Let p be a convex functional on a real linear space X and let [f.sub.0] be a linear functional on a subspace Xo of X such that [f.sub.0](x) [less than or equal to] p(x) for all x [member of] [X.sub.0].
Clearly, [p.sub.[alpha]] is a convex functional on [X.sub.0].
Similarly we say the map [beta] is a nonnegative continuous convex functional on a cone P of a real Banach space E if [beta] : P [right arrow] [0,[infinity]) is continuous and
First we define the nonnegative continuous concave functionals [alpha] and [phi] and nonnegative continuous convex functionals [gamma], [beta] and [theta] on P respectively by
Assume that [phi] : H [right arrow] R is a lower semicontinuous and convex functional, that [THETA], [[THETA].sub.1], [[THETA].sub.2] : H x H [right arrow] R satisfy conditions (H1)-(H4), and that A, [A.sub.1], [A.sub.2] : H [right arrow] H are inverse- strongly monotone.
Let f : C [right arrow] R be a convex functional with L-Lipschitz continuous gradient [nabla]f.
Let f : C [right arrow] R be a convex functional with L-Lipschitz continuous gradient [nabla]f; for instance, putting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and f(x) = (1/2)<Sx, x>, we obtain that [parallel]S[parallel] = 1 and [nabla]f = S with Lipschitz constant L = 4.
Let [THETA],[[THETA].sub.1], [[THETA].sub.2] be three bifunctions from C x C to R satisfying (H1)-(H4) and let [phi] : C [right arrow] R be a lower semicontinuous and convex functional. Let [R.sub.i] : C [right arrow] [2.sup.H] be a maximal monotone mapping and let A,[A.sub.k] : H [right arrow] H and [B.sub.i] : C [right arrow] H be [zeta]-inverse strongly monotone, [[zeta].sub.k]inverse strongly monotone, and [[eta].sub.i]-inverse strongly monotone, respectively, for k = 1,2 and i = 1,2.
BREDIES, A forward-backward splitting algorithm for the minimization of non-smooth convex functionals
in Banach space, Inverse Problems, 25 (2009), 015005 (20 pages).