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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 728452, 15 pages doi:10.1155/2010/728452 Research Article A Generalized Nonlinear Random Equations with Random Fuzzy Mappings in Uniformly Smooth Banach Spaces Nawitcha Onjai-Uea 1, 2 and Poom Kumam 1, 2 1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand 2 Centre of Excellence in Mathematics, CHE, Sriayudthaya Road, Bangkok 10400, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 26 July 2010; Accepted 31 October 2010 Academic Editor: Yeol J. E. Cho Copyright q 2010 N. Onjai-Uea and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce and study the general nonlinear random H, η-accretive equations with random fuzzy mappings. By using the resolvent technique for the H, η-accretive operators, we prove the existence theorems and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in q-uniformly smooth Banach spaces. Our result in this paper improves and generalizes some known corresponding results in the literature. 1. Introduction Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965 with a view to reconcile mathematical modeling and human knowledge in the engineering sciences. The concept of fuzzy sets is incredible wide range of areas, from mathematics and logics to traditional and advanced engineering methodologies. Applications are found in many contexts, from medicine to finance, from human factors to consumer products, and from vehicle control to computational linguistics. Random variational inequality theories is an important part of random function anal- ysis. These topics have attracted many scholars and exports due to the extensive applications of the random problems see, e.g., 1–17. In 1997, Huang 3 first introduced the concept of random fuzzy mapping and studied the random nonlinear quasicomplementarity problem for random fuzzy mappings. Further, Huang studied the random generalized nonlinear variational inclusions for random fuzzy mappings in Hilbert spaces. Ahmad and Baz ´ an 18 2 Journal of Inequalities and Applications studied a class of random generalized nonlinear mixed variational inclusions for random fuzzy mappings and constructed an iterative algorithm for solving such random problems. Very recently, Lan et al. 11, introduced and studied a class of general nonlinear random multivalued operator equations involving generalized m-accretive mappings in Banach spaces and an iterative algorithm with errors for this nonlinear random multivalued operator equations. Inspired and motivated by recent works in these fields see 2, 13, 14, 18–29,inthis paper, we introduce and study a class of general nonlinear random equations with random fuzzy mappings in Banach spaces. By using Chang’s lemma and the resolvent operator technique for H, η-accretive mapping. We prove the existence and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in q-uniformly smooth Banach spaces. Our results improve and extend the corresponding results of recent works. 2. Preliminaries Throughout this paper, let Ω, A,μ be a complete σ-finite measure space and E be a separable real Banach space. We denote by BE, ·, · and ·the class of Borel σ-fields in E, the inner product and the norm on E, respectively. In the sequel, we denote 2 E ,CBE and  H by 2 E  {A : A ∈ E},CBE{A ⊂ E : A is nonempty, bounded and closed} and the Hausdorff metric on CBE, respectively. Next, we will use the following definitions and lemmas. Definition 2.1. An operator x : Ω → E is said to be measurable if, for any B ∈BE, {t ∈ Ω : xt ∈ B}∈A. Definition 2.2. A operator F : Ω × E → E is called a random operator if for any x ∈ E, Ft, x yt is measurable. A random operator F is said to be continuous resp. linear, bounded if for any t ∈ Ω, the operator Ft, · : E → E is continuous resp. linear, bounded. Similarly, we can define a random operator a : Ω × E × E → E. We will write F t x Ft, xt and a t x, yat, xt,yt for all t ∈ Ω and xt,yt ∈ E. It is well known that a measurable operator i s necessarily a random operator. Definition 2.3. A multivalued operator G : Ω → 2 E is said to be measurable if, for any B ∈ BE, G −1 B{t ∈ Ω : Gt ∩ B /  ∅} ∈ A. Definition 2.4. A operator u : Ω → E is called a measurable selection of a multivalued measurable operator Γ : Ω → 2 E if u is measurable and for any t ∈ Ω, ut ∈ Γt. Lemma 2.5 see 19. Let M : Ω × E → CBE be a H-continuous random multivalued operator. Then, for any measurable operator x : Ω → E, the multivalued operator M·,x· : Ω → CBE is measurable. Lemma 2.6 see 19. Let M, V : Ω × E → CBE be two measurable multivalued operators, >0 be a constant and x : Ω → E be a measurable selection of M. Then there exists a measurable selection y : Ω → E of V such that, for any t ∈ Ω ,   x  t  − y  t    ≤  1     H  M  t  ,V  t  . 2.1 Journal of Inequalities and Applications 3 Definition 2.7. A multivalued operator F : Ω × E → 2 E is called a random multivalued operator if, for any x ∈ E, F·,x is measurable. A random multivalued operator F : Ω × E → CBE is said to be H-continuous if, for any t ∈ Ω, Ft, · is continuous in  H·, ·, where  H·, · is the Hausdorff metric on CBE defined as follows: for any given A, B ∈ CBE,  H  A, B   max  sup x∈A inf y∈B d  x, y  , sup y∈B inf x∈A d  x, y   . 2.2 Let FE be the family of all fuzzy sets over E. A mapping F : E →FE is called a fuzzy mapping over E. If F is a fuzzy mapping over E, then Fxdenoted by F x  is fuzzy set on E,andF x y is the membership degree of the point y in F x .LetA ∈FE, α ∈ 0, 1. Then the set  A  α  { x ∈ E : A  x  ≥ α } 2.3 is called a α-cut set of fuzzy set A. i A fuzzy mapping F : Ω →FE is called measurable if, for any given α ∈ 0, 1, F· α : Ω → 2 E is a measurable multivalued mapping. ii A fuzzy mapping F : Ω × E →FE is called a random fuzzy mapping if, for any x ∈ E, F·,x : Ω →FE is a measurable fuzzy mapping. Let K, T, G : Ω × E →FE be three random fuzzy mappings satisfying the following condition C: there exists three mappings a, b, c : E → 0, 1, such that  K t,x  ax ∈ CB  E  ,  T t,x  bx ∈ CB  E  ,  G t,x  cx ∈ CB  E  , ∀  t, x  ∈ Ω × E. 2.4 By using the random fuzzy mappings K, T and G, we can define the three multivalued mappings  K,  T and  G as follows, respectively.  K : Ω × E −→ CB  E  ,  t, x  −→  K t,x  ax , ∀  t, x  ∈ Ω × E,  T : Ω × E −→ CB  E  ,  t, x  −→  T t,x  bx , ∀  t, x  ∈ Ω × E,  G : Ω × E −→ CB  E  ,  t, x  −→  G t,x  cx , ∀  t, x  ∈ Ω × E. 2.5 It means that  K  t, x    K t,x  ax  { z ∈ E,  K t,x  z  ≥ a  x  } ∈ CB  E  ,  T  t, x    T t,x  bx  { z ∈ E,  T t,x  z  ≥ b  x  } ∈ CB  E  ,  G  t, x    G t,x  cx  { z ∈ E,  G t,x  z  ≥ c  x  } ∈ CB  E  . 2.6 It easy to see that  K,  T and  G are the random multivalued mappings. We call  K,  T and  G are random multivalued mappings induced by fuzzy mappings K, T and G, respectively. 4 Journal of Inequalities and Applications Suppose that p, S : Ω×E → E and M : Ω×E×E → 2 E with Imp ∩ domMt, ·,s /  ∅, H : Ω × E → E and N : Ω × E × E × E → E be two single-valued mappings. Let K, T, G : Ω×E →FE be three random fuzzy mappings satisfying the condition C. Given mappings a, b, c : E → 0, 1. Now, we consider the following problem: Find measurable mappings x, u, v, w : Ω → E such that for all t ∈ Ω, xt ∈ E, K t,xt ut ≥ axt, T t,xt vt ≥ bxt, G t,xt wt ≥ cxt and 0 ∈ N  t, S  t, x  t  ,u  t  ,v  t   M  t, p  t, x  t  ,w  t   . 2.7 The problem 2.7 is called random variational inclusion problem for random fuzzy mappings in Banach spaces. The set of measurable mappings x, u, v, w is called a random solution of 2.7. Some special cases of 2.7: 1 If G is a single-valued operator, p ≡ I, where I is the identity mapping and Nt, x, y, zft, zgt, x, y for all t ∈ Ω and x, y, z ∈ E, then 2.7 is equivalent to finding x, v : Ω → E such that vt ∈ Tt, xt and 0 ∈ f  t, v  t   g  t, S  t, x  t  ,u  t   M  t, x  t  ,G  t, x  t  , 2.8 for all t ∈ Ω and u ∈ Mt, xt. The problem 2.8 was considered and studied by Agarwal et al. 1, when G ≡ I. If Mt, x, sMt, x for all t ∈ Ω, x, s ∈ E and, for all t ∈ Ω, Mt, · : E → 2 E is a H t ,η-accretive mapping, then 2.7 reduces to the following generalized nonlinear random multivalued operator equation involving H t ,η-accretive mapping in Banach spaces: Find x, v : Ω → E such that vt ∈ Tt, xt and 0 ∈ N  t, S  t, x  t  ,u  t  ,v  t   M  t, g  t, x  t   , 2.9 for all t ∈ Ω and ut ∈ Mt, xt. The generalized duality mapping J q : E → 2 E ∗ is defined by J q  x    f ∗ ∈ E ∗ :  x, f ∗    x  q ,   f ∗     x  q−1  , 2.10 for all x ∈ E, where q>1 is a constant. In particular, J 2 is the usual normalized duality mapping. It is well known that, in general, J q xx q−2 J 2 x for all x /  0andJ q is single- valued if E ∗ is strictly convex see, e.g., 29.IfE  H is a Hilbert space, then J 2 becomes the identity mapping of H. In what follows we will denote the single-valued generalized duality mapping by j q . The modules of smoothness of E is the function ρ E : 0, ∞ → 0, ∞ defined by ρ E  t   sup  1 2   x  y      x − y   − 1:  x  ≤ 1,   y   ≤ t  . 2.11 A Banach space E is called uniformly smooth if lim t → 0 ρ E t/t0andE is called q-uniformly smooth if there exists a constant c>0 such that ρ Et ≤ ct q , where q>1 is a real number. Journal of Inequalities and Applications 5 It is well known that Hilbert spaces, L p or l p  spaces, 1 <p<∞ and the Sobolev spaces W m,p ,1<p<∞,areallq-uniformly smooth. In the study of characteristic inequalities in a q-uniformly smooth Banach space, Xu 30 proved the following result. Lemma 2.8. Let q>1 be a given real number and E be a real uniformly smooth Banach space. Then E is q-uniformly smooth if and only if there exists a constant c q > 0 such that, for all x, y ∈ E and j q x ∈ J q x, the following inequality holds:   x  y   q ≤  x  q  q  y, j q  x    c q   y   q . 2.12 Definition 2.9. A random operator p : Ω × E → E is said to be: a α-strongly accretive if there exists j 2 xt − yt ∈ J 2 xt − yt such that  g t  x  − g t  y  ,j 2  x  t  − y  t   ≥ α  t    x  t  − y  t    2 , 2.13 for all xt,yt ∈ E and t ∈ Ω, where αt > 0 is a real-valued random variable; b β-Lipschitz continuous if there exists a real-valued random variable βt > 0 such that   g t  x  − g t  y    ≤ β  t    x  t  − y  t    , 2.14 for all xt,yt ∈ E and t ∈ Ω. Definition 2.10. Let S : Ω × E → E be a random operator. A operator N : Ω × E × E × E → E is said to be: a -strongly accretive with respect to S in the first argument if there exists j 2 xt − yt ∈ J 2 xt − yt such that  N t  S t  x  , ·, ·  − N t  S t  y  , ·, ·  ,j 2  x  t  − y  t   ≥   t    x  t  − y  t    2 , 2.15 for all xt,yt ∈ E and t ∈ Ω, where t > 0 is a real-valued random variable; b -Lipschitz continuous in the first argument if there exists a real-valued random variable t > 0 such that   N t  x, ·, ·  − N t  y, ·, ·    ≤   t    x  t  − y  t    , 2.16 for all xt,yt ∈ E and t ∈ Ω. Similarly, we can define the Lipschitz continuity in the second argument and third argument of N·, ·, ·. 6 Journal of Inequalities and Applications Definition 2.11. Let η : Ω × E × E → E ∗ be a random operator H : Ω × E → E be a random operator and M : Ω × E → 2 E be a random multivalued operator. Then M is said to be: a η-accretive if  u  t  − v  t  ,η t  x, y  ≥ 0, 2.17 for all xt,yt ∈ E, ut ∈ M t x and vt ∈ M t y where M t zMt, zt,for all t ∈ Ω; b strictly η-accretive if  u  t  − v  t  ,η t  x, y  ≥ 0, 2.18 for all xt,yt ∈ E, ut ∈ M t x, vt ∈ M t y and t ∈ Ω and the equality holds if and only if utvt for all t ∈ Ω; c r-strongly η-accretive if there exists a real-valued random variable rt > 0 such that, for any t ∈ Ω,  u  t  − v  t  ,η t  x, y  ≥ r  t    x  t  − y  t    2 , 2.19 for all xt,yt ∈ E, ut ∈ M t x, vt ∈ M t y and t ∈ Ω. Definition 2.12. Let η : Ω × E × E → E be a single-valued mapping, A : Ω × E → E be a single-valued mapping, M : Ω × E → 2 E be a multivalued mapping. i M t is said to be m-accretive if M t is accretive and I  ρtMt, ·EE for all t ∈ Ω and ρt > 0, where I is identity operator on E; ii M t is said to be generalized m-accretive if M t is η-accretive and I ρtMt, ·EE for all t ∈ Ω and ρt > 0; iii M t is said to be H t -accretive if M t is accretive and H t  ρtMt, ·EE for all t ∈ Ω and ρt > 0; iv M t is said to be H t ,η-accretive if M t is η-accretive and H t  ρtMt, ·EE for all t ∈ Ω and ρt > 0. Remark 2.13. If E  E ∗  H is a Hilbert space, then a–c of Definition 2.11 reduce to the definition of η-monotonicity, strict η-monotonicity and strong η-monotonicity, respectively, if E is uniformly smooth and ηx, yj 2 x − y ∈ J 2 x − y, then a–c of Definition 2.11 reduce to the definitions of accretive, strictly accretive and strongly accretive in uniformly smooth Banach spaces, respectively. Definition 2.14. The operator η : Ω × E × E → E ∗ is said to be: τ-Lipschitz continuous if there exists a real-valued random variable τt > 0 such that   η t  x, y    ≤ τ  t    x  t  − y  t    , 2.20 for all xt,yt ∈ E and t ∈ Ω. Journal of Inequalities and Applications 7 Definition 2.15. A multivalued measurable operator T : Ω × E → CBE is said to be γ-  H- Lipschitz continuous if there exists a measurable function γ : Ω → 0, ∞ such that, for any t ∈ Ω,  H  T t  x  ,T t  y  ≤ γ  t    x  t  − y  t    , 2.21 for all xt,yt ∈ E. Definition 2.16. Let M : Ω × E × E → 2 E be a H t ,η-accretive random operator and H : Ω × E → E be r-strongly monotone random operator. Then the proximal operator J ρt,H t M t·,x is defined as follows: J ρt,H t M t·,x  z    H t  ρ  t  M t  −1  z  , 2.22 for all t ∈ Ω and z ∈ E, where ρ : Ω → 0, ∞ is a measurable function and η : Ω×E×E → E ∗ is a strictly monotone operator. Lemma 2.17 see 31. Let η : E × E → E be a τ-Lipschitz continuous operator, H : Ω × E → E be a r-strongly η-accretive operator and M : Ω × E → 2 E be an H t ,η-accretive operator. Then, the proximal operator J ρt,H t M t : E → E is τ q−1 /r-Lipschitz continuous, that is,    J ρt,H t M t  x  − J ρt,H t M t  y     ≤ τ q−1 r   x − y   , ∀x, y ∈ E, t ∈ Ω. 2.23 3. Random Iterative Algorithms In this section, we suggest and analyze a new class of iterative methods and construct some new random iterative algorithms for solving 2.7. Lemma 3.1. The set of measurable mapping x, u, v, w : Ω → E a random solution of problem 2.7 if and only if for all t ∈ Ω, and p t  x   J ρt,H t M t·,w  H t  p t  x   − ρ  t  N t  S t  x  ,u,v   . 3.1 Proof. The proof directly follows from the definition of J ρt,H t M t·,w as follows: p t  x   J ρt,H t M t·,w  H t  p t  x   − ρ  t  N t  S t  x  ,u,v   ⇐⇒ p t  x    H t  ρ  t  M t  −1  H t  p t  x   − ρ  t  N t  S t  x  ,u,v   ⇐⇒ H t  p t  x   − ρ  t  N t  S t  x  ,u,v  ∈ H t  p t  x    ρ  t  M t  p t  x  ,w  ⇐⇒ 0 ∈ M t  p t  x  ,w   N t  S t  x  ,u,v  . 3.2 8 Journal of Inequalities and Applications Algorithm 3.2. Suppose that K, T, G : Ω × E →FE be three random fuzzy mappings satisfying the condition C.Let  K,  T,  G : Ω × E → CBE be  H-continuous random multivalued mappings induced by K, T,andG, respectively. Let S, p : Ω × E → E, η : Ω × E × E → E and N : Ω × E × E × E → E be random single-valued operators. Let M : Ω × E × E → 2 E be a random multivalued operator such that for each fixed t ∈ Ω and s ∈ E, Mt, ·,s : E → 2 E be a H t ,η-accretive mapping and Rangep  dom Mt, ·,s /  ∅. For any given measurable mapping x 0 : Ω → E, the multivalued mappings  K·,x 0 ·,  T·,x 0 ·,  G·,x 0 · : Ω → E are measurable by Lemma 2.5. Then, there exists measurable selections u 0 · ∈  K·,x 0 ·,v 0 · ∈  T·,x 0 · and w 0 · ∈  G·,x 0 · by Himmelberg 6.Set x 1  t   x 0  t  − λ  t   p t  x 0  − J ρt,H t M t  ·,w 0   H t  p t  x 0   − ρ  t  N t  S t  x 0  ,u 0 ,v 0    , 3.3 where λtρt and H t are the same as in Lemma 3.1. Then it is easy to know that x 1 : Ω → E is measurable. Since u 0 t ∈  K t x 0  ∈ CBE,v 0 t ∈  T t x 0  ∈ CBE and w 0 t ∈  G t x 0  ∈ CBE,byLemma 2.6, there exist measurable selections u 1 t ∈  K t x 1 ,v 1 t ∈  T t x 1  and w 1 t ∈  G t x 1  such that for all t ∈ Ω,  u 0  t  − u 1  t   ≤  1  1 1   H   K t  x 0  ,  K t  x 1   ,  v 0  t  − v 1  t   ≤  1  1 1   H   T t  x 0  ,  T t  x 1   ,  w 0  t  − w 1  t   ≤  1  1 1   H   G t  x 0  ,  G t  x 1   . 3.4 By induction, we can define the sequences {x n t}, {u n t}, {v n t} and {w n t} inductively satisfying x n1  t   x n  t  − λ  t   p t  x n  − J ρt,H t M t·,w n   H t  p t  x n   − ρ  t  N t  S t  x n  ,u n ,v n    , u n  t  ∈  M t  x n  ,  u n  t  − u n1  t   ≤  1  1 n  1   H   K t  x n  ,  K t  x n1   , v n  t  ∈  T t  x n  ,  v n  t  − v n1  t   ≤  1  1 n  1   H   T t  x n  ,  T t  x n1   , w n  t  ∈  G t  x n  ,  w n  t  − w n1  t   ≤  1  1 n  1   H   G t  x n  ,  G t  x n1   . 3.5 From Algorithm 3.2 , we can get the following algorithms. Journal of Inequalities and Applications 9 Algorithm 3.3. Suppose that E, M, η, S,  K,  T and λ are the same as in Algorithm 3.2.Let  G : Ω × E → E be a random single-valued operator, p ≡ I and Nt, x, y, zft, zgt, x, y for all t ∈ Ω and x, y, z ∈ E. Then, for given measurable x 0 : Ω → E, we have x n1  t    1 − λ  t  x n  t   λ  t  J ρt,H t M t·,G t x n   x n  t  − ρ  t   f t  v n   g t  S t  x n  ,u n   , u n  t  ∈  K t  x n  ,  u n  t  − u n1  t   ≤  1  1 n  1   H   K t  x n  ,  K t  x n1   , v n  t  ∈  T t  x n  ,  v n  t  − v n1  t   ≤  1  1 n  1   H   T t  x n  ,  T t  x n1   . 3.6 Algorithm 3.4. Let M : Ω×E → 2 E be a random multivalued operator such that for each fixed t ∈ Ω, Mt, · : E → 2 E is a H, η-accretive mapping and Rangep  dom Mt, · /  ∅.IfS, p, η, N,  K,  T and λ are the same as in Algorithm 3.2, then for given measurable x 0 : Ω → E, we have x n1  t   x n  t  − λ  t   p t  x n  − J ρt,H t M t·,w n   p t  x n  − ρ  t  N t  S t  x n  ,u n ,v n    , u n  t  ∈  K t  x n  ,  u n  t  − u n1  t   ≤  1  1 n  1   H   K t  x n  ,  K t  x n1   , v n  t  ∈  T t  x n  ,  v n  t  − v n1  t   ≤  1  1 n  1   H   T t  x n  ,  T t  x n1   . 3.7 4. Existence and Convergence Theorems In this section, we prove the existence and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in q- uniformly smooth Banach spaces. Theorem 4.1. Suppose that E is a q-uniformly smooth and separable real Banach space, p : Ω ×E → E is α-strongly accretive and β-Lipschitz continuous, η : Ω × E × E → E be τ-Lipschitz continuous, H : Ω × E → E is r-strongly η-accretive and μ A -Lipschitz continuous and M : Ω × E × E → 2 E is a random multivalued operator such that for each fixed t ∈ Ω and s ∈ E, Mt, ·,s : E → 2 E is a H t ,η-accretive mapping and Rangep  dom Mt, ·,s /  ∅.LetS : Ω × E → E be a σ-Lipschitz continuous random operator and N : Ω × E × E × E → E be -Lipschitz continuous in the first argument, μ-Lipschitz continuous in the second argument and ν-Lipschitz continuous in the third argument, respectively. Let K, T, G : Ω × E →FE be three random fuzzy mappings satisfying the condition (C),  K,  T,  G : Ω × E → CBE be three random multivalued mappings induced by the mappings K, T, G, respectively, K, T and G are  H-Lipschitz continuous with constants μ  K t, μ  T t and μ  G t, respectively. If for each real-valued random variables ρt > 0 and πt > 0 such that, for any t ∈ Ω, x, y, z ∈ E,    J ρt,H t M t·,x  z  − J ρt,H t M t·,y  z     ≤ π  t    x − y   4.1 10 Journal of Inequalities and Applications and the following conditions hold: l  t    1 − qαtc q β  t  q  1/q  π  t  μ  G  t  < 1, μ A  t  β  t   ρ  t    t  σ  t   μ  t  μ  K  t   ν  t  μ  T  t  < r  t  1 − l  t  τ  t  q−1 , 4.2 where c q isthesameasinLemma 2.8 for any t ∈ Ω. If there exist real-valued random variables λt,ρt > 0 then there exist x ∗ t ∈ E, u ∗ t ∈ K t x ∗ ,v ∗ t ∈ T t x ∗  and w ∗ t ∈ G t x ∗  such that x ∗ t,u ∗ t,v ∗ t,w ∗ t is solution of 2.7 and x n  t  −→ x ∗  t  ,u n  t  −→ u ∗  t  ,v n  t  −→ v ∗  t  ,w n  t  −→ w ∗  t  4.3 as n →∞,where{x n t}, {u n t}, {v n t} and {w n t} are iterative sequences generated by Algorithm 3.2. Proof. From Algorithm 3.2, Lemma 2.17 and 4.1, we compute  x n1  t  − x n  t       x n  t  − λ  t   p t  x n  − J ρt,H t M t·,w n   H t  p t  x n   − ρ  t  N t  S t  x n  ,u n ,v n    − x n−1  t   λ  t   p t  x n−1  − J ρt,H t M t·,w n−1   H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1       ≤   x n  t  − x n−1  t  − λ  t   p t  x n  − p t  x n−1      λ  t     J ρt,H t M t·,w n   H t  p t  x n   − ρ  t  N t  S t  x n  ,u n ,v n   − J ρt,H t M t·,w n−1   H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1      ≤   x n  t  − x n−1  t  − λ  t   p t  x n  − p t  x n−1      λ  t     J ρt,H t M t·,w n   H t  p t  x n   − ρ  t  N t  S t  x n  ,u n ,v n   − J ρt,H t M t·,w n   H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1       λ  t     J ρt,H t M t·,w n   H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1   − J ρt,H t M t·,w n−1   H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1      ≤   x n  t  − x n−1  t  − λ  t   p t  x n  − p t  x n−1      λ  t  τ  t  q−1 r  t    H t  p t  x n   − ρ  t  N t  S t  x n  ,u n ,v n  −  H t  p t  x n−1   − ρ  t  N t  S t  x n−1  ,u n−1 ,v n−1      λ  t  π  t   w n − w n−1  [...]... 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