On nonlinear implicit fractional differential equations without compactness

Volume 10, Issue 10, pp 5528--5539 http://dx.doi.org/10.22436/jnsa.010.10.33

Publication Date: October 31, 2017 Submission Date: May 24, 2017

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Authors

Samia Bushnaq - Department of Science, Princess Sumaya University for Technology, Amman 11941, Jordan. Wajid Hussain - Department of Mathematics, University of Qurtuba, Peshawar, Khyber Pakhtunkhwa, Pakistan. Kamal Shah - Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan.

Abstract

The main purpose of this research paper is to develop some sufficient conditions for the existence of solution of a nonlinear problem of implicit fractional differential equations (IFDEs) with boundary conditions, using prior estimate method. The distinction of the method applied here is, it does not require compactness of the operator. This idea is the result of motivation from the book of O'Regan and other [R. P. Agarwal, D. O'Regan, Y. J. Cho, Y.-Q. Chen, Taylor and Francis Group, New York, (2006)]. Devising the respective conditions, we also developed some conditions for Hyers-Ulam type stability to the solution of the said problem. To justify the relevant results a suitable example is provided.

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Keywords

• Implicit fractional differential equations
• Brouwer degree
• topological degree theory
• boundary conditions
• Banach contraction theorem
• Lebesgue dominated convergence theorem

MSC

•  34A08
•  35R11
•  26A33

References

• [1] S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in fractional differential equations, Springer, New York (2012)
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. P. Agarwal, D. Baleanu, S. Rezapour, S. Salehi , The existence of solutions for some fractional finite difference equations via sum boundary conditions, Adv. Difference Equ., 2014 (2014), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] R. P. Agarwal, M. Benchohra, S. Hamani , Boundary value problems for fractional differential equations, Georgian Math. J., 16 (2009), 401–411.
  • View Article
  • Google Scholar


• [4] R. P. Agarwal, D. O’Regan, Y. J. Cho, Y.-Q. Chen, Topological Degree Theory and its Applications, Taylor and Francis Group, New York (2006)
  • Google Scholar


• [5] Z. Bai , On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916–924.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan, H. Jafari , On the existence of solution for fractional differential equations of order \(3 < \delta\leq 4\) , Adv. Difference Equ., 2015 (2015), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] D. Baleanu, R. P. Agarwal, H. Mohammad, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 8 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal., 87 (2008), 851–863.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12.
  • Google Scholar


• [10] M. Benchohra, J. E. Lazreg, Existence results for nonlinear implicit fractional differential equations, Surv. Math. Appl., 9 (2014), 79–92.
  • Google Scholar


• [11] K. Diethelm, N. J. Ford, Analysis of Fractional Differential Equations, J. Math. Anal. Appl., 265 (2002), 229–248.
  • View Article
  • Google Scholar


• [12] C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050–1055.
  • View Article
  • Google Scholar


• [13] D. H. Hyers , On the stability of the linear functional equations, Proc. Nat. Acad. Sci., 27 (1941), 222–224.
  • View Article
  • Google Scholar


• [14] F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian., 75 (2006), 233–240.
  • MathSciNet
  • Google Scholar


• [15] S.-M. Jung , Hyers-Ulam Stability of linear differential equations of first order , Appl. Math. Lett., 17 (2004), 1135–1140.
  • View Article
  • Google Scholar


• [16] R. A. Khan, M. Rehman, J. Henderson , Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, Fract. Differ. Calc., 1 (2011), 29–43.
  • Google Scholar


• [17] R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal., 19 (2015), 515–526.
  • Google Scholar


• [18] A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives , (Theory and Applications), Gordon and Breach, Switzerland (1993)
  • Google Scholar


• [19] V. Lakshmikantham, S. Leela, J. V. Devi , Theory of Fractional Dynamic Systems, CSP, UK (2009)
  • Google Scholar


• [20] L. Lv, J.-R. Wang, W. Wei, Existence and uniqueness results for fractional differential equations with boundary value conditions, Opuscula Math., 31 (2011), 629–643.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] R. L. Magin, Fractional calculus in bioengineering-part 2, Crit. Rev. Biomed. Eng., 32 (2004), 105–193.
  • Google Scholar


• [22] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Am. Math. Soc., United States of America (1979)
  • MathSciNet
  • Google Scholar


• [23] K. S. Miller, B. Ross, An Introduction to the fractional Calculus and fractional Differential Equations, Wiley, New York (1993)
  • MathSciNet
  • Google Scholar


• [24] N. Nyamoradi, D. Baleanu, R. P. Agarwal , Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions, Adv. Difference Equ., 2013 (2013), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [25] N. Nyamoradi, D. Baleanu, R. P. Agarwal, On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval , Adv. Math. Phys., 2013 (2013), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, , San Diego (1999)
  • Google Scholar


• [27] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23–130.
  • View Article
  • MathSciNet
  • Google Scholar


• [29] W. Soedel , Vibrations of Shells and Plates, Dekker, New York (2004)
  • Google Scholar


• [30] I. M. Stamova, Mittag-Leffler stability of impulsive differential equations of fractional order, Quart. Appl. Math., 73 (2015), 525–535.
  • View Article
  • MathSciNet
  • Google Scholar


• [31] S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Co., New York (1961)


• [32] J. C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 91 (2011), 437–445.
  • View Article
  • Google Scholar


• [33] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York (1960)
  • Google Scholar


• [34] J.-R. Wang, X.-Z. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72–83.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] J.-R. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] J.-R. Wang, Y. Zhou, W. Wei , Study in fractional differential equations by means of topological degree methods , Numer. Func. Anal. Optim., 33 (2012), 216–238.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] Y. Zhao, S. Sun, Z. Han, Q.-P. Li , The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2086–2097.
  • View Article
  • MathSciNet
  • Google Scholar