Previous |  Up |  Next

Article

Keywords:
nonlinear functional differential equation; initial value problem; non–Volterra’s type operator
Summary:
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.
References:
[1] N. V. Azbelev, V. P. Maksimov and L. F.  Rakhmatullina: Introduction to the Theory of Functional Differential Equations. Nauka, Moscow, 1991. (Russian) MR 1144998
[2] S. R.  Bernfeld and V.  Lakshmikantham: An Introduction to Nonlinear Boundary Value Problems. Academic Press Inc., New York and London, 1974. MR 0445048
[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. MR 0166459 | Zbl 0129.07702
[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 10.1023/A:1021767411094 | MR 1923257
[5] R. D.  Driver: Existence theory for a delay-differential system. Contrib. Diff. Equations 1 (1963), 317–336. MR 0150421 | Zbl 0126.10102
[6] J.  Hale: Theory of Functional Differential Equations. Springer-Verlag, New York-Heidelberg-Berlin, 1977. MR 0508721 | Zbl 0352.34001
[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 1415806
[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 10.1023/A:1022829931363 | MR 1452425
[9] I.  Kiguradze and B.  Půža: On boundary value problems for functional differential equations. Mem. Differential Equations Math. Phys. 12 (1997), 106–113. MR 1636865
[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 1431622
[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 1607273
[13] I.  Kiguradze and Z.  Sokhadze: On singular functional differential inequalities. Georgian Math. J. 4 (1997), 259–278. DOI 10.1023/A:1022901729928 | MR 1443538
[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 10.1023/A:1022994513010 | MR 1457927
[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 1643553
[16] V.  Lakshmikantham: Lyapunov function and a basic inequality in delay-differential equations. Arch. Rational Mech. Anal. 10 (1962), 305–310. DOI 10.1007/BF00281197 | MR 0144044 | Zbl 0109.31203
[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 0184048
[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 0140787 | Zbl 0114.04201
[19] A. D.  Myshkis: General theory of differential equations with retarded argument. Uspekhi Mat. Nauk 4 (1949), 99–141. (Russian) MR 0032913
[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 0248426
[22] W. Rzymowski: Delay effects on the existence problems for differential equations in Banach space. J.  Differential Equations 32 (1979), 91–100. DOI 10.1016/0022-0396(79)90053-6 | MR 0532765 | Zbl 0423.34090
[23] Š.  Schwabik, M.  Tvrdý and O.  Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha, 1979. MR 0542283
[24] Z.  Sokhadze: On a theorem of Myshkis-Tsalyuk. Mem. Differential Equations Math. Phys. 5 (1995), 131–132. Zbl 0866.34053
Partner of
EuDML logo