เอกสารประชุมวิชาการระดับขาติมหาวิทยาลัยทักษิณ ครั้งที่ 28 2561

1095 งานวิจัยและนวัตกรรมเพื่อสังคมที่มั่นคง มั่งคั่ง และยั่งยืน (Research and Innovation for Social Stability, Prosperity and Sustainability) By edge-preserving of 2 T , then 2 1 L T [ V E (  . Since 2 1 1 L O T [ V x V E (  and E ( is convex, we get 2 1 1 2 1 2 1 1 1 1 1 1 1 L L L L L L L L L x V T [ V x T [ V Z V E ( D D D D  . (6) By induction, we get O O O x V Z V [ V E (  for all 1 O t . For 0 0 0 V x V Z V [ E (  . Similarly, by using the same proof, we have O O V x V Z O V [ E (  . From the transitivity of ( , that O O O O O O x Z Z [ x [ E (  . Next we show that ^ ` O x is bounded. For V F  and 0 0 0 0 0 0 x V Z V [ V V x V Z V [ E (  . Notice that 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 2 2 2 1 1 2 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O x V x T Z V x V T Z V x V Z V x V x T [ V x V x V T [ V x V x V [ V x V x D D D D D D D D D D D D D D D D D D D D D d d d d 1 2 1 1 2 1 2 3 3 3 1 1 2 1 2 3 3 3 1 1 2 1 2 3 3 1 1 2 1 1 1 1 1 1 1 1 1 1 1 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O V [ V x V x V x T x V x V x V x V T x V x V x V x V x V x V D D D D D D D D D D D D D D D D D D D D D D D D D D d d 1 2 1 1 1 O O O O O O O O O x V x V x V x V x V D D D D (7) It mean that, O O MJN x V f exists. Then ^ ` O x is bounded so are ^ ` O Z and ^ ` O [ . Next we show that 0 O J O O MJN T x x f , for 1 2 3 J . Let V F  , there exists 0 S ! such that O O x V S Z V S d d and O [ V S d for all 1 O t .