Abstract
The purpose of this paper is to introduce a new iterative scheme for approximating the solution of a triple hierarchical variational inequality problem. Under some requirements on parameters, we study the convergence analysis of the proposed iterative scheme for the considered triple hierarchical variational inequality problem which is defined over the set of solutions of a variational inequality problem defined over the intersection of the set of common fixed points of a sequence of nearly nonexpansive mappings and the set of solutions of the classical variational inequality. Our strong convergence theorems extend and improve some known corresponding results in the contemporary literature for a wider class of nonexpansive type mappings in Hilbert spaces.
MSC:47J20, 47J25.
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1 Introduction
The classical variational inequality problem initially studied by Stampacchia [1] for a nonlinear operator is a problem which provides us such which satisfies
where C is a nonempty closed convex subset of a real Hilbert space H and D is a nonempty closed convex subset of C. The variational inequality (1.1) is denoted by . The set of solutions of (1.1) is denoted by , that is,
For , we use and .
In the framework of variational inequality problems, various problems arising in several branches of pure and applied sciences can be studied (see [2, 3]).
The equivalence relation between the variational inequality and fixed point problems can be seen by projection technique which plays an important role in developing an important role in developing some efficient methods for solving variational inequality problems and related optimization problems. The problem of finding the fixed points of a nonexpansive mapping is the subject of current interest related to variational inequality problems in functional analysis.
Over the set of fixed points of a nonexpansive mapping, several authors (see [4–7]etc.) have studied the variational inequality problem in a particular manner. This kind of variational inequality is named a hierarchical variational inequality; it is defined as follows:
where T and S are two nonexpansive mappings from a nonempty closed convex subset C of a real Hilbert space H into itself, and denotes the set of fixed points of the mapping S. One can easily observe that is equivalent to the fixed point problem , that is, is a fixed point of the nonexpansive mapping , where is the metric projection from H onto a nonempty closed convex subset of H.
After all, in the scenario of variational inequality problem, we eagerly discuss such kind of variational inequality problem which is defined over the set of solutions of a variational inequality and the set of fixed points of a nonexpansive mapping, having a triple structure in contrast with bilevel programming problems or hierarchical constrained optimization problems or hierarchical fixed point problems. This kind of variational inequality is called the triple hierarchical variational inequality (see [8, 9]), which is also called the triple hierarchical constrained optimization problem (see [8]), and it is defined as follows:
where is the set of solutions of , and mappings A, F, and S are inverse strongly monotone, strongly monotone and Lipschitz continuous, and nonexpansive from a nonempty closed convex subset C of a real Hilbert space H into itself, respectively.
If is nonempty, then the metric projection is well defined. The minimum norm solution of exists uniquely and is exactly the nearest point projection of the origin to , that is, . Alternatively, is the unique solution of the quadratic minimization problem:
Finding of this minimum norm solution is an interesting problem. In this context, Yao et al. [10] proposed two iterative schemes in an implicit and an explicit both ways to find the minimum norm solution of . They proved two strong convergence results by regularizing the nonexpansive mapping T using contractions.
Recently, Ceng et al. [11], motivated by the results of Yao et al. [10] introduced and studied two iterative schemes, one of which was an implicit while other was an explicit one. They proved two strong convergence results by the considered iterative schemes under suitable conditions on parameters for considered triple hierarchical variational inequalities for both cases. Some hybrid steepest-descent-like methods with variable parameters for triple hierarchical variational inequalities are also studied in Ceng et al. [12]. The importance of the triple hierarchical variational inequalities and a nice survey on this topic is given in [13].
In 2005, the first author introduced the class of nearly nonexpansive mappings [14, 15] which is an important generalization of the class of nonexpansive mappings. Let C be a nonempty subset of a Banach space X. Fix a sequence in with . A mapping is said to be nearly nonexpansive with respect to the sequence if for each ,
We now discuss the notion of the sequence of nearly nonexpansive mappings.
Let C be a nonempty subset of a Banach space X. Let be a sequence of mappings from C into itself. We denote by the set of common fixed points of the sequence , that is, . Fix a sequence in with , and let be a sequence of mappings from C into X. Then the sequence is called a sequence of nearly nonexpansive mappings (see [16]) with respect to a sequence if
Clearly, the sequence of nearly nonexpansive mappings can easily be seen to be a wider class of sequence of nonexpansive mappings.
Motivated and inspired by the works mentioned above, we introduce an explicit iterative scheme that generates a sequence and prove that this sequence converges strongly to a unique solution of the considered triple hierarchical variational inequality problem defined over the set of solutions of a variational inequality problem which is defined over the intersection of the set of common fixed points of a sequence of nearly nonexpansive mappings and the set of solutions of the classical variational inequality problem. Our results generalize the result of Ceng et al. [11] in the context of the sequence of nearly nonexpansive mappings and in some other remarkable senses. Our results also extend the result of Yao et al. [10] and many other related works.
2 Preliminaries
Throughout this paper, we denote by → and ⇀ the strong convergence and weak convergence, respectively. The symbol ℕ stands for the set of all natural numbers and denotes the set of all weak limits of the sequence .
Let C be a nonempty subset of a real Hilbert space H with inner product and norm , respectively. A mapping is called
-
(1)
monotone if
-
(2)
η-strongly monotone if there exists a positive real number η such that
-
(3)
α-inverse strongly monotone if there exists a positive real number α such that
-
(4)
k-Lipschitzian if there exists a constant such that
-
(5)
ρ-contraction if there exists a constant such that
-
(6)
nonexpansive if for all ,
-
(7)
λ-strictly pseudocontractive if there exists such that
where I is the identity mapping. Note that if is λ-strictly pseudocontractive, then the mapping is -inverse strongly monotone.
Let C be a nonempty closed convex subset of H. Then, for any , there exists a unique nearest point in C, denoted by , such that
The mapping is called the metric projection from H onto C (see Agarwal et al. [14] for some other information related to ).
Let be a monotone and k-Lipschitz continuous mapping and let be the normal cone to C at , i.e.,
Define
Then T is a maximal monotone and if and only if .
Let C be a nonempty subset of a real Hilbert space H and let be two mappings. We denote , the collection of all bounded subsets of C. The deviation between and on [16], denoted by , is defined by
The following lemmas will be needed to prove our main results.
Lemma 2.1 ([17])
The metric projection mapping is characterized by the following properties:
-
(i)
for all ;
-
(ii)
for all and ;
-
(iii)
for all and ;
-
(iv)
for all .
Lemma 2.2 ([18])
Let C be a nonempty subset of a real Hilbert space H. Suppose that and . Let be a k-Lipschitzian and η-strongly monotone operator on C. Define the mapping by
Then W is a contraction provided . More precisely, for ,
where .
Lemma 2.3 ([14])
Let T be a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. Then is demiclosed at zero, i.e., if is a sequence in C weakly converging to some and the sequence strongly converges to 0, then .
Lemma 2.4 ([19])
Assume is a sequence of nonnegative real numbers such that
where and are sequences of nonnegative real numbers which satisfy the conditions:
-
(i)
and ;
-
(ii)
, or
(ii)′ is convergent.
Then .
Lemma 2.5 ([20])
Let C be a nonempty closed convex subset of a real Hilbert space H and let () such that . Let be nonexpansive mappings with and let . Then T is nonexpansive from C into itself and .
Proposition 2.1 ([21])
Let C be a nonempty subset of a real Hilbert space H. Let be an α-inverse strongly monotone mapping. Then, the mapping is nonexpansive from C into H, if .
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a ρ-contraction mapping. Let be a nonexpansive mapping and be an α-inverse strongly monotone mapping. Let be a sequence of nearly nonexpansive mappings from C into itself with respect to a sequence such that for all and and let T be a mapping from C into itself defined by for all . Suppose that , and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where , are sequences in , and is a sequence in (for some a, b with ) satisfying the following conditions:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
for each and ;
-
(iv)
, and ;
-
(v)
there are constants and satisfying
If the generated sequence is bounded and , then it converges strongly to the point , where is the unique solution of the triple hierarchical variational inequality of finding such that
Proof First of all, we assume that is bounded and . We divide the proof into several steps.
Step 1. .
Set and . Then, we have
From (3.2), we have
where M is a constant such that
Set and . Since is bounded, it follows that . Now, we have
From (3.4) and (3.5), we obtain
where . Note that and . Therefore, from conditions (ii), (iii), and Lemma 2.4, we have .
Step 2. for and as .
Set and . One can observe that
We also have
From (3.2), we have
Thus, we get
Since , and as , we obtain as . From (3.6), we have
Noticing that , , and . Thus, using conditions (ii) and (iii), and applying Lemma 2.4, we have
Step 3. as .
Let . Then using Lemma 2.1(iv), we have
It follows that
From (3.7) and (3.9), we have
which gives
We have , , , and as . Therefore, we have as .
Step 4. as .
Since , we get
It follows that
Also, we get
Note that
Thus,
Step 5. .
Note that A is an α-inverse strongly monotone mapping so that it is -Lipschitz continuous. Therefore, we have
Since is a bounded sequence in C, there exists a subsequence of which converges weakly to some in C. Since , it follows, from the demiclosedness principle of nonexpansive mappings, that . Now, let us show that . Let
Note that is maximal monotone and if and only if . Let , the graph of . Then, we have and hence . Thus, we have
On the other hand, from and , we have
and hence
Therefore, from and , we have
Letting limit we obtain . Thus, together with the maximal monotonicity of imply .
Step 6. .
From (3.2), we have
Therefore, we have
Set , . Note . Then, we have
Let . Observe that
The first and fourth terms in (3.11) are nonnegative due to the property of the projection operator given in Lemma 2.1(ii), and the monotonicity of , respectively. Note . Thus, from (3.11), we have
Noticing, from (3.10), that , we have . It is clear from (3.8) that . By assumption and the sequence is bounded; we see that is bounded. Thus, from (3.12), we have
This is sufficient to guarantee that , i.e., every weak limit point of the sequence solves the hierarchical variational inequality (3.1). In fact, if is a subsequence of such that , then, from (3.13), we have
that is,
Note that . Moreover, is monotone and Lipschitz continuous, and is closed and convex. Therefore, the inequality (3.14) is equivalent to the inequality (3.1) by the Minty lemma (see [22]). Thus, we have .
Now, we choose a subsequence of satisfying
Without loss of generality, we may further assume that . Note that . As is a solution of the triple hierarchical variational inequality (3.3), we obtain
Step 7. as .
Noticing that , , and . Set , where and . From (3.2), we have
It follows that
for some . Since , by using condition (v) we have
Note that
We observe that
Hence from (3.16), we get
for some constant . Therefore from (3.15) and (3.17), we have
where
Note that and . Using Lemma 2.4, we obtain . This completes the proof. □
If we put in (3.3), then this triple hierarchical variational inequality reduces to the variational inequality (3.18). Thus, the following is the direct consequence of Theorem 3.1.
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a nonexpansive mapping. Let be an α-inverse strongly monotone mapping and be a sequence of nearly nonexpansive mappings from C into itself with respect to a sequence such that for all and and let T be a mapping from C into itself defined by for all . Suppose that , , and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality (3.1), is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , to be bounded and , where , , and are sequences mentioned in Theorem 3.1 satisfying all the conditions of Theorem 3.1. Then the sequence converges strongly to a unique solution of the variational inequality of finding such that
Take and in Theorem 3.1, we have the following.
Corollary 3.1 (Ceng et al. [[11], Theorem 4.1])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a ρ-contraction mapping. Let S and T be nonexpansive mappings from C into itself such that . Suppose that and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where and are sequences in satisfying the conditions (i)-(ii) of Theorem 3.1. Suppose that , and , , where and are constants. Then the following hold:
-
(a)
If the generated sequence is bounded, then the sequence converges strongly to the point , where is the unique solution of the triple hierarchical variational inequality of finding such that
-
(b)
If the sequence in C for arbitrary , generated by the following iterative process:
for all , is bounded, then the sequence converges strongly to the unique solution of the variational inequality of finding such that
We now derive the result of Yao et al. [[10], Theorem 4.1] as a corollary.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a ρ-contraction mapping. Let S and T be nonexpansive mappings from C into itself such that . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where and are sequences in satisfying the conditions (i)-(ii) of Theorem 3.1. Suppose that and , , where and are constants. Then:
-
(a)
If the generated sequence is bounded, then the sequence converges strongly to the point , where is the unique solution of the variational inequality of finding such that
-
(b)
If the sequence in C for arbitrary , generated by the following iterative process:
for all , is bounded, then the sequence converges strongly to a minimum norm solution of the hierarchical variational inequality (3.19).
Again, we derive the following result as a corollary for S and T being two nonexpansive mappings.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a ρ-contraction mapping. Let S and T be nonexpansive mappings from C into itself and be an α-inverse strongly monotone mapping such that . Suppose that and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where , are sequences in and is a sequence in (for some a, b with ) satisfying the conditions (i)-(ii) of Theorem 3.1. Suppose that , , and , , where and are constants. Then the following hold:
-
(a)
If the generated sequence is bounded and , then the sequence converges strongly to the point , where is the unique solution of the triple hierarchical variational inequality of finding such that
-
(b)
If the sequence in C for arbitrary , generated by the following iterative process:
for all , is bounded and , then the sequence converges strongly to the unique solution of the variational inequality of finding such that
4 Applications
In this section, we present two applications of Theorem 3.1. The first application is concerned with the image recovery problem which is equivalent to finding a common fixed point of finitely many nonexpansive self mappings. The first application improves a number of results related to this context. The second application deals with a strictly pseudocontractive mapping.
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a ρ-contraction mapping. Let be a nonexpansive mapping and be an α-inverse strongly monotone mapping. Let such that . Let be nonexpansive mappings such that and assume that . Suppose that and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where , are sequences in and is a sequence in (for some a, b with ) satisfying all the conditions of Corollary 3.3. Then the following hold:
-
(a)
If the generated sequence is bounded and , then the sequence converges strongly to the point , where is the unique solution of the triple hierarchical variational inequality of finding such that
-
(b)
If the sequence in C for arbitrary , generated by the following iterative process:
for all , is bounded and , then the sequence converges strongly to the unique solution of the variational inequality of finding such that
Proof Lemma 2.5 implies that T is nonexpansive from C into itself and . Hence, the result follows from Corollary 3.3. □
Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a k-Lipschitzian and η-strongly monotone operator, and be a ρ-contraction mapping. Let be a nonexpansive mapping. Let be a λ-strictly pseudocontractive mapping and be a sequence of nearly nonexpansive mappings from C into itself with respect to a sequence such that for all and and let T be a mapping from C into itself defined by for all . Suppose that , , and , where . Assume that Ω, the set of solutions of the hierarchical variational inequality of finding such that
is nonempty. Consider the sequence in C for arbitrary , generated by the following iterative process:
for all , where , are sequences in and is a sequence in (for some a, b with ) satisfying the conditions (i)-(iv) of Theorem 3.1. Suppose that , and , where and are constants. Then the following hold:
-
(a)
If the generated sequence is bounded and , then the sequence converges strongly to the point , where is the unique solution of the triple hierarchical variational inequality of finding such that
-
(b)
If the sequence in C for arbitrary , generated by the following iterative process:
for all , is bounded and , then the sequence converges strongly to the unique solution of the variational inequality of finding such that
Proof Put in Theorem 3.1, then A is -inverse strongly monotone. We also have and . Therefore, the conclusion follows from Theorem 3.1 and Theorem 3.2. □
5 Numerical example
In this section, we discuss the following example which shows the effectiveness and convergence of iteratively generated sequence by the considered scheme (3.2) of Theorem 3.1.
Example 5.1 Let and . Let A, S, and T be mappings defined by , , and for all .
Let be mappings defined by and for all . Define , , and in by , , and , where . It is clear that S and T are nonexpansive self mappings, and A is 2-inverse strongly monotone. Note F is a 2-Lipschitzian and 2-strongly monotone, and g is a -contraction mapping. Here, , , and . We take , , , , and . Note that . Observe that with for all and . The iterative algorithm (3.2) can be written as
where
We observe that
and
Let . For , we have
Thus, .
Next we show that for all . Note . Suppose that for some . Now, for , we have
Thus, by mathematical induction, we get for all . Therefore, for all . It can be seen from Table 1 and Figure 1 that converges to 0 (see also (5.3)). Thus, all the assumptions of Theorem 3.1 are satisfied. Therefore, the iteratively generated sequence defined by (3.2) converges strongly to , which is also the unique solution of the triple hierarchical variational inequality (3.3).
The numerical values of up to have been calculated in Table 2 and convergence of sequence is given in Figure 2. Finally, mathematically, we show that and as . Note
which implies that
Therefore, from (5.2) we get as .
From (5.1), we have
which implies that
We have and as . Thus, we obtain
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Sahu, D., Kang, S.M., Sagar, V. et al. Iterative methods for triple hierarchical variational inequalities and common fixed point problems. Fixed Point Theory Appl 2014, 244 (2014). https://doi.org/10.1186/1687-1812-2014-244
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DOI: https://doi.org/10.1186/1687-1812-2014-244