A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities

Journal of Inequalities and Applications, Sep 2017

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).

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A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities

Saddeek Journal of Inequalities and Applications A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities Ali Mohamed Saddeek 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). 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 sup Aun, un – u ≤  n→∞ imply lim inf Aun, un – η ≥ Aη, u – η ; n→∞ (ii) coercive, if there exists a function ρ : R+ → R+ with limξ→∞ ρ(ξ ) = +∞ such that Au, u ≥ ρ u V u V ; (iii) potential, if   (iv) bounded Lipschitz continuous, if Au – Aη V ∗ ≤ μ(R) u – η V , A t(u + η) , u + η – A(tu), u dt = A(u + tη), η dt;   where R = max{ u V , η V }, 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 ≥ (vi) inverse strongly monotone, if there exists a constant γ >  such that 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, (.) (.) 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 such that IM(u) = +,∞, uo.∈wM,, then problem (.) is equivalent to finding u ∈ M such that (.) (.) Au, η – u ≥  ∀η ∈ M, Au = f . 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 mapping J : V → V ∗ is called a duality mapping with gauge function if, for every u ∈ V , Ju, u = ( u V ) u V and Ju V ∗ = ( u V ). 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+ 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 > . J(un+ – un) = τ (f – Aun), n ≥ , 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 ≥ , h(u) = inf α ∈ [, ] : αu ∈ M ∀u ∈ M. where τ > , u is an arbitrary point in M, and h : M → [, ] is a function defined by He and Zhu [] as follows: In [], Saddeek and Ahmed considered the following two-layer iteration method for solving the nonlinear operator equation (.) in a Banach space V : 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 (.). 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 (.) (.) (.) (.) (.) (.) 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 – η V ∀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 where  < τ < min , μ ,  μ = μ R + –(R + γ ) , (.) (.) R = sup u V , u∈S R = sup Au – f V ∗ , u∈S S = u ∈ M : F(u) ≤ F(u) . 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 []. 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 sup wn, un – u ≤  n→∞ imply that for every η ∈ V there exists w ∈ A(u) such that lim inf wn, un – η ≥ w, u – η ; n→∞ w, u ≥ ρ u V u V ∀u ∈ V , w ∈ A(u); (ii) coercive, if there exists a function ρ : R+ → R+ with limξ→∞ ρ(ξ ) = +∞ such that (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 for all u, η ∈ V , w ∈ A(u), w´ ∈ A(η), where μ(R) and (ξ ) as above; (v) inverse strongly monotone, if there exists a constant γ >  such that w – w´ V ∗ ≤ μ(R) w – w´ , u – η ≥ γ w – w´ V for all u, η ∈ V , w ∈ A(u), w´ ∈ A(η). 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) = u V –  Jη, u + Jη V ∗ + 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 – η H , ∀u ∈ M, η ∈ H, and the generalized F-projection reduces to the projection The following two lemmas are also useful in the sequel. M from H to C(M). 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 Jη – Ju¯ , u¯ – v + (F(v) – F(u¯ ) ≥  ∀v ∈ M; (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) = u V –  Jη, u + η V , ∀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 (i) G(u, u¯ ) + G(u¯ , η) ≤ G(u, η) ∀u ∈ M; (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 → u V together imply that limn→∞ un – u V = . 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: where wn ∈ A(un), and τ , J , h are defined as above. Observe that iteration (.) is a modification and generalization of iterations (.) and (.). 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 ≥ , 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 u V , u∈S˜ R˜ = sup h(u)w – f V ∗ , u∈S˜ S˜ = u ∈ M : F˜ (u) ≤ 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 ⊂ int(D(F)). Let F˜ , R˜, R˜, S˜, and μ˜  be defined by (.), (.), and (.). Assume that  < 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 to u˜ = SFOL(M,F,h(w)–f ) (i.e., the minimum norm element in SOL(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˜. 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 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, wn ∈ A(un+ + t(un – un+)) wn – wn, un+ – un ≤ μ(R∗) ( – t)(un+ – un) V where R∗ = max{ un+ + t(un – un+) V , un V }. Since 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 – wn – wn, un+ – un ≥ –μ˜  Moreover, since A is potential, we have F˜ (un) – F˜ (un+) = h(un)wn, un – un+ dt – f , un – un+ + F(un) – F(un+) = h(un) wn – wn , un – un+ dt – f – h(un)wn, un – un+    ≥ –   + F(un) – F(un+)  + F(un) – F(un+) . h(un) wn – wn , un – un+ dt + τ – τ f – h(un)wn, un+ – un (.) (.) (.) (.) (.) (.) (.) (.) Setting η = un in (.) and using the uniform monotonicity of J , (.), (.), it results that F˜ (un) – F˜ (un+) ≥ –μ˜  This implies that F˜ (un+) ≤ F˜ (un) ≤ F˜ (u) and so un+ ∈ S˜. So {un} is bounded. Step . We prove that limn→∞ un+ – un V =  and limn→∞ Jun+ – Jun V ∗ = . It follows from (.) that the sequence {F˜ (un)} is bounded and monotone, and thus we have that limn→∞ F˜ (un) exists. This together with (.) implies that lim λ n→∞ lim n→∞ un+ – un V = . Since J is bounded Lipschitz continuous, from (.) that lim Jun+ – Jun V ∗ = . n→∞ Step . We show that there exists a subsequence {unk } of {un} such that unk u¯ ∈ V , limk→∞ F(unk ) ≥ F(u¯ ), and lim supk→∞ h(unk ) wnk , unk – 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. Setting η = u¯ in (.) and using the weak lower semicontinuity of F, (.), (.), we have F(unk+ ) – F(u¯ ) ≥ Junk , unk+ – u¯ , k ≥ . lim sup h(unk ) wnk , unk – u¯ ≤ lim sup Cu¯ Junk+ – Junk V ∗ + unk+ – unk V k→∞ k→∞ + lim sup F(u¯ ) – F(unk ) + lim sup f , unk – 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→∞ ≥ likm→i∞nf h(unk ) wnk , unk – η + lim inf F(unk) – F(η) + lim inf f , η – unk k→∞ k→∞ ≥ h( w¯), u¯ – η + F(u¯ ) – F(η) + f , η – u¯ , where w¯ ∈ A u¯. This means that u¯ ∈ SOL(M, F, h(w) – f ). Step . We prove that lim sup –Ju¯ , unk+ – u¯ + F(u¯ ) – F(unk+ ) ≤ , k→∞ where u¯ = SFOL(M,F,h(w)–f ). Indeed take a subsequence {unk+ } of {un} such that unk+ u¯ . Note that u¯ = SFOL(M,F,h(w)–f ). Then from u¯ ∈ SOL(M, F, h(w) – f ), the weak lower semicontinuity of F, and Lemma .(ii), the desired inequality (.) follows immediately. Step . We show that limn→∞ un – u¯ V = . Since unk+ u¯ , it follows from the weak lower semicontinuity of · V that lim inf unk+ V ≥ u¯ V . k→∞ F(u) – F(u¯ ) ≥ u∗, u – u¯ , and hence 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 (.) (.) (.) (.) (.) In view of u n k+ = F  SOL(M,F ,h(w)–f )  – J (Ju n k τ – h(u )(f – w )), we have n n k k G u  n k+ , Ju n k τ – h(u )(f – w ) n n k k ≤ G u, Ju  ¯ n k τ – h(u )(f – w ) . n n k k η By using (.) with J = Ju n k τ – h(u )(f – w ), we have n n k k u n k+ V  u ≤ ¯  V  V –  Ju n k τ – h(u )(f – w ), u n n ¯ k k + Ju n k τ – h(u )(f – w ) n n k k  V ∗ + F (u)  ¯ = u ¯ –  Ju n k τ – h(u )(f – w ), u n n k k n k+ +  Ju n k τ – h(u )(f – w ), u n n k k n k+ – u ¯ + Ju n k τ – (u )(f – w ) n n k k  V ∗ + F (u),  ¯ –  Ju n k τ – h(u )(f – w ), u n n k k n k+ + Ju n k τ – h(u )(f – w ) n n k k + F (u  n k+ )  V ∗ u n k+ V  u ≤ ¯  V +  Ju , u n k n k+ – u + F (u) – F (u ¯  ¯  n k+ ) τ +  h(u ) w , u n n k k n k+ τ – u +  h(u ) f , u – u ¯ n ¯ k n k+ . (.) on the both sides of (.) and using u n k+ u, (.)-(.), ¯ which implies that Taking the lim sup and (.) yields k →∞ which implies that lim sup k →∞ lim sup k →∞ u n k+ V  u , ≤ ¯ V  u n k+ V ≤ ¯ V u . This shows that k lim →∞ u n k+ V = u . ¯ V k lim →∞ u n k+ = u. ¯ Combining (.) and (.), we have u ¯ V ≤ lim inf k →∞ u n k+ V ≤ lim sup k →∞ u n k+ V ≤ ¯ V u . Since V is a uniformly convex Banach space, then it has the Kadec-Klee property, and so from u n k+ u and (.) we obtain ¯ Let us now show that the whole sequence converges strongly to u. ¯ (.) (.) (.) (.) 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 that limj→∞ unj+ = uˆ , then by Lemma .(i) we obtain ≤ k,lji→m∞ G(unk+ , ) – G  ≤ G(u¯ , uˆ ) = lim G(unk+ , unj+ ) = lim G unk+ , SFOL(M,F,h(w)–f ) k,j→∞ k,j→∞ F SOL(M,F,h(w)–f ),  = lim k,j→∞ G(unk+ , ) – G(unj+ , ) = G(u¯ , ) – G(u¯ , ) = , which means that G(u¯ , uˆ) =  and hence, by Lemma .(ii), it results that uˆ = u¯ . Consequently, limn→∞ un = u¯ . This completes the proof of Theorem .. 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, F˜ , S˜, μ˜  and F˜, F, R˜ , R˜  be the same as in Theorem .. 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+ ∀η ∈ M, (.) with  < τ < τ = hγ , h > , converges strongly to u˜ = F SOL(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 {un} such that unk+ u¯ ∈ SOL(M, F, h(w) – f ) and limk→∞ unk+ H = u¯ H . 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 ). Now, we prove that un u¯ and limn→∞ un H = u¯ H . From u¯ ∈ SOL(M, F, h(w) – f ), we have τ F(η) – F(u¯ ) ≥ τ f – h( w¯), η – u¯ , ∀η ∈ M. Setting η = un+ in (.) and η = u¯ in (.), we have τ F(un+) – F(u¯ ) ≥ τ f – h(w¯ ), un+ – u¯ and τ F(u¯ ) – F(un+) ≥ (un+ – un, un+ – u¯ ) + τ f – h(un)wn, u¯ – un+ . (.) (.) (.) Adding (.) and (.), we have (un+ – u¯ , un+ – u¯ ) ≤ (un – u¯ , un+ – u¯ ) – τ h(un)wn – h( w¯), un+ – u¯ = un – u¯ – τ h(un)wn – h(w¯ ) , un+ – u¯ , which implies that un+ – u¯ H ≤ un – u¯ – τ h(un)wn – h(w¯ ) H . Then, by the inverse strong monotonicity of A, we obtain for all sufficiently large n  un+ – u¯ 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 τ h ≤ un – u¯ H – τ h  – γ wn – w¯ H . Since  – τγh > , it follows that un+ – u¯ H ≤ un – u¯ H and so limn→∞ un – u¯ H = σu¯. 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 lim inf un H > u¯ H . n→∞ lim sup un H ≤ u¯ H . n→∞ Analogically to the proof of step  with obvious modifications, we have (.) (.) This, together with (.), implies that limn→∞ un H = u¯ H . 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 = pp– , p > . Set u(s) = (u(s), u(s)), η(s) = (η(s), η(s)), M = {u ∈ V : u(s) +  ≥  ∀s ∈ (, l)}, and λ = [( + ddus ) + ( ddus )]  , λ =  + u. 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 []. Consider the mappings A, B, C, D : V → V ∗ defined by    l l l  Au, η = Bu, η = Cu, η = l T(λ) λ   dη u   ds  + du ds  + du du , ds ds , dη ds ds; +  + η + du ds du dη ds ds uη ds; ds; Du, η = T(λ)η ds. 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 results. 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 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. 1. 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Ali Mohamed Saddeek. A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities, Journal of Inequalities and Applications, 2017, 216, DOI: 10.1186/s13660-017-1482-0