# Multivalued Variational Inequalities withDJ-Pseudomonotone Mappings in Reflexive Banach Spaces

Authors: A. M. Saddeek; S. A. Ahmed
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the study of a class of variational inequalities with multivaluedDJ-pseudomonotone mappings in reflexive Banach spaces by using the DJ-antiresolvent technique. An application to the multivalued nonlinear DJ-complementarity problem is also presented. The results coincide with the corresponding results announced by many others for the gradient state.


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## Body

## 1. Introduction and Preliminaries

Variational inequalities give a convenient mathematical framework for discussing a large variety of interesting problems appearing in pure and applied sciences. It is well known that the theory of pseudomonotone mappings plays an important part in the study of the above-mentioned variational inequalities.In recent years, pseudomonotone theory has become an attractive field for many mathematicians (see [1–8]).In a very recent paper [9], by using the DJ-antiresolvent technique (where J is the duality mapping) devised by the first author, the author introduced a new concept of monotonicity, which is called the DJ-pseudomonotone type.In the present paper, the concept of multivaluedDJ-pseudomonotone mappings in reflexive Banach spaces is used to study a wide class of variational inequalities, called the multivalued DJ-pseudomonotone variational inequalities.Moreover, the results obtained in this paper can be applied to the multivalued nonlinearDJ-complementarity problem. This problem contains, in particular, a mathematical model arising in the study of the postcritical equilibrium state of a thin plate resting, without friction on a flat rigid support (see [10–12]). The results coincide with the corresponding results (see [2, 13–15]) in the case of gradient mappings.Unless otherwise stated,V stands for a real reflexive Banach space with norm ∥·∥V and V⋆ stands for the uniformly convex dual of V with the dual norm ∥·∥V⋆. The duality pairing between V and V⋆ is denoted by 〈·,·〉. The set of all nonnegative integers is denoted by ℕ. The field of real (resp., positive real) numbers is denoted by ℝ (resp., ℝ+). Notation “→” stands for strong convergence and “⇀” for weak convergence.A mappingJ:V→V⋆ is said to be a duality mapping (see, e.g., [16]) with gauge function Φ (i.e., Φ is continuous strictly increasing real-valued function satisfying Φ(0)=0 and limt→+∞Φ(t)=+∞) if for every u∈V,〈Ju,u〉=∥Ju∥V⋆∥u∥V=Φ(∥u∥V)(∥u∥V). If V=H is a Hilbert space, then J≡I, the identity mapping.Assume thatV⋆ has a weakly sequentially continuous duality mapping J (i.e., if {un}n∈ℕ is a sequence in V which weakly convergent to a point u, then the sequence {J(un)}n∈ℕ converges to J(u) (see, e.g., [17])).Letg:V→ℝ∪{+∞} be a function. The domain of g is domg={u∈V:g(u)<+∞}. When domg≠ϕ, g is called proper (see, e.g., [18]). The interior of the domain of g is denoted by intdomg. The function g is said to be Gâteaux differentiable at u∈intdomg (see, e.g., [18]), if
(1)g′(u,η)=limt→0g(u+tη)-g(u)t
exists for all η∈V.Letg be proper, convex, lower semicontinuous, and Gâteaux differentiable at u∈intdomg; then the gradient of g at u is the function ∇g(u) which is defined by 〈∇g(u),η〉=g′(u,η) for any η∈V. It is known (see, e.g., [19]) that the conjugate g⋆:V⋆→ℝ∪{+∞} is also proper, convex, and lower semicontinuous.The convex functiong is said to be of Legendre type (see, e.g., [20]) if the following conditions hold: (L1)
intdom(g)≠ϕ,  g is Gâteaux differentiable on intdom(g) and dom∇g=intdomg;(L2)
intdom(g⋆)≠ϕ,  g⋆ is Gâteaux differentiable on intdom(g⋆) and dom∇g⋆=intdom(g⋆).It is well known (see, e.g., [21]) that if g is a proper, convex, lower semicontinuous, and Legendre type, then ∇g⋆=(∇g)-1 and range ∇g⋆=dom∇g.Throughout this paper, the functiong:V→ℝ+∪{+∞} is proper, convex, and lower semicontinuous which is also Legendre on intdom(g).The Bregman distance (see, e.g., [22]) is the function Dg:V×intdom(g)→ℝ∪{+∞} defined by
(2)Dg(ν,u)=g(ν)-g(u)-〈∇g(u),ν-u〉,
with
(3)domDg=(domg)×intdomg.
It should be pointed out that if V=H is a Hilbert space and g=(1/2)∥·∥H2, then ∇g=I (the identity mapping) and DJ(ν,u)=(1/2)∥ν-u∥H2.For a multivalued mappingT:V→2V⋆, the associated DJ-antiresolvent (where J:V→V⋆ is the duality mapping) of J-T (see [9]) is the mapping TJ:V→2V, defined by
(4)TJ=∇g⋆(J-T).
Such a mapping is known as (see [23]) a Dg-antiresolvent mapping of T when J=∇g (in this case, the mapping TJ is denoted by Tg).In light of the above-mentioned discussion, we note that ifJ-T=∇g, then TJ is the identity mapping I.Following [9], the mapping J-T:V→2V⋆ is said to be DJ-pseudomonotone, if for every u,η∈domJ∩domT,ν∈(J-T)(u),ω∈(J-T)(η), and every sequence {un}n∈ℕ⊂domJ∩domT and νn∈(J-T)(un) the conditions
(5)∇g⋆(νn)⇀∇g⋆(ν),limsupn→∞〈νn,∇g⋆(νn)-∇g⋆(ν)〉≤0
imply that
(6)liminfn→∞〈νn,∇g⋆(νn)-∇g⋆(ω)〉≥〈ω,∇g⋆(ν)-∇g⋆(ω)〉.
As remarked in [9], the DJ-pseudomonotonicity of the mapping J-T coincides with the pseudomonotonicity (or the Dg-pseudomonotonicity in the sense of Bregman distance Dg) of the mapping ∇g, if J-T=∇g.The multivalued variational inequality defined by theDJ-mapping (or multivalued DJ-variational inequality) J-T:V→2V⋆ and the set K⊂V is to find u∈K such that
(7)∃ν∈(J-T)(u),ω∈(J-T)(η):〈ν-f,∇g⋆(ω)-∇g⋆(ν)〉≥0,∀η∈K,
where f∈2V⋆.The multivalued nonlinear complementarity problem defined by theDJ-mapping (or multivalued nonlinear DJ-complementarity problem) J-T:V→2V⋆ and the set K is to find u∈K such that
(8)〈ν-f,∇g⋆(ν)〉=0,〈ν-f,∇g⋆(ω)-∇g⋆(ν)〉≥0,∀η∈K,
where f∈2V⋆, ν∈(J-T)(u), and ω∈(J-T)(η).The multivaluedDJ-variational inequality and multivalued nonlinear DJ-complementarity problem are very general in the sense that they include, as special cases, multivalued variational inequality and multivalued nonlinear complementarity problem.The following definition and results will be used in the sequel.Definition 1 (see, e.g., [15, p. 84]).
The mappingA:V→2V⋆ is continuous on finite dimensional subspaces if for any finite dimensional subspace V0⊂V, the restriction of A to V0∩dom(A) is weakly continuous.Corollary 2 (see [24]).
Letj:V0⊂V→V be the injection mapping. Let j⋆:V⋆→V0⋆ be its dual mapping. Then, j⋆Aj:dom(A)∩jV0→2V0⋆ is continuous.Corollary 3 (see [25]).
LetK be a nonempty compact convex set of ℝn and let S:K→2K be continuous. Then S admits a fixed point.


## 2. Main Results

Theorem 4.
LetK be a closed convex set in H and let T:K→2H be a multivalued mapping. Then the following are equivalent:
(9)(1)∇g⋆(ν′)∈PrK(x)=argmin{DI(∇g⋆(ω′),x)=12∥∇g⋆(ω′)-x∥H2:∇g⋆(ω′)∈K,ω′∈(I-T)(η),η∈K{(ω′)}},
the multivalued projection for K;
(10)(2)∇g⋆(ν′)∈K:〈∇g⋆(ν′)-x,∇g⋆(ω′)-∇g⋆(ν′)〉≥0∀∇g⋆(ω′)∈K,whereω′∈(I-T)(η),ν′∈(I-T)(y),y,η∈K.Proof.
Assume that (1) holds. Letx∈2H and ∇g⋆(ν′)∈PrK(x)⊂K. For every ∇g⋆(ω′)∈K and t∈(0,1], we have
(11)DI(x,∇g⋆(ν′))≤DI(x,(1-t)∇g⋆(ν′)+t∇g⋆(ω′))=DI(x,∇g⋆(ν′))-t〈x-∇g⋆(ν′),∇g⋆(ω′)-∇g⋆(ν′)〉+t2DI(∇g⋆(ω′),∇g⋆(ν′)).
This implies
(12)〈x-∇g⋆(ν′),∇g⋆(ω′)-∇g⋆(ν′)〉≤tDI(∇g⋆(ω′),∇g⋆(ν′)).
Hence, t→0+ implies (2).
On the other hand, assume that (2) holds. For every∇g⋆(ω′)∈K, we have
(13)DI(∇g⋆(ω′),x)=DI(∇g⋆(ω′),∇g⋆(ν′))+〈∇g⋆(ω′)-x,∇g⋆(ω′)-∇g⋆(ν′)〉+DI(∇g⋆(ν′),x)≥DI(∇g⋆(ν′),x).
This implies (1).Remark 5.
In the particular situation whenI-T=∇g Theorem 4 coincides (in gradient setting) with Theorem 2.3 in [15] and also with Proposition 2.1 (1) and (2) in [2].Theorem 6.
In addition to conditions onV,V⋆,g⋆, and J, one assumes that V is separable, K⊂V is nonempty closed and convex, T:K→2V⋆,∇g⋆ are weakly continuous mappings, and either T or ∇g⋆ is continuous. Moreover, assume that the mapping J-T:K→2V⋆ is a bounded DJ-pseudomonotone mapping and that, for each η∈K, there exist η0∈K,ω0∈(J-T)(η0),ω∈(J-T)(η), and r>0 such that
(14)〈ω,∇g⋆(ω)-∇g⋆(ω0)〉>〈f,∇g⋆(ω)-∇g⋆(ω0)〉,f∈2V⋆,∥∇g⋆(ω)∥V≥r.
Then there exists a solution to the multivalued DJ-variational inequality (7).Proof.
Suppose that{x1,x2,x3,…} is an infinite dense set in K and Vm,m∈ℕ, is the linear span of {x1,x2,x3,…,xm}.
LetKm=con{x1,x2,x3,…,xm}={∑i=1m‍λixi,λi≥0,∑i=1m‍λi=1}. Let jm:Vm→V be the injection mapping and let jm⋆:V⋆→Vm⋆ be its restriction dual. Observe that ∪Km is dense in K, for m∈ℕ.
Now, fix an integerm≥1 and consider the finite dimensional problem.
Findum∈Km such that for each η∈Km, there exist sm∈jm⋆(J-T)jm(um),νm∈(J-T)(um),  and  ω∈(J-T)(η) such that
(15)〈sm-jm⋆f,∇g⋆(ω)-∇g⋆(νm)〉≥0.
The equivalent form of problem (15) is to find um∈Km such that for each η∈Km, there exist sm∈jm⋆(J-T)jm(um),νm∈(J-T)(um),   and  ω∈(J-T)(η) such that
(16)〈∇g⋆(νm),∇g⋆(ω)-∇g⋆(νm)〉≥〈∇g⋆(νm)+jm⋆f-sm,∇g⋆(ω)-∇g⋆(νm)〉.
Using the identification of Vm with ℝm and Vm⋆ and Theorem 4 (with ℝm=HandJ=I), we see that (16) is equivalent to ∇g⋆(νm)∈PrKm(∇g⋆(νm)+jm⋆f-sm).
LetB-r(0) be any closed ball containing Km. It is well known (see, e.g., [26, p. 54, 224]) that B-r(0) is compact and convex in Km; thus it is weakly closed.
From Corollary2, j⋆(J-T)j:B-r(0)→2B-r(0) is continuous; hence the function ∇g⋆(ν)↦PrB-r(0)(∇g⋆(ν)+jm⋆f-sm) is continuous from B-r(0) into 2B-r(0).
Hence, by Corollary3, this equation admits a solution. If the closed convex set K is assumed to be bounded, then by the reflexivity of V it is weakly compact (by employing the Banach -Alaoglu theorem (see, e.g., [16, p. 3]).
Then we have a subsequence denoted by{um}m∈ℕ such that um⇀u∈K. Since J-T is bounded, we have ∥νm∥V⋆≤Mfor all m∈ℕ. Since J-T is weakly continuous and since either J-T or ∇g⋆ is continuous by hypothesis, it follows that TJ is weakly continuous by [27, Lemma 1]. So, we have ∇g⋆(νm)⇀∇g⋆(ν).
Now, we prove that(17)limsupm→∞〈νm,∇g⋆(νm)-∇g⋆(ν)〉≤0.
For any ϵ>0, choose N so large and u~∈KN such that
(18)∥∇g⋆(ν)-∇g⋆(ν~)∥V<ϵforν~∈(J-T)(u~).
Therefore, we have
(19)〈νm-f,∇g⋆(νm)-∇g⋆(ν~)〉≤0form≥N.
Since KN⊂Km, we have
(20)limsupm→∞〈νm,∇g⋆(νm)-∇g⋆(ν)〉=limsupm→∞[〈νm,∇g⋆(νm)-∇g⋆(ν~)〉+〈νm,∇g⋆(ν~)-∇g⋆(ν)〉]≤(∥f∥V⋆+M)ϵ.
Since ϵ is arbitrary, this shows the desired inequality.
By theDJ-pseudomonotonicity of J-T, it follows that
(21)liminfm→∞〈νm,∇g⋆(νm)-∇g⋆(ω)〉≥〈ν,∇g⋆(ν)-g⋆(ω)〉
for all η∈domJ∩domT and ω∈(J-T)(η).
Ifη∈Kn,m≥n, we have
(22)〈νm,∇g⋆(νm)-∇g⋆(ω)〉≤〈f,∇g⋆(νm)-∇g⋆(ω)〉.
Hence
(23)〈ν,∇g⋆(ν)-∇g⋆(ω)〉≤〈f,∇g⋆(ν)-∇g⋆(ω)〉
for every η in Kn,n∈ℕ,ω∈(J-T)(η).
Since∪nKn is dense in Kn, so we have that u is a solution to (7).
Now, to complete the proof, we consider the case whenK is unbounded.
In this case we consider the setKρ={η∈K:∥∇g⋆(ω)∥V≤ρ}, where ρ=max{∥∇g⋆(ω0)∥V,r}.
SinceKρ is bounded, there exists at least one uρ∈Kρ:
(24)〈νρ-f,∇g⋆(ω)-∇g⋆(νρ)〉≥0
for νρ∈(J-T)(uρ) and η∈Kρ.
Sinceη0∈Kρ, we have
(25)〈νρ-f,∇g⋆(νρ)-∇g⋆(ω0)〉≤0forω0∈(J-T)(η0).
This, together with (14), implies that ∥∇g⋆(νρ)∥V<ρ.
To clarify thatuρ is also a solution to original problem on K, for any η∈K, set ∇g⋆(ωt) = (1-t)∇g⋆(νρ)+t∇g⋆(ω) for t>0 is sufficiently small, where ωt∈(J-T)(ηt) and ηt∈Kρ. Consequently
(26)uρ∈Kρ⊂K:0≤〈νρ-f,∇g⋆(ωt)-∇g⋆(νρ)〉=t〈νρ-f,∇g⋆(ω)-∇g⋆(νρ)〉forη∈K.
This completes the proof.Remark 7.
In the particular situation whenJ-T=∇g, Theorem 6 coincides with the Brezis Theorem (see, e.g., [13, 14]) for the case of gradient mapping.
We are now in a position to state and prove the following theorem.Theorem 8.
Let all assumptions of Theorem6 hold, except for condition (14) let it be replaced by the DJ-coercive condition: for ω∈(J-T)(η),
(27)lim∥∇g⋆(ω)∥V→∞[〈ω,∇g⋆(ω)-∇g⋆(ω0)〉∥∇g⋆(ω)∥V]=+∞,ω0∈(J-T)(η0),η,η0∈K.
Suppose further that K has the following property (W): α∇g⋆(ω)∈K for all ∇g⋆(ω)∈K and α≥0.
Then for everyf∈2V⋆ there exist u∈K, ν∈(J-T)(u) such that
(28)〈ν-f,∇g⋆(ν)〉=0,〈ν-f,∇g⋆(ω)-∇g⋆(ν)〉≥0
for all η∈K,ω∈(J-T)(η).Proof.
Letf∈2V⋆ satisfy ∥f∥V⋆<M and
(29)∥∇g⋆(ω0)∥V<2M-∥f∥V⋆∥f∥V⋆ρ.
The DJ-coercivity of J-T implies that there exists ρ>0 such that
(30)〈ω,∇g⋆(ω)-∇g⋆(ω0)〉≥2M∥∇g⋆(ω)∥V
for ω∈(J-T)(η),η∈K with ∥∇g⋆(ω)∥V≥ρ.
So we conclude(31)〈ω-f,∇g⋆(ω)-∇g⋆(ω0)〉≥2M∥∇g⋆(ω)∥V-∥f∥V⋆∥∇g⋆(ω)∥V-∥f∥V⋆∥∇g⋆(ω0)∥V≥(2M-∥f∥V⋆)ρ-∥f∥V⋆∥∇g⋆(ω0)∥V>0forη∈K,ω∈(J-T)(η)with∥∇g⋆(ω)∥V≥ρ.
The second part of (28) thus follows from Theorem 6.
To prove the first part of (28), observe that we can choose a point η in K and ω∈(J-T)(η) and assume that ∇g⋆(ω)=0.
Therefore, from Theorem6, we have
(32)〈ν-f,∇g⋆(ν)〉≤0
for all ν∈(J-T)(u),u∈K.
On the other hand, setting∇g⋆(ω)=α∇g⋆(ν), where α>1, we get
(33)0≤〈ν-f,∇g⋆(ω)-∇g⋆(ν)〉=(α-1)〈ν-f,∇g⋆(ν)〉.
This implies
(34)〈ν-f,∇g⋆(ν)〉≥0.
So,
(35)〈ν-f,∇g⋆(ν)〉=0.
This completes the proof of (28).The following proposition gives a characterization of the sum of twoDJ-Pseudomonotone mappings.Proposition 9.
LetV,V⋆, and J be as above and let Ti:V→2V⋆,i=1,2,and∇g⋆ be weakly continuous mappings. If J-Ti:V→2V⋆,i=1,2, are DJ-pseudomonotone mappings such that domJ∩domTi≠ϕ,i=1,2, then ∑i=12‍(J-Ti) is DJ-pseudomonotone.Proof.
Lety~n∈∑i=12‍(J-Ti)(ηn),y~∈∑i=12‍(J-Ti)(η),ηn,η∈domJ∩domTi,i=1,2, with ∇g⋆(y~n)⇀∇g⋆(y~) and
(36)limsupn→∞〈y~n,∇g⋆(y~n)-∇g⋆(y~)〉≤0.
Now, we prove for yn(i)∈(J-Ti)(ηn),y(i)∈(J-Ti)(η),i=1,2, that
(37)limsupn→∞〈yn(1),∇g⋆(yn(1))-∇g⋆(y(1))〉≤0,limsupn→∞〈yn(2),∇g⋆(yn(2))-∇g⋆(y(2))〉≤0.
If
(38)limsupn→∞〈yn(2),∇g⋆(yn(2))-∇g⋆(y(2))〉=ϵ>0,
(note that otherwise, by symmetry), then there exists a subsequence {ynk(2)}k∈ℕ⊂{yn(2)}n∈ℕ such that
(39)limsupk→∞〈ynk(2),∇g⋆(ynk(2))-∇g⋆(y(2))〉=ϵ.
This implies that
(40)limsupk→∞〈ynk(1),∇g⋆(ynk(1))-∇g⋆(y(1))〉=limsupk→∞[{(ynk(2))}〈y~nk,∇g⋆(y~nk)-∇g⋆(y~)〉-〈ynk(2),∇g⋆(ynk(2))-∇g⋆(y(2))〉]≤0-ϵ.
From the DJ-pseudomonotonicity of J-T1, we get for all y′∈(J-T1)(η′),η′∈domJ∩domT1(41)〈y(1),∇g⋆(y(1))-∇g⋆(y′)〉≤liminfk→∞〈ynk(1),∇g⋆(ynk(1))-∇g⋆(y′)〉.
Letting y′=y(1), we obtain
(42)0≤liminfk→∞〈ynk(1),∇g⋆(ynk(1))-∇g⋆(y(1))〉≤0-ϵ,
a contradiction.
Hence,(43)limsupk→∞〈ynk(1),∇g⋆(ynk(1))-∇g⋆(y(1))〉≤0,limsupk→∞〈ynk(2),∇g⋆(ynk(2))-∇g⋆(y(2))〉≤0.
This holds for any subsequence, so (37) holds and the proof follows immediately by the superadditivity of the liminf.


## 3. Application to Multivalued NonlinearDJ-Complementarity Problem

As applications of Theorem8 we consider the multivalued nonlinear DJ-complementarity problem (8) with T=λT1+T2-λJ, where Ti,i=1,2 are two nonlinear multivalued mappings from K to 2V⋆, and λ∈(0,∞).Theorem 10.
LetV,V⋆,g⋆,J, and K be the same as in Theorem 6, and suppose that K has the property (W). Let the mappings ∇g⋆ and Ti,i=1,2 be weakly continuous and let either ∇g⋆ or both Ti,i=1,2 be continuous. Let J-Ti:K→2V⋆,i=1,2 be two bounded DJ-pseudomonotone mappings.Let(44)1ρ1=infn∈K∥∇g⋆(λw1+w2)∥V≠0({(∥∇g⋆(λw1+w2)∥V2)-1}〈{(λw0(1)+w0(2))}w1,∇g⋆(λw1+w2)-∇g⋆(λw0(1)+w0(2))〉×(∥∇g⋆(λw1+w2)∥V2)-1{(λw0(1)+w0(2))}),a=liminfn∈K∥∇g⋆(λw1+w2)∥V→∞({(∥∇g⋆(λw1+w2)∥V2)-1}〈{(λw0(1)+w0(2))}w2,∇g⋆(λw1+w2)-∇g⋆(λw0(1)+w0(2))〉×(∥∇g⋆(λw1+w2)∥V2)-1{(λw0(1)+w0(2))}),b=liminft→∞Φ(t)t
Be such that a<b, where wi∈(J-Ti)(η),w0(i)∈(J-Ti)(η0),i=1,2,η,η0∈K,ρ1>0 and Φ is the gauge function. Then for every λ>ρ1(b-a) problem (8) with T=λT1+T2-λJ has a solution in K.Proof.
By Proposition9, J-T=λ(J-T1)+(J-T2) is DJ-pseudomonotone for every λ≥0. Set λ>ρ(b-a). Then
(45)liminfη∈K∥∇g⋆(λw1+w2)∥V→∞({(∥∇g⋆(λw1+w2)∥V2)-1}〈{(λw0(1)+w0(2))}λw1+w2,∇g⋆(λw1+w2)-∇g⋆(λw0(1)+w0(2))〉×(∥∇g⋆(λw1+w2)∥V2)-1{(λw0(1)+w0(2))})≥λρ1+a>b>0.
This implies that the mapping J-T=λ(J-T1)+(J-T2) is DJ-coercive.The conclusion follows from Theorem8.



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