Existence of solutions for a mixed fractional boundary value problem. (English) Zbl 1422.34041
Summary: In this paper, we prove the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo-type fractional derivatives. For this, we convert the posed problem to a sum of two integral operators, then we apply Krasnoselskii’s fixed point theorem to conclude the existence of nontrivial solutions.
Keywords:
boundary value problem; fractional derivative; fixed point theorem; existence of solution; integral equationReferences:
| [1] | Agrawal, OP, Fractional variational calculus and transversality condition, J. Phys. A, Math. Gen., 39, 10375-10384, (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008 |
| [2] | Atanackovic, TM; Stankovic, B, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal., 10, 139-150, (2007) · Zbl 1131.34003 |
| [3] | Błaszczyk, T; Ciesielski, M, Numerical solution of fractional Sturm-Liouville equation in integral form, Fract. Calc. Appl. Anal., 17, 307-320, (2014) · Zbl 1305.34008 |
| [4] | Blaszczyk, T; Ciesielski, M, Fractional oscillator equation - transformation into integral equation and numerical solution, Appl. Math. Comput., 257, 428-435, (2015) · Zbl 1338.34012 |
| [5] | Blaszczyk, T, A numerical solution of a fractional oscillator equation in a non-resisting medium with natural boundary conditions, Rom. Rep. Phys., 67, 350-358, (2015) |
| [6] | Blaszczyk, T; Ciesielski, M, Numerical solution of Euler-Lagrange equation with Caputo derivatives, Adv. Appl. Math. Mech., 9, 173-185, (2017) · Zbl 1499.65742 · doi:10.4208/aamm.2015.m970 |
| [7] | Guezane-Lakoud, A, Khaldi, R, Torres, DFM: On a fractional oscillator equation with natural boundary conditions. https://arxiv.org/pdf/1701.08962. To appear in Prog. Frac. Diff. Appl. |
| [8] | Guezane Lakoud, A; Khaldi, R; Kılıçman, A, Solvability of a boundary value problem at resonance, SpringerPlus, 5, (2016) · doi:10.1186/s40064-016-3172-7 |
| [9] | Khaldi, R; Guezane-Lakoud, A, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal., 2017, (2017) |
| [10] | Guezane-Lakoud, A; Kılıçman, A, Unbounded solution for a fractional boundary value problem, Adv. Differ. Equ., 2014, (2014) · Zbl 1419.34022 · doi:10.1186/1687-1847-2014-154 |
| [11] | Agarwal, RP, Bohner, M, Li, W-T: Nonoscillation and Oscillation Theory for Functional Differential Equations. CRC Press, Boca Raton (2004) · Zbl 1068.34002 · doi:10.1201/9780203025741 |
| [12] | Ahmad, B, Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions, Differ. Equ. Dyn. Syst., 21, 387-401, (2013) · Zbl 1296.34009 · doi:10.1007/s12591-012-0154-2 |
| [13] | Franco, D; Nieto, JJ; O’Regan, D, Upper and lower solutions for first order problems with nonlinear boundary conditions, Extr. Math., 18, 153-160, (2003) · Zbl 1086.34509 |
| [14] | Khaldi, R; Guezane-Lakoud, A, Upper and lower solutions method for higher order boundary value problems, Prog. Fract. Differ. Appl., 3, 53-57, (2017) · doi:10.18576/pfda/030105 |
| [15] | Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier Science, Amsterdam (2006) · Zbl 1092.45003 |
| [16] | Ntouyas, SK; Tariboon, J; Sudsutad, W, Boundary value problems for Riemann-Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions, Mediterr. J. Math., 13, 939-954, (2016) · Zbl 1408.34023 · doi:10.1007/s00009-015-0543-1 |
| [17] | Zhang, L; Ahmad, B; Wang, G, The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett., 31, 1-6, (2014) · Zbl 1315.34016 · doi:10.1016/j.aml.2013.12.014 |
| [18] | Podlubny, I: Fractional Differential Equation. Academic Press, Sain Diego (1999) · Zbl 0924.34008 |
| [19] | Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993) · Zbl 0818.26003 |
| [20] | Smart, DR: Fixed Point Theorems. Cambridge Tracts in Mathematics, vol. 66. Cambridge University Press, London (1974) · Zbl 0297.47042 |
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