Lemma 3.1 Let E be a strictly convex and smooth Banach space, and D a nonempty closed subset of E. Suppose T : D → N(D) is a quasi-ϕ-nonexpansive multi-valued mapping. Then F(T) is closed and convex.
Proof. First, we show F(T) is closed. Let {x
n
} be a sequence in F(T) such that x
n
→ x*. Since T is quasi-ϕ-nonexpansive, we have
for all z ∈ T(x*) and for all n ∈ ℕ. Therefore,
By Lemma 2.1(c), we obtain x* = z. Hence, T(x*) = {x*}. So, we have x* ∈ F(T). Next, we show F(T) is convex. Let x, y ∈ F(T) and t ∈ (0,1), put p = tx + (1 - t)y. We show p ∈ F(T). Let w ∈ F(p), we have
By Lemma 2.1(c), we obtain p = w. Hence, T(p) = {p}. So, we have p ∈ F(T). Therefore, F(T) is convex.
Lemma 3.2. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : D → N(D) be a relatively nonexpansive multi-valued mapping. If {x
n
} is a bounded sequence such that limn→∞d(x
n
,Tx
n
) and x* = ΠF(T)x, then
Proof. From (S3) of the mapping T, we choose a subsequence of {x
n
} such that and
By Lemma 2.1(b), we immediately obtain that
Lemma 3.3. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : D → N(D) be a relatively nonexpansive multi-valued mapping. Let {x
n
} be a sequence in D defined as follows: u ∈ E, x1 ∈ D and
(3.1)
where w
n
∈ Tx
n
for all n ∈ ℕ, {α
n
}, {β
n
} are sequences in [0,1]. Then {x
n
} is bounded.
Proof. Let p ∈ F(T) and for all n ∈ ℕ. Then
for all n ∈ ℕ. By using (1.4), we have
and
This implies that {x
n
} is bounded.
Theorem 3.4 Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : D → N(D) be a relatively nonexpansive multivalued mapping. Let {α
n
} and {β
n
} be sequences in (0,1) satisfying
(C1) limn→∞, α
n
= 0;
(C2) ;
(C3) lim infn→∞β
n
(1- β
n
) > 0.
Then {x
n
} defined by (3.1) converges strongly to ΠF(T)u, where ΠF(T)is the generalized projection from E onto F(T).
Proof. By Lemma 3.1, F(T) is closed and convex. So, we can define the generalized projection ΠF(T)onto F(T). Putting u* = ∏F(T)u, by Lemma 3.3 we know that {x
n
} is bounded and hence, {w
n
} is bounded. Let g : [0,2r] → [0,∞) be a function satisfying the properties of Lemma 2.3, where r = sup{∥u∥, ∥x
n
∥, ∥w
n
∥ : n ∈ ℕ}. Put
Then
(3.2)
and
(3.3)
for all n ∈ ℕ. Put
It follows from (3.3) that
(3.4)
Let . Then for all n ∈ ℕ. It follows from (2.3) and (3.2) that
(3.5)
The rest of proof will be divided into two parts:
Case (1). Suppose that there exists n0 ∈ ℕ such that is nonincreasing. In this situation, {ϕ(u*, x
n
)} is then convergent. Then limn→∞(ϕ(u*, x
n
) - ϕ(u*, xn+1)) = 0. This together with (C1), (C3), and (3.4), we obtain
Therefore,
Since J-1 is uniformly norm-to-norm continuous on every bounded subset of E, we have
(3.6)
Since , we obtain
(3.7)
Then,
(3.8)
and
(3.9)
From (3.8), (3.9) and Lemma 2.3, we have
and
This together with (3.6) gives
(3.10)
From (3.7), (3.10) and invoking Lemma 3.2, we have
Hence the conclusion follows by Lemma 2.5.
Case (2). Suppose that there exists a subsequence {n
i
} of {n} such that
for all i ∈ ℕ. Then, by Lemma 2.7, there exists a nondecreasing sequence {m
k
} ⊂ ℕ, m
k
→ ∞ such that
for all k ∈ ℕ. This together with (3.4) gives
for all k ∈ N. Then, by conditions (C1) and (C3)
By the same argument as Case (1), we get
(3.11)
From (3.5), we have
(3.12)
Since , we have
In particular, since , we get
It follows from (3.11) that . This together with (3.12) gives
But for all k ∈ ℕ. We conclude that x
k
→ u*.
This implies that limn→∞x
n
= u* and the proof is finished.
Letting β
n
= β gives the following result.
Corollary 3.5. Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : D → N(D) be a relatively nonexpansive multivalued mapping. Let {x
n
} be a sequence in D defined as follows: u ∈ E,x1 ∈ D and
where w
n
∈ Tx
n
for all n ∈ ℕ, {α
n
} is a sequence in [0,1] satisfying condition (C1) and (C2), and β ∈ (0,1). Then {x
n
} converges strongly to ∏F(T)u.