A New Fixed Point Theorem and Applications

Definition 1. Let X be a topological space, Y a nonempty set, and Φ a family of continuous mappings φ : Δ_n → X, n ∈ N. A triple (X, Y, Φ) is said to be a generalized finitely continuous topological space (GFC-space) if and only if for each finite subset N = {y₀, y₁, …, y_n} of Y, there is φ_N : Δ_n → X of the family Φ.

In the sequel, we also use (X, Y, {φ_N}) to denote (X, Y, Φ).

Definition 2. Let S : Y → 2^X be a multivalued mapping. A subset D of Y is called an S-subset of Y if and only if for each N = {y₀, y₁, …, y_n}⊆Y and each $$, one has φ_N(Δ_k) ⊂ S(D), where Δ_k is the face of Δ_n corresponding to $$, that is, the simplex with vertices $$. Roughly speaking, if D is an S-subset of Y, then (S(D), D, Φ) is a GFC-space.

The class of GFC-space contains a large number of spaces with various kinds of generalized convexity structures such as FC-space and G-convex space (see [15–17]).

Definition 3 (see [8].)Let (X, Y, {φ_N}) be a GFC-space and Z a nonempty set. Let T : X → 2^Z and F : Y → 2^Z be two set-valued mappings; F is called a weak KKM mapping with respect to T, shortly, weak T-KKM mapping if and only if for each N = {y₀, y₁, …, y_n}⊆Y, $$ and x ∈ φ_N(Δ_k), $$.

Definition 4 (see [8].)Let X be a Hausdorff space, (X, Y, {φ_N}) a GFC-space, Z a topological space, T : X → 2^Z, f : Y × Z → R ∪ {−∞, +∞}, and g : X × Z → R ∪ {−∞, +∞}. Let λ ∈ R. f is called (λ, T, g)-GFC quasiconvex if and only if for each x ∈ X, z ∈ T(x), N = {y₀, y₁, …, y_n}∈〈Y〉, and $$, one has the implication $$, for all j = 0,1, …, k implies that g(x^′, z) < λ for all x^′ ∈ φ_N(Δ_k).

For λ ∈ R, define β ∈ R and H_λ : Y → 2^Z by β = inf _x∈Xsup _z∈T(x)g(x, z) and H_λ(y) = {z ∈ Z : f(y, z) ≥ λ}, respectively.

Lemma 5 (see [8].)For λ < β, if f is (λ, T, g)-GFC quasiconvex, then H_λ is a weak T-KKM mapping.

Lemma 6. Let $$ be a family of GFC-spaces and X = ∏_i∈I  X_i a compact Hausdorff space. For each i ∈ I, let $$ and $$ be such that the conditions hold as follows:

• (i)

  for each x ∈ X, each $$ and each $$, one has $$ for all i ∈ I,

• (ii)

  $$ for all i ∈ I.

Then, there exists $$ such that $$ for all i ∈ I.

Theorem 7. Let $$ be a family of GFC-spaces and X = ∏_i∈I  X_i a Hausdorff space. For each i ∈ I, let $$, $$, and $$ with the following properties:

• (i)

  for each x ∈ X, $$, and $$, one has $$ for all i ∈ I,

• (ii)

  for each compact subset K of X and each i ∈ I, $$;

• (iii)

  there exists a nonempty compact subset K_i of X_i and for each N_i ∈ 〈Y_i〉, there exists an S_i-subset $$ of Y_i containing N_i with $$ being compact such that

  $$ ()

where $$, K = ∏_i∈IK_i, and $$.

Then, there exists $$ such that $$ for all i ∈ I.

Proof. As K is a compact subset of X, by the condition (ii), there exists a finite set $$, such that

By the condition (iii), there exists an S_i-subset $$ of Y_i containing N_i such that and it follows that We observe that the family $$ is a family of GFC-space and $$ is compact for each i ∈ I, defining set-valued mapping $$ and $$ as follows: We check assumptions (i) and (ii) of Lemma 6 for replaced G_i and F_i by $$ and $$, respectively. By (i) and the definition of S-subset, for each $$, each $$ and each $$, we have then assumption (i) of Lemma 6 is satisfied.

By (4), we have

On the other hand, for all $$, Hence, Thus, (ii) of Lemma 6 is also satisfied. According to Lemma 6, there exists a point $$ such that $$ for all i ∈ I.

Remark 8. Theorem 7 generalizes Theorem  3.4 of Ding [6] from FC-space to GFC-space, and our condition (iii) is different from its condition (iii). Theorem 7 also extends Theorem  3 in [18]. Note that Theorem 7 is the variation of Theorem  3.2 in [8].

As a special case of Theorem 7, we have the following fixed point theorem that will be used to prove a weak KKM theorem in Section 4.

Corollary 9. Let X be the Hausdorff space, (X, Y, {φ_N}) a GFC-space, G : X → 2^X, F : X → 2^Y, and S : Y → 2^X with the following properties:

• (i)

  for each x ∈ X, N = {y₀, y₁, …, y_n}⊆Y, and $$, one has φ_N(Δ_k)⊆G(x),

• (ii)

  for each compact subset K of X, K⊆⋃_y∈Y int F⁻¹(y),

• (iii)

  for each N ∈ 〈Y〉, there exists an S-subset L_N of Y containing N with S(L_N) being compact such that

  $$ ()

Then, there exists $$ such that $$.

Theorem 10. Let X be a Hausdorff space, (X, Y, {φ_N}) a GFC-space, Z a nonempty set, T : X → 2^Z, H : Y → 2^Z, and S : Y → 2^X; assume that

• (i)

  H is a weak T-KKM mapping,

• (ii)

  for each y ∈ Y, the set {x ∈ X : T(x)∩H(y) ≠ ∅} is compactly closed,

• (iii)

  there exists a compact K of X, and, for any N ∈ 〈Y〉, there exists an S-subset L_N of   Y containing N with S(L_N) being compact such that

  $$ ()

Then, there exists a point $$ such that $$ for each y ∈ Y.

Proof. Define F : X → 2^Y and G : X → 2^X by

Suppose the conclusion does not hold. Then, for each x ∈ X, there exists a y ∈ Y such that

It is easy to see that F has nonempty values. By (ii), for each y ∈ Y, is compactly open. Then, Since K is a compact subset of X, then there exists N ∈ 〈Y〉 such that Then, assumption (ii) of Corollary 9 is satisfied.

It follows from (iii) that there exists a compact K of X and for any N ∈ 〈Y〉, there exists a S-subset L_N of Y containing N with S(L_N) being compact such that

Therefore, assumption (iii) of Corollary 9 is also satisfied.

Furthermore, G has no fixed point. Indeed, if x ∈ G(x), then there exists y ∈ F(x) such that

which contracts the definition of F. Thus, assumption (i) of Corollary 9 must be violated; that is, there exist an $$, $$, and such that That is, for each $$, Hence,

On the other hand, since H is a weak T-KKM mapping and $$, we have

which is contradict. This completes the proof.

Remark 11. (1) Theorem 10 extends Theorem  1 in [13] from the G-convex space to GFC-space, and our proof techniques are different. Theorem 10 also generalizes Theorem  4.1 of [8] from the compactness assumption to noncompact situation.

(2) If Z is a topological space, condition (ii) in Theorem 10 is fulfilled in any of the following cases (see [13]):

• (i)

  H has closed values, and T is u.s.c, on each compact subset of X.

• (ii)

  H has compactly closed values, and T is u.s.c, on each compact of subset of X and its values are compact.

Theorem 12. Let X be a Hausdorff space, (X, Y, {φ_N}) a GFC-space, Z a topological space, T : X → 2^Z u.s.c., f : Y × Z → R ∪ {−∞, +∞}. and S : Y → 2^X; assume that

• (i)

  for each y ∈ Y, f(y, ·) is u.s.c. on each compact subset of Z,

• (ii)

  f is (λ, T, g)-GFC quasiconvex for all λ < β sufficiently close to β,

• (iii)

  there exists a compact K of X, and, for any N ∈ 〈Y〉, there exists an S-subset L_N of  Y containing N with S(L_N) being compact such that

  $$ ()

Then,

Proof. Let λ < β be arbitrary. By Lemma 5 and condition (ii), H_λ is a weak T-KKM mapping. It follows from condition (i) that H_λ has closed values. Hence, the set {x ∈ X : T(x)∩H_λ(y) ≠ ∅} is compactly closed for all y ∈ Y (see Remark 11 (2)). Thus, all the conditions of Theorem 10 are satisfied, and so there exists an $$ such that

This implies that $$ and so Since λ < β is arbitrary, we get the conclusion. This completes the proof.

Remark 13. Theorem 12 improves Theorem  4.2 of [8] from the compactness assumption to noncompact situation. Theorem 12 also extends Theorem  4 of [12] from compact G-convex space to noncompact GFC-space. Our result includes corresponding earlier Fan-type minimax inequalities due to Tan [19], Park [20], Liu [21], and Kim [22].