Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2008)

  • Volume: 28, Issue: 1, page 83-93
  • ISSN: 1509-9407

Abstract

top
In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type ( D α ₙ - ∑ i = 1 n - 1 a i D α i ) x ( t ) ∈ F ( t , x ( φ ( t ) ) ) , a.e. on (0,1), I 1 - α ₙ x ( 0 ) = c , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

How to cite

top

Hussein A.H. Salem. "Set-valued fractional order differential equations in the space of summable functions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 28.1 (2008): 83-93. <http://eudml.org/doc/271184>.

@article{HusseinA2008,
abstract = {In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^\{αₙ\} - ∑_\{i=1\}^\{n-1\} a_i D^\{α_i\})x(t) ∈ F(t,x(φ(t)))$, a.e. on (0,1), $I^\{1 - αₙ\} x(0) = c$, αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.},
author = {Hussein A.H. Salem},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {fractional calculus; set-valued problem},
language = {eng},
number = {1},
pages = {83-93},
title = {Set-valued fractional order differential equations in the space of summable functions},
url = {http://eudml.org/doc/271184},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Hussein A.H. Salem
TI - Set-valued fractional order differential equations in the space of summable functions
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2008
VL - 28
IS - 1
SP - 83
EP - 93
AB - In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{αₙ} - ∑_{i=1}^{n-1} a_i D^{α_i})x(t) ∈ F(t,x(φ(t)))$, a.e. on (0,1), $I^{1 - αₙ} x(0) = c$, αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.
LA - eng
KW - fractional calculus; set-valued problem
UR - http://eudml.org/doc/271184
ER -

References

top
  1. [1] A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), 434-442. Zbl1027.34003
  2. [2] Z. Artstein and K. Prikry, Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl. 127 (1987), 540-547. Zbl0649.28011
  3. [3] M. Bassam, Some existence theorems on D.E. of generalized order, J. fur die Reine und Angewandte Mathematik 218 (1965), 70-78. 
  4. [Bc] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86. Zbl0677.54013
  5. [5] M. Cichoń, Multivalued perturbations of m-accretive differential inclusions in a non-separable Banach space, Commentationes Math. 32 (1992), 11-17. Zbl0770.34015
  6. [6] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. 
  7. [4] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. 302 (2005), 56-64. Zbl1064.34004
  8. [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. 
  9. [8] J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mathematyczne, PWN, Warsaw, 1982. 
  10. [9] A.M.A. El-Sayed and A.G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15-25. Zbl0830.34012
  11. [AAA] A.M.A. El-Sayed and A.G. Ibrahim, Definite integral of fractional order for set-valued functions, J. Frac. Calculus 11 (1996), 15-25. 
  12. [11] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary order, Nonlinear Anal. 33 (2) (1998), 181-186. Zbl0934.34055
  13. [12] A.M.A. El-Sayed and A.G. Ibrahim, Set-valued integral equations of arbitrary (fractional) order, Appl. Math. Comput. 118 (2001), 113-121. Zbl1024.45003
  14. [14] V.G. Gutev, Selection theorems under an assumption weaker than lower semi-continuity, Topol. Appl. 50 (1993), 129-138. Zbl0809.54015
  15. [15] S.B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Frac. Calculus 7 (1995), 101-105. Zbl0839.34003
  16. [kil] A.A. Kilbas and J.J. Trujillo, Differential equations of fractional order: methods, results and problems, I, Appl. Anal. 78 (2001), 153-192. Zbl1031.34002
  17. [kst] A.A. Kilbas, H.M. Srivastava and J.J Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, 2006. Zbl1092.45003
  18. [16] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. Zbl0789.26002
  19. [17] K. Przesławski and L.E. Rybiński, Michael selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109 (1990), 537-543. Zbl0706.54018
  20. [18] D. Repovs and P.V. Semenov, Continuous Selection of Multivalued Mappings, Kluwer Academic Press, 1998. Zbl0915.54001
  21. [19] S. Samko, A. Kilbas and O. Marichev, Fraction Integrals and Drivatives, Gordon and Breach Science Publisher, 1993. Zbl0818.26003
  22. [sz] J.-P. Sun and Y.-H. Zhao, Multiplicity of positive solutions of a class of nonlinear fractional differential equations, J. Computer and Math. Appl. 49 (2005), 73-80. Zbl1085.34501
  23. [20] C. Swartz, Measure, Integration and Function Spaces, World Scientific, Singapore, 1994. 

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.