On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 2, page 183-215
  • ISSN: 0862-7959

Abstract

top
The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense d x ( t ) = d A 0 ( t ) · x ( t ) + d f 0 ( t ) , x ( t 0 ) = c 0 ( t I ) with a unique solution x 0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems d x ( t ) = d A k ( t ) · x ( t ) + d f k ( t ) , x ( t k ) = c k ( k = 1 , 2 , ) to have a unique solution x k for any sufficiently large k such that x k ( t ) x 0 ( t ) uniformly on I . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.

How to cite

top

Ashordia, Malkhaz. "On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations." Mathematica Bohemica 141.2 (2016): 183-215. <http://eudml.org/doc/276997>.

@article{Ashordia2016,
abstract = {The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\{\rm d\} x(t)=\{\rm d\} A_0(t)\cdot x(t)+\{\rm d\} f_0(t)$, $x(t_\{0\})=c_0$$(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\{\rm d\} x(t)=\{\rm d\} A_k(t)\cdot x(t)+\{\rm d\} f_k(t)$, $x(t_\{k\})=c_k$$(k=1,2,\dots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k(t)\rightarrow x_0(t)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.},
author = {Ashordia, Malkhaz},
journal = {Mathematica Bohemica},
keywords = {linear system of generalized ordinary differential equations in the Kurzweil sense; Cauchy problem; well-posedness; Opial type necessary condition; Opial type sufficient condition; efficient sufficient condition},
language = {eng},
number = {2},
pages = {183-215},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations},
url = {http://eudml.org/doc/276997},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Ashordia, Malkhaz
TI - On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 183
EP - 215
AB - The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense ${\rm d} x(t)={\rm d} A_0(t)\cdot x(t)+{\rm d} f_0(t)$, $x(t_{0})=c_0$$(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems ${\rm d} x(t)={\rm d} A_k(t)\cdot x(t)+{\rm d} f_k(t)$, $x(t_{k})=c_k$$(k=1,2,\dots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k(t)\rightarrow x_0(t)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.
LA - eng
KW - linear system of generalized ordinary differential equations in the Kurzweil sense; Cauchy problem; well-posedness; Opial type necessary condition; Opial type sufficient condition; efficient sufficient condition
UR - http://eudml.org/doc/276997
ER -

References

top
  1. Ashordia, M., 10.1016/j.camwa.2004.04.041, Comput. Math. Appl. 50 (2005), 957-982. (2005) Zbl1090.34043MR2165650DOI10.1016/j.camwa.2004.04.041
  2. Ashordia, M., On the general and multipoint boundary value problems for linear systems of generalized ordinary differential equations, linear impulse and linear difference systems, Mem. Differ. Equ. Math. Phys. 36 (2005), 1-80. (2005) Zbl1098.34010MR2196660
  3. Ashordia, M., Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations, Czech. Math. J. 46(121) (1996), 385-404. (1996) Zbl0879.34037MR1408294
  4. Ashordia, M., 10.1007/BF02259778, Georgian Math. J. 3 (1996), 501-524. (1996) Zbl0876.34021MR1419831DOI10.1007/BF02259778
  5. Ashordia, M., On the stability of solutions of the multipoint boundary value problem for the system of generalized ordinary differential equations, Mem. Differ. Equ. Math. Phys. 6 (1995), 1-57. (1995) MR1415807
  6. Ashordia, M., 10.1007/BF02307443, Proc. Georgian Acad. Sci. Math. 1 (1993), 385-394 this paper also appears in Georgian Math. J. 1 (1994), 343-351. (1994) Zbl0808.34015MR1262572DOI10.1007/BF02307443
  7. Ashordia, M., 10.1007/BF02254726, Proc. Georgian Acad. Sci. Math. 1 (1993), 129-141 this paper also appears in Georgian Math. J. 1 (1994), 115-126. (1994) MR1251497DOI10.1007/BF02254726
  8. Ashordiya, M. T., 10.1023/B:DIEQ.0000035786.69859.d8, Differ. Equ. 40 477-490 (2004), translation from Differ. Uravn. 40 443-454 (2004). (2004) Zbl1087.34002MR2153643DOI10.1023/B:DIEQ.0000035786.69859.d8
  9. Ashordiya, M. T., Well-posedness of the Cauchy-Nicoletti boundary value problem for systems of nonlinear generalized ordinary differential equations, Differ. Equations 31 (1995), 352-362 translation from Differ. Uravn. 31 382-392 (1995). (1995) Zbl0853.34018MR1373033
  10. Halas, Z., Continuous dependence of inverse fundamental matrices of generalized linear ordinary differential equations on a parameter, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 44 (2005), 39-48. (2005) Zbl1092.34003MR2218566
  11. Hildebrandt, T. H., On systems of linear differentio-Stieltjes-integral equations, Ill. J. Math. 3 (1959), 352-373. (1959) Zbl0088.31101MR0105600
  12. Kiguradze, I., The Initial Value Problem and Boundary Value Problems for Systems of Ordinary Differential Equations. Vol. 1. Linear Theory, Metsniereba, Tbilisi (1997), Russian. (1997) MR1484729
  13. Kiguradze, I. T., 10.1007/BF01100360, J. Sov. Math. 43 (1988), 2259-2339 translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30 3-103 (1987). (1987) Zbl0631.34020MR0925829DOI10.1007/BF01100360
  14. Krasnosel'skiĭ, M. A., Kreĭn, S. G., On the principle of averaging in nonlinear mechanics, Uspekhi Mat. Nauk (N.S.) 10 147-152 Russian (1955). (1955) MR0071596
  15. Kurzweil, J., Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions, Series in Real Analysis 11 World Scientific Publishing, Hackensack (2012). (2012) Zbl1248.34001MR2906899
  16. Kurzweil, J., Generalized ordinary differential equations, Czech. Math. J. 8 (83) (1958), 360-388. (1958) Zbl0102.07003MR0111878
  17. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (82) (1957), 418-449. (1957) Zbl0090.30002MR0111875
  18. Kurzweil, J., Vorel, Z., Continuous dependence of solutions of differential equations on a parameter, Czech. Math. J. 7 (82) 568-583 (1957), Russian. (1957) Zbl0090.30001MR0111874
  19. Monteiro, G. A., Tvrdý, M., Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight, Bound. Value Probl. (electronic only) 2014 (2014), Article ID 71, 18 pages. (2014) Zbl1303.45001MR3352635
  20. Opial, Z., 10.1016/0022-0396(67)90018-6, J. Differ. Equations 3 (1967), 580-594. (1967) Zbl0161.06102MR0216068DOI10.1016/0022-0396(67)90018-6
  21. Schwabik, Š., Generalized Ordinary Differential Equations, Series in Real Analysis 5 World Scientific Publishing, Singapore (1992). (1992) Zbl0781.34003MR1200241
  22. Schwabik, {Š}., Tvrdý, M., Vejvoda, O., Differential and Integral Equations. Boundary Value Problems and Adjoints, Czechoslovak Academy of Sciences Reidel Publishing, Dordrecht-Boston (1979). (1979) Zbl0417.45001MR0542283
  23. Tvrd{ý}, M., Differential and integral equations in the space of regulated functions, Mem. Differ. Equ. Math. Phys. 25 (2002), 1-104. (2002) Zbl1081.34504MR1903190

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.