One has (i) (z=P_{C}x Leftrightarrowlangle x-z,y-zrangleleq0), (forall yin C); (ii) (z=P_{C}x Leftrightarrow|x-z|^{2}leq|x-y|^{2}-|y-z|^{2}), (forall yin C); (iii) (langle P_{C}x-P_{C}y,x-yranglegeq|P_{C}x-P_{C}y|^{2}), (forall yin H), which hence implies that (P_{C}) is nonexpansive and monotone. .
(z=P_{C}xLeftrightarrowlangle x-z, y-zrangleleq0) for all (yin C); (z=P_{C}xLeftrightarrow|x-z|^{2}leq|x-y|^{2}-|y-z|^{2}) for all (yin C); (langle x-y, P_{C}x-P_{C}y ranglegeq Vert P_{C}x-P_{C}y Vert ^{2}) for all (yinmathcal{H}), which hence implies that (P_{C}) is nonexpansive.
(z=P_{C}x Leftrightarrowlangle x-z,y-zrangleleq0), (forall yin C); (z=P_{C}x Leftrightarrow|x-z|^{2}leq|x-y|^{2}-|y-z|^{2}), (forall yin C); (langle P_{C}x-P_{C}y,x-yranglegeq|P_{C}x-P_{C}y|^{2}), (forall yin H), which hence implies that (P_{C}) is nonexpansive and monotone.
(z=P_{C}x Leftrightarrow langle x-z,y-zrangleleq0), (forall yin C); (z=P_{C}x Leftrightarrow |x-z|^{2}leq|x-y|^{2}-|y-z|^{2}), (forall yin C); (langle P_{C}x-P_{C}y,x-yranglegeq|P_{C}x-P_{C}y|^{2}), (forall yin H), which hence implies that (P_{C}) is nonexpansive and monotone.
Given (xinmathcal{H}) and (zin C), (1) (z=P_{C}xLeftrightarrowlangle x-z, y-zrangleleq0) for all (yin C); (2) (z=P_{C}xLeftrightarrow|x-z|^{2}leq|x-y|^{2}-|y-z|^{2}) for all (yin C); (3) (langle x-y, P_{C}x-P_{C}y ranglegeq Vert P_{C}x-P_{C}y Vert ^{2}) for all (yinmathcal{H}), which hence implies that (P_{C}) is nonexpansive. .
Given any (xin H) and (zin C), one has (i) (z=P_{C}x Leftrightarrowlangle x-z,y-zrangleleq0), (forall yin C); (ii) (z=P_{C}x Leftrightarrow|x-z|^{2}leq|x-y|^{2}-|y-z|^{2}), (forall yin C); (iii) (langle P_{C}x-P_{C}y,x-yranglegeq|P_{C}x-P_{C}y|^{2}), (forall yin H), which hence implies that (P_{C}) is nonexpansive and monotone. .
