dc.description.abstract | • Let H be a real Hilbert space. Let K, F : H → H be bounded,
continuous and monotone mappings. Let {un }∞ and {vn }∞ be
n=1
n=1
sequences in E defined iteratively from arbitrary u1 , v1 ∈ H by
un+1 = un − βn (F un − vn ) − βn (un − u1 ),
vn+1 = vn − βn (Kvn + un ) − βn (vn − v1 ), n ≥ 1
2
where {βn }∞ is a real sequence in (0, 1) such that ∞ βn < ∞ and
n=1
n=1
∞
∞
n=1 βn = +∞. Then, the sequence {un }n=1 converges strongly to
∗.
u
• Let H be a real Hilbert space. For each i = 1, 2, ...m, let Fi , Ki : H → H be bounded and monotone mappings. Let {un }∞ , {vi,n }∞ , i =
n=1
n=1
1, 2, . . . , m be sequences in H defined iteratively from arbitrary u1 , vi,1 ∈
H by
un+1 = un − λn αn un +
m
i=1 Ki vi,n
− λn θn (un − u1 ),
vi,n+1 = vi,n − λn αn (Fi un − vi,n ) − λn θn (vi,n − vi,1 ), i = 1, 2, . . . , m
where {λn }∞ , {αn }∞ and {θn }∞ are real sequences in (0, 1)
n=1
n=1
n=1
such that λn = o(θn ), αn = o(θn ) and ∞ λn θn = +∞. Suppose i=1 that u + m Ki Fi u = 0 has a solution in H. Then, there exist real i=1 constants ε0 , ε1 > 0 and a set Ω ⊂ W such that if αn ≤ ε0 θn and λn ≤ ε1 θn , ∀n ≥ n0 , for some n0 ∈ N and w∗ := (u∗ , x∗ , x∗ , . . . , x∗ ) ∈
m
1 2
Ω (where x∗ = Fi u∗ , i = 1, 2, . . . , m), the sequence {un }∞ converges
n=1
i
strongly to u∗ .
• Let E be a reflexive real Banach space with uniformly Gˆteaux dif-
a
ferentiable norm. Let K be a nonempty closed convex subset of E and {Tn }∞ be a sequence of Ln -Lipschitzian mappings of K into itself with Ln ≥ 1,
∞
(Ln − 1) < ∞. Let
n=1
F (Tn ) = ∅. For
n=1
a fixed δ ∈ (0, 1) and each n ∈ N, define Sn : K → K by Sn x := (1 − δ)x + δTn x, ∀x ∈ K. Let {αn }∞ be a sequence in [0, 1] such
n=1
∞
that lim αn = 0 and
n→∞
n=1
αn = ∞ . Let {xn }∞ be a sequence in K
n=1
defined by x1 = u ∈ K and
xn+1 = αn u + (1 − αn )Sn xn ,
∞
for all n ∈ N. Suppose that (Kmin )
F (Tn ) = ∅ and lim ||Tn+1 xn −
n=1
n→∞
Tn xn || = 0. Then, {xn }∞ converges strongly to some common fixed
n=1
points of {Tn }∞ .
n=1
• Let K be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bi-function from K × K satisfying (A1) − (A4), ψ a μ-inverse-strongly monotone mapping of K into H, A an α-inverse- strongly monotone mapping of K into H and M : H → 2H a maximal monotone mapping. Let T : H → H be a nonexpansive mapping such that Ω := F (T ) ∩ I(A, M ) ∩ EP = ∅ and suppose f : H → H is a contraction map with constant γ ∈ (0, 1). Suppose {xn }∞ and
n=1
{un }∞ are generated by x1 ∈ H,
n=1
F (un , y) + ψxn , y − un +
1
rn
y − un , un − xn ≥ 0 ∀y ∈ K,
xn+1 = βn xn + (1 − βn )T αn f (xn ) + (1 − αn )JM,λ (un − λAun ) , for all n ≥ 1, where {αn }∞ and {βn }∞ are sequences in [0,1] and
n=1
n=1
{rn }∞ ⊂ (0, ∞) satisfying:
n=1
(i) 0 < c ≤ βn ≤ d < 1,
∞
(ii)
αn = ∞,
lim αn = 0,
n→∞
n=1
(iii) λ ∈ (0, 2α],
(iv) 0 < a ≤ rn ≤ b < 2μ,
{xn }∞
n=1
lim |rn+1 − rn | = 0,
n→∞
then
converges strongly to z, where z := PΩ f (z) and PΩ f (z) is the metric projection of f (z) onto Ω. • Let K be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bi-function from K × K to R satisfying (A1) − (A4), ψ a μ-inverse-strongly monotone mapping of K into H, A an α-inverse-
strongly monotone mapping of K into H and M : H → 2H a maximal monotone mapping. Let := {T (u) : 0 ≤ u < ∞} be a one-parameter nonexpansive semigroup on H such that Ω := F ( )∩I(A, M )∩EP = ∅ and suppose f : H → H is a contraction mapping with a constant γ ∈
(0, 1). Let {tn } ⊂ (0, ∞) be a real sequence such that limn→∞ tn = ∞.
Suppose {xn }∞ and {un }∞ are generated by x1 ∈ H,
n=1
n=1
F (un , y) + ψxn , y − un + r1 y − un , un − xn ≥ 0 ∀y ∈ K
n
wn = JM,λ (un − λAun )
xn+1 = βn xn + (1 − βn ) 1 tn T (u)[αn f (xn ) + (1 − αn )wn ]du ,
tn 0
for all n ≥ 1, where {αn }∞ and {βn }∞ are sequences in (0,1) and
n=1
n=1
{rn }∞ ⊂ (0, ∞) satisfying:
n=1
∞
(i) lim βn = 0,
n=1 |βn+1 − βn | < ∞,
n→∞
(ii)
∞
αn = ∞,
lim αn = 0,
n→∞
n=1
(iii) λ ∈ (0, 2α],
(iv) 0 < a ≤ rn ≤ b < 2μ,
1
(v) lim |tn −tn−1 | αn (1−βn ) = 0,
tn
∞
n=1 |αn+1
∞
n=1 |rn+1
− αn | < ∞,
− rn | < ∞,
n→∞
then {xn }∞ converges strongly to z, where z := PΩ f (z) and PΩ f (z)
n=1
is the metric projection of f (z) onto Ω.
• Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed and convex subset of E. Let F be a bifunction from C × C satisfying (A1) − (A4). Suppose {Tn }∞ n=0 is a countable family of relatively nonexpansive mappings of C into E such that Ω := (∩∞ F (Tn )) ∩ EP (F ) = ∅. Let {αn }, {βn } and {γn }
n=0
be sequences in (0, 1) such that αn + βn + γn = 1. Suppose {xn }∞
n=0
is iteratively generated by u, u0 ∈ E,
xn = Trn un ,
un+1 = J −1 (αn Ju + βn Jxn + γn JTn xn ), n ≥ 0,
with the conditions
∞
(i) lim αn = 0,
αn = ∞;
n→∞
n=0
(ii) 0 < b ≤ βn γn ≤ 1;
(iii) {rn }∞ ⊂ (0, ∞) satisfying lim inf rn > 0.
n=0
n→∞
Then, {xn }∞ converges strongly to ΠΩ u, where ΠΩ u is the general-
n=0
ized projection of u onto Ω.
• Let E be a 2-uniformly convex real Banach space which is also uni- formly smooth. Let C be a nonempty, closed and convex subset of E. Suppose B : C → E ∗ is an operator satisfying (B1) − (B3) and {Tn }∞ is an infinite family of relatively-quasi nonexpansive mappings
n=1 of C into itself such that F := V I(C, B) ∩
∩∞ F (Tn )
n=1
∩m GM EP (Fk , φk ) ∩
k=1
= ∅. Let {xn }∞ be iteratively generated by x0 ∈
n=0
C, C1 = C, x1 = ΠC1 x0 ,
υn = ΠC J −1 (Jxn − λn Bxn )
y = J −1 (α Jυ + (1 − α )JT υ )
n
n
n
n
n n
z = J −1 (β Jx + (1 − β )Jy )
n
n
0
n
n
G2 G1
Gm Gm−1
un = Trm,n Trm−1,n ...Tr2,n Tr1,n zn
C
2
n+1 = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + βn (||x0 || + 2 w, Jxn − Jx0 )}
x
n+1 = ΠCn+1 x0 , n ≥ 1,
where J is the duality mapping on E. Suppose {αn }∞ , {βn }∞
n=1
n=1
and {γn }∞ are sequences in (0, 1) such that lim inf αn (1 − αn ) >
n=1
n→∞
0,
lim βn = 0 and {λn }∞ ⊂ [a, b] for some a, b with 0 < a <
n=1
n→∞
2
b < c 2α , where 1 is the 2-uniformly convexity constant of E and c {rk,n }∞ ⊂ (0, ∞), (k = 1, 2, ..., m) satisfying lim inf rk,n > 0, (k =
n=1
n→∞
1, 2, ..., m). Suppose that for each bounded subset D of C, the ordered pair ({Tn }, D) satisfies either condition AKTT or condition ∗ AKTT.
Let T be the mapping from C into E defined by T x := lim Tn x for all n→∞
x ∈ C and suppose that T is closed and F (T ) = ∩∞ F (Tn ). Then,
n=1
{xn }∞ converges strongly to ΠF x0 . | en_US |