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Systems of general nonlinear setvalued mixed variational inequalities problems in Hilbert spaces
Fixed Point Theory and Applications volume 2011, Article number: 31 (2011)
Abstract
In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear setvalued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semicontinuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear setvalued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g.
2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.
1. Introduction and preliminaries
Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉, and ·, respectively. Let CB(H) be the family of all nonempty, closed, and bounded sets in H. Let A, B : H → CB(H) be nonlinear setvalued mappings, g : H → H be a singlevalued mapping, and φ : H → (∞, +∞] be a proper convex lower semicontinuous function on H. For each fixed positive real numbers, ρ and η, we consider the following socalled system of general nonlinear setvalued mixed variational inequalities problems:
Find x*, y*∈ H, u* ∈ Ay*, v* ∈ Bx*, such that
We denote by SGNSM(A, B, g, φ, ρ, η), the set of all solutions (x*, y*, u*, v*) of the problem (1.1).
We shall now discuss several special cases of the problem (1.1).
Special cases of the problem (1.1) are as follows:

(I)
If g = I (: the identity operator), then, from the problem (1.1), we have the following system of nonlinear setvalued mixed variational inequalities problems:
Find x*, y* ∈ H, u* ∈ Ay*, v* ∈ Bx*, such that
We denote by SNSM(A, B, φ, ρ, η), the set of all solutions (x*, y*, u*, v*) of the problem (1.2).

(II)
If K is a closed convex subset of H and φ (x) = δ_{ K } (x) for all x ∈ K, where δ_{ K } is the indicator function of K defined by
then, from the problem (1.1), we have the following system of general nonlinear setvalued variational inequalities problems:
Find x*, y* ∈ K, u* ∈ Ay*, v* ∈ Bx*, such that
We denote by SGNS(A, B, g, K, ρ, η), the set of all solutions (x*, y*, u*, v*) of the problem (1.3).
The problem (1.3) was recently introduced and studied by Noor [1], when A and B are singlevalued mappings. Consequently, it was pointed out that such a problem includes a wide class of the system of variational inequalities problems and related optimization problems as special cases, and hence the results announced in [1] is very interesting.

(III)
If A, B : H → H are singlevalued mappings, then, from the problem (1.1), we have the following system of general nonlinear mixed variational inequalities problems:
Find x*, y* ∈ H, such that
We denote by SGNM(A, B, g, φ, ρ, η), the set of all solutions (x*, y*) of the problem (1.4).
This means, generally speaking, the class of system general nonlinear setvalued variational inequalities problems is more general and has had a great impact and influence in the development of several branches of pure, applied, and engineering sciences. For more information and results on the general variational inequalities problems, one may consult [2–18].
Inspired and motivated by the recent research going on in this area, in this paper, we consider the existence theorem and a method for finding solutions for the systems of nonlinear general setvalued mixed variational inequalities problems (1.1). Our results extend the results announced by Noor [1], from singlevalued mappings to setvalued mappings, and hence include several related problems as spacial cases.
We need the following basic concepts and wellknown results:
Definition 1.1. A mapping g : H → H is said to be:

(1)
monotone if

(2)
νstrongly monotone if there exists a constant ν > 0, such that
Definition 1.2. A setvalued mapping A : H → 2 ^{H} is said to be νstrongly monotone if there exists a constant ν > 0, such that,
Definition 1.3. A setvalued mapping A : H → CB(H) is said to be τLipschitzian continuous if there exists a constant τ > 0, such that,
where H(·,·) is the Hausdorff metric on CB(H).
Definition 1.4. A singlevalued mapping T : H → H is said to be a κLipschitzian continuous mapping if there exists a positive constant κ, such that,
In the case of κ = 1, the mapping T is known as a nonexpansive mapping.
Definition 1.5. [19] If M is a maximal monotone operator on H, then, for any λ > 0, the resolvent operator associated with M is defined as
It is wellknown that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. Furthermore, the resolvent operator is singlevalued and nonexpansive. In particular, it is wellknown that the subdifferential ∂φ of a proper convex lower semicontinuous function φ : H → (∞, +∞] is a maximal monotone operator.
Moreover, we have the following interesting characterization:
Lemma 1.6. [19]The points u, z ∈ H satisfy the inequality
if and only if u = J_{ φ } (z), where J_{ φ } (I + λ∂φ)^{1}is the resolvent operator and λ > 0 is a constant.
The property of the resolvent operator J_{ φ } presented in Lemma 1.6 plays an important role in developing the numerical methods for solving the system of general nonlinear setvalued mixed variational inequalities problems. In fact, assuming that g : H → H is a surjective mapping and by applying Lemma 1.6, one can easily prove the following result:
Lemma 1.7. If g : H → H is a surjective mapping, then the problem (1.1) is equivalent to the following problem:
Find x*, y* ∈ H, u* ∈ Ay*, v* ∈ Bx*, such that,
where J_{ φ } = (I + ∂φ)^{1}.
The equivalent formulation (1.5) enables us to suggest an explicit iterative method for solving the system of general nonlinear setvalued mixed variational inequalities problem (1.1), as we show in the next section. Of course, we hope to use the Lemma 1.7 to obtain our results in this paper, and hence, from now on, we assume that the mapping g : H → H is a surjection.
In order to prove our main results, the next lemma is very important.
Lemma 1.8. [20]Let B_{1}, B_{2} ∈ CB(H) and r > 1 be any real number. Then, for all b_{1} ∈ B_{1}, there exists b_{2} ∈ B_{2}, such that b_{1}  b_{2} ≤ rH(B_{1}, B_{2}).
2. Main results
We begin with some observations that are guidelines to a method for proving the main results in this paper.
Remark 2.1. If (x*, y*, u*, v*) ∈ SGNSM(A, B, g, φ, ρ, η), then it follows from (1.5) that
From Remark 2.1, we suggest a method for finding a solution for the problem (2.1), as following iterative procedures:
Let {ε_{ n } } be a sequence of positive real numbers with ε_{ n } → 0 as n → ∞ and t ∈ (0, 1] be fixed. For any x_{0}, y_{0} ∈ H, pick u_{0} ∈ Ay_{0} and let
Then take v_{1} ∈ Bx_{1} and let
Now, by Lemma 1.8, there exists u_{1} ∈Ay_{1}, such that
Take
Similarly, by Lemma 1.8, there exists v_{2} ∈ Bx_{2}, such that
Take
Inductively, we have the following algorithm:
Algorithm 1. Let {ε_{ n } } be a sequence of nonnegative real numbers with ε_{ n } → 0 as n → ∞ and t ∈ (0, 1] be a fixed constant. For any x_{0}, y_{0} ∈ H, compute the sequences {x_{ n } }, {y_{ n } } ⊂ H, and generated by the iterative processes:
We now state and prove the existence theorem of a solution for the problem (1.1).
Theorem 2.2. Let H be a real Hilbert space. Let A : H → CB(H) be ν_{ A }strongly monotone and Lipschitz continuous mapping with a constant τ_{ A } and B : H → CB(H) be ν_{ B }strongly monotone and Lipschitz continuous mapping with a constant τ_{ B }. Let g : H → H be ν_{ g }strongly monotone and Lipschitz continuous mapping with a constant τ_{ g }. Put
If the following conditions are satisfied:

(i)
p ∈ [0, δ ), where ,

(ii)
and ,
then SGNSM(A, B, g, φ, ρ, η) ≠ ∅. Moreover, the sequence {x_{ n } }, {y_{ n } }, {u_{ n } }, and {v_{ n } } defined by (2.1) converge strongly to x*, y*, u*, and v*, respectively, where (x*, y*, u*, v*) ∈ SGNSM(A, B, g, φ, ρ, η).
Proof. Firstly, by (2.1), we have
Now, we compute
and
where . Substituting (2.3) and (2.4) into (2.2), we have
Now, since y_{n+1}= J_{ φ } [g(x_{n+1})  ηv_{n+1}] and the resolvent operator J_{ φ } is nonexpansive, we have
Using the same lines as in (2.3) and (2.4), we know that
where ^{.} Substituting (2.6) into (2.5), we have
Observe that
and
Consequently, by the conditions (i) and (ii), we have Δ =: (p + q)(p + r) < 1.
Now, let s ∈ (Δ, 1) be a fixed real number. Then, by (2.8) and (2.9), there exists a positive integer, N, such that (p + q_{ n } )(p + r_{ n } ) < s for all n ≥ N. Then, by (2.7), we have
where κ : = 1  t(1  s). Then it follows from (2.10) that
Hence it follows that
Since κ < 1, it follows from (2.11) that x_{ m }  x_{ n }  → 0 as n → ∞, which implies that {x_{ n } } is a Cauchy sequence in H. Consequently, by (2.6), it follows that {y_{ n } } is a Cauchy sequence in H. Moreover, since A is a τ_{ A }  Lipschitz continuous mapping, and B is a τ_{ B } Lipschitz continuous mapping, we also know that {u_{ n } } and {v_{ n } } are Cauchy sequences, respectively. Thus there exist x*, y*, u*, v* ∈ H, such that x_{ n } → x*, y_{ n } → y*, u_{ n } → u*, and v_{ n } → v* as n → ∞. Moreover, by applying the continuity of the mappings A, B, g, and J_{ φ } to (2.1), we have
Hence, from Lemma 1.7, it follows that (x*, y*, u*, v*) ∈ SGNSM(A, B, g, φ, ρ, η).
Finally, we prove that u* ∈ Ay* and v* ∈ Bx*. Indeed, we have
That is, d(u*, Ay*) = 0. Hence, since Ay* ∈ CB(H), we must have u* ∈ Ay*.
Similarly, we can show that v* ∈ Bx*. This completes the proof.
Remark 2.3. Theorem 2.2 not only gives the conditions for the existence of a solution for the problem (1.1) but also provides an iterative algorithm to find such a solution for any initial points x_{0}, y_{0} ∈ H.
Using Theorem 2.2, we can obtain the following results:

(I)
If g = I (: the identity mapping), then from Algorithm 1, we have the following:
Algorithm 2. Let {ε_{ n } } be a sequence of nonnegative real numbers with ε_{ n } → 0. Let t ∈ (0, 1] be a fixed constant. For any x_{0}, y_{0} ∈ H, compute the sequences {x_{ n } }, {y_{ n } } ⊂ H, and generated by the iterative processes:
where u_{ n } ∈ Ay_{ n } and v_{ n } ∈ Bx_{ n } satisfy the following:
Corollary 2.4. Let H be a real Hilbert space. Let A : H → CB(H) be ν_{ A } ^{}strongly monotone and Lipschitz continuous mapping with a constant τ_{ A }, and B : H → CB(H) be ν_{ B }strongly monotone and Lipschitz continuous mapping with a constant τ_{ B }. If
then SNSM(A, B, φ, ρ, η) ≠ ∅. Moreover, the sequences {x_{ n } }, {y_{ n } }, {u_{ n } }, and {v_{ n } } defined by (2.12) converge strongly to x*, y*, u* and v*, respectively, where (x*, y*, u*, v*) ∈ SNSM(A, B, φ, ρ, η).
Proof. If g = I (: the identity operator), we know that the constant p defined in Theorem 2.2 is vanished. Thus the result follows immediately.

(II)
If the function φ(·) is the indicator function of a closed convex set K in H, then it is wellknown that J_{ φ } = P_{ K } , the projection operator of H onto the closed convex set K (see [2]). Then, from Algorithm 1, we have the following:
Algorithm 3. Let {ε_{ n } } be a sequence of nonnegative real numbers with ε_{ n } → 0 as n → ∞. Let t ∈ (0, 1] be a fixed constant. For any x_{0}, y_{0} ∈ K, compute the sequences {x_{ n } }, {y_{ n } } ⊂ K, , and generated by the iterative processes:
Corollary 2.5. Let K be a closed convex subset of a real Hilbert space H. Let A : K → CB(H) be ν_{ A }strongly monotone and Lipschitz continuous mapping with a constant τ_{ A } , and B : K → CB(H) be ν_{ B }strongly monotone and Lipschitz continuous mapping with a constant τ_{ B }. Let g : K → K be a ν_{ g }strongly monotone and Lipschitz continuous mapping with a constant τ_{ g } and satisfying K ⊂ g(H).
Put
If the following conditions are satisfied:

(i)
p ∈ [0, δ), where ,

(ii)
, and ,
then SGNS(A, B, g, K, ρ, η) ≠ ∅. Moreover, the sequence {x_{ n } }, {y_{ n } }, {u_{ n } }, and {v_{ n } } defined by (2.13) converge strongly to x*, y*, u* and v*, respectively, where (x*, y*, u*, v*) ∈ SGNS(A, B, g, K, ρ, η).
Remark 2.6. Corollary 2.5 is an extension of the results announced by Noor [1] from singlevalued mappings to setvalued mappings.

(III)
If A, B : H → H are singlevalued mappings, then, from Algorithm 1, we have the following:
Algorithm 4. Let t ∈ (0, 1] be a fixed constant. For any x_{0}, y_{0} ∈ H, compute the sequences {x_{ n } }, {y_{ n } } ⊂ H by the iterative processes:
Corollary 2.7. Let H be a real Hilbert space. Let A : H → H be ν_{ A }strongly monotone and Lipschitz continuous mapping with a constant τ_{ A }, and B : H → H be ν_{ B }strongly monotone and Lipschitz continuous mapping with a constant τ_{ B }. Let g : H → H be ν_{ g }strongly monotone and Lipschitz continuous mapping with a constant τ_{ g }. Put
If the following conditions are satisfied:

(i)
p ∈ [0, δ), where ,

(ii)
, and ,
then SGNM(A, B, g, φ, ρ, η) ≠ ∅. Moreover, the sequences {x_{ n } } and {y_{ n } } defined by (2.14) converge strongly to x* and y*, respectively, where (x*, y*) ∈ SGNM(A, B, g, φ, ρ, η).
Remark 2.8. Under the assumption of Corollary 2.7, the solution of SGNM(A, B, g, φ, ρ, η) is unique, that is, there is a unique (x*, y*) ∈ H×H such that (x*, y*) ∈ SGNM(A, B, g, φ, ρ, η). Indeed, if (x*, y*) and (x', y') are elements of SGNM(A, B, g, φ, ρ, η). Put
Replacing x_{n+1}by x*, x_{ n } by x', y_{ n } by y*, and y_{n1}by y', then, following the lines proof given in Theorem 2.2, we know that
and
By the conditions (i), (ii), and (2.16), we must have x* = x'. Consequently, by (2.15), we also have y* = y'.
Remark 2.9. Recall that a mapping A : H → H is said to be:

(1)
μcocoercive if there exists a constant μ > 0 such that

(2)
relaxed μcocoercive if there exists a constant μ > 0 such that

(3)
relaxed (μ, ν)cocoercive if there exist constants μ, ν > 0 such that
It is easy to see that the class of the relaxed (μ, ν) cocoercive mappings is the most general one. However, it is worth noting that if the mapping A is relaxed (μ, ν)cocoercive, and τLipschitz continuous mapping satisfying ν  μτ^{2}> 0, then A is a (ν  μτ^{2})strongly monotone. Hence, the result appeared in Corollary 2.7 can be also applied to the class of the relaxed cocoercive mappings. In the conclusion, for a suitable and appropriate choice of the mappings A, B, g, and φ, Theorem 2.2 includes many important known results given by some authors as special cases.
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Acknowledgements
Yeol Je Cho was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00050). Narin Petrot was supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.
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Agarwal, R.P., Cho, Y.J. & Petrot, N. Systems of general nonlinear setvalued mixed variational inequalities problems in Hilbert spaces. Fixed Point Theory Appl 2011, 31 (2011). https://doi.org/10.1186/16871812201131
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DOI: https://doi.org/10.1186/16871812201131