Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions
Abstract
:1. Introduction
2. Preliminaries
- (i)
- for,
- (ii)
- for.
3. Proof of Theorem 1
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Jackson, F. On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 1908, 46, 253–281. [Google Scholar] [CrossRef]
- Jackson, F. On q-definite integrals. Quart J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Ferreira, R.A.C. Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70, 1–10. [Google Scholar] [CrossRef]
- Ferreira, R.A.C. Positive solutions for a class of boundary value problems with fractional q-differences. Comput. Math. Appl. 2011, 61, 367–373. [Google Scholar] [CrossRef]
- Yang, C. Positive solutions for a three-point boundary value problem of fractional q-difference equations. Symmetry 2018, 10, 358. [Google Scholar] [CrossRef] [Green Version]
- Zhao, Y.; Chen, H.; Zhang, Q. Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivatives. Bound. Value Probl. 2013, 2013, 103. [Google Scholar] [CrossRef] [Green Version]
- Ma, K.; Han, Z.; Sun, S. Existence and uniqueness of solutions for fractional q-difference Schrödinger equations. J. Appl. Math. Comput. 2020, 62, 611–620. [Google Scholar] [CrossRef]
- Guo, F.; Kang, S. Positive solutions for a class of fractional boundary value problem with q-derivatives. Mediterr. J. Math. 2019, 16, 113. [Google Scholar] [CrossRef]
- Jin, N.; Sun, S.; Chen, G. Existence of solutions for a class of the boundary value problem of fractional q-difference inclusions. J. Appl. Math. Comput. 2017, 55, 409–420. [Google Scholar] [CrossRef]
- Kang, S.; Chen, H.; Li, L.; Cui, Y.; Ma, S. Existence of three positive solutions for a class of Riemann-Liouville fractional q-difference equation. J. Anal. Appl. Comput. 2019, 9, 590–600. [Google Scholar]
- Zhai, C.; Ren, J. The unique solution for a fractional q-difference equation with three-point boundary conditions. Indag. Math. 2018, 29, 948–961. [Google Scholar] [CrossRef]
- Zhai, C.; Ren, J. Nonlocal q-fractional boundary value problem with Stieltjes integral conditions. Nonlinear Anal. Model. Control 2019, 24, 582–602. [Google Scholar]
- Ren, J.; Zhai, C. Unique solutions for fractional q-difference boundary value problems via a fixed point method. Bull. Malays. Math. Sci. Soc. 2019, 42, 1507–1521. [Google Scholar] [CrossRef]
- Almeida, R.; Martins, N. Existence results for fractional q-difference equations of order alpha is an element of ]2,3[ with three-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 1675–1685. [Google Scholar] [CrossRef]
- Ahmad, B.; Nieto, J.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. 2014, 351, 2890–2909. [Google Scholar] [CrossRef]
- Ahmad, B.; Nieto, J.J.; Alsaedi, A.; Al-Hutami, H. Boundary value problems of nonlinear fractional q-difference (integral) equations with two fractional orders and four-point nonlocal integral boundary conditions. Filomat 2014, 28, 1719–1736. [Google Scholar] [CrossRef] [Green Version]
- Mao, J.; Zhao, Z.; Wang, C. The unique iterative positive solution of fractional boundary value problem with q-difference. Appl. Math. Lett. 2020, 100, 106002. [Google Scholar] [CrossRef]
- Yang, W. Existence results for nonlinear fractional q-difference equations with nonlocal Riemann-Liouville q-integral boundary conditions. Filomat 2016, 30, 2521–2533. [Google Scholar] [CrossRef]
- Yang, W.; Qin, Y. Positive solutions for nonlinear Caputo type fractional q-difference equations with integral boundary conditions. Mathematics 2016, 4, 63. [Google Scholar] [CrossRef]
- Wang, G. Twin iterative positive solutions of fractional q-difference Schrödinger equations. Appl. Math. Lett. 2018, 76, 103–109. [Google Scholar] [CrossRef]
- Wang, G.; Bai, Z.; Zhang, L. Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem. J. Anal. Appl. Comput. 2019, 9, 1204–1215. [Google Scholar] [CrossRef]
- Nasiruzzaman, M.; Mukheimer, A.; Mursaleen, M. A Dunkl-type generalization of Szász-Kantorovich operators via Post-Quantum calculus. Symmetry 2019, 11, 232. [Google Scholar] [CrossRef] [Green Version]
- Bai, C.; Yang, D. The iterative positive solution for a system of fractional q-difference equations with four-point boundary conditions. Discret. Dyn. Nat. Soc. 2020, 2020, 3970903. [Google Scholar] [CrossRef] [Green Version]
- Suantai, S.; Ntouyas, S.K.; Asawasamrit, S.; Tariboon, J. A coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions. Adv. Differ. Equ. 2015, 2015, 124. [Google Scholar] [CrossRef] [Green Version]
- Jiang, M.; Zhong, S. Existence of extremal solutions for a nonlinear fractional q-difference system. Mediterr. J. Math. 2016, 13, 279–299. [Google Scholar] [CrossRef]
- Yang, W. Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions. Appl. Math. Comput. 2014, 244, 702–725. [Google Scholar] [CrossRef]
- Zhao, Q.; Yang, W. Positive solutions for singular coupled integral boundary value problems of nonlinear higher-order fractional q-difference equations. Adv. Differ. Equ. 2015, 2015, 290. [Google Scholar] [CrossRef] [Green Version]
- Cheng, W.; Xu, J.; Cui, Y. Positive solutions for a system of nonlinear semipositone fractional q-difference equations with q-integral boundary conditions. J. Nonlinear Sci. Appl. 2017, 10, 4430–4440. [Google Scholar] [CrossRef] [Green Version]
- Fu, Z.; Bai, S.; O’Regan, D.; Xu, J. Nontrivial solutions for an integral boundary value problem involving Riemann-Liouville fractional derivatives. J. Inequal. Appl. 2019, 2019, 104. [Google Scholar] [CrossRef]
- Xu, J.; Goodrich, C.S.; Cui, Y. Positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mater. 2019, 113, 1343–1358. [Google Scholar] [CrossRef]
- Zhang, H.; Li, Y.; Xu, J. Positive solutions for a system of fractional integral boundary value problems involving Hadamard-type fractional derivatives. Complexity 2019, 2019, 2671539. [Google Scholar] [CrossRef]
- Xu, J.; Wei, Z.; O’Regan, D.; Cui, Y. Infinitely many solutions for fractional Schrödinger-Maxwell equations. J. Anal. Appl. Comput. 2019, 9, 1165–1182. [Google Scholar] [CrossRef]
- Cheng, W.; Xu, J.; O’Regan, D.; Cui, Y. Positive solutions for a nonlinear discrete fractional boundary value problems with a p-Laplacian operator. J. Anal. Appl. Comput. 2019, 9, 1959–1972. [Google Scholar]
- Cheng, W.; Xu, J.; Cui, Y.; Ge, Q. Positive solutions for a class of fractional difference systems with coupled boundary conditions. Adv. Differ. Equ. 2019, 2019, 249. [Google Scholar] [CrossRef]
- Xu, J.; Jiang, J.; O’Regan, D. Positive solutions for a class of p-Laplacian Hadamard fractional-order three-point boundary value problems. Mathematics 2020, 8, 308. [Google Scholar] [CrossRef] [Green Version]
- Guo, D.; Lakshmikantham, V. Nonlinear Problems in Abstract Cones; Academic Press: Orlando, FL, USA, 1988. [Google Scholar]
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Li, Y.; Liu, J.; O’Regan, D.; Xu, J. Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions. Mathematics 2020, 8, 828. https://doi.org/10.3390/math8050828
Li Y, Liu J, O’Regan D, Xu J. Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions. Mathematics. 2020; 8(5):828. https://doi.org/10.3390/math8050828
Chicago/Turabian StyleLi, Yaohong, Jie Liu, Donal O’Regan, and Jiafa Xu. 2020. "Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions" Mathematics 8, no. 5: 828. https://doi.org/10.3390/math8050828