New iterative schemes for solving variational inequality and fixed points problems involving demicontractive and quasi-nonexpansive mappings in Banach spaces
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Introduction
Let H be a real Hilbert space with inner product 〈⋅, ⋅〉H and norm ∥⋅∥H and K is a nonempty closed convex subset of H. A mapping A : H → H is said to be k-strongly monotone if there exists k ∈ (0, 1) such that for all x, y ∈ D(A),
if L < 1, T is called contraction and if L = 1, T is called nonexpansive. We denote by Fix(T) the set of fixed points of the mapping T, that is Fix(T) := {x ∈ D(T) : x = Tx}. We assume that Fix(T) is nonempty. If T is nonexpansive mapping, it is well known Fix(T) is closed and convex. A map T is called quasi-nonexpansive if ∥Tx – p∥ ≤ ∥x – p∥ holds for all x in K and p ∈ Fix(T). The mapping T : K → K is said to be firmly nonexpansive, if
(Example of a Demicontractive Function which is not Quasi-nonexpansive and is not Pseudocontractive). Letfbea real function defined byf(x) = –x2 – x; it can be seen thatf : [–2, 1] → [–2, 1]. This function is demicontractive on [–2, 1] and continuous. It is not quasi-nonexpansive and is not pseudocontractive on [–2, 1] (check for instance the condition of pseudocontractivity forx = –1 .5 andy = –0 . 6).
Fixed point thoery is one of the most powerful and important tools of modern mathematics and may be considered a core subject of nonlinear analysis. In the last few decades, the problem of nonlinear analysis with its relation to fixed point theory has emerged as a rapidly growing area of research because of its applications in game theory, optimization problem, control theory, integral and differential equations and inclusions, dynamic systems theory, signal and image processing, and so on. The crucial key of this success is due to the possibility of representing various problems arising in the above disciplines, in the form of an equivalent fixed point problem. Until now there have been many effective algorithms for solving fixed point problem, the reader can consult [5, 8, 11, 14, 17, 18, 22, 23].
Recently, iterative methods for nonexpansive mappings have been applied to solve convex minimization see, e.g., [10, 14, 19] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H
converges strongly to the unique solution of the minimization problem (4), where T is a nonexpansive mapping in H and A a strongly positive bounded linear operator. Marino and Xu [10] extended Moudafi’s results [14] and Xu’s results [19] via the following general iteration x0 ∈ H and
where {αn}n∈ℕ ⊂ (0, 1), A is bounded linear operator on H and T is a nonexpansive. Under suitable conditions, they proved the sequence {xn} defined by (6) converges strongly to x∗ ∈ Fix(T), which is the unique solution of the following variational inequality
As far as we know, all the recent and important results regarding approximation of solutions to variational inequality problems over the set of fixed points of nonlinear operators in the literature have been done for monotone operators over the set of fixed points of nonexpansive mappings. Furthermore, it is well known that accretive operators is an extension of monotone operators in Banach spaces and the class of demicontractive mappings contains those of nonexpansive, quasi-nonexpansive and strictly pseudo-contractive mappings with nonempty fixed point sets as subclasses.Thus, it is natural to extend the known results on variational inequality problems over the set of fixed points of nonexpansive mappings to variational inequality problems involving accretive operators over the set of common fixed points of demicontractive and quasi-nonexpansive mappings. This leads to this important natural question.
Question 3
Can we construct an iterative method with a strongly accretive and Lipschitzian operator for solving a variational inequality problem with quasi-nonexpansive and demicontractive mappings in real Banach spaces?
Our aim in this paper is to answer the above question in the affirmative. Thus, we introduce an iterative algorithm for solving variational inequality problems involving accretive operators over the set of common fixed points of demicontractive and quasi-nonexpansive mappings in Banach spaces. The results obtained here extend and unify the result of Marino and Xu [10], Sow [18] and most of the recent results in this direction. Our technique of proof is of independent interest.
Preliminairies
Let E be a Banach space with norm ∥⋅∥ and dual E∗. Let φ : [0, +∞) → [0, ∞) be a strictly increasing continuous function such that φ(0) = 0 and φ(t) → +∞ as t → ∞. Such a function φ is called gauge. Associed to a gauge a duality map Jφ : E → 2E∗ defined by:
If the gauge is defined by φ(t) = t, then the corresponding duality map is called the normalized duality map and is denoted by J. Hence the normalized duality map is given by
exists for each x, y ∈ S, (see e.g., [4] for more details on duality maps).
Remark 2
Note also that a duality mapping exists in each Banach space. We recall from [1] some of the examples of this mapping in lp, Lp, Wm,p-spaces, 1 < p < ∞.
[7] LetEbe a Banach space satisfying Opial’s property, Kbe a closed convex subset ofE, andT : K → Kbe a nonexpansive mapping such thatF(T) ≠ ∅. Then I – T is demiclosed; that is,
$$\begin{array}{}
\displaystyle
\langle x - y\:,\: J_{q}(x) - J_{q}(y)\rangle \: \leq c_{1}\|x - y\|^{q},\;\;
~\forall~ \: x,y \in E.
\end{array}$$
Lemma 8
[20] LetEbe a uniformly convex real Banach space. For arbitraryr > 0, letB(0)r := {x ∈ E : ||x|| ≤ r} and λ ∈ [0, 1]. Then there exists a continuous, strictly increasing and convex function
[21] Assume that {an} is a sequence of nonnegative real numbers such thatan+1 ≤ (1 – αn)an + αnσn + βn, n ≥ 0, where {αn}, {βn} and {σn} satisfy the conditions:
[13] Let tn be a sequence of real numbers that does not decrease at infinity in a sense that there exists a subsequence tni of tn such that tni such that tni ≤ tni+1for alli ≥ 0. For sufficiently large numbersn ∈ ℕ, an integer sequence {τ(n)} is defined as follows:
([12], Proposition 2.1). AssumeKis a closed convex subset of a Hilbert spaceH. LetT : K → Kbe a self-mapping ofC. IfTis ak-demicontractive mapping, then the fixed point setFix(T) is closed and convex.
Lemma 12
[12] LetKbe a nonempty closed convex subset of a real Hilbert spaceHandT : K → Kbe a mapping.
IfTis ak-strictly pseudo-contractive mapping, thenTsatisfies the Lipschitzian condition
IfTis ak-strictly pseudo-contractive mapping, then the mapping I – T is demiclosed at 0.
Lemma 13
[18] Letq > 1 be a fixed real number andEbe aq-uniformly smooth real Banach space with constantdq. LetKbe a nonempty, closed convex subset ofEandA : K → Ebe ak-strongly accretive andL-Lipschitzian operator withk > 0, L > 0. Assume that$\begin{array}{}
\displaystyle
0 \lt \eta \lt
\Big(\dfrac{kq}{ d_qL{^q}}\Big)^{\frac{1}{q-1}}~~
and ~~\tau=\eta\Big(k-\dfrac{d_q L^q \eta^{q-1}}{q}\Big).
\end{array}$Then for eacht ∈ $\begin{array}{}
\displaystyle
\Big(0, min\{1,\,\, \dfrac{1}{\tau}\}\Big),
\end{array}$we have
In this section, we present our explicit iterative method for solving a variational inequality problem with quasi-nonexpansive and demicontractive mappings in a real Banach space.
Theorem 14
Letq > 1 be a fixed real number andEbe aq-uniformly smooth and uniformly convex real Banach space having a weakly continuous duality mapJφandf : E → Ebe anb-Lipschitzian mapping with a constantb ≥ 0. LetA : E → Ebe anμ-strongly accretive andL-Lipschitzian operator with$\begin{array}{}
\displaystyle
0 \lt \eta \lt \Big(\dfrac{\mu q}{d_qL^q}\Big)^{\frac{1}{q-1}}
\end{array}$and 0 ≤ γb < τ, where$\begin{array}{}
\displaystyle
\tau=\eta\Big(\mu-\dfrac{d_q L^q \eta^{q-1}}{q}\Big).
\end{array}$LetT1 : E → Ebe ak-demicontractive mapping andT2 : E → Ebe a quasi-nonexpansive mapping such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Let {xn} be a sequence defined as follows:
$\begin{array}{}
\displaystyle
\lim_{n\rightarrow \infty}\inf\beta_n(1-\beta_n) \gt 0.
\end{array}$Assume that I – T1and I – T2are demiclosed at origin. Then, the sequence {xn} generated by(10)converges strongly tox∗ ∈ Γ, which is the unique solution of the following variational inequality:
From the choice of η and γ, properties of QΓ, (ηA – γf) is strongly accretive, then the variational inequality (11) has a unique solution in Γ. Without loss of generality, we can assume $\begin{array}{}
\displaystyle
\alpha_n\in\Big(0, min\{1\,\, , \dfrac{1}{\tau}\}\Big).
\end{array}$ In what follows, we denote x∗ to be the unique solution of (11). Fixing p ∈ Γ. We prove that the sequence {xn} is bounded. Using (10), inequality (ii) of Theorem 7 and inequality (8), we can compute
We show that the sequence {xn} converges strongly to a point x∗. Now we divide the rest of the proof into two cases. Case 1. Assume that the sequence {∥xn – p∥} is monotonically decreasing. Then {∥xn – p∥} is convergent. Clearly, we have
We show that $\begin{array}{}
\displaystyle
\limsup_{n\to +\infty}
\end{array}$ 〈ηAx∗ – γf(x∗), Jφ(x∗ – xn)〉 ≤ 0. First, we note that there exists a subsequence {xnj} of {xn} such that xnj converges weakly to a in E and
Since {xnj} is bounded, there exists a subsequence {xnji} of {xnj} which converges weakly to a. Without loss of generality, we can assume that {xnj} converges weakly to the point a. From (19) and I – T1 is demiclosed, we obtain a ∈ Fix(T1). From Lemma 8, the fact that T2 is quasi-nonexpansive and (15), we have
Since znj ⇀ a, it follows from (24) and I – T2 is demiclosed that a ∈ Fix(T2). Therefore, a ∈ Γ. On the other hand, by using x∗ solves (11) and the assumption that the duality mapping Jφ is weakly continuous, we have,
Finally, we show that xn → x∗. In fact, since Φ(t) = $\begin{array}{}
\int_ {0}^{t}
\end{array}$φ(σ)dσ, ∀t ≥ 0, and φ is a gauge function, then for 1 ≥ k ≥ 0, Φ(kt) ≤ kΦ(t). From (10), Lemmas 6 and 13, observe that
Hence, by Lemma 9, we conclude that the sequence {xn} converge strongly to the point x∗ ∈ Γ.
Case 2. Assume that the sequence {∥xn – x∗∥} is not monotonically decreasing. Set Bn = ∥ xn – x∗∥ and τ : ℕ → ℕ be a mapping for all n ≥ n0 (for some n0 large enough) by τ(n) = max{k ∈ ℕ : k ≤ n, Bk ≤ Bk+1}. Obviously, {τ(n)} is a non-decreasing sequence such that τ(n) → ∞ as n → ∞ and Bτ(n) ≤ Bτ(n)+1 for n ≥ n0. From (16), we have
By same argument as in case 1, we can show that xτ(n) and yτ(n) are bounded in E and $\begin{array}{}
\displaystyle
\limsup_{\tau(n)\to +\infty}
\end{array}$ 〈ηAx∗ – γf(x∗), Jφ(x∗ – xτ(n))〉 ≤ 0. We have for all n ≥ n0,
Hence, $\begin{array}{}
\displaystyle
\lim_{n\rightarrow \infty}
\end{array}$Bn = 0, that is {xn} converges strongly to x∗. This completes the proof.□
We now apply Theorem 14 for solving variational inequality problems over the set of common fixed points of two nonexpansive mappings. In that case the demiclosedness assumption is not necessary.
Theorem 15
Letq > 1 be a fixed real number andEbe aq-uniformly smooth and uniformly convex real Banach space having a weakly continuous duality map andf : E → Ebe anb-Lipschitzian mapping with a constantb ≥ 0. LetA : E → Ebe anμ-strongly accretive andL-Lipschitzian operator with$\begin{array}{}
\displaystyle
0 \lt \eta \lt \Big(\dfrac{\mu q}{d_qL^q}\Big)^{\frac{1}{q-1}}
\end{array}$and 0 ≤ γb < τ, where$\begin{array}{}
\displaystyle
\tau=\eta\Big(\mu-\dfrac{d_q L^q \eta^{q-1}}{q}\Big).
\end{array}$LetT1 : E → EandT2 : E → Etwo nonexpansive mappings such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Let {xn} be a sequence defined as follows:
$\begin{array}{}
\displaystyle
\lim_{n\rightarrow \infty}\inf\beta_n(1-\beta_n) \gt 0.
\end{array}$Then, the sequence {xn} generated by(26)converges strongly tox∗ ∈ Γ, which is the unique solution of the following variational inequality:
Since every nonexpansive mapping is quasi-nonexpansive and 0-demicontractive. The proof follows Lemma 5 and Theorem 14.□
We now apply Theorem 14 for solving fixed point problems with demicontractive and quasi-nonexpansive mappings in E = lq, 1 < q < ∞.
Theorem 16
Assume thatE = lq, 1 < q < ∞ and f : E → Ebe anb-Lipschitzian mapping with a constantb ≥ 0. LetA : E → Ebe anμ-strongly accretive andL-Lipschitzian operator with$\begin{array}{}
\displaystyle
0 \lt \eta \lt \Big(\dfrac{\mu q}{d_qL^q}\Big)^{\frac{1}{q-1}}
\end{array}$and 0 ≤ γb < τ, where$\begin{array}{}
\displaystyle
\tau=\eta\Big(\mu-\dfrac{d_q L^q \eta^{q-1}}{q}\Big).
\end{array}$LetT1 : E → Ebe ak-demicontractive mapping andT2 : E → Ebe a quasi-nonexpansive mapping such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Let {xn} be a sequence defined as follows:
$\begin{array}{}
\displaystyle
\lim_{n\rightarrow \infty}\inf\beta_n(1-\beta_n) \gt 0.
\end{array}$Assume thatI – T1andI – T2are demiclosed at origin. Then, the sequence {xn} generated by(28)converges strongly tox∗ ∈ Γ, which is the unique solution of the following variational inequality:
Theorem 14 extends and generalizes the main result of Moudafi [14], Xu [19], Marino and Xu [10], Sow [18] and most of the recent results in this direction. In the following ways:
From a real Hilbert space to a real q-uniformly smooth and uniformly convex Banach space which admits a weakly sequentially continuous generalized duality mapping.
From nonexpensive mappings to a class of demicontractive and quasi-nonexpansive mapping.
Application to constrained optimization problems
Convex optimization theory is a powerful tool for solving many practical problems in operational research. In particular, it has been widely used to solve practical minimization problems over complicated constraints [3, 16], e.g., convex optimization problems with a fixed point constraint and with a variational inequality constraint. Consider the following constrained optimization problem: let H be a real Hilbert space and T1 : H → H be a k-demicontractive mapping and T2 : H → H be a quasi-nonexpansive mapping such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Given a convex objective function g : H → ℝ, the problem can be expressed as
The set of solutions of (17) is denoted by Sol(g).
Definition 1
Let H be a real Hilbert space. A function g : H → ℝ is said to be α-strongly convex if there exists α > 0 such that for every x, y ∈ H with x ≠ y and β ∈ (0, 1), the following inequality holds:
LetHbe a real Hilbert space andg : H → ℝ a real-valued differentiable convex function. Assume thatgis strongly convex. Then the differential map ∇g : H → His strongly monotone, i.e., there exists a positive constantμsuch that
LetKbe a nonempty, closed convex subset ofEbe normed linear space and letg : K → ℝ a real valued differentiable convex function. Thenx∗is a minimizer ofgoverKif and only ifx∗solves the following variational inequality 〈∇ g(x∗), y – x∗〉 ≥ 0 for ally ∈ K.
Remark 6
By Lemma 19, x∗ ∈ Sol(g) if and only if x∗ solves the following variational inequality problem:
$$\begin{array}{}
\displaystyle
\langle \nabla g (x^*), x^*-p \rangle \leq 0,\,\, \forall p \in \Gamma.
\end{array}$$
Hence, one has the following result.
Theorem 20
LetHbe a real Hilbert space andg : H → ℝ be a differentiable, strongly convex real-valued function withL-Lipschitz contin-uous gradient ∇g. LetT1 : H → Hbe ak-demicontractive mapping andT2 : H → Hbe a quasi-nonexpansive mapping such thatFix(T1) ∩ Fix(T2) ≠ ∅. Assume that 0 < η < $\begin{array}{}
\displaystyle
\dfrac{2\mu}{ L^2}
\end{array}$, I – T1andI – T2are demiclosed at origin. Let {xn} be a sequence defined as follows:
Now, we consider the following quadratic optimization problem:
$$\begin{array}{}
\displaystyle
\min_{x\in \Gamma}\,g(x):= \dfrac{1}{2}\langle Ax, x \rangle,
\end{array}$$
where A : H → H be a strongly positive bounded linear operator.
Applying Theorem 20, we obtain the following result.
Theorem 21
LetHbe a real Hilbert space andA : H → Hbe strongly bounded linear operator with coefficientμ > 0. LetT1 : H → Hbe ak-demicontractive mapping andT2 : H → Hbe a quasi-nonexpansive mapping such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Assume that 0 < η < $\begin{array}{}
\displaystyle
\dfrac{2\mu}{ \| A\|^2}
\end{array}$andI – T1andI – T2are demiclosed at origin. Let {xn} be a sequence defined as follows:
Here, α > 0 is the regularization parameter, g be a convex real-valued function with L-Lipschitz continuous gradient ∇g. We can see that the gradient ∇gα = ∇g + αI is (L + α)-Lipschitzian and α-strongly monotone.
Applying Theorem 20, we obtain the following result.
Theorem 22
LetHbe a real Hilbert space andgbe a convex real-valued function withL-Lipschitz continuous gradient ∇ g. LetT1 : H → Hbe ak-demicontractive mapping andT2 : H → Hbe a quasi-nonexpansive mapping such that Γ := Fix(T1) ∩ Fix(T2) ≠ ∅. Assume that 0 < η < $\begin{array}{}
\displaystyle
\dfrac{2\alpha}{{(L+\alpha)}^2},
\end{array}$I – T1andI – T2are demiclosed at origin. Let {xn} be a sequence defined as follows:
Then, the sequence {xn} generated by(38)converges strongly to a solution of(37).
Conclusion
In this work, we introduce and analyze a new iterative method based on a general iterative method with strongly accretive operator for approximating a common fixed points of quasi-nonexpansive and demicontractive mappings which is also the solution of some variational inequality problems in real Banach spaces. Our results are used to solve some constrained optimization problems. The class of demicontractive mappings contains those of quasi-nonexpansive, strictly pseudo-contractive and nonexpansive mappings as subclasses. The results obtained here extend and unify the result of Moudafi [14], Xu [19], Marino and Xu [10], Sow [18] and most of the recent results in this direction.