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

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.

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

Let 𝑋 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→2^𝑋∗ 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): (𝑥),𝜙ELP𝐹,𝜙,𝐾∶find𝑥∈𝐾suchthatforsome𝑢∈𝐹𝑢,𝑥,𝑦≤0,∀𝑦∈𝐾. (1)

Definition 1. Let 𝐴,𝐵 be nonempty subsets of 𝑋. The Hausdorff metric ℋ(⋅,⋅) between 𝐴 and 𝐵 is defined by ℋ(𝐴,𝐵)=max{𝑒(𝐴,𝐵),𝑒(𝐵,𝐴)}, (2) 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 mapping 𝐹∶𝑋→2^𝑋∗ 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. Let 𝑋 and 𝑌 be two topological spaces and 𝑥∈𝑋. A set-valued mapping 𝐹∶𝑋→2^𝑌 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]). Let 𝐴 be a nonempty subset of 𝑋. The measure of noncompactness 𝜇 of the set 𝐴 is defined by 𝜇(𝐴)=inf𝜖>0∶𝐴⊂^𝑛_𝑖=1𝐴_𝑖,diam𝐴_𝑖,<𝜖,𝑖=1,2,…,𝑛 (3) where diam𝐴_𝑖 denotes the diameter of the set 𝐴_𝑖, for 𝑖=1,2,…,𝑛.

Definition 6. Let 𝑋 be a real reflexive Banach space with its dual 𝑋^∗ and let 𝐹∶𝑋→2^𝑋∗ 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. If 𝜙(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. Let 𝐾 be a nonempty, closed, and convex subset of 𝑋  and let 𝐹∶𝑋→2^𝑋∗ 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 exists 𝑢∈𝐹(𝑥) such that 𝜙(𝑢,𝑥,𝑦)≤0, for all 𝑦∈𝐾;(ii)𝜙(𝑣,𝑥,𝑦)≤0, for all 𝑦∈𝐾, 𝑣∈𝐹(𝑦).

Proof. First, we assume that for some 𝑢∈𝐹(𝑥), 𝜙(𝑢,𝑥,𝑦)≤0, for all 𝑦∈𝐾. Because 𝜙 is monotone with respect to 𝐹, we have 𝜙𝑣,𝑥,𝑦≤0,∀𝑦∈𝐾,𝑣∈𝐹𝑦. (4)
Conversely, suppose that for all 𝑦∈𝐾, 𝑣∈𝐹(𝑦), we obtain 𝜙𝑣,𝑥,𝑦≤0. (5)
For any given 𝑦∈𝐾, we define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥 for all 𝑡∈(0,1). Replacing 𝑦 by 𝑦_𝑡 in the left-hand side of the last inequality, we have that, for each 𝑣_𝑡∈𝐹(𝑦_𝑡), 0≥𝜙𝑣_𝑡,𝑥,𝑦_𝑡=𝜙𝑣_𝑡()𝑥𝑣,𝑥,𝑡𝑦+1−𝑡≥𝑡𝜙_𝑡,𝑥,𝑦+(1−𝑡)𝜙𝑣_𝑡𝑣,𝑥,𝑥=𝑡𝜙_𝑡,𝑥,𝑦. (6) This implies that 𝜙𝑣_𝑡,𝑥,𝑦≤0,∀𝑣_𝑡∈𝐹𝑦_𝑡,𝑡∈(0,1). (∗) Since 𝐹∶𝑋→2^𝑋∗ is a nonempty compact-valued mapping, 𝐹(𝑦_𝑡) 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 𝑣_𝑡∈𝐹(𝑦_𝑡), there exists a 𝑢_𝑡∈𝐹(𝑥) such that 𝑣_𝑡−𝑢_𝑡≤(1+𝑡)ℋ𝐹𝑦_𝑡,𝐹(𝑥). (7) Since 𝐹(𝑥) is compact, without loss of generality, we assume that 𝑢_𝑡→𝑢∈𝐹(𝑥) as 𝑡→0⁺. Since 𝐹 is ℋ-hemicontinuous, we get that as 𝑡→0⁺, 𝑣_𝑡−𝑢_𝑡≤(1+𝑡)ℋ𝐹𝑦_𝑡,𝐹(𝑥)⟶0. (8) This implies that 𝑣_𝑡→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 𝜙𝑢,𝑥,𝑦≤0,∀𝑦∈𝐾. (9) This completes the proof.

3. Levitin-Polyak Well-Posedness of ELP(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. Let 𝛼≥0 be a given number, and let 𝑋, 𝐾, 𝐹, and 𝜙 be defined as the previous section.

Definition 9. A sequence {𝑥_𝑛}⊂𝑋 is called an LP 𝛼-approximating sequence for ELP(𝐹,𝜙,𝐾), if there exist 𝑤_𝑛∈𝑋 with 𝑤_𝑛→0 and 0<𝜖_𝑛→0 such that 𝑥_𝑛+𝑤_𝑛∈𝐾 for all 𝑛∈𝑁 and there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖_𝑛,∀𝑦∈𝐾,𝑛∈𝑁. (10)
If 𝛼₁>𝛼₂≥0, then every LP 𝛼₂-approximating sequence is LP 𝛼₁-approximating. When 𝛼=0, we say that {𝑥_𝑛} 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. If 𝜙(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 𝑤_𝑛≡0, Definitions 10 and 11 reduce to Definitions 3.2 and 3.3 of [11], respectively.

To obtain the metric characterizations of LP 𝛼-well-posedness, we consider the following LP 𝛼-approximating solution set of ELP(𝐹,𝜙,𝐾): Ω_𝛼(𝜖)=𝑑()(𝑥)≤𝛼𝑥∈dom𝜙∶𝑥,𝐾≤𝜖,andthereexists𝑢∈𝐹suchthat∀𝑦∈𝐾,𝜙𝑢,𝑥,𝑦2𝑥−𝑦²+𝜖,∀𝜖≥0. (11)

Theorem 13. Let 𝐾 be a nonempty, closed, and convex subset of 𝑋 and let 𝐹∶𝑋→2^𝑋∗ 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 Ω_𝛼(𝜖)≠0/,∀𝜖>0𝑎𝑛𝑑diamΩ_𝛼(𝜖)⟶0as𝜖⟶0. (12)

Proof. First, we assume that ELP(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^∗∈Ω_𝛼(𝜖). If diam(Ω_𝛼(𝜖))↛0 as 𝜖→0, then there exist constant 𝑙>0 and sequences {𝜖_𝑛}⊂ℝ₊ with 𝜖_𝑛→0 and {𝑥_𝑛⁽¹⁾},{𝑥_𝑛⁽²⁾} with 𝑥_𝑛⁽¹⁾,𝑥_𝑛⁽²⁾∈Ω_𝛼(𝜖_𝑛) such that 𝑥_𝑛⁽¹⁾−𝑥_𝑛⁽²⁾>𝑙,∀𝑛∈𝑁. (13) Because of 𝑥_𝑛⁽¹⁾,𝑥_𝑛⁽²⁾∈Ω_𝛼(𝜖_𝑛), by the definition of Ω_𝛼(𝜖_𝑛), for 𝑥_𝑛⁽¹⁾, we obtain 𝑑𝑥_𝑛⁽¹⁾,𝐾≤𝜖_𝑛<𝜖_𝑛+1𝑛, (14) and there exists 𝑢_𝑛∈𝐹(𝑥_𝑛⁽¹⁾) such that 𝜙𝑢_𝑛,𝑥_𝑛⁽¹⁾≤𝛼,𝑦2𝑥_𝑛⁽¹⁾−𝑦²+𝜖_𝑛,∀𝑦∈𝐾. (15) Since 𝐾 is closed and convex, then there exists 𝑥_𝑛⁽¹⁾∈𝐾 such that ‖𝑥_𝑛⁽¹⁾−𝑥_𝑛⁽¹⁾‖<𝜖_𝑛+(1/𝑛). Let 𝑤_𝑛=𝑥_𝑛⁽¹⁾−𝑥_𝑛⁽¹⁾; we get 𝑤_𝑛+𝑥_𝑛⁽¹⁾=𝑥_𝑛⁽¹⁾∈𝐾 and ‖𝑤_𝑛‖=‖𝑥_𝑛⁽¹⁾−𝑥_𝑛⁽¹⁾‖→0. This implies that 𝑤_𝑛→0. Thus, {𝑥_𝑛⁽¹⁾} is an LP approximating sequence for ELP(𝐹,𝜙,𝐾). By the similar argument, we obtain that {𝑥_𝑛⁽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 {𝑥_𝑛}⊂𝑋 be an LP 𝛼-approximating sequence for ELP(𝐹,𝜙,𝐾). Then, there exists 𝑤_𝑛∈𝑋 with 𝑤_𝑛→0 such that 𝑥_𝑛+𝑤_𝑛∈𝐾, and there exist 0<𝜖^′_𝑛→0 and 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖^′_𝑛,∀𝑦∈𝐾,𝑛∈𝑁. (16) Since 𝑥_𝑛+𝑤_𝑛∈𝐾, then there exists 𝑥_𝑛∈𝐾 such that 𝑥_𝑛+𝑤_𝑛=𝑥_𝑛. It is obvious that 𝑑(𝑥_𝑛,𝐾)≤‖𝑥_𝑛−𝑥_𝑛‖=‖𝑤_𝑛‖→0. Suppose that 𝜖_𝑛=max{𝜖^′_𝑛,‖𝑤_𝑛‖}; we get that 𝑥_𝑛∈Ω_𝛼(𝜖_𝑛). From (12), we have that {𝑥_𝑛} 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), 𝜙𝑣,𝑥,𝑦≤liminf_𝑛→∞𝜙𝑣,𝑥_𝑛,𝑦≤liminf_𝑛→∞𝜙𝑢_𝑛,𝑥_𝑛,𝑦≤liminf_𝑛→∞𝛼2𝑥_𝑛−𝑦²+𝜖^′_𝑛=𝛼2𝑥−𝑦². (17) For any 𝑦∈𝐾, let 𝑦_𝑡=𝑥+𝑡(𝑦−𝑥), for all 𝑡∈[0,1]. Since 𝐾 is a nonempty, closed, and convex subset, we have that 𝑦_𝑡∈𝐾. Then, (17) implies that 𝜙𝑣_𝑡,𝑥,𝑦_𝑡≤𝛼2𝑥−𝑦_𝑡²,∀𝑣_𝑡∈𝐹𝑦_𝑡. (18) Since 𝜙 is concave in third argument and 𝜙(𝑢,𝑥,𝑥)=0, for all 𝑢∈𝑋^∗, 𝑥∈𝐾, 𝜙𝑣_𝑡,𝑥,𝑦≤𝛼𝑡2𝑥−𝑦²,∀𝑣_𝑡∈𝐹𝑦_𝑡,𝑦∈𝐾. (19) Since 𝐹 is a nonempty compact-valued mapping and ℋ-hemicontinuous, by Lemma 2, for each fixed 𝑣_𝑡∈𝐹(𝑦_𝑡) and each 𝑡∈(0,1), there exists a 𝑢_𝑡∈𝐹(𝑥) such that ‖𝑣_𝑡−𝑢_𝑡‖≤ℋ(𝐹(𝑦_𝑡),𝐹(𝑥)). Since 𝐹 is ℋ-hemicontinuous, we get that ‖𝑣_𝑡−𝑢_𝑡‖≤ℋ(𝐹(𝑦_𝑡),𝐹(𝑥))→0 as 𝑡→0⁺. Since 𝐹 is compact, without loss of generality, we assume that 𝑢_𝑡→𝑢∈𝐹(𝑥) as 𝑡→0⁺. Thus, we obtain that 𝑣_𝑡−𝑢≤𝑣_𝑡−𝑢_𝑡+𝑢_𝑡𝐹𝑦−𝑢≤ℋ_𝑡,𝐹(𝑥)+𝑢_𝑡−𝑢⟶0as𝑡⟶0⁺. (20) This implies that 𝑣_𝑡→𝑢 as 𝑡→0⁺. It follows from (19) that 𝜙𝑢,𝑥,𝑦≤0,∀𝑦∈𝐾. (21) Therefore, 𝑥 solves ELP(𝐹,𝜙,𝐾).
To complete the proof, we only need to prove that ELP(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∈Ω_𝛼(𝜖) for all 𝜖>0 and 0<𝑥₁−𝑥₂≤diamΩ_𝛼(𝜖)⟶0, (22) a contradiction to (12). This completes the proof.

Theorem 14. Let 𝐾 be a nonempty, closed, and convex subset of 𝑋 and let 𝐹∶𝑋→2^𝑋∗ 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 Ω_𝛼(𝜖)≠0/,∀𝜖>0𝑎𝑛𝑑𝜇Ω_𝛼(𝜖)⟶0𝑎𝑠𝜖⟶0. (23)

Proof. Assume that ELP(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 {𝑥_𝑛} be any sequence in 𝑆. Then, {𝑥_𝑛} 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, {𝑥_𝑛} has a subsequence which converges strongly to some point of 𝑆. Thus, 𝑆 is compact. It is easy to see that Ω_𝛼(𝜖)⊃𝑆≠0/ for all 𝜖>0. Now we show that 𝜇Ω_𝛼(𝜖)⟶0as𝜖⟶0. (24) It is easy to see that, for every 𝜖>0, ℋΩ_𝛼(𝜖),𝑆=max𝑒Ω_𝛼(𝜖),𝑆,𝑒𝑆,Ω_𝛼(𝜖)=𝑒Ω_𝛼(𝜖),𝑆. (25) Taking into account the compactness of 𝑆, we obtain 𝜇Ω_𝛼(𝜖)≤2ℋΩ_𝛼(𝜖),𝑆+𝜇(𝑆)=2𝑒Ω_𝛼(𝜖),𝑆. (26) To prove (23), it is sufficient to show that 𝑒Ω_𝛼(𝜖),𝑆⟶0as𝜖⟶0. (27) Indeed, if 𝑒(Ω_𝛼(𝜖),𝑆)↛0 as 𝜖→0, then there exist 𝑙>0 and {𝜖_𝑛}⊂ℝ⁺ with 𝜖_𝑛→0, and 𝑥_𝑛∈Ω_𝛼(𝜖_𝑛) such that 𝑥_𝑛∉𝑆+𝐵(0,𝑙),∀𝑛∈𝑁, (28) where 𝐵(0,𝑙) is the closed ball centered at 0 with radius 𝑙. By the definition of Ω_𝛼(𝜖_𝑛), we know that 𝑑(𝑥_𝑛,𝐾)≤𝜖_𝑛<𝜖_𝑛+(1/𝑛), and there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖_𝑛,∀𝑦∈𝐾. (29) Thus, there exists 𝑥_𝑛∈𝐾 such that ‖𝑥_𝑛−𝑥_𝑛‖<𝜖_𝑛+(1/𝑛). Let 𝑤_𝑛=𝑥_𝑛−𝑥_𝑛; then, we have 𝑤_𝑛+𝑥_𝑛∈𝐾 with 𝑤_𝑛→0. So {𝑥_𝑛} 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 {𝑥_𝑛𝑘} of {𝑥_𝑛} which converges strongly to some point of 𝑆. This contradicts (28) and so 𝑒Ω_𝛼(𝜖),𝑆⟶0as𝜖⟶0. (30)
Conversely, suppose that (23) holds. We first show that Ω_𝛼(𝜖) is closed for all 𝜖>0. Let {𝑥_𝑛}⊂Ω_𝛼(𝜖) with 𝑥_𝑛→𝑥; then, there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝑑(𝑥_𝑛,𝐾)≤𝜖 and 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖,∀𝑦∈𝐾,𝑛∈𝑁. (31) Since 𝐹 is an upper semicontinuous and nonempty compact-valued mapping, there exist a sequence {𝑢_𝑛𝑘} of {𝑢_𝑛} and some 𝑢∈𝐹(𝑥) such that 𝑢_𝑛𝑘→𝑢. Therefore, it follows from (31) and the lower semicontinuity of 𝜙 that 𝜙≤𝛼𝑢,𝑥,𝑦2𝑥−𝑦²+𝜖,∀𝑦∈𝐾. (32) It is obvious that 𝑑(𝑥,𝐾)≤𝜖. This implies that 𝑥∈Ω_𝛼(𝜖) and so Ω_𝛼(𝜖) is nonempty closed for all 𝜖>0. Observe that 𝑆=_𝜖>0Ω_𝛼(𝜖). (33) Since 𝜇(Ω_𝛼(𝜖))→0, the theorem in page 412 of [15] can be applied and one concludes that 𝑆 is nonempty and compact with 𝑒Ω_𝛼(𝜖),𝑆=ℋΩ_𝛼(𝜖),𝑆⟶0. (34) Let {𝑥_𝑛}⊂𝑋 be an LP 𝛼-approximating sequence for ELP(𝐹,𝜙,𝐾). Then, there exists 𝑤_𝑛∈𝑋 with 𝑤_𝑛→0 such that 𝑥_𝑛+𝑤_𝑛∈𝐾, and there exist 𝑢_𝑛∈𝐹(𝑥_𝑛) and 0<𝜖^′_𝑛→0 such that 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖^′_𝑛,∀𝑦∈𝐾,𝑛∈𝑁. (35) Since 𝑥_𝑛+𝑤_𝑛∈𝐾, then there exists 𝑥_𝑛∈𝐾 such that 𝑥_𝑛+𝑤_𝑛=𝑥_𝑛. It follows that 𝑑𝑥_𝑛,𝐾≤𝑥_𝑛−𝑥_𝑛=𝑤_𝑛⟶0. (36) Set 𝜖_𝑛=max{‖𝑤_𝑛‖,𝜖^′_𝑛}; we get 𝑥_𝑛∈Ω_𝛼(𝜖_𝑛). From (23) and the definition of Ω_𝛼(𝜖_𝑛), we obtain 𝑑𝑥_𝑛,𝑆≤𝑒Ω_𝛼𝜖_𝑛,𝑆⟶0. (37) Since 𝑆 is compact, there exists 𝑝_𝑛∈𝑆 such that 𝑝_𝑛−𝑥_𝑛=𝑑𝑥_𝑛,𝑆⟶0. (38) From the compactness of 𝑆, there exists a subsequence {𝑝_𝑛𝑘} of {𝑝_𝑛} 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 the ELP(𝐹,𝜙,𝐾) in Banach spaces is Levitin-Polyak well-posed.

For any 𝛿₀≥0, we denote 𝑀(𝛿₀)={𝑥∈𝑋∶𝑑_𝐾(𝑥)≤𝛿₀}. We have the following result.

Theorem 15. Let 𝐾 be a nonempty, closed, and convex subset of 𝑋 and let 𝐹∶𝑋→2^𝑋∗ 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 {𝑥_𝑛} be an LP approximating sequence for ELP(𝐹,𝜙,𝐾). Then, there exist 0<𝜖^′_𝑛→0 and 𝑤_𝑛∈𝑋 with 𝑤_𝑛→0 such that 𝑥_𝑛+𝑤_𝑛∈𝐾, (39) and there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) satisfying 𝜙𝑢_𝑛,𝑥_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜖^′_𝑛,∀𝑦∈𝐾,𝑛∈𝑁. (40) Since 𝑥_𝑛+𝑤_𝑛∈𝐾, then there exists 𝑥_𝑛∈𝐾 such that 𝑥_𝑛+𝑤_𝑛=𝑥_𝑛. Thus, 𝑑𝑥_𝑛,𝐾≤𝑥_𝑛−𝑥_𝑛=𝑤_𝑛⟶0. (41) Let 𝜖_𝑛=max{𝜖^′_𝑛,‖𝑤_𝑛‖}; we can get 𝑑(𝑥_𝑛,𝐾)≤𝜖_𝑛. Without loss of generality, suppose that {𝑥_𝑛}⊂𝑀(𝛿₀) for 𝑛 is sufficiently large. By the compactness of 𝑀(𝛿₀), there exist a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} and 𝑥∈𝑀(𝛿₀) such that 𝑥_𝑛𝑘→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 {𝑢_𝑛𝑘} of {𝑢_𝑛} and some 𝑢∈𝐹(𝑥) such that 𝑢_𝑛𝑘→𝑢. Since 𝜙 is lower semicontinuous in first and second arguments, it follows from (40) that 𝜙𝑢,≤𝛼𝑥,𝑦2𝑥−𝑦²,∀𝑦∈𝐾. (42) For any 𝑦∈𝐾, let 𝑦_𝑡=𝑥+𝑡(𝑦−𝑥), for all 𝑡∈(0,1); it is obvious that 𝑦_𝑡∈𝐾. Now, from (42), we have 𝜙𝑢,𝑥,𝑦_𝑡≤𝛼2𝑥−𝑦_𝑡². (43) By the convexity of 𝜙, it follows that, for each 𝑡∈(0,1), we obtain 𝜙𝑢,𝑥,𝑦≤𝛼𝑡2𝑥−𝑦²,∀𝑦∈𝐾. (44) Let 𝑡→0⁺ in the last inequality; then, we have 𝜙𝑢,𝑥,𝑦≤0,∀𝑦∈𝐾. (45) This shows that 𝑥 solves ELP(𝐹,𝜙,𝐾). Thus, ELP(𝐹,𝜙,𝐾) is strongly LP 𝛼-well-posed in the generalized sense.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.