Existence of solutions of impulsive boundary value problems for singular fractional differential systems

Yuji Liu

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 4, page 405-444
  • ISSN: 0862-7959

Abstract

top
A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point theorem: Leray-Schauder Nonlinear Alternative (Lemma 2.1). An example is given to illustrate the efficiency of the main theorems, see Example 5.1.

How to cite

top

Liu, Yuji. "Existence of solutions of impulsive boundary value problems for singular fractional differential systems." Mathematica Bohemica 142.4 (2017): 405-444. <http://eudml.org/doc/294874>.

@article{Liu2017,
abstract = {A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point theorem: Leray-Schauder Nonlinear Alternative (Lemma 2.1). An example is given to illustrate the efficiency of the main theorems, see Example 5.1.},
author = {Liu, Yuji},
journal = {Mathematica Bohemica},
keywords = {singular fractional differential system; impulsive boundary value problem; fixed point theorem},
language = {eng},
number = {4},
pages = {405-444},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of solutions of impulsive boundary value problems for singular fractional differential systems},
url = {http://eudml.org/doc/294874},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Liu, Yuji
TI - Existence of solutions of impulsive boundary value problems for singular fractional differential systems
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 4
SP - 405
EP - 444
AB - A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point theorem: Leray-Schauder Nonlinear Alternative (Lemma 2.1). An example is given to illustrate the efficiency of the main theorems, see Example 5.1.
LA - eng
KW - singular fractional differential system; impulsive boundary value problem; fixed point theorem
UR - http://eudml.org/doc/294874
ER -

References

top
  1. Arara, A., Benchohra, M., Hamidi, N., Nieto, J. J., 10.1016/j.na.2009.06.106, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 580-586. (2010) Zbl1179.26015MR2579326DOI10.1016/j.na.2009.06.106
  2. Belmekki, M., Nieto, J. J., Rodríguez-López, R., 10.1155/2009/324561, Bound. Value Probl. (electronic only) 2009 (2009), Article ID 324561, 18 pages. (2009) Zbl1181.34006MR2525590DOI10.1155/2009/324561
  3. Belmekki, M., Nieto, J. J., Rodríguez-López, R., 10.14232/ejqtde.2014.1.16, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2014 (2014), Article No. 16, 27 pages. (2014) Zbl06439060MR3199694DOI10.14232/ejqtde.2014.1.16
  4. Dehghani, R., Ghanbari, K., Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bull. Iran. Math. Soc. 33 (2007), 1-14. (2007) Zbl1148.34008MR2374531
  5. Karakostas, G. L., Positive solutions for the Φ -Laplacian when Φ is a sup-multiplicative- like function, Electron. J. Differ. Equ. (electronic only) 2004 (2004), Article No. 68, 12 pages. (2004) Zbl1057.34009MR2057655
  6. Kaufmann, E. R., Mboumi, E., 10.14232/ejqtde.2008.1.3, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2008 (2008), Article No. 3, 11 pages. (2008) Zbl1183.34007MR2369417DOI10.14232/ejqtde.2008.1.3
  7. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., 10.1016/S0304-0208(06)80001-0, North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). (2006) Zbl1092.45003MR2218073DOI10.1016/S0304-0208(06)80001-0
  8. Kilbas, A. A., Trujillo, J. J., 10.1080/00036810108840931, Appl. Anal. 78 (2001), 153-192. (2001) Zbl1031.34002MR1887959DOI10.1080/00036810108840931
  9. Lakshmikantham, V., Baĭnov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics 6. World Scientific, Singapore (1989). (1989) Zbl0719.34002MR1082551
  10. Liu, Y., Positive solutions for singular FDEs, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 73 (2011), 89-100. (2011) Zbl1240.34113MR2799434
  11. Liu, Y., 10.1515/tmj-2015-0003, Tbil. Math. J. (electronic only) 8 (2015), 1-22. (2015) Zbl06418062MR3323916DOI10.1515/tmj-2015-0003
  12. Mainardi, F., 10.1007/978-3-7091-2664-6_7, Fractals and Fractional Calculus in Continuum Mechanics, Udine, 1996 CISM Courses and Lectures 378. Springer, New York (1997), 291-348 A. Carpinteri et al. (1997) MR1611587DOI10.1007/978-3-7091-2664-6_7
  13. Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, Expository lectures held at Harvey Mudd College, Claremont, Calif., 1977 CBMS Regional Conference Series in Mathematics 40. American Mathematical Society, Providence (1979). (1979) Zbl0414.34025MR0525202
  14. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience Publication. John Wiley & Sons, New York (1993). (1993) Zbl0789.26002MR1219954
  15. Miller, K. S., Samko, S. G., 10.1080/10652460108819360, Integral Transforms Spec. Funct. 12 (2001), 389-402. (2001) Zbl1035.26012MR1872377DOI10.1080/10652460108819360
  16. Nahušev, A. M., The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Sov. Math., Dokl. 18 666-670 (1977), translation from Dokl. Akad. Nauk SSSR 234 1977 308-311. (1977) Zbl0376.34015MR0454145
  17. Nieto, J. J., 10.1016/j.aml.2010.06.007, Appl. Math. Lett. 23 (2010), 1248-1251. (2010) Zbl1202.34019MR2665605DOI10.1016/j.aml.2010.06.007
  18. Nieto, J. J., dx.doi.org/10.7153/fdc-01-05, Fract. Differ. Calc. 1 (2011), 99-104. (2011) MR2995577DOIdx.doi.org/10.7153/fdc-01-05
  19. Podlubny, I., Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications Mathematics in Science and Engineering 198. Academic Press, San Diego (1999). (1999) Zbl0924.34008MR1658022
  20. Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367-386 correction ibid 6 2003 109-110. (2002) Zbl1042.26003MR1967839
  21. Rida, S. Z., El-Sherbiny, H. M., Arafa, A. A. M., 10.1016/j.physleta.2007.06.071, Phys. Lett., A 372 (2008), 553-558. (2008) Zbl1217.81068MR2378723DOI10.1016/j.physleta.2007.06.071
  22. Su, X., 10.1016/j.aml.2008.03.001, Appl. Math. Lett. 22 (2009), 64-69. (2009) Zbl1163.34321MR2483163DOI10.1016/j.aml.2008.03.001
  23. Wang, X., Bai, C., 10.14232/ejqtde.2011.1.3, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 3, 15 pages. (2011) Zbl1207.35029MR2756028DOI10.14232/ejqtde.2011.1.3
  24. Wei, Z., Dong, W., 10.14232/ejqtde.2011.1.87, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 87, 13 pages. (2011) Zbl06528091MR2854026DOI10.14232/ejqtde.2011.1.87
  25. Wei, Z., Dong, W., Che, J., 10.1016/j.na.2010.07.003, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 3232-3238. (2010) Zbl1202.26017MR2680017DOI10.1016/j.na.2010.07.003
  26. Yuan, C., Multiple positive solutions for ( n - 1 , 1 ) -type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 13, 12 pages. (2011) Zbl1261.34014MR2771149
  27. Yuan, C., Jiang, D., O'Regan, D., Agarwal, R. P., 10.14232/ejqtde.2012.1.13, Electron. J. Qual. Theory Differ. Equ. (electronic only) 2012 (2012), Article No. 13, 17 pages. (2012) Zbl06476163MR2889755DOI10.14232/ejqtde.2012.1.13
  28. Zhang, S., Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ. (electronic only) 2006 (2006), Article No. 36, 12 pages. (2006) Zbl1096.34016MR2213580
  29. Zhao, Y., Sun, S., Han, Z., Zhang, M., 10.1016/j.amc.2011.01.103, Appl. Math. Comput. 217 (2011), 6950-6958. (2011) Zbl1227.34011MR2775685DOI10.1016/j.amc.2011.01.103

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.