C i C i i = 1, 2, . . . T i A i G i (u, v) cardG = 2 C 1 C 2 H x ∈ H {x k } ∞ k=1 {y k } ∞ k=1 y 0 = x, x k = P C 1 (y k−1 ), y k = P C 2 (x k ), k = 1, 2, , P C (x) k → ∞ C = C 1 ∩ C 2 P C H C C 1 C 2 H P C (x) k → ∞ cardG ≥ 2 C i T i {T i } i≥2 {T i } ∞ i=1 λ i C H H F =  ∞ i=1 F ix(T i ) = ∅ F ix(T i ) T i u ∗ ∈ F u ∗ ∈ C F (u ∗ ), v − u ∗  ≥ 0 v ∈ C, F : C → H C H F u α ∈ C u α ∈ C  ∞  i=1 γ i A i (u α ) + αu α , v − u α  ≥ 0 ∀v ∈ C, A i = I − T i α > 0 {γ i } γ i > 0; ∞  i=1 γ i  λ i = γ < ∞,  λ i = 1 − λ i 2 . ϕ : H −→ R ϕ  { n } ∞ n=0 {α n } ∞ n=0 • (i) k = 0 z 0 ∈ C  0 α 0 (ii) k = n z ∈ C min z∈C  ϕ(z) +   n  ∞  i=1 γ i A i (z n ) + α n z n  − ϕ  (z n ), z  . z n+1 (iii) z n+1 − z n  n ← n + 1 (ii) { n } ∞ n=0 {α n } ∞ n=0 0 <  n ≤ 1; 0 < α n+1 ≤ α n ≤ 1; α n → 0 n → ∞; ∞  n=0  n α n = ∞; ∞  n=0  2 n < ∞; ∞  n=0 (α n − α n+1 ) 2 α 3 n  n < ∞, {u α } {z n } u ∗ B =  ∞ i=1 γ i A i W T n T n−1 T 1 γ n γ n−1 γ 1 {T i } N i=1 u 0 ∈ H, u k+1 = T [k+1] u k − λ k+1 µF (T [k+1] u k ), T [n] = T n mod N µ ∈  0, 2η L 2  η L F {u k } ∞ k=0 u ∗ C =  N i=1 F ix(T i ) x 0 ∈ H {x n } ∞ n=0 x 0 ∈ H, y 0 0 = x 0 , y i k = (1 − β i k )y i−1 k + β i k T i (y i−1 k ), i = 1, 2, · · · , N, x k+1 = (1 − β 0 k )x k + β 0 k (I − λ k µF )y N k , k ≥ 0. λ k β i k i = 0, . . . , N λ k ∈ (0, 1) β i k ∈ (α, β) α, β ∈ (0, 1) k ≥ 0 lim k→∞ λ k = 0; ∞  k=0 λ k = ∞; lim k→∞   β i k+1 − β i k   = 0. u ∗ W n H C H F : C −→ H x ∗ ∈ C F (x ∗ ), x − x ∗  ≥ 0 ∀x ∈ C. V I(F, C) C H F : C −→ H C H C C H F : C −→ H U C u ∈ C \U v ∈ U F (u), u − v > 0. • x ∗ ∈ C x ∗ ∈ C x ∗ = P C (x ∗ − λF (x ∗ )) λ > 0 x 0 ∈ C x n+1 = P C (x n − λF (x n )), n = 0, 1, 2, · · · {x n } • F x 0 = x ∈ C, y n = P C (x n − λF (x n )), x n+1 = P C (x n − λF (y n )), n = 0, 1, 2, · · · λ ∈ (0, 1/L) L F • J H u ∗ ∈ C J(u ∗ ) = min u∈C J(u). ϕ : H → R v ∈ C  > 0 G : u → ϕ(u) + J  (v) − ϕ  (v), u . G  (v) = J  (v). v ∈ C v min u∈C {ϕ(u) + J  (v) − ϕ  (v), u}. { n } n∈N (i) k = 0  0 u 0 ∈ C (ii) k = n u ∈ C min u∈C {ϕ(u) +  n J  (u n ) − ϕ  (u n ), u}. u n+1 (iii) u n+1 − u n  n ← n + 1 (ii) J  F u 0 ∈ C  0 > 0 min u∈C {ϕ(u) +  0 F (u 0 ) − ϕ  (u 0 ), u}. u 1 u 0  0 u 1  1 u 2 z n ∈ C z n+1 min u∈C {ϕ(u) +  n F (u n ) − ϕ  (u n ), u}; C H F : C −→ H a C u ∗ ϕ : C −→ R ϕ  b C u n+1 F L C 0 <  n < 2ab/L 2 {u n } u ∗ F { n } ∞ n=0 {α n } ∞ n=0 • (i) k = 0 z 0 ∈ C  0 α 0 (ii) k = n z ∈ C min z∈C {ϕ(z) +  n (F (z n ) + α n z n ) − ϕ  (z n ), z}. z n+1 (iii) z n+1 − z n  n ← n + 1 (ii) { n } ∞ n=0 {α n } ∞ n=0 Ψ (i) 0 <  n ≤ 1 0 < α n+1 ≤ α n ≤ 1 α n → 0 n → ∞ (ii)  ∞ n=0  n α n = ∞  ∞ n=0  2 n < ∞ (iii)  ∞ n=0 (α n − α n+1 ) 2 α 3 n  n < ∞ H C H F : C −→ H ϕ : H −→ R ϕ  n ∈ N z n+1 Ψ F lim z n+1 − u ∗  = 0, u ∗ E C E T C E k x y ∈ D(T ) T k > 0 j(x − y) ∈ J(x − y) T (x) − T (y), j(x − y) ≤ x − y 2 − k(x − y) − (T (x) − T (y)) 2 , j(x) E I E (I − T )(x) − (I − T )(y), j(x − y) ≥ k(I − T )(x) − (I − T )(y) 2 , x, y ∈ D(T ) j(x − y) ∈ J(x − y) H  T (x) − T (y)  2 ≤ x − y  2 +λ  (I − T )(x) − (I − T )(y)  2 , x, y ∈ D(T ) λ = 1 − k T λ 0 ≤ λ < 1 [...]... auxiliary problem algorithm for a countably infinite family of non-self strictly pseudocontractive mappings in Hilbert spaces Constructed a procedure to determine a common fixed point for a countably infinite family of non-self strictly pseudocontractive and prove the convergence of regularization solution 2 Established the regularization auxiliary problem algorithm to determine a common element of the set... methods are constructed based on an infinite sum of mappings and Wn -mapping, introduced by Takahashi W Using these methods, we constructed a procedure for determing a common fixed point of a countably infinite family of non-self strictly pseudocontractive mappings, as well as, for finding a common element of the set of solutions for a variational inequality involving a monotone Lipschitz continuous... algorithm for com- puting a common fixed point for a countably infinite family of non-seld strictly pseudocontractive mappings in Hilbert space Let in H H Let be a real Hilbert space and let {Ti }∞ i=1 be a nonempty closed convex subset be a countably infinite family of non-self mappings of problem: C C into H such that F = λi -strictly ∞ i=1 F ix(Ti ) pseudocontractive = ∅ Consider the find an element... continuous mapping and the set of common fixed points for a countably infinite family of nonexpansive mappings on a closed convex subset in Hilbert spaces An illustrative example is considered Ch­¬ng 3 Variational inequality problem over the set of common fixed points for a finite family of nonexpansive mappings 3.1 Variational inequality problem over the set of common fixed points for a finite family... we applied a regularization auxiliary problem algorithm for finding a common fixed point for a countably infinite family of non-self strictly pseudocontractive mappings in Hilbert space At each iteration step, we solve a regularization problem with monotone mapping B= ∞ i=1 γi Ai , which is very complicated in computation To overcome this difficulty, in the next section, we introduce a new method,... regularization auxiliary problem algorithm for finding common fixed points of a countably infinite family of non-self strictly pseudocontractive or nonexpansive mappings in Banach spaces 2 Find a solution for a mixed variational inequality problem over the set of common fixed points of a countably infinite family of nonexpansive mappings in Banach spaces 22 23 3 Could we use mapping where 4 Sn = n i=1 γi Ai... and such that, for some positive constants η -strongly F = monotone N i=1 F ix(Ti ) Let = ∅ {Ti }N i=1 be L and N If Then the sequence S : H −→ H be a be L-Lipschitz be a mapping continuous and nonexpansive self-maps of converges strongly to the unique element Comment 3.1 η, F F : H → H p∗ of γ -strictly {xk }k∈N , defined by H such that (3.2) − (3.3), (3.1) pseudocontractive H, the mapping T , defined... set of common fixed points of a finite family of nonexpansive mappings in Hilbert spaces We introduced a new method, which is a combination of the hybrid stepest descent method for variational inequalities with the Krasnoselskij - Mann type algorithm for fixed point problems A strong convergence theorem is proved under simpler conditions and some numerical example are also given for illustration Conclusion... let from C into condition H (2.3) C be a nonempty closed convex subset of a real Hilbert space be a sequence of non-self such that ∞ i=1 F ix(Ti ) F := and the functional differentiable on H, = ∅ ϕ : H −→ R and its Gateaux derivative schitz continuous Then, for every (2.5) λi -strictly Moreover, if Assumption n ≥ 0, H pseudocontractive mappings Assume that {γi }∞ i=1 satisfies is proper, convex, and Gateaux... variational inequality problem and the set of common fixed points for an infinite family of nonexpansive or strictly pseudocontractive mappings in Hilbert spaces 3 Constructed an explicit iterative algorithm to determine a solution for a class of variational inequality problem over the set of common fixed points for a finite family of nonexpansive mappings or strictly pseudocontractive mappings in Hilbert spaces