existence of a periodic solution for some partial functional differential equations with infinite delay

Journal of Mathematical Analysis and Applications 256, 257᎐280 Ž2001 doi:10.1006rjmaa.2000.7321, available online at http:rrwww.idealibrary.com on Existence of a Periodic Solution for Some Partial Functional Differential Equations with Infinite Delay Rachid Benkhalti Department of Mathematics, Pacific Lutheran Uni¨ ersity, Tacoma, Washington 98447 and Hassane Bouzahir and Khalil Ezzinbi Departement de Mathematiques, Uni¨ ersite´ Cadi Ayyad, Faculte´ des Sciences Semlalia, ´ ´ B.P 2390, Marrakech, 40000 Morocco Submitted by J Henderson Received April 17, 2000 This paper deals with the existence of periodic solutions for some partial functional differential equations with infinite delay We suppose that the linear part is nondensely defined and satisfies the Hille᎐Yosida condition In the nonlinear case we give several criteria to ensure the existence of a periodic solution In the nonhomogeneous linear case, we prove the existence of a periodic solution under the existence of a bounded solution ᮊ 2001 Academic Press INTRODUCTION The purpose of this work is to discuss the existence of a periodic solution of the following partial functional differential equation with infinite delay ¡d ¢x s ␾ g B , ~ dt x Ž t s Ax Ž t q F Ž t , x , t t G 0, Ž 1 This work is supported by Moroccan Grant PARS MI 36 257 0022-247Xr01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved 258 BENKHALTI, BOUZAHIR, AND EZZINBI where A is a nondensely defined linear operator on a Banach space Ž E, < [...]... 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Kellermann and H Hieber, Integrated semigroup, J Funct Anal 15 Ž1989., 160᎐180 15 S Murakami, Stability for functional differential equations with infinite delay in Banach space, preprint 16 R D Nussbaum, The radius of essential spectrum, Duke Math J 37 Ž1970., 473᎐478 17 B N Sadovskii, On a fixed point principle, Funct Anal Appl 1 Ž1967., 74᎐76 18 J S Shin, Comparison theorems and uniqueness of mild solutions... and a uniform fading memory space; that means that B satisfies the extra axioms ŽC and ŽD2 5 S0 Ž t 5 ª 0 as t ª qϱ We will show the existence of a periodic solution without condition Ž13 Before doing so let us recall the following results Let H : Z ª Z be a closed linear operator with a dense domain DŽ H in a Banach space Z We denote by ␴ Ž H the spectrum of H We define the essential spectrum of. .. , f be the bounded solution Then Ž P␻n␾ nG 0 s Ž yn ␻ Ž , ␾ , f nG 0 is bounded Hence, by Theorem 7, the map P␻ has a fixed point in B, which gives an ␻ -periodic solution of Eq Ž11 FUNCTIONAL DIFFERENTIAL EQUATIONS 277 5 APPLICATION In order to apply the abstract result of the previous section, we consider the following partial functional differential equation with infinite delay ¡Ѩ Ѩt wŽ t, ␰ s ... and remarks led to several improvements in the exposition 280 BENKHALTI, BOUZAHIR, AND EZZINBI REFERENCES M Adimy, H Bouzahir, and K Ezzinbi, Existence for a class of partial functional differential... lence classes B For a deeper discussion on the phase space B, we refer to w12x 260 BENKHALTI, BOUZAHIR, AND EZZINBI DEFINITION We say that a function x: Žyϱ, ax ª E, a ) 0, is an integral solution... y Ž t s T0 Ž t q ␻ ␾ Ž q lim tq ␻ H ␭ªqϱ T0 Ž t q ␻ y s B␭ F Ž s, x s ds, 262 BENKHALTI, BOUZAHIR, AND EZZINBI which implies that y Ž t s T0 Ž t T0 Ž ␻ ␾ Ž q lim ␻ H ␭ªqϱ q lim tq ␻