# Article

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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
Summary:
The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense ${\rm d} x(t)={\rm d} A_0(t)\cdot x(t)+{\rm d} f_0(t)$, $x(t_{0})=\nobreak c_0$ $(t\in I)$ 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 $x_k(t)\to x_0(t)$ 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.
References:
[1] Ashordia, M.: Lyapunov stability of systems of linear generalized ordinary differential equations. Comput. Math. Appl. 50 (2005), 957-982. DOI 10.1016/j.camwa.2004.04.041 | MR 2165650 | Zbl 1090.34043
[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. MR 2196660 | Zbl 1098.34010
[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. MR 1408294 | Zbl 0879.34037
[4] Ashordia, M.: On the correctness of nonlinear boundary value problems for systems of generalized ordinary differential equations. Georgian Math. J. 3 (1996), 501-524. DOI 10.1007/BF02259778 | MR 1419831 | Zbl 0876.34021
[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. MR 1415807
[6] Ashordia, M.: On the correctness of linear boundary value problems for systems of generalized ordinary differential equations. Proc. Georgian Acad. Sci. Math. 1 (1993), 385-394 this paper also appears in Georgian Math. J. 1 (1994), 343-351. DOI 10.1007/BF02307443 | MR 1262572 | Zbl 0808.34015
[7] Ashordia, M.: On the stability of solutions of linear boundary value problems for the system of ordinary differential equations. Proc. Georgian Acad. Sci. Math. 1 (1993), 129-141 this paper also appears in Georgian Math. J. 1 (1994), 115-126. DOI 10.1007/BF02254726 | MR 1251497
[8] Ashordiya, M. T.: Well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations on an infinite interval. Differ. Equ. 40 477-490 (2004), translation from Differ. Uravn. 40 443-454 (2004). DOI 10.1023/B:DIEQ.0000035786.69859.d8 | MR 2153643 | Zbl 1087.34002
[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). MR 1373033 | Zbl 0853.34018
[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. MR 2218566 | Zbl 1092.34003
[11] Hildebrandt, T. H.: On systems of linear differentio-Stieltjes-integral equations. Ill. J. Math. 3 (1959), 352-373. DOI 10.1215/ijm/1255455257 | MR 0105600 | Zbl 0088.31101
[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. MR 1484729
[13] Kiguradze, I. T.: Boundary-value problems for systems of ordinary differential equations. J. Sov. Math. 43 (1988), 2259-2339 translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30 3-103 (1987). DOI 10.1007/BF01100360 | MR 0925829 | Zbl 0631.34020
[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). MR 0071596
[15] Kurzweil, J.: Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions. Series in Real Analysis 11 World Scientific Publishing, Hackensack (2012). MR 2906899 | Zbl 1248.34001
[16] Kurzweil, J.: Generalized ordinary differential equations. Czech. Math. J. 8 (83) (1958), 360-388. MR 0111878 | Zbl 0102.07003
[17] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (82) (1957), 418-449. MR 0111875 | Zbl 0090.30002
[18] Kurzweil, J., Vorel, Z.: Continuous dependence of solutions of differential equations on a parameter. Czech. Math. J. 7 (82) 568-583 (1957), Russian. MR 0111874 | Zbl 0090.30001
[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. MR 3352635 | Zbl 1303.45001
[20] Opial, Z.: Linear problems for systems of nonlinear differential equations. J. Differ. Equations 3 (1967), 580-594. DOI 10.1016/0022-0396(67)90018-6 | MR 0216068 | Zbl 0161.06102
[21] Schwabik, Š.: Generalized Ordinary Differential Equations. Series in Real Analysis 5 World Scientific Publishing, Singapore (1992). MR 1200241 | Zbl 0781.34003
[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). MR 0542283 | Zbl 0417.45001
[23] Tvrd{ý}, M.: Differential and integral equations in the space of regulated functions. Mem. Differ. Equ. Math. Phys. 25 (2002), 1-104. MR 1903190 | Zbl 1081.34504

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