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Existence of solutions for a fractional boundary value problem via the mountain pass method and an iterative technique. (English) Zbl 1268.34027

Summary: We consider the existence of a solution to the following fractional boundary value problem: \[ \begin{cases} \frac{d}{dx}(p_0D^{-\beta}_x(u'(x))+q_xD^{-\beta}_1(u'(x)))+f(x,u(x))=0,\quad x\in(0,1),\\ u(0)=u(1)=0, \end{cases} \] where the constants \(\beta\in(0,1)\), \(_0D^{-\beta}_x\) and \(_xD^{-\beta}_1\) denote left and right Riemann-Liouville fractional integrals of order \(\beta\) respectively, \(0<p=1-q<1\) and \(f\colon[0,1]\times \mathbb R\to\mathbb R\) is continuous. Due to the general assumption on the constants \(p\) and \(q\), the problem does not have a variational structure. Despite that, here we study it by performing variational methods, combined with an iterative technique, and give an existence criterion for the solution of the problem under suitable assumptions. The results extend the results in [F. Jiao and Y. Zhou, ibid. 62, No. 3, 1181–1199 (2011; Zbl 1235.34017)].

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35R11 Fractional partial differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1235.34017
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References:

[1] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 973-1033 (2010) · Zbl 1198.26004
[2] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 1403-1412 (2000)
[3] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., The fractional-order governing equation of Lévy motion, Water Resour. Res., 36, 1413-1423 (2000)
[4] Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer · Zbl 0917.73004
[5] Carreras, B. A.; Lynch, V. E.; Zaslavsky, G. M., Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8, 5096-5103 (2001)
[6] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, vol. 204 (2006), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[7] Podlubny, I., Fractional Differential Equations (1999), Academic press: Academic press New York · Zbl 0918.34010
[8] Shlesinger, M. F.; West, B. J.; Klafter, J., Lévy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Lett., 58, 1100-1103 (1987)
[9] Ervin, V. J.; Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22, 558-576 (2006) · Zbl 1095.65118
[10] Jiao, F.; Zhou, Y., Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62, 1181-1199 (2011) · Zbl 1235.34017
[11] De Figueiredo, D.; Girardi, M.; Matzeu, M., Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential Integral Equations, 17, 119-126 (2004) · Zbl 1164.35341
[12] Servadei, R., A semilinear elliptic PDE not in divergence form via variational methods, J. Math. Anal. Appl., 383, 190-199 (2011) · Zbl 1221.35159
[13] Rabinowitz, P. H., (Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0609.58002
[14] Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34 (2000), Springer Verlag · Zbl 0939.49001
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