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Periodic solutions in fading memory spaces

Pages: 250 - 257, Issue Special, August 2005

 Abstract        Full Text (201.7K)              

Khalil Ezzinbi - Department de Mathematiques, Faculte des Sciences Semlalia, B.P. 2390, Marrakech, Morocco, Morocco (email)
James H. Liu - Department of Mathematics, James Madison University, Harrisonburg, VA 22807, United States (email)
Nguyen Van Minh - Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, United States (email)

Abstract: For $A(t)$ and $f(t,x,y)$ $T$-periodic in $t$, consider the following evolution equation with infinite delay in a general Banach space $X$, $$u^\prime (t)+ A(t)u(t)=f(t,u(t),u_t),\;\; t> 0,\;\;u(s) =\phi (s),\;\;s \leq 0, $$ where the resolvent of the unbounded operator $A(t)$ is compact, and $u_t (s)=u(t+s),\; s\leq 0$. We will work with general fading memory phase spaces satisfying certain axioms, and derive periodic solutions. We will show that the related Poincar\'{e} operator is condensing, and then derive periodic solutions using the boundedness of the solutions and some fixed point theorems. This way, the study of periodic solutions for equations with infinite delay in general Banach spaces can be carried to fading memory phase spaces. In doing so, we will improve a condition of [4] and extend the results of [7,8].

Keywords:  Infinite delay, fading memory phase space, bounded and periodic solutions, condensing operators, Hale and Lunel¡¯s fixed point theorem.
Mathematics Subject Classification:  34G

Received: September 2004;      Revised: March 2005;      Published: September 2005.