Topological Methods in Nonlinear Analysis

Existence and stability of fractional differential equations with Hadamard derivative

Abstract

In this paper, we study nonlinear fractional differential equations with Hadamard derivative and Ulam stability in the weighted space of continuous functions. Firstly, some new nonlinear integral inequalities with Hadamard type singular kernel are established, which can be used in the theory of certain classes of fractional differential equations. Secondly, some sufficient conditions for existence of solutions are given by using fixed point theorems via a prior estimation in the weighted space of the continuous functions. Meanwhile, a sufficient condition for nonexistence of blowing-up solutions is derived. Thirdly, four types of Ulam-Hyers stability definitions for fractional differential equations with Hadamard derivative are introduced and Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability results are presented. Finally, some examples and counterexamples on Ulam-Hyers stability are given.

Article information

Source
Topol. Methods Nonlinear Anal. Volume 41, Number 1 (2013), 113-133.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461253858

Mathematical Reviews number (MathSciNet)
MR3086536

Zentralblatt MATH identifier
06212165

Citation

Wang, JinRong; Zhou, Yong; Medveď, Milan. Existence and stability of fractional differential equations with Hadamard derivative. Topol. Methods Nonlinear Anal. 41 (2013), no. 1, 113--133.https://projecteuclid.org/euclid.tmna/1461253858

References

• 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. Appl. Math., 109 (2010), 973–1033 \ref\key 2
• B. Ahmad and J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory , Topol. Methods Nonlinear Anal., 35 (2010), 295–304 \ref\key 3
• H. Amann, Ordinary differential equations , Walter de Gruyter, Berlin (1990) \ref\key 4
• Z.B. Bai, On positive solutions of a nonlocal fractional boundary value problem , Nonlinear Anal., 72 (2010), 916–924 \ref\key 5
• M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay , J. Math. Anal. Appl., 338 (2008), 1340–1350 \ref\key 6
• G. Butler and T. Rogers, A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations , J. Math. Anal. Appl., 33 (1971), 77–81 \ref\key 7
• L. Cădariu, Stabilitatea Ulam–Hyers–Bourgin Pentru Ecuatii Functionale, Ed. Univ. Vest Timişoara, Timişara (2007) \ref\key 8
• Y.-K. Chang, V. Kavitha and M. Mallika Arjunanb, Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order , Nonlinear Anal., 71 (2009), 5551–5559 \ref\key 9
• C. Chicone, Ordinary differential equations with applications , Springer, New York (2006) \ref\key 10
• C. Corduneanu, Principles of Differential and Integral Equations, Chelsea Publ. Company, New York (1971) \ref\key 11
• W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations , Nonlinear Anal., 72 (2010), 1768–1777 \ref\key 12
• K. Diethelm, The analysis of fractional differential equations , Lecture Notes in Mathematics (2010) \ref\key 13
• S.-B. Hsu, Ordinary Differential Equations with Applications, World Scientific, New Jersey (2006) \ref\key 14
• D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser (1998) \ref\key 15
• S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001) \ref\key 16
• A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations , Mathematics Studies, 204 , North-Holland, Elsevier Science B.V., Amsterdam (2006) \ref\key 17
• V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers (2009) \ref\key 18
• Y. Li, Y. Chen and I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems , Automatica, 45 (2009), 1965–1969 \ref\key 19 ––––, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability , Comput. Math. Appl., 59 (2010), 1810–1821 \ref\key 20
• M. Medve\ud, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions , J. Math. Anal. Appl., 214 (1997), 349–366 \ref\key 21 ––––, Integral inequalities and global solutions of semilinear evolution equations , J. Math. Anal. Appl., 267 (2002), 643–650 \ref\key 22 ––––, Singular integral inequalities with several nonlinearities and integral equations with singular kernels , Nonlinear Oscil., 11 (2007), 70–79 \ref\key 23
• M. Medve\ud, M. Pospí\usil and L. Škripková, Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices , Nonlinear Anal., 74 (2011), 3903–3911 \ref\key 24
• K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York (1993) \ref\key 25
• G. M. Mophou, G.M. N'Guérékata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay , Appl. Math. Comput., 216(2010), 61–69 \ref\key 26
• L.C. Piccinini, G. Stampacchia and G. Vidossich, Ordinary Differential Equations in $\mathbb{R}^n$, Springer, Berlin (1984) \ref\key 27
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999) \ref\key 28
• I.A. Rus, Ecuaţii Diferenţiale, Ecuaţii Integrale şi Sisteme Dinamice, Transilvania Press, Cluj-Napoca (1996) \ref\key 29 ––––, Ulam stability of ordinary differential equations, , Studia Univ. Babeş-Bolyai Math., 54 (2009), 125–133 \ref\key 30
• V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP (2010) \ref\key 31
• J. Wang and L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative , E.J. Qualitative Theory of Differential Equations, 2011 , no. 63, e1–e10 \ref\key 32 ––––, New concepts and results in stability of fractional differential equations, , Commun. Nonlinear Sci. Numer. Simulat. (2011). doi:10.1016/j.cnsns.2011.09.030 \ref\key 33
• J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls , Nonlinear Anal., 12 (2011), 262–272 \ref\key 34 ––––, Analysis of nonlinear fractional control systems in Banach spaces , Nonlinear Anal., 74 (2011), 5929–5942 \ref\key 35 ––––, Existence and controllability results for fractional semilinear differential inclusions , Nonlinear Anal., 12 (2011), 3642–3653 \ref\key 36
• S. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations , J. Math. Anal. Appl., 278 (2003), 136–148 \ref\key 37
• Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations , Nonlinear Anal., 11 (2010), 4465–4475 \ref\key 38
• Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for $p$-type fractional neutral differential equations , Nonlinear Anal., 71 (2009), 2724–2733 \ref\key 39 ––––, Existence and uniqueness for fractional neutral differential equations with infinite delay , Nonlinear Anal., 71 (2009), 3249–3256