Knaster-Kuratowski-Mazurkiewicz Theorem in Generalized Metric Spaces with Applications

We study Knaster-Kuratowski-Mazurkiewicz theorem in the setting of generalized metric spaces. We establish some results on fixed points of Knaster-Kuratowski-Mazurkiewicz (KKM) mappings. Fan’s matching and Schauder’s type fixed point theorem in generalized metric spaces are also proved as interesting consequences of our main results. Examples are given to validate our results. We use these results to prove existence result for a given Atangana-Baleanu-Caputo fractional boundary value problem.


Introduction
In the large spectrum of mathematical problems, a fixed point of a specific map provides the solution of a mathematical problem. Therefore, fixed point theory is of pronounced significance in numerous fields of mathematics and other disciplines of science and engineering. Fixed point results describe the conditions under which a mathematical problem has a solution. Some fixed point results can be seen in [1][2][3].
The notion of metric space is fundamental in mathematics and has a major role in understanding and applying the topological concepts in different domains of analysis. This idea has pulled a substantial consideration from mathematicians owing to the notion of fixed point theory in metric spaces. Numerous extension and generalizations of the notion of metric space have been done in the literature. Czerwik [4] in 1993 presented the idea of b-metric spaces.
In 1998, Czerwik [5] reintroduced this idea in which the constant 2 associated with the triangular inequality was replaced by a constant k ≥ 1. In 2010, Khamsi and Hussain [6] generalized the idea of b-metric and named it metric type spaces. After that Hitzler and Seda [7] in 2000 gave the idea about dislocated metric spaces in which distance of a point from itself may or may not be zero. The idea of generalized metric spaces was given by Jleli and Samet [8], which covers distinctive notable structures including metric type spaces, metric spaces, and dislocated metric spaces, among others. KKM map was presented by Knaster et al. [9] in 1929 and it established the framework for some notable existing results like Ky Fan Browder's fixed point theorem, Nash's equilibrium theorem, and Ky Fan's minimax inequality theorem [10][11][12][13][14]. The Fan's theorem [15] is a significant result for KKM mappings and is being implemented as a useful technique in the modern nonlinear analysis. The first endeavor to stretch out these types of theorems in metric spaces was done in [16], where the author studied the case of hyperconvex metric spaces. Using the work of Chang and Yen [17] and the idea of Khamsi [16], Amini et al. introduced KKM mappings in metric spaces [18]. After that, Khamsi and Hussain [6] extended those results (of [18]) in the settings of metric type spaces.
In Section 2 of this article, we give some basic definitions and notions for the sake of completeness. We also define and study open set in generalized metric spaces and show that an open ball needs not to be an open set in general. In Section 3, we extend the theorems and the related results of [6] to generalized metric spaces. We also prove some results related to the fixed points of KKM mappings in generalized metric spaces in this section. Fan's matching and Schauder's type fixed point theorem as interesting consequences of our main results are furnished in Section 4.

Basic Definitions and Results
In this section, we recall some basic definitions, which we use to prove our main results.
Definition 1 (see [8]). Let W ≠ ϕ and d : W × W ⟶ ½0,∞ be the given mapping. For every λ ∈ W, define the following set: Definition 2 (see [8]). Let W ≠ ϕ and d : W × W ⟶ ½0,∞ be the given mapping. Then, d is known as a generalized metric on W, if for all β, λ ∈ W, the following conditions are satisfied: (1) dðλ, βÞ = dðβ, λÞ The pair (W, d) is known as a generalized metric space. Clearly, if the set Cðd, W, λÞ = ϕ for every λ ∈ W, then (W, d) is a generalized metric space if and only if Equations (35) and (44) are satisfied.
Definition 3 (see [8]). Let fλ n g be the sequence in the generalized metric space (W, d). Then, fλ n g is d-Cauchy sequence if Definition 4 (see [8]). Consider the generalized metric space (W, d) and λ ∈ W. The sequence fλ n g in W is d which converges to λ if Recall that open and closed balls in the generalized metric space (W, d) are, respectively, defined as Bðα, rÞ = fβ ∈ W : dðα, βÞ < rgand B½α, r = fβ ∈ W : dðα, βÞ ≤ rgfor any α ∈ W and r > 0. Now, we define open set in a generalized metric space similar as defined in [6].
In this case, we denote κ ∈ Y ∘ and read it as κ which is the interior point of Y. The collection of all such subsets of W will be denoted by τ, which defines topology on (W, d).
The complement of an open set is called closed set and if κ ∈ Y is called closure of Y; then, κ ∈ Y or there exists fκ n g ∈ Y such that fκ n g ∈ Cðd, W, κÞ.
Remark 6. It is not necessary for an open ball in a generalized metric space to be an open set.
Then, (W, d) is the generalized metric space. Now, the open ball having center α ∘ = 1/2 and radius δ = 1, denoted by the set A, is given by the following: Choose 1 = γ ∈ A and δ ′ > 0, then For any δ ′ > 0, we have infinite many α ∈ P such that α ∈ Bð1, δ′Þ. So, there does not exist δ′ > 0 satisfying Thus, the open ball A = Bð1/2, 1Þ is not an open set.
Definition 7. Consider the nonempty subset Y of the generalized metric space (W, d). Then, Y is referred as sequentially compact if there exists convergent subsequence fα n k g for every sequence

Main Results
We start with some useful notions which are essential to establish our main results. Let Y and Z be two topological spaces and G : Y ⟶ 2 Z be the nonempty set-valued mapping, where 2 Z represents the collection of all nonempty subsets of Z.
The set-valued mapping G : Y ⟶ 2 Z is referred to be as follows: 2 Journal of Function Spaces The set of all nonempty finite subsets of a set W is denoted by <W > . For a nonempty bounded subset Y of the generalized metric space ðW, dÞ, we define the following: for any A ∈ hYi. More generally, for the topological space Z, consider the two set-valued mappings G : Y ⟶ 2 Z and H : Y ⟶ 2 Z such that for any A ∈ hYi; in this case, G is referred as a generalized KKM mapping with reference to H. If the set-valued mapping H : Y ⟶ 2 Z satisfies the condition that for any generalized KKM mapping G : Y ⟶ 2 Z with reference to H, the class fGðyÞ,y ∈ Yg has finite intersection property, then H has KKM property, and we write it as follows: Consider the generalized metric space ðW, dÞ and ϕ ≠ Y ⊂ W. Then, H : Y ⟶ 2 W is called to have approximate fixed point property if for any ε > 0, there exists y ∈ Y such that We now present approximate fixed point property of KKM type mapping on subadmissible subset of a generalized metric space and generalize the main results of [6,18]. Theorem 8. Consider the nonempty subadmissible subset Y of the generalized metric space ðW, dÞ. Let H ∈ KKMðY, YÞ be such that HðYÞ is totally bounded. Then, H has an approximate fixed point property.
where Z is totally bounded. Thus, for any ε > 0, Y has a finite subset C such that where Bðc, εÞ is an open ball having radius ε and center c. Now, we define a map G : Y ⟶ 2 Y by the following: where S represents the constant associated with inequality and B c ðy, εÞ denotes the complement of Bðy, εÞ in W for any ε > 0 and y ∈ Y . Obviously, GðyÞ is closed. Now we prove ∩ c∈C GðcÞ = ϕ. On contrary assume that ∩ c∈C GðcÞ ≠ ϕ, then we have the following: So, there exists fκ n g ∈ B c ðc, SεÞ such that dðκ n , κÞ < ε for all ε > 0 and n ≥ N. So, Thus, Now, choose fz m n g ∈ Bðκ n , εÞ ∩ Bðc, εÞ for all ε > 0 and n ≥ N. Then, fz m n g ∈ Cðd, W, cÞ and which contradicts to fκ n g ∈ B c ðc, SεÞ. Thus, we have ∩ c∈C G ðcÞ = ϕ. Hence, G is not a generalized KKM mapping with reference to H. As H ∈ KKMðY, YÞ, so there is a finite nonempty subset D ⊆ Y such that i.e., we have y ∘ ∈ HðcoðDÞÞ such that y ∘ ∉ GðρÞ for any ρ ∈ D.
As y ∘ ∉ GðρÞ = Z ∩ B c ðρ, SεÞ, so 3 Journal of Function Spaces for any ρ ∈ D. Now, for any ρ ∈ D. We may write it as D ⊆ Bðy ∘ , SεÞ. As For y ∘ ∈ HðcoðDÞÞ, we have y ε ∈ coðDÞ such that y ∘ ∈ Hðy ε Þ. And Thus, we have the following: As ε is arbitrary, so, H has an approximate fixed point property.

Applications of KKM Maps
As the consequence of Theorem 8, we deduce the following fixed point theorem.

Theorem 9.
Consider the nonempty subset Y which is subadmissible in the generalized metric space ðW, dÞ and H ∈ KKMðY, YÞ be such that H is compact and closed. Then, H has a fixed point.
Proof. As H is compact, hence, HðYÞ is compact. So, HðYÞ is totally bounded. Hence by Theorem 8, H has an approximate fixed point property; i.e., for any ε > 0, there exists y ε ∈ Y such that In particular for n ≥ 1 and ε = 1/n, we have y n ∈ Y such that Since (z n ) is a sequence in HðYÞ for n ≥ 1 and HðYÞ is compact, so, there exists convergent subsequence ðz n k Þ in HðYÞ and suppose it converges to z.
Also for n ≥ 1, we have the following: As the sequence ðz n k Þ in ðW, dÞ is convergent to z, so, there exists S > 0 such that So, (y n ) also converges to z.
Since fðy n , z n Þg ∈ GrðHÞ and H is closed, so, z ∈ HðzÞ, which is our required result.
As d is continuous in first variable from Remark 5.1 in [19], Proposition 3.11 in [20] states that closed ball in ð½0,∞Þ, dÞ is a closed set. Define H : Y ⟶ 2 Y by the following: Then, H is a KKM map and H ∈ KKMðY, YÞ. Also, Hð YÞ is closed and bounded and clearly HðYÞ = ½0, 1 is totally bounded. So, by Theorem 8, H has an approximate fixed point property. Further, H is closed and compact, so by Theorem 9, H has a fixed point.
The next result will be helpful to present Schauder's type fixed point theorem for generalized metric spaces.

Lemma 10.
Let Y be the nonempty subadmissible subset of the generalized metric space ðW, dÞ and Z be the topological space. Suppose that g : Z ⟶ Y is continuous and H ∈ KKMðY, ZÞ. Then, g ∘ H ∈ KKMðY, YÞ.
Proof. Consider the generalized KKM mapping G : Y ⟶ 2 Y with reference to g ∘ H such that GðyÞ is closed for every y ∈ Y. Since G is a generalized KKM mapping with reference to g ∘ H, so, for any nonempty finite subset A of Y, we have the following: Thus, g -1 ðGÞ is a generalized KKM mapping with reference to H.
Also, g is continuous and Thus, the collection fGðyÞ, y ∈ Yg has finite intersection property, which gives g ∘ H ∈ KKMðY, YÞ.

Journal of Function Spaces
As the consequence of Theorem 9 and Lemma 10, we obtain Schauder's type fixed point theorem in generalized metric spaces.
Theorem 11. Let Y be the nonempty subadmissible subset of the generalized metric space ðW, dÞ. Suppose that I ∈ KKM ðY, YÞ. Then any continuous map S : Y ⟶ Y such that SðYÞ is compact has a fixed point.
Proof. From Lemma 10, we have the following: As S is continuous and SðYÞ is compact, so S is closed and compact. Hence, from Theorem 9, S has a fixed point. Now, we present the generalized Fan's matching theorem in generalized metric spaces by using KKM property.
Proof. Assume that H ∈ KKMðY, ZÞ and define the multivalued map G : Y ⟶ 2 Z by GðyÞ = HðYÞ ∩ K c ðyÞfor y ∈ Y. Then, GðyÞ is closed for every y ∈ Y. On the contrary, assume that HðcoðAÞÞ ∩ ð ∩ y∈A KðyÞÞ = ϕ for any A ∈ hYi. Since A ∈ hYi and Y is admissible, so, Hence, G is a generalized KKM mapping with reference to H. As H ∈ KKMðY, ZÞ, thus, the class fGðyÞ: y ∈ Yg has a finite intersection property. So, which is contrary to the fact HðYÞ ⊆ KðYÞ. So, there exists A ∈ hYi such that Now, we present an application of Theorem 11 to find the existence of solutions to the following AB-Caputo fractional BVP: with boundary conditions where ABC 0 D α represents the AB-Caputo fractional derivative and g : ½0, 1 × ℝ ⟶ ℝ. Also, λ, γ > 0, 0 ≤ t ≤ η ≤ 1.
The following lemma will be crucial for the proof of our next result. Lemma 16. Assume that K : ½0, 1 ⟶ ℝ is continuous. Then, the solution of linear AB-Caputo fractional BVP with boundary Equation (45) is given by the following: