In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.
We study the well-posedness of the second order degenerate integro-differential equations （P2）： （Mu）t＇（t） ＋ a（Mu）＇（t） = Au（t） ＋ ft_c~ a（t - s）Au（s）ds ＋ f（t）, 0 ≤ t ≤ 27r, with periodic boundary conditions Mu（O） = Mu（27r）, （Mu）＇（O） = （Mu）＇（2π）, in periodic Lebesgue-Bochner spaces LP（T,X）, periodic Besov spaces BBp,q（T, X） and periodic Triebel-Lizorkin spaces F~,q（＇F, X）, where A and M are closed linear operators on a Banach space X satisfying D（A） C D（M）, a C LI（R＋） and a is a scalar number. Using known operator- valued Fourier multiplier theorems, we completely characterize the well-posedness of （P2） in the above three function spaces.
