## Topological Methods in Nonlinear Analysis

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

#### Abstract

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} \displaystyle M \bigg (\int_{\mathbb{R}} (|{}_{- \infty} D_t^\alpha u (t)|^2 + b (t) |u(t)|^2 )\, d t \bigg) \\ \qquad \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, $u \in \mathbb{R}$, $b \colon \mathbb{R} \to \mathbb{R}$, $\inf\limits_{t \in \mathbb{R}} b (t) \gt 0$, $f \colon \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M\colon \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 617-630.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588654

Digital Object Identifier
doi:10.12775/TMNA.2015.061

Mathematical Reviews number (MathSciNet)
MR3494961

Zentralblatt MATH identifier
1360.34017

#### Citation

Nyamoradi, Nemat; Zhou, Yong. Existence of solutions for a Kirchhoff type fractional differential equations via minimal principle and Morse theory. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 617--630. doi:10.12775/TMNA.2015.061. https://projecteuclid.org/euclid.tmna/1458588654

#### References

• 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).