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

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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 
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     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},
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     volume = {52},
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     mrnumber = {1940049},
     zbl = {1023.34054},
     language = {en},
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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/