Fractional q -difference equations on the half line

Saïd Abbas; Mouffak Benchohra; Nadjet Laledj; Yong Zhou

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 4, page 207-223
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Abbas, Saïd, et al. "Fractional ${q}$-difference equations on the half line." Archivum Mathematicum 056.4 (2020): 207-223. <http://eudml.org/doc/297403>.

@article{Abbas2020,
abstract = {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.},
author = {Abbas, Saïd, Benchohra, Mouffak, Laledj, Nadjet, Zhou, Yong},
journal = {Archivum Mathematicum},
keywords = {fractional $q$-difference equation; attractivity; diagonalization; bounded solution; Banach space; Fréchet space; fixed point},
language = {eng},
number = {4},
pages = {207-223},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Fractional $\{q\}$-difference equations on the half line},
url = {http://eudml.org/doc/297403},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Abbas, Saïd
AU - Benchohra, Mouffak
AU - Laledj, Nadjet
AU - Zhou, Yong
TI - Fractional ${q}$-difference equations on the half line
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 4
SP - 207
EP - 223
AB - 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.
LA - eng
KW - fractional $q$-difference equation; attractivity; diagonalization; bounded solution; Banach space; Fréchet space; fixed point
UR - http://eudml.org/doc/297403
ER -

References

top
  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. (2012) MR2977716
  2. Abbas, S., Benchohra, M., 10.7151/dmdico.1141, Discuss. Math. Differ. Incl. Control Optim. 33 (1) (2013), 1–17. (2013) MR3136582DOI10.7151/dmdico.1141
  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. (2013) MR3161669
  4. Abbas, S., Benchohra, M., Graef, J.R., Henderson, J., Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018. (2018) MR3791511
  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. (2013) MR3058403
  6. Abbas, S., Benchohra, M., N’Guérékata, G.M., Topics in Fractional Differential Equations, Springer, New York, 2012. (2012) MR2962045
  7. Abbas, S., Benchohra, M., N’Guérékata, G.M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. (2015) Zbl1314.34002MR3309582
  8. Adams, C.R., 10.2307/1968274, Annals Math. 30 (1928), 195–205. (1928) MR1502876DOI10.2307/1968274
  9. Agarwal, R., Certain fractional q - integrals and q -derivatives, Proc. Cambridge Philos. Soc. 66 (1969), 365–370. (1969) MR0247389
  10. Ahmad, B., Boundary value problem for nonlinear third order q -difference equations, Electron. J. Differential Equations 2011 (94) (2011), 1–7. (2011) MR2832270
  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. (2012) MR3016054
  12. Almezel, S., Ansari, Q.H., Khamsi, M.A., Topics in Fixed Point Theory, Springer-Verlag, New York, 2014. (2014) MR3411798
  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. (2012) MR2955875
  14. Bothe, D., 10.1007/BF02783044, Israel J. Math. 108 (1998), 109–138. (1998) MR1669396DOI10.1007/BF02783044
  15. Carmichael, R.D., 10.2307/2369887, American J. Math. 34 (1912), 147–168. (1912) MR1506145DOI10.2307/2369887
  16. Corduneanu, C., Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. (1973) Zbl0273.45001MR0358245
  17. Dudek, S., 10.2298/AADM1702340D, Appl. Anal. Discrete Math. 11 (2017), 340–357. (2017) MR3719830DOI10.2298/AADM1702340D
  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. (2015) MR3319202
  19. El-Shahed, M., Hassa, H.A., 10.1090/S0002-9939-09-10185-5, Proc. Amer. Math. Soc. 138 (2010), 1733–1738. (2010) MR2587458DOI10.1090/S0002-9939-09-10185-5
  20. Etemad, S., Ntouyas, S.K., Ahmad, B., 10.3390/math7080659, Mathematics 7 (2019), 1–15. (2019) DOI10.3390/math7080659
  21. Granas, A., Dugundji, J., Fixed Point Theory, Springer-Verlag, New York, 2003. (2003) Zbl1025.47002MR1987179
  22. Kac, V., Cheung, P., Quantum Calculus, Springer, New York, 2002. (2002) Zbl0986.05001MR1865777
  23. Kilbas, A.A., Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (6) (2001), 1191–1204. (2001) MR1858760
  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. (2006) Zbl1092.45003MR2218073
  25. Mönch, H., 10.1016/0362-546X(80)90010-3, Nonlinear Anal. 4 (1980), 985–999. (1980) MR0586861DOI10.1016/0362-546X(80)90010-3
  26. Olszowy, L., 10.1016/j.na.2012.11.001, Nonlinear Anal. 81 (2013), 211–223. (2013) MR3016450DOI10.1016/j.na.2012.11.001
  27. Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S., 10.2298/AADM0701311R, Appl. Anal. Discrete Math. 1 (2007), 311–323. (2007) MR2316607DOI10.2298/AADM0701311R
  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. (2007) MR2378985
  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. (1987) MR1347689
  30. Tarasov, V.E., Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010. (2010) MR2796453
  31. Tenreiro Machado, J.A., Kiryakova, V., The chronicles of fractional calculus, Fract. Calc. Appl. Anal. 20 (2017), 307–336. (2017) MR3657873
  32. Zhou, Y., Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. (2014) MR3287248

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.