| Title:
|
Fractional ${q}$-difference equations on the half line (English) |
| Author:
|
Abbas, Saïd |
| Author:
|
Benchohra, Mouffak |
| Author:
|
Laledj, Nadjet |
| Author:
|
Zhou, Yong |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
207-223 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This article deals with some results about the existence of solutions and bounded solutions and the attractivity for a class of fractional ${q}$-difference equations. Some applications are made of Schauder fixed point theorem in Banach spaces and Darbo fixed point theorem in Fréchet spaces. We use some technics associated with the concept of measure of noncompactness and the diagonalization process. Some illustrative examples are given in the last section. (English) |
| Keyword:
|
fractional $q$-difference equation |
| Keyword:
|
attractivity |
| Keyword:
|
diagonalization |
| Keyword:
|
bounded solution |
| Keyword:
|
Banach space |
| Keyword:
|
Fréchet space |
| Keyword:
|
fixed point |
| MSC:
|
26A33 |
| idZBL:
|
Zbl 07285961 |
| idMR:
|
MR4173075 |
| DOI:
|
10.5817/AM2020-4-207 |
| . |
| Date available:
|
2020-10-30T16:39:51Z |
| Last updated:
|
2021-02-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148389 |
| . |
| Reference:
|
[1] Abbas, S., Benchohra, M.: On the existence and local asymptotic stability of solutions of fractional order integral equations.Comment. Math. 52 (1) (2012), 91–100. MR 2977716 |
| Reference:
|
[2] Abbas, S., Benchohra, M.: Existence and attractivity for fractional order integral equations in Fréchet spaces.Discuss. Math. Differ. Incl. Control Optim. 33 (1) (2013), 1–17. MR 3136582, 10.7151/dmdico.1141 |
| Reference:
|
[3] Abbas, S., Benchohra, M., Diagana, T.: Existence and attractivity results for some fractional order partial integro-differential equations with delay.Afr. Diaspora J. Math. 15 (2) (2013), 87–100. MR 3161669 |
| Reference:
|
[4] Abbas, S., Benchohra, M., Graef, J.R., Henderson, J.: Implicit Fractional Differential and Integral Equations: Existence and Stability.De Gruyter, Berlin, 2018. MR 3791511 |
| Reference:
|
[5] Abbas, S., Benchohra, M., Henderson, J.: Asymptotic attractive nonlinear fractional order Riemann-Liouville integral equations in Banach algebras.Nonlinear Stud. 20 (1) (2013), 1–10. MR 3058403 |
| Reference:
|
[6] Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations.Springer, New York, 2012. MR 2962045 |
| Reference:
|
[7] Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations.Nova Science Publishers, New York, 2015. Zbl 1314.34002, MR 3309582 |
| Reference:
|
[8] Adams, C.R.: In the linear ordinary $q$-difference equation.Annals Math. 30 (1928), 195–205. MR 1502876, 10.2307/1968274 |
| Reference:
|
[9] Agarwal, R.: Certain fractional $q$- integrals and $q$-derivatives.Proc. Cambridge Philos. Soc. 66 (1969), 365–370. MR 0247389 |
| Reference:
|
[10] Ahmad, B.: Boundary value problem for nonlinear third order $q$-difference equations.Electron. J. Differential Equations 2011 (94) (2011), 1–7. MR 2832270 |
| Reference:
|
[11] Ahmad, B., Ntouyas, S.K., Purnaras, L.K.: Existence results for nonlocal boundary value problems of nonlinear fractional $q$-difference equations.Adv. Difference Equ. 2012 (2012), 14 pp. MR 3016054 |
| Reference:
|
[12] Almezel, S., Ansari, Q.H., Khamsi, M.A.: Topics in Fixed Point Theory.Springer-Verlag, New York, 2014. MR 3411798 |
| Reference:
|
[13] Benchohra, M., Berhoun, F., N’Guérékata, G.M.: Bounded solutions for fractional order differential equations on the half-line.Bull. Math. Anal. Appl. 146 (4) (2012), 62–71. MR 2955875 |
| Reference:
|
[14] Bothe, D.: Multivalued perturbations of m-accretive differential inclusions.Israel J. Math. 108 (1998), 109–138. MR 1669396, 10.1007/BF02783044 |
| Reference:
|
[15] Carmichael, R.D.: The general theory of linear $ q$-difference equations.American J. Math. 34 (1912), 147–168. MR 1506145, 10.2307/2369887 |
| Reference:
|
[16] Corduneanu, C.: Integral Equations and Stability of Feedback Systems.Academic Press, New York, 1973. Zbl 0273.45001, MR 0358245 |
| Reference:
|
[17] Dudek, S.: Fixed point theorems in Fréchet Algebras and Fréchet spaces and applications to nonlinear integral equations.Appl. Anal. Discrete Math. 11 (2017), 340–357. MR 3719830, 10.2298/AADM1702340D |
| Reference:
|
[18] Dudek, S., Olszowy, L.: Continuous dependence of the solutions of nonlinear integral quadratic Volterra equation on the parameter.J. Funct. Spaces (2015), 9 pp., Article ID 471235. MR 3319202 |
| Reference:
|
[19] El-Shahed, M., Hassa, H.A.: Positive solutions of ${q}$-difference equation.Proc. Amer. Math. Soc. 138 (2010), 1733–1738. MR 2587458, 10.1090/S0002-9939-09-10185-5 |
| Reference:
|
[20] Etemad, S., Ntouyas, S.K., Ahmad, B.: Existence theory for a fractional $q$-integro-difference equation with $q$-integral boundary conditions of different orders.Mathematics 7 (2019), 1–15. 10.3390/math7080659 |
| Reference:
|
[21] Granas, A., Dugundji, J.: Fixed Point Theory.Springer-Verlag, New York, 2003. Zbl 1025.47002, MR 1987179 |
| Reference:
|
[22] Kac, V., Cheung, P.: Quantum Calculus.Springer, New York, 2002. Zbl 0986.05001, MR 1865777 |
| Reference:
|
[23] Kilbas, A.A.: Hadamard-type fractional calculus.J. Korean Math. Soc. 38 (6) (2001), 1191–1204. MR 1858760 |
| Reference:
|
[24] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. Zbl 1092.45003, MR 2218073 |
| Reference:
|
[25] Mönch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.Nonlinear Anal. 4 (1980), 985–999. MR 0586861, 10.1016/0362-546X(80)90010-3 |
| Reference:
|
[26] Olszowy, L.: Existence of mild solutions for the semilinear nonlocal problem in Banach spaces.Nonlinear Anal. 81 (2013), 211–223. MR 3016450, 10.1016/j.na.2012.11.001 |
| Reference:
|
[27] Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S.: Fractional integrals and derivatives in $q$-calculus.Appl. Anal. Discrete Math. 1 (2007), 311–323. MR 2316607, 10.2298/AADM0701311R |
| Reference:
|
[28] Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S.: On $q$-analogues of Caputo derivative and Mittag-Leffler function.Fract. Calc. Appl. Anal. 10 (2007), 359–373. MR 2378985 |
| Reference:
|
[29] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications.Gordon and Breach, Amsterdam, 1987, Engl. Trans. from the Russian. MR 1347689 |
| Reference:
|
[30] Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media.Springer, Heidelberg; Higher Education Press, Beijing, 2010. MR 2796453 |
| Reference:
|
[31] Tenreiro Machado, J.A., Kiryakova, V.: The chronicles of fractional calculus.Fract. Calc. Appl. Anal. 20 (2017), 307–336. MR 3657873 |
| Reference:
|
[32] Zhou, Y.: Basic Theory of Fractional Differential Equations.World Scientific, Singapore, 2014. MR 3287248 |
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