Geodesic
New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations
Mathematica Bohemica, Tome 127 (2002) no. 4, pp. 509-524

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The nonimprovable sufficient conditions for the unique solvability of the problem \[ u^{\prime }(t)=\ell (u)(t)+q(t),\qquad u(a)=c, \] where $\ell \: C(I;\mathbb{R})\rightarrow L(I;\mathbb{R})$ is a linear bounded operator, $q\in L(I;\mathbb{R})$, $c\in \mathbb{R}$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell $ is not of Volterra’s type with respect to the point $a$.
The nonimprovable sufficient conditions for the unique solvability of the problem \[ u^{\prime }(t)=\ell (u)(t)+q(t),\qquad u(a)=c, \] where $\ell \: C(I;\mathbb{R})\rightarrow L(I;\mathbb{R})$ is a linear bounded operator, $q\in L(I;\mathbb{R})$, $c\in \mathbb{R}$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell $ is not of Volterra’s type with respect to the point $a$.
DOI : 10.21136/MB.2002.133950
Classification : 34K05, 34K06, 34K10, 65L05
Keywords: linear functional differential equations; differential equations with deviating arguments; initial value problems 
Hakl, R.; Lomtatidze, A.; Půža, B. New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. Mathematica Bohemica, Tome 127 (2002) no. 4, pp. 509-524. doi: 10.21136/MB.2002.133950
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[1] Azbelev, N. V.; Maksimov, V. P.; Rakhmatullina, L. F.: Introduction to the Theory of Functional Differential Equations. Nauka, Moskva, 1991. (Russian) | MR

[2] Bravyi, E.; Hakl, R.; Lomtatidze, A.: Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. Czechoslovak Math. J (to appear). | MR

[3] Bravyi, E.: A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differential Equations Math. Phys. 20 (2000), 133–135. | MR | Zbl

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

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

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

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