A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities
- Ali Mohamed Saddeek^{1}Email author
https://doi.org/10.1186/s13660-017-1482-0
© The Author(s) 2017
Received: 2 December 2016
Accepted: 25 August 2017
Published: 12 September 2017
Abstract
Most mathematical models arising in stationary filtration processes as well as in the theory of soft shells can be described by single-valued 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 two-layer 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. 37:934-942, 2001).
Keywords
modified two-layer iteration multivalued pseudomonotone mapping generalized mixed variational inequalities strong convergence uniformly convex spacesMSC
49J40 90C331 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 → and ⇀ strong and weak convergence, respectively. Let \(A_{0}: V\rightarrow V^{\ast}\) be a nonlinear single-valued mapping.
Definition 1.1
- (i)pseudomonotone, if it is bounded and for every sequence \(\{u_{n}\}\subset V\) such thatimply$$\begin{aligned} u_{n}\rightharpoonup u \in V\quad\mbox{and}\quad \limsup _{n\rightarrow \infty} \langle A_{0}u_{n},u_{n}-u \rangle\leq0 \end{aligned}$$$$\begin{aligned} \liminf_{n\rightarrow\infty} \langle A_{0}u_{n},u_{n}- \eta \rangle\geq \langle A_{0}\eta,u-\eta\rangle; \end{aligned}$$
- (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, ifwhere \(R= \max\{ \Vert u \Vert _{V}, \Vert \eta \Vert _{V}\}\), μ is a nondecreasing function on \([0,+\infty)\), and Φ is the gauge function (i.e., it is a strictly increasing continuous function on \([0,+\infty)\) such that \(\Phi(0)=0\) and \(\lim_{\xi\rightarrow\infty} {\Phi(\xi)=+\infty}\));$$\begin{aligned} \Vert A_{0}u-A_{0}\eta \Vert _{V^{\ast}} \leq\mu(R) \Phi\bigl( \Vert u-\eta \Vert _{V}\bigr), \end{aligned}$$
- (v)uniformly monotone, if there exists a gauge Φ such that$$\begin{aligned} \langle A_{0}u-A_{0}\eta,u-\eta\rangle\geq \Phi \bigl( \Vert u-\eta \Vert _{V}\bigr) \Vert u-\eta \Vert _{V}; \end{aligned}$$
- (vi)inverse strongly monotone, if there exists a constant \(\gamma >0\) such that$$\begin{aligned} \langle A_{0}u-A_{0}\eta,u-\eta\rangle\geq \gamma \Vert A_{0}u-A_{0}\eta \Vert _{V}^{2}. \end{aligned}$$
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, 7–9] and the references cited therein) and the problem of finding the equilibrium of soft shells (see, for example, [1, 7, 10–12] 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]).
It is well known (see, for example, [3, 14]) that \(J(0)=0\), J is odd, single-valued, 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 single-valued, 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 single-valued duality mapping J is bounded Lipschitz continuous and uniformly monotone.
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]).
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 two-layer iterations (1.7) and (1.8) so that strong convergence is guaranteed have recently been made.
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.
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 two-layer iteration (1.5).
Theorem 1.1
see [1], Theorem 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 two-layer 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 two-layer 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
- (i)pseudomonotone, if it is bounded and, for every sequence \(\{u_{n}\}\subset V\), \(\{w_{n}\}\subset A_{0}(u_{n})\), the conditionsimply that for every \(\eta\in V\) there exists \(w \in A_{0}(u)\) such that$$\begin{aligned} u_{n}\rightharpoonup u \in V\quad\mbox{and}\quad \limsup _{n\rightarrow \infty} \langle w_{n},u_{n}-u\rangle\leq0 \end{aligned}$$$$\begin{aligned} \liminf_{n\rightarrow\infty} \langle w_{n},u_{n}- \eta \rangle\geq \langle w,u-\eta\rangle; \end{aligned}$$
- (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, iffor all \(u,\eta\in V\), \(w^{1}\in A_{0}(t(u+\eta))\), \(w^{2}\in A_{0}(t u)\), \(w^{3}\in A_{0}(u+t\eta)\), \(t\in[0,1]\);$$\begin{aligned} \int^{1}_{0}\bigl(\bigl\langle w^{1},u+ \eta\bigr\rangle -\bigl\langle w^{2},u\bigr\rangle \bigr) \,dt= \int^{1}_{0}\bigl\langle w^{3},\eta\bigr\rangle \,dt \end{aligned}$$
- (iv)bounded Lipschitz continuous, iffor all \(u,\eta\in V\), \(w\in A_{0}(u)\), \(\acute{w}\in A_{0}(\eta)\), where \(\mu(R)\) and \(\Phi(\xi)\) as above;$$\begin{aligned} \Vert w-\acute{w} \Vert _{V^{\ast}} \leq\mu(R) \Phi\bigl( \Vert u-\eta \Vert _{V}\bigr) \end{aligned}$$
- (v)inverse strongly monotone, if there exists a constant \(\gamma >0\) such thatfor all \(u,\eta\in V\), \(w\in A_{0}(u)\), \(\acute{w}\in A_{0}(\eta)\).$$\begin{aligned} \langle w-\acute{w},u-\eta\rangle\geq \gamma \Vert w-\acute{w} \Vert _{V}^{2} \end{aligned}$$
Definition 2.1 is an extension of Definition 1.1((i)-(iv), (vi)) of single-valued mappings to multivalued mappings.
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 \(\Pi^{F_{1}}_{M}(\eta)=\arg\min_{u\in M} G_{1}(u,J\eta )\), \(\forall\eta\in V\).
The following two lemmas are also useful in the sequel.
Lemma 2.1
see [28]
- (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$$\begin{aligned} \langle J\eta-J\bar{u},\bar{u}-v\rangle+ (F_{1}(v)-F_{1}( \bar{u})\geq0\quad \forall v\in M; \end{aligned}$$
- (iii)
if V is strictly convex, then \(\Pi^{F_{1}}_{M}(\eta)\) is a single-valued 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]
- (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 Kadec-Klee 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 Kadec-Klee property.
3 Main results
3.1 The modified two-layer iteration
Observe that iteration (3.3) is a modification and generalization of iterations (1.11) and (1.9).
If \(V=H\), \(A_{0}\) is a single-valued mapping in (3.3) and \(h(u_{n})=1\) \(\forall n\geq0\), we have iteration (1.8).
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 γ-Lipschitzian functional with \(M\subset \operatorname{int} (D(F_{1}))\). Let F̃, \(\tilde{{R}_{0}}\), \(\tilde{{R}_{1}}\), \(\tilde{{S}}_{0}\), 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 γ-Lipschitzian, and \(A_{0}\) is coercive and bounded, it results from [1] and [2] that \(F_{1}\) is weakly lower semicontinuous and F̃ 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. □
Now we divide the proof into steps.
Step 2. We prove that \(\lim_{n\rightarrow\infty} \Vert u_{n+1}-u_{n} \Vert _{V}=0\) and \(\lim_{n\rightarrow\infty} \Vert Ju_{n+1}-Ju_{n} \Vert _{V^{\ast}}=0\).
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 \(\lim_{k\rightarrow\infty}{F_{1}(u_{n_{k}})}\geq F_{1}(\bar{u})\).
Step 4. We show that \(\bar{u}\in \mathit{SOL}(M,F_{1},h(w)-f)\).
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 \(\{G_{2}(u_{n+1},0)\}\) 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.
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, F̃, \(\tilde{{S}_{0}}\), \(\tilde{\mu}_{0}\) and \(\tilde{F}_{0}\), \(F_{1}\), \(\tilde{R}_{0}\), \(\tilde{R}_{1}\) be the same as in Theorem 3.1.
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 Kadec-Klee 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 \(\lim_{n\rightarrow\infty} \Vert u_{n} \Vert _{H} = \Vert \bar{u} \Vert _{H}\).
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 \(n\rightarrow\infty\).
Applying again the virtue of the Kadec-Klee property of H, we obtain \(\lim_{n\rightarrow\infty} u_{n}= \bar{u}\). This completes the proof of Theorem 3.2. □
Example 3.1
Axisymmetric shell problem
A quintessential example of a single-valued 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 \(V^{\ast}=[\overset{\circ}{W}{}_{q}^{(-1)}(0,l)]^{2}\), \(q=\frac{p}{p-1}\), \(p>1\). 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].
4 Conclusion
A generalized multivalued pseudomonotone mixed variational inequality is considered, and a modified two-layer 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.
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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