Boundary value problems of fractional q-difference equations on the half-line
Tóm tắt
In this paper, we consider the boundary value problem of a class of nonlinear fractional q-difference equations involving the Riemann–Liouville fractional q-derivative on the half-line. By means of Schauder fixed point theorem and Leggett–Williams fixed point theorem, some results on existence and multiplicity of solutions are obtained.
Tài liệu tham khảo
Jackson, F.: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)
Jackson, F.: On q-definite integrals. Pure Appl. Math. Q. 41, 193–203 (1910)
Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)
Agarwal, R.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)
Miao, F., Liang, S.: Uniqueness of positive solutions for fractional q-difference boundary value problems with p-Laplacian operator. Electron. J. Differ. Equ. 2013, 174, 1–11 (2013)
Graef, J., Kong, L.: Positive solutions for a class of higher order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218, 9682–9689 (2012)
Ahmad, B., Nietoa, J., Alsaedi, A., Al-Hutami, H.: Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 351, 2890–2909 (2014)
Han, Z.L., Pan, Y.Y., Yang, D.W.: The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter. Appl. Math. Lett. 36, 1–6 (2014)
Li, X.H., Han, Z.L., Li, X.C.: Boundary value problems of fractional q-difference Schrodinger equations. Appl. Math. Lett. 46, 100–105 (2015)
Chidouh, A., Torres, D.: Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities. Opusc. Math. 38(1), 31–40 (2018)
Zhao, X., Ge, W.: Unbounded solutions for a fractional boundary value problems on infinite interval. Acta Appl. Math. 109, 495–505 (2010)
He, L., Dong, X., Bai, Z., Chen, B.: Solvability of some two-point fractional boundary value problems under barrier strip conditions. J. Funct. Spaces 2017, Article ID 1465623 (2017)
Liang, S., Zhang, J.: Existence of three positive solutions of m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval. Comput. Math. Appl. 61, 3343–3354 (2011)
Zhao, X., Zhao, D., Ge, W.: Solvability for fractional multi-point boundary value problems on half line. Ann. Differ. Equ. 27(3), 395–400 (2011)
Liu, X., Jia, M.: Multiple solutions for fractional differential equations with nonlinear boundary conditions. Comput. Math. Appl. 59, 2880–2886 (2010)
Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23, 31–39 (2018)
Cui, Y., Sun, Q., Su, X.: Monotone iterative technique for nonlinear boundary value problems of fractional order \(p\in (2,3]\). Adv. Differ. Equ. 2017, 248, 1–15 (2017)
Zhang, X., Zhong, Q.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 20(6), 1471–1484 (2017)
Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)
Xu, M., Han, Z.: Positive solutions for integral boundary value problem of two-term fractional differential equations. Bound. Value Probl. 2018, 100, 1–13 (2018)
Song, Q., Bai, Z.: Positive solutions of fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, 183 (2018)
Gamze Düzgün, F., Iannizzotto, A.: Three nontrivial solutions for nonlinear fractional Laplacian equations. Adv. Nonlinear Anal. 7(2), 211–226 (2018)
Giacomoni, J., Mukherjee, T., Sreenadh, K.: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6(3), 327–354 (2017)
Lyons, J., Neugebauer, J.: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opusc. Math. 37(3), 421–434 (2017)
Annaby, M., Mansour, Z.: q-Fractional Calculus and Equations. Springer, New York (2012)
Rajković, P., Marinković, S., Stanković, M.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)
Li, X., Han, Z., Sun, S.: Existence of positive solutions of nonlinear fractional q-difference equation with parameter. Adv. Differ. Equ. 2013, 260 (2013)
Granans, A., Guenther, P., Lee, J.: Some general existence principle in Carathéodory theory of nonlinear systems. J. Math. Pures Appl. 70, 153–196 (1991)
Su, X., Zhang, S.: Unbounded solutions to a boundary value problem of fractional order on the half-line. Comput. Math. Appl. 61, 1079–1087 (2011)