Let us consider the net iterative scheme as follows:
(16)
where V
i
=k
i
I+(1−k
i
)T
i
, f:C→H is a ρ-contraction mapping, S:C→H is a nonexpansive mapping, is a countable family of k
i
-strict pseudo-contraction mappings and . Set α
0=1, {α
n
}⊂(0,1) is a strictly decreasing sequence and {β
n
}⊂(0,1). As we will see the convergence of the scheme depends on the choice of the parameters {α
n
} and {β
n
}. We list some possible hypotheses on them:
-
(H1)
there exists ? > 0 such that ß
n
=? a
n
;
-
(H2)
lim
n?8
ß
n
/a
n
=t ? [ 0, 8);
-
(H3)
lim
n?8
a
n
=0 and ;
-
(H4)
;
-
(H5)
;
-
(H6)
lim
n?8
|a
n
-a
n-1|/a
n
=0;
-
(H7)
lim
n?8
|ß
n
-ß
n-1|/ß
n
=0;
-
(H8)
lim
n?8
[ |a
n
-a
n-1|+|ß
n
-ß
n-1|]/a
n
ß
n
=0;
-
(H9)
there exists a constant K > 0 such that?
.
Proposition 1
Assume that (H1) holds. Then {x
n
} and {y
n
} are bounded.
Proof
Let Then we have
(17)
So, by induction, one can obtain that
(18)
Hence {x
n
} is bounded. Of course {y
n
} is bounded too. □
Proposition 2
Suppose that (H1) and (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then
-
(1)
{x
n
} is asymptotically regular, that is,
(19)
-
(2)
the weak cluster points set .
Proof
Set . From (16) and since P
C
is a nonexpansive mapping, we have
(20)
(21)
By definition of y
n
one obtain that
(22)
So, substituting (22) in (21), we obtain
(23)
By Proposition 1, we say
So, we have
(24)
So, if (H4) and (H5) hold, we obtain the asymptotic regularity by Lemma 5, if instead, (H6) and (H7) hold, from (H1), we can write
(25)
By Lemma 5, we obtain the asymptotics regularity.
In order to prove (2), since V
i
x
n
∈ C for each i ≥ 1 and , we have
(26)
Now, fixing , from (16), we have
It follows that
(27)
Now, from Lemma 9 and (27), we get
By (H1) and (H3), it follows that β
n
→0, as n→∞, so that
(28)
Since for each i≥1 and {α
n
} is strictly decreasing, one has
(29)
Hence, we obtain
Since {x
n
} is asymptotically regular and demiclosedness principle, we obtain the proposition. □
Corollary 1
Suppose that the hypotheses of Proposition 2 hold. Then
-
(i)
lim
n?8
?x
n
-y
n
?=0;
-
(ii)
lim
n?8
?x
n
-V
i
y
n
?=0, ?i=1;
-
(iii)
lim
n?8
?y
n
-V
i
y
n
?=0, ?i=1.
Proof
To prove (i), we can observe that
Since β
n
→0 as n→∞, we obtain (i).To prove (i i), we observe that
and
Since ∥y
n
−x
n
∥→0 and ∥x
n
−V
i
x
n
∥→0 as n→∞, ∀i≥1, then ∥y
n
−V
i
x
n
∥→0, that is, we obtain (i i). To prove (i i i), we can observe that
By (i) and (i i), we obtain (i i i). □
Theorem 1
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and be a countable family of k
i
-strict pseudo-contraction mappings and . Let α
0=1, and x
1 ∈ C and define the sequence {x
n
} by
(30)
where {α
n
}⊂(0,1) and {α
n
} is a strictly decreasing sequence, V
i
=k
i
I+(1−k
i
)T
i
, {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
(31)
Proof
First of all, since is a contraction. By Banach contraction principle, so there exists a unique such that , Moreover, from Lemma 3(1), we have
Since (H2) implies (H1), thus {x
n
} is bounded. Moreover, since either (H4) and (H5) or (H6) and (H7) then {x
n
} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of x
n
, that is, ω
w
(x
n
), is a subset of
.
Let be a subsequence of {x
n
} such that
and ’. So, it follows that . Then, we also have
Set , we obtain
(32)
By Lemma 3(1), we have
(33)
From (32) and (33), it follows that
∥x
n+1 − z∥2
Let and for all n≥1. Since
and , we have
Hence, by Lemma 5, we conclude that x
n
→z as n→∞. □
Remark 1
In the iterative scheme (30), if we set f≡0, then we get . In this case, from (31), it follows that
That is
Therefore, the point z is the unique solution to the following quadratic minimization problem:
By changing the restrictions on parameters in Theorem 1, we obtain the following results.
Theorem 2
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and be a countable family of k
i
-strict pseudo-contraction mappings and . Let α
0 = 1, and x
1 ∈ C and define the sequence {x
n
} by
(34)
where {α
n
}⊂(0,1) and {α
n
} is a strictly decreasing sequence, V
i
=k
i
I+(1−k
i
)T
i
, {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ ∈ (0, ∞), (H3), (H8) and (H9). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
(35)
Proof
First, we shows that (49) has the unique solution. Let x
′ and x
∗ be two solutions. Then, since x
′ is solution, for y=x
∗ one has
(36)
and
(37)
Adding (36) and (37), we obtain
so x
′=x
∗. Also now the condition (H2) with 0<τ<∞ implies (H1) so the sequence {x
n
} is bounded. Moreover, since (H8) implies (H6) and (H7), then {x
n
} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of x
n
, i.e., ω
w
(x
n
), is a subset of
.
From (20)-(24), we observe that
Let γ
n
=(1−ρ)α
n
and . From condition (H3) and (H8), we have
By Lemma 5, we obtain
From (34), we have
It follows that
Let . For all , we get
(38)
By Lemma 4, we have
(39)
(40)
and
(41)
By Lemma 3(1), we obtain
(42)
Now, from (38)-(42), it follows that
(43)
since v
n
→0 and (I−V
i
)y
n
→0, as n→∞, then every weak cluster point of {x
n
} is also a strong cluster point. By Proposition 2, {x
n
} is bounded, thus there exists a subsequence converging to x
∗. For all by (38), we compute
(44)
Since v
n
→0,(I−V
i
)y
n
→0 for all i≥1, and ∥u
n
−u
n−1∥/α
n
→0, letting k→∞ in (44), we obtain
Since (49) has the unique solution, it follows that ω
w
(x
n
)={x
∗}. Since every weak cluster point of {x
n
} is also a strong cluster point, we conclude that x
n
→x
∗ as n→∞. This completes the proof.
If we take T
i
=T, for all i≥1, where T:C→C is a k-strict pseudo-contraction mapping in Theorem 1, then we get the following result: □
Corollary 2
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and T:C→C be a k-strict pseudo-contraction mapping such that F(T)≠∅. Let x
1 ∈ C and define the sequence {x
n
} by
(45)
where V=k I+(1−k)T,{α
n
}⊂(0,1) and {β
n
}⊂(0,1) are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {x
n
} converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality:
Taking k
i
=0, for all i≥1 in Theorem 1, then we get the following result:
Corollary 3
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and be a countable family of nonexpansive mappings and . Let α
0 = 1, x
1 ∈ C and define the sequence {x
n
} by
(46)
where {α
n
}⊂(0,1) and {α
n
} is a strictly decreasing sequence, {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
If we take k=0 in Corollary 2, then we get the following result:
Corollary 4
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and T:C→C be a nonexpansive mapping such that F(T)≠∅. Let x
1∈C and define the sequence {x
n
} by
(47)
where {α
n
}⊂(0,1),{β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {x
n
} converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality:
If we take T
i
=T, for all i≥1, where T:C→C is a k-strict pseudo-contraction mapping in Theorem 2, then we obtain the following result:
Corollary 5
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and T:C→C be a k-strict pseudo-contraction mapping and . Let x
1∈C and define the sequence {x
n
} by
(48)
where V=k I+(1−k)T, {α
n
}⊂(0,1), {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ ∈ (0, ∞), (H3), (H8) and (H9). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
(49)
If we take k
i
=0, for all i≥1 in Theorem 2, then we get the following result:
Corollary 6
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and be a countable family of nonexpansive mappings and . Let α
0 = 1, x
1 ∈ C and define the sequence {x
n
} by
(50)
where {α
n
}⊂(0,1) and {α
n
} is a strictly decreasing sequence, {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ ∈ (0, ∞), (H3), (H8) and (H9). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
(51)
If k=0 in Corollary 5, then we get the following Corollary:
Corollary 7
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S, T:C→C be nonexpansive mappings and . Let x
1∈C and define the sequence {x
n
} by
(52)
where {α
n
}⊂(0,1), {β
n
}⊂(0,1) and {α
n
} and {β
n
} are sequences satisfying the conditions (H2) with τ ∈ (0, ∞), (H3), (H8) and (H9). Then the sequence {x
n
} converges strongly to a point , which is the unique solution of the variational inequality:
(53)
Remark 2
Prototypes for the iterative parameters are, for example, α
n
=n
−θ and β
n
=n
−ω (with θ, ω > 0). Since |α
n
−α
n−1|≈n
−θ and |β
n
−β
n−1|≈n
−ω, it is not difficult to prove that (H 8) is satisfied for 0<θ, ω<1and (H 9) is satisfied if θ+ω≤1.
Remark 3
Theorem 1 and Theorem 2 extend and improve the result of Gu et al. (2011) from the countable family of nonexpansive mappings to more general the countable family of strictly pseudo contraction mappings.