# Levitin-Polyak Well-Posedness of an Equilibrium-Like Problem in Banach Spaces

Authors: Ru-liang Deng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The concept of Levitin-Polyak well-posedness of an equilibrium-like problem in Banach spaces is introduced. Under suitable conditions, some characterizations of its Levitin-Polyak well-posedness are established. Some conditions under which an equilibrium-like problem in Banach spaces is Levitin-Polyak well-posed are also derived.


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## Body

## 1. Introduction

In 1966, Tykhonov [1] first established the well-posedness of a minimization problem, which has been known as Tykhonov well-posedness. Since it is important in optimization problems, various concepts of well-posedness have been introduced and studied in past decades. For more about the well-posedness, we refer to [2–4] and the references therein.The Tykhonov well-posedness of a constrained minimization problem requires that every minimizing sequence should lie in the constraint set. In many situations, the minimizing sequence produced by a numerical optimization method usually fails to be feasible but gets closer and closer to the constraint set. Levitin and Polyak [5] generalized the concept of Tykhnov well-posedness by requiring the existence and uniqueness of minimizer and the convergence of every generalized minimizing sequence toward the unique minimizer, which has been known as Levitin and Polyak well-posedness. There are a lot of results concerned with Tykhonov well-posedness, LP well-posedness, and their generalizations for minimization problems. For details, we refer to [1–3, 5–7].Recently, the concept of well-posedness has been extended to many other fields, including Nash equilibrium [8], inclusion problems, and fixed point problems [9–13]. Lemaire [12, 13] studied the relations between the well-posedness of minimization problems, inclusion problems, and fixed point problems. Fang et al. [11] proved that the well-posedness of a general mixed variational inequality is equivalent to the existence and the uniqueness of its solution in the Hilbert space. Recently, Ceng and Yao [9] got some results for the well-posedness of the generalized mixed variational inequality, the corresponding inclusion problem, and the corresponding fixed point problem. On the other hand, Li and Xia [14] considered the Levitin-Polyak well-posedness of a generalized variational inequality in Banach space. And they showed that the Levitin-Polyak well-posedness of a generalized variational inequality is equivalent to the uniqueness and existence of its solutions. However, there has been no result for the Levitin-Polyak well-posedness of an equilibrium-like problem.Motivated and inspired by the research work going on in this field, in this paper, we extend the notion of Levitin-Polyak well-posedness to an equilibrium-like problem in Banach spaces and give some metric characterizations of its Levitin-Polyak well-posedness. Finally, we derive some conditions under which an equilibrium-like problem is Levitin-Polyak well-posed.


## 2. Preliminaries

LetX be a real reflexive Banach space with its dual X* and let K be a nonempty, closed, and convex subset of X. Let F:X→2X* be a set-valued mapping, and let ϕ:X*×X×X→R be a functional. In this paper, we consider the following equilibrium-like problem associated with (F,ϕ,K):
(1)ELP(F,ϕ,K):findx∈Ksuchthatforsomeu∈F(x),ϕ(u,x,y)≤0,∀y∈K.Definition 1.
LetA,B be nonempty subsets of X. The Hausdorff metric H(·,·) between A and B is defined by
(2)H(A,B)=max⁡{e(A,B),e(B,A)},
where e(A,B)=sup⁡a∈Ad(a,B) with d(a,B)=inf⁡b∈B∥a-b∥.Lemma 2 (Nadler’s theorem [7]).
Let(X,∥·∥) be a normed vector space and let H(·,·) be the Hausdorff metric on the collection CB(X) of all nonempty, closed, and bounded subsets of X, induced by a metric d in terms of d(u,v)=∥u-v∥, which is defined by H(U,V)=max⁡{e(U,V),e(V,U)}, for U and V in CB(X), where e(U,V)=sup⁡x∈Ud(x,V) with d(x,V)=inf⁡y∈V∥x-y∥. If U and V lie in CB(X), then, for any ϵ>0 and any u∈U, there exists v∈V such that ∥u-v∥≤(1+ϵ)H(U,V). In particular, whenever U and V are compact subsets in X, one has ∥u-v∥≤H(U,V).Definition 3 (see [9]).
A nonempty set-valued mappingF:X→2X* is said to be(i)
H-hemicontinuous if, for any x,y∈X, the function t↦H(F(x+t(y-x),F(x))) from [0,1] into R+=[0,+∞) is continuous at 0+, where H(·,·) is the Hausdorff metric defined on CB(X);(ii)
H-uniformly continuous if, for all ϵ>0, there exists δ>0 such that for all x,y∈X with ∥x-y∥<δ, one has H(F(x),F(y))<ϵ, where H(·,·) is the Hausdorff metric defined on CB(X).Definition 4.
LetX and Y be two topological spaces and x∈X. A set-valued mapping F:X→2Y is said to be upper semicontinuous (u.s.c. in short) at x, if for any neighbourhood V of F(x), there exists a neighbourhood U of x such that F(y)⊂V, for all y∈U. If F is u.s.c. at each point of X, we say that F is u.s.c. on X.Definition 5 (see [15]).
LetA be a nonempty subset of X. The measure of noncompactness μ of the set A is defined by
(3)μ(A)=inf⁡{A⊂⋃i=1n‍Ai,diam⁡Ai<ϵϵ>0:A⊂⋃i=1n‍Ai,diam⁡Ai<ϵ,i=1,2,…,nA⊂⋃i=1n‍Ai,diam⁡Ai<ϵ},
where diam⁡Ai denotes the diameter of the set Ai, for i=1,2,…,n.Definition 6.
LetX be a real reflexive Banach space with its dual X* and let F:X→2X* be a set-valued mapping. A functional ϕ:X*×X×X→R is said to be monotone with respect to F, if for any x,y∈X and u∈F(x),v∈F(y), ϕ(u,x,y)≥ϕ(v,x,y).Remark 7.
Ifϕ(u,x,y)=〈u,x-y〉, for all x,y∈X and u∈F(x), it is easy to know that ϕ is monotone with respect to F which reduces to F  being monotone.We first prove the following proposition.Proposition 8.
LetK be a nonempty, closed, and convex subset of X  and let F:X→2X* be a nonempty compact-valued mapping which is H-hemicontinuous. Let ϕ:X*×X×X→R be monotone with respect to F, continuous in first argument, and concave in third argument. Moreover, ϕ(u,x,x)=0, for all u∈X*, x∈K. Then, for a given x∈K, the following statements are equivalent:(i)
there existsu∈F(x)such thatϕ(u,x,y)≤0,for ally∈K;(ii)
ϕ(v,x,y)≤0,for ally∈K, v∈F(y).Proof.
First, we assume that for someu∈F(x), ϕ(u,x,y)≤0, for all y∈K. Because ϕ is monotone with respect to F, we have
(4)ϕ(v,x,y)≤0,∀y∈K,v∈F(y).
Conversely, suppose that for ally∈K, v∈F(y), we obtain
(5)ϕ(v,x,y)≤0.
For any giveny∈K, we define yt=ty+(1-t)x for all t∈(0,1). Replacing y by yt in the left-hand side of the last inequality, we have that, for each vt∈F(yt),
(6)0≥ϕ(vt,x,yt)=ϕ(vt,x,ty+(1-t)x)≥tϕ(vt,x,y)+(1-t)ϕ(vt,x,x)=tϕ(vt,x,y).
This implies that
(*)ϕ(vt,x,y)≤0,∀vt∈F(yt),t∈(0,1).
Since F:X→2X* is a nonempty compact-valued mapping, F(yt) and F(x) are nonempty compact and hence lie in CB(X). From Lemma 2, we get that, for each t∈(0,1) and for each fixed vt∈F(yt), there exists a ut∈F(x) such that
(7)∥vt-ut∥≤(1+t)H(F(yt),F(x)).
Since F(x) is compact, without loss of generality, we assume that ut→u∈F(x) as t→0+. Since F is H-hemicontinuous, we get that as t→0+,
(8)∥vt-ut∥≤(1+t)H(F(yt),F(x))⟶0.
This implies that vt→u∈F(x) as t→0+. Since ϕ is continuous in first argument, by (*) we obtain that there exists an u∈F(x) such that
(9)ϕ(u,x,y)≤0,∀y∈K.
This completes the proof.


## 3. Levitin-Polyak Well-Posedness ofELP(F,ϕ,K)

In this section, we extend the concepts of Levitin-Poylak well-posedness to the equilibrium-like problem and establish its metric characterizations. Letα≥0 be a given number, and let X, K, F, and ϕ be defined as the previous section.Definition 9.
A sequence{xn}⊂X is called an LP α-approximating sequence for ELP(F,ϕ,K), if there exist wn∈X with wn→0 and 0<ϵn→0 such that xn+wn∈K for all n∈N and there exists un∈F(xn) such that
(10)ϕ(un,xn,y)≤α2∥xn-y∥2+ϵn,∀y∈K,n∈N.
Ifα1>α2≥0, then every LP α2-approximating sequence is LP α1-approximating. When α=0, we say that {xn} is an LP approximating sequence for ELP(F,ϕ,K).Definition 10.
ELP
(
F
,
ϕ
,
K
) is strongly LP α-well-posed if ELP(F,ϕ,K) has an unique solution and every LP α-approximating sequence converges strongly to the unique solution. In the sequel, strong LP 0-well-posedness is always called as strong LP well-posedness. If α1>α2≥0, then strong LP α1-well-posedness implies strong LP α2-well-posedness.Definition 11.
ELP
(
F
,
ϕ
,
K
) is strongly LP α-well-posed in the generalized sense if ELP(F,ϕ,K) has nonempty solution set S and every LP α-approximating sequence has a subsequence which converges strongly to some point of S. In the sequel, strong LP 0-well-posedness in the generalized sense is always called as strong LP well-posedness in the generalized sense. If α1>α2≥0, then strong LP α1-well-posedness in the generalized sense implies strong LP α2-well-posedness in the generalized sense.Remark 12.
Ifϕ(u,x,y)=〈u,x-y〉+φ(x)-φ(y), for all x,y∈X, u∈F(x), then Definitions 10 and 11 reduce to Definitions 3.3 and 3.4 of [14], respectively. Moreover, when X is a Hilbert space, K=X, and wn≡0, Definitions 10 and 11 reduce to Definitions 3.2 and 3.3 of [11], respectively.To obtain the metric characterizations of LPα-well-posedness, we consider the following LP α-approximating solution set of ELP(F,ϕ,K):
(11)Ωα(ϵ)={α2∥x-y∥2+ϵx∈dom⁡ϕ:ffd(x,K)≤ϵ,ffandthereexistsu∈F(x)ffsuchthat∀y∈K,ϕ(u,x,y)≤α2∥x-y∥2+ϵ},fffff∀ϵ≥0.Theorem 13.
LetK be a nonempty, closed, and convex subset of X and let F:X→2X* be a H-hemicontinuous and nonempty compact-valued mapping. Let ϕ:X*×X×X→R be monotone with respect to F, lower semicontinuous in second argument, and concave in third argument. Moreover, ϕ(u,x,x)=0, for all u∈X*, x∈K. Then, ELP(F,ϕ,K) is strongly LP α-well-posed if and only if
(12)Ωα(ϵ)≠∅,∀ϵ>0anddiam⁡(Ωα(ϵ))⟶0asϵ⟶0.Proof.
First, we assume thatELP(F,ϕ,K) is strongly LP α-well-posed and x*∈K is the unique solution of ELP(F,ϕ,K). It is easy to see that x*∈Ωα(ϵ). If diam⁡(Ωα(ϵ))↛0 as ϵ→0, then there exist constant l>0 and sequences {ϵn}⊂R+ with ϵn→0 and {xn(1)},{xn(2)} with xn(1),xn(2)∈Ωα(ϵn) such that
(13)∥xn(1)-xn(2)∥>l,∀n∈N.
Because of xn(1),xn(2)∈Ωα(ϵn), by the definition of Ωα(ϵn), for xn(1), we obtain
(14)d(xn(1),K)≤ϵn<ϵn+1n,
and there exists un∈F(xn(1)) such that
(15)ϕ(un,xn(1),y)≤α2∥xn(1)-y∥2+ϵn,∀y∈K.
Since K is closed and convex, then there exists x¯n(1)∈K such that ∥xn(1)-x¯n(1)∥<ϵn+(1/n). Let wn=x¯n(1)-xn(1); we get wn+xn(1)=x¯n(1)∈K and ∥wn∥=∥xn(1)-x¯n(1)∥→0. This implies that wn→0. Thus, {xn(1)} is an LP approximating sequence for ELP(F,ϕ,K). By the similar argument, we obtain that {xn(2)} is an LP approximating sequence for ELP(F,ϕ,K). So they have to converge strongly to the unique solution of ELP(F,ϕ,K), which contradicts condition (13).
Conversely, suppose that condition (12) holds. Let {xn}⊂X be an LP α-approximating sequence for ELP(F,ϕ,K). Then, there exists wn∈X with wn→0 such that xn+wn∈K, and there exist 0<ϵn′→0 and un∈F(xn) such that
(16)ϕ(un,xn,y)≤α2∥xn-y∥2+ϵn′,∀y∈K,n∈N.
Since xn+wn∈K, then there exists x¯n∈K such that xn+wn=x¯n. It is obvious that d(xn,K)≤∥xn-x¯n∥=∥wn∥→0. Suppose that ϵn=max⁡{ϵn′,∥wn∥}; we get that xn∈Ωα(ϵn). From (12), we have that {xn} is a Cauchy sequence and converges strongly to a point x¯∈K. Since ϕ is monotone with respect to F and lower semicontinuous in second argument, it follows from (16) that, for any y∈K, v∈F(y),
(17)ϕ(v,x¯,y)≤liminf⁡n→∞{ϕ(v,xn,y)}≤liminf⁡n→∞{ϕ(un,xn,y)}≤liminf⁡n→∞{α2∥xn-y∥2+ϵn′}=α2∥x¯-y∥2.
For any y∈K, let yt=x¯+t(y-x¯), for all t∈[0,1]. Since K is a nonempty, closed, and convex subset, we have that yt∈K. Then, (17) implies that
(18)ϕ(vt,x¯,yt)≤α2∥x¯-yt∥2,∀vt∈F(yt).
Since ϕ is concave in third argument and ϕ(u,x,x)=0, for all u∈X*, x∈K,
(19)ϕ(vt,x¯,y)≤αt2∥x¯-y∥2,∀vt∈F(yt),y∈K.
Since F is a nonempty compact-valued mapping and H-hemicontinuous, by Lemma 2, for each fixed vt∈F(yt) and each t∈(0,1), there exists a ut∈F(x¯) such that ∥vt-ut∥≤H(F(yt),F(x¯)). Since F is H-hemicontinuous, we get that ∥vt-ut∥≤H(F(yt),F(x¯))→0 as t→0+. Since F is compact, without loss of generality, we assume that ut→u∈F(x¯) as t→0+. Thus, we obtain that
(20)∥vt-u∥≤∥vt-ut∥+∥ut-u∥≤H(F(yt),F(x¯))+∥ut-u∥⟶0ast⟶0+.
This implies that vt→u as t→0+. It follows from (19) that
(21)ϕ(u,x¯,y)≤0,∀y∈K.
Therefore, x¯ solves ELP(F,ϕ,K).
To complete the proof, we only need to prove thatELP(F,ϕ,K) has a unique solution. Suppose that ELP(F,ϕ,K) has two distinct solutions x1 and x2. Then, it is obvious that x1,x2∈Ωα(ϵ) for all ϵ>0 and
(22)0<∥x1-x2∥≤diam⁡(Ωα(ϵ))⟶0,
a contradiction to (12). This completes the proof.Theorem 14.
LetK be a nonempty, closed, and convex subset of X and let F:X→2X* be a H-hemicontinuous and nonempty compact-valued mapping. Let ϕ:X*×X×X→R be monotone with respect to F and lower semicontinuous in second argument. Moreover, ϕ(u,x,x)=0, for all u∈X*, x∈K. Then, ELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense if and only if
(23)Ωα(ϵ)≠∅,∀ϵ>0andμ(Ωα(ϵ))⟶0asϵ⟶0.Proof.
Assume thatELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense. Let S be the solution set of ELP(F,ϕ,K). Then, S is nonempty and compact. Indeed, let {xn} be any sequence in S. Then, {xn} is an LP α-approximating sequence for ELP(F,ϕ,K). Since ELP(F,ϕ,K) is strongly α-well-posed in the generalized sense, {xn} has a subsequence which converges strongly to some point of S. Thus, S is compact. It is easy to see that Ωα(ϵ)⊃S≠∅ for all ϵ>0. Now we show that
(24)μ(Ωα(ϵ))⟶0asϵ⟶0.
It is easy to see that, for every ϵ>0,
(25)H(Ωα(ϵ),S)=max⁡{e(Ωα(ϵ),S),e(S,Ωα(ϵ))}=e(Ωα(ϵ),S).
Taking into account the compactness of S, we obtain
(26)μ(Ωα(ϵ))≤2H(Ωα(ϵ),S)+μ(S)=2e(Ωα(ϵ),S).
To prove (23), it is sufficient to show that
(27)e(Ωα(ϵ),S)⟶0asϵ⟶0.
Indeed, if e(Ωα(ϵ),S)↛0 as ϵ→0, then there exist l>0 and {ϵn}⊂R+ with ϵn→0, and xn∈Ωα(ϵn) such that
(28)xn∉S+B(0,l),∀n∈N,
where B(0,l) is the closed ball centered at 0 with radius l. By the definition of Ωα(ϵn), we know that d(xn,K)≤ϵn<ϵn+(1/n), and there exists un∈F(xn) such that
(29)ϕ(un,xn,y)≤α2∥xn-y∥2+ϵn,∀y∈K.
Thus, there exists x¯n∈K such that ∥x¯n-xn∥<ϵn+(1/n). Let wn=x¯n-xn; then, we have wn+xn∈K with wn→0. So {xn} is an LP α-approximating sequence for ELP(F,ϕ,K). Since ELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense, there exists a subsequence {xnk} of {xn} which converges strongly to some point of S. This contradicts (28) and so
(30)e(Ωα(ϵ),S)⟶0asϵ⟶0.
Conversely, suppose that (23) holds. We first show that Ωα(ϵ) is closed for all ϵ>0. Let {xn}⊂Ωα(ϵ) with xn→x; then, there exists un∈F(xn) such that d(xn,K)≤ϵ and
(31)ϕ(un,xn,y)≤α2∥xn-y∥2+ϵ,∀y∈K,n∈N.
Since F is an upper semicontinuous and nonempty compact-valued mapping, there exist a sequence {unk} of {un} and some u∈F(x) such that unk→u. Therefore, it follows from (31) and the lower semicontinuity of ϕ that
(32)ϕ(u,x,y)≤α2∥x-y∥2+ϵ,∀y∈K.
It is obvious that d(x,K)≤ϵ. This implies that x∈Ωα(ϵ) and so Ωα(ϵ) is nonempty closed for all ϵ>0. Observe that
(33)S=⋂ϵ>0‍Ωα(ϵ).
Since μ(Ωα(ϵ))→0, the theorem in page 412 of [15] can be applied and one concludes that S is nonempty and compact with
(34)e(Ωα(ϵ),S)=H(Ωα(ϵ),S)⟶0.
Let {x^n}⊂X be an LP α-approximating sequence for ELP(F,ϕ,K). Then, there exists wn∈X with wn→0 such that x^n+wn∈K, and there exist u^n∈F(x^n) and 0<ϵn′→0 such that
(35)ϕ(u^n,x^n,y)≤α2∥x^n-y∥2+ϵn′,∀y∈K,n∈N.
Since x^n+wn∈K, then there exists x¯n∈K such that x^n+wn=x¯n. It follows that
(36)d(x^n,K)≤∥x^n-x¯n∥=∥wn∥⟶0.
Set ϵn=max⁡{∥wn∥,ϵn′}; we get x^n∈Ωα(ϵn). From (23) and the definition of Ωα(ϵn), we obtain
(37)d(x^n,S)≤e(Ωα(ϵn),S)⟶0.
Since S is compact, there exists pn∈S such that
(38)∥pn-x^n∥=d(x^n,S)⟶0.
From the compactness of S, there exists a subsequence {pnk} of {pn} which converges strongly to p¯∈S. Hence, the corresponding subsequence {x^nk} of {x^n} converges strongly to p¯∈S. Thus, ELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense. The proof is complete.


## 4. Conditions for Levitin-Polyak Well-Posedness

In this section, we get some conditions under which theELP(F,ϕ,K) in Banach spaces is Levitin-Polyak well-posed.For anyδ0≥0, we denote M(δ0)={x∈X:dK(x)≤δ0}. We have the following result.Theorem 15.
LetK be a nonempty, closed, and convex subset of X and let F:X→2X* be a H-hemicontinuous and nonempty compact-valued mapping. Let ϕ:X*×X×X→R be monotone with respect to F, lower semicontinuous in first and second arguments, and concave in third argument. Moreover, ϕ(u,x,x)=0, for all u∈X*, x∈K. If there exists some δ0 with δ0>0 such that M(δ0) is compact, then ELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense.Proof.
Let{xn} be an LP approximating sequence for ELP(F,ϕ,K). Then, there exist 0<ϵn′→0 and wn∈X with wn→0 such that
(39)xn+wn∈K,
and there exists un∈F(xn) satisfying
(40)ϕ(un,xn,y)≤α2∥xn-y∥2+ϵn′,∀y∈K,n∈N.
Since xn+wn∈K, then there exists x¯n∈K such that xn+wn=x¯n. Thus,
(41)d(xn,K)≤∥xn-x¯n∥=∥wn∥⟶0.
Let ϵn=max⁡{ϵn′,∥wn∥}; we can get d(xn,K)≤ϵn. Without loss of generality, suppose that {xn}⊂M(δ0) for n is sufficiently large. By the compactness of M(δ0), there exist a subsequence {xnk} of {xn} and x¯∈M(δ0) such that xnk→x¯. It is easy to see that x¯∈K. Furthermore, by the u.s.c. of F at x¯ and compactness of F(x¯), there exist a subsequence {unk} of {un} and some u¯∈F(x¯) such that unk→u¯. Since ϕ is lower semicontinuous in first and second arguments, it follows from (40) that
(42)ϕ(u¯,x¯,y)≤α2∥x¯-y∥2,∀y∈K.
For any y∈K, let yt=x¯+t(y-x¯), for all t∈(0,1); it is obvious that yt∈K. Now, from (42), we have
(43)ϕ(u¯,x¯,yt)≤α2∥x¯-yt∥2.
By the convexity of ϕ, it follows that, for each t∈(0,1), we obtain
(44)ϕ(u¯,x¯,y)≤αt2∥x¯-y∥2,∀y∈K.
Let t→0+ in the last inequality; then, we have
(45)ϕ(u¯,x¯,y)≤0,∀y∈K.
This shows that x¯ solves ELP(F,ϕ,K). Thus, ELP(F,ϕ,K) is strongly LP α-well-posed in the generalized sense.



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*Source: 368098-2014-05-21.xml*