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A modified twolayer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities
Journal of Inequalities and Applications volumeÂ 2017, ArticleÂ number:Â 216 (2017)
Abstract
Most mathematical models arising in stationary filtration processes as well as in the theory of soft shells can be described by singlevalued or generalized multivalued pseudomonotone mixed variational inequalities with proper convex nondifferentiable functionals. Therefore, for finding the minimum norm solution of such inequalities, the current paper attempts to introduce a modified twolayer iteration via a boundary point approach and to prove its strong convergence. The results here improve and extend the corresponding recent results announced by Badriev, Zadvornov and Saddeek (Differ. Equ. 37:934942, 2001).
1 Introduction
Let V be a real Banach space, \(V^{\ast}\) be its dual space, \(\Vert \cdot \Vert _{V^{\ast}}\) be the dual norm of the given norm \(\Vert \cdot \Vert _{V}\), and \(\langle\cdot,\cdot \rangle\) be the duality pairing between \(V^{\ast}\) and V. Let M be a nonempty closed convex subset of V. Let \(C(V^{\ast})\) be the family of nonempty compact subsets of \(V^{\ast}\). Let H be a real Hilbert space with the inner product \((\cdot,\cdot )\) and the norm \(\Vert \cdot \Vert _{H}\), respectively.
We denote by â†’ and â‡€ strong and weak convergence, respectively. Let \(A_{0}: V\rightarrow V^{\ast}\) be a nonlinear singlevalued mapping.
Definition 1.1
For all \(u, \eta\in V\), the mapping \(A_{0}:V\rightarrow V^{\ast}\) is said to be as follows:

(i)
pseudomonotone, if it is bounded and for every sequence \(\{u_{n}\}\subset V\) such that
$$\begin{aligned} u_{n}\rightharpoonup u \in V\quad\mbox{and}\quad \limsup _{n\rightarrow \infty} \langle A_{0}u_{n},u_{n}u \rangle\leq0 \end{aligned}$$imply
$$\begin{aligned} \liminf_{n\rightarrow\infty} \langle A_{0}u_{n},u_{n} \eta \rangle\geq \langle A_{0}\eta,u\eta\rangle; \end{aligned}$$ 
(ii)
coercive, if there exists a function \(\rho: \mathbb {R^{+}}\rightarrow\mathbb{R^{+}}\) with \(\lim_{\xi\rightarrow\infty} {\rho(\xi)=+\infty}\) such that
$$\begin{aligned} \langle A_{0}u,u\rangle\geq \rho\bigl( \Vert u \Vert _{V}\bigr) \Vert u \Vert _{V}; \end{aligned}$$ 
(iii)
potential, if
$$\begin{aligned} \int^{1}_{0}\bigl(\bigl\langle A_{0} \bigl(t(u+ \eta)\bigr),u+\eta\bigr\rangle \bigl\langle A_{0}(tu),u\bigr\rangle \bigr) \,dt= \int^{1}_{0}\bigl\langle A_{0}(u+t\eta), \eta\bigr\rangle \,dt; \end{aligned}$$ 
(iv)
bounded Lipschitz continuous, if
$$\begin{aligned} \Vert A_{0}uA_{0}\eta \Vert _{V^{\ast}} \leq\mu(R) \Phi\bigl( \Vert u\eta \Vert _{V}\bigr), \end{aligned}$$where \(R= \max\{ \Vert u \Vert _{V}, \Vert \eta \Vert _{V}\}\), Î¼ is a nondecreasing function on \([0,+\infty)\), and Î¦ is the gauge function (i.e., it is a strictly increasing continuous function on \([0,+\infty)\) such that \(\Phi(0)=0\) and \(\lim_{\xi\rightarrow\infty} {\Phi(\xi)=+\infty}\));

(v)
uniformly monotone, if there exists a gauge Î¦ such that
$$\begin{aligned} \langle A_{0}uA_{0}\eta,u\eta\rangle\geq \Phi \bigl( \Vert u\eta \Vert _{V}\bigr) \Vert u\eta \Vert _{V}; \end{aligned}$$ 
(vi)
inverse strongly monotone, if there exists a constant \(\gamma >0\) such that
$$\begin{aligned} \langle A_{0}uA_{0}\eta,u\eta\rangle\geq \gamma \Vert A_{0}uA_{0}\eta \Vert _{V}^{2}. \end{aligned}$$
If \(\Phi(\xi)=\xi\) and \(\mu(R)=\gamma>0\), in (iv), the mapping \(A_{0}\) is called Î³Lipschitzian mapping, and if there exists \(\alpha>0\) such that \(\Phi(\xi)=\alpha\xi\), in (v), the mapping \(A_{0}\) is called strongly monotone mapping. It is obvious that any inverse strongly monotone mapping is \(\frac{1}{\gamma}\)Lipschitzian mapping.
The singlevalued pseudomonotone mixed variational inequality problem is formulated as finding a point \(u\in M\) such that
where \(A_{0}:V\rightarrow V^{\ast}\) is a singlevalued pseudomonotone mapping, \(F_{1}:V\rightarrow\mathbb{R}\cup \{+\infty\}\) is a proper convex and lower semicontinuous (but, in general, nondifferentiable) functional, and \(f\in V^{\ast}\) is a given element.
Problem (1.1) is equivalent to finding \(u\in V\) such that
where \(\partial F_{1}(u)\) is the subdifferential of \(F_{1}\), i.e.,
The interior of the domain of \(F_{1}\) is denoted by \(\operatorname{int} (D(F_{1}))\).
Such problems appear in many fields of physics (e.g., in hydrodynamics, elasticity or plasticity), more specifically, when describing or analyzing the steady state filtration (see, for example, [1, 7â€“9] and the references cited therein) and the problem of finding the equilibrium of soft shells (see, for example, [1, 7, 10â€“12] and the references cited therein).
The existence of at least one solution to problem (1.1) can be guaranteed by imposing pseudomonotonicity and coercivity conditions on the mapping \(A_{0}\) (see, for example, [2, 3]).
If \(f=0\) and \(F_{1}(u)=I_{M}(u)\) \(\forall u\in M\), where \(I_{M}\) is the indicator functional of M defined by \(u\in M\) such that \(I_{M}(u)= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 0, & u\in M,\\ +\infty,& \mbox{o.w}, \end{array}}\) then problem (1.1) is equivalent to finding \(u\in M\) such that
which is known as the classical variational inequality problem firstly introduced and studied by Stampacchia [13]. Problem (1.3) is equivalent to the following nonlinear operator equation: find \(u\in M\) such that
A mapping \(J:V\rightarrow V^{\ast}\) is called a duality mapping with gauge function Î¦ if, for every \(u\in V\), \(\langle Ju,u \rangle=\Phi( \Vert u \Vert _{V}) \Vert u \Vert _{V}\) and \(\Vert Ju \Vert _{V^{\ast }}=\Phi( \Vert u \Vert _{V})\). If \(V=H\), then the duality mapping with the gauge function \(\Phi(\xi)=\xi\) can be identified with the identity mapping of H into itself.
It is well known (see, for example, [3, 14]) that \(J(0)=0\), J is odd, singlevalued, bijective and is uniformly continuous on bounded sets if V is a reflexive Banach space and \(V^{\ast}\) is uniformly convex; moreover, \(J^{1}\) is also singlevalued, bijective, and \(JJ^{1}=I_{V^{\ast}}\), \(J^{1}J=I_{V}\).
Therefore, we always assume that the dual space of a reflexive Banach space is uniformly convex.
Remark 1.1
see, for example, [15]
The singlevalued duality mapping J is bounded Lipschitz continuous and uniformly monotone.
In order to find a solution of problem (1.1), Badriev et al. [1] suggested the following twolayer iteration method: for an arbitrary \(u_{0}\in M\), define \(u_{n+1} \in M\) as follows:
where \(\tau>0\) is an iteration parameter and \(n\geq0\).
In this way the original variational inequality problem (1.1) is thus reduced to another variational inequality problem involving the duality mapping J instead of the original pseudomonotone mapping \(A_{0}\). Such a problem can then be solved by known methods (see, for example, [16, 17]).
If \(V=H\), then the iteration generated by (1.5) can be written in the following form:
for an arbitrary \(u_{0}\in M\) and \(\tau>0\).
In [18], Saddeek and Ahmed considered the following twolayer iteration method for solving the nonlinear operator equation (1.4) in a Banach space V:
where \(u_{0}\) is an arbitrary point in M and \(\tau>0\).
In the case when \(V=H\), iteration (1.7) can be written as follows:
for \(\tau>0\) and \(u_{0}\) is an arbitrary point in M.
Saddeek and Ahmed [18] proved some weak convergence theorems of iterations (1.7) and (1.8) for approximating the solution of nonlinear equation (1.4).
Attempts to modify the twolayer iterations (1.7) and (1.8) so that strong convergence is guaranteed have recently been made.
In [19], Saddeek introduced the following modification of (1.8) in a Hilbert space H (boundary point method):
where \(\tau>0\), \(u_{0}\) is an arbitrary point in M, and \(h:M \rightarrow[0,1]\) is a function defined by He and Zhu [20] as follows:
He obtained strong convergence results for finding the minimum norm solution of nonlinear equation (1.4).
In [20], He and Zhu have observed that, if \(0 \notin M\), calculating \(h(u_{n})\) implies determining \(h(u_{n})u_{n}\), a boundary point of M, so iteration (1.9) is known as the boundary point method.
In [21], Saddeek extended the results of Saddeek [19] to a uniformly convex Banach space and introduced the following modification of the twolayer iteration (1.7) (boundary point method):
where \(\tau>0\), \(u_{0}\) is an arbitrary point in M, \(\tau>0\), and h is defined by (1.10).
In [22], Noor introduced and studied the following generalized multivalued pseudomonotone mixed variational inequality problem: find \(u\in M\), \(w\in A_{0}(u)\) such that
where \(A_{0}:V\rightarrow C(V^{\ast})\) is a multivalued pseudomonotone mapping (see definition below), \(F_{1}:V\rightarrow \mathbb{R}\cup\{+\infty\}\) is a functional as above, and \(f\in V^{\ast}\) is a given element.
Clearly, problems (1.1) and (1.3) are special cases of problem (1.12).
The set of all \(u\in M\) satisfying (1.12) is denoted by \(\mathit{SOL}(M,F_{1},A_{0}f)\).
In [1], Badriev et al. obtained the following weak convergence theorems using the twolayer iteration (1.5).
Theorem 1.1
see [1], Theorem 1
Let V be a real reflexive Banach space with a uniformly convex dual space \(V^{\ast}\), and let \(J:V \rightarrow V^{\ast}\) be the duality mapping. Let M be a nonempty closed convex subset of V. Let \(A_{0}:V \rightarrow V^{\ast}\) be a pseudomonotone, coercive, potential, and bounded Lipschitz continuous mapping. Let \(F_{1}:V\rightarrow\mathbb{R}\cup \{+\infty\}\) be a proper convex and Î³Lipschitzian (i.e., \(\mid F_{1}(u)F_{1}(\eta)\mid \leq\gamma \Vert u\eta \Vert _{V}\) \(\forall u,\eta\in V, \gamma>0\)) functional. Define a functional \(F:V\rightarrow\mathbb{R}\cup\{+\infty\}\) by
Assume also that
where
Then the sequence \(\{u_{n}\}\) defined by (1.5) is bounded in V, and all of its weak limit points are solutions of problem (1.1).
Badriev et al. [1] have remarked that, due to the reflexivity of V, the mixed variational inequality (1.1) is solvable by Theorem 1.1.
In Theorem 1.1, the assumption that V is reflexive can be dropped. Indeed, if \(V^{\ast}\) is uniformly convex, then V is uniformly smooth (and hence V is reflexive).
Theorem 1.2
see [1], Theorem 2
Let \(V=H\) be a real Hilbert space, and let M be a nonempty closed convex subset of H. Let \(A_{0}:H \rightarrow H\) be a pseudomonotone, coercive, potential, and inverse strongly monotone mapping. Let \(F_{i}:H\rightarrow\mathbb{R}\cup\{+\infty\}\), \(i=0,1\), be the same as in Theorem 1.1.
Then the sequence \(\{u_{n}\}\) defined by (1.6) with \(0<\tau<\tau_{0}=2\gamma\) converges weakly in H to a solution of problem (1.1).
Some attempts to prove the weak convergence of the whole sequence in the framework of Banach spaces have been made by Saddeek and Ahmed [23] and Saddeek [24, 25].
Although the above mentioned theorems and all their extensions are unquestionably interesting, only weak convergence theorems are obtained unless very strong assumptions are made.
This suggests an important question: can the twolayer iteration method (1.5) be modified to prove its strong convergence to the minimum norm solution of problem (1.12).
In this paper, inspired by [20, 21], and [22], a generalized multivalued pseudomonotone mixed variational inequality is considered, and a modified twolayer iteration via a boundary point approach to find the minimum norm solution of such inequalities is introduced, and its strong convergence is proved in the framework of uniformly convex spaces. The results obtained in this paper improve and generalize the corresponding recent results announced by [1].
2 Definitions and preliminary
Definition 2.1
A multivalued mapping \(A_{0}:V\rightarrow C(V^{\ast})\) is called

(i)
pseudomonotone, if it is bounded and, for every sequence \(\{u_{n}\}\subset V\), \(\{w_{n}\}\subset A_{0}(u_{n})\), the conditions
$$\begin{aligned} u_{n}\rightharpoonup u \in V\quad\mbox{and}\quad \limsup _{n\rightarrow \infty} \langle w_{n},u_{n}u\rangle\leq0 \end{aligned}$$imply that for every \(\eta\in V\) there exists \(w \in A_{0}(u)\) such that
$$\begin{aligned} \liminf_{n\rightarrow\infty} \langle w_{n},u_{n} \eta \rangle\geq \langle w,u\eta\rangle; \end{aligned}$$ 
(ii)
coercive, if there exists a function \(\rho: \mathbb {R^{+}}\rightarrow\mathbb{R^{+}}\) with \(\lim_{\xi\rightarrow\infty} {\rho(\xi)=+\infty}\) such that
$$\begin{aligned} \langle w,u\rangle\geq \rho\bigl( \Vert u \Vert _{V} \bigr) \Vert u \Vert _{V} \quad\forall u\in V, w\in A_{0}(u); \end{aligned}$$ 
(iii)
potential, if
$$\begin{aligned} \int^{1}_{0}\bigl(\bigl\langle w^{1},u+ \eta\bigr\rangle \bigl\langle w^{2},u\bigr\rangle \bigr) \,dt= \int^{1}_{0}\bigl\langle w^{3},\eta\bigr\rangle \,dt \end{aligned}$$for all \(u,\eta\in V\), \(w^{1}\in A_{0}(t(u+\eta))\), \(w^{2}\in A_{0}(t u)\), \(w^{3}\in A_{0}(u+t\eta)\), \(t\in[0,1]\);

(iv)
bounded Lipschitz continuous, if
$$\begin{aligned} \Vert w\acute{w} \Vert _{V^{\ast}} \leq\mu(R) \Phi\bigl( \Vert u\eta \Vert _{V}\bigr) \end{aligned}$$for all \(u,\eta\in V\), \(w\in A_{0}(u)\), \(\acute{w}\in A_{0}(\eta)\), where \(\mu(R)\) and \(\Phi(\xi)\) as above;

(v)
inverse strongly monotone, if there exists a constant \(\gamma >0\) such that
$$\begin{aligned} \langle w\acute{w},u\eta\rangle\geq \gamma \Vert w\acute{w} \Vert _{V}^{2} \end{aligned}$$for all \(u,\eta\in V\), \(w\in A_{0}(u)\), \(\acute{w}\in A_{0}(\eta)\).
Definition 2.1 is an extension of Definition 1.1((i)(iv), (vi)) of singlevalued mappings to multivalued mappings.
Let \(G_{1}:M\times V^{\ast} \rightarrow\mathbb{R}\cup\{+\infty\}\) be a functional defined as follows:
where \(u\in M\), \(\eta\in V\), \(J\eta\in V^{\ast}\).
Definition 2.2
see, for example, [28]
The mapping \(\Pi^{F_{1}}_{M}: V\rightarrow C(M)\) is called generalized \(F_{1}\)projection mapping if \(\Pi^{F_{1}}_{M}(\eta)=\arg\min_{u\in M} G_{1}(u,J\eta )\), \(\forall\eta\in V\).
If \(V=H\) and \(F_{1}(u)=0\) \(\forall u\in M\), then (2.1) reduces to the following simple form:
and the generalized \(F_{1}\)projection reduces to the projection \(\Pi_{M}\) from H to \(C(M)\).
The following two lemmas are also useful in the sequel.
Lemma 2.1
see [28]
The generalized \(F_{1}\)projection \(\Pi^{F_{1}}_{M}(\eta)\) has the following properties:

(i)
\(\Pi^{F_{1}}_{M}(\eta)\) is a nonempty closed convex subset of M for all \(\eta\in V\);

(ii)
for all \(\eta\in V\), \(\bar{u}\in\Pi^{F_{1}}_{M}(\eta)\) if and only if
$$\begin{aligned} \langle J\etaJ\bar{u},\bar{u}v\rangle+ (F_{1}(v)F_{1}( \bar{u})\geq0\quad \forall v\in M; \end{aligned}$$ 
(iii)
if V is strictly convex, then \(\Pi^{F_{1}}_{M}(\eta)\) is a singlevalued mapping.
Let \(G_{2}:V\times V \rightarrow\mathbb{R^{+}}\cup\{0\}\) be a functional defined as follows:
$$\begin{aligned} G_{2}(u,\eta)= \Vert u \Vert _{V}^{2}2 \langle J\eta,u\rangle+ \Vert \eta \Vert _{ V}^{2}, \quad\forall u, \eta\in V. \end{aligned}$$(2.2)
Lemma 2.2
see [29]
Let V be a real Banach space with a uniformly convex dual space \(V^{\ast}\), let M be a nonempty closed convex subset of V, and let \(\eta\in V\), \(\bar{u}\in\Pi^{F_{1}}_{M}(\eta)\). Then

(i)
\(G_{2}(u,\bar{u})+G_{2}(\bar{u},\eta)\leq G_{2}(u,\eta)\) \(\forall u\in M\);

(ii)
for \(u, \eta\in V\), \(G_{2}(u,\eta)=0\) iff \(u=\eta\).
A Banach space V is said to have the KadecKlee property (see, for example, [30]) if, for every sequence \(\{u_{n}\}\) in V with \(u_{n}\rightharpoonup u\) and \(\Vert u_{n} \Vert _{V}\rightarrow \Vert u \Vert _{V}\) together imply that \(\lim_{n\rightarrow\infty} \Vert u_{n}u \Vert _{V}=0\).
Every Hilbert space is uniformly convex, and every uniformly convex Banach space has the KadecKlee property.
3 Main results
In this section, we propose a modification of the twolayer iteration method (1.5) by the boundary point method to establish strong convergence theorems of the modified iteration for finding the minimum norm solution of the following generalized pseudomonotone mixed variational inequality in uniformly convex spaces: find \(u\in M\), \(w\in A_{0}(u)\) such that
where \(A_{0}\), \(F_{1}\), f are defined as above and h is a positive constant.
3.1 The modified twolayer iteration
For an arbitrary point \(u_{0} \in M\), define \(u_{n+1}\in M\) as follows:
where \(\tau>0\) is the iteration parameter, \(n\geq0\), J is the duality mapping, \(w_{n}\in A_{0}(u_{n})\) and h is defined by (1.10).
For \(M=V\), \(F_{1}(u)=0\) \(\forall u\in M\), and \(\eta=u_{n+1}\pm z\), \(z\in M\), (3.2) is equivalent to
where \(w_{n}\in A_{0}(u_{n})\), and Ï„, J, h are defined as above.
Observe that iteration (3.3) is a modification and generalization of iterations (1.11) and (1.9).
If \(V=H\), \(A_{0}\) is a singlevalued mapping in (3.3) and \(h(u_{n})=1\) \(\forall n\geq0\), we have iteration (1.8).
Iteration (3.3) can be considered as a modified method for solving the following operator inclusion problem: find \(u\in V\) such that
For each \(u\in V\), \(w^{2}\in A_{0}(t u)\), let \(\tilde{F}:V\rightarrow\mathbb{R}\cup\{+\infty\}\) be a functional defined by
Let us assume also that
where \(w\in A_{0}(u)\).
Let \(\tilde{\mu}_{0}\) be a positive constant such that
Theorem 3.1
Let V be a real uniformly convex Banach space with a uniformly convex dual space \(V^{\ast}\), \(J:V \rightarrow V^{\ast}\) be the duality mapping, and let M be a nonempty closed convex subset of V. Let \(A_{0}:V \rightarrow C(V^{\ast})\) be a multivalued mapping. Suppose that \(A_{0}\) is pseudomonotone, coercive, potential, and bounded Lipschitz continuous mapping. Let \(F_{1}:V\rightarrow\mathbb{R}\cup\{+\infty\}\) be a proper convex (not necessarily differentiable) and Î³Lipschitzian functional with \(M\subset \operatorname{int} (D(F_{1}))\). Let FÌƒ, \(\tilde{{R}_{0}}\), \(\tilde{{R}_{1}}\), \(\tilde{{S}}_{0}\), and \(\tilde{\mu}_{0}\) be defined by (3.5), (3.6), and (3.7). Assume that \(0<\tau=\min\{1,\frac{1}{\tilde{\mu}_{0}}\}\). Let \(\{h(u_{n})\}\) be an increasing and bounded real sequence in \([0,1]\).
Then, for an arbitrary \(u_{0}=u\in M\), the sequence \(\{u_{n}\}\) defined by (3.2) converges strongly to \(\tilde{u}=\Pi_{\mathit{SOL}(M,F_{1},h(w)f)}^{F_{1}}{0}\) (i.e., the minimum norm element in \(\mathit{SOL}(M,F_{1},h(w)f)\)).
Proof
Since \(F_{1}\) is supposed to be convex and Î³Lipschitzian, and \(A_{0}\) is coercive and bounded, it results from [1] and [2] that \(F_{1}\) is weakly lower semicontinuous and FÌƒ is coercive; moreover, \(\tilde{R}_{0}<+\infty\) and \(\tilde{R}_{1}<+\infty\). Hence \(\tilde{\mu}_{0}<+\infty\). This means that the iterative sequence (3.2) is well defined.â€ƒâ–¡
Now we divide the proof into steps.
Step 1. We prove that \(\{u_{n}\}\) is bounded. To this end, it suffices to prove that
Let us prove (3.8) by induction on n. For \(n=0\), we have \(u_{0}\in \tilde{S}_{0}\). Suppose now that \(u_{n}\in\tilde{S}_{0}\). We will show that \(u_{n+1}\in\tilde{S}_{0}\).
Setting \(\eta=u_{n}\) in (3.2) and taking into account that the functional \(F_{1}\) is Î³Lipschitzian and J is uniformly monotone, and the inequality \(\tau\leq1\), we obtain
Now, using (3.9) together with the strict monotonicity of Î¦, we have
Furthermore, it follows from the bounded Lipschitz continuity of \(A_{0}\) that, for any \(t\in[0,1]\), \(w_{n}\in A_{0}u_{n}\), \(w_{n}^{3}\in A_{0}(u_{n+1}+t(u_{n}u_{n+1}))\)
where \(R_{{\ast}}=\max\{ \Vert u_{n+1}+t(u_{n}u_{n+1}) \Vert _{V}, \Vert u_{n} \Vert _{V}\}\).
Since
it follows from the definition of \(R_{\ast}\) that
Since Î¼Ìƒ is an increasing function, we must have
Consequently, it follows from (3.11) and (3.14) that
Moreover, since \(A_{0}\) is potential, we have
Setting \(\eta=u_{n}\) in (3.2) and using the uniform monotonicity of J, (3.15), (3.16), it results that
This implies that \(\tilde{F}(u_{n+1})\leq\tilde{F}(u_{n})\leq \tilde{F}(u_{0})\) and so \(u_{n+1}\in\tilde{S}_{0}\). So \(\{u_{n}\}\) is bounded.
Step 2. We prove that \(\lim_{n\rightarrow\infty} \Vert u_{n+1}u_{n} \Vert _{V}=0\) and \(\lim_{n\rightarrow\infty} \Vert Ju_{n+1}Ju_{n} \Vert _{V^{\ast}}=0\).
It follows from (3.17) that the sequence \(\{\tilde{F}(u_{n})\}\) is bounded and monotone, and thus we have that \(\lim_{n\rightarrow \infty} \tilde{F}(u_{n})\) exists. This together with (3.17) implies that
Since Î¦ is continuous and strictly increasing, it follows from (3.18) that
Since J is bounded Lipschitz continuous, Î¦ is continuous and \(\Phi(0)=0\), it follows from (3.19) that
Step 3. We show that there exists a subsequence \(\{u_{n_{k}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{k}}\rightharpoonup \bar{u} \in V\), \(\lim_{k\rightarrow\infty}{F_{1}(u_{n_{k}})}\geq F_{1}(\bar{u})\), and \(\limsup_{k\rightarrow\infty} h(u_{n_{k}})\langle w_{n_{k}},u_{n_{k}}\bar{u}\rangle\leq0\).
Since \(\{u_{n}\}\) is bounded and V is reflexive, we can choose a subsequence \(\{u_{n_{k}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{k}}\rightharpoonup\bar{u} \in V\) as \(k\rightarrow\infty\).
This together with the weak lower semicontinuity of \(F_{1}\) implies that \(\lim_{k\rightarrow\infty}{F_{1}(u_{n_{k}})}\geq F_{1}(\bar{u})\).
Since \(F_{1}\) is Î³Lipschitzian, \(\{h(u_{n})\}\subset[0,1]\), it follows from (3.2) that, for arbitrary \(\eta\in M\),
where \(C_{\eta}\) is a positive constant depending on Î·.
Setting \(\eta=\bar{u}\) in (3.21) and using the weak lower semicontinuity of \(F_{1}\), (3.19), (3.20), we have
Step 4. We show that \(\bar{u}\in \mathit{SOL}(M,F_{1},h(w)f)\).
Since \(\{h(u_{n})\}\subset[0,1]\) is bounded and monotone increasing, it follows that
By (3.19)(3.23), the lower semicontinuity of \(F_{1}\) and by the pseudomonotonicity of \(A_{0}\), we have
where \(\bar{w}\in A_{0}\bar{u}\). This means that \(\bar{u}\in \mathit{SOL}(M,F_{1},h(w)f)\).
Step 5. We prove that
where \(\bar{u}=\Pi_{\mathit{SOL}(M, F_{1}, h(w)f)}^{F_{1}} 0\).
Indeed take a subsequence \(\{u_{n_{k+1}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{k+1}}\rightharpoonup\bar{u}\).
Note that \(\bar{u}=\Pi_{\mathit{SOL}(M, F_{1}, h(w)f)}^{F_{1}} 0\). Then from \(\bar{u}\in \mathit{SOL}(M, F_{1}, h(w)f)\), the weak lower semicontinuity of \(F_{1}\), and Lemma 2.1(ii), the desired inequality (3.24) follows immediately.
Step 6. We show that \(\nonumber\lim_{n\rightarrow\infty} \Vert u_{n}\bar{u} \Vert _{V}=0\).
Since \(u_{n_{k+1}}\rightharpoonup\bar{u}\), it follows from the weak lower semicontinuity of \(\Vert \cdot \Vert _{V}\) that
From the convexity of \(D(F_{1})\), \(F_{1}\) and from the weak lower semicontinuity of \(F_{1}\), we obtain that \(F_{1}\) is subdifferentiable in \(\operatorname{int} (D(F_{1}))\). Thus, for all \(u\in D(F_{1})\), there exists an element \(u^{\ast}\in V^{\ast}\) such that
and hence
In view of \(u_{n_{k+1}}=\Pi_{\mathit{SOL}(M, F_{1}, h(w)f)}^{F_{1}} J^{1}(Ju_{n_{k}}\tau h(u_{n_{k}})(fw_{n_{k}}))\), we have
By using (2.1) with \(J\eta=Ju_{n_{k}}\tau h(u_{n_{k}})(fw_{n_{k}})\), we have
which implies that
Taking the \(\limsup_{k\rightarrow\infty}\) on the both sides of (3.27) and using \(u_{n_{k+1}}\rightharpoonup\bar{u}\), (3.22)(3.24), and (3.26) yields
which implies that
Combining (3.25) and (3.28), we have
This shows that
Since V is a uniformly convex Banach space, then it has the KadecKlee property, and so from \(u_{n_{k+1}}\rightharpoonup\bar{u}\) and (3.30) we obtain
Let us now show that the whole sequence converges strongly to Å«.
Since \(\{G_{2}(u_{n+1},0)\}\) is bounded and nondecreasing (indeed, by Lemma 2.2(i), we have \(G_{2}(u_{n+1},u_{n})+G_{2}(u_{n+1},0)\leq G_{2}(u_{n},0)\) and \(G_{2}(u_{n+1},u_{n})\geq ( \Vert u_{n+1} \Vert _{V} \Vert u_{n} \Vert _{V})^{2}\geq 0\)), it follows that \(\{G_{2}(u_{n+1},0)\}\) is convergent.
This together with (3.31) implies that
Now, following to [31], we suppose that there exists some subsequence \(\{u_{n_{j+1}}\}\) of \(\{u_{n}\}\) such that \(\lim_{j\rightarrow \infty} u_{n_{j+1}} = \hat{u}\), then by Lemma 2.2(i) we obtain
which means that \(G_{2}(\bar{u}, \hat{u})=0\) and hence, by Lemma 2.2(ii), it results that \(\hat{u}=\bar{u}\).
Consequently, \(\lim_{n\rightarrow\infty} u_{n}=\bar{u}\). This completes the proof of Theorem 3.1.
Theorem 3.2
Let \(V=H\) be a real Hilbert space, and let M be a nonempty closed convex subset of H. Let \(A_{0}:H \rightarrow C(H)\) be a multivalued mapping. Suppose that \(A_{0}\) is a pseudomonotone, coercive, potential, and inverse strongly monotone mapping. Let \(\{h(u_{n})\}\), M, FÌƒ, \(\tilde{{S}_{0}}\), \(\tilde{\mu}_{0}\) and \(\tilde{F}_{0}\), \(F_{1}\), \(\tilde{R}_{0}\), \(\tilde{R}_{1}\) be the same as in Theorem 3.1.
Then, for arbitrary \(u_{0}=u\in M\), the sequence \(\{u_{n}\}\) defined by
with \(0<\tau<\tau_{0}=\frac{2\gamma}{h}\), \(h>0\), converges strongly to \(\tilde{u}=\Pi_{\mathit{SOL}(M,F_{1},h(w)f)}^{F_{1}}{0}\).
Proof
Since any inverse strongly monotone mapping is \(\frac{1}{\gamma}\)Lipschitzian mapping, i.e., bounded Lipschitz continuous with \(\mu(\xi)=\frac{1}{\gamma_{0}}\) and \(\Phi(\xi )=\xi\), then by simple modifications of the proof of Theorem 3.1, we can easily show that there exists a subsequence \(\{u_{n_{k+1}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{k+1}} \rightharpoonup\bar{u}\in \mathit{SOL}(M,F_{1}, h(w)f)\) and \(\lim_{k\rightarrow\infty} \Vert u_{n_{k+1}} \Vert _{H} = \Vert \bar{u} \Vert _{H}\).
Since every Hilbert space is uniformly convex, by virtue of the KadecKlee property of H, we have \(\lim_{k\rightarrow\infty} u_{n_{k+1}}=\bar{u}\in \mathit{SOL}(M,F_{1}, h(w)f)\).
Now, we prove that \(u_{n}\rightharpoonup\bar{u}\) and \(\lim_{n\rightarrow\infty} \Vert u_{n} \Vert _{H} = \Vert \bar{u} \Vert _{H}\).
From \(\bar{u}\in \mathit{SOL}(M,F_{1}, h(w)f)\), we have
Setting \(\eta=u_{n+1}\) in (3.34) and \(\eta=\bar{u}\) in (3.33), we have
and
Adding (3.35) and (3.36), we have
which implies that
Then, by the inverse strong monotonicity of \(A_{0}\), we obtain for all sufficiently large n
Since \(2\frac{\tau h}{\gamma}>0\), it follows that \(\Vert u_{n+1}\bar{u} \Vert _{H}\leq \Vert u_{n}\bar{u} \Vert _{H}\) and so \(\lim_{n\rightarrow \infty} \Vert u_{n}\bar{u} \Vert _{H}=\sigma_{\bar{u}}\).
By following the same arguments as in [1] and [32], we can readily claim that all weak limit points of the sequence \(\{u_{n}\}\) coincide, and hence \(u_{n}\rightharpoonup\bar{u}\) as \(n\rightarrow\infty\).
By the weak lower semicontinuity of \(\Vert \cdot \Vert _{H}\), this implies that
Analogically to the proof of step 6 with obvious modifications, we have
This, together with (3.37), implies that \(\lim_{n\rightarrow \infty} \Vert u_{n} \Vert _{H} = \Vert \bar{u} \Vert _{H}\).
Applying again the virtue of the KadecKlee property of H, we obtain \(\lim_{n\rightarrow\infty} u_{n}= \bar{u}\). This completes the proof of Theorem 3.2.â€ƒâ–¡
Remark 3.1
Theorems 3.1 and 3.2 extend and improve the corresponding Theorems 1.1 and 1.2.
Example 3.1
Axisymmetric shell problem
A quintessential example of a singlevalued mapping satisfying all the assumptions contemplated in Theorems 3.1 and 3.2 which appears in determining the axisymmetric equilibrium position of a soft netlike rotation shell is as follows:
The shell surface (in a strainless state) is assumed to be a cylinder of length l and radius 1. Let s be a Lagrangian coordinate in the longitudinal direction such that \(0< s< l\).
Let \(V=[\overset{\circ}{W}{}_{p}^{(1)}(0,l)]^{2}\) and \(V^{\ast}=[\overset{\circ}{W}{}_{q}^{(1)}(0,l)]^{2}\), \(q=\frac{p}{p1}\), \(p>1\). Set \(u(s)=(u_{1}(s),u_{2}(s))\), \(\eta(s)=(\eta_{1}(s),\eta_{2}(s))\), \(M=\{u\in V:u_{2}(s)+1\geq0\ \forall s\in(0,l)\}\), and \(\lambda_{1}=[(1+\frac{du_{1}}{ds})^{2}+(\frac {du_{2}}{ds})^{2}]^{\frac{1}{2}}\), \(\lambda_{2}=1+u_{2}\).
Consider the surface force is characterized by a known constant function \(\mathbb{P}\). Let \(T_{i}(\lambda_{i})\), \(i=1,2\), be two functions (tightening force) satisfying conditions (3)(5) in Badriev and Banderov [33].
Consider the mappings \(A,B,C,D: V\rightarrow V^{\ast}\) defined by
If \(A_{0}=(A+D)+\mathbb{P}(B+C)\), then by Theorems 2 and 3 in [33] it follows that the mapping \(A_{0}\) satisfies all the assumptions postulated in Theorems 3.1 and 3.2.
4 Conclusion
A generalized multivalued pseudomonotone mixed variational inequality is considered, and a modified twolayer iteration via a boundary point approach to find the minimum norm solution of such inequalities is introduced, and its strong convergence is proved in the framework of uniformly convex spaces. The results develop the corresponding recent results.
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Acknowledgements
The author would like to extend his sincere gratitude to the two referees for their laudable comments and precious suggestions. I am also profoundly grateful to professor doctor IB Badriev for many valuable discussions.
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Saddeek, A.M. A modified twolayer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities. J Inequal Appl 2017, 216 (2017). https://doi.org/10.1186/s1366001714820
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DOI: https://doi.org/10.1186/s1366001714820