- Open Access
Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance
© Hu et al.; licensee Springer. 2013
- Received: 24 May 2013
- Accepted: 2 October 2013
- Published: 8 November 2013
In this paper, by using the extension of Mawhin’s continuation theorem due to Ge, we study the existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained.
- fractional p-Laplacian equation
- coupled system
- boundary value problem
- degree theory
In the recent years, fractional differential equations have played an important role in many fields such as physics, electrical circuits, biology, control theory, etc. (see [1–9]). Recently, many scholars have paid more attention to boundary value problems for fractional differential equations (see [10–25]).
where , are real numbers, and is the Riemann-Liouville fractional derivative.
where is a real number, , are given constants such that , and , are the Riemann-Liouville differentiation and integration.
where , , , , , , , D is the standard Riemann-Liouville differentiation and are given continuous functions.
where , , , , , .
where , . Obviously, is invertible and its inverse operator is , where is a constant such that .
In the past few decades, many important results relative to a p-Laplacian equation with certain boundary value conditions have been obtained. We refer the reader to [35–38] and the references cited therein. We noticed that is a quasi-linear operator. So, Mawhin’s continuation theorem is not suitable for a p-Laplacian operator. In , Ge and Ren extended Mawhin’s continuation theorem, which is used to deal with more general abstract operator equations.
where , , , are the standard Caputo fractional derivatives, , , and is continuous.
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of solutions for BVP (1.1) under nonlinear growth restriction of f and g, based on the extension of Mawhin’s continuation theorem due to Ge (see ). Finally, in Section 4, an example is given to illustrate the main result.
In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper.
is a closed subset of Y,
is linearly homeomorphic to , .
Definition 2.2 Let X be a real Banach space and . The operator is said to be a projector provided , for , . The operator is said to be a semi-projector provided .
Definition 2.3 ()
Let and be the complement space of in X, then . On the other hand, suppose that is a subspace of Y and is the complement space of in Y so that . Let be a projector and be a semi-projector, and let be an open and bounded set with origin , where θ is the origin of a linear space.
Lemma 2.1 (, Ge-Mawhin’s continuation theorem)
is M-compact in . In addition, if
(C1) , ,
(C2) , for ,
where and is a homeomorphism with , then the equation has at least one solution in .
provided that the right-hand side integral is pointwise defined on .
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on .
Lemma 2.2 
where , , here n is the smallest integer greater than or equal to α.
Lemma 2.3 
In this paper, we denote with the norm , with the norm and with the norm , where . Then we denote with the norm and with the norm . Obviously, both and are Banach spaces.
In this section, a theorem on the existence of solutions for BVP (1.1) will be given.
Theorem 3.1 Let be continuous. Assume that
where , , ();
Then BVP (1.1) has at least one solution.
In order to prove Theorem 3.1, we need to prove some lemmas below.
and M is a quasi-linear operator.
and is a quasi-linear operator. Then the proof is complete. □
Lemma 3.2 Let be an open and bounded set, then is M-compact in .
where and .
Similar proof can show that . Thus, we have .
Let be an open and bounded set with . For each , we can get . Thus, . Take any in the type . Since , we can get . So (2.1) holds. It is easy to verify (2.2).
Since is uniformly continuous on , so is equicontinuous. Similarly, we can get is equicontinuous. Considering that is uniformly continuous on , we have is also equicontinuous. So, we can obtain that is compact.
Similarly, we can get that is compact. So, we can obtain that is compact.
It is easy to verify that is the zero operator. Similarly, we can get and is the zero operator. So (2.3) holds.
Similarly, we have . So, (2.4) holds. Then we have that is M-compact in . The proof is complete. □
Then, by the integral mean value theorem, there exist constants such that and . So, from (H2), we get and .
So, is bounded. The proof is complete. □
Hence, is bounded. The proof is complete. □
which contradicts (3.11) or (3.12). Therefore, is bounded. The proof is complete. □
Proof of Theorem 3.1 Set . It follows from Lemmas 3.1 and 3.2 that M is a quasi-linear operator and is M-compact on . By Lemmas 3.3 and 3.4, we get that the following two conditions are satisfied:
(C1) , ,
(C2) , for .
So, condition (C3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that has at least one solution in . Therefore BVP (1.1) has at least one solution. The proof is complete. □
Then (H1) and the first part of (H2) hold.
By Theorem 3.1, we obtain that BVP (4.1) has at least one solution.
The authors are grateful to those who gave useful suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013QNA33).
- Metzler R, Klafter J: Boundary value problems for fractional diffusion equations. Physica A 2000, 278: 107-125. 10.1016/S0378-4371(99)00503-8MathSciNetView ArticleMATHGoogle Scholar
- Scher H, Montroll E: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 1975, 12: 2455-2477.View ArticleGoogle Scholar
- Mainardi F: Fractional diffusive waves in viscoelastic solids. In Nonlinear Waves in Solids. Edited by: Wegner JL, Norwood FR. ASME/AMR, Fairfield; 1995:93-97.Google Scholar
- Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voss H, Werther J. Springer, Heidelberg; 1999:217-224.Google Scholar
- Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5: 81-88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar
- Glockle WG, Nonnenmacher TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
- Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainardi F. Springer, Wien; 1997:291-348.View ArticleGoogle Scholar
- Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103: 7180-7186. 10.1063/1.470346View ArticleGoogle Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATHGoogle Scholar
- Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Hu L: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
- Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 3: 1-11.MathSciNetView ArticleMATHGoogle Scholar
- Jafari H, Gejji VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MathSciNetView ArticleMATHGoogle Scholar
- Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 2009, 71: 2391-2396. 10.1016/j.na.2009.01.073MathSciNetView ArticleMATHGoogle Scholar
- Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 2009, 71: 5545-5550. 10.1016/j.na.2009.04.045MathSciNetView ArticleMATHGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36: 1-12.View ArticleMathSciNetGoogle Scholar
- Kosmatov N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135: 1-10.MathSciNetMATHGoogle Scholar
- Wei Z, Dong W, Che J: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 2010, 73: 3232-3238. 10.1016/j.na.2010.07.003MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Zhang Y: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 2011, 218(5):1719-1725. 10.1016/j.amc.2011.06.051MathSciNetView ArticleMATHGoogle Scholar
- Bai Z: Solvability for a class of fractional m -point boundary value problem at resonance. Comput. Math. Appl. 2011, 62(3):1292-1302. 10.1016/j.camwa.2011.03.003MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 2010, 217: 480-487. 10.1016/j.amc.2010.05.080MathSciNetView ArticleMATHGoogle Scholar
- Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 2011, 74: 792-804. 10.1016/j.na.2010.09.030MathSciNetView ArticleMATHGoogle Scholar
- Yang L, Chen H: Unique positive solutions for fractional differential equation boundary value problems. Appl. Math. Lett. 2010, 23: 1095-1098. 10.1016/j.aml.2010.04.042MathSciNetView ArticleMATHGoogle Scholar
- Hu Z, Liu W: Solvability for fractional order boundary value problems at resonance. Bound. Value Probl. 2011, 20: 1-10.MathSciNetMATHGoogle Scholar
- Jiang W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 2011, 74: 1987-1994. 10.1016/j.na.2010.11.005MathSciNetView ArticleMATHGoogle Scholar
- Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 2009, 22: 64-69. 10.1016/j.aml.2008.03.001MathSciNetView ArticleMATHGoogle Scholar
- Bai C, Fang J: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 2004, 150: 611-621. 10.1016/S0096-3003(03)00294-7MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Alsaedi A: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl. 2010, 2010: 1-17.MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto J: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2009, 58: 1838-1843. 10.1016/j.camwa.2009.07.091MathSciNetView ArticleMATHGoogle Scholar
- Rehman M, Khan R: A note on boundary value problems for a coupled system of fractional differential equations. Comput. Math. Appl. 2011, 61: 2630-2637. 10.1016/j.camwa.2011.03.009MathSciNetView ArticleMATHGoogle Scholar
- Su X: Existence of solution of boundary value problem for coupled system of fractional differential equations. Eng. Math. 2009, 26: 134-137.Google Scholar
- Yang W: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 2012, 63: 288-297. 10.1016/j.camwa.2011.11.021MathSciNetView ArticleMATHGoogle Scholar
- Jiang W: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal. 2012, 13: 2285-2292. 10.1016/j.nonrwa.2012.01.023View ArticleMathSciNetMATHGoogle Scholar
- Leibenson LS: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 1945, 9: 7-10.MathSciNetGoogle Scholar
- Pang H, Ge W, Tian M: Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p -Laplacian. Comput. Math. Appl. 2008, 56: 127-142. 10.1016/j.camwa.2007.11.039MathSciNetView ArticleMATHGoogle Scholar
- Liu B, Yu J: On the existence of solutions for the periodic boundary value problems with p -Laplacian operator. J. Syst. Sci. Math. Sci. 2003, 23: 76-85.MathSciNetMATHGoogle Scholar
- Lian L, Ge W: The existence of solutions of m -point p -Laplacian boundary value problems at resonance. Acta Math. Appl. Sin. 2005, 28: 288-295.MathSciNetGoogle Scholar
- Chen T, Liu W, Hu Z: A boundary value problem for fractional differential equation with p -Laplacian operator at resonance. Nonlinear Anal. 2012, 75: 3210-3217. 10.1016/j.na.2011.12.020MathSciNetView ArticleMATHGoogle Scholar
- Ge W, Ren J: An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p -Laplacian. Nonlinear Anal. 2004, 58: 477-488. 10.1016/j.na.2004.01.007MathSciNetView ArticleMATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
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