Some Existence Theorems for Nonconvex Variational Inequalities Problems

Abstract

By using nonsmooth analysis knowledge, we provide the conditions for existence solutions of the variational inequalities problems in nonconvex setting. We also show that the strongly monotonic assumption of the mapping may not need for the existence of solutions. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature.

1. Introduction

Variational inequalities theory, which was introduced by Stampacchia [1], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems.

It should be pointed out that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting; see [2–5] for examples. Moreover, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping. In fact, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined, even in the nonconvex case (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.).

The main aim of this paper is intending to consider the conditions for the existence solutions of some variational inequalities problems in nonconvex setting. We will make use of some recent nonsmooth analysis techniques to overcome the difficulties that arise from the nonconvexity. Also, it is worth mentioning that we have considered when the mapping may not satisfy the strongly monotonic assumption. In this sense, our result represents an improvement and refinement of the known results.

2. Preliminaries

Let ℋ be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed subset of ℋ. We denote by 𝑑_𝐶(⋅) the usual distance function to the subset 𝐶; that is, 𝑑_𝐶(𝑢)=inf_𝑣∈𝐶‖𝑢−𝑣‖. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis.

Definition 2.1. Let 𝑢∈ℋ be a point not lying in 𝐶. A point 𝑣∈𝐶 is called a closest point or a projection of 𝑢 onto 𝐶 if 𝑑_𝐶(𝑢)=‖𝑢−𝑣‖. The set of all such closest points is denoted by proj_𝐶(𝑢); that is, proj_𝐶(𝑢)=𝑣∈𝐶∶𝑑_𝐶(𝑢)=‖𝑢−𝑣‖. (2.1)

Definition 2.2. Let 𝐶 be a subset of ℋ. The proximal normal cone to 𝐶 at 𝑥 is given by 𝑁^𝑃_𝐶(𝑥)=𝑧∈ℋ∶∃𝜌>0;𝑥∈proj_𝐶𝑥+𝜌𝑧. (2.2)

The following characterization of 𝑁^𝑃_𝐶(𝑥) can be found in [6].

Lemma 2.3. Let 𝐶 be a closed subset of a Hilbert space ℋ. Then, 𝑧∈𝑁^𝑃_𝐶(𝑥)⟺∃𝜎>0,⟨𝑧,𝑦−𝑥⟩≤𝜎𝑦−𝑥²,∀𝑦∈𝐶. (2.3)

Clarke et al. [7] and Poliquin et al. [8] have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets. This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.

Definition 2.4. For a given 𝑟∈(0,+∞], a subset 𝐶 of ℋ is said to be uniformly prox-regular with respect to 𝑟 if, for all 𝑥∈𝐶 and for all 0≠𝑧∈𝑁^𝑃_𝐶(𝑥), one has 𝑧‖𝑧‖,𝑥−𝑥≤1‖2𝑟𝑥−𝑥‖²,∀𝑥∈𝐶. (2.4) We make the convention 1/𝑟=0 for 𝑟=+∞.

It is well known that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radius 𝑟>0. Thus, in view of Definition 2.4, for the case of 𝑟=∞, the uniform 𝑟-prox-regularity 𝐶 is equivalent to the convexity of 𝐶. Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class 𝑝-convex sets, 𝐶^1,1 submanifolds (possibly with boundary) of ℋ, the images under a 𝐶^1,1 diffeomorphism of convex sets, and many other nonconvex sets; see [6, 8].

Now, let us state the following facts, which summarize some important consequences of the uniform prox-regularity. The proof of this result can be found in [7, 8].

Lemma 2.5. Let 𝐶 be a nonempty closed subset of ℋ,𝑟∈(0,+∞] and set 𝐶_𝑟∶={𝑥∈ℋ;𝑑(𝑥,𝐶)<𝑟}. If 𝐶 is uniformly 𝑟-uniformly prox-regular, then the following hold: (1)for all 𝑥∈𝐶_𝑟, proj_𝐶(𝑥)≠∅, (2)for all 𝑠∈(0,𝑟), proj_𝐶 is Lipschitz continuous with constant 𝑟/(𝑟−𝑠) on 𝐶_𝑠, (3)the proximal normal cone is closed as a set-valued mapping.

In this paper, we are interested in the following classes of nonlinear mappings.

Definition 2.6. A mapping 𝑇∶𝐶→ℋ is said to be (a)𝛾-strongly monotone if there exists a constant 𝛾>0 such that 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝛾𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (2.5)(b)𝜇-Lipschitz if there exist a constants 𝜇>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝜇‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶. (2.6)

3. System of Nonconvex Variational Inequalities Involving Nonmonotone Mapping

Let ℋ be a real Hilbert space, and let 𝐶 be a nonempty closed subset of ℋ. In this section, we will consider the following problem: find 𝑥^∗,𝑦^∗∈𝐶 such that𝑦^∗−𝑥^∗−𝜌𝑇𝑦^∗∈𝑁^𝑃_𝐶𝑥^∗,𝑥^∗−𝑦^∗−𝜂𝑇𝑥^∗∈𝑁^𝑃_𝐶𝑦^∗, (3.1) where 𝜌 and 𝜂 are fixed positive real numbers, 𝐶 is a closed subset of ℋ, and 𝑇∶𝐶→ℋ is a mapping.

The iterative algorithm for finding a solution of the problem (3.1) was considered by Moudafi [9], when 𝐶 is 𝑟-uniformly prox-regular and 𝑇 is a strongly monotone mapping. He also remarked that two-step models (3.1) for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities since it can be applied to problems arising, especially from complementarity problems, convex quadratic programming, and other variational problems. In this section, we will generalize such result by considering the conditions for existence solution of problem (3.1) when 𝑇 is not necessary strongly monotone. To do so, we will use the following algorithm as an important tool.

Algorithm 3.1. Let 𝐶 be an 𝑟-uniformly prox-regular subset of ℋ. Assume that 𝑇∶𝐶→ℋ is a nonlinear mapping. Letting 𝑥₀ be an arbitrary point in 𝐶, we consider the following two-step projection method: 𝑦_𝑛=proj_𝐶𝑥_𝑛−𝜂𝑇𝑥_𝑛,𝑥_𝑛+1=proj_𝐶𝑦_𝑛−𝜌𝑇𝑦_𝑛, (3.2) where 𝜌, 𝜂 are positive reals number, which were appeared in problem (3.1).

Remark 3.2. The projection algorithm above has been introduced in the convex case, and its convergence was proved see [10]. Observe that (3.2) is well defined provided the projection on 𝐶 is not empty. Our adaptation of the projection algorithm will be based on Lemma 2.5.

Now we will prove the existence theorems of problem (3.1), when 𝐶 is a closed uniformly 𝑟-prox-regular. Moreover, from now on, the number 𝑟 will be understood as a finite positive real number (if not specified otherwise). This is because, as we know, if 𝑟=∞, then such a set 𝐶 is nothing but the closed convex set.

We start with an important remark.

Remark 3.3. Let 𝐶 be a uniformly 𝑟-prox-regular closed subset of ℋ. Let 𝑇₁,𝑇₂∶𝐶→ℋ be such that 𝑇₁ is a 𝜇₁-Lipschitz continuous, 𝛾-strongly monotone mapping and 𝑇₂ is a 𝜇₂-Lipschitz continuous mapping. If 𝜉=𝑟[𝜇²₁−𝛾𝜇₂−(𝜇²₁−𝛾𝜇₂)²−𝜇²₁(𝛾−𝜇₂)²]/𝜇²₁, then for each 𝑠∈(0,𝜉) we have 𝛾𝑡_𝑠−𝜇₂>𝜇²₁−𝜇²₂𝑡²_𝑠−1, (3.3) where 𝑡_𝑠=𝑟/(𝑟−𝑠).

It is worth to point out that, in Remark 3.3, we have to assume that 𝜇₂<𝜇₁. Thus, from now on, without loss of generality we will always assume that 𝜇₂<𝜇₁.

Theorem 3.4. Let 𝐶 be a uniformly 𝑟-prox-regular closed subset of a Hilbert space ℋ, and let 𝑇∶𝐶→ℋ be a nonlinear mapping. Let 𝑇₁,𝑇₂∶𝐶→ℋ be such that 𝑇₁ is a 𝜇₁-Lipschitz continuous and 𝛾-strongly monotone mapping, 𝑇₂ is a 𝜇₂-Lipschitz continuous mapping. If 𝑇=𝑇₁+𝑇₂ and the following conditions are satisfied: (a)𝑀^𝜌,𝜂𝛿_𝑇(𝐶)<𝜉, where 𝛿_𝑇(𝐶)=sup{‖𝑢−𝑣‖;𝑢,𝑣∈𝑇(𝐶)};(b) there exists 𝑠∈(𝑀^𝜌,𝜂𝛿_𝑇(𝐶),𝜉) such that 𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂−𝜁<𝜌,𝜂<𝑚𝑖𝑛𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂1+𝜁,𝑡_𝑠𝜇₂, (3.4)where 𝑀^𝜌,𝜂=max{𝜌,𝜂}, 𝑡_𝑠=𝑟/(𝑟−𝑠), and 𝜁=(𝑡_𝑠𝛾−𝜇₂)²−(𝜇²₁−𝜇²₂)(𝑡²_𝑠−1)/𝑡_𝑠(𝜇²₁−𝜇²₂).
Then the problem (3.1) has a solution. Moreover, the sequence (𝑥_𝑛,𝑦_𝑛) which is generated by (3.2) strongly converges to a solution (𝑥^∗,𝑦^∗)∈𝐶×𝐶 of the problem (3.1).

Proof. Firstly, by condition (b), we can easily check that 𝑦_𝑛−𝜌𝑇𝑦_𝑛 and 𝑥_𝑛−𝜂𝑇𝑥_𝑛 belong to the set 𝐶_𝑠, for all 𝑛=1,2,3,…. Thus, from Lemma 2.5 (1), we know that (3.2) is well defined. Consequently, from (3.2) and Lemma 2.5 (2), we have ‖𝑥_𝑛+1−𝑥_𝑛‖=‖proj_𝐶𝑦_𝑛−𝜌𝑇𝑦_𝑛−proj_𝐶𝑦_𝑛−1−𝜌𝑇𝑦_𝑛−1‖≤𝑡_𝑠‖𝑦_𝑛−𝑦_𝑛−1−𝜌𝑇𝑦_𝑛−𝑇𝑦_𝑛−1‖≤𝑡_𝑠‖𝑦_𝑛−𝑦_𝑛−1−𝜌𝑇₁𝑦_𝑛−𝑇₁𝑦_𝑛−1‖+𝜌‖𝑇₂𝑦_𝑛−𝑇₂𝑦_𝑛−1‖. (3.5) Since the mapping 𝑇₁ is 𝛾-strongly monotone and 𝜇₁-Lipschitz continuous, we obtain 𝑦_𝑛−𝑦_𝑛−1−𝜌𝑇₁𝑦_𝑛−𝑇₁𝑦_𝑛−1²=𝑦_𝑛−𝑦_𝑛−1²−2𝜌⟨𝑦_𝑛−𝑦_𝑛−1,𝑇₁𝑦_𝑛−𝑇₁𝑦_𝑛−1⟩+𝜌²𝑇₁𝑦_𝑛−𝑇₁𝑦_𝑛−1²≤𝑦_𝑛−𝑦_𝑛−1²−2𝜌𝛾‖𝑦_𝑛−𝑦_𝑛−1‖+𝜌²𝜇²₁𝑦_𝑛−𝑦_𝑛−1²=1−2𝜌𝛾+𝜌²𝜇²₁𝑦_𝑛−𝑦_𝑛−1². (3.6) On the other hand, since 𝑇₂ is 𝜇₂-Lipschitz continuous, we have ‖𝑇₂𝑦_𝑛−𝑇₂𝑦_𝑛−1‖≤𝜇₂‖𝑦_𝑛−𝑦_𝑛−1‖. (3.7) Thus, by (3.5), (3.6), and (3.7), we obtain ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝑡_𝑠𝜌𝜇₂+1−2𝜌𝛾+𝜌²𝜇²₁‖𝑦_𝑛−𝑦_𝑛−1‖. (3.8) Similarly, we have ‖𝑦_𝑛−𝑦_𝑛−1‖=‖proj_𝐶𝑥_𝑛−𝜂𝑇𝑥_𝑛−proj_𝐶𝑥_𝑛−1−𝜂𝑇𝑥_𝑛−1‖≤𝑡_𝑠‖𝑥_𝑛−𝑥_𝑛−1−𝜂𝑇𝑥_𝑛−𝑇𝑥_𝑛−1‖≤𝑡_𝑠‖𝑥_𝑛−𝑥_𝑛−1−𝜂𝑇₁𝑥_𝑛−𝑇₁𝑥_𝑛−1‖+𝜂‖𝑇₂𝑥_𝑛−𝑇₂𝑥_𝑛−1‖≤𝑡_𝑠𝜂𝜇₂+1−2𝜂𝛾+𝜂²𝜇²₁‖𝑥_𝑛−𝑥_𝑛−1‖. (3.9) Combining (3.8) and (3.9), we get ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝑡²_𝑠𝜃_𝜌𝜃_𝜂‖𝑥_𝑛−𝑥_𝑛−1‖, (3.10) where 𝜃_𝜌∶=𝜌𝜇₂+1−2𝜌𝛾+𝜌²𝜇²₁ and 𝜃_𝜂∶=𝜂𝜇₂+1−2𝜂𝛾+𝜂²𝜇²₁. Moreover, by (3.4), we know that 𝑡_𝑠𝜃_𝜌 and 𝑡_𝑠𝜃_𝜂 are elements of the interval (0,1). Thus, from (3.10), it follows that ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝜅^𝑛‖𝑥₁−𝑥₀‖ (3.11) for all 𝑛=1,2,3,…, where 𝜅∶=𝑡²_𝑠𝜃_𝜌𝜃_𝜂. Hence, for any 𝑚≥𝑛>1, it follows that ‖𝑥_𝑚−𝑥_𝑛‖≤^𝑚−1_𝑖=𝑛‖𝑥_𝑖+1−𝑥_𝑖‖≤^𝑚−1_𝑖=𝑛𝜅^𝑖‖𝑥₁−𝑥₀𝜅‖≤^𝑛1−𝜅‖𝑥₁−𝑥₀‖. (3.12) Since 𝜅<1, it follows that 𝜅^𝑛→0 as 𝑛→∞, and this implies that {𝑥_𝑛}⊂𝐶 is a Cauchy sequence. Consequently, from (3.9), we also have that {𝑦_𝑛} is a Cauchy sequence in 𝐶. Thus, by Lemma 2.5 (3), the closedness property of 𝐶 implies that there exists (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that (𝑥_𝑛,𝑦_𝑛)→(𝑥^∗,𝑦^∗) as 𝑛→∞.
We claim that (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of the problem (3.1). Indeed, by the definition of the proximal normal cone, from (3.2), we have 𝑥_𝑛−𝑦_𝑛−𝜂𝑇𝑥_𝑛∈𝑁^𝑃_𝐶𝑦_𝑛,𝑦_𝑛−𝑥_𝑛+1−𝜌𝑇𝑦_𝑛∈𝑁^𝑃_𝐶𝑥_𝑛+1. (3.13) By letting 𝑛→∞, using the closedness property of the proximal cone together with the continuity of 𝑇, we have 𝑥^∗−𝑦^∗−𝜂𝑇𝑥^∗∈𝑁^𝑃_𝐶𝑦^∗,𝑦^∗−𝑥^∗−𝜌𝑇𝑦^∗∈𝑁^𝑃_𝐶𝑥^∗. (3.14) This completes the proof.

Immediately, by setting 𝑇₂=0, we have the following result.

Theorem 3.5. Let 𝐶 be a uniformly 𝑟-prox-regular closed subset of a Hilbert space ℋ. Let 𝑇∶𝐶→ℋ be a 𝜇-Lipschitz continuous and 𝛾-strongly monotone mapping. If the following conditions are satisfied: (a)𝑀^𝜌,𝜂𝛿_𝑇(𝐶)<𝜉, where 𝛿_𝑇(𝐶)=sup{‖𝑢−𝑣‖;𝑢,𝑣∈𝑇(𝐶)}; (b)there exists 𝑠∈(𝑀^𝜌,𝜂𝛿_𝑇(𝐶),𝜉) such that 𝛾𝜇²𝛾−𝜁<𝜌,𝜂<𝜇²+𝜁, (3.15)where 𝜁=(𝑡_𝑠𝛾)²−(𝜇²₁)(𝑡²_𝑠−1)/𝑡_𝑠(𝜇²₁) and 𝑡_𝑠=𝑟/(𝑟−𝑠).
Then the problem (3.1) has a solution. Moreover, the sequence (𝑥_𝑛,𝑦_𝑛) which is generated by (3.2) strongly converges to a solution (𝑥^∗,𝑦^∗)∈𝐶×𝐶 of the problem (3.1).

In view of proving Theorem 3.4, we can obtain the following result, which contains a recent result presented by Moudafi [9] as a special case.

Theorem 3.6. Let 𝐶 be a uniformly 𝑟-prox-regular closed subset of a Hilbert space ℋ, and let 𝑇∶𝐶→ℋ be a mapping. Let 𝑇₁,𝑇₂∶𝐶→ℋ be such that 𝑇₁ is a 𝜇₁-Lipschitz continuous and 𝛾-strongly monotone mapping, 𝑇₂ is a 𝜇₂-Lipschitz continuous mapping. If 𝑇=𝑇₁+𝑇₂ and there exists 𝑠∈(0,𝜉) such that 𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂−𝜁<𝜌<𝑚𝑖𝑛𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂1+𝜁,𝑡_𝑠𝜇₂,𝑠1+‖𝑇𝑦_𝑛‖,𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂−𝜁<𝜂<𝑚𝑖𝑛𝛾𝑡_𝑠−𝜇₂𝑡_𝑠𝜇²₁−𝜇²₂1+𝜁,𝑡_𝑠𝜇₂,𝑠1+‖𝑇𝑥_𝑛‖ (3.16) for all 𝑛=1,2,3,…, where 𝑡_𝑠=𝑟/(𝑟−𝑠), 𝜁=(𝑡_𝑠𝛾−𝜇₂)²−(𝜇²₁−𝜇²₂)(𝑡²_𝑠−1)/𝑡_𝑠(𝜇²₁−𝜇²₂) and the sequence (𝑥_𝑛,𝑦_𝑛) was generated by (3.2), then the sequence (𝑥_𝑛,𝑦_𝑛) strongly converges to a solution (𝑥^∗,𝑦^∗)∈𝐶×𝐶 of the problem (3.1).

Remark 3.7. (i) An inspection of Theorem 3.6 shows that the sequences {𝑇𝑥_𝑛} and {𝑇𝑦_𝑛} are bounded.
(ii) By setting 𝑇₂=∶0, we see that Theorem 3.6 reduces to a result presented by Moudafi [9].

Remark 3.8. If 𝐶 is a convex set, by the definition of the proximal normal cone, we can reformulate (3.1) as follows: find 𝑥^∗,𝑦^∗∈𝐶×𝐶 such that 𝜌𝑇𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝑥≥0,∀𝑥∈𝐶,𝜂𝑇^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶. (3.17) The problem (3.17) was introduced and studied by Verma [10], when 𝑇 is a strong monotone mapping. Hence, Theorem 3.4 extends and improves the results presented by Verma [10]. For further recent results related to the problem (3.17), see also [2, 3, 5, 11–13].

4. Further Results

By using the techniques as in Theorem 3.4, we can also obtain an existence theorem of the following problem: find 𝑥^∗∈𝐶 such that−𝑇𝑥^∗∈𝑁^𝑃_𝐶𝑥^∗. (4.1) The problem of type (4.1) was studied by Noor [14] but in a finite dimension Hilbert space setting. In this section, we intend to consider the problem (4.1) in an infinite dimension Hilbert space. To do this, the following remark is useful.

Remark 4.1. Let 𝑇∶𝐶→𝐶 be a 𝛾-strongly monotone and 𝜇-Lipschitz continuous mapping. Then, the function 𝑓∶(1,𝜇²/(𝜇²−𝛾²))→(0,∞) which is defined by 𝑓(𝑡)=𝑡²𝛾²−𝜇²+𝜇²𝑡𝜇²𝜇,∀𝑡∈1,²𝜇²−𝛾², (4.2) is a continuous decreasing function on its domain.

We now close this section by proving an existence theorem to the problem (4.1) in a nonconvex infinite dimensional setting.

Theorem 4.2. Let 𝐶 be a uniformly 𝑟-prox-regular closed subset of a Hilbert space ℋ, and let 𝑇∶𝐶→ℋ be a 𝛾-strongly monotone and 𝜇-Lipschitz continuous mapping. If 0<𝛿_𝑇(𝐶)≤𝛾𝑟, then the problem (4.1) has a solution.

Proof. Firstly, by using an elementary calculation, we know that the function ℎ∶[1,𝜇²/(𝜇²−𝛾²))→(0,∞) which is defined by ℎ(𝑡)=𝑟(𝑡−1)𝑡𝛿_𝑇(𝐶)(𝑡)𝜇+𝑓,∀𝑡∈1,²𝜇²−𝛾², (4.3) is a continuous increasing function on [1,(𝜇²𝑟²−𝛿²_𝑇(𝐶))/𝑟²(𝜇²−𝛾²)]. Moreover, we see that the net {𝑡_𝑠}_{𝑠∈(0,𝑟)} which is defined by 𝑡_𝑠=∶𝑟/(𝑟−𝑠) converges to 1 as 𝑠↓0. Using these observations, together with the fact that ℎ(𝑡)↓𝛾/𝜇² as 𝑡↓1, we can find 𝑠^∗∈(0,𝑟(𝑟²𝛾²−𝛿²_𝑇(𝐶))/(𝜇²𝑟²−𝛿²_𝑇(𝐶))) such that 𝜇²ℎ(𝑡_𝑠∗)>𝛾. It is worth to notice that, from the choice of 𝑠^∗, we have 𝛾/𝜇²−𝑓(𝑡_𝑠∗)<𝑠^∗/𝛿_𝑇(𝐶).
Now, we choose a fixed positive real number 𝜌 such that 𝛾𝜇²−𝑓𝑡_𝑠∗𝛾<𝜌<𝑚𝑖𝑛𝜇²+𝑓𝑡_𝑠∗,𝑠^∗𝛿_𝑇(𝐶). (4.4) Next, let us start with an element 𝑥₀∈𝐶 and use an induction process to obtain a sequence {𝑥_𝑛}⊂𝐶 satisfying 𝑥_𝑛+1=proj_𝐶𝑥_𝑛−𝜌𝑇𝑥_𝑛,∀𝑛=0,1,2,…. (4.5) Note that, because of the choice of 𝜌, we can easily check that 𝑥_𝑛−𝜌𝑇𝑥_𝑛∈𝐶_𝑠∗ for all 𝑛=1,2,3,…. Following the proof of Theorem 3.4, we know that {𝑥_𝑛} is a Cauchy sequence in 𝐶. If 𝑥_𝑛→𝑥^∗ as 𝑛→∞, the closedness property of the proximal cone together with the continuity of 𝑇, from (4.5), we see that 𝑥^∗ is a solution of the problem (4.1). This completes the proof.

Remark 4.3. Theorems 3.4, 3.5, and 4.2 not only give the conditions for the existence solution of the problems (3.1) and (4.1), respectively, but also provide the algorithm to find such solutions for any initial vector 𝑥₀∈𝐶.

Acknowledgments

This study was supported by the Commission on Higher Education and the Thailand Research Fund (Project no. MRG5380247).