Existence of solutions for a Kirchhoff type fractional differential equations via minimal principle and Morse theory

Nemat Nyamoradi, Yong Zhou

DOI: http://dx.doi.org/10.12775/TMNA.2015.061


In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff type fractional differential equation: \begin{equation*} \begin{cases} M (\int_{\mathbb{R}} (|{}_{- \infty} D_t^\alpha u (t)|^2 + b (t) |u(t)|^2 )\, d t) \cdot ({}_tD_\infty^{\alpha} ({}_{- \infty} D_t^\alpha u (t) ) + b(t) u (t)) = f (t, u (t)), t \in \mathbb{R}, u \in H^\alpha (\mathbb{R}), \end{cases} \end{equation*} where $\alpha \in ({1}/{2},1)$, ${}_tD_\infty^{\alpha}$ and ${}_{- \infty} D_t^\alpha$ are the right and left inverse operators of the corresponding Liouville--Weyl fractional integrals of order $\alpha$ respectively, $H^\alpha$ is the classical fractional Sobolev Space, $u \in \mathbb{R}$, $b \colon \mathbb{R} \to \mathbb{R}$, $\inf\limits_{t \in \mathbb{R}} b (t) \ge 0$, $f \colon \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ Caratheodory function and $M\colon \mathbb{R}^+ \to \mathbb{R}^+$ is a~function that satisfy some suitable conditions.


Fractional differential equations; minimal principle; Morse theory; solutions; Critical point theory

Full Text:



O. Agrawal, J. Tenreiro Machado and J. Sabatier, Fractional Derivatives and their Application: Nonlinear Dynamics, Springer-Verlag, Berlin, 2004.

C. O. Alves, F. S. J. A. Correa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85-93.

S. Aouaoui, Existence of three solutions for some equation of Kirchhoff type involving variable exponents, Appl. Math. Comput. 218 (2012), 7184-7192.

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305-330.

T. Bartsch and S.J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), 419-441.

K. C. Chang, A variant of mountain pass lemma, Sci. Sinica Ser. A 26 (1983), 1241-1255.

M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Methods Appl. Sci. 22 (5) (1999), 375-388.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

F. Jiao and Y. Zhou, Existence results fro fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos 22 (4) (2012), 1-17.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, (2006).

G. Kirchhoff, Vorlesungen uber Mathematische Physik, Mechanik, Teubner, Leipzig (1883).

J. Q. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci. 2 (1989), 32-39.

J. Q. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl. 258 (2001), 209-222.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989.

K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

P. Rabinowitz, Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMS Amer. Math. Soc., No 65, 1986.

J. Sabatier, O. Agrawal and J. Tenreiro Machado, Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag, Berlin, 2007.

S. Spagnolo, The Cauchy problem for the Kirchhoff equations, Rend. Sem. Fis. Mat. Milano 62 (1992), 17-51.

C. Torres, Existence of solution for a class of fractional Hamiltonian systems, Electronic J. Differential Equations 259 (2013), 1-12.

G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005.

Z. Zhang and R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems. Math. Method. Appl. Sci. (2013) (Preprint).


  • There are currently no refbacks.

Partnerzy platformy czasopism