Strong convergence of an new iterative method for a zero of accretive operator and nonexpansive mapping

Let E be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence {x_n} by the algorithm , where {a_n} and {r_n} are two sequences satisfying certain conditions, denotes the resolvent (I + r_nA)^-1 for r_n > 0, F be a strongly positive bounded linear operator on E is , and ϕ be a MKC on E. Strong convergence of the algorithm {x_n} is proved assuming E either has a weakly continuous duality map or is uniformly smooth.

MSC: 47H09; 47H10

Let E be a real Banach space, C a nonempty closed convex subset of E, and T : C → C a mapping. Recall that T is nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ for all x, y ∈ C. A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ C, Tx = x}.

It is assumed throughout the paper that T is a nonexpansive mapping such that . The normalized duality mapping J from a Banach space E into is given by J(x) = {f ∈ E* : 〈x, f〉 = ∥x∥² = ∥f∥²}, x ∈ E, where E* denotes the dual space of E and 〈.,.〉 denotes the generalized duality pairing.

Theorem 1.1. (Banach [1]). Let (X, d) be a complete metric space and let f be a contraction on X, that is, there exists r ∈ (0, 1) such that d(f(x), f(y)) ≤ rd(x, y) for all x, y ∈ X. Then f has a unique fixed point.

Theorem 1.2. (Meir and Keeler [2]). Let (X, d) be a complete metric space and let ϕ be a Meir-Keeler contraction (MKC, for short) on X, that is, for every ε > 0, there exists δ > 0 such that d(x, y) < ε + δ implies d(ϕ(x), ϕ(y)) < ε for all x, y ∈ X. Then ϕ has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

Let F be a strongly positive bounded linear operator on E, that is, there exists a constant such that

where I is the identity mapping and J is the normalized duality mapping.

Let D be a subset of C. Then Q : C → D is called a retraction from C onto D if Q(x) = x for all x ∈ D. A retraction Q : C → D is said to be sunny if Q(x + t(x - Q(x))) = Q(x) for all x ∈ C and t ≥ 0 whenever x + t(x - Q(x)) ∈ C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. In a smooth Banach space E, it is known (cf. [[3], p. 48]) that Q : C → D is a sunny nonexpansive retraction if and only if the following condition holds:

Recall that an operator A with domain D(A) and range R(A) in E is said to be accretive, if for each x_i ∈ D(A) and y_i ∈ Ax_i, i = 1, 2, there is a j ∈ J(x₂ - x₁) such that

An accretive operator A is m-accretive if R(I + λA) = E for all λ > 0. Denote by N(A) the zero set of A; i.e.,

Throughout the rest of this paper it is always assumed that A is m-accretive and N(A) is nonempty. Denote by J_r the resolvent of A for r > 0:

Note that if A is m-accretive, then J_r : E → E is nonexpansive and F(J_r) = N(A) for all r > 0. We also denote by A_r the Yosida approximation of A, i.e., . It is well known that J_r is a nonexpansive mapping from E to C := D(A).

Recall that a gauge is a continuous strictly increasing function φ : [0, ∞) → [0, ∞) such that φ(0) = 0 and φ(t) → ∞ as t → ∞. Associated to a gauge φ is the duality mapping J_φ : E → E* defined by

Following Browder [4], we say that a Banach space E has a weakly continuous duality map if there exists a gauge φ for which the duality map J_φ is single-valued and weak-to-weak* sequentially continuous(i.e., if {x_n} is a sequence in E weakly convergent to a point x, then the sequence J_φ(x_n) converges weakly* to J_φ(x)). It is known that l^p has a weakly continuous duality map for all 1 < p < ∞, with gauge φ(t) = t^p-1. Set

Then

where ∂ denotes the subdifferential in the sense of convex analysis.

Recently, Hong-Kun Xu [5] introduced the following iterative scheme: for x₁ = x ∈ C,

where {a_n} and {r_n} are two sequences satisfying certain conditions, and denotes the resolvent (I + r_nA)^-1 for r_n > 0. He proved the strong convergence of the algorithm {x_n} assuming E either has a weakly continuous duality map or is uniformly smooth.

Motivated and inspired by the results of Hong-Kun Xu, we introduce the following iterative scheme: for any x₀ ∈ E,

where {a_n} and {r_n} are two sequences satisfying certain conditions, denotes the resolvent (I + r_nA)^-1 for r_n > 0, F be a strongly positive bounded linear operator on E is , and ϕ be a MKC on E. Strong convergence of the algorithm {x_n} is proved assuming E either has a weakly continuous duality map or is uniformly smooth. Our results extend and improve the corresponding results of Hong-Kun Xu [5] and many others.

In order to prove our main results, we need the following lemmas.

Lemma 2.1. [5]. Assume that E has a weakly continuous duality map J_φ with gauge φ,

(i) For all x, y ∈ E, there holds the inequality

(ii) Assume a sequence {x_n} in E is weakly convergent to a point x, then there holds the equality

Lemma 2.2. [6,7]. Let {s_n} be a sequence of nonnegative real numbers satisfying

where {λ_n}, {δ_n} and {γ_n} satisfy the following conditions:

(i) {λ_n} ⊂ [0,1] and ,

(ii) lim sup_n→∞ δ_n ≤ 0 or (iii) . Then lim_n→∞ s_n = 0.

Lemma 2.3. (The Resolvent Identity [8,9]). For λ > 0 and ν > 0 and x ∈ E,

Lemma 2.4. (see [ [10], Lemma 2.3]). Assume that F is a strongly positive linear bounded operator on a smooth Banach space E with coefficient and 0 < ρ ≤ ∥F∥^-1. Then,

Lemma 2.5. (see [ [11], Lemma 2.3]). Let ϕ be a MKC on a convex subset C of a Banach space E. Then for each ε > 0, there exists r ∈ (0,1) such that

Lemma 2.6. Let E be a reflexive Banach space which admits a weakly continuous duality map J_φ with gauge φ. Let T : E → E be a nonexpansive mapping. Now given ϕ : E → E be a MKC, F be a strongly positive linear bounded operator with coefficient . Assume that , the sequence {x_t} defined by x_t = tγϕ(x_t) + (I - tF)Tx_t. Then T has a fixed point if and only if {x_t} remains bounded as t → 0⁺, and in this case, {x_t} converges as t → 0⁺ strongly to a fixed point of T. If , then uniquely solves the variational inequality

Proof. The definition of {x_t} is well defined. Indeed, from the definition of MKC, we can see MKC is also a nonexpansive mapping. Consider a mapping S_t on E defined by

It is easy to see that S_t is a contraction. Indeed, by Lemma 2.4, we have

for all x, y ∈ E. Hence S_t has a unique fixed point, denoted as x_t, which uniquely solves the fixed point equation

We next show the sequence {x_t} is bounded. Indeed, we may assume and with no loss of generality t < ∥F∥^-1. Take p ∈ F(T) to deduce that, for t ∈ (0, 1),

Hence

and {x_t} is bounded.

Next assume that {x_t} is bounded as t → 0⁺. Assume t_n → 0⁺ and is bounded. Since E is reflexive, we may assume that for some z ∈ E. Since J_φ is weakly continuous, we have by Lemma 2.1,

Put

It follows that

Since

we obtain

On the other hand, however,

Combining Equations (2.2) and (2.3) yields

Hence, Tz = z and z ∈ F(T).

Finally, we prove that {x_t} converges strongly to a fixed point of T provided it remains bounded when t → 0.

Let {t_n} be a sequence in (0, 1) such that t_n → 0 and as n → ∞. Then the argument above shows that z ∈ F(T). We next show that . By contradiction, there is a number ε₀ > 0 such that . Then by Lemma 2.8, there is a number r ∈ (0, 1) such that

It follows that

Therefore,

Now observing that implies , we conclude from the last inequality that

It contradicts . Hence .

We finally prove that the entire net {x_t} converges strongly. Towards this end, we assume that two null sequences {t_n} and {s_n} in (0, 1) are such that

We have to show . Indeed, for p ∈ F(T). Since

we derive that

Notice

It follows that,

Now replacing t in (2.5) with t_n and letting n → ∞, noticing for z ∈ F(T), we obtain 〈(F - γϕ)z, J_φ(z - p)〉 ≤ 0. In the same way, we have .

Thus, we have

Adding up (2.6) gets

On the other hand, without loss of generality, we may assume there is a number ε such that , then by Lemma 2.5 there is a number r₁ such that . Noticing that

Hence and {x_t} converges strongly. Thus we may assume . Since we have proved that, for all t ∈ (0, 1) and p ∈ F(T),

letting t → 0, we obtain that

This implies that

Lemma 2.7. (see [12]). Assume that C₂ ≥ C₁ > 0. Then for all x ∈ E.

Lemma 2.8. [13]. Let C be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial's condition, and suppose T : C → E is a nonexpansive mapping. Then the mapping I - T is demiclosed at zero, that is x_n ⇀ x and ∥x_n - Tx_n∥ → 0, then x = Tx.

Lemma 2.9. In a smooth Banach space E there holds the inequality

Theorem 3.1. Suppose that E is reflexive which admits a weakly continuous duality map J_φ with gauge φ and A is an m-accretive operator in E such that . Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient . Assume

(i) ;

(ii) r_n → ∞.

Then {x_n} defined by (1.4) converges strongly to a point in F*.

Proof. First notice that {x_n} is bounded. Indeed, take p ∈ F* to get

By induction, we have

This implies that {x_n} is bounded and hence

We next prove that

lim sup_n→∞〈γϕ(p) - Fp, J_φ(x_n - p)〉 ≤ 0, where p = lim_t→0 x_t with .

Since {x_n} is bounded, take a subsequence of {x_n} such that

Since E is reflexive, we may further assume that . Moreover, since

we obtain

Taking the limit as k → ∞ in the relation

we get . That is, . Hence by (3.1) and Lemma 2.6 we have

Finally to prove that x_n → p, we apply Lemma 2.1 to get

An application of Lemma 2.2 yields that Φ(∥x_n - p∥) → 0. That is, ∥x_n - p∥ → 0, i.e., x_n → p. The proof is complete.

Theorem 3.2. Suppose that E is reflexive which admits a weakly continuous duality map J_φ with gauge φ and A is an m-accretive operator in E such that . Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient . Assume

(i) , and ;

(ii) r_n ≥ ε for all n and .

Then {x_n} defined by (1.4) converges strongly to a point in F*.

Proof. We only include the differences. We have

Thus,

If r_n-1 ≤ r_n, using the resolvent identity

we obtain

It follows from (3.2) that

where M > 0 is some appropriate constant. Similarly we can prove (3.3) if r_n-1 ≥ r_n. By assumptions (i) and (ii) and Lemma 2.2, we conclude that

This implies that

since . It follows that

Now if is a subsequence of {x_n} converging weakly to a point , then taking the limit as k → ∞ in the relation

we get ; i.e., . We therefore conclude that all weak limit points of {x_n} are zeros of A.

The rest of the proof follows that of Theorem 3.1.

Finally, we consider the framework of uniformly smooth Banach spaces. Assume r_n ≥ ε for some ε > 0 (not necessarily r_n → ∞), A is an m-accretive operator in E. Moreover let ϕ : E → E be a MKC and F be a strongly positive linear bounded operator on E. Since is nonexpansive, the map is a contraction and for each integer n ≥ 1 it has a unique fixed z_t,n ∈ E. Hence the scheme

is well defined.

Note that {z_t,n} is uniformly bounded; indeed, for all t ∈ (0, 1), n ≥ 1 and p ∈ F*. A key component of the proof of the next theorem is the following lemma.

Lemma 3.1. The limit is uniform for all n ≥ 1.

Proof. It suffices to show that for any positive integer n_t (which may depend on t ∈ (0, 1)), if is the unique point in E that satisfies the property

then converges as t → 0 to a point in F*. For simplicity put

It follows that

Note that Fix(V_t) = F* for all t. Note also that {w_t} is bounded; indeed, we have for all t ∈ (0, 1) and p ∈ F*. Since {V_t w_t} is bounded, it is easy to see that

Since r_n ≥ ε for all n, by Lemma 2.7, we have

Let {t_k} be a sequence in (0,1) such that t_k → 0 as k → ∞. Define a function f on E by

where LIM denotes a Banach limit on l^∞. Let

Then K is a nonempty closed convex bounded subset of E. We claim that K is also invariant under the nonexpansive mapping J_ε. Indeed, noting (3.8), we have for w ∈ K,

Since a uniformly smooth Banach space has the fixed point property for nonexpansive mappings and since J_ε is a nonexpansive self-mapping of E, J_ε has a fixed point in K, say w'. Now since w' is also a minimizer of f over E, it follows that, for w ∈ E,

Since E is uniformly smooth, the duality map J is uniformly continuous on bounded sets, letting λ → 0⁺ in the last equation yields

Since

we obtain

It follows that

Upon letting w = γϕ(w') - Fw' + w' in (3.9), we see that the last equation implies

Therefore, contains a subsequence, still denoted , converging strongly to w₁ (say). By virtue of (3.8), w₁ is a fixed point of J_ε; i.e., a point in F*.

To prove that the entire net {w_t} converges strongly, assume {s_k} is another null subsequence in (0, 1) such that strongly. Then w₂ ∈ F*.

Repeating the argument of (3.10) we obtain

In particular,

and

Adding up the last two equations gives

That is, w₁ = w₂. This concludes the proof.

Theorem 3.3. Suppose that E is a uniformly smooth Banach space and A is an m-accretive operator in E such that . Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient . Assume

(i) , and ;

(ii) lim_n→∞ = r_n = r,r ∈ R⁺, r_n ≥ ε for all n and .

Then {x_n} defined by (1.4) converges strongly to a point in F*.

Proof. Since

Thus

We next claim that , where with z_t,n = tγϕ(z_t,n) + (I - tF)J_rz_t,n.

For this purpose, let be a subsequence chosen in such a way that and . Moreover, since ∥x_n - J_rx_n∥ → 0, using Lemma 2.8, we know . Hence by Lemma 2.6, we have

Finally to prove that strongly, we write

Apply Lemma 2.9 to get

It follows that

where . By Lemma 2.2 and (3.16), we see that .

Remark 3.4. If γ = 1, F is the identity operator and ϕ(x_n) = u in our results, we can obtain Theorems 3.1, 4.1, 4.2, 4.4 and Lemma 4.3 of Hong-Kun Xu [5].

The authors declare that they have no competing interests.

The main idea of this paper is proposed by Meng Wen. All authors read and approved the final manuscript.

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