Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Theorem B. Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping on C. Fix u ∈ C and define z_t ∈ C  as  z_t = tu + (1 − t)Tz_t for t ∈ (0,1). Then as t → 0, {z_t} converges strongly to a element of F(T) nearest to u.

Theorem H. Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping on C. Define a real sequence {α_n} in [0,1] by α_n = n^−θ,   0 < θ < 1. Define a sequence {x_n} by (1.3). Then {x_n} converges strongly to the element of F(T) nearest to u.

Lemma 1.1 (see [16].)Let D be a nonempty bounded closed convex subset of a Hilbert space H and let S = {T(t) : 0 ≤ t < ∞} be a nonexpansive semigroup on D. Then, for any 0 ≤ h < ∞,

Lemma 1.2 (see [17].)Let H be a Hilbert space, C a closed convex subset of H, and T : C → C a nonexpansive mapping with F(T) ≠ ∅. Then I − T is demiclosed, that is, if {x_n} is a sequence in C weakly converging to x and if {(I − T)x_n} strongly converges to y, then (I − T)x = y.

Lemma 1.3 (see [18].)Let C be a nonempty closed convex subset of a real Hilbert space H and let P_C be the metric projection from H onto C( i.e., for x ∈ H,   P_Cx is the only point in C such that ∥x − P_Cx∥ = inf {∥x − z∥:z ∈ C}). Given x ∈ H and z ∈ C. Then z = P_Cx if and only if there holds the relations

Lemma 1.4. Let H be a Hilbert space, f a α-contraction, and A a strongly positive linear bounded self-adjoint operator with the coefficient $$. Then, for $$,

That is, A − γf is strongly monotone with coefficient $$.

Proof. From the definition of strongly positive linear bounded operator, we have

On the other hand, it is easy to see Therefore, we have for all x, y ∈ H. This completes the proof.

Remark 1.5. Taking γ = 1 and A = I, the identity mapping, we have the following inequality:

Furthermore, if f is a nonexpansive mapping in Remark 1.5, we have

Lemma 1.6 (see [9].)Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with coefficient $$ and 0 < ρ ≤ ∥A∥⁻¹. Then $$.

Lemma 1.7 (see [12].)Let {α_n} be a sequence of nonnegative real numbers satisfying the following condition:

where {γ_n} is a sequence in (0,1) and {σ_n} is a sequence of real numbers such that

• (i)

  lim _n→∞γ_n = 0 and $$,

• (ii)

  either limsup _n→∞σ_n ≤ 0 or $$.

Then $$ converges to zero.

Lemma 2.1. Let H a real Hilbert space and S = {T(s) : 0 ≤ s < ∞} a nonexpansive semigroup on H such that F(S) ≠ ∅. Let {λ_t} _0<t<1 be a continuous net of positive real numbers such that lim _t→0λ_t = ∞. Let f : H → H be an α-contraction, A a strongly positive linear bounded self-adjoint operator of H into itself with coefficient $$. Assume that $$. Let {x_t} be a sequence defined by (1.13). Then

• (i)

  {x_t} is bounded for all t ∈ (0, ∥A∥⁻¹);

• (ii)

  lim _t→0∥T(τ)x_t − x_t∥ = 0 for all 0 ≤ τ < ∞;

• (iii)

  x_t defines a continuous curve from (0, ∥A∥⁻¹) into H.

Proof. (i) Taking p ∈ F(S), we have

It follows that This implies that {x_t} is not only bounded, but also that {x_t} is contained in $$ of center p and radius $$, for all fixed p ∈ F(S). Moreover for p ∈ F(S) and t ∈ (0, ∥A∥⁻¹),

(ii) Observe that

Taking $$ as D in Lemma 1.1 and passing to lim _t→0 in (2.4), we can obtain (ii) immediately.

(iii) Taking t₁, t₂ ∈ (0, ∥A∥⁻¹) and fixing p ∈ F(S), we see that

Thus applying (2.3), we arrive at It follows that where and This inequality, together with the continuity of the net {λ_t}, gives the continuity of the curve {x_t}.

Theorem 2.2. Let H be a real Hilbert space H and S = {T(s) : 0 ≤ s < ∞} a nonexpansive semigroup such that F(S) ≠ ∅. Let {λ_t} _0<t<1 be a net of positive real numbers such that lim_t→0λ_t = ∞. Let f be an α-contraction and let A be a strongly positive linear bounded self-adjoint operator on H with the coefficient $$. Assume that $$. Then sequence {x_t} defined by (1.13) strongly converges as t → 0 to x^* ∈ F(S), which solves the following variational inequality:

Equivalently, one has

Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of A − γf (Lemma 1.4) and it was proved in [9]. Next, we will use x^* ∈ F(S) to denote the unique solution of (2.10). To prove that x_t → x^*  (t → 0), we write, for a given p ∈ F(S),

Using x_t − p to make inner product, we obtain that It follows that which yields that Since H is a Hilbert space and {x_t} is bounded as t → 0, we have that if {t_n} is a sequence in (0,1) such that t_n → 0 and $$. By (2.15), we see $$. Moreover, by (ii) of Lemma 2.1 we have $$. We next prove that $$ solves the variational inequality (2.10). From (1.13), we arrive at For p ∈ F(S), it follows from (1.22) that Passing to lim _t→0, since {x_t} is a bounded sequence, we obtain that is, $$ satisfies the variational inequality (2.10). By the uniqueness it follows $$. In a summary, we have shown that each cluster point of {x_t} (as t → 0) equals x^*. Therefore, x_t → x^* as t → 0. The variational inequality (2.10) can be rewritten as This, by Lemma 1.3, is equivalent to This completes the proof.

Remark 2.3. Theorem 2.2 which include the corresponding results of Shioji and Takahashi [15] as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng [14] when A = I, the identity mapping and γ = 1.

Theorem 2.4. Let H be a real Hilbert space H and S = {T(s) : 0 ≤ s < ∞} a nonexpansive semigroup such that F(S) ≠ ∅. Let {s_n} be a positive real divergent sequence and let {α_n} and {β_n} be sequences in (0,1) satisfying the following conditions lim _n→∞α_n = lim _n→∞β_n = 0 and $$. Let f be an α-contraction and let A be a strongly positive linear bounded self-adjoint operator with the coefficient $$. Assume that $$. Then sequence {x_n} defined by (1.14) strongly converges to x^* ∈ F(S), which solves the variational inequality (2.10).

Proof. We divide the proof into three parts.

By using Lemma 1.7, we can obtain the desired conclusion easily.

Remark 2.5. If γ = 1 and A = I, the identity mapping, then Theorem 2.4 is reduced to Theorem 3.3 of Plubtieng and Punpaeng [14].

If the sequence {β_n} ≡ 0, then Theorem 2.4 is reduced to the following.

Corollary 2.6. Let H be a real Hilbert space H and S = {T(s) : 0 ≤ s < ∞} a nonexpansive semigroup such that F(S) ≠ ∅. Let {s_n} be a positive real divergent sequence and let {α_n} be a sequence in (0,1) satisfying the following conditions lim _n→∞α_n = 0 and $$. Let f be a α-contraction and let A be a strongly positive linear bounded self-adjoint operator with the coefficient $$. Assume that $$. Let {x_n} be a sequence generated by the following manner:

Then the sequence {x_n} defined by above iterative algorithm converges strongly to x^* ∈ F(S), which solves the variational inequality (2.10).

Remark 2.7. Corollary 2.6 includes Theorem 2 of Shioji and Takahashi [15] as a special case.

Remark 2.8. Theorem 2.2 and Corollary 2.6 improve Theorem 3.2 and Theorem 3.4 of Marino and Xu [9] from a single nonexpansive mapping to a nonexpansive semigroup, respectively.