New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness

and Applied Analysis 3 (w3) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε. A function p : X×X → [0,∞) is said to be a τ-function [5, 10, 14, 15, 27–29], first introduced and studied by Lin and Du, if the following conditions hold: (τ1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z ∈ X; (τ2) if x ∈ X and {yn} in X with limn→∞yn = y such that p(x, yn) ≤ M for some M = M(x) > 0, then p(x, y) ≤ M; (τ3) for any sequence {xn} in X with limn→∞ sup{p(xn, xm) : m > n} = 0, if there exists a sequence {yn} in X such that limn→∞p(xn, yn) = 0, then limn→∞d(xn, yn) = 0; (τ4) for x, y, z ∈ X, p(x, y) = 0 and p(x, z) = 0 imply y = z. Note that not either of the implications p(x, y) = 0 ⇔ x = y necessarily holds and p is nonsymmetric in general. It is well known that the metric d is a w-distance and any wdistance is a τ-function, but the converse is not true; see [27, 29] for more detail. Definition 7 (see [4]). Let (X, d) be a metric space, p be a τfunction, g : X → X be a single-valued self-map and T : X → N(X) be a multivalued map. (1) The maps g and T are said to have the p-approximate coincidence point property onX provided inf x∈X p (gx, Tx) = 0. (8) (2) The map T is said to have the p-approximate fixed point property onX provided inf x∈X p (x, Tx) = 0. (9) The following results are crucial in this paper. Lemma 8 (see [29, Lemma 2.1]). Let (X, d) be a metric space and p : X × X → [0,∞) be a function. Assume that p satisfies the condition (τ3). If a sequence {xn} in X with limn→∞ sup{p(xn, xm) : m > n} = 0, then {xn} is a Cauchy sequence inX. For each x ∈ X and A ⊆ X, we denote p(x, A) = infy∈Ap(x, y). Lemma 9 (see [10]). Let A be a closed subset of a metric space (X, d) and p : X×X → [0,∞) be any function. Suppose that p satisfies (τ3) and there exists u ∈ X such that p(u, u) = 0. Then p(u, A) = 0 if and only if u ∈ A. The concepts of τ-functions and τ-metrics were introduced in [10] as follows. Definition 10 (see [10]). Let (X, d) be a metric space. A function p : X × X → [0,∞) is called a τ-function if it is a τ-function onX with p(x, x) = 0 for all x ∈ X. Remark 11. If p is a τ-function, then, from (τ4), p(x, y) = 0 if and only if x = y. Example 12 (see [10]). Let X = R with the metric d(x, y) = |x−y| and 0 < a < b. Define the functionp : X×X → [0,∞) by p (x, y) = max {a (y − x) , b (x − y)} . (10) Then p is nonsymmetric and hence p is not ametric. It is easy to see that p is a τ-function. Definition 13 (see [10]). Let (X, d) be a metric space and p be a τ-function. For any A, B ∈ CB(X), define a functionDp : CB(X) × CB(X) → [0,∞) by Dp (A, B) = max {δp (A, B) , δp (B, A)} , (11) where δp(A, B) = supx∈Ap(x, B), then Dp is said to be the τ-metric on CB(X) induced by p. Clearly, any Hausdorff metric is a τ-metric, but the reverse is not true. Lemma 14 (see [10]). Let (X, d) be a metric space andDp be a τ-metric on CB(X) induced by a τ-function p. Then every τ0-metricDp is a metric on CB(X). The following characterizations ofMT-functions is quite useful for proving our main results. Lemma 15 (see [18]). Let φ : [0,∞) → [0, 1) be a function. Then the following statements are equivalent. (a) φ is anMT-function. (b) For each t ∈ [0,∞), there exist r t ∈ [0, 1) and ε t > 0 such that φ(s) ≤ r t for all s ∈ (t, t + ε t ). (c) For each t ∈ [0,∞), there exist r t ∈ [0, 1) and ε t > 0 such that φ(s) ≤ r t for all s ∈ [t, t + ε t ]. (d) For each t ∈ [0,∞), there exist r t ∈ [0, 1) and ε t > 0 such that φ(s) ≤ r t for all s ∈ (t, t + ε t ]. (e) For each t ∈ [0,∞), there exist r t ∈ [0, 1) and ε t > 0 such that φ(s) ≤ r t for all s ∈ [t, t + ε t ). (f) For any nonincreasing sequence {xn}n∈N in [0,∞), one has 0 ≤ sup n∈Nφ(xn) < 1. (g) φ is a function of contractive factor [12]; that is, for any strictly decreasing sequence {xn}n∈N in [0,∞), one has 0 ≤ sup n∈Nφ(xn) < 1. 3. New Nonlinear Conditions for p-Approximate Coincidence Point Property In Section 3, we will establish some new existence theorems concerning approximate coincidence point property, approximate fixed point property, coincidence point and fixed point for various types of nonlinear maps in metric spaces without global completeness. 4 Abstract and Applied Analysis Theorem 16. Let (X, d) be a metric space, p be a τ-function, T : X → N(X) be a multivalued map, and f : X → X be a self-map. Suppose that (S1) there exist a nondecreasing function τ : [0,∞) → [0,∞) and anMT-functionφ : [0,∞) → [0, 1) such that for each x ∈ X, if y ∈ X with fy ̸ = fx and fy ∈ Tx, then it holds p (fy, Ty) ≤ φ (τ (p (fx, fy))) p (fx, fy) . (12) (S2) T(X) = ⋃ x∈X T(x) ⊆ f(X). Then the following statements hold. (a) There exists a sequence {xn}n∈N inX such that inf n∈N p (fxn, fxn+1) = lim n→∞ p (fxn, fxn+1) = lim n→∞ d (fxn, fxn+1) = inf n∈N d (fxn, fxn+1) = 0. (13) (b) infx∈Xp(fx, Tx) = infx∈Xd(fx, Tx) = 0; that is, f and T have the p-approximate coincidence point property and approximate coincidence point property onX. (c) If one further assumes the following conditions hold: (L1) f(X) is a complete subspace ofX, (L2) for each sequence {xn} inXwithfxn+1 ∈ Txn, n ∈ N and limn→∞fxn = fw, one hasTw as a closed subset ofX and limn→∞p(fxn, Tw) = 0, then COP(f, T) ̸ = 0. Proof. Let x1 ∈ X. By (S2), there exists x2 ∈ X such that fx2 ∈ Tx1. If fx1 = fx2, then fx1 ∈ Tx1 and so inf x∈X p (fx, Tx) ≤ p (fx1, Tx1) ≤ p (fx1, fx1) = 0, (14) which implies infx∈Xp(fx, Tx) = 0. Clearly, infx∈Xd(fx, Tx) = 0. Let wn = x1 for all n ∈ N. Then lim n→∞ p (fwn, fwn+1) = inf n∈N p (fwn, fwn+1) = p (fx1, fx1)=0, lim n→∞ d (fwn, fwn+1) = inf n∈N d (fwn, fwn+1) = d (fx1, fx1)=0. (15) So, the conclusions (a) and (b) hold in this case. Otherwise, if fx2 ̸ = fx1, since p is a τ -function, p(fx1, fx2) > 0. Let μ : [0,∞) → [0, 1) be defined by μ(t) = (1+φ(t))/2. Clearly, 0 ≤ φ(t) < μ(t) < 1 for all t ∈ [0,∞). By [3, Lemma 2.1], we know that μ is also anMT-function. From (S1), we get p (fx2, Tx2) ≤ φ (τ (p (fx1, fx2))) p (fx1, fx2) < μ (τ (p (fx1, fx2))) p (fx1, fx2) . (16) Since μ(τ(p(fx1, fx2)))p(fx1, fx2) > 0, there exists ξ ∈ Tx2 such that p (fx2, ξ) < μ (τ (p (fx1, fx2))) p (fx1, fx2) . (17) Using (S2) again, there exists x3 ∈ X such thatfx3 = ξ ∈ Tx2. Hence, from (17), we have p (fx2, fx3) < μ (τ (p (fx1, fx2))) p (fx1, fx2) . (18) If fx2 = fx3 ∈ Tx2, then, following a similar argument as above, we can prove the conclusions (a) and (b). Otherwise, if fx3 ̸ = fx2, then there exists x4 ∈ X such that fx4 ∈ Tx3 and p (fx3, fx4) < μ (τ (p (fx2, fx3))) p (fx2, fx3) . (19) By induction, we can obtain a sequences {xn} inX satisfying fxn+1 ∈ Txn, (20) p (fxn+1, fxn+2) < μ (τ (p (fxn, fxn+1))) p (fxn, fxn+1) , for each n ∈ N. (21) Since μ(t) < 1 for all t ∈ [0,∞), we deduces from the inequality (21) that the sequence {p(fxn,fxn+1)}n∈N is strictly decreasing in [0,∞). Hence lim n→∞ p (fxn, fxn+1) = inf n∈N p (fxn, fxn+1) ≥ 0 exists. (22) Since τ is nondecreasing, {τ(p(fxn, fxn+1))}n∈N is a nonincreasing sequence in [0,∞). Since μ is an MT-function, by (f) of Lemma 15, we have 0 ≤ sup n∈N μ (τ (p (fxn, fxn+1))) < 1. (23) Let γ := sup n∈N μ(τ(p(fxn, fxn+1))). So γ ∈ [0, 1). Put λ := (1 + γ)/2. Then 0 ≤ γ < λ < 1. By (21), we get p (fxn+1, fxn+2) < μ (τ (p (fxn, fxn+1))) p (fxn, fxn+1)