- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 594802, 14 pages
Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations with Nonlocal and Integral Boundary Conditions
1Department of Mathematics, Fatih University, 34500 Buyucekmece, Turkey
2ITTU, Ashgabat, Turkmenistan
3Institute of Cybernetics, ANAS, and Baku State University, 1141 Baku, Azerbaijan
Received 20 March 2012; Accepted 6 May 2012
Academic Editor: Ravshan Ashurov
Copyright © 2012 Allaberen Ashyralyev and Yagub A. Sharifov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. I. Bagley, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983.
- G. Sorrentinos, “Fractional derivative linear models for describing the viscoelastic dynamic behavior of polymeric beams,” in Proceedings of the IMAS Conference and Exposition on Structural Dynamics, St. Louis, Mo, USA, 2006.
- G. Sorrentinos, “Analytic modeling and experimental identification of viscoelastic mechanical systems,” in Advances in Fractional Calculus, Springer, 2007.
- F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
- R. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004.
- M. Ortigueira, “Special issue on fractional signal processing and applications,” Signal Processing, vol. 83, no. 11, pp. 2285–2480, 2003.
- B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 231–248, 2000.
- K. B. Oldham, “Fractional differential equations in electrochemistry,” Advances in Engineering Software, vol. 41, no. 1, pp. 9–12, 2010.
- R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000.
- M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011.
- E. M. Rabei, T. S. Alhalholy, and A. Rousan, “Potentials of arbitrary forces with fractional derivatives,” International Journal of Modern Physics A, vol. 19, no. 17-18, pp. 3083–3092, 2004.
- O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.
- S. Tu, K. Nishimoto, S. Jaw, and S. D. Lin, “Applications of fractional calculus to ordinary and partial differential equations of the second order,” Hiroshima Mathematical Journal, vol. 23, no. 1, pp. 63–77, 1993.
- K. Dichelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg, Germany, 2004.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, London, UK, 1993.
- J. L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976.
- R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
- M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order,” Surveys in Mathematics and its Applications, vol. 3, pp. 1–12, 2008.
- R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1–10, 2007.
- V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677–2682, 2008.
- S. Xinwei and L. Landong, “Existence of solution for boundary value problem of nonlinear fractional differential equation,” Applied Mathematics A, vol. 22, no. 3, pp. 291–298, 2007.
- S. Zhang, “Existence of solution for a boundary value problem of fractional order,” Acta Mathematica Scientia B, vol. 26, no. 2, pp. 220–228, 2006.
- R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401–411, 2009.
- A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011.
- A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011.
- A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations,” AIP Conference Proceeding, vol. 1389, pp. 617–620, 2011.
- A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011.
- C. Yuan, “Two positive solutions for -type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 930–942, 2012.
- M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,” Advances in Difference Equations, vol. 2011, Article ID 748608, 32 pages, 2011.
- M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2011, Article ID 867932, 19 pages, 2011.
- C. Yuan, “Multiple positive solutions for semipositone -type boundary value problems of nonlinear fractional differential equations,” Analysis and Applications, vol. 9, no. 1, pp. 97–112, 2011.
- C. Yuan, “Multiple positive solutions for -type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 12, pp. 1–12, 2011.
- R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009.
- R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750, 25 pages, 2010.
- R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532–3554, 2010.
- A. S. Berdyshev, A. Cabada, and E. T. Karimov, “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,” Nonlinear Analysis, vol. 75, no. 6, pp. 3268–3273, 2011.
- B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 708576, 11 pages, 2009.
- A. Bouncherif, “Second order boundary value problems with integral boundary conditions,” Nonlinear Analysis, vol. 70, no. 1, pp. 368–379, 2009.
- R. A. Khan, “Existence and approximation of solutions of nonlinear problems with integral boundary conditions,” Dynamic Systems and Applications, vol. 14, no. 2, pp. 281–296, 2005.
- R. A. Khan, M. U. Rehman, and J. Henderson, “Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions,” Fractional Differential Equations, vol. 1, no. 1, pp. 29–43, 2011.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.
- A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.