, an existence result was obtained by using the Schauder fixed point theorem. In addition, Bai and Fang (see [12]) discussed the existence of a positive solution to the singular coupled system of the form
where \(0< s,p<1\), \(D^{s}\) is the standard Riemann-Liouville fractional derivative, \(f,g:(0,1]\times[0,+\infty)\rightarrow[0,+\infty)\) are two given continuous functions, and \(\lim_{t\rightarrow0^{+}}f(t,\cdot)=\lim_{t\rightarrow0^{+}}g(t,\cdot )=+\infty\). A nonlinear alternative of Leray-Schauder type and the Krasnoselskii fixed point theorem in a cone were applied to establish the existence results on a positive solution.
The anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes (see [17, 18]) and recently received considerable attention. For an example and details of the anti-periodic boundary value problems, see [19, 20] and the references therein.
The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [21]) introduced the p-Laplacian equation as follows:
where \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\). Obviously, \(\phi_{p}\) is invertible and its inverse operator is \(\phi_{q}\), where \(q>1\) is a constant such that \(1/p+1/q=1\). In the past few decades, many important results as regards (1.1) with certain boundary value conditions have been obtained. We refer the readers to [22–25] and the references cited therein. However, as far as we know, there are relatively few results on the anti-periodic boundary value problems (ABVPs for short) for coupled systems of the fractional p-Laplacian equations.
Motivated by the works mentioned previously, in this paper, we investigate the existence of solutions for ABVP for a coupled system of the fractional p-Laplacian equation of the form
where \(0<\alpha,\beta,\gamma,\delta\leq1\), \(D_{0^{+}}^{\alpha}\) is a Caputo fractional derivative of order α, and \(f,g:[0,1]\times\mathbb {R}^{2}\rightarrow\mathbb{R}\) are continuous. Note that the nonlinear operator \(D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha})\) reduces to the linear operator \(D_{0^{+}}^{\beta}D_{0^{+}}^{\alpha}\) when \(p=2\) and the additive index law
holds under some reasonable constraints on the function u (see [26]).
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, based on the Schaefer fixed point theorem, we establish one theorem on the existence of solutions for ABVP (1.2) (Theorem 3.1). Finally, in Section 4, an explicit example is given to illustrate the main result.
2 Preliminaries
For convenience of the readers, we present here some necessary basic knowledge and definitions as regards the fractional calculus theory, which can be found, for instance, in [27, 28].
Definition 2.1
The Riemann-Liouville fractional integral operator of order \(\alpha>0\) of a function \(u:(0,+\infty )\rightarrow\mathbb{R}\) is given by
provided that the right side integral is pointwise defined on \((0,+\infty)\).
Definition 2.2
The Caputo fractional derivative of order \(\alpha>0\) of a continuous function \(u:(0,+\infty)\rightarrow \mathbb{R}\) is given by
where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on \((0,+\infty)\).
Lemma 2.1
(see [28])
Let \(\alpha>0\). Assume that \(u,D_{0^{+}}^{\alpha}u\in L([0,1],\mathbb{R})\). Then the following equality holds:
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), and n is the smallest integer greater than or equal to α.
Next, we will give the Schaefer fixed point theorem (see for example [25]), which will be used in this paper.
Lemma 2.2
Let X be a Banach space and \(T:X\rightarrow X\) is a completely continuous operator. If the set \(\Omega=\{u\in X|u=\lambda Tu,\lambda\in(0,1)\}\) is bounded, then T has at least one fixed point in X.
In this paper, we take \(Z=C([0,1],\mathbb{R})\) with the norm \(\|z\| _{0}=\max_{t\in[0,1]}|z(t)|\), \(X=\{u|u,D_{0^{+}}^{\alpha}u\in Z\}\) with the norm \(\|u\|_{X}=\max\{\|u\|_{0},\|D_{0^{+}}^{\alpha}u\|_{0}\}\), and \(Y=\{ v|v,D_{0^{+}}^{\gamma}v\in Z\}\) with the norm \(\|v\|_{Y}=\max\{\|v\|_{0},\| D_{0^{+}}^{\gamma}v\|_{0}\}\). For \((u,v)\in X\times Y\), let \(\|(u,v)\|_{X\times Y}=\max\{\|u\|_{X}, \|v\| _{Y}\}\). Obviously, \((X\times Y,\|\cdot\|_{X\times Y})\) is a Banach space.
3 Existence result
In this section, a theorem on the existence of solutions for ABVP (1.2) will be given under the nonlinear growth restrictions of f and g.
As a consequence of Lemma 2.1, we have the following result, which is useful in what follows.
Lemma 3.1
Given \((h_{1},h_{2})\in Z \times Z\), the unique solution of
is
where
and \(\phi_{q}\) is understood as the operator \(\phi_{q}:Z\rightarrow Z\) defined by \(\phi_{q}(z)(t)=\phi_{q}(z(t))\).
Proof
Assume that \((u,v)\) satisfies the equations of ABVP (3.1), then Lemma 2.1 implies that
From the boundary value condition \(D_{0^{+}}^{\alpha}u(0)=-D_{0^{+}}^{\alpha}u(1)\), one has
Thus we have
By the condition \(u(0)=-u(1)\), we get
A similar proof can show that
where
The proof is complete. □
Define the operator \(\mathcal{T}:X\times Y\rightarrow X\times Y\) by
where
and \(N_{1}:Y\rightarrow Z\), \(N_{2}:X\rightarrow Z\) are Nemytskii operators defined by
Clearly, the fixed points of \(\mathcal{T}\) are the solutions of ABVP (1.2).
Our main result, based on the Schaefer fixed point theorem and Lemma 3.1, is stated as follows.
Theorem 3.1
Let \(f,g:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous. Assume that
-
(H)
for \(\forall(u,v)\in\mathbb{R}^{2}\), \(t\in[0,1]\), there exist nonnegative functions \(a_{1},b_{1},c_{1},a_{2}, b_{2},c_{2}\in Z\) such that
$$\begin{aligned}& \bigl\vert f(t,u,v)\bigr\vert \leq a_{1}(t)+b_{1}(t)|u|^{p-1}+c_{1}(t)|v|^{p-1}, \\& \bigl\vert g(t,u,v)\bigr\vert \leq a_{2}(t)+b_{2}(t)|u|^{p-1}+c_{2}(t)|v|^{p-1}. \end{aligned}$$
Then ABVP (1.2) has at least one solution, provided that
where
Proof
The proof will be given in the following two steps.
Step 1: \(\mathcal{T}:X\times Y\rightarrow X\times Y\) is completely continuous.
By the definitions of \(T_{1}\) and \(T_{2}\), we obtain
Obviously, the operators \(T_{1}\), \(D_{0^{+}}^{\alpha}T_{1}\), \(T_{2}\), \(D_{0^{+}}^{\gamma}T_{2}\) are compositions of the continuous operators. So \(T_{1}\), \(D_{0^{+}}^{\alpha}T_{1}\), \(T_{2}\), \(D_{0^{+}}^{\gamma}T_{2}\) are continuous in Z. Hence, \(\mathcal{T}\) is a continuous operator in \(X \times Y\).
Let \(\Omega:=\Omega_{1}\times\Omega_{2}\subset X\times Y\) be an open bounded set, then \(T_{1}(\overline{\Omega_{2}})\), \(T_{2}(\overline{\Omega _{1}})\), and \(D_{0^{+}}^{\alpha}T_{1}(\overline{\Omega_{2}})\), \(D_{0^{+}}^{\gamma}T_{2}(\overline{\Omega_{1}})\) are bounded. Moreover, for \(\forall(u,v)\in \overline{\Omega}\), \(t\in[0,1]\), there exist constants \(L_{1},L_{2},L_{3}>0\) such that
Thus, in view of the Arzelà -Ascoli theorem, we need only to prove that \(\mathcal{T}(\overline{\Omega})\subset X\times Y\) is equicontinuous.
For \(0\leq t_{1}< t_{2}\leq1\), \((u,v)\in\overline{\Omega}\), we have
Similarly, one has
Since \(t^{\alpha}\) is uniformly continuous in \([0,1]\), we see that \((T_{1}(\overline{\Omega_{2}}),T_{2}(\overline{\Omega_{1}}))\subset Z\times Z\) is equicontinuous. A similar proof can show that \((I_{0^{+}}^{\beta}N_{1}(\overline{\Omega_{2}}),I_{0^{+}}^{\delta}N_{2}(\overline{\Omega _{1}}))\subset Z\times Z\) is equicontinuous. This, together with the uniformly continuity of \(\phi_{q}(s)\) on \([-L_{3},L_{3}]\), shows that \((D_{0^{+}}^{\alpha}T_{1}(\overline{\Omega_{2}}), D_{0^{+}}^{\gamma }T_{2}(\overline{\Omega_{1}}))\subset Z\times Z\) is also equicontinuous. Thus, we find that \(\mathcal{T}:X\times Y\rightarrow X\times Y\) is compact.
Step 2: A priori bounds.
Set
Now it remains to show that the set Ω is bounded.
Since \(0<\alpha\leq1\), by Lemma 2.1, we have
So we get
Hence, from the anti-periodic boundary value condition \(u(0)=-u(1)\), one has
Thus we obtain
which together with
yields
Similarly, we can get
For \((u,v)\in\Omega\), we have
Thus we get
which together with \(\phi_{q}(\lambda)=\lambda^{q-1}\) (\(\lambda\in(0,1)\)) yields
From the hypothesis (H), for \(\forall t\in[0,1]\), we get
which together with \(|\phi_{p}(D_{0^{+}}^{\alpha}u(t))|=|D_{0^{+}}^{\alpha}u(t)|^{p-1}\) yields
Repeating arguments similar to the above we can arrive at
So we have
Hence, in view of (3.2), we can get
where
Thus, from (3.3) and (3.4), one has
Therefore, combining (3.7) and (3.9) with (3.8) and (3.10), we have
As a consequence of the Schaefer fixed point theorem, we deduce that \(\mathcal{T}\) has at least one fixed point which is the solution of ABVP (1.2). The proof is complete. □
4 An example
In this section, we will give an example to illustrate our main result.
Example 4.1
Consider the following ABVP for the coupled system of the fractional p-Laplacian equation:
Corresponding to ABVP (1.2), we get \(p=3\), \(\alpha=1/2\), \(\beta =3/4\), \(\gamma=3/4\), \(\delta=1/2\), and
Choose \(a_{1}(t)=10\), \(b_{1}(t)=1/10\), \(c_{1}(t)=0\), \(a_{2}(t)=2\), \(b_{2}(t)=1/4\), \(c_{2}(t)=0\). By a simple calculation, we obtain \(\|b_{1}\|_{0}=1/10\), \(\|c_{1}\|_{0}=0\), \(\|b_{2}\| _{0}=1/4\), \(\|c_{2}\|_{0}=0\), and
Obviously, ABVP (4.1) satisfies all assumptions of Theorem 3.1. Hence, ABVP (4.1) has at least one solution.
References
Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keilet, F, Mackens, W, Voß, F, Werther, J (eds.) Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Berlin (1999)
Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81-88 (1991)
Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995)
Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974)
Bai, Z, Hu, L: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005)
Bai, Z, Zhang, Y: Solvability of fractional three point boundary value problems with nonlinear growth. Appl. Math. Comput. 218, 1719-1725 (2011)
Jafari, H, Gejji, VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180, 700-706 (2006)
Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135 (2010)
Liang, S, Zhang, J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71, 5545-5550 (2009)
Yang, W: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 63, 288-297 (2012)
Ahmad, B, Nieto, J: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838-1843 (2009)
Bai, C, Fang, J: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611-621 (2004)
Jiang, W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 74, 1987-1994 (2011)
Liu, Y, Chen, H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011, Article ID 404917 (2011)
Su, H, Wang, B, Wei, Z, Zhang, X: Positive solutions of four point boundary value problems for higher order p-Laplacian operator. J. Math. Anal. Appl. 330, 836-851 (2007)
Su, X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64-69 (2009)
Ahn, C, Rim, C: Boundary flows in general coset theories. J. Phys. A 32, 2509-2525 (1999)
Kleinert, H, Chervyakov, A: Functional determinants from Wronski Green function. J. Math. Phys. 40, 6044-6051 (1999)
Liu, B: Anti-periodic solutions for forced Rayleigh-type equations. Nonlinear Anal., Real World Appl. 10, 2850-2856 (2009)
Liu, W, Zhang, J, Chen, T: Anti-symmetric periodic solutions for the third order differential systems. Appl. Math. Lett. 22, 668-673 (2009)
Leibenson, LS: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSR 9, 7-10 (1983) (in Russian)
Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012)
Chen, T, Liu, W, Yang, C: Antiperiodic solutions for Liénard-type differential equation with p-Laplacian operator. Bound. Value Probl. 2010, Article ID 194824 (2010)
Liu, B, Yu, J: On the existence of solutions for the periodic boundary problems with p-Laplacian operator. J. Syst. Sci. Math. Sci. 23, 76-85 (2003)
Smart, DR: Fixed Point Theorems. Cambridge University Press, London (1974)
Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
Acknowledgements
The author would like to thank the referee and the associate editor for their very helpful suggestions. This work was supported by the National Natural Science Foundation of China (11271364).
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Jiang, J. Solvability of anti-periodic boundary value problem for coupled system of fractional p-Laplacian equation. Adv Differ Equ 2015, 305 (2015). https://doi.org/10.1186/s13662-015-0629-x
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DOI: https://doi.org/10.1186/s13662-015-0629-x