Geodesic
On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 673-690 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

On the segment $I=[a,b]$ consider the problem \[ u^{\prime }(t)=f(u)(t) , \quad u(a)=c, \] where $f\:C(I,\mathbb{R})\rightarrow L(I,\mathbb{R})$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in \mathbb{R}$. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.
On the segment $I=[a,b]$ consider the problem \[ u^{\prime }(t)=f(u)(t) , \quad u(a)=c, \] where $f\:C(I,\mathbb{R})\rightarrow L(I,\mathbb{R})$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in \mathbb{R}$. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.
Classification : 34K05, 34K10, 34K99
Keywords: nonlinear functional differential equation; initial value problem; non–Volterra’s type operator 
@article{CMJ_2002_52_4_a0,
     author = {Bravyi, E. and Hakl, R. and Lomtatidze, A.},
     title = {On {Cauchy} problem for first order nonlinear functional differential equations of {non-Volterra{\textquoteright}s} type},
     journal = {Czechoslovak Mathematical Journal},
     pages = {673--690},
     year = {2002},
     volume = {52},
     number = {4},
     mrnumber = {1940049},
     zbl = {1023.34054},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a0/}
}
TY  - JOUR
AU  - Bravyi, E.
AU  - Hakl, R.
AU  - Lomtatidze, A.
TI  - On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type
JO  - Czechoslovak Mathematical Journal
PY  - 2002
SP  - 673
EP  - 690
VL  - 52
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a0/
LA  - en
ID  - CMJ_2002_52_4_a0
ER  - 
%0 Journal Article
%A Bravyi, E.
%A Hakl, R.
%A Lomtatidze, A.
%T On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type
%J Czechoslovak Mathematical Journal
%D 2002
%P 673-690
%V 52
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a0/
%G en
%F CMJ_2002_52_4_a0
Bravyi, E.; Hakl, R.; Lomtatidze, A. On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 673-690. http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a0/

[1] N. V. Azbelev, V. P. Maksimov and L. F.  Rakhmatullina: Introduction to the Theory of Functional Differential Equations. Nauka, Moscow, 1991. (Russian) | MR

[2] S. R.  Bernfeld and V.  Lakshmikantham: An Introduction to Nonlinear Boundary Value Problems. Academic Press Inc., New York and London, 1974. | MR

[3] J.  Blaz: Sur l’existence et l’unicité de la solution d’une equation differentielle á argument retardé. Ann. Polon. Math. 15 (1964), 9–14. | DOI | MR | Zbl

[4] E.  Bravyi, R.  Hakl and A.  Lomtatidze: Optimal conditions on unique solvability of the Cauchy problem for the first order linear functional differential equations. Czechoslovak Math. J 52(127) (2002), 513–530. | DOI | MR

[5] R. D.  Driver: Existence theory for a delay-differential system. Contrib. Diff. Equations 1 (1963), 317–336. | MR | Zbl

[6] J.  Hale: Theory of Functional Differential Equations. Springer-Verlag, New York-Heidelberg-Berlin, 1977. | MR | Zbl

[7] Sh.  Gelashvili and I.  Kiguradze: On multi-point boundary value problems for systems of functional differential and difference equations. Mem. Differential Equations Math. Phys. 5 (1995), 1–113. | MR

[8] I.  Kiguradze and B.  Půža: On boundary value problems for systems of linear functional differential equations. Czechoslovak Math. J. 47(122) (1997), 341–373. | DOI | MR

[9] I.  Kiguradze and B.  Půža: On boundary value problems for functional differential equations. Mem. Differential Equations Math. Phys. 12 (1997), 106–113. | MR

[10] I.  Kiguradze and Z.  Sokhadze: Concerning the uniqueness of solution of the Cauchy problem for functional differential equations. Differentsial’nye Uravneniya 31 (1995), 1977–1988. (Russian) | MR

[11] I.  Kiguradze and Z.  Sokhadze: Existence and continuability of solutions of the initial value problem for the system of singular functional differential equations. Mem. Differential Equations Math. Phys. 5 (1995), 127–130.

[12] I.  Kiguradze and Z.  Sokhadze: On the Cauchy problem for singular evolution functional differential equations. Differentsial’nye Uravneniya 33 (1997), 48–59. (Russian) | MR

[13] I.  Kiguradze and Z.  Sokhadze: On singular functional differential inequalities. Georgian Math. J. 4 (1997), 259–278. | DOI | MR

[14] I.  Kiguradze and Z.  Sokhadze: On global solvability of the Cauchy problem for singular functional differential equations. Georgian Math. J. 4 (1997), 355–372. | DOI | MR

[15] I.  Kiguradze and Z.  Sokhadze: On the structure of the set of solutions of the weighted Cauchy problem for evolution singular functional differential equations. Fasc. Math. (1998), 71–92. | MR

[16] V.  Lakshmikantham: Lyapunov function and a basic inequality in delay-differential equations. Arch. Rational Mech. Anal. 10 (1962), 305–310. | DOI | MR | Zbl

[17] A. I.  Logunov and Z. B.  Tsalyuk: On the uniqueness of solution of Volterra type integral equations with retarded argument. Mat. Sb. 67 (1965), 303–309. (Russian) | MR

[18] W. L.  Miranker: Existence, uniqueness and stability of solutions of systems of nonlinear difference-differential equations. J. Math. Mech. 11 (1962), 101–107. | MR | Zbl

[19] A. D.  Myshkis: General theory of differential equations with retarded argument. Uspekhi Mat. Nauk 4 (1949), 99–141. (Russian) | MR

[20] A. D.  Myshkis and L. E.  Elsgolts: State and problems of theory of differential equations with deviated argument. Uspekhi Mat. Nauk 22 (1967), 21–57. (Russian)

[21] A. D.  Myshkis and Z. B.  Tsalyuk: On nonlocal continuability of solutions to differential equaitons with retarded argument. Differentsial’nye Uravneniya 5 (1969), 1128–1130. (Russian) | MR

[22] W. Rzymowski: Delay effects on the existence problems for differential equations in Banach space. J.  Differential Equations 32 (1979), 91–100. | DOI | MR | Zbl

[23] Š.  Schwabik, M.  Tvrdý and O.  Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha, 1979. | MR

[24] Z.  Sokhadze: On a theorem of Myshkis-Tsalyuk. Mem. Differential Equations Math. Phys. 5 (1995), 131–132. | Zbl