Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.


Introduction
Variational inequality is a powerful tool for studying problems arising in optimization, economics, differential equations, engineering and structural analysis, etc. For details, we refer to [3,5,16] and the references therein. In 1988, Cohen [6] extended an auxiliary principle technique to study the existence of solutions for a class of variational inequalities. In 1994, Yao [16] obtained the existence of solutions for generalized variational inequalities in Banach spaces. Later, Chang-Xiang [5] investigated the existence of solutions for a class of quasibilinear variational inequalities by making use of the minimax inequality due to themselves in Hilbert spaces. In 2012, Yao-Postolache [17] introduced an iterative scheme for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings, and showed a few necessary and sufficient conditions for strong convergence of the sequences generated by the proposed scheme. In 2016, Yao-Postolache-Liou-Yao [19] introduced a monotone variational inequality in Hilbert spaces, suggested an implicit algorithm, and proved its convergence hierarchical to the solution of the monotone variational inequality. Recently, Yao-Postolache-Yao [23] considered the fixed point and variational inequality problems in Hilbert spaces, suggested an extragradient algorithm, and proved strong convergence of the proposed algorithm, while Yao-Postolache-Yao [21] introduced a generalized variational inequality in Hilbert spaces, constructed an iterative algorithm for solving the generalized variational inequality, and obtained strong convergence of the algorithm.
Variational-like inequality and generalized variational-like inequality, known as useful and important generalized forms of variational inequalities, were also discussed and analyzed by many authors. For details, we refer to [1,7,8,11,13,24,25] and the references therein. Especially, Liu-Ume-Kang [13] and Zeng [24] established some existence and uniqueness theorems of solutions for generalized nonlinear variational-like inequalities in reflexive Banach spaces by applying the minimax inequality due to Ding-Tan [9].
Stimulated and inspired by the recent results in , we introduce a new generalized nonlinear variational-like inequality which includes these variational inequalities and variational-like inequalities in [6-8, 16, 24] as special cases. Next, the Banach's fixed point theorem, Schauder's fixed point theorem, and FanKKM theorem are applied to prove the existence of a solution for the generalized nonlinear variational-like inequality. Moreover, in order to suggest an iterative algorithm for computing the approximate solutions of the generalized nonlinear variational-like inequality, an auxiliary variational-like inequality is introduced and the existence and uniqueness of the solution for the auxiliary variationallike inequality is proved by using the minimax inequality due to Ding-Tan [9]. Finally, both the existence and uniqueness of solutions for the generalized nonlinear variationallike inequality and the convergence of iterative sequences presented in the algorithm are discussed under certain conditions.

Preliminaries
Throughout this paper, unless other specified, we always assume that R = (-∞, +∞), N and ω stand for the sets of all positive and nonnegative integers, respectively, D is a nonempty bounded closed convex subset of a reflexive Banach space B with the dual space B * and u, v is the dual pairing between u ∈ B * and v ∈ B. Assume that the functional b : D × D → R satisfies the following conditions: (b1) b is linear in the first argument and convex in the second argument; It follows that b is continuous in the second argument on D.
Special Cases: (A) If N(x, y) = xy, ∀x, y ∈ B * and g = I (the identity mapping in D), then problem (2.2) collapses to finding u ∈ D such that which was introduced and studied by Ding [8].
which is known as a mixed nonlinear variational-like inequality and was discussed by Ding [7] and Zeng [24] in Banach and Hilbert spaces, respectively. (2.5) Yao [16] investigated problem (2.5) which included the variational inequalities introduced by Cohen [6] as a special case. In brief, there are a number of special cases of problems (2.2)-(2.5) for suitable choices of the mappings N , E, F, η, g, and the functional b, which can be found in [6-8, 16, 24] and the references cited therein.
We need the following definitions and results which will be used in the paper. (1) N is said to be Lipschitz continuous in the first argument if there exists a constant α > 0 such that (2) N is said to be η-strongly monotone with respect to F in the second argument if there exists a constant β > 0 such that (3) η is said to be Lipschitz continuous if there exists a constant δ > 0 such that (4) g is said to be Lipschitz continuous if there exists a constant r > 0 such that

Definition 2.3
Let X and Y be topological spaces. A mapping f : X → Y is said to be compact if it is continuous and has a relatively compact range.

Definition 2.4 ([1]) Let D be a nonempty convex subset of a Banach space
(2) η-strongly convex if there exists a constant μ > 0 satisfying For D ⊆ B, we define by conv(D) the convex hull of B. The set-valued mapping P : D → 2 D is said to be a KKM mapping, if for any finite subset {v 1 , . . . , v m } of D, there exist a nonempty compact convex subset X 0 of D and a nonempty compact subset Then there existsv ∈ K such that ϕ(u,v) ≤ 0 for all u ∈ D.

Existence of solutions for the generalized nonlinear variational-like inequality (2.2)
This section is devoted to the existence result of solutions for the generalized nonlinear variational-like inequality (2.2) by employing the Banach's fixed point theorem, Schauder's fixed point theorem, and FanKKM theorem.
is continuous on D, and the mappings F, N(Eu 1 , ·), and η have the 0-diagonally concave relation on D; (c2) η is Lipschitz continuous with a constant δ > 0 and for any v ∈ D, η(·, v) is continuous from the weak topology to the weak topology; (c3) N is Lipschitz continuous and η-strongly monotone with respect to F in the first and second argument with constants α > 0, β > 0, respectively, and g is Lipschitz continuous with a constant r > 0; Then the generalized nonlinear variational-like inequality (2.2) has a solution in D.
Proof First of all, let u 1 be an arbitrary fixed element in D. For each u 0 ∈ D, we show that there exists a uniqueŵ ∈ D such that Define a set-valued mapping P : D → 2 D by Obviously, v ∈ P(v) = ∅, ∀v ∈ D. Next we claim that P is a KKM mapping. Otherwise, there exists a finite set That is, which implies that by (b1), (c1), and (c4) which is a contradiction. Hence P : D → 2 D is a KKM mapping. Since P(v) w is a weakly closed subset of the bounded set D, it is weakly compact. It follows from Lemma 2.2 that and further By (c1), (c2), and (2.1), we gain that (3.1) holds. Namely, (3.1) possesses a solutionŵ ∈ D for any u 0 ∈ D. Now we prove the uniqueness of solution for (3.1) with respect to u 0 ∈ D.
Suppose that w ∈ D \ {ŵ} is also a solution of (3.1) with respect to u 0 ∈ D. It follows that Taking v = w in (3.1), v =ŵ in (3.2) and adding them together, we get that in view of (c3), which implies that β ≤ 0, a contradiction. That is,ŵ is the unique solution of (3.1) with respect to u 0 ∈ D. It follows that there exists a mapping f : D → D such that for each u 0 ∈ D, fu 0 is the unique solution of (3.1). Secondly, for each u 1 ∈ D, we show that there exists a unique w 0 ∈ D satisfying In fact, for every x, y ∈ D, there exist w 1 = f (x), w 2 = f (y) such that for each v ∈ D. Taking v = w 2 in (3.4), v = w 1 in (3.5) and adding these two inequalities, we know that by (b2), (b3), (c3), and (c4), which means that It follows from (c4) that f is a contraction mapping on D and so it has a unique fixed point w 0 ∈ D satisfying (3.3) according to the Banach's fixed point theorem. We now verify that w 0 is the unique solution of (3.3) relative to u 1 ∈ D. Suppose that w 0 ∈ D \ {w 0 } is another solution of (3.3) relative to u 1 ∈ D, that is, Take v = w 0 in (3.3), v = w 0 in (3.6) and add (3.3) and (3.6). Based on (b2), (b3), (c3), and (c4), we conclude that which contradicts (c4). Therefore, w 0 is the unique solution of (3.3) relative to u 1 ∈ D.
Hence for each u 1 ∈ D, there exists a mapping h : D → D such that hu 1 is the unique solution of (3.3). Finally, we show that h is a compact mapping on D. By the definition of h, we obtain that Since E is a compact mapping, it is easy to verify that h is also compact. The Schauder's fixed point theorem yields that there exists some u ∈ D such that hu = u. Consequently, u ∈ D is a solution of problem (2.2). This completes the proof.
Remark 3.1 Theorem 3.1 extends and improves the corresponding results in [1, 6-8, 16, 24]. Not only these variational and variational-like inequalities in [6-8, 16, 24] are replaced by the more generalized nonlinear variational-like inequality (2.2), but also Theorem 3.1 first combines the Banach's fixed point theorem, Schauder's fixed point theorem, and FanKKM theorem to establish the existence of solutions for the generalized nonlinear variational-like inequality (2.2).

Existence and uniqueness of solutions for the auxiliary variational-like inequality and iterative algorithm
In this section, we introduce an auxiliary variational-like inequality and establish an existence and uniqueness theorem of the solution for the auxiliary variational-like inequality by applying the minimax inequality due to Ding-Tan [9]. Based on this theorem, we suggest a new iterative algorithm to compute the approximate solutions of the generalized nonlinear variational-like inequality (2.2). Let K : D → R ∪ {+∞} be a given Fréchet differentiable η-strongly convex functional and ρ > 0 be a constant. Let θ be a constant in [0, 1] and x be any fixed element in D. For each u ∈ D, we consider the following auxiliary variational-like inequality: Find w ∈ D such that (d1) N is η-strongly monotone with respect to F in the second argument with a constant (d3) K is η-strongly convex with a constant μ > 0 and K is sequentially continuous from the weak topology to the strongly topology, then the auxiliary variational-like inequality (4.1) processes a solution in D with respect to u ∈ D.
Proof Let x be any fixed element in D. For each u ∈ D, define a functional ϕ : It is not difficult to verify that condition (a) of Lemma 2.4 is satisfied. Next, we claim that ϕ(v, w) satisfies condition (b) of Lemma 2.4. If it were false, there would exist a finite set > 0 for any i ∈ {1, 2, . . . , m}. It follows from this that which is impossible. Hence condition (b) of Lemma 2.4 is fulfilled. For a given v * ∈ D, put X = {v * } and Y = {w ∈ D : v *w ≤ R}, where R = θρδ N(Ex, Fv * ) + θργ gu ]. It is clear that X and Y are both weakly compact convex subsets of D. By virtue of (b2), (b3), (c2), (d1), and (d3), we gain that for each w ∈ D \ Y , there is a v * ∈ co(X ∪ {w}) such that Thus, condition (c) of Lemma 2.4 holds as well. As a result, Lemma 2.4 ensures that there exists some w ∈ D such that ϕ(v, w) ≤ 0 for all v ∈ D, namely, the auxiliary problem (4.1) has a solution w ∈ D with respect to u ∈ D. Now we prove the uniqueness of the solution for the auxiliary problem (4.1) with respect to u ∈ D. Suppose that w ∈ D \ {w} is another solution of the auxiliary problem (4.1) relative to u ∈ D. It follows that (4.2) Taking v = w in (4.1), v = w in (4.2) and adding them together, we obtain by (d1)-(d3) that which implies that (1θ )μ + θρβ ≤ 0, a contradiction. Therefore, w is the unique solution of the auxiliary problem (4.1) relative to u ∈ D. This completes the proof.