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The existence of solutions for general variational inequality and applications in FC spaces
Journal of Inequalities and Applications volume 2012, Article number: 49 (2012)
Abstract
In this article, we prove the existence of solutions for the general variational inequality ϕ(x, y) ≥ f(x)  f(y) and Minty type theorem by using the generalized KKM theorem in topological spaces without linear structure. Some properties of solutions set for the general variational inequality are studied by Minty type theorem. As applications, equivalence between Browder fixed point theorem and Ky Fan's minimax inequality are studied in topological spaces.
1 Introduction
The existence of solutions and the properties of solutions set for general variational inequality were studied by many authors in linear topologcal spaces (see [1–7]). These results depend deeply on the linear structure of spaces. In 1987, Horvath [8] used the concept of retractile subset family to replace the convexity assumptions and established the KKM theorem in Hspace. Motivated by Horvath's work, the various extensions of KKM theorem and their applications in many aspects were investigated by various authors under the assumption without convexity. Recently, Ding [9, 10] and Deng and Xia [11] introduced the class of finitely continuous topological spaces (in short FCspaces) without any convexity and linear structure. The class of FCspaces includes the classes of Hspaces, Gconvex spaces, GHconvex spaces, Lconvex spaces and many topological spaces with convexity structure. The purpose of this article is to study the existence of solutions for the following general variational inequality (in short VI(1.1))
Minty type theorem and the properties of the solutions set for general variational inequality(1.1) in FCspaces.
2 Preliminaries
Let E be a topological space, X be a nonempty subset of E. Throughout this article, we assume that 〈X〉 denotes all nonempty finite subset of X, Δ_{ n }denotes the standardnsimplex (e_{0},..., e_{n}) in R^{n+1}, Δ_{ k }denotes ksubsimplex of Δ_{ n }with vectors \left({e}_{{i}_{0}},\dots ,{e}_{{i}_{k}}\right), (0 ≤ k ≤ n), R = (∞, +∞), \stackrel{\u0304}{R}=\left(\infty ,+\infty \right]. The following definitions can be found in [4, 7, 9, 11].
Definition 2.1. (E,φ_{ N }) is said to be a finitely continuous space (in short FCspace) if E is a topological space and for each N = {x_{0},..., x_{ n }} ∈ 〈E〉, there exists a continuous mapping φ_{ N }: Δ_{ n }→ E. A subset X of (E, φ_{ N }) is said to be an FCsubspace of E if for each N = {x_{0},...,x_{ n }} ∈ 〈E〉 and for any \left\{{x}_{{i}_{0}},\dots ,{x}_{{i}_{k}}\right\}\subset X\cap N, φ_{ N }(Δ_{ k }) ⊂ X, where {\Delta}_{k}=co\left\{{e}_{{i}_{j}}:j=0,\dots ,k\right\}.
Definition 2.2. Let X be an FCsubspace of (E, φ_{ N }).

(1)
f : X → R is said to be an Rconvex function if for each N = {x _{0}, ..., x _{n}} ∈ 〈X〉, f\left({x}_{t}\right)\le \sum _{i=0}^{n}{t}_{i}f\left({x}_{i}\right), where {x}_{t}={\phi}_{N}\left(\sum _{i=0}^{n}{t}_{i}{e}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}{t}_{i}\in \left[0,1\right],\sum _{i=0}^{n}{t}_{i}=1.

(2)
f : X → R is said to be a strictly Rconvex function if for each N = {x _{0}, x _{1}} ∈ 〈X〉 and x _{0} ≠ x _{1}, f(x_{ t } ) < tf(x _{0}) + (1  t)f(x _{1}), where x_{ t } = φ_{ N } (te _{0} + (1  t)e _{1}), t ∈ (0, 1).

(3)
\varphi :X\times X\to \overline{R} is said to be λquasiconvex related to the second variant if subset {y ∈ X; ϕ(x, y) < λ} is an FCsubspace for any x ∈ X. ϕ is said to be Rquasiconvex related to the second variant if ϕ is λquasiconvex related to the second variant for any x ∈ X and λ ∈ R.

(4)
\varphi :X\times X\to \overline{R} is said to be an Rsemicontinuous function if for any N = {x _{0}, x _{1}} ∈ 〈X〉,
\underset{t\to {0}^{+}}{\overline{\mathrm{lim}}}\varphi ({x}_{t},{x}_{1})\le \varphi ({x}_{0},{x}_{1}),
where x_{ t } = φ_{ N } ((1  t)e_{0} + te_{1}), t ∈ [0, 1].
Definition 2.3. Let X be a nonempty subset of E. A function \varphi :X\times X\to \overline{R} is said to be monotone if ϕ(x, y)+ϕ(y, x) ≤ 0 for any x, y ∈ X. In addition, if ϕ also satisfies the condition ϕ(x, y) + ϕ(y, x) = 0 if and only if x = y, then ϕ is said to be strictly monotone.
Definition 2.4. Let X be a nonempty set and (Y, φ_{ N } ) be an FCspace. For λ ∈ R, a function \varphi :X\times Y\to \overline{R} is said to be λgeneralized Rdiagonally quasiconvex (resp., Rdiagonally quasiconcave) related to the first variant if, for each {x_{0}, ..., x_{ n } } ∈ 〈X〉, there exists N = {y_{0}, ..., y_{ n } } ∈ 〈Y〉 such that, for each subset {i_{0}, ..., i_{ k } } ⊂ {0, ..., n} and \u0233\in {\phi}_{N}\left({\Delta}_{k}\right),
where Δ_{ k }denotes ksubsimplex of with vectors \left({e}_{{i}_{0}},\dots ,{e}_{{i}_{k}}\right), (0 ≤ k ≤ n) related to \left\{{y}_{{i}_{0}},\dots ,{y}_{{i}_{k}}\right\}.
Definition 2.5. Let X be a nonempty set and (Y, φ_{ N } ) be an FCspace. A mapping T : X → 2^{Y}is said to be a generalized RKKM mapping if, for each finite subset {x_{0}, ..., x_{ n } } ∈ 〈X〉, there exists N = {y_{0}, ..., y_{ n } } ∈ 〈Y〉 such that, for each subset {i_{0}, ..., i_{ k } } ⊂ {0, ..., n}, {\phi}_{N}\left({\Delta}_{k}\right)\subset {\cup}_{j=0}^{k}T\left({x}_{{i}_{j}}\right).
Lemma 2.1. ϕ is λgeneralized Rdiagonally quasiconvex (resp., Rdiagonally quasiconcave) related to the first variant if and only if the mapping G : X → 2^{Y}defined by
is a generalized RKKM mapping.
Proof If ϕ is λgeneralized Rdiagonally quasiconvex related to the first variant, then for each {x_{0}, ..., x_{ n } } ∈ 〈X〉, there exists N = {y_{0}, ..., y_{ n } } ∈ 〈Y〉 such that, for each subset {i_{0}, ..., i_{ k } } ∈ {0, ..., n} and \u0233\in {\phi}_{N}\left({\Delta}_{k}\right), \underset{0\le j\le k}{\text{max}}\varphi \left({x}_{{i}_{j}},\u0233\right)\ge \lambda. Hence, there exists some j_{0} ∈ {0, ..., k} such that \varphi \left({x}_{{i}_{{j}_{0}}},\u0233\right)\ge \lambda, i.e., \u0233\in G\left({x}_{{i}_{{j}_{0}}}\right), therefore {\phi}_{N}\left({\Delta}_{k}\right)\subset {\cup}_{j=0}^{k}G\left({x}_{{i}_{j}}\right), i.e., G is a generalized RKKM mapping.
Conversely, suppose that G is a generalized RKKM mapping. Then for each {x_{0}, ..., x_{ n } } ∈ 〈X〉, there exists N = {y_{0}, ..., y_{ n } } ∈ 〈Y〉 such that, for each subset {i_{0}, ..., i_{ k } } ⊂ {0, ..., n}, {\phi}_{N}\left({\Delta}_{k}\right)\subset {\cup}_{j=0}^{k}G\left({x}_{{i}_{j}}\right). Thus for any \u0233\in {\phi}_{N}\left({\Delta}_{k}\right), there exists some j_{0} ∈ {0, ..., k} such that \u0233\in G\left({x}_{{i}_{{j}_{0}}}\right), i.e., \varphi \left({x}_{{i}_{{j}_{0}}},\u0233\right)\ge \lambda. It follows that \underset{0\le j\le k}{\text{max}}\varphi \left({x}_{{i}_{j}},\u0233\right)\ge \lambda. This completes the proof.
Definition 2.6. Let X, Y be two nonempty subsets of E. For λ ∈ R, a function \varphi :X\times Y\to \overline{R} is said to be λtransfer compactly lower (resp., upper) semicontinuous relate to the second variant if, for each nonempty compact subset K ⊂ Y and each y ∈ K, there exists x ∈ X such that ϕ(x, y) > λ (resp., ϕ(x, y) < λ) implies that there exists an open neighborhood U(y) of y in K and a point \stackrel{\u0304}{x}\in X such that \varphi \left(\stackrel{\u0304}{x},z\right)>\lambda (resp., \varphi \left(\stackrel{\u0304}{x},y\right)<\lambda) for all z ∈ U(y).
Definition 2.7. Let X, Y be two nonempty subsets of E. A mapping T : X → 2^{Y}is said to be transfer compactly closed valued on X if, for x ∈ X and for each nonempty compact subset K of Y, y ∉ T(x) ⋂ K implies that there exists a point \stackrel{\u0304}{x}\in X such that y\notin c{l}_{K}\left(T\left(\stackrel{\u0304}{x}\right)\cap K\right), where c{l}_{K}\left(T\left(\stackrel{\u0304}{x}\right)\cap K\right) is the closure of T\left(\stackrel{\u0304}{x}\right)\cap K in K.
Lemma 2.2. Let T : X → 2^{Y}be a setvalued mapping defined by
Then T is transfer compactly closed valued if and only if ϕ is λtransfer compactly lower (resp., upper) semicontinuous related to second variant.
Proof. Suppose that T is transfer compactly closed valued. If for each nonempty compact subset K ⊂ Y and each y ∈ K, there exists x ∈ X such that ϕ(x, y) > λ (resp., \varphi \left(\stackrel{\u0304}{x},y\right)<\lambda), then y ∉ T(x) ⋂ K. Thus, there exists a point \stackrel{\u0304}{x}\in X such that y\notin c{l}_{K}\left(T\left(\stackrel{\u0304}{x}\right)\cap K\right). Hence, there exists an open neighborhood U(y) of y in K such that U\left(y\right)\cap T\left(\stackrel{\u0304}{x}\right)=\mathrm{0\u0338}, i.e., \varphi \left(\stackrel{\u0304}{x},z\right)>\lambda (resp., \varphi \left(\stackrel{\u0304}{x},z\right)<\lambda) for all z ∈ U(y).
Conversely, for each nonempty compact subset K ⊂ Y and for any y ∉ T(x) ⋂ K, we have (i) if y ∉ K then y\notin c{l}_{K}\left(T\left(\stackrel{\u0304}{x}\right)\cap K\right); (ii) if y ∈ K and y ∉ T(x) then ϕ(x, y) > λ (resp., \varphi \left(\stackrel{\u0304}{x},z\right)<\lambda). Hence, there exists an open neighborhood U(y) of y in K and \stackrel{\u0304}{x}\in X such that \varphi \left(\stackrel{\u0304}{x},z\right)>\lambda (resp. \varphi \left(\stackrel{\u0304}{x},z\right)<\lambda) for all z ∈ U(y), therefore y\notin c{l}_{K}\left(T\stackrel{\u0304}{x}\right). This implies that T is transfer compactly closed valued. The proof is completed.
3 Solutions of VI(1.1)
In the Sections 3 and 4, we assume that (E, φ_{ N } ) is an FCspace, X is a nonempty FCsubspace of E,\phantom{\rule{2.77695pt}{0ex}}f:X\to \stackrel{\u0304}{R} is a function with f ≢ +∞, and ϕ : X × X → R is a function with ϕ(x, x) ≥ 0 for all x ∈ X.
Theorem 3.1. Suppose that

(i)
there exists a compact subset K of E and x* ∈ X ⋂ K such that
f\left(x\right)>\varphi \left(x,{x}^{*}\right)+f\left({x}^{*}\right)\phantom{\rule{1em}{0ex}}\left(\forall x\in X\backslash K\right); 
(ii)
f(y) + ϕ(x, y)  f(x) is 0generalized Rdiagonally quasiconvex related to the variant y;

(iii)
f(x)ϕ(x, y)f(y) is 0transfer compactly lower semicontinuous related to the variant x.
Then VI(1.1) has a solution in X ⋂ K, i.e., there exists \stackrel{\u0304}{x}\in X\cap K such that
Proof. For any y ∈ X, let
Then G is a multivalued mapping from X into itself. It is easy to see that, if \stackrel{\u0304}{x}\in {\cap}_{y\in X}G\left(y\right)\ne \mathrm{0\u0338}, then \stackrel{\u0304}{x} is a solution of VI(1.1).
We now show that {\cap}_{y\in X}G\left(y\right)\ne \mathrm{0\u0338}. In fact, condition (ii) and Lemma 2.1 imply that G is a generalized RKKM mapping, condition (iii) and Lemma 2.2 imply that G is transfer compactly closed valued. It follows from condition (i) that x ∉ G(x*) for all x ∈ X\K, and so G(x*) ⊂ K, i.e., \overline{G\left({x}^{*}\right)} is a compact subset of E. By Theorem 3.4 in [11], {\cap}_{y\in X}G\left(y\right)\ne \mathrm{0\u0338}. In addition, {\cap}_{y\in X}G\left(y\right)\subset G\left({x}^{*}\right)\subset K, and so the solutions of VI(1.1) is in X ⋂ K. This completes the proof.
Theorem 3.2. Suppose that

(i)
there exist a compact subset K of E and x* ∈ X ⋂ K such that
f\left(x\right)>\varphi \left(x,{x}^{*}\right)+f\left({x}^{*}\right)\phantom{\rule{1em}{0ex}}\left(\forall {x}^{*}\in X\backslash K\right); 
(ii)
f(y) + ϕ(x, y) is Rquasiconvex related to the variant y;

(iii)
f(x)  ϕ(x, y) is lower semicontinuous related to the variant x.
Then, VI(1.1) has a solution in X ⋂ K.
Proof. We first show the multivalued mapping G defined by (3.1) is a generalized RKKM mapping. Suppose that G is not a generalized RKKM mapping, then there exist N = {x_{0}, ..., x_{ n } } ∈ 〈X〉, \left\{{e}_{{i}_{0}},\dots ,{e}_{{i}_{k}}\right\}\subset \left\{{e}_{0},\dots ,{e}_{n}\right\} and y' ∈ φ_{ N } (Δ_{ k }) such that {y}^{\prime}\notin \bigcup _{j=0}^{k}G\left({x}_{{i}_{j}}\right). Since f(x) + ϕ(x, y) is Rquasiconvex related to the variant y and \left\{{x}_{{i}_{0}},\dots ,{x}_{{i}_{k}}\right\}\subset \left\{y\in X:f\left(y\right)+\phi \left({y}^{\prime},y\right)<f\left({y}^{\prime}\right)\right\}, then φ_{ N } (Δ_{ k }) ⊂ {y ∈ X : f(y) + φ(y', y) < f(y')}. And y' ∈ φ_{ N } (Δ_{ k }), then f(y') + φ(y', y') < f(y'), i.e., φ(y', y') < 0, this contracts φ(y', y') ≥ 0. Therefore, G is a generalized RKKM mapping.
Condition (i) implies that x ∉ G(x*) for all x ∈ X\K, so, G(x*) ⊂ K, i.e., \overline{G\left({x}^{*}\right)} is a compact subset of E. It follows from condition (iii) that G(y) is a closed subset for each y ∈ X. By Theorem 3.2 in [11], {\cap}_{y\in X}G\left(y\right)\ne \mathrm{0\u0338}, and so VI(1.1) has a solution in X ⋂ K. This completes the proof.
By Theorems 3.1 and 3.2, it is easy to get Ky Fan's minimax inequality in FCspaces.
Corollary 3.1. Assume that (E, φ_{ N } ), X, f, ϕ satisfy the conditions (ii) and (iii) in Theorem 3.1.
Then there exists \stackrel{\u0304}{x}\in X such that
Corollary 3.2 Assume that (E, φ_{ N } ), X, f, ϕ satisfy the conditions (ii) and (iii) in Theorem 3.2.
Then there exists \stackrel{\u0304}{x}\in X such that
Remark 3.1 Theorems 3.1 and 3.2, Corollaries 3.1 and 3.2 extend some results of Gwinner [1] and Ky Fan minimax inequality to FCspaces without linear structure.
4 Minty theorem and monotone variational inequality
In this section, we present Minty type theorem in FCspaces.
Theorem 4.1. Suppose that

(i)
ϕ: X × X → R is a monotone and Rsemicontinuous mapping;

(ii)
for each x ∈ X, f(y) + ϕ(x, y) is an Rconvex function related to the variant y;

(iii)
for any N = {x _{0}, x _{1}} ∈ 〈X〉, \underset{t\to {0}^{+}}{\text{lim inf}}f\left({x}_{t}\right)\ge f\left({x}_{0}\right), where x_{ t } = φ_{ N } ((1  t)e _{0} + te _{1}), t ∈ [0, 1].
Then there exists \stackrel{\u0304}{x}\in X such that f\left(y\right)+\varphi \left(\stackrel{\u0304}{x},y\right)\ge f\left(\stackrel{\u0304}{x}\right) for all y ∈ X if and only if f\left(y\right)\varphi \left(y,\stackrel{\u0304}{x}\right)\ge f\left(\stackrel{\u0304}{x}\right) for all y ∈ X.
Proof. For each y ∈ X, let
It is sufficient to prove that
Since ϕ is monotone, it is easy to see that {\cap}_{y\in X}M\left(y\right)\subset {\cap}_{y\in X}N\left(y\right). We claim that
Suppose that (4.4) is not true. Then there exists \stackrel{\u0304}{x}\in {\cap}_{y\in X}N\left(y\right) and \stackrel{\u0304}{x}\notin {\cap}_{y\in X}M\left(y\right) i.e.,
and there exists \u0233\in X such that
Let A=\left\{\stackrel{\u0304}{x},\u0233\right\}, Setting x_{ t } = φ_{ A } ((1  t)e_{0} + te_{1}) for each t ∈ [0, 1]. By condition (iii), we have
Since ϕ is Rsemicontinuous, \underset{t\to {\sigma}^{+}}{\text{lim sup}}\varphi \left({x}_{t},\u0233\right)\le \varphi \left(\stackrel{\u0304}{x},\u0233\right), we have
Thus, there exists t* ∈ [0, 1] such that
therefore, we have
Setting y = x_{ t } in (4.5), we get
Combining (4.7) and (4.8), we have
It follows from condition (ii) that
Combining (4.9) and (4.10), we have ϕ(x_{ t }, x_{ t } ) < 0, this contradicts to ϕ(x_{ t }, x_{ t } ) ≥ 0. Hence, (4.4) is true. The proof is completed.
Remark 4.1. Theorem 4.1 extends famous Minty theorem in FCspaces without linear structure.
By using Minty type theorem (Theorem 4.1), we now discuss the properties of the solutions set for VI(1.1).
Theorem 4.2. Suppose that

(i)
there exist a compact subset K of E and x* ∈ X ⋂ K such that
f\left(x\right)>\varphi \left(x,{x}^{*}\right)+f\left({x}^{*}\right)\phantom{\rule{1em}{0ex}}\left(\forall x\in X\backslash K\right); 
(ii)
for each x ∈ X, f(y) + ϕ(x, y) is a Rconvex function related to the variant y;

(iii)
f(x)  ϕ(x, y) is lower semicontinuous related to the variant x;

(iv)
ϕ(x, y) is a monotone and Rsemicontinuous mapping;

(v)
for any N = {x _{0}, x _{1}} ∈ 〈X〉, \underset{t\to {0}^{+}}{\text{lim inf}}f\left({x}_{t}\right)\ge f\left({x}_{0}\right), where x_{ t } = φ_{ N } ((1  t)e _{0} + te _{1}), t ∈ [0, 1].
Then the solution set for VI(1.1) is a nonempty compact FCsubspace of E in X ⋂ K.
Furthermore, either (I) ϕ is a strictly monotone function in (iv) or (II) f(y)+ϕ(x, y) is strictly Rconvex function in (ii), then the solution of VI(1.1) is unique in X ⋂ K.
Proof Let S be the solution set for VI(1.1) in X ⋂ K. It is clear that S is nonempty by Theorem 3.2 and S={\cap}_{y\in X}M\left(y\right), where M(y) is defined by (4.1). We show that S is a compact FCsubspace of E in X ⋂ K. In fact, from Theorem 4.1, we have S={\cap}_{y\in X}M\left(y\right)={\cap}_{y\in X}N\left(y\right), where N(y) is defined by (4.2). Since f(y) + ϕ(x, y) is a Rconvex function related to the variant y, it is easy to see that N(y) is an FCsubspace of E, then S={\cap}_{y\in X}N\left(y\right) is an FCsubspace of E. By condition (i) and (iii), we have S ⊂ K and S={\cap}_{y\in X}M\left(y\right) is closed. Therefore, S is a nonempty compact FCsubspace of E in X ⋂ K.
We now prove that the solution of VI(1.1) is unique in X ⋂ K under the condition (I). In fact, let x_{1}, x_{2} ∈ X ⋂ K be two solutions of VI(1.1) with x_{1} ≠ x_{2}. Then
and
Putting y = x_{2} in (4.11) and y = x_{1} in (4.12), respectively, we have
Adding (4.13) and (4.14), we have ϕ(x_{1}, x_{2}) + ϕ(x_{2}, x_{1}) ≥ 0. This implies that ϕ(x_{1}, x_{2}) + ϕ(x_{2}, x_{1}) = 0 and so x_{1} = x_{2}.
Now we prove that the solution of VI(1.1) is unique in X ⋂ K under condition (II). In fact, suppose that x_{1}, x_{2} ∈ X ⋂ K be two solutions of VI(1.1) with x_{1} ≠ x_{2}. Let N = {x_{1}, x_{2}} and \stackrel{\u0304}{x}={\phi}_{N}\left(\frac{1}{2}{e}_{0}+\frac{1}{2}{e}_{1}\right)\in X. Then
Therefore, we have
Since ϕ(x, y) is monotone, we know that ϕ(x, y) + ϕ(y, x) ≤ 0 for any x, y ∈ X. Especially setting x = y, we have ϕ(x, x) ≤ 0. Since ϕ(x, x) ≥ 0 for all x ∈ X, it follows that ϕ(x, x) = 0. Therefore (4.15) implies that ϕ(x_{1}, x_{2}) > f(x_{1})f(x_{2}). Analogously, we have ϕ(x_{2}, x_{1}) > f(x_{2})f(x_{1}) and so ϕ(x_{1}, x_{2}) + ϕ(x_{2}, x_{1}) > 0, this contradicts to the fact that ϕ is monotone. Thus, the solution of VI(1.1) is unique. This completes the proof.
5 Applications
In 1968, Browder [12] established the fixed point theorem for setvalued mappings in Hausdorff topological vector spaces. This theorem plays an important role in solving quasivariational inequality (see [3, 13]). As applications, we now generalize the Browder fixed point theorem to FCspaces by using Corollary 3.4 and prove the equivalent relation between Browder fixed point theorem and Ky Fan's minimax inequality in FCspaces without linear structure.
Theorem 5.1. Let (E, φ_{ N } ) be an FCspace and X be a compact FCsubspace of E. Suppose that T : X → 2^{X}is a setvalued mapping satisfying one of the following conditions

(i)
for each x ∈ X, T(x) is a nonempty FCsubspace of X and for each y ∈ X, T ^{1}(y) is an open subset of X;

(ii)
for each x ∈ X, T(x) is an open subset of X, and for each y ∈ X, T ^{1}(y) is a nonempty FCsubspace of X.
Then T has a fixed point in X.
Proof. Suppose that condition (i) is satisfied and T has no fixed point in X. For x, y ∈ X, let
where graph(T) = {(x, y);y ∈ T(x)}. Since for each x ∈ X, x ∉ T(x) we have ϕ(x, x) = 0. Now we verify that all conditions of Corollary 3.2 are satisfied. In fact, for λ ∈ R and x ∈ X, we have
This implies that ϕ(x, y) is Rquasiconvex related to the variant y. For each y ∈ X and λ ∈ R,
This implies that ϕ(x,y) is upper semicontinuous related to the variant x. From Corollary 3.2, there exists \stackrel{\u0304}{x}\in X such that \varphi \left(\stackrel{\u0304}{x},y\right)\ge 0 for all y ∈ X.
Since ϕ(x, y) ≤ 0 for all x, y ∈ X, we know that \varphi \left(\stackrel{\u0304}{x},y\right)=0 for all y ∈ X, i.e., T\left(\stackrel{\u0304}{x}\right)=\mathrm{0\u0338}, which is a contradiction.
Now we suppose that the condition (ii) is satisfied. Let H(x) = T^{1}(x) = {y ∈ X; x ∈ T(y)} for x ∈ X. Then H satisfies condition (i). Hence, H has a fixed point in X and so T has a fixed point in X. This completes the proof.
Theorem 5.2. Browder type fixed point theorem (Theorem 5.1) is equivalent to Ky Fan's minimax inequality (Corollary 3.2).
Proof. Using the Ky Fan's minimax inequality to prove the Browder fixed point theorem has been shown by Theorem 5.1. Now we prove that Ky Fan's minimax inequality using Browder fixed point theorem. In fact, if the conclusion of Corollary 3.2 is not true, then for each x ∈ X, there exists y ∈ X such that f(y) + ϕ(x, y) < f(x). Set
Then T(x) is a nonempty FCsubspace of X for any x ∈ X and for any y ∈ X.
is an open subset of X. Therefore there exists \stackrel{\u0304}{x}\in X such that \stackrel{\u0304}{x}\in T\left(\stackrel{\u0304}{x}\right), i.e., f\left(\stackrel{\u0304}{x}\right)+\varphi \left(\stackrel{\u0304}{x},\stackrel{\u0304}{x}\right)<f\left(\stackrel{\u0304}{x}\right) and so \varphi \left(\stackrel{\u0304}{x},\stackrel{\u0304}{x}\right)<0. This is a contraction. The proof is completed.
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Wang, J. The existence of solutions for general variational inequality and applications in FC spaces. J Inequal Appl 2012, 49 (2012). https://doi.org/10.1186/1029242X201249
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DOI: https://doi.org/10.1186/1029242X201249