Saddeek Journal of Inequalities and Applications (2017) 2017:216 DOI 10.1186/s13660-017-1482-0
RESEARCH
Open Access
A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities Ali Mohamed Saddeek* *
Correspondence:
[email protected] Faculty of Science, Department of Mathematics, Assiut University, Assiut, Egypt
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). MSC: 49J40; 90C33 Keywords: modified two-layer iteration; multivalued pseudomonotone mapping; generalized mixed variational inequalities; strong convergence; uniformly convex spaces
1 Introduction Let V be a real Banach space, V ∗ be its dual space, · V ∗ be the dual norm of the given norm · V , and ·, · be the duality pairing between V ∗ and V . Let M be a nonempty closed convex subset of V . Let C(V ∗ ) be the family of nonempty compact subsets of V ∗ . Let H be a real Hilbert space with the inner product (·, ·) and the norm · H , respectively. We denote by → and strong and weak convergence, respectively. Let A : V → V ∗ be a nonlinear single-valued mapping. Definition . (see [–]) For all u, η ∈ V , the mapping A : V → V ∗ is said to be as follows: (i) pseudomonotone, if it is bounded and for every sequence {un } ⊂ V such that un u ∈ V
and
lim supA un , un – u ≤ n→∞
imply lim infA un , un – η ≥ A η, u – η; n→∞
© The Author(s) 2017. 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.
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 2 of 16
(ii) coercive, if there exists a function ρ : R+ → R+ with limξ →∞ ρ(ξ ) = +∞ such that A u, u ≥ ρ uV uV ; (iii) potential, if
A t(u + η) , u + η – A (tu), u dt =
A (u + tη), η dt;
(iv) bounded Lipschitz continuous, if A u – A ηV ∗ ≤ μ(R) u – ηV , where R = max{uV , ηV }, μ is a nondecreasing function on [, +∞), and is the gauge function (i.e., it is a strictly increasing continuous function on [, +∞) such that () = and limξ →∞ (ξ ) = +∞); (v) uniformly monotone, if there exists a gauge such that A u – A η, u – η ≥ u – ηV u – ηV ; (vi) inverse strongly monotone, if there exists a constant γ > such that A u – A η, u – η ≥ γ A u – A ηV . If (ξ ) = ξ and μ(R) = γ > , in (iv), the mapping A is called γ -Lipschitzian mapping, and if there exists α > such that (ξ ) = αξ , in (v), the mapping A is called strongly monotone mapping. It is obvious that any inverse strongly monotone mapping is γ -Lipschitzian mapping. The single-valued pseudomonotone mixed variational inequality problem is formulated as finding a point u ∈ M such that A u, η – u + F (η) – F (u) ≥ f , η – u ∀η ∈ M,
(.)
where A : V → V ∗ is a single-valued pseudomonotone mapping, F : V → R ∪ {+∞} is a proper convex and lower semicontinuous (but, in general, nondifferentiable) functional, and f ∈ V ∗ is a given element. Problem (.) is equivalent to finding u ∈ V such that ∈ A u – f + ∂F (u),
(.)
where ∂F (u) is the subdifferential of F , i.e., ∂F (u) = u∗ ∈ V ∗ : F (η) – F (u) ≥ u∗ , η – u ∀η ∈ V . The interior of the domain of F is denoted by int(D(F )). 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,
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 3 of 16
for example, [, –] and the references cited therein) and the problem of finding the equilibrium of soft shells (see, for example, [, , –] and the references cited therein). The existence of at least one solution to problem (.) can be guaranteed by imposing pseudomonotonicity and coercivity conditions on the mapping A (see, for example, [, ]). If f = and F (u) = IM (u) ∀u ∈ M, where IM is the indicator functional of M defined by u ∈ M, u ∈ M such that IM (u) = ,+∞, o.w , then problem (.) is equivalent to finding u ∈ M such that A u, η – u ≥ ∀η ∈ M,
(.)
which is known as the classical variational inequality problem firstly introduced and studied by Stampacchia []. Problem (.) is equivalent to the following nonlinear operator equation: find u ∈ M such that A u = f .
(.)
A mapping J : V → V ∗ is called a duality mapping with gauge function if, for every u ∈ V , Ju, u = (uV )uV and JuV ∗ = (uV ). If V = H, then the duality mapping with the gauge function (ξ ) = ξ can be identified with the identity mapping of H into itself. It is well known (see, for example, [, ]) that J() = , J is odd, single-valued, bijective and is uniformly continuous on bounded sets if V is a reflexive Banach space and V ∗ is uniformly convex; moreover, J – is also single-valued, bijective, and JJ – = IV ∗ , J – J = IV . Therefore, we always assume that the dual space of a reflexive Banach space is uniformly convex. Remark . (see, for example, []) The single-valued duality mapping J is bounded Lipschitz continuous and uniformly monotone. In order to find a solution of problem (.), Badriev et al. [] suggested the following two-layer iteration method: for an arbitrary u ∈ M, define un+ ∈ M as follows: J(un+ – un ), η – un+ + τ F (η) – F (un+ ) ≥ τ f – A un , η – un+ ∀η ∈ M,
(.)
where τ > is an iteration parameter and n ≥ . In this way the original variational inequality problem (.) is thus reduced to another variational inequality problem involving the duality mapping J instead of the original pseudomonotone mapping A . Such a problem can then be solved by known methods (see, for example, [, ]). If V = H, then the iteration generated by (.) can be written in the following form: (un+ – un , η – un+ ) + τ F (η) – F (un+ ) ≥ τ (f – A un , η – un+ ) ∀η ∈ M, for an arbitrary u ∈ M and τ > .
(.)
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 4 of 16
In [], Saddeek and Ahmed considered the following two-layer iteration method for solving the nonlinear operator equation (.) in a Banach space V : n ≥ ,
J(un+ – un ) = τ (f – A un ),
(.)
where u is an arbitrary point in M and τ > . In the case when V = H, iteration (.) can be written as follows: un+ = un – τ (A un – f ),
n ≥ ,
(.)
for τ > and u is an arbitrary point in M. Saddeek and Ahmed [] proved some weak convergence theorems of iterations (.) and (.) for approximating the solution of nonlinear equation (.). Attempts to modify the two-layer iterations (.) and (.) so that strong convergence is guaranteed have recently been made. In [], Saddeek introduced the following modification of (.) in a Hilbert space H (boundary point method): un+ = un – τ h(un )(A un – f ),
n ≥ ,
(.)
where τ > , u is an arbitrary point in M, and h : M → [, ] is a function defined by He and Zhu [] as follows: h(u) = inf α ∈ [, ] : αu ∈ M
∀u ∈ M.
(.)
He obtained strong convergence results for finding the minimum norm solution of nonlinear equation (.). In [], He and Zhu have observed that, if ∈/ M, calculating h(un ) implies determining h(un )un , a boundary point of M, so iteration (.) is known as the boundary point method. In [], Saddeek extended the results of Saddeek [] to a uniformly convex Banach space and introduced the following modification of the two-layer iteration (.) (boundary point method): Jun+ = Jun – τ h(un )(A un – f ),
n ≥ ,
(.)
where τ > , u is an arbitrary point in M, τ > , and h is defined by (.). In [], Noor introduced and studied the following generalized multivalued pseudomonotone mixed variational inequality problem: find u ∈ M, w ∈ A (u) such that w, η – u + F (η) – F (u) ≥ f , η – u ∀η ∈ M,
(.)
where A : V → C(V ∗ ) is a multivalued pseudomonotone mapping (see definition below), F : V → R ∪ {+∞} is a functional as above, and f ∈ V ∗ is a given element. Clearly, problems (.) and (.) are special cases of problem (.). The set of all u ∈ M satisfying (.) is denoted by SOL(M, F , A – f ). In [], Badriev et al. obtained the following weak convergence theorems using the twolayer iteration (.).
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 5 of 16
Theorem . (see [], Theorem ) Let V be a real reflexive Banach space with a uniformly convex dual space V ∗ , and let J : V → V ∗ be the duality mapping. Let M be a nonempty closed convex subset of V . Let A : V → V ∗ be a pseudomonotone, coercive, potential, and bounded Lipschitz continuous mapping. Let F : V → R ∪ {+∞} be a proper convex and γ -Lipschitzian (i.e., | F (u) – F (η) |≤ γ u – ηV ∀u, η ∈ V , γ > ) functional. Define a functional F : V → R ∪ {+∞} by F(u) = F (u) + F (u) – f , u,
F (u) =
A t(u) , u dt,
f ∈ V ∗.
(.)
Assume also that
< τ < min , , μ
μ = μ R + – (R + γ ) ,
(.)
where R = sup uV , u∈S
R = sup A u – f V ∗ ,
S = u ∈ M : F(u) ≤ F(u ) .
u∈S
Then the sequence {un } defined by (.) is bounded in V , and all of its weak limit points are solutions of problem (.). Badriev et al. [] have remarked that, due to the reflexivity of V , the mixed variational inequality (.) is solvable by Theorem .. In Theorem ., the assumption that V is reflexive can be dropped. Indeed, if V ∗ is uniformly convex, then V is uniformly smooth (and hence V is reflexive). Theorem . (see [], Theorem ) Let V = H be a real Hilbert space, and let M be a nonempty closed convex subset of H. Let A : H → H be a pseudomonotone, coercive, potential, and inverse strongly monotone mapping. Let Fi : H → R ∪ {+∞}, i = , , be the same as in Theorem .. Then the sequence {un } defined by (.) with < τ < τ = γ converges weakly in H to a solution of problem (.). Some attempts to prove the weak convergence of the whole sequence in the framework of Banach spaces have been made by Saddeek and Ahmed [] and Saddeek [, ]. 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 (.) be modified to prove its strong convergence to the minimum norm solution of problem (.). In this paper, inspired by [, ], and [], 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 [].
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 6 of 16
2 Definitions and preliminary Definition . (see [, , ]) A multivalued mapping A : V → C(V ∗ ) is called (i) pseudomonotone, if it is bounded and, for every sequence {un } ⊂ V , {wn } ⊂ A (un ), the conditions un u ∈ V
and
lim supwn , un – u ≤ n→∞
imply that for every η ∈ V there exists w ∈ A (u) such that lim infwn , un – η ≥ w, u – η; n→∞
(ii) coercive, if there exists a function ρ : R+ → R+ with limξ →∞ ρ(ξ ) = +∞ such that w, u ≥ ρ uV uV
∀u ∈ V , w ∈ A (u);
(iii) potential, if
w , u + η – w , u dt =
w , η dt
for all u, η ∈ V , w ∈ A (t(u + η)), w ∈ A (tu), w ∈ A (u + tη), t ∈ [, ]; (iv) bounded Lipschitz continuous, if ´ V ∗ ≤ μ(R) u – ηV w – w ´ ∈ A (η), where μ(R) and (ξ ) as above; for all u, η ∈ V , w ∈ A (u), w (v) inverse strongly monotone, if there exists a constant γ > such that ´ u – η ≥ γ w – w ´ V w – w, ´ ∈ A (η). for all u, η ∈ V , w ∈ A (u), w Definition . is an extension of Definition .((i)-(iv), (vi)) of single-valued mappings to multivalued mappings. Let G : M × V ∗ → R ∪ {+∞} be a functional defined as follows: G (u, Jη) = uV – Jη, u + JηV ∗ + F (u),
(.)
where u ∈ M, η ∈ V , Jη ∈ V ∗ . Definition . (see, for example, []) The mapping FM : V → C(M) is called generalized F -projection mapping if FM (η) = arg minu∈M G (u, Jη), ∀η ∈ V . If V = H and F (u) = ∀u ∈ M, then (.) reduces to the following simple form: G (u, Jη) = u – ηH ,
∀u ∈ M, η ∈ H,
and the generalized F -projection reduces to the projection M from H to C(M). The following two lemmas are also useful in the sequel.
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 7 of 16
Lemma . (see []) The generalized F -projection FM (η) has the following properties: (i) FM (η) is a nonempty closed convex subset of M for all η ∈ V ; (ii) for all η ∈ V , u¯ ∈ FM (η) if and only if ¯ ≥ ∀v ∈ M; ¯ u¯ – v + (F (v) – F (u) Jη – J u, (iii) if V is strictly convex, then FM (η) is a single-valued mapping. Let G : V × V → R+ ∪ {} be a functional defined as follows: G (u, η) = uV – Jη, u + ηV ,
∀u, η ∈ V .
(.)
Lemma . (see []) Let V be a real Banach space with a uniformly convex dual space V ∗ , let M be a nonempty closed convex subset of V , and let η ∈ V , u¯ ∈ FM (η). Then ¯ + G (u, ¯ η) ≤ G (u, η) ∀u ∈ M; (i) G (u, u) (ii) for u, η ∈ V , G (u, η) = iff u = η. A Banach space V is said to have the Kadec-Klee property (see, for example, []) if, for every sequence {un } in V with un u and un V → uV together imply that limn→∞ un – uV = . Every Hilbert space is uniformly convex, and every uniformly convex Banach space has the Kadec-Klee property.
3 Main results In this section, we propose a modification of the two-layer iteration method (.) 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 ∈ M, w ∈ A (u) such that h(w), η – u + F (η) – F (u) ≥ f , η – u
∀η ∈ M,
(.)
where A , F , f are defined as above and h is a positive constant.
3.1 The modified two-layer iteration For an arbitrary point u ∈ M, define un+ ∈ M as follows: Jun+ – Jun , η – un+ + τ F (η) – F (un+ ) ≥ τ f – h(un )wn , η – un+ ∀η ∈ M, (.) where τ > is the iteration parameter, n ≥ , J is the duality mapping, wn ∈ A (un ) and h is defined by (.). For M = V , F (u) = ∀u ∈ M, and η = un+ ± z, z ∈ M, (.) is equivalent to Jun+ = Jun – τ h(un )wn – f ,
∀n ≥ ,
(.)
where wn ∈ A (un ), and τ , J, h are defined as above. Observe that iteration (.) is a modification and generalization of iterations (.) and (.).
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 8 of 16
If V = H, A is a single-valued mapping in (.) and h(un ) = ∀n ≥ , we have iteration (.). Iteration (.) can be considered as a modified method for solving the following operator inclusion problem: find u ∈ V such that f ∈ A u,
f ∈ V ∗.
(.)
For each u ∈ V , w ∈ A (tu), let F˜ : V → R ∪ {+∞} be a functional defined by ˜ F(u) = F˜ (u) + F (u) – f , u,
F˜ (u) =
h(u)w , u dt,
f ∈ V ∗.
(.)
Let us assume also that R˜ = sup uV , u∈S˜
R˜ = sup h(u)w – f V ∗ , u∈S˜
˜ ˜ ) , ≤ F(u S˜ = u ∈ M : F(u)
(.)
where w ∈ A (u). Let μ˜ be a positive constant such that μ˜ = μ R˜ + – (R˜ + γ ) .
(.)
Theorem . Let V be a real uniformly convex Banach space with a uniformly convex dual space V ∗ , J : V → V ∗ be the duality mapping, and let M be a nonempty closed convex subset of V . Let A : V → C(V ∗ ) be a multivalued mapping. Suppose that A is pseudomonotone, coercive, potential, and bounded Lipschitz continuous mapping. Let F : V → R ∪ {+∞} be a proper convex (not necessarily differentiable) and γ -Lipschitzian functional with M ⊂ ˜ R˜ , R˜ , S˜ , and μ˜ be defined by (.), (.), and (.). Assume that < int(D(F )). Let F, τ = min{, μ˜ }. Let {h(un )} be an increasing and bounded real sequence in [, ]. Then, for an arbitrary u = u ∈ M, the sequence {un } defined by (.) converges strongly (i.e., the minimum norm element in SOL(M, F , h(w) – f )). to u˜ = FSOL(M,F ,h(w)–f ) Proof Since F is supposed to be convex and γ -Lipschitzian, and A is coercive and bounded, it results from [] and [] that F is weakly lower semicontinuous and F˜ is coercive; moreover, R˜ < +∞ and R˜ < +∞. Hence μ˜ < +∞. This means that the iterative sequence (.) is well defined. Now we divide the proof into steps. Step . We prove that {un } is bounded. To this end, it suffices to prove that {un } ⊂ S˜ ,
un V ≤ R˜ ,
n ≥ .
(.)
Let us prove (.) by induction on n. For n = , we have u ∈ S˜ . Suppose now that un ∈ S˜ . We will show that un+ ∈ S˜ .
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 9 of 16
Setting η = un in (.) and taking into account that the functional F is γ -Lipschitzian and J is uniformly monotone, and the inequality τ ≤ , we obtain un+ – un V un+ – un V ≤ Jun+ – Jun , un+ – un
≤ τ f – h(un )wn , un+ – un + F (un ) – F (un+ ) ≤ [R˜ + γ ]un+ – un V .
(.)
Now, using (.) together with the strict monotonicity of , we have un+ – un V ≤ – (R˜ + γ ).
(.)
Furthermore, it follows from the bounded Lipschitz continuity of A that, for any t ∈ [, ], wn ∈ A un , wn ∈ A (un+ + t(un – un+ )) w – wn , un+ – un ≤ μ(R∗ ) ( – t)(un+ – un ) un+ – un V n V ≤ μ(R∗ ) un+ – un V un+ – un V ,
(.)
where R∗ = max{un+ + t(un – un+ )V , un V }. Since un+ + t(un – un+ ) – un V ≤ ( – t)(un+ – un ) ≤ un+ – un V , V V
(.)
it follows from the definition of R∗ that R∗ ≤ R˜ + – (R˜ + γ ).
(.)
Since μ˜ is an increasing function, we must have μ(R ˜ ∗ ) ≤ μ˜ .
(.)
Consequently, it follows from (.) and (.) that – wn – wn , un+ – un ≥ –μ˜ un+ – un V un+ – un V .
(.)
Moreover, since A is potential, we have ˜ n+ ) = ˜ n ) – F(u F(u
h(un )wn , un – un+ dt – f , un – un+ + F (un ) – F (un+ )
=
h(un ) wn – wn , un – un+ dt – f – h(un )wn , un – un+
+ F (un ) – F (un+ ) ≥ – h(un ) wn – wn , un – un+ dt + τ – τ f – h(un )wn , un+ – un
+ F (un ) – F (un+ ) .
(.)
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 10 of 16
Setting η = un in (.) and using the uniform monotonicity of J, (.), (.), it results that ˜ n+ ) ≥ –μ˜ un+ – un V un+ – un V ˜ n ) – F(u F(u + τ – J(un+ ) – J(un ), un+ – un ≥ λ un+ – un V un+ – un V , λ = τ – – μ˜ > .
(.)
˜ n ) ≤ F(u ˜ ) and so un+ ∈ S˜ . So {un } is bounded. ˜ n+ ) ≤ F(u This implies that F(u Step . We prove that limn→∞ un+ – un V = and limn→∞ Jun+ – Jun V ∗ = . ˜ n )} is bounded and monotone, and thus we It follows from (.) that the sequence {F(u ˜ n ) exists. This together with (.) implies that have that limn→∞ F(u lim λ un+ – un V un+ – un V = .
n→∞
(.)
Since is continuous and strictly increasing, it follows from (.) that lim un+ – un V = .
n→∞
(.)
Since J is bounded Lipschitz continuous, is continuous and () = , it follows from (.) that lim Jun+ – Jun V ∗ = .
n→∞
(.)
Step . We show that there exists a subsequence {unk } of {un } such that unk u¯ ∈ V , ¯ and lim supk→∞ h(unk )wnk , unk – u ¯ ≤ . limk→∞ F (unk ) ≥ F (u), Since {un } is bounded and V is reflexive, we can choose a subsequence {unk } of {un } such that unk u¯ ∈ V as k → ∞. This together with the weak lower semicontinuity of F implies that limk→∞ F (unk ) ≥ ¯ F (u). Since F is γ -Lipschitzian, {h(un )} ⊂ [, ], it follows from (.) that, for arbitrary η ∈ M, h(unk )wnk , unk – η = h(unk )wnk , unk – unk+ + h(unk )wnk , unk+ – η ≤ h(unk )wnk , unk – unk+ + τ – Junk+ – Junk , η – unk+ + F (η) – F (unk ) + F (unk ) – F (unk+ ) + f , unk+ – unk + f , unk – η ≤ wnk V ∗ + f V ∗ + γ unk+ – unk V + τ – Junk+ – Junk V ∗
× η – unk+ V + F (η) – F (unk ) + f , unk – η ≤ Cη Junk+ – Junk V ∗ + unk+ – unk V (.) + F (η) – F (unk ) + f , unk – η, where Cη is a positive constant depending on η.
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 11 of 16
Setting η = u¯ in (.) and using the weak lower semicontinuity of F , (.), (.), we have ¯ ≤ lim sup Cu¯ Junk+ – Junk V ∗ + unk+ – unk V lim sup h(unk )wnk , unk – u k→∞
k→∞
¯ – F (unk ) + lim supf , unk – u ¯ + lim sup F (u) k→∞
k→∞
≤ .
(.)
Step . We show that u¯ ∈ SOL(M, F , h(w) – f ). Since {h(un )} ⊂ [, ] is bounded and monotone increasing, it follows that lim h(un ) = h > .
(.)
n→∞
By (.)-(.), the lower semicontinuity of F and by the pseudomonotonicity of A , we have = lim inf Cη Junk+ – Junk V ∗ + unk+ – unk V k→∞
≥ lim inf h(unk )wnk , unk – η + lim inf F (unk ) – F (η) + lim inff , η – unk k→∞
k→∞
¯ – F (η) + f , η – u, ¯ ¯ u¯ – η + F (u) ≥ h(w),
k→∞
¯ This means that u¯ ∈ SOL(M, F , h(w) – f ). ¯ ∈ A u. where w Step . We prove that
¯ unk+ – u ¯ + F (u) ¯ – F (unk+ ) ≤ , lim sup –J u,
(.)
k→∞
where u¯ = FSOL(M,F . ,h(w)–f ) ¯ Indeed take a subsequence {unk+ } of {un } such that unk+ u. ¯ Note that u¯ = FSOL(M,F . Then from u ∈ SOL(M, F , h(w) – f ), the weak lower ,h(w)–f ) semicontinuity of F , and Lemma .(ii), the desired inequality (.) follows immediately. ¯ V = . Step . We show that limn→∞ un – u ¯ it follows from the weak lower semicontinuity of · V that Since unk+ u,
¯ V. lim inf unk+ V ≥ u
(.)
k→∞
From the convexity of D(F ), F and from the weak lower semicontinuity of F , we obtain that F is subdifferentiable in int(D(F )). Thus, for all u ∈ D(F ), there exists an element u∗ ∈ V ∗ such that ¯ ≥ u∗ , u – u¯ , F (u) – F (u) and hence ¯ ≥ Junk , unk+ – u, ¯ F (unk+ ) – F (u)
k ≥ .
(.)
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 12 of 16
In view of unk+ = FSOL(M,F J – (Junk – τ h(unk )(f – wnk )), we have ,h(w)–f )
¯ Junk – τ h(unk )(f – wnk ) . G unk+ , Junk – τ h(unk )(f – wnk ) ≤ G u, By using (.) with Jη = Junk – τ h(unk )(f – wnk ), we have unk+ V – Junk – τ h(unk )(f – wnk ), unk+ + Junk – τ h(unk )(f – wnk ) V ∗ + F (unk+ ) ¯ V – Junk – τ h(unk )(f – wnk ), u¯ ≤ u ¯ + Junk – τ h(unk )(f – wnk ) V ∗ + F (u) ¯ V – Junk – τ h(unk )(f – wnk ), unk+ = u + Junk – τ h(unk )(f – wnk ), unk+ – u¯ ¯ + Junk – τ (unk )(f – wnk ) V ∗ + F (u), which implies that ¯ V + Junk , unk+ – u ¯ + F (u) ¯ – F (unk+ ) unk+ V ≤ u ¯ + τ h(unk )f , u¯ – unk+ . + τ h(unk )wnk , unk+ – u
(.)
¯ (.)-(.), Taking the lim supk→∞ on the both sides of (.) and using unk+ u, and (.) yields ¯ V , lim sup unk+ V ≤ u k→∞
which implies that ¯ V. lim sup unk+ V ≤ u
(.)
k→∞
Combining (.) and (.), we have ¯ V ≤ lim inf unk+ V ≤ lim sup unk+ V ≤ u ¯ V. u k→∞
(.)
k→∞
This shows that ¯ V. lim unk+ V = u
k→∞
(.)
Since V is a uniformly convex Banach space, then it has the Kadec-Klee property, and so from unk+ u¯ and (.) we obtain ¯ lim unk+ = u.
k→∞
¯ Let us now show that the whole sequence converges strongly to u.
(.)
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 13 of 16
Since {G (un+ , )} is bounded and nondecreasing (indeed, by Lemma .(i), we have G (un+ , un ) + G (un+ , ) ≤ G (un , ) and G (un+ , un ) ≥ (un+ V – un V ) ≥ ), it follows that {G (un+ , )} is convergent. This together with (.) implies that ¯ ). lim G (un+ , ) = G (u,
(.)
n→∞
Now, following to [], we suppose that there exists some subsequence {unj+ } of {un } such ˆ then by Lemma .(i) we obtain that limj→∞ unj+ = u, ¯ u) ˆ = lim G (unk+ , unj+ ) = lim G unk+ , FSOL(M,F ≤ G (u, ,h(w)–f ) k,j→∞
k,j→∞
, ≤ lim G (unk+ , ) – G FSOL(M,F ,h(w)–f ) k,j→∞
= lim G (unk+ , ) – G (unj+ , ) k,j→∞
¯ ) – G (u, ¯ ) = , = G (u, ¯ u) ˆ = and hence, by Lemma .(ii), it results that uˆ = u. ¯ which means that G (u, ¯ This completes the proof of Theorem .. Consequently, limn→∞ un = u. Theorem . Let V = H be a real Hilbert space, and let M be a nonempty closed convex subset of H. Let A : H → C(H) be a multivalued mapping. Suppose that A is a pseudomonotone, coercive, potential, and inverse strongly monotone mapping. Let {h(un )}, M, ˜ S˜ , μ˜ and F˜ , F , R˜ , R˜ be the same as in Theorem .. F, Then, for arbitrary u = u ∈ M, the sequence {un } defined by (un+ – un , η – un+ ) + τ F (η) – F (un+ ) ≥ τ f – h(un )wn , η – un+ with < τ < τ =
γ h
∀η ∈ M, (.)
, h > , converges strongly to u˜ = FSOL(M,F . ,h(w)–f )
Proof Since any inverse strongly monotone mapping is γ -Lipschitzian mapping, i.e., bounded Lipschitz continuous with μ(ξ ) = γ and (ξ ) = ξ , then by simple modifications of the proof of Theorem ., we can easily show that there exists a subsequence {unk+ } of ¯ H. {un } such that unk+ u¯ ∈ SOL(M, F , h(w) – f ) and limk→∞ unk+ H = u Since every Hilbert space is uniformly convex, by virtue of the Kadec-Klee property of H, we have limk→∞ unk+ = u¯ ∈ SOL(M, F , h(w) – f ). ¯ H. Now, we prove that un u¯ and limn→∞ un H = u From u¯ ∈ SOL(M, F , h(w) – f ), we have ¯ ≥ τ f – h(w), ¯ η – u¯ , τ F (η) – F (u)
∀η ∈ M.
(.)
Setting η = un+ in (.) and η = u¯ in (.), we have ¯ ≥ τ f – h(w), ¯ un+ – u¯ τ F (un+ ) – F (u)
(.)
¯ – F (un+ ) ≥ (un+ – un , un+ – u) ¯ + τ f – h(un )wn , u¯ – un+ . τ F (u)
(.)
and
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 14 of 16
Adding (.) and (.), we have ¯ un+ – u) ¯ ≤ (un – u, ¯ un+ – u) ¯ – τ h(un )wn – h(w), ¯ un+ – u¯ (un+ – u, ¯ , un+ – u¯ , = un – u¯ – τ h(un )wn – h(w) which implies that ¯ H ≤ un – u¯ – τ h(un )wn – h(w) ¯ H . un+ – u Then, by the inverse strong monotonicity of A , we obtain for all sufficiently large n ¯ H ≤ un – u ¯ H – τ h(un )wn – h(w), ¯ un – u¯ + τ h(un )wn – h(w) ¯ H un+ – u ¯ H – τ h(wn – w, ¯ un – u) ¯ + τ h wn – w ¯ H = un – u τh ¯ H – τ h – ¯ H . wn – w ≤ un – u γ ¯ H ≤ un – u ¯ H and so limn→∞ un – u ¯ H = σu¯ . Since – τγh > , it follows that un+ – u By following the same arguments as in [] and [], we can readily claim that all weak limit points of the sequence {un } coincide, and hence un u¯ as n → ∞. By the weak lower semicontinuity of · H , this implies that ¯ H. lim inf un H > u
(.)
n→∞
Analogically to the proof of step with obvious modifications, we have ¯ H. lim sup un H ≤ u
(.)
n→∞
¯ H. This, together with (.), implies that limn→∞ un H = u ¯ Applying again the virtue of the Kadec-Klee property of H, we obtain limn→∞ un = u. This completes the proof of Theorem .. Remark . Theorems . and . extend and improve the corresponding Theorems . and .. Example . (Axisymmetric shell problem) A quintessential example of a single-valued mapping satisfying all the assumptions contemplated in Theorems . and . 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 . Let s be a Lagrangian coordinate in the longitudinal direction such that < s < l. ◦
◦
(–) ∗ Let V = [W () p (, l)] and V = [W q (, l)] , q =
p , p–
p > . Set u(s) = (u (s), u (s)), η(s) =
(η (s), η (s)), M = {u ∈ V : u (s) + ≥ ∀s ∈ (, l)}, and λ = [( + du ) + ( du ) ] , λ = + u . ds ds Consider the surface force is characterized by a known constant function P. Let Ti (λi ), i = , , be two functions (tightening force) satisfying conditions ()-() in Badriev and Banderov [].
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 15 of 16
Consider the mappings A, B, C, D : V → V ∗ defined by du du dη T (λ ) + Au, η = , , ds; λ ds ds ds l dη du u + + u η ds; Bu, η = ds ds l du dη du Cu, η = + η + ds; ds ds ds l T (λ )η ds. Du, η =
l
If A = (A + D) + P(B + C), then by Theorems and in [] it follows that the mapping A satisfies all the assumptions postulated in Theorems . and ..
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. 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. Competing interests The author declares that he has no competing interests. Authors’ contributions I am the only author. I have read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 2 December 2016 Accepted: 25 August 2017 References 1. Badriev, IB, Zadvornov, OZ, Saddeek, AM: Convergence analysis of iterative methods for some variational inequalities with pseudomonotone operators. Differ. Equ. 37(7), 934-942 (2001) 2. Ekeland, I, Temam, R: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976) 3. Lions, JL: Quelques Méthodes de Résolution des Problémes aux Limites Nonlinéaires. Dunod and Gauthier-Villars, Paris (1969) 4. Gajewskii, H, Gröger, K, Zacharias, K: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie Verlag, Berlin (1974) 5. Marcotte, P, Wu, JH: On the convergence of projection methods. J. Optom. Theory Appl. 85, 347-362 (1995) 6. Zeidler, E: Nonlinear Functional Analysis and Its Applications II(B). Nonlinear Monotone Operators. Springer, Berlin (1990) 7. Badriev, IB, Zadvornov, OA, Saddeek, AM: On the iterative methods for solving some variational inequalities of the second kind. In: Contemporary Problems of Mathematical Modeling (Materials of the IX All-Russian School-Seminar, 8-13 September 2001, Abrau-Dyrso), pp. 36-41. Rostov University Publishers, Rostov-Un-Don (2001) 8. Badriev, IB: On the solving of variational inequalities of stationary problems of two-phase flow in porous media. Appl. Mech. Mater. 392, 183-187 (2013) 9. Badriev, IB, Nechaeva, LA: Mathematical simulation of steady filtration with multivalued law. PNRPU Mech. Bull. 3, 37-65 (2013) 10. Badriev, IB, Shagidulin, RR: A study of the convergence of a recursive process for solving a stationary problem of the theory of soft shells. J. Math. Sci. 73(5), 519-525 (1995) 11. Berezhnoi, DV, Paimushin, VN, Shalashilin, VI: Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements. Mech. Solids 44, 837-851 (2010)
Saddeek Journal of Inequalities and Applications (2017) 2017:216
Page 16 of 16
12. Davydov, RL, Sultanov, LU: Mathematical modeling of large elastic-plastic deformations. Appl. Math. Sci. 8(60), 2991-2996 (2014) 13. Stampacchia, G: Formes bilinéaires coercitivives sur les ensembles convexes. Comptes Rendus de l’Academie des Sciences 258, 4413-4416 (1994) 14. Istratescu, VI: Fixed Point Theory. Reidel, Dordrecht (1981) 15. Ciarlet, P: The Finite Element Method Foe Elliptic Problems. North-Holland, New York (1978) 16. Glowinski, R, Lions, JL, Tremolieres, R: Analyse Numerique des Inequations Variationnelles. Dunod, Paris (1976) 17. Fortin, M, Glowinski, R: In: Resolution Numeriques de Problémmes aux Limites Par des Mé thodes de Lagrangien Augmente, Paris (1983) 18. Saddeek, AM, Ahmed, SA: Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces. Arch. Math. 44, 273-281 (2008) 19. Saddeek, AM: A strong convergence theorem for a modified Krasnoselskii iteration method and its application to seepage theory in Hilbert spaces. J. Egypt. Math. Soc. 22, 476-480 (2014) 20. He, S, Zhu, W: A modified Mann iteration by boundary point method for finding minimum norm fixed point of nonexpansive mappings. Abstr. Appl. Anal. 2013, Article ID 768595 (2013) 21. Saddeek, AM: On the convergence of a generalized modified Krasnoselskii iterative process for generalized strictly pseudo-contractive mappings in uniformly convex Banach spaces. Fixed Point Theory Appl. 2016, 60 (2016) 22. Noor, MA: Generalized mixed variational inequalities and resolvent equations. Positivity 1, 145-154 (1997) 23. Saddeek, AM, Ahmed, SA: On the convergence of some iteration processes for J-pseudomonotone mixed variational inequalities in uniformly smooth Banach spaces. Math. Comput. Model. 46, 557-572 (2007) 24. Saddeek, AM: Generalized iterative process and associated regularization for J-pseudomonotone mixed variational inequalities. Appl. Math. Comput. 213, 8-17 (2009) 25. Saddeek, AM: Convergence analysis of generalized iterative methods for some variational inequalities involving pseudomonotone operators in Banach spaces. Appl. Math. Comput. 217, 4856-4865 (2011) 26. Pascali, D, Sburlan, S: Nonlinear Mapping of Monotone Type. Editura Academiei, Romania (1978) 27. Chang, SS, Wang, L, Tang, YK, Zhao, YH, Ma, Z: Strong convergence theorems of nonlinear operator equations for countable family of multi-valued quasi-ϕ -asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012, 69 (2012) 28. Wu, KQ, Huang, NJ: The generalized f -projection operator with an application. Bull. Aust. Math. Soc. 73, 307-317 (2006) 29. Li, X, Huang, N, O’Regan, D: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 60, 1322-1331 (2010) 30. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) 31. Qian, S: Strong convergence theorem for totally quasi-ϕ -asymptotically nonexpansive multivalued mappings under relaxed conditions. Fixed Point Theory Appl. 2015, 213 (2015) 32. Maruster, S: The solution by iteration of nonlinear equations in Hilbert spaces. Proc. Am. Math. Soc. 36(1), 69-73 (1977) 33. Badriev, IB, Banderov, VV: Numerical solution of the equilibrium of axisymmetric soft shells. Vestnik 1, 29-35 (2015)
