Electron. J. Diff. Equ., Vol. 2013 (2013), No. 06, pp. 1-6.

Asymptotically periodic solutions for differential and difference inclusions in Hilbert spaces

Gheorghe Morosanu, Figen Ozpinar

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 $f \in L_{loc}^2(\mathbb{R}_+,H)$ 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 8, 2013.
Math Subject Classifications: 39A10, 39A11, 47H05, 34G25.
Key Words: Differential inclusion; difference inclusion; subdifferential; maximal monotone operator; weak convergence; strong convergence.

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Gheorghe Morosanu
Department of Mathematics and its Applications
Central European University
Budapest, Hungary
email: morosanug@ceu.hu
Figen Ozpinar
Bolvadin Vocational School
Afyon Kocatepe University
Afyonkarahisar, Turkey
email: fozpinar@aku.edu.tr

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