# Iterative Schemes for Nonconvex Quasi-Variational Problems withV-Prox-Regular Data in Banach Spaces

Authors: M. Bounkhel; Dj. Bounekhel
Journal: Journal of Function Spaces
 (2017)

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## Abstract

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.


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## Body

## 1. Introduction

LetX be a Banach space and let X∗ be the dual space of X. Let 〈·,·〉 denote the duality pairing of X∗ and X. Let C:X⇉X be a set-valued mapping with nonempty closed values and let F:X×X→R be a bifunction satisfying F(x,x)=0 for all x∈Fix(C)≔{x∈X:x∈C(x)}. We associate with aclosed convex valued set-valued mapping C and aconvex bifunction F the following well known quasi-equilibrium problem:QEPC,FFindx¯∈Cx¯,such  thatFx¯,x≥0,∀x∈Cx¯.In this paper we propose the following appropriate extensions of QEPC,F from the convex case to the nonconvex case in Banach spaces setting. We associate with C and F the following nonconvex quasi-variational problem equilibrium problems:NQVPC,FFindx¯∈Cx¯,s.t.-∂πFx¯,·x¯∩NπCx¯;x¯≠∅,where ∂π (resp. Nπ) is the V-proximal subdifferential (resp. V-proximal normal cone) introduced and studied in [1].The proposed nonconvex quasi-variational problem extends many existing quasi-equilibrium problems and quasi-variational inequalities from the convex case to the nonconvex case and from Hilbert spaces setting to Banach spaces setting.(1)
IfX is a Hilbert space, the proposed NQVPC,F becomes (1)Findx¯∈Cx¯,such  that-∂PFx¯,·x¯∩NPCx¯;x¯≠∅,where∂P and NP are the usual proximal subdifferential and proximal normal cone in Hilbert spaces. This problem has been introduced and studied in Bounkhel et al. [2]. Since then it has been studied and extended in various ways in Hilbert spaces by the authors in [3] and in Noor [4] and many works (see for instances Noor et al. [5, 6]).(2)
IfX is a Hilbert space, C is a convex closed set in X, F is a convex bifunction, and ρ=0, then NQVPC,F becomes the following well known convex equilibrium problem: (2)Findx¯∈C,such  thatFx¯,x≥0,∀x∈C,which has been studied in various works (see for instance Moudafi [7], M. A. Noor and K. I. Noor [5], and the references therein).(3)
IfF(x,y)=〈T(x),y-x〉, with T:X→X∗, is a nonlinear operator then NQVPC,F reduces to (3)Findx¯∈Cx¯,s.t.-Tx¯∈NπCx¯;x¯which will be shown in Section4 to be equivalent in the uniform V-prox-regular case, for some ρ≥0, to the following quasi-variational inequality: (4)Findx¯∈Cx¯,s.t.Tx¯,x-x¯+ρVJx¯,x≥0,∀x∈Cx¯.This inequality is new in Banach spaces. However, it has been studied, in Hilbert spaces, in Bounkhel et al. [2], when C is a uniformly V-prox-regular set (see also Bounkhel and Al-Sinan [8] and Noor et al. [5, 6]).Whenρ=0 and C(x)≡C the last inequality becomes (5)Findx¯∈C,such  thatTx¯,x-x¯≥0,∀x∈C,which is known as the classical variational inequality introduced and studied in Stampacchia [9].Our main objective of the present paper is to prove the convergence of some algorithms to solutions of the proposed nonconvex quasi-variational problem NQVPC,F.


## 2. Preliminaries

In order to prepare the framework of our study we need to state some concepts and results. First we recall (see for instance [1, 10]) the definition of p-uniformly convex and q-uniformly smooth Banach spaces. The space X is said to be p-uniformly convex (resp. q-uniformly smooth) if there is a constant c>0 such that (6)δXϵ≥cϵpresp.ρXt≤ctq,where δX and ρX are defined, respectively, by (7)δXϵ=inf⁡1-x+y2:x=y=1,x-y=ϵ,0≤ϵ≤2,ρXt=sup⁡12x+y+x-y-1:x=1,y=t,t>0.Notice that the constants p and q in the previous definition always satisfy p≥2 and q∈(1,2]. Also we need to recall from [1] the concept of V-proximal subdifferential∂πf(x) (called in [1]generalised proximal subdifferential). An element x∗∈X∗ belongs to ∂πf(x) provided that there exists σ>0 so that(8)x∗,x′-x≤fx′-fx+σVJx,x′,for x′ very close to x, where J:X→X∗ is the normalised duality mapping and V:X∗×X→R is a functional defined by(9)Vx∗,x=x∗2-2x∗,x+x2,for  anyx∗∈X∗,x∈X.For a closed nonempty setS in X and x¯∈S, the authors in [1] defined the concept of V-proximal normal coneNπ(S;x¯) (called in [1]generalised proximal normal cone) by Nπ(S;x¯)=∂πψS(x¯), where ψS denotes the indicator function associated with S, that is, ψS(x)=0 if x∈S and ψS(x)=+∞ if x∉S. We recall, respectively, the concepts of limiting Fréchet subdifferential ∂LF and limiting V-proximal subdifferential ∂Lπ (see [11]): (10)∂Lπfx=limsupx′→x∂πfx′≔w-limnxn∗:xn∗∈∂πfxnwithxn⟶fx,∂LFfx=limsupx′→x∂Ffx′≔w-limnxn∗:xn∗∈∂Ffxnwithxn⟶fx,where xn→fx means xn→x with f(xn)→f(x) and (11)∂Ffx=x∗∈X∗:∀ϵ>0,∃δ>0:x∗,x′-x≤fx′-fx+ϵx′-x,∀x′∈x+δB.The limiting Fréchet normal cone is defined similarly, that is,(12)∂LFNS;x=limsupx′→x∂FNS;x′≔w-limnxn∗:xn∗∈NFS;xnwithxn⟶Sx,where xn→Sx denotes xn→x with xn∈S and NF(S;x) is the Fréchet normal cone which is defined by NF(S;x¯)=∂FψS(x¯).These all nonconvex objects coincide with their analogues defined in Convex Analysis whenever the data are convex as the following proposition shows (see [1]).Proposition 1.
LetX be a reflexive Banach space. (1)
Letf:X→R∪+∞ be a l.s.c. convex function and x¯∈X with f(x¯)<∞. Then (13)∂πfx=∂Con.fx≔x∗∈X∗:x∗;x-x¯≤fx-fx¯,∀x∈X.(2)
LetS be a closed convex subset in of X and x¯∈S. Then (14)NπS;x¯=NCon.S;x¯≔x∗∈X∗:x∗;x-x¯≤0,∀x∈S.The following result is needed in our study. It has been proved in [11].Theorem 2.
LetX be a q-uniformly smooth and p-uniformly convex Banach space. Assume that X admits an equivalent norm · such that ·s (for some s≥2) is C2-differentiable on X∖{0} and let V be the functional associated with that norm ·. (1)
Letf:X→R∪{∞}, be a l.s.c. function at x¯∈dom⁡f. Then(15)∂Lπfx¯=∂FLfx¯.(2)
LetS be any closed nonempty set of X. Then (16)NFLS;x¯=NLπS;x¯.We notice that the class of spaces satisfying the assumptions of the previous theorem is very large; it contains obviously any Hilbert space andLp spaces and Sobolev spaces Wp,m with p≥2 (see Theorem 1.1 in Section 5 in [10, 12]) and for more examples and discussions we refer to [10, 12]. We close this section with the following two concepts of uniform V-prox-regularity for functions and sets (see [13]).Definition 3.
LetX be a reflexive smooth Banach space. For a given r∈(0,∞], a subset S is V-uniformly prox-regular with respect to r provided that for all x∈S and all nonzero x∗∈Nπ(S;x) we have(17)x∗x∗,x′-x≤12rVJx,x′,∀x′∈S.We use the convention 1/r=0 for r=+∞.Obviously, this class contains the class of uniformly prox-regular sets ([14, 15]) from Hilbert spaces to Banach spaces since in Hilbert spaces we have VJx,x′=x-x′2 and the V-proximal normal cone Nπ(S;x) coincides with the usual proximal normal cone NP(S;x).Definition 4.
LetX be a reflexive smooth Banach space. Let f:X→R∪{+∞} be a l.s.c. function and let S⊂dom⁡f≔{x∈X:f(x)<∞} be a nonempty closed set in X. We recall from [13] that f is said to be uniformly V-prox-regular over S provided that for all x∈S and all x∗∈∂πf(x) we have(18)x∗,x′-x≤fx′-fx+12rVJx,x′,∀x′∈S.We say that f is uniformly V-prox-regular around x¯∈dom⁡f provided that f is uniformly V-prox-regular over some closed neighborhood of x¯; that is, there exists a closed neighborhood Vx¯ of x¯ such that ∀x∈Vx¯,∀x∗∈∂πf(x) the inequality (18) holds for any x′∈Vx¯.The following example is quoted from [13]. For its proof we refer the reader to [13].Example 5.
(1) Any l.s.c. proper convex function is uniformlyV-prox-regular over any nonempty closed set S in its domain with r=+∞.
(2) Both the indicator functionψS and the distance function dS of uniformly V-prox-regular set S are uniformly V-prox-regular over S with respect to the same constant r.
(3) Any lower-C2 function f over convex strongly compact K in X is uniformly V-prox-regular over K with some r∈(0,+∞] (see [13] for the definition of lower-C2 functions).The following two lemmas are needed in our proofs in Section4. The proof of the first one is proved in [1]. The second one is proved in [16].Lemma 6.
LetX be a p-uniformly convex and q-uniformly smooth Banach space and S be a bounded set. Then for some η,κ>0 we have (19)η-1x-yp≤VJx,y≤κ-1x-yq,∀x,y∈S.Lemma 7.
IfX is a uniformly convex Banach space, then the inequality (20)VJx,y≥8C2δXx-y4Cholds for all x and y in X, where C=x2+y2/2.


## 3. Main Results

First we show that in the convex caseNQVPC,F coincides with the quasi-equilibrium problem QEPC,F.Proposition 8.
LetX be a reflexive Banach space. Assume that C is a closed convex set-valued mapping and F is a convex bifunction satisfying F(x,x)=0 for any x∈Fix(C). Then we have NQVPC,F  ⇔  QEPC,F.Proof.
⁡
⇒ ?. Let x¯ be a solution of NQVPC,F; that is, there exists y∗∈∂πF(x¯,·)(x¯) such that -y∗∈Nπ(C(x¯),x¯). Since C(x¯) is a closed convex set, the V-proximal normal cone Nπ(C(x¯),x¯) coincides with the convex normal cone NCon.(C(x¯),x¯) (by Proposition 1) and so (21)y∗;x-x¯≥0,∀x∈Cx¯.On the other hand, the convexity of the bifunction F and Proposition 1 yield (22)y∗;x-x¯≤Fx¯,x-Fx¯,x¯,∀x∈X.Since x¯∈C(x¯) we have F(x¯,x¯)=0 (by assumption) and hence the previous two inequalities ensure (23)Fx¯,x≥0,∀x∈Cx¯;that is, x¯ is a solution of QEPC,F.
⇐ ?. Let x¯ be a solution of NQEPC,F, that is, F(x¯,x)≥0,∀x∈C(x¯). Since C(x¯) is a closed convex set and F(x¯,·) is a convex function, the function x↦h(x)≔F(x¯,x)+ψC(x¯)(x) admits at x¯ a global minimum on X. It follows that (24)0∈∂Con.hx¯=∂Con.Fx¯,·x¯+∂Con.ψCx¯x¯=∂Con.Fx¯,·x¯+NCon.Cx¯;x¯.which is equivalent to [-∂Con.F(x¯,·)(x¯)]∩NCon.(C(x¯);x¯)≠∅ and hence the proof is complete since ∂πF(x¯,·)(x¯)=∂Con.F(x¯,·)(x¯) and Nπ(C(x¯),x¯)=NCon.(C(x¯),x¯).In the next proposition we establish an inequality characterisation of the proposed nonconvex quasi-variational problemNQVPC,F whenever the data C and F are uniformly V-prox-regular.Proposition 9.
LetX be a reflexive Banach space and x¯∈X. Assume that C(x¯) is uniformly V-prox-regular with ratio r∈(0,∞] and that F(x¯,·) is uniformly V-prox-regular over C(x¯) with ratio r′∈(0,∞]. Assume also that F(x¯,·) is γ-Lipschitz around x¯ and F(x,x)=0 for any x∈Fix(C). If x¯ is a solution of NQVPC,F, then x¯ is a solution of the following nonconvex quasi-equilibrium problem. Find x¯∈C(x¯) such that NQEPC,FFx¯,x+ρVJx¯,x≥0,∀x∈Cx¯,for some nonnegative ρ≥0.Proof.
Assume thatx¯ is a solution of NQVPC,F; that is, y∗∈∂πF(x¯,·)(x¯) such that -y∗∈Nπ(C(x¯);x¯). By uniform V-prox-regularity of the set C(x¯) we have (25)-y∗,x-x¯≤y∗2rVJx¯,x,∀x∈Cx¯.The γ-Lipschitz continuity of F(x¯,·) ensures that y∗≤γ and so we obtain(26)-y∗,x-x¯≤γ2rVJx¯,x,∀x∈Cx¯.On the other hand the uniform V-prox-regularity of F(x¯,·) over C(x¯) with ratio r′>0; we have(27)y∗,x-x¯≤12r′VJx¯,x+Fx¯,x-Fx¯,x¯,∀x∈Cx¯.Combining this inequality (27) with (26) we obtain(28)Fx¯,x-Fx¯,x¯+12r′VJx¯,x≥-γ2rVJx¯,x∀x∈Cx¯.Since x¯∈C(x¯) we have F(x¯,x¯)=0 and so (28) becomes (29)Fx¯,x+ρVJx¯,x≥0∀x∈Cx¯,with ρ≔γ/2r+1/2r′≥0. Thus the proof is complete.It is a natural question to ask whether the converse in the previous proposition is true or not. The answer is positive provided that the spaceX and the data C and F satisfy some additional assumptions as the following proposition shows.Proposition 10.
LetX be a q-uniformly smooth and p-uniformly convex Banach space. Assume that X admits an equivalent norm · such that ·s (for some s≥2) is C2-differentiable on X∖{0} and let V be the functional associated with that norm ∥·∥. Assume that C(x¯) is V-proximal normally regular at x¯, that is, Nπ(C(x¯),x¯)=NLπ(C(x¯),x¯) and that F(x¯,·) is V-proximal subdifferentially regular at x¯, that is, ∂πF(x¯,·)(x¯)=∂LπF(x¯,·)(x¯). Assume that F(x,x)=0 for any x∈Fix(C). If x¯ is a solution of NQEPC,F for some ρ≥0, then x¯ is a solution of NQVPC,F.Proof.
Letx¯ be a solution of NQEPC,F for some ρ≥0; that is, (30)Fx¯,x+ρVJx¯,x≥0∀x∈Cx¯.Then x¯ is a global minimum of the function x↦h(x)=F(x¯,x)+ρV(Jx¯,x)+ψC(x¯)(x) over X and hence (31)0∈∂πhx¯⊂∂Lπhx¯=∂LπFx¯,·+ρVJx¯,·+ψCx¯·x¯.Note that the function x↦V(J(x¯),x) is differentiable and its gradient is given by grad⁡(V(J(x¯),·))(x)=2(J(x)-J(x¯)). Using the fact that the limiting V-proximal subdifferential coincides with the limiting Fréchet subdifferential (by Theorem 2) and the exact sum rules for the limiting Fréchet subdifferential (see for instance [17]) we can write(32)0∈∂LπFx¯,·+ρVJx¯,·+ψCx¯·x¯∈∂LFFx¯,·+ρVJx¯,·+ψCx¯·x¯∈∂LFFx¯,·x¯+∂LFρVJx¯,·x¯+∂LFψCx¯·x¯∈∂LFFx¯,·x¯+2ρJx¯-Jx¯+NLFCx¯;x¯∈∂LFFx¯,·x¯+NLFCx¯;x¯.This is equivalent to say that [-∂LFF(x¯,·)(x¯)]∩NLF(C(x¯);x¯)≠∅. Thus completing the proof since ∂πF(x¯,·)(x¯)=∂LπF(x¯,·)(x¯)=∂LFF(x¯,·)(x¯) and Nπ(C(x¯),x¯)=NLπ(C(x¯);x¯)=NLF(C(x¯);x¯).The following proposition has its own interest and is needed to prove the equivalence betweenNQVPC,F and NQEPC,F whenever C and F are uniformly V-prox-regular.Proposition 11.
LetX be a reflexive Banach space and let f:X→R∪{∞} be a l.s.c. function and let x¯∈dom⁡f. If f is uniformly V-prox-regular around x¯, then ∂πf(x¯)=∂Lπf(x¯); that is, f is V-proximal subdifferentially regular at x¯. Consequently, for any uniformly V-prox-regular closed set S at x¯∈S we have Nπ(S,x¯)=NLπ(S;x¯); that is, S is V-proximal normally regular at x¯.Proof.
We only prove the first assertion; the second one follows directly from the first one and Example5 Part (2). Since we always have the inclusion ∂πf(x¯)⊂∂Lπf(x¯), it is enough to prove the reverse one, that is, ∂Lπf(x¯)⊂∂πf(x¯). Let x∗∈∂Lπf(x¯); that is, there exists xn→fx and xn∗∈∂πf(xn) such that x∗=w-limnxn∗. By the uniform V-prox-regularity of f around x¯, there exist r>0 and δ>0 such that for any x∈x¯+δB and any y∗∈∂πf(x)(33)y∗,x′-x≤12rVJx,x′+fx′-fx,∀x′∈x+δB.Since xn→x¯ we can write for n large enough that xn∈x¯+(δ/2)B and hence by (33) we have(34)xn∗,x′-xn≤12rVJxn,x′+fx′-fxn,∀x′∈xn+δB.Fix any y∈x¯+(δ/2)B. Clearly y∈xn+(δ/2)B+(δ/2)B⊂xn+δB and hence (34) ensures (35)x∗,y-x¯=x∗-xn∗,y-x¯+xn∗,y-xn+xn∗;xn-x¯≤x∗-xn∗,y-x¯+xn∗;xn-x¯+12rVJxn,y+fy-fxn.Using now the fact that xn→fx¯, the continuity of J and V, and the weak convergence of xn∗ to x∗ to pass to the limit as n goes to ∞ and to get (36)x∗,y-x¯≤12rVJx¯,y+fy-fx¯,for any y∈x¯+(δ/2)B, this means by definition that x∗∈∂πf(x¯) and the proof is complete.Using this result together with Propositions9 and 10 we obtain the equivalence between NQVPC,F and NQEPC,F.Proposition 12.
LetX be a q-uniformly smooth and p-uniformly convex Banach space and x¯∈X. Assume that X admits an equivalent norm · such that ·s (for some s≥2) is C2-differentiable on X∖{0} and let V be the functional associated with that norm ·. Assume that C(x¯) is uniformly V-prox-regular with ratio r∈(0,∞] and that F(x¯,·) is uniformly V-prox-regular over C(x¯) with ratio r′∈(0,∞]. Assume also that F(x¯,·) is γ-Lipschitz around x¯ and F(x,x)=0 for any x∈Fix(C). Then NQVPC,F is equivalent to NQEPC,F for some ρ≥0.


## 4. Convergence Analysis

### 4.1. Case 1:C Is a Constant Set-Valued Mapping

In this case the proposed problem becomes as follows:NVPC,FFindx¯∈Cs.t.-∂πFx¯,·x¯∩NπC;x¯≠∅.In this subsection we propose the following algorithm.Algorithm 13.
Letρ≥0 and λn>0 for all n≥1; (1)
Selectx0∈C;(2)
Forn≥1 select xn+1∈C such that (37)λn-1Jxn-Jxn+1,x-xn+1≤Fxn,x+ρVJxn,x,∀x∈C.Theorem 14.
LetX be a q-uniformly convex Banach space. Let C be a closed nonempty subset of X and let F:C×C→R be a bifunction satisfying F(x,x)=0 for any x∈Fix(C). Let {xn}n be a sequence generated by Algorithm 13. Assume that (1)
C is V-uniformly prox-regular with some r∈(0,∞];(2)
C is ball compact; that is, C∩ηB is compact for any η>0;(3)
The solution set ofNQVPC,F is nonempty;(4)
F is W-strongly monotone over C for some σ≥0; that is, (38)Fx,y+Fy,x≤-σWx,y,∀x,y∈C,whereW(x,y)≔(1/2)[V(J(x),y)+V(J(y),x)];(5)
F is upper semicontinuous with respect to the first variable over C; that is, (39)limsupx′→x⁡Fx′,y≤Fx,y∀x,y∈C;(6)
The bifurcationF is γ-Lipschitz with respect to the second variable and F(xn+1,·) is V-uniformly prox-regular over C with some r′∈(0,+∞];(7)
There existsλ>0 such that λn≥λ for all n;(8)
The parametersr,r′,γ,ρ,σ satisfy the inequalities 2ρ≤γ/2r+1/2r′≤σ/3.Then, there exists subsequence of {xn} converges to x~∈C which solves NVPC,F.Proof.
Letx¯∈C be a solution of NVPC,F. Then by Proposition 9 we have (40)Fx¯,x+ρ0VJx¯,x≥0,∀x∈C,for ρ0≔γ/2r+1/2r′. By the W-strong monotonicity of F over C we have (41)Fx,x¯+Fx¯,x≤-σWx,x¯,∀x∈C.By setting x=xn in these two inequalities we get (42)Fxn,x¯+Fx¯,xn≤-σWxn,x¯,-Fx¯,xn≤ρ0VJx¯,xn.Combining these two inequalities we obtain (43)Fxn,x¯≤ρ0VJx¯,xn-σWxn,x¯≤2ρ0-σWxn,x¯.Using the 8th assumption in Theorem 14 we have 2ρ0-σ≤-ρ0 and hence (44)Fxn,x¯≤-ρ0Wxn,x¯.This combined with Algorithm 13 gives(45)xn+1∗,x¯-xn+1≤Fxn,x¯+ρVJxn,x¯≤-ρ0Wxn,x¯+ρVJxn,x¯≤2ρ-ρ0Wxn,x¯,with xn+1∗≔λn-1[J(xn)-J(xn+1)]. Therefore,(46)Jxn-Jxn+1,x¯-xn+1≤λn2ρ-ρ0Wxn,x¯.Define now a sequence of nonnegative real numbers ϕn=(1/2)V(J(xn),x¯). It is not hard to verify that(47)2ϕn+1-ϕn+VJxn,xn+1=2Jxn-Jxn+1,x¯-xn+1.Indeed,(48)2ϕn+1-ϕn=VJxn+1,x¯-VJxn,x¯=Jxn+12-2Jxn+1,x¯+x¯2-Jxn2-2Jxn,x¯+x¯2=Jxn+12+2Jxn-Jxn+1,x¯-Jxn2=2Jxn-Jxn+1,x¯-Jxn+12-Jxn2+2Jxn+1,xn+1=2Jxn-Jxn+1,x¯-VJxn,xn+1-2Jxn,xn+1+2Jxn+1,xn+1=2Jxn-Jxn+1,x¯-VJxn,xn+1-2Jxn-Jxn+1,xn+1=2Jxn-Jxn+1,x¯-xn+1-VJxn,xn+1.It follows that (49)ϕn+1-ϕn≤Jxn-Jxn+1,x¯-xn+1,which ensures with (46) that (50)ϕn+1-ϕn≤λn2ρ-ρ0Wxn,x¯.Using the assumption ρ0≥2ρ in the 8th assumption we obtain (51)ϕn+1≤ϕn.Therefore, the sequence {ϕn} is a nonincreasing converging sequence to some limit and so it is bounded by some α>0. Thus by the properties of the functional V we obtain (52)x¯-xn2≤VJxn,x¯=2ϕn≤2αand so (53)xn≤x¯+2α;that is, {xn} is bounded and so by the q′-uniform convexity of X∗ (by Lemma 6) we have for some η>0 depending on α and on the space X∗ the inequality (54)Jxn+1-Jxnq′≤ηV∗J∗Jxn+1,Jxn=ηVJxn,xn+1,where J∗:X∗→X∗∗(=X) is the normalised duality mapping on X∗ and V∗:X∗∗×X∗→R is the functional defined by (55)V∗x∗∗;x∗≔x∗∗2-2x∗∗;x∗+x∗2,∀x∗∈X∗,x∗∗∈X∗∗.Using now (46) and (47) and the assumption ρ0≥2ρ we obtain (56)12VJxn,xn+1≤ϕn-ϕn+1.Therefore, it follows from the 7th assumption of Theorem 14 that(57)xn+1∗q′=λn-q′Jxn+1-Jxnq′≤λ-q′Jxn+1-Jxnq′≤λ-q′ηVJxn,xn+1≤2ηλq′ϕn-ϕn+1⟶0asn⟶∞,which ensures that limn→∞xn+1∗=0. On the other hand, since {xn} is bounded in C and C is ball compact then there exists a subsequence {xnk} which converges to some limit x~∈C. By Algorithm 13 this subsequence satisfies(58)xnk+1∗,x-xnk+1≤Fxnk,x+ρVJxnk,x,∀k,∀x∈C.Thus, by letting k→∞ in the inequality (58) and by taking into account the upper semicontinuity of F and the continuity of V and J, we obtain (59)0≤Fx~,x+ρVJx~,x,∀x∈C.This means that x~ is a solution of NEPC,F. Finally, using now Proposition 12 we get x~ is a solution of NVPC,F and so the proof is complete.


### 4.2. Case 2:C Is a General Set-Valued Mapping

In this general case we propose the following algorithm.Algorithm 15.
Letρ≥0 and λn>0 for all n≥1; (1)
Selectx0∈C(x0);(2)
Forn≥1 select xn+1∈C(xn) such that (60)λn-1Jxn-Jxn+1,x-xn+1≤Fxn,x+ρVJxn,x,∀x∈ImC,whereM>0 is a given positive number and ImC is the image of C, that is, ImC≔{y∈X:∃x∈Xsuch  thaty∈C(x)}.Obviously Algorithm 15 coincides with Algorithm 13 when C is a constant set-valued mapping. However the assumptions assumed on F in the previous subsection are not sufficient to prove the convergence of the sequence {xn} generated by Algorithm 15 to a solution of NQVPC,F. We need to replace the W-strong monotonicity by a relaxed W-strong monotonicity of the bifunction F over ImC and we do not assume the nonemptiness of the solution set of the proposed problem. We will say that F is relaxed W-strongly monotone over ImC provided that for some σ≥0 we have(61)Fx,y≤-σWx,y,∀x,y∈ImC.By symmetry of W, it is clear that any W-relaxed strongly monotone bifunction with respect to σ≥0 is W-strongly monotone with respect to 2σ. This relaxed assumption on F has been used in Hilbert spaces in [4] and in Banach spaces in [13]. The following theorem is our main result in this subsection.Theorem 16.
LetX be a q-uniformly convex Banach space. Let C be a closed nonempty subset of X and let F:C×C→R be a bifunction satisfying F(x,x)=0 for any x∈Fix(C). Let {xn}n be a sequence generated by Algorithm 15. Assume that (1)
The values ofC are V-uniformly prox-regular with some ratio r∈(0,∞];(2)
The image ofC is ball compact in X and its graph is closed;(3)
F is relaxed W-strongly monotone over ImC with some σ>0;(4)
F is upper semicontinuous with respect to the first variable over ImC;(5)
F(xn,·) is V-uniformly prox-regular over ImC with some r′∈(0,+∞];(6)
There existsλ>0 such that λn≥λ for all n;(7)
The nonnegative parameterρ is taken in the interval [0,σ/2].Then, there exists subsequence of {xn} converging to a solution of NQVPC,F.Proof.
Letx¯∈ImC. By the relaxed W-strong monotonicity of F over ImC we have (62)Fxn,x¯≤-σWxn,x¯,∀n≥1.By Algorithm 15 we have (63)xn+1∗,x¯-xn+1≤Fxn,x¯+ρVJxn,x¯,with xn+1∗≔λn-1[J(xn)-J(xn+1)]. Combining these two inequalities we get(64)xn+1∗,x¯-xn+1≤ρVJxn,x¯-σWxn,x¯≤2ρ-σWxn,x¯.Therefore,(65)Jxn-Jxn+1,x¯-xn+1≤λn2ρ-σWxn,x¯.Define now the same nonnegative real sequence ϕn=(1/2)V(J(xn),x¯) used in the proof of Theorem 14. Then we have (66)ϕn+1-ϕn≤Jxn-Jxn+1,x¯-xn+1,which ensures with (65) that (67)ϕn+1-ϕn≤λn2ρ-σWxn+1,x¯.Using the assumption σ≥2ρ yields (68)ϕn+1≤ϕn.Following the same reasoning in the proof of Theorem 14 and the ball compactness of the image of C, we get a subsequence {xnk} which converges to some limit x~ satisfying x~∈C(x~) by closedness of the graph of C. By Algorithm 15 this subsequence satisfies(69)xnk+1∗,x-xnk+1≤Fxnk+1,x+ρVJxnk+1,x,∀k,∀x∈ImC.Thus, by letting k→∞ in the inequality (69) and by taking into account the upper semicontinuity of F and the continuity of V and J, we obtain (70)0≤Fx~,x+ρVJx~,x,∀x∈Cx~.This means that x~ is a solution of NQEPC,F which ensures by Proposition 12 that under the assumptions of our theorem the solution x~ is also a solution of NQVPC,F. Thus completing the proof.


### 4.3. Case 3:F Has the Form: F(x,y)=〈T(x),y-x〉

In this subsection we restrict our attention to the following form of the bifunctionF: (71)Fx,y=Tx,y-x,where T:X→X∗ is a nonlinear operator. In this case ∂πF(x¯,·)(x¯)={T(x¯)} and so NQVPC,F becomes: NQVPC,TFindx¯∈Cx¯,such  thatTx¯∈-NπCx¯,x¯.We suggest the following algorithm to solve NQVPC,T under some natural and appropriate assumptions on C and T.Algorithm 17.
Letδn↓0 with δ0 be too small. (i)
Selectx0∈C(x0),y0∗=T(x0) and ρ>0;(ii)
Forn≥0,
(a)
Computezn+1≔J∗(J(xn)-ρyn∗);(b)
Computexn+1≔πC(xn)(J(zn+1)) and yn+1∗≔T(xn+1),whereπS is the generalised projection defined in terms of the functional V instead of the norm square (introduced and studied in the convex case in [16] and for the nonconvex case we refer to the recent paper [11]). A point x¯∈S is called the generalised projection of a given x∗∈X∗ provided that (72)Vx∗,x¯=infs∈S⁡Vx∗,s.The following characterisation of the V-proximal normal cone in terms of the generalised projection is proved in [1].Proposition 18.
For any closed nonempty setS in a reflexive Banach space X and for any point x¯∈S we have (73)NπS;x¯=x∗∈X∗:∃λ>0such  thatx¯∈πSJx¯+λx∗.We need the following lemma:Lemma 19.
LetS be a closed set in X, x¯∈S,y∗∈X∗, and r>0. If  x¯∈πS(J(x¯)-ry∗); then x¯∈πS(J(x¯)-ρy∗), for any ρ∈[0,r].Proof.
Letr>0,y∗∈X∗, and let x¯ be a point satisfying x¯∈πS(J(x¯)-ry∗). Assume that ρ∈[0,r]. Let λ≔ρ/r∈[0,1]. We claim that (74)VJx¯-ρy∗,x¯=infs∈S⁡VJx¯-ρy∗,s.First, observe that for any s∈S we have (75)2Jx¯-ρy∗-Jx¯;s-x¯=2λJx¯-ry∗+1-λJx¯-Jx¯;s-x¯=2λJx¯-ry∗-Jx¯;s-x¯.If 〈(J(x¯)-ry∗)-J(x¯);s-x¯〉<0, then obviously we have(76)2Jx¯-ρy∗-Jx¯;s-x¯<0≤VJx¯,s.Otherwise, we have Jx¯-ry∗-Jx¯;s-x¯≥0. Then since 0≤λ≤1 we have (77)2λJx¯-ry∗-Jx¯;s-x¯≤2Jx¯-ry∗-Jx¯;s-x¯and so we obtain (78)2Jx¯-ρy∗-Jx¯;s-x¯≤2Jx¯-ry∗-Jx¯;s-x¯≤Jx¯-ry∗2-2Jx¯-ry∗;x¯+x¯2+2Jx¯-ry∗;s-Jx¯-ry∗2-s2+s2-2Jx¯;s-x¯-x¯2≤VJx¯-ry∗,x¯-VJx¯-ry∗,s+VJx¯,s≤infz∈S⁡VJx¯-ry∗,z-VJx¯-ry∗,s+VJx¯,s≤VJx¯,s;that is,(79)2Jx¯-ρy∗-Jx¯;s-x¯≤VJx¯,s.
Therefore, from (76) and (79) we have in both cases (80)2Jx¯-ρy∗-Jx¯;s-x¯≤VJx¯,s,∀s∈S.Hence (81)2Jx¯-ρy∗-Jx¯;s-x¯-VJx¯,s≤0,∀s∈S.On the other hand we have the decomposition (82)2Jx¯-ρy∗-Jx¯;s-x¯-VJx¯,s=2Jx¯-ρy∗;s-2Jx¯-ρy∗;x¯+2x¯2-2Jx¯;s-x¯2-2Jx¯;s+s2=Jx¯-ρy∗2-2Jx¯-ρy∗;x¯+x¯2-Jx¯-ρy∗2-2Jx¯-ρy∗;s+s2=VJx¯-ρy∗,x¯-VJx¯-ρy∗,s.Consequently, we have (83)VJx¯-ρy∗,x¯-VJx¯-ρy∗,s≤0,for  anys∈S,that is, (84)VJx¯-ρy∗,x¯=infs∈S⁡VJx¯-ρy∗,s;which means that x¯∈πS(J(x¯)-ρy∗) and hence the proof is complete.Now, we state and prove our main theorem forNQVPC,T.Theorem 20.
LetX be a 2-uniformly smooth Banach space. Let C:X⇉X be a set-valued mapping with closed nonempty values and T:X→X∗. Let {xn}n be a sequence generated by Algorithm 17. Assume that(1)
The solution set ofNQVPC,T is nonempty;(2)
T is bounded by some constant L>0;(3)
T is J-Lipschitz, with constant β>0; that is, (85)Tx1-Tx2≤βJx1-Jx2,∀xi∈X,i=1,2;(4)
T is J-strongly monotone with constant α>0; that is, (86)J∗Tx1-Tx2;Jx1-Jx2≥αJx1-Jx22,∀x1,x2∈X;(5)
The values ofC satisfy for some r∈(0,∞]: (87)u¯∈πCu¯Ju¯+ru∗,∀u∗∈X∗for any unit vectoru∗ in X∗ and any u¯ solution of NQVPC,T;(6)
There exists some constantk∈(0,1) and ξ>0 such that(88)JπCx1x1∗-JπCx2x2∗≤ξx1∗-x2∗+kJx1-Jx2,for allxi∈X,xi∗∈X∗,i=1,2;(7)
The positive constantsα and β satisfy the inequality α>β1-(1-k)2/cξ2;(8)
The parameterρ in Algorithm 17 satisfies (89)αβ2-ϵ¯<ρ<min⁡μ-δ0L,αβ2+ϵ¯,ϵ¯≔α2-β21-1-k2/cξ2β2.Then, the sequence {xn}n generated by Algorithm 17 converges to a solution of NQVPC,T.Proof.
Letx¯∈C(x¯) be a solution of NQVPC,T, that is, -T(x¯)∈Nπ(C(x¯);x¯). Then by the characterisation of the V-proximal normal cone in Proposition 18, there exists λ>0 such that x¯∈πC(x¯)(J(x¯)-λT(x¯)). Using Lemma 19 we obtain x¯∈πC(x¯)(J(x¯)-τT(x¯)), for any τ∈[0,λ]. By assumption (5) we may assume that λ≤r/L and so we get ρ≤r/L. Hence x¯∈πC(x¯)(J(z¯)) for z¯≔J∗(J(x¯)-ρT(x¯)). Since X is 2-uniformly smooth we have X∗ is 2-uniformly convex; that is, (90)δX∗ϵ≥2c-1ϵ2,for some constant c>0 (depending only on the space X∗) and so by Lemma 7 we get (91)V∗J∗x∗,y∗≥8C2δX∗x∗-y∗4C≥c-1x∗-y∗2,∀x∗,y∗∈X∗.Thus we can write (92)ρTxn-Tx¯-Jxn-Jx¯2≤cV∗ρJ∗Txn-Tx¯;Jxn-Jx¯.Therefore, (93)Jzn+1-Jz¯2=Jxn-ρTxn-Jx¯+ρTx¯2≤cV∗ρJ∗Txn-Tx¯;Jxn-Jx¯≤cρ2Txn-Tx¯2+Jxn-Jx¯2-2cρJ∗Txn-Tx¯;Jxn-Jx¯.Using the J-Lipschitz continuity of T with ratio β we have (94)Txn-Tx¯2≤β2Jxn-Jx¯2and by the J-strong monotonicity of T with ratio α we have (95)J∗Txn-Tx¯;Jxn-Jx¯≥αJxn-Jx¯2.Thus, we get (96)Jzn+1-Jz¯2≤cρ2β2Jxn-Jx¯2+Jxn-Jx¯2-2cραJxn-Jx¯2≤c1+ρ2β2-2ραJxn-Jx¯2and so (97)Jzn+1-Jz¯≤c1+ρ2β2-2ραJxn-Jx¯.On the other hand we have by the 6th assumption (98)Jxn+1-Jx¯=JπCxnJzn+1-JπCx¯Jz¯≤ξJzn+1-Jz¯+kJxn-Jx¯.
Thus(99)Jxn+1-Jx¯≤k+ξc1+ρ2β2-2ραJxn-Jx¯.Our assumptions and the choice of ρ ensure that k+ξc(1+ρ2β2-2ρα)<1 and hence J(xn)-J(x¯)→0 which means that xn→x¯ by the uniform continuity of J∗ and thus completing the proof.Remark 21.
A simple inspection of the proof of the previous theorem shows that the result is valid in the case whenT is taken a general set-valued mapping instead of a single-valued operator defined from X to X∗ and of course the assumptions on T should be adapted naturally for the set-valued case.


## 4.1. Case 1:C Is a Constant Set-Valued Mapping

In this case the proposed problem becomes as follows:NVPC,FFindx¯∈Cs.t.-∂πFx¯,·x¯∩NπC;x¯≠∅.In this subsection we propose the following algorithm.Algorithm 13.
Letρ≥0 and λn>0 for all n≥1; (1)
Selectx0∈C;(2)
Forn≥1 select xn+1∈C such that (37)λn-1Jxn-Jxn+1,x-xn+1≤Fxn,x+ρVJxn,x,∀x∈C.Theorem 14.
LetX be a q-uniformly convex Banach space. Let C be a closed nonempty subset of X and let F:C×C→R be a bifunction satisfying F(x,x)=0 for any x∈Fix(C). Let {xn}n be a sequence generated by Algorithm 13. Assume that (1)
C is V-uniformly prox-regular with some r∈(0,∞];(2)
C is ball compact; that is, C∩ηB is compact for any η>0;(3)
The solution set ofNQVPC,F is nonempty;(4)
F is W-strongly monotone over C for some σ≥0; that is, (38)Fx,y+Fy,x≤-σWx,y,∀x,y∈C,whereW(x,y)≔(1/2)[V(J(x),y)+V(J(y),x)];(5)
F is upper semicontinuous with respect to the first variable over C; that is, (39)limsupx′→x⁡Fx′,y≤Fx,y∀x,y∈C;(6)
The bifurcationF is γ-Lipschitz with respect to the second variable and F(xn+1,·) is V-uniformly prox-regular over C with some r′∈(0,+∞];(7)
There existsλ>0 such that λn≥λ for all n;(8)
The parametersr,r′,γ,ρ,σ satisfy the inequalities 2ρ≤γ/2r+1/2r′≤σ/3.Then, there exists subsequence of {xn} converges to x~∈C which solves NVPC,F.Proof.
Letx¯∈C be a solution of NVPC,F. Then by Proposition 9 we have (40)Fx¯,x+ρ0VJx¯,x≥0,∀x∈C,for ρ0≔γ/2r+1/2r′. By the W-strong monotonicity of F over C we have (41)Fx,x¯+Fx¯,x≤-σWx,x¯,∀x∈C.By setting x=xn in these two inequalities we get (42)Fxn,x¯+Fx¯,xn≤-σWxn,x¯,-Fx¯,xn≤ρ0VJx¯,xn.Combining these two inequalities we obtain (43)Fxn,x¯≤ρ0VJx¯,xn-σWxn,x¯≤2ρ0-σWxn,x¯.Using the 8th assumption in Theorem 14 we have 2ρ0-σ≤-ρ0 and hence (44)Fxn,x¯≤-ρ0Wxn,x¯.This combined with Algorithm 13 gives(45)xn+1∗,x¯-xn+1≤Fxn,x¯+ρVJxn,x¯≤-ρ0Wxn,x¯+ρVJxn,x¯≤2ρ-ρ0Wxn,x¯,with xn+1∗≔λn-1[J(xn)-J(xn+1)]. Therefore,(46)Jxn-Jxn+1,x¯-xn+1≤λn2ρ-ρ0Wxn,x¯.Define now a sequence of nonnegative real numbers ϕn=(1/2)V(J(xn),x¯). It is not hard to verify that(47)2ϕn+1-ϕn+VJxn,xn+1=2Jxn-Jxn+1,x¯-xn+1.Indeed,(48)2ϕn+1-ϕn=VJxn+1,x¯-VJxn,x¯=Jxn+12-2Jxn+1,x¯+x¯2-Jxn2-2Jxn,x¯+x¯2=Jxn+12+2Jxn-Jxn+1,x¯-Jxn2=2Jxn-Jxn+1,x¯-Jxn+12-Jxn2+2Jxn+1,xn+1=2Jxn-Jxn+1,x¯-VJxn,xn+1-2Jxn,xn+1+2Jxn+1,xn+1=2Jxn-Jxn+1,x¯-VJxn,xn+1-2Jxn-Jxn+1,xn+1=2Jxn-Jxn+1,x¯-xn+1-VJxn,xn+1.It follows that (49)ϕn+1-ϕn≤Jxn-Jxn+1,x¯-xn+1,which ensures with (46) that (50)ϕn+1-ϕn≤λn2ρ-ρ0Wxn,x¯.Using the assumption ρ0≥2ρ in the 8th assumption we obtain (51)ϕn+1≤ϕn.Therefore, the sequence {ϕn} is a nonincreasing converging sequence to some limit and so it is bounded by some α>0. Thus by the properties of the functional V we obtain (52)x¯-xn2≤VJxn,x¯=2ϕn≤2αand so (53)xn≤x¯+2α;that is, {xn} is bounded and so by the q′-uniform convexity of X∗ (by Lemma 6) we have for some η>0 depending on α and on the space X∗ the inequality (54)Jxn+1-Jxnq′≤ηV∗J∗Jxn+1,Jxn=ηVJxn,xn+1,where J∗:X∗→X∗∗(=X) is the normalised duality mapping on X∗ and V∗:X∗∗×X∗→R is the functional defined by (55)V∗x∗∗;x∗≔x∗∗2-2x∗∗;x∗+x∗2,∀x∗∈X∗,x∗∗∈X∗∗.Using now (46) and (47) and the assumption ρ0≥2ρ we obtain (56)12VJxn,xn+1≤ϕn-ϕn+1.Therefore, it follows from the 7th assumption of Theorem 14 that(57)xn+1∗q′=λn-q′Jxn+1-Jxnq′≤λ-q′Jxn+1-Jxnq′≤λ-q′ηVJxn,xn+1≤2ηλq′ϕn-ϕn+1⟶0asn⟶∞,which ensures that limn→∞xn+1∗=0. On the other hand, since {xn} is bounded in C and C is ball compact then there exists a subsequence {xnk} which converges to some limit x~∈C. By Algorithm 13 this subsequence satisfies(58)xnk+1∗,x-xnk+1≤Fxnk,x+ρVJxnk,x,∀k,∀x∈C.Thus, by letting k→∞ in the inequality (58) and by taking into account the upper semicontinuity of F and the continuity of V and J, we obtain (59)0≤Fx~,x+ρVJx~,x,∀x∈C.This means that x~ is a solution of NEPC,F. Finally, using now Proposition 12 we get x~ is a solution of NVPC,F and so the proof is complete.


## 4.2. Case 2:C Is a General Set-Valued Mapping

In this general case we propose the following algorithm.Algorithm 15.
Letρ≥0 and λn>0 for all n≥1; (1)
Selectx0∈C(x0);(2)
Forn≥1 select xn+1∈C(xn) such that (60)λn-1Jxn-Jxn+1,x-xn+1≤Fxn,x+ρVJxn,x,∀x∈ImC,whereM>0 is a given positive number and ImC is the image of C, that is, ImC≔{y∈X:∃x∈Xsuch  thaty∈C(x)}.Obviously Algorithm 15 coincides with Algorithm 13 when C is a constant set-valued mapping. However the assumptions assumed on F in the previous subsection are not sufficient to prove the convergence of the sequence {xn} generated by Algorithm 15 to a solution of NQVPC,F. We need to replace the W-strong monotonicity by a relaxed W-strong monotonicity of the bifunction F over ImC and we do not assume the nonemptiness of the solution set of the proposed problem. We will say that F is relaxed W-strongly monotone over ImC provided that for some σ≥0 we have(61)Fx,y≤-σWx,y,∀x,y∈ImC.By symmetry of W, it is clear that any W-relaxed strongly monotone bifunction with respect to σ≥0 is W-strongly monotone with respect to 2σ. This relaxed assumption on F has been used in Hilbert spaces in [4] and in Banach spaces in [13]. The following theorem is our main result in this subsection.Theorem 16.
LetX be a q-uniformly convex Banach space. Let C be a closed nonempty subset of X and let F:C×C→R be a bifunction satisfying F(x,x)=0 for any x∈Fix(C). Let {xn}n be a sequence generated by Algorithm 15. Assume that (1)
The values ofC are V-uniformly prox-regular with some ratio r∈(0,∞];(2)
The image ofC is ball compact in X and its graph is closed;(3)
F is relaxed W-strongly monotone over ImC with some σ>0;(4)
F is upper semicontinuous with respect to the first variable over ImC;(5)
F(xn,·) is V-uniformly prox-regular over ImC with some r′∈(0,+∞];(6)
There existsλ>0 such that λn≥λ for all n;(7)
The nonnegative parameterρ is taken in the interval [0,σ/2].Then, there exists subsequence of {xn} converging to a solution of NQVPC,F.Proof.
Letx¯∈ImC. By the relaxed W-strong monotonicity of F over ImC we have (62)Fxn,x¯≤-σWxn,x¯,∀n≥1.By Algorithm 15 we have (63)xn+1∗,x¯-xn+1≤Fxn,x¯+ρVJxn,x¯,with xn+1∗≔λn-1[J(xn)-J(xn+1)]. Combining these two inequalities we get(64)xn+1∗,x¯-xn+1≤ρVJxn,x¯-σWxn,x¯≤2ρ-σWxn,x¯.Therefore,(65)Jxn-Jxn+1,x¯-xn+1≤λn2ρ-σWxn,x¯.Define now the same nonnegative real sequence ϕn=(1/2)V(J(xn),x¯) used in the proof of Theorem 14. Then we have (66)ϕn+1-ϕn≤Jxn-Jxn+1,x¯-xn+1,which ensures with (65) that (67)ϕn+1-ϕn≤λn2ρ-σWxn+1,x¯.Using the assumption σ≥2ρ yields (68)ϕn+1≤ϕn.Following the same reasoning in the proof of Theorem 14 and the ball compactness of the image of C, we get a subsequence {xnk} which converges to some limit x~ satisfying x~∈C(x~) by closedness of the graph of C. By Algorithm 15 this subsequence satisfies(69)xnk+1∗,x-xnk+1≤Fxnk+1,x+ρVJxnk+1,x,∀k,∀x∈ImC.Thus, by letting k→∞ in the inequality (69) and by taking into account the upper semicontinuity of F and the continuity of V and J, we obtain (70)0≤Fx~,x+ρVJx~,x,∀x∈Cx~.This means that x~ is a solution of NQEPC,F which ensures by Proposition 12 that under the assumptions of our theorem the solution x~ is also a solution of NQVPC,F. Thus completing the proof.


## 4.3. Case 3:F Has the Form: F(x,y)=〈T(x),y-x〉

In this subsection we restrict our attention to the following form of the bifunctionF: (71)Fx,y=Tx,y-x,where T:X→X∗ is a nonlinear operator. In this case ∂πF(x¯,·)(x¯)={T(x¯)} and so NQVPC,F becomes: NQVPC,TFindx¯∈Cx¯,such  thatTx¯∈-NπCx¯,x¯.We suggest the following algorithm to solve NQVPC,T under some natural and appropriate assumptions on C and T.Algorithm 17.
Letδn↓0 with δ0 be too small. (i)
Selectx0∈C(x0),y0∗=T(x0) and ρ>0;(ii)
Forn≥0,
(a)
Computezn+1≔J∗(J(xn)-ρyn∗);(b)
Computexn+1≔πC(xn)(J(zn+1)) and yn+1∗≔T(xn+1),whereπS is the generalised projection defined in terms of the functional V instead of the norm square (introduced and studied in the convex case in [16] and for the nonconvex case we refer to the recent paper [11]). A point x¯∈S is called the generalised projection of a given x∗∈X∗ provided that (72)Vx∗,x¯=infs∈S⁡Vx∗,s.The following characterisation of the V-proximal normal cone in terms of the generalised projection is proved in [1].Proposition 18.
For any closed nonempty setS in a reflexive Banach space X and for any point x¯∈S we have (73)NπS;x¯=x∗∈X∗:∃λ>0such  thatx¯∈πSJx¯+λx∗.We need the following lemma:Lemma 19.
LetS be a closed set in X, x¯∈S,y∗∈X∗, and r>0. If  x¯∈πS(J(x¯)-ry∗); then x¯∈πS(J(x¯)-ρy∗), for any ρ∈[0,r].Proof.
Letr>0,y∗∈X∗, and let x¯ be a point satisfying x¯∈πS(J(x¯)-ry∗). Assume that ρ∈[0,r]. Let λ≔ρ/r∈[0,1]. We claim that (74)VJx¯-ρy∗,x¯=infs∈S⁡VJx¯-ρy∗,s.First, observe that for any s∈S we have (75)2Jx¯-ρy∗-Jx¯;s-x¯=2λJx¯-ry∗+1-λJx¯-Jx¯;s-x¯=2λJx¯-ry∗-Jx¯;s-x¯.If 〈(J(x¯)-ry∗)-J(x¯);s-x¯〉<0, then obviously we have(76)2Jx¯-ρy∗-Jx¯;s-x¯<0≤VJx¯,s.Otherwise, we have Jx¯-ry∗-Jx¯;s-x¯≥0. Then since 0≤λ≤1 we have (77)2λJx¯-ry∗-Jx¯;s-x¯≤2Jx¯-ry∗-Jx¯;s-x¯and so we obtain (78)2Jx¯-ρy∗-Jx¯;s-x¯≤2Jx¯-ry∗-Jx¯;s-x¯≤Jx¯-ry∗2-2Jx¯-ry∗;x¯+x¯2+2Jx¯-ry∗;s-Jx¯-ry∗2-s2+s2-2Jx¯;s-x¯-x¯2≤VJx¯-ry∗,x¯-VJx¯-ry∗,s+VJx¯,s≤infz∈S⁡VJx¯-ry∗,z-VJx¯-ry∗,s+VJx¯,s≤VJx¯,s;that is,(79)2Jx¯-ρy∗-Jx¯;s-x¯≤VJx¯,s.
Therefore, from (76) and (79) we have in both cases (80)2Jx¯-ρy∗-Jx¯;s-x¯≤VJx¯,s,∀s∈S.Hence (81)2Jx¯-ρy∗-Jx¯;s-x¯-VJx¯,s≤0,∀s∈S.On the other hand we have the decomposition (82)2Jx¯-ρy∗-Jx¯;s-x¯-VJx¯,s=2Jx¯-ρy∗;s-2Jx¯-ρy∗;x¯+2x¯2-2Jx¯;s-x¯2-2Jx¯;s+s2=Jx¯-ρy∗2-2Jx¯-ρy∗;x¯+x¯2-Jx¯-ρy∗2-2Jx¯-ρy∗;s+s2=VJx¯-ρy∗,x¯-VJx¯-ρy∗,s.Consequently, we have (83)VJx¯-ρy∗,x¯-VJx¯-ρy∗,s≤0,for  anys∈S,that is, (84)VJx¯-ρy∗,x¯=infs∈S⁡VJx¯-ρy∗,s;which means that x¯∈πS(J(x¯)-ρy∗) and hence the proof is complete.Now, we state and prove our main theorem forNQVPC,T.Theorem 20.
LetX be a 2-uniformly smooth Banach space. Let C:X⇉X be a set-valued mapping with closed nonempty values and T:X→X∗. Let {xn}n be a sequence generated by Algorithm 17. Assume that(1)
The solution set ofNQVPC,T is nonempty;(2)
T is bounded by some constant L>0;(3)
T is J-Lipschitz, with constant β>0; that is, (85)Tx1-Tx2≤βJx1-Jx2,∀xi∈X,i=1,2;(4)
T is J-strongly monotone with constant α>0; that is, (86)J∗Tx1-Tx2;Jx1-Jx2≥αJx1-Jx22,∀x1,x2∈X;(5)
The values ofC satisfy for some r∈(0,∞]: (87)u¯∈πCu¯Ju¯+ru∗,∀u∗∈X∗for any unit vectoru∗ in X∗ and any u¯ solution of NQVPC,T;(6)
There exists some constantk∈(0,1) and ξ>0 such that(88)JπCx1x1∗-JπCx2x2∗≤ξx1∗-x2∗+kJx1-Jx2,for allxi∈X,xi∗∈X∗,i=1,2;(7)
The positive constantsα and β satisfy the inequality α>β1-(1-k)2/cξ2;(8)
The parameterρ in Algorithm 17 satisfies (89)αβ2-ϵ¯<ρ<min⁡μ-δ0L,αβ2+ϵ¯,ϵ¯≔α2-β21-1-k2/cξ2β2.Then, the sequence {xn}n generated by Algorithm 17 converges to a solution of NQVPC,T.Proof.
Letx¯∈C(x¯) be a solution of NQVPC,T, that is, -T(x¯)∈Nπ(C(x¯);x¯). Then by the characterisation of the V-proximal normal cone in Proposition 18, there exists λ>0 such that x¯∈πC(x¯)(J(x¯)-λT(x¯)). Using Lemma 19 we obtain x¯∈πC(x¯)(J(x¯)-τT(x¯)), for any τ∈[0,λ]. By assumption (5) we may assume that λ≤r/L and so we get ρ≤r/L. Hence x¯∈πC(x¯)(J(z¯)) for z¯≔J∗(J(x¯)-ρT(x¯)). Since X is 2-uniformly smooth we have X∗ is 2-uniformly convex; that is, (90)δX∗ϵ≥2c-1ϵ2,for some constant c>0 (depending only on the space X∗) and so by Lemma 7 we get (91)V∗J∗x∗,y∗≥8C2δX∗x∗-y∗4C≥c-1x∗-y∗2,∀x∗,y∗∈X∗.Thus we can write (92)ρTxn-Tx¯-Jxn-Jx¯2≤cV∗ρJ∗Txn-Tx¯;Jxn-Jx¯.Therefore, (93)Jzn+1-Jz¯2=Jxn-ρTxn-Jx¯+ρTx¯2≤cV∗ρJ∗Txn-Tx¯;Jxn-Jx¯≤cρ2Txn-Tx¯2+Jxn-Jx¯2-2cρJ∗Txn-Tx¯;Jxn-Jx¯.Using the J-Lipschitz continuity of T with ratio β we have (94)Txn-Tx¯2≤β2Jxn-Jx¯2and by the J-strong monotonicity of T with ratio α we have (95)J∗Txn-Tx¯;Jxn-Jx¯≥αJxn-Jx¯2.Thus, we get (96)Jzn+1-Jz¯2≤cρ2β2Jxn-Jx¯2+Jxn-Jx¯2-2cραJxn-Jx¯2≤c1+ρ2β2-2ραJxn-Jx¯2and so (97)Jzn+1-Jz¯≤c1+ρ2β2-2ραJxn-Jx¯.On the other hand we have by the 6th assumption (98)Jxn+1-Jx¯=JπCxnJzn+1-JπCx¯Jz¯≤ξJzn+1-Jz¯+kJxn-Jx¯.
Thus(99)Jxn+1-Jx¯≤k+ξc1+ρ2β2-2ραJxn-Jx¯.Our assumptions and the choice of ρ ensure that k+ξc(1+ρ2β2-2ρα)<1 and hence J(xn)-J(x¯)→0 which means that xn→x¯ by the uniform continuity of J∗ and thus completing the proof.Remark 21.
A simple inspection of the proof of the previous theorem shows that the result is valid in the case whenT is taken a general set-valued mapping instead of a single-valued operator defined from X to X∗ and of course the assumptions on T should be adapted naturally for the set-valued case.



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*Source: 8708065-2017-01-30.xml*