A local existence theorem for a class of delay differential equations

Ioan I. Vrabie

DOI: http://dx.doi.org/10.12775/TMNA.2016.023


The goal of this paper is to show that some classes of partial differential functional equations admit a natural formulation as ordinary functional differential equations in infinite dimensional Banach spaces. Moreover, the equations thus obtained are driven by continuous right-hand sides satisfying the compactness assumptions required by the infinite-dimensional version of a Peano-like existence theorem. Two applications, one to a semilinear wave equation with delay and another one to a pseudoparabolic PDE in Mechanics, are included.


Delay differential equations; local existence; metric fixed point arguments; topological fixed point arguments; semilinear wave equation; pseudoparabolic equation

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R.A. Adams, Sobolev spaces, Academic Press, Boston San Diego New York London Sidney Tokyo Toronto, 1978.

R.I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl. 82 (1981), 33–48.

H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations 24 (1977), 412–425.

M.D. Burlică and D. Roşu, A class of nonlinear delay evolution equations with nonlocal initial conditions, Proc. Amer. Math. Soc. 142 (2014), 2445–2458.

O. Cârjă, M. Necula and I.I. Vrabie, Viability, Invariance and Applications, Elsevier Horth-Holland Mathematics Studies 207, 2007.

R.D. Driver, Ordinary and delay differential equations, Appl. Math. Sci. 20, Springer Verlag, New York Hedelberg Berlin, (1977).

N. Dunford and J.T. Schwartz, Linear Operators Part I: General Theory, Interscience Publishers, Inc. New York, 1958.

M. Frigon and D. O’Reagan, Existence results for initial value problems in Banach spaces, Differ. Equ. Dyn. Syst. 2(1994), 41-48.

A. Halanay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York and London 1966.

J. Hale, Functional differential equations, Applied Mathematical Sciences 3, Springer Verlag, 1971.

M. A. Krasnosel’skiĭ, Two remarks on the method of successive approximations, Uspehi Mat. Nauk 10 (1955), 123–127 (Russian).

E. Mitidieri and I.I. Vrabie, Existence for nonlinear functional differential equations, Hiroshima Math. J. 17 (1987), 627–649.

E. Mitidieri and I.I. Vrabie, A class of strongly nonlinear functional differential equations, Ann. Mat. Pura Appl. (4) CLI (1988), 125–147.

J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.

I.I. Vrabie, $C_0$-semigroups and applications, North-Holland Publishing Co. Amsterdam, 2003.


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