Adv Syst Sci Appl 2020; 01:91–103 Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/854 On Properties of Coincidence Points of Mappings between (q1, q2)-Quasimetric Spaces Richik Sengupta1, Zukhra T. Zhukovskaya2, Sergey E. Zhukovskiy2,3* 1Peoples’ Friendship University of Russia. 2 V.A. Trapeznikov Institute of Control Sciences of RAS. 3 Moscow Institute of Physics and Technology. Abstract: In this paper, the properties of coincidence points of mappings acting between (q1, q2)- quasimetric spaces are studied. For a pair of mappings, we obtain estimates for the distance from a point to the coincidence points set and intersection of the respective graphs of the mappings. In addition, the stability of coincidence points is studied. A generalization of Lim’s lemma is obtained. Keywords: (q1, q2)-quasimetric spaces, coincidence points, set-valued mappings 1. INTRODUCTION AND STATEMENT OF THE PROBLEM The paper is devoted to the investigation of coincidence points of pairs of set-valued mappings acting between (q1, q2)-quasimetric spaces. In order to proceed to the statement of the problem, let us recall the definitions of the concepts in use. Let X be a nonempty set, numbers q0 ≥ 1, q1 ≥ 1, q2 ≥ 1 be given. A function ρX : X ×X → R+ is called a (q1, q2)-quasimetric if • ρX(x, y) = 0⇔ x = y ∀ x, y ∈ X (the identity axiom); • ρX(x, z) ≤ q1ρX(x, y) + q2ρX(y, z) ∀x, y, z ∈ X (the (q1, q2)-generalized triangle inequality). If ρX is a (q1, q2)-quasimetric, then the space (X, ρX) is called a (q1, q2)-quasimetric space. The concept of the (q1, q2)-quasimetric space was introduced in [1]. If q1 = q2 = 1 then this concept coincides with the concept of a quasimetric space. If we additionally assume that a quasimetric satisfies the symmetry axiom, i.e. ρX(x, y) ≡ ρX(y, x), it becomes a metric. The detailed description of topological properties of (q1, q2)-quasimetric spaces was provided in [2]. Recall some basic definitions. A sequence {xi} ⊂ X is said to converge to a point x ∈ X if ρX(x, xi)→ 0 as i→∞. The point x is called a limit of {xi}. A subset of X is said to be closed if every limit of every convergent sequence from this set belongs to this set. A sequence {xi} ⊂ X is said to be a Cauchy sequence if ∀ ε > 0 ∃N ∈ N : ρX(xj, xi) < ε ∀ i > j > N. Let (X, ρX) and (Y, ρY ) be (q1, q2)-quasimetric spaces, Φ,Ψ : X ⇒ Y be set-valued mappings that map points x ∈ X to non-empty subsets of Y. ∗Corresponding author: s-e-zhuk@yandex.ru 92 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY A point ξ ∈ X is called a coincidence point of the set-valued mappings Φ,Ψ if Φ(ξ) ∩Ψ(ξ) 6= ∅. Below we use the following notation: Coin(Ψ,Φ) stands for the set of all coincidence points of mappings Ψ and Φ; gphΨ stands for the graph of Ψ, i.e. gphΨ = {(x, y) ∈ X × Y : y ∈ Ψ(x)}; Γ(Ψ,Φ) stands for the intersection of the graphs of Ψ and Φ, i.e. Γ(Ψ,Φ) := {(x, y) ∈ X × Y : y ∈ Ψ(x) ∩ Φ(x)}. It is obvious that Coin(Ψ,Φ) 6= ∅ ⇔ Γ(Ψ,Φ) 6= ∅. For non-empty sets U, V ⊂ X, denote dist(U, V ) = inf{ρX(x1, x2) : x1 ∈ U, x2 ∈ V }, h+ X(U, V ) = sup u∈U dist(u, V ), hX(U, V ) := max{h+ X(U, V ), h+ X(V, U)}. The function h+ X is called the Hausdorff deviation; the function hX is called the Hausdorff (q̂1, q̂2)-quasimetric (note that (q̂1, q̂2) may differ from (q1, q2)). For Hausdorff deviation h+ X , the (q1, q2)-generalized triangle inequality holds (see [3, page 25]), i.e. h+ X(U,W ) ≤ q1h + X(U, V ) + q2h + X(V,W ) ∀U, V,W ⊂ X. Moreover, the definitions above directly imply that hX(U, V ) ≥ h+ X(U, V ) ≥ dist(U, V ) ∀U, V ⊂ X. Let us recall now some definitions related to set-valued mappings. Definition 1.1: ( [3, Definition 5.5]) Given a number β ≥ 0, the set-valued mapping Φ : X ⇒ Y is called β-Lipschitz if h(Φ(x1),Φ(x2)) ≤ βρX(x1, x2) ∀x1, x2 ∈ X. If (X, ρX) = (Y, ρY ) and β < 1 then the β-Lipschitz set-valued mapping Φ is said to be a contraction. Definition 1.2: ( [3, Definition 4.4]) The set-valued mapping Φ : X ⇒ Y is said to be closed, if for all sequences {xi} ⊂ X, {yi} ⊂ Y and points x ∈ X, y ∈ Y such that xi → x, yi → y and (xi, yi) ∈ gph(Φ) for all i, we have (x, y) ∈ gph(Φ). Definition 1.3: We will say that the graph of the set-valued mapping Φ : X ⇒ Y is complete, if for all Cauchy sequences {xi} ⊂ X and {yi} ⊂ Y such that {(xi, yi)} ⊂ gph(Φ), there exists a point (x, y) ∈ gph(Φ) such that xi → x and yi → y. Denote by BX(x0, r) a closed ball in X centered at x ∈ X with the radius r > 0, i.e. BX(x0, r) = {x ∈ X : ρX(x0, x) ≤ r}. Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 93 Definition 1.4: ( [3, definition 5.4]) Given a number α > 0, the set-valued mapping Ψ : X ⇒ Y is called α-covering if ⋃ y∈Ψ(x) BY (y, αr) ⊆ Ψ(BX(x, r)) ∀r ≥ 0,∀x ∈ X. Let us now recall the coincidence point existence theorem from [3]. Let numbers α > 0, β ∈ [0, α) and set-valued mappings Ψ,Φ : X ⇒ Y be given. Denote MΨ,Φ(x, r) := {y ∈ Φ(x) : dist(Ψ(x), y) < r}, x ∈ X, r > 0, S(θ, n) := 1− θn 1− θ , θ ∈ [0, 1), n = 0, 1, 2... , m0 := min { j ∈ N : q2 ( β α )j < 1 } . Theorem 1.1: ( [3, Theorem 5.7]) Let numbers α > 0 and β ∈ [0, α) be given. Assume that • the set-valued mapping Ψ : X ⇒ Y is α-covering and its graph is closed; • set-valued mapping Φ : X ⇒ Y is β-Lipschitz; • at least one of the graphs gph(Ψ) or gph(Φ) is complete. Then for all x0 ∈ X, r0 > dist(Ψ(x0),Φ(x0)), y1 ∈MΨ,Φ(x0, r0) there exists ξ ∈ X such that Ψ(ξ) ∩ Φ(ξ) 6= ∅, lim λ→ξ ρX(x0, λ) ≤ q2 1α m0−1S(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0. This assertion not only provides the sufficient conditions for the existence of a coincidence point but also an estimate of the distance from a point x0 ∈ X to a coincidence point ξ of the given mappings. A problem to obtain an estimate of distance from the point y1 to a point η ∈ Ψ(ξ) ∩ Φ(ξ) naturally arises. For the case when (X, ρX) and (Y, ρY ) are metric spaces, this problem was solved in [4, 5]. The main goal of our paper is to obtain results analogous to those in [4,5] for set-valued mappings acting between (q1, q2)-quasimetric spaces. We also discuss a similar problem for fixed points of set-valued mappings and derive propositions on fixed points properties similar to those in [6]. The results of this paper may have applications in the investigation of various nonlinear equations. One of the possible applications of the results is the investigation of nonlinear equations in Banach spaces equipped with an additional bimodule structure over a group ring based on the theory developed in [7]. Note that the results on coincidence points and their analogs (see, for example, [8], [9]) are applied in the study of equations appearing in economic models (see [10]), differential inclusions (see [11], [12], [13]), and other problems. 2. MAIN RESULTS. ESTIMATES OF DISTANCE FROM A POINT TO THE INTERSECTION SET OF TWO GRAPHS Let q1 ≥ 1, q2 ≥ 1 be given numbers, (X, ρX), (Y, ρY ) be (q1, q2)-quasimetric spaces. Given α > 0, β ∈ [0, α), denote byFα,β the set of all ordered pairs of set-valued mappings (Ψ,Φ), Ψ,Φ : X ⇒ Y, such that Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) 94 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY • the set-valued mapping Ψ : X ⇒ Y is α-covering; • set-valued mapping Φ : X ⇒ Y is β-Lipschitz; • either gph(Ψ) is complete and Φ(x) is a closed set for every x ∈ X or Ψ is closed and gphΦ is complete. Theorem 2.1: Let numbers α > 0, β ∈ [0, α) and an arbitrary ordered pair of set-valued mappings (Ψ,Φ) ∈ Fα,β be given. Then for all x0 ∈ X, r0 > dist(Ψ(x0),Φ(x0)), y1 ∈MΨ,Φ(x0, r0) there exist ξ ∈ X and η ∈ Y such that η ∈ Ψ(ξ) ∩ Φ(ξ), lim λ→ξ ρX(x0, λ) ≤ q2 1α m0−1S(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0, (2.1) lim κ→η ρY (y1, κ) ≤ β q2 1α m0−1S(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0. (2.2) Before proving Theorem 2.1, let us prove the following lemma. Lemma 2.1: Let the set-valued mapping Ψ : X ⇒ Y be α-covering, the set-valued mapping Φ : X ⇒ Y be β-Lipschitz. Then for arbitrary δ > 0, x0 ∈ X, y1 ∈MΨ,Φ(x0, αδ + dist(Ψ(x0),Φ(x0))) there exist sequences {xi} ⊂ X and {yi} ⊂ Y such that ρX(x0, x1) ≤ δ + dist(Ψ(x0),Φ(x0)) α , (2.3) ρX(xi−1, xi) ≤ (β α + δ ) ρX(xi−2, xi−1) ∀i ≥ 2, (2.4) yi ∈ Ψ(xi) ∩ Φ(xi−1) ∀i ≥ 1, (2.5) ρY (yi−1, yi) ≤ (β + αδ)ρX(xi−2, xi−1) ∀i ≥ 2. (2.6) Proof Let us take an arbitrary x0 ∈ X and δ > 0. Set r0 = αδ + dist(Ψ(x0),Φ(x0)). Let us take an arbitrary point y1 ∈MΨ,Φ(x0, r0). Since the mapping Ψ is α-covering, there exists a point x1 ∈ BX(x0, r0) such that y1 ∈ Ψ(x1). Therefore, y1 ∈ Ψ(x1) ∩ Φ(x0). Let us construct the sought sequences by induction. If x0 = x1 then set x2 := x1, y2 := y1. Assume that x0 6= x1. Set r1 := (β + αδ)ρX(x0, x1). Since the mapping Φ is β-Lipschitz, we have hY (Φ(x0),Φ(x1)) < r1. Therefore, there exists a point y2 ∈ Φ(x1) such that ρY (y1, y2) < r1. Since y1 ∈ Ψ(x1), we have y2 ∈ ⋃ y∈Ψ(x1) BY (y, r1). Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 95 Therefore, since the set-valued mapping Ψ is α-covering, there exists a point x2 ∈ BX(x1, r1) such that y2 ∈ Ψ(x2) and ρX(x1, x2) ≤ r1 α . The sought x2, y2 are constructed. Let us now assume that for a certain j, the sought points xi, yi, i = 1, j, are constructed. Let us construct xj+1, yj+1. If xj = xj−1 then set xj+1 := xj , yj+1 := yj . Assume that xj 6= xj−1. Set rj := (β + αδ)ρX(xj−1, xj). Since the mapping Φ is β-Lipschitz, we have hY (Φ(xj−1),Φ(xj)) < rj. Therefore, there exists a point yj+1 ∈ Φ(xj) such that ρY (yj, yj+1) ≤ rj. Since yj ∈ Ψ(xj), we have yj+1 ∈ ⋃ y∈Ψ(xj) BY (y, rj). Since the set-valued mapping Ψ is α-covering, there exists a point xj+1 ∈ BX(xj, rj) such that yj+1 ∈ Ψ(xj+1) and ρX(xj, xj+1) ≤ rj α . The sought xj+1, yj+1 are constructed. Proof of Theorem 2.1. Without loss of generality, we assume α = 1. Take a δ > 0 such that min { j ∈ N : q2 ( β α + δ )j < 1 } = m0, r0 > dist(Ψ(x0),Φ(x0)) + δ. Let us consider the corresponding sequences {xi} and {yi}, that were constructed in Lemma 2.1. Let us show that {xi} is a Cauchy sequence. For integers i, j ≥ 0, we have ρX(xi, xi+j) ≤ q1ρX(xi, xi+1) + q2ρX(xi+1, xi+j) ≤ ≤ q1r0(β + δ)i + q2 ( q1ρX(xi+1, xi+2) + q2ρX(xi+2, xi+j) ) ≤ ≤ q1r0(β + δ)i + q1q2r0(β + δ)i+1 + q2 2 ( q1ρX(xi+2, xi+3) + q2ρX(xi+3, xi+j) ) ≤ ≤ · · · ≤ q1r0(β + δ)i ( 1 + q2(β + δ) + · · ·+ qj−2 2 (β + δ)j−2 + qj−1 2 (β + δ)j−1q−1 1 ) = = q1r0(β + δ)iS̃(j). Here S̃(j) = S(q2(β + δ), j − 1) + qj−1 2 (β + δ)j−1q−1 1 , j ∈ N, S̃(0) = 0. Thus, for any non-negative integer i and k, we have ρX(xi, xi+k) ≤ q1ρX(xi, xi+m0) + q2ρX(xi+m0 , xi+k) ≤ ≤ q1ρX(xi, xi+m0) + q2 ( q1ρX(xi+m0 , xi+2m0)+ +q2ρX(xi+2m0 , xi+k) ) ≤ q1ρX(xi, xi+m0) + q1q2ρX(xi+m0 , xi+2m0)+ +q2 2 ( q1ρX(xi+2m, xi+3m0) + q2ρX(xi+3m0 , xi+k) ) ≤ Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) 96 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY ≤ q1ρX(xi, xi+m0) + q1q2ρX(xi+m0 , xi+2m0) + q1q 2 2ρX(xi+2m0 , xi+3m0) + q2 3ρX(xi+3m0 , xi+k) ≤ ≤ · · · ≤ q1ρX(xi, xi+m0) + q1q2ρX(xi+m0 , xi+2m0)+ + · · ·+ q1q p−1 2 ρX(xi+(p−1)m0 , xi+pm0) + qp2ρX(xi+pm0 , xi+k) ≤ ≤ q2 1r0β iS̃(m0) ( 1 + q2(β + δ)m0 + q2 2(β + δ)2m0 + · · ·+ qp−1 2 (β + δ)(p−1)m0 ) + +qp2q1r0(β + δ)i+pm0S̃(k − pm0) = = q2 1r0(β + δ)iS̃(m0)S(q2(β + δ)m0 , p) + qp2(β + δ)i+pm0q1r0S̃(k − pm0), where p is the integer part of k/m0. Since, q2(β + δ)m0 < 1, we have ρX(xi, xi+k) ≤ ≤ q2 1r0(β + δ)i ( S̃(m0)S(q2(β + δ)m0 , p) + qp2r0(β + δ)pm0q−1 1 S̃(k − pm0) ) ≤ ≤ q2 1r0(β + δ)i ( S̃(m0) 1− q2(β + δ)m0 + q−1 1 S̃(k − pm0) ) . (2.7) Since, 0 ≤ k − pm0 ≤ m0, the value q−1 1 S(k − pm0) is uniformly bounded for all k. Therefore, {xi} is a Cauchy sequence. Let us show that {yi} is also a Cauchy sequence. According to the lemma, ρY (yi+1, yi+j+1) ≤ (β + δ)ρX(xi, xi+j) for every i and j. Therefore, repeating the arguments above, we get ρX(yi+1, yi+k+1) ≤ (β + δ)[q2 1r0(β + δ)i ( S̃(m0)S(q2(β + δ)m0 , p)+ +qp2(β + δ)pm0q−1 1 S̃(k − pm0) ) ] ≤ ≤ q2 1r0(β + δ)i+1 ( S̃(m0) 1−q2(β+δ)m0 + q−1 1 S̃(k − pm0) ) . (2.8) Therefore, {yi} is a Cauchy sequence. Consider now two cases. At first, assume that the gph(Ψ) is complete and each value of Φ is a closed set. Then the Cauchy sequences {xi}, {yi} converge to points ξ ∈ X, η ∈ Ψ(ξ) respectively as (xi, yi) ∈ gph(Ψ). We have hY (Φ(xi),Φ(ξ)) = hY (Φ(ξ),Φ(xi)) ≤ βρX(ξ, xi), and thus h+ Y (yi+1,Φ(ξ)) ≤ βρX(ξ, xi). Therefore, h+ Y (η,Φ(ξ)) ≤ q1ρY (η, yi+1) + q2h + Y (yi+1,Φ(ξ)) ≤ q1ρY (η, yi+1) + q2βρX(ξ, xi). Since {yi+1} tends to η and {xi} tends to ξ, we have h+ Y (η,Φ(ξ)) = 0. This equality and the closedness of Φ(ξ) imply η ∈ Φ(ξ). Assume now that gph(Φ) is complete and Ψ is closed. Then the Cauchy sequences {xi}, {yi} converge to some points ξ ∈ X , η ∈ Φ(ξ) respectively, since (xi, yi+1) ∈ gph(Φ). The set gph(Ψ) is closed, therefore η ∈ Ψ(ξ). So, it is proved that η ∈ Φ(ξ) ∩Ψ(ξ). Passing to the limit in (2.7) as k → +∞ and putting i = 0 we obtain lim ζ→ξ ρX(x0, ζ) ≤ q2 1α m0S ( q2 β+αδ α ,m0 − 1 ) + q1q m0−1 2 (β + αδ)m0−1 αm0 − q2(β + αδ)m0 ( δ + dist(Ψ(x0),Φ(x0)) α ) Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 97 Hence, as the choice of δ is arbitrary, it implies (2.1). Analogously passing to the limit as k →∞ in (2.8) and substituting i = 0 by virtue of the choice of δ we obtain (2.2). 2 Let us now obtain an estimate of the distance from a point (x, y) ∈ X × Y to the set Γ(Ψ,Φ). Put K(m0) := q2 1α m0−1S(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 . For vectors z = (x, y) ∈ X × Y, A = (AX , AY ) ∈ R2 and a subset Γ ⊂ X × Y, we write D(z,Γ) ≤ A if ∀ ε > 0 ∃(ξ, η) ∈ Γ : lim λ→ξ ρX(x, λ) ≤ AX + ε, lim κ→η ρY (y, κ) ≤ AY + ε. Theorem 2.2: Let α > 0 and β ∈ [0, α) be given. If (Ψ,Φ) ∈ Fα,β then the set Γ(Ψ,Φ) is non-empty and, moreover, for arbitrary x ∈ X, y ∈ Y, the inequality D ( (x, y),Γ(Ψ,Φ) ) ≤ A(x, y, yφ) ∀ yφ ∈ Φ(x), (2.9) holds. Here A(x, y, yφ) := ( K(m0)dist(Ψ(x), yφ), q1ρY (y, yφ) + q2βK(m0)dist(Ψ(x), yφ) ) . Proof Theorem 2.1 implies that Γ(Ψ,Φ) 6= ∅. Let us take an arbitrary x ∈ X, y ∈ Y, yφ ∈ Φ(x), ε > 0. Set r := dist(Ψ(x), yφ). The definition of dist(·, ·) implies dist(Ψ(x),Φ(x)) ≤ r. Therefore, yφ ∈MΨ,Φ(x, r + ε). It follows from Theorem 2.1 that there exist ξ ∈ X and η ∈ Y such that (ξ, η) ∈ Γ(Ψ,Φ) and lim λ→ξ ρX(x, λ) ≤ K(m0)(r + ε), lim κ→η ρY (yφ, κ) ≤ βK(m0)(r + ε). Since ρY (y, κ) ≤ q1ρY (y, yφ) + q2ρY (yφ, κ), we have lim κ→η ρY (y, κ) ≤ q1ρY (y, yφ) + q2 lim κ→η (yφ, κ) ≤ q1ρY (y, yφ) + q2βK(m0)(r + ε). The arbitrariness of ε > 0 implies that (2.9) holds. Theorem 2.3: Let α > 0 and β ∈ [0, α) be given. If (Ψ,Φ) ∈ Fα,β then the set Γ(Ψ,Φ) is non-empty and, moreover, for arbitrary x ∈ X, y ∈ Y, the inequality D((x, y),Γ(Ψ,Φ)) ≤ A(x, y) (2.10) holds. Here A(x, y) := ( K(m0)(r1 + r2), q1r2 + q2βK(m0)(q1r1 + q2r2) ) , r1 := dist(Ψ(x), y), r2 := dist(y,Φ(x)). Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) 98 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY Proof Take arbitrary points x ∈ X, y ∈ Y, yψ ∈ Ψ(x), yφ ∈ Φ(x) and a number ε > 0 such that ρY (yψ, y) < dist(Ψ(x), y) + ε 2 , ρY (y, yφ) < dist(y,Φ(x)) + ε 2 . Set ε1 := ε(q1 + q2) 2 , r := q1r1 + q2r2. Let us show that yφ ∈MΨ,Φ(x, r + ε1). Indeed, by the assumption yφ ∈ Φ(x). Moreover, ρY (yψ, yφ) < q1r1 + q2r2 + ε(q1 + q2) 2 ≤ r + ε1 and dist(Ψ(x),Φ(x)) ≤ dist(Ψ(x), yφ) ≤ ρY (yψ, yφ). Hence, dist(Ψ(x), y) < r + ε1. Therefore, yφ ∈MΨ,Φ(x, r + ε1). Theorem 2.1 implies that there exist ξ ∈ X and η ∈ Y such that (ξ, η) ∈ Γ(Ψ,Φ) and lim λ→ξ ρX(x, λ) ≤ K(m0)(r + ε1), lim κ→η ρY (yφ, κ) ≤ K(m0)β(r + ε1). (2.11) Since ρY (y, κ) ≤ q1ρY (y, yφ) + q2(yφ, κ) the inequality (2.11) implies lim κ→η ρY (y, κ) ≤ q1ρY (y, yφ) + q2 lim κ→η (yφ, κ) ≤ q1r2 + q2K(m0)β(r + ε1). Moreover, it follows from (2.11) that lim λ→ξ ρX(x, λ) ≤ K(m0)(r + ε1). The arbitrariness of ε > 0 implies that the inequality (2.10) holds. Remark 2.1: In the case when (X, ρX) and (Y, ρY ) are metric spaces, Theorem 2.1 coincides with [4, Theorem 1] and Theorem 2.2 coincides with [4, Theorem 3]. 3. COROLLARIES. ESTIMATES OF DISTANCE BETWEEN INTERSECTIONS OF GRAPHS AND SETS OF COINCIDENCE POINTS Let q1 ≥ 1, q2 ≥ 1 be given numbers, (X, ρX), (Y, ρY ) be (q1, q2)-quasimetric spaces. Let us define one more function which characterizes a distance between subsets of (q1, q2)-quasimetric spaces. For U, V ⊂ X set e+(U, V ) := sup v∈V dist(U, v). Even though the definitions of e+ and h+ look quite similar, these functions are actually different as ρX is not necessarily symmetric. Let us describe some properties of the function e+. Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 99 Proposition 3.1: For arbitrary sets U, V, W ⊂ X, the following inequalities hold e+(U,W ) ≤ q1e +(U, V ) + q2e +(V,W ), (3.12) dist(U,W ) ≤ q1e +(U, V ) + q2e +(V,W ), (3.13) Proof Let us prove (3.12). Take arbitrary sets U, V, W ⊂ X, a point w ∈ W and a number ε > 0. According to the definition of e+ there exists v ∈ V and such that ρX(v, w) ≤ e+(V,W ). Moreover, there exists u ∈ U such that ρX(u, v) ≤ e+(U, V ). We have, dist(U,w) ≤ ρX(u,w) ≤ q1ρX(u, v) + q2ρX(v, w) ≤ q1e +(U, V ) + q2e +(V,W ). Due to the arbitrariness of w ∈ W, the above inequality implies (3.12). Inequality (3.13) follows from (3.12) as dist(U,W ) ≤ e+(U,W ). We will also use the following inequality which was proved in ( [3, Property 5.1]): dist(U,W ) ≤ q1dist(U, V ) + q2h +(V,W ). (3.14) Theorem 3.1: Let α > 0 and β ∈ [0, α) be given. If (Ψ,Φ) ∈ Fα,β then for arbitrary (ξ̃, η̃) ∈ Γ(Ψ̃, Φ̃) and ε > 0 there exists (ξ, η) ∈ Γ(Ψ,Φ) such that lim λ→ξ ρX(ξ̃, λ) ≤ K(m0) ( q1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2h +(Φ̃(ξ̃),Φ(ξ̃)) ) +ε, lim κ→η ρY (η̃, κ) ≤ q1q2h +(Φ̃(ξ̃),Φ(ξ̃)) + q2βK(m0) ( q2 1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2 2h +(Φ̃(ξ̃),Φ(ξ̃)) ) + ε. Proof Fix an arbitrary pair (ξ̃, η̃) ∈ Γ(Ψ̃, Φ̃) and take an arbitrary ε > 0. Applying Theorem 2.3 to the mappings Φ and Ψ we obtain that there exists (ξ, η) ∈ Γ(Ψ,Φ) such that lim λ→ξ ρX(ξ̃, λ) ≤ K(m0)(r1 + r2) + ε, (3.15) lim κ→η ρY (η̃, κ) ≤ q1r2 + q2βK(m0)(q1r1 + q2r2) + ε. (3.16) Here r1 := dist(Ψ(ξ̃), η̃), r2 := dist(η̃,Φ(ξ̃)). It follows from (3.13) that dist(Ψ(ξ̃), η̃) ≤ q1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2e +(Ψ̃(ξ̃), η̃). Since η̃ ∈ Ψ̃(ξ̃), we have r1 = dist(Ψ(ξ̃), η̃) ≤ q1e +(Ψ(ξ̃), Ψ̃(ξ̃)). Since η̃ ∈ Φ̃(ξ̃), we have dist(η̃, Φ̃(ξ̃)) = 0. So, it follows from (3.14) that r2 = dist(η̃,Φ(ξ̃)) ≤ q2h +(Φ̃(ξ̃),Φ(ξ̃)). Hence, the desired inequalities follow from (3.15) and (3.16). Let us introduce one more notation. Given arbitrary sets Γ̃,Γ ⊂ X × Y and a vector A = (AX , AY ) ∈ R2, the notation H+(Γ̃,Γ) ≤ A Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) 100 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY means that for arbitrary (x, y) ∈ Γ̃ we have D((x, y),Γ) ≤ A. Let arbitrary set-value mappings Ψ,Φ, Φ̃, Ψ̃ : X ⇒ Y and numbers α > 0, β ∈ [0, α) be given. Set AX(x) := K(m0) ( q1e +(Ψ(x), Ψ̃(x)) + q2h +(Φ̃(x),Φ(x)) ) , AY (x) := q1q2h +(Φ̃(x),Φ(x)) + q2βK(m0) ( q2 1e +(Ψ(x), Ψ̃(x)) + q2 2h +(Φ̃(x),Φ(x)) ) , A(x) := (AX(x), AY (x)), x ∈ X. Given a set Ξ ⊂ X, denote sup ξ∈Ξ A(ξ) = {Λ ∈ R2 : Λ ≥ A(ξ) ∀ ξ ∈ Ξ}. Here the inequality is understood in the coordinate-wise sense. Theorem 3.2: Let α > 0 and β ∈ [0, α) be given. If (Ψ,Φ) ∈ Fα,β then H+(Γ(Ψ̃, Φ̃),Γ(Ψ,Φ)) ≤ Λ ∀Λ ∈ sup ξ̃∈Coin(Ψ̃,Φ̃) A(ξ̃). (3.17) If, in addition, ρX is lower semicontinuous with respect to the second argument, then h+(Coin(Ψ̃, Φ̃),Coin(Ψ,Φ)) ≤ sup ξ̃∈Coin(Ψ̃,Φ̃) AX(ξ̃) (3.18) Proof Take arbitrary Λ ∈ R2 such that Λ ≥ A(ξ) for all ξ̃ ∈ Coin(Ψ̃, Φ̃). Take arbitrary (ξ̃, η̃) ∈ Γ(Ψ̃, Φ̃). Theorem 3.1 implies that D((ξ̃, η̃),Γ(Ψ,Φ)) ≤ Λ. Hence, (3.17) is proved. Let us prove (3.18). Assume now that ρX is lower semicontinuous with respect to the second argument. Theorem 2.3 implies that for every pair (ξ̃, η̃) ∈ Γ(Ψ̃, Φ̃) and every ε > 0 there exists a point ξ ∈ Coin(Ψ,Φ) such that ρX(ξ̃, ξ) ≤ K(m0)(dist(Ψ(ξ̃), η̃) + dist(η̃,Φ(ξ̃))) + ε, (3.19) It follows from (3.13) that dist(Ψ(ξ̃), η̃) ≤ q1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2e +(Ψ̃(ξ̃), η̃). Since η̃ ∈ Ψ̃(ξ̃), we have dist(Ψ(ξ̃), η̃) ≤ q1e +(Ψ(ξ̃), Ψ̃(ξ̃)). Since η̃ ∈ Φ̃(ξ̃), we have dist(η̃, Φ̃(ξ̃)) = 0. Thus, (3.14) implies dist(η̃,Φ(ξ̃)) ≤ q2h +(Φ̃(ξ̃),Φ(ξ̃)). Substituting these estimates into (3.19) we obtain ρX(ξ̃, ξ) ≤ AX(ξ̃) + ε. Hence, h+(Coin(Ψ̃, Φ̃),Coin(Ψ,Φ)) = = sup ξ̃∈Coin(Ψ̃,Φ̃) dist(ξ̃,Coin(Ψ,Φ)) ≤ sup ξ̃∈Coin(Ψ̃,Φ̃) AX(ξ̃). Inequality (3.18) is proved. Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 101 Corollary 3.1: Let assumptions of Theorem 3.1 hold and ρX be lower semi-continuous with respect to the second argument. Then for arbitrary (ξ̃, η̃) ∈ Γ(Ψ̃, Φ̃) and ε > 0 there exists (ξ, η) ∈ Γ(Ψ,Φ) such that ρX(ξ̃, ξ) ≤ K(m0)(q1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2h +(Φ̃(ξ̃),Φ(ξ̃))) + ε, (3.20) ρY (η̃, η) ≤ q1q2h +(Φ̃(ξ̃),Φ(ξ̃)) + q2βK(m0)(q2 1e +(Ψ(ξ̃), Ψ̃(ξ̃)) + q2 2h +(Φ̃(ξ̃),Φ(ξ̃))) + ε. Recall Lim’s lemma (see [14]). LetX be a complete metric space, β ∈ [0, 1), Φ, Φ̃ : X ⇒ X be β-contractive set-valued mappings such that Φ(x), Φ̃(x) are closed for every x. h(Fix(Φ),Fix(Φ̃)) ≤ 1 1− β sup x∈X h(Φ(x), Φ̃(x)). Here Fix(Φ) is the set of fixed points of the mapping Φ. Let us now derive a generalization of Lim’s lemma for coincidence points of mappings between (q1, q2)-quasimetric spaces. Corollary 3.2: Let α > 0 and β ∈ [0, α) be given. Assume that ρX is lower semicontinuous with respect to the second argument. If (Ψ,Φ), (Ψ̃, Φ̃) ∈ Fα,β then h+(Coin(Ψ̃, Φ̃),Coin(Ψ,Φ)) ≤ K(m0) sup x∈X ( q1e +(Ψ(x), Ψ̃(x)) + q2h +(Φ̃(x),Φ(x)) ) (3.21) Proof It follows from Theorem 3.2 that h+(Coin(Ψ̃, Φ̃),Coin(Ψ,Φ)) ≤ sup ξ̃∈Coin(Ψ̃,Φ̃) AX(ξ̃) ≤ sup x∈X AX(x) = = K(m0) sup x∈X ( q1e +(Ψ(x), Ψ̃(x)) + q2h +(Φ̃(x),Φ(x)) ) . Hence, (3.21) holds. Let us now derive a generalization of Lim’s lemma for fixed points of self-mappings of (q1, q2)-quasimetric spaces. Corollary 3.3: Assume that (X, ρX) is a complete (q1, q2)-quasimetric space and ρX is lower semicontinuous with respect to the second argument. Given a number β ∈ [0, 1), assume that mappings Φ, Φ̃ : X → X are β-contractions and closed. Then h+(Fix(Φ̃),Fix(Φ)) ≤ q2 q2 1S(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 sup x∈X h+(Φ̃(x),Φ(x)). Here n0 := min{j ∈ N : q2β j < 1}. Proof Set Ψ(x) := {x}, Ψ̃(x) := {x}, α := 1. Then (Ψ,Φ), (Ψ̃, Φ̃) ∈ Fα,β, m0 = n0, e+(Ψ(x), Ψ̃(x)) = 0, and K(m0) = q2 1S(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 . Hence, applying Corollary 3.2 we obtain the desired inequality. Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) 102 R. SENGUPTA, Z.T. ZHUKOVSKAYA, S.E. ZHUKOVSKIY Corollary 3.4: Let (X, ρX) be a complete (q1, q2)-quasimetric space, ρX be lower semicontinuous in the second argument, (Σ, ρΣ) be a (q1, q2)-quasimetric space, Φ : X × Σ→ X be given. Given numbers β ∈ [0, 1) and l ≥ 0, assume that mapping Φ(·, σ) is a β-contraction and closed for every σ ∈ Σ, Φ(x, ·) is l-Lipschitz. Then the set-valued mapping σ 7→ Fix(Φ(·, σ)) is Lipschitz. Proof Take arbitrary σ, σ̃ ∈ Σ. It follows from Corollary 3.3 that h+ ( Fix(Φ(·, σ̃)),Fix(Φ(·, σ)) ) ≤ c sup x∈X h+(Φ(x, σ̃),Φ(x, σ)), where c = q2 q2 1S(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 . Hence, h ( Fix(Φ(·, σ̃)),Fix(Φ(·, σ)) ) ≤ c sup x∈X h(Φ(x, σ̃),Φ(x, σ)).Moreover, since Φ(x, ·) is l-Lipschitz, we have sup x∈X h(Φ(x, σ̃),Φ(x, σ)) ≤ lρΣ(σ̃, σ). Thus, h ( Fix(Φ(·, σ̃)),Fix(Φ(·, σ)) ) ≤ lcρΣ(σ̃, σ), which completes the proof. Let us now discuss the question of the stability of coincidence points. Given a sequence of pairs of set-valued mappings (Ψn,Φn),Ψn,Φn : X ⇒ Y,which tend in a certain sense to a pair of set-valued mappings (Ψ,Φ), Ψn,Φn : X ⇒ Y, and a point ξ ∈ Coin(Ψ,Φ). Our goal is to derive conditions for the existence of points ξn ∈ Coin(Ψn,Φn) such that ξn → ξ. Corollary 3.5: Assume that ρX is lower semicontinuous with respect to the second argument, {(Ψn,Φn)} ⊂ Fα,β for every n, and there exists a point ξ ∈ Coin(Ψ,Φ) such that e+(Ψn(ξ),Ψ(ξ))→ 0, h+(Φ(ξ),Φn(ξ))→ 0. Then there exists a sequence {ξn}, such that ξn ∈ Coin(Ψn,Φn) ∀n, ξn → ξ as n→∞. Proof Take an arbitrary point η ∈ Ψ(ξ) ∩ Φ(ξ). Corollary 3.1 implies that for every n there exists a point ξn ∈ Coin(Ψn,Φn) such that ρX(ξ, ξn) ≤ K(m0) ( q1e +(Ψn(ξ),Ψ(ξ)) + q2h +(Φ(ξ),Φn(ξ)) ) + 2−n. Since e+(Ψn(ξ),Ψ(ξ))→ 0 and h+(Φ(ξ),Φn(ξ))→ 0, we have ρX(ξ, ξn)→ 0. Therefore, ξn → ξ. ACKNOWLEDGEMENTS The research is supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye) and Russian Foundation for Basic Research (project no. 20-31- 70013). The results in Section 3 were obtained with the support of the Russian Science Foundation (project No. 17-11-01168). Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) PROPERTIES OF COINCIDENCE POINTS IN (Q1, Q2)-QUASIMETRIC SPACES 103 REFERENCES 1. Arutyunov A.V. & Greshnov A.V. (2016) Theory of (q1, q2)-quasimetric spaces and coincidence points, Doklady Mathematics, 94(1), 434–437. 2. 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(1985) On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations, Journal of Mathematical Analysis and Applications, 110, 436–441. Copyright © 2020 ASSA. Adv Syst Sci Appl (2020) Introduction and statement of the problem Main Results. Estimates of distance from a point to the intersection set of two graphs Corollaries. Estimates of distance between intersections of graphs and sets of coincidence points