We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems (FHS):
where \(\alpha \in \left(\left.\frac{1}{2},1\right]\right.,t\in {\mathbb{R}},x\in {\mathbb{R}}^{N},\) and \({-}_{t}{D}_{t}^{\alpha }\) and \({}_{t}{D}_{\infty }^{\alpha }\) are the left and right Liouville–Weyl fractional derivatives of order α on the entire axis ℝ, respectively. The novelty of our results is that, under the assumption that the nonlinearity \(W\in {C}^{1}\left({\mathbb{R}}\times {\mathbb{R}}^{N},{\mathbb{R}}\right)\) involves a combination of superquadratic and subquadratic terms, for the first time, we show that the FHS possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of the FHS goes to infinity and zero, respectively. Some recent results available in the literature are generalized and significantly improved.
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References
O. Agrawal, J. Tenreiro Machado, and J. Sabatier, “Fractional derivatives and their application,” Nonlin. Dynam., No. 1-4, Springer-Verlag, Berlin (2004).
A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” J. Funct. Anal., 14, No. 4, 349–381 (1973).
A. Ambrosetti and V. C. Zelati, “Multiple homoclinic orbits for a class of conservative systems,” Rend. Semin. Mat. Univ. Padova, 89, 177–194 (1993).
A. Bahri, “Critical points at infinity in some variational problems,” Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow (1989).
Z. Bai and H. L¨u, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” J. Math. Anal. Appl., 311, No. 2, 495–505 (2005).
A. Benhassine, “Multiplicity of solutions for nonperiodic perturbed fractional Hamiltonian equations,” Electron. J. Different. Equat., 93, 1–15 (2017).
A. Benhassine, “Multiple of homoclinic solutions for a perturbed dynamical systems with combined nonlinearities,” Mediterr. J. Math., 14, No. 3, 1–20 (2017).
A. Benhassine, “Existence and multiplicity of periodic solutions for a class of the second order Hamiltonian systems,” Nonlin. Dyn. Syst. Theory, 14, No. 3, 257–264 (2014).
A. Benhassine, “Existence of infinitely many solutions for nonperiodic fractional Hamiltonian systems,” Different. Integr. Equat. (to appear).
D. Benson, S. Wheatcraft, and M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resour. Res., 36, No. 6, 1403–1412 (2000).
D. Benson, S. Wheatcraft, and M. Meerschaert, “The fractional-order governing equation of lvy motion,” Water Resour. Res., 36, No. 6, 1413–1423 (2000).
P. C. Carriao and O. H. Miyagaki, “Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems,” J. Math. Anal. Appl., 230, No. 1, 157–172 (1999).
P. Chen, X. He, and X. H. Tang, “Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory,” Math. Meth. Appl. Sci., 39, No. 5, 1005–1019 (2016).
Y. Ding, “Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,” Nonlin. Anal., 25, No. 11, 1095–1113 (1995).
V. Ervin and J. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numer. Meth. Partial Different. Equat., 22, 58–76 (2006).
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ., River Edge, NJ (2000).
W. Jang, “The existence of solutions for boundary-value problems of fractional differential equations at resonance,” Nonlin. Anal., 74, No. 5, 1987–1994 (2011).
F. Jiao and Y. Zhou, “Existence results for fractional boundary-value problem via critical point theory,” Internat. J. Bifur. Chaos Appl. Sci. Eng., 22, No. 4, 1–17 (2012).
F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problem via critical point theory,” Comput. Math. Appl., 62, No. 3, 1181–1199 (2011).
A. Kilbas, H. Srivastava, and J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006).
W. Omana and M. Willem, “Homoclinic orbits for a class of Hamiltonian systems,” Different. Integr. Equat., 5, No. 5, 1115–1120 (1992).
H. Poincaré, Les M´ethodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris (1897–1899).
P. H. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” Amer. Math. Soc., Providence, RI, 65, 45–60 (1986).
P. H. Rabinowitz and K. Tanaka, “Some results on connecting orbits for a class of Hamiltonian systems,” Math. Z., 206, No. 3, 473–499 (1991).
C. Torres, “Existence of solution for a class of fractional Hamiltonian systems,” Electron. J. Different. Equat., 2013, No. 259, 1–12 (2013).
Z. Zhang and R. Yuan, “Variational approach to solutions for a class of fractional Hamiltonian systems,” Math. Meth. Appl. Sci., 37, No. 13, 1873–1883 (2014).
Z. Zhang and R. Yuan, “Existence of solutions to fractional Hamiltonian systems with combined nonlinearities,” Electron. J. Different. Equat., 2016, No. 40, 1–13 (2016).
W. Zou, “Variant fountain theorems and their applications,” Manuscripta Math., 104, 343–358 (2001).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 2, pp. 155–167, February, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i2.328.
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Benhassine, A. Two Different Sequences of Infinitely Many Homoclinic Solutions for a Class of Fractional Hamiltonian Systems. Ukr Math J 75, 175–189 (2023). https://doi.org/10.1007/s11253-023-02192-9
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DOI: https://doi.org/10.1007/s11253-023-02192-9