Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 06, pp. 1-6.
Title: Asymptotically periodic solutions for differential and
difference inclusions in Hilbert spaces
Authors: Gheorghe Morosanu (Central European Univ., Budapest, Hungary)
Figen Ozpinar (Afyon Kocatepe Univ., Turkey)
Abstract:
Let $H$ be a real Hilbert space and let $A:D(A)\subset H\to H$ be a
(possibly set-valued) maximal monotone operator. We
investigate the existence of asymptotically periodic solutions to
the differential equation (inclusion)
$u'(t) + Au(t)\ni f(t) + g(t)$, $t>0$, where
$f \in L_{\rm loc}^2(\mathbb{R}_+,H)$ is a $T$-periodic
function ($T>0$) and $g \in L^1(\mathbb{R}_+,H)$. Consider also the
following difference inclusion (which is a discrete analogue of the
above inclusion): $ \Delta u_n + c_n A u_{n+1}\ni f_n + g_n
, \ n=0,1, \dots$, where $(c_n)\subset (0,+\infty)$,
$(f_n)\subset H$ are $p$-periodic sequences for a positive integer
$p$ and $(g_n)\in \ell^{1}(H)$. We investigate the weak or strong
convergence of its solutions to $p$-periodic sequences. We show that
the previous results due to Baillon, Haraux (1977) and
Djafari Rouhani, Khatibzadeh (2012) corresponding to $g\equiv 0$,
respectively $g_n=0$, $n=0,1,\dots$, remain valid for
$g\in L^1(\mathbb{R}_+,H)$, respectively $(g_n)\in l^1(H)$.
Submitted October 18, 2012. Published January 08, 2013.
Math Subject Classifications: 39A10, 39A11, 47H05, 34G25.
Key Words: Differential inclusion; difference inclusion; subdifferential;
maximal monotone operator; weak convergence; strong convergence.