STRONG CONVERGENCE OF STRICT PSEUDO-CONTRACTIONS IN Q-UNIFORMLY SMOOTH BANACH SPACES

Abstract

In this paper, we introduce a general iterative algorithm for finding a common element of the common fixed point set of an infinite family of ${\lambda}_i$-strict pseudo-contractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in q-uniformly smooth Banach spaces. Then, we analyze the strong convergence of the iterative sequence generated by the proposed iterative algorithm under mild conditions.

1. Introduction

Throughout this paper, we denote by E and E∗ a real Banach space and the dual space of E respectively. Let C be a subset of E and T be a mapping on C. We use F(T) to denote the set of fixed points of T. Let q > 1 be a real number. The (generalized) duality mapping Jq : E → 2E∗ is defined by

for all x ∈ E, where ⟨·, ·⟩ denotes the generalized duality pairing between E and E∗. In particular, J = J2 is called the normalized duality mapping and Jq(x) = ∥x∥q−2 J2(x) for x ≠ 0. If E is a Hilbert space, then J = I where I is the identity mapping. It is well known that if E is smooth, then Jq is single-valued, which is denoted by jq. Among nonlinear mappings, nonexpansive mappings and strict pseudo-contractions are two kinds of the most important nonlinear mappings. The study of them has a very long history (see [1-16,19-31] and the references therein). Recall that a mapping T : C → E is nonexpansive if

A mapping T : C → E is λ-strict pseudo-contractive in the terminology of Browder and Petryshyn (see [2,3,4]), if there exists a constant λ > 0 such that

for every x, y ∈ C and for some jq(x − y) ∈ Jq(x − y). It is clear that (1.1) is equivalent to the following inequality

Remark 1.1. The class of strictly pseudo-contractive mappings has been studied by several authors (see, e.g., [2,3,4,20,22]). However, their iterative methods are far less developed though Browder and Petryshyn [24] initiated their work in 1967. As a matter of fact, strictly pseudo-contractive mappings have more powerful applications in solving inverse problems (see, e.g., [32]). Therefore it is interesting to develop the theory of iterative methods for strictly pseudo-contractive mappings.

In the early sixties, Stampacchia [33] first introduced variational inequality theory, which has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences (see [7,8,9,10,11,34] and the references therein). In 1968, Brezis [34] initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximal-point mappings due to Moreau [35]. Variational inclusions include variational, quasi-variational, variational-like inequalities as special cases. It can be viewed as innovative and novel extension of the variational principles and thus, has wide applications in the fields of optimization and control, economics and transportation equilibrium and engineering sciences. Recently, some new and interesting problems, which are called to be system of variational inequalitys/inclusions received many attentions. System of variational inequalitys/inclusions can be viewed as natural and innovative generalizations of the variational inequalities/inclusions and it can provide new insight regarding problems being studied and can stimulate new and innovative ideas for solving problem.

Ceng et al. [26] proposed the following new system of variational inequality problem in a Hilbert space H: find x∗, y∗ ∈ C such that

where λ, μ > 0 are two constants, A,B : E → E are two nonlinear mappings. This is called the new system of variational inequalities. If we add up the requirement that x∗ = y∗ and A = B, then problem (1.3) reduces to the classical variational inequality problem: find x∗ ∈ C such that

In order to find the solutions of the system of variational inequality problem (1.3), Ceng et al. [26] studied the following approximation method. Let the mappings A,B : C → H be inverse-strongly monotone, S : C → C be nonexpansive. Suppose that x1 = u ∈ C and {xn} is generated by

They proved that the iterative sequence defined by the relaxed extragradient method (1.5) converges strongly to a fixed point of S, which is a solution of the system of variational inequality (1.3).

On the other hand, in order to find the common element of the solutions set of a variational inclusion and the set of fixed points of a nonexpansive mapping T, Zhang et al. [6] introduced the following new iterative scheme in a Hilbert space H. Starting with an arbitrary point x1 = x ∈ H, define sequence {xn} by

where A : H → H is an α-cocoercive mapping, M : H → 2H is a maximal monotone mapping, S : H → H is a nonexpansive mapping and {αn} is a sequence in [0,1]. Under mild conditions, they obtained a strong convergence theorem.

Motivated by Zhang et al. [6] and Zeng et al. [26], Qin et al. [8] considered the following new system of variational inclusion problem in a uniformly convex and 2-uniformly smooth Banach space: find (x∗, y∗) ∈ E × E such that

The following problems are special cases of problem (1.7).

(1) If A = B and M1 = M2 = M, then problem (1.7) reduces to the problem: find (x∗, y∗) ∈ E × E such that

(2) If x∗ = y∗, problem (1.7) reduces to the problem: find x∗ ∈ E such that

Qin et al. [8] also introduced the following scheme for finding a common element of the solution set of the general system (1.7) and the fixed point set of a λ-strict pseudo-contraction. Starting with an arbitrary point x1 = u ∈ E, define sequences {xn} by

And they proved a strong convergence theorem under mild conditions.

One question arises naturally: Can we extend Theorem 2.1 of Zhang et al. [6], Theorem 3.1 of Qin et al. [8], Theorem 3.1 of Zeng et al. [26] from Hilbert spaces or 2-uniformly smooth Banach spaces to more broad q-uniformly smooth Banach spaces? We put forth another question: Can we get some more general results even without the condition of uniform convexity of Banach spaces ? However, the condition of uniform convexity of Banach spaces is necessary in Theorem 3.1 of Qin et al. [8], Yao et al. [36] and so on.

The purpose of this article is to give the affirmative answers to these questions mentioned above. Motivated by Zhang et al. [6], Qin et al. [8], Yao et al. [9], Hao [10], J. C. Yao [11], and Takahashi et al. [12], we consider a relaxed extragradient-type method for finding common elements of the solution set of a general system of variational inclusions for inverse-strongly accretive mappings and the common fixed point set of an infinite family of λi-strict pseudocontractions. Furthermore, we obtain strong convergence theorems under mild conditions to improve and extend the corresponding results.

 

2. Preliminaries

The norm of a Banach space E is said to be Gâteaux differentiable if the limit

exists for all x, y on the unit sphere S(E) = {x ∈ E : ∥x∥ = 1}. If, for each y ∈ S(E), the above limit is uniformly attained for x ∈ S(E), then the norm of E is said to be uniformly Gâteaux differentiable. The norm of E is said to be Fréchet differentiable if, for each x ∈ S(E), the above limit is attained uniformly for y ∈ S(E).

Let ρE : [0, 1) → [0, 1) be the modulus of smoothness of E defined by

A Banach space E is said to be uniformly smooth if Let q be a fixed real number with 1 < q ≤ 2. Then a Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρE(t) ≤ ctq. It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is q-uniformly smooth, then q ≤ 2 and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable. In particular, the norm of E is Fréchet differentiable.

Recall that, a mapping T : C → E is said to be L-Lipschitz if for all x, y ∈ C, there exists a constant L > 0 such that

In particular, if 0 < L < 1, then T is called contractive and if L = 1, then T reduces to a nonexpansive mapping.

For some η > 0, T : C → E is said to be η-strongly accretive, if for all x, y ∈ C, there exists η > 0, jq(x − y) ∈ Jq(x − y) such that

For some μ > 0, T : C → E is said to be μ-inverse strongly accretive, if for all x, y ∈ C there exists jq(x − y) ∈ Jq(x − y) such that

A set-valued mapping T : D(T) ⊆ E → 2E is said to be m-accretive if for any x, y ∈ D(T), there exists j(x − y) ∈ J(x − y), such that for all u ∈ T(x) and v ∈ T(y),

A set-valued mapping T : D(T) ⊆ E → 2E is said to be m-accretive if T is accretive and (I + ρT )(D(T)) = E for every (equivalently, for some scalar ρ > 0), where I is the identity mapping.

Let M : D(M) → 2E be m-accretive. Denote by JM,ρ the resolvent of M for ρ > 0:

It is known that JM,ρ is a single-valued and nonexpansive mapping from E to which will be assumed convex (this is so provided E is uniformly smooth and uniformly convex).

Let {Tn} be a family of mappings from a subset C of a Banach space E into itself with We say that {Tn} satisfies the AKTT-condition (see [17]), if for each bounded subset B of C,

The following proposition supports {Tn} satisfying AKTT-condition.

Proposition 2.1. Let C be a nonempty convex subset of a real q-uniformly smooth Banach space E. Assume that is a countable family of λi -strict pseudo-contractions with {λi} ⊂ (0, 1) and inf{λi : i ≥ 1} > 0 such that For each n ∈ ℕ, define Tn : C → C by

Let be a family of nonnegative numbers with k ≤ n such that

Then the following results hold:

Proof. (1) and (2) can be deduced directly from Lemma 2.11 in [20]. And the argument of (3) and (4) is similar to the section 4 (Applications) in [17] and so it is omitted. □

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

Lemma 2.2 ([16]). Let C be a closed convex subset of a strictly convex Banach space E. Let T1 and T2 be two nonexpansive mappings from C into itself with F(T1) ∩ F(T2) ≠ ∅. Define a mapping S by

where λ is a constant in (0, 1). Then S is nonexpansive and F(S) = F(T1) ∩ F(T2).

Lemma 2.3 ([19]). Let {αn}be a sequence of nonnegative numbers satisfying the property:

where {γn}, {bn}, {cn} satisfy the restrictions:

Then limn→∞ αn = 0.

Lemma 2.4 ([18]). Let q > 1. Then the following inequality holds:

for arbitrary positive real numbers a, b.

Lemma 2.5 ([19]). Let E be a real q-uniformly smooth Banach space, then there exists a constant Cq > 0 such that

In particular, if E is a real 2-uniformly smooth Banach space, then there exists a best smooth constant K > 0 such that

Lemma 2.6 ([17,23]). Suppose that {Tn} satisfy the AKTT-condition such that

Then limn→∞ supω∈B ∥Tω − Tnω∥ = 0 for each bounded subset B of C.

Lemma 2.7 ([22]). Let C be a nonempty convex subset of a real q-uniformly smooth Banach space E and T : C → C be a λ-strict pseudo-contraction. For α ∈ (0, 1), we define Tαx = (1 − α)x + αTx. Then, as α ∈ (0, μ], is nonexpansive such that F(Tα) = F(T).

Lemma 2.8 ([23]). Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E which admits weakly sequentially continuous generalized duality mapping jq from E into E∗ ( i.e., if for all {xn} ⊂ E with xn ⇀ x, implies that Let T : C → C be a nonexpansive mapping. Then, for all {xn} ⊂ C, if xn ⇀ x and xn − Txn → 0, then x = Tx.

Lemma 2.9 ([23]). Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E. Let V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants and Then for each the mapping S : C → E defined by S := (I − tμV) is a contraction with a constant 1 − tτ.

Lemma 2.10 ([23]). Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E. Let QC be a sunny nonexpansive retraction from E onto C, V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E be a L-Lipschitzian mapping with constant L ≥ 0 and T : C → C be a nonexpansive mapping such that F(T) ≠ ∅. Let Then the sequence {xt} defined by

has following properties:

Lemma 2.11 ([1]). Let C be a closed convex subset of a smooth Banach space E. Let be a nonempty subset of C. Let be a retraction and let j, jq be the normalized duality mapping and generalized duality mapping on E, respectively. Then the following are equivalent:

Lemma 2.12. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping jq from E into E∗. Let QC be a sunny nonexpan-sive retraction from E onto C, V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E be a L-Lipschitzian mapping with constant L ≥ 0 and T : C → C be a nonexpansive mapping such that F(T) ≠ ∅. Suppose that Let {xt} be defined by (2.1) for each Then {xt} converges strongly to x∗ ∈ F(T), which is the unique solution of the following variational inequality:

Proof. We firstly show the uniqueness of a solution of the variational inequality (2.2). Suppose that both are solutions of (2.2). It follows that

Adding up (2.3) and (2.4), we have

Notice that

Therefore and the uniqueness is proved. We use x∗ to denote the unique solution of (2.2).

Next, we prove that xt → x∗ as t → 0.

Since E is reflexive and {xt} is bounded due to Lemma 2.10 (i), there exists a subsequence {xtn} of {xt} and some point such that By Lemma 2.10 (ii), we have limt→0 ∥xtn − Txtn∥ = 0. Together with Lemma 2.8, we can get that Setting yt = tγfxt+(I −tμV)Txt, where Then, we can rewrite (2.1) as xt = QCyt. We claim that Thanks to Lemma 2.11, we have that

It follows from (2.5) and Lemma 2.9 that

Thus,

which implies that

Using that the duality map jq is weakly sequentially continuous from E to E∗ and noticing (2.6), we get that

Next, we shall prove that solves the variational inequality (2.2).

Since xt = QCyt = QCyt − yt + tγfxt + (I − tμV)Txt, we derive that

Note that for ∀z ∈ F(T),

It thus follows from Lemma 2.11 and (2.8) that

where M = supn≥0 {μk ∥xt − z∥q−1} < ∞. Now replacing t in (2.9) with tn and letting n → ∞, noticing (2.7) and Lemma 2.10 (ii), we obtain That is, is a solution of (2.2); Hence by uniqueness. Therefore xtn → x∗ as n → ∞. And consequently, xt → x∗ as t → 0. □

Lemma 2.13 ([20]).Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let A : C → E be a α-inverse-strongly accretive operator. Then the following inequality holds:

In particular, if then I − λA is nonexpansive.

Lemma 2.14. Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let Mi : D(Mi) → 2E be m-accretive with for i=1,2 and ρ1, ρ2 be two arbitrary positive constants. Let A,B : C → E be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let G : C → C be a mapping defined by

If then G : C → C is nonexpansive.

Proof. We have by Lemma 2.13 that for all x, y ∈ C,

which implies that G : C → C is nonexpansive. This completes the proof. □

Lemma 2.15. Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let Mi : D(Mi) → 2E be m-accretive with for i=1,2 and ρ1, ρ2be two arbitrary positive constants. Then, (x∗, y∗) ∈ C × C is a solution of general system (1.7) if and only if x∗ = Gx∗, where G is defined by Lemma 2.14.

Proof. Note that

and the above system is equivalent to

This completes the proof. □

 

3. Main results

Theorem 3.1. Let C be a nonempty closed convex subset of a strictly convex and real q-uniformly smooth Banach space E, which admits a weakly sequentially continuous generalized duality mapping jq : E → E∗. Let QC be a sunny nonex-pansive retraction from E onto C. Assume A,B : C → E are α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let Mi : D(Mi) → 2E be m-accretive with Suppose that V : C → E is k-Lipschitz and η-strongly accretive with constants k, η > 0, f : C → E is L-Lipschitz with constant L ≥ 0. be an infinite family of λn-strict pseudo-contractions with {λn} ⊂ (0, 1) and inf{λn : n ≥ 0} = λ > 0, such that and Define a mapping Tnx := (1 − σ)x + σSnx for all x ∈ C and n ≥ 0. For arbitrarily given x0 ∈ C and δ ∈ (0, 1), let {xn} be the sequence generated iteratively by

Assume that {αn} and {γn} are two sequences in (0,1) satisfying the following conditions:

Suppose in addition that satisfies the AKTT-condition. Let S : C → C be the mapping defined by Sx = limn→∞ Snx for all x ∈ C and suppose that Then {xn} converges strongly to x∗ ∈ F, which is the unique solution of the following variational inequality

Proof. Step 1. We show that sequences {xn} is bounded. By condition (ii) there is a positive number b such that lim supn→∞ γn < b < 1. Applying condition (i) and (ii), we may assume, without loss of generality, that {γn} ⊂ (0, b] and From Lemma 2.9, we deduce that ∥((1 − γn)I − αnμV)x − ((1 − γn)I − αnμV)y∥ ≤ ((1 − γn) − αnτ) ∥x − y∥ for ∀ x, y ∈ C. For x∗ ∈ F, it follows from Lemma 2.15 that

Putting y∗ = JM2,ρ2(x∗−ρ2Bx∗), then we can get that x∗ = JM1,ρ1 (y∗−ρ1Ay∗).

By Lemma 2.13, we obtain

It follows from (3.3) that

Combining Lemma 2.7 and the condition of we can deduce that

Substituting (3.5) into (3.4) and simplifying, we have that

It follows that

Hence, {xn} is bounded. {yn}, {kn} and {zn} are also bounded.

Step 2. We shall claim that ∥xn+1 − xn∥ → 0, as n → ∞. We observe that

This together with Lemma 2.7 implies that

At the same time, we observe that

Substituting (3.7) into (3.8), we have that

where satisfying the AKTT-condition, we deduce that

From (i), (ii), (3.9), (3.10) and Lemma 2.3, we deduce that

Notice that

which implies that

Combining conditions (i), (ii), (3.11) and (3.12), we deduce that

For any bounded subset B of C, we observe that

Since {Sn} satisfies the AKTT-condition, we have that

That is, {Tn} satisfies the AKTT-condition. Define a mapping T : C → C by Tx = limn→∞ Tnx for all x ∈ C. It follows that

Noticing that

we deduce that S : C → C is a λ-strict pseudo-contraction. In view of (3.14), Lemma 2.6 and the condition of 0 < σ ≤ d, where we have that T : C → C is a nonexpansive and F(T) = F(S). Hence we have

Let W : C → C be the mapping defined by

In view of Lemma 2.2, we see that W is nonexpansive such that

Noting that

we obtain

From Lemma 2.6, we can get that

Combing (3.13), (3.16) and (3.17), we deduce that

Define xt = QC[tγfxt + (I − tμV)Wxt]. From Lemma 2.11, we deduce that {xt} converges strongly to x∗ ∈ F(W) = F, which is the unique solution of the variational inequality of (3.2).

Step 3. We show that

where x∗ is the solution of the variational inequality of (3.2). To show this, we take a subsequence {xni} of {xn} such that

Without loss of generality, we may further assume that xni ⇀ z for some point z ∈ C due to reflexivity of the Banach space E and boundness of {xn}. It follows from (3.18) and Lemma 2.8 that z ∈ F(W). Since the Banach space E has a weakly sequentially continuous generalized duality mapping jq : E → E∗, we obtain that

Step 4. Finally we prove that limn→∞ ∥xn − x∗∥. Setting hn = αnγfxn + γnxn+[(1−γn)I −αnμV]yn, ∀n ≥ 0. Then by (3.1) we can write xn+1 = QChn. It follows from Lemmas 2.3 and Lemmas 2.10 that

which implies

Put γn = αn(τ − γL) and Applying Lemma 2.2 to (3.19), we obtain that xn → x∗ ∈ F as n → ∞. This completes the proof. □

Remark 3.1. Compared with the known results in the literature, our results are very different from those in the following aspects:

Remark 3.2. The variational inequality problem in a q-uniformly smooth Banach space E: finding x∗ such that

is also very interesting and important. As we can see that:

Corollary 3.2. Let C be a nonempty closed convex subset of a strictly con-vex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j : E → E∗. Let QC be a sunny non-expansive retraction from E onto C. Assume the mappings A,B : C → E are α-inverse-strongly accretive and β-inverse-strongly accretive, respectively.Let Mi : D(Mi) → 2E be m-accretive with Suppose V : C → E is a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E is a L-Lipschitzian with constant L ≥ 0. Let and 0 ≤ γL < τ where τ = μ(η − K2μk2). Let T : C → C be a nonexpansive with F = F(T) ∩ F(G) ≠ ∅. For arbitrarily given δ ∈ (0, 1) and x0 ∈ C , let {xn} be the sequence generated iteratively by

Assume that {αn} and {γn} are two sequences in (0, 1) satisfying the following conditions:

Then {xn} defined by (3.21) converges strongly to x∗ ∈ F, which is the unique solution of the following variational inequality:

 

4. Conclusion

In this research, a general iterative algorithm is proposed for finding a common element of the common fixed point set of an infinite family of λi-strict pesudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in q-uniformly smooth Banach spaces. Then we analyzed the strong convergence of the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, and extension of the corresponding results in some references.