CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 207–224, August 2021 DOI: 10.4067/S0719-06462021000200207 Coincidence point results of nonlinear contractive mappings in partially ordered metric spaces K. Kalyani 1 N. Seshagiri Rao 2 1 Department of Mathematics, Vignan’s Foundation for Science, Technology & Research, Vadlamudi-522213, Andhra Pradesh, India. kalyani.namana@gmail.com 2 Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No.1888, Adama, Ethiopia. seshu.namana@gmail.com ABSTRACT In this paper, we proved some coincidence point results for f- nondecreasing self-mapping satisfying certain rational type contractions in the context of a metric space endowed with a partial order. Moreover, some consequences of the main result are given by involving integral type contractions in the space. Some numerical examples are illustrated to sup- port our results. As an application, we have discussed the existence of a unique solution of integral equation. RESUMEN En este art́ıculo, probamos algunos resultados sobre puntos de coincidencia para un auto-mapeo no decreciente f satisfa- ciendo ciertas contracciones de tipo racional en el contexto de un espacio métrico dotado de un orden parcial. Más aún, se entregan algunas consecuencias del resultado principal que involucran contracciones de tipo integral en el espacio. Se ilustran algunos ejemplos numéricos en apoyo a nuestros re- sultados. Como una aplicación, discutimos la existencia de una única solución de una ecuación integral. Keywords and Phrases: Ordered metric spaces; rational contractions; compatible mappings; weakly compatible mappings; coupled fixed point; common fixed point. 2020 AMS Mathematics Subject Classification: 41A50, 47H10. Accepted: 01 April, 2021 Received: 25 Nov, 2020 c©2021 K. Kalyani et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200207 https://orcid.org/0000-0002-4531-5976 https://orcid.org/0000-0003-2409-6513 208 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) 1 Introduction A remarkable fixed point theorem was first introduced by Banach [4] in 1922, which is one of the most influential results in analysis. It is being used widely in many different areas of mathematics and its applications. It needs the structure of complete metric spaces together with a contractive condition on the self map which is easy to test in many circumstances. Basically this principle gives a sequence of approximate solutions and also give a valuable information about the convergence rate of a fixed point. This kind of iteration process has been used both in mathematics and computer science. In particular, fixed point iterations together with monotone iterative techniques are the central methods when solving a large class of problems in theoretical and applied mathematics and play an important role in many algorithms. Many authors have extended this theorem by introducing more generalized contractive conditions, which imply the existence of a fixed point [6, 7, 8, 9, 11, 12, 13, 14, 15, 16]. The existence of fixed point results for self-mappings in partially ordered sets have been considered first by Ran and Reurings [36] and presented some applications to matrix equations therein. These results were again generalized and extended by Nieto et al. [32, 33] in partially ordered sets and applied their results to study the ordinary differential equations. Prominent works on various existence and uniqueness theorems on fixed point and common fixed point for monotone mappings in cone metric spaces, partially ordered metric spaces and others spaces, refer the readers to [5, 10, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], which generate natural interest to establish usable fixed point theorems by weakening its hypothesis. Various types of contraction conditions have been used to find a fixed point of a single and multivalued mappings on metric spaces by Altun et al. [1], Aslantas et al. [2, 3], and Sahin et al. [37]. It is well known that a powerful technique for proving existence results for nonlinear problems is the method of upper and lower solutions. In many cases it is possible to find a minimal and a maximal solution between the lower and the upper solution by an iterative scheme: the monotone iterative technique. This method provides a constructive procedure for the solutions and it is also useful for the investigation of qualitative properties of solutions. This method has been used to acquire the unique solution of periodic boundary value problems of ordinary and partial differential equations, integro ordinary and partial differential equations by several authors, some of which are in [23, 32, 33]. The aim of this paper is to prove the coincidence point and common fixed point results for f- nondecreasing self-mapping satisfying generalized contractive conditions of rational type in the context of partially ordered metric spaces. These results generalize and extend the result of [7, 12, 14, 25, 26] in partially ordered metric spaces. Some consequences of the main results are given in terms of integral type contractions in the same space. Further, some examples and an application for the existence of the unique solution for an integral equation are presented at the end. CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 209 2 Preliminaries The following definitions are frequently used in our study. Definition 2.1. [40] The triple (X,d,≤) is called a partially ordered metric space, if (X,≤) is a partially ordered set together with (X,d) is a metric space. Definition 2.2. [40] If (X,d) is a complete metric space, then the triple (X,d,≤) is called a complete partially ordered metric space. Definition 2.3. [38] Let (X,≤) be a partially ordered set. A mapping f : X → X is said to be strictly increasing (strictly decreasing), if f(x) < f(y) (f(x) > f(y)) for all x,y ∈ X with x < y. Definition 2.4. [40] A point x ∈ A, where A is a non-empty subset of a partially ordered set (X,≤) is called a common fixed (coincidence) point of two self-mappings f and T, if fx = Tx = x (fx = Tx). Definition 2.5. [39] The two self-mappings f and T defined over a subset A of a partially ordered metric space (X,d,≤) are called commuting, if fTx = Tfx for all x ∈ A. Definition 2.6. [39] Two self-mappings f and T defined over A ⊂ X are compatible, if for any sequence {xn} with lim n→+∞ fxn = lim n→+∞ Txn = µ for some µ ∈ A, then lim n→+∞ d(Tfxn,fTxn) = 0. Definition 2.7. [40] Two self-mappings f and T defined over A ⊂ X are said to be weakly compatible, if they commute only at their coincidence points (i.e., if fx = Tx, then fTx = Tfx). Definition 2.8. [40] Let f and T be two self-mappings defined over a partially ordered set (X,≤). A mapping T is called monotone f-nondecreasing, if fx ≤ fy implies Tx ≤ Ty, for all x,y ∈ X. Definition 2.9. [38] Let A be a non-empty subset of a partially ordered set (X,≤). If every two elements of A are comparable, then it is called a well ordered set. Definition 2.10. [39] A partially ordered metric space (X,d,≤) is called an ordered complete, if for each convergent sequence {xn} ∞ n=0 ⊂ X, one of the following conditions holds: • if {xn} is a non-decreasing sequence in X such that xn → x implies xn ≤ x, for all n ∈ N that is, x = sup{xn} or, • if {xn} is a non-increasing sequence in X such that xn → x implies x ≤ xn, for all n ∈ N that is, x = inf{xn}. 210 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) 3 Main Results We start this section with the following coincidence point theorem in the context of a partially ordered metric space. Theorem 3.1. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the self- mappings f and T on X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying the following condition d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + β d(fx,Tx) d(fy,Ty) d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.1) for all x, y in X for which fx 6= fy are comparable, and for some α,β,γ,δ,λ ∈ [0,1) with 0 ≤ α+β + 2(γ +δ)+λ < 1. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings f and T are compatible, then f and T have a coincidence point in X. Proof. Suppose for some x0 ∈ X such that fx0 ≤ Tx0. From the hypothesis, we have T(X) ⊆ f(X), then choose a point x1 ∈ X such that fx1 = Tx0. But Tx1 ∈ f(X), then there exists another point x2 ∈ X such that fx2 = Tx1. As by a similar argument above, we obtain a sequence {xn} in X such that fxn+1 = Txn for all n ≥ 0. Since, fx0 ≤ Tx0 = fx1 and T is monotone f-nondecreasing mapping, then we have that Tx0 ≤ Tx1. Similarly, we get Tx1 ≤ Tx2 as fx1 ≤ fx2. Continuing the same process, we obtain that Tx0 ≤ Tx1 ≤ ... ≤ Txn ≤ Txn+1 ≤ ... . Now, we discuss the following two cases. Case 1: If d(Txn0,Txn0+1) = 0 for some n0 ∈ N, then Txn0+1 = Txn0 and by the above argument, we have Txn0+1 = Txn0 = fxn0+1. Therefore, xn0+1 is a coincidence point of T and f, and so we have the result. Case 2: If d(Txn,Txn+1) > 0 for all n ∈ N, then from contraction condition (3.1), we have d(Txn+1,Txn) ≤ α d(fxn+1,Txn+1) [1 + d(fxn,Txn)] 1 + d(fxn+1,fxn) + β d(fxn+1,Txn+1) d(fxn,Txn) d(fxn+1,fxn) + γ [d(fxn+1,Txn+1) + d(fxn,Txn)] + δ [d(fxn+1,Txn) + d(fxn,Txn+1)] + λd(fxn+1,fxn), which implies that d(Txn+1,Txn) ≤ αd(Txn,Txn+1) + βd(Txn,Txn+1) + γ [d(Txn,Txn+1) + d(Txn−1,Txn)] + δ [d(Txn,Txn) + d(Txn−1,Txn+1)] + λd(Txn,Txn−1). CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 211 Therefore, we arrive at d(Txn+1,Txn) ≤ ( γ + δ + λ 1 − α − β − γ − δ ) d(Txn,Txn−1). Continuing the same process up to n times, we obtain that d(Txn+1,Txn) ≤ ( γ + δ + λ 1 − α − β − γ − δ )n d(Tx1,Tx0). Let k = γ+δ+λ 1−α−β−γ−δ < 1. Moreover, from the triangular inequality for m ≥ n, we have d(Txm,Txn) ≤ d(Txm,Txm−1) + d(Txm−1,Txm−2) + ... + d(Txn+1,Txn) ≤ ( km−1 + km−2 + ... + kn ) d(Tx1,Tx0) ≤ kn 1 − k d(Tx1,Tx0), as m,n → +∞, d(Txm,Txn) → 0, this shows that the sequences {Txn} is a Cauchy sequence in X. So, by the completeness of X, there exists a point µ ∈ X such that Txn → µ as n → +∞. The continuity of T implies that lim n→+∞ T(Txn) = T ( lim n→+∞ Txn ) = Tµ. Since, fxn+1 = Txn then fxn+1 → µ as n → +∞. Further, the compatibility of T and f, we have lim n→+∞ d(Tfxn,fTxn) = 0. From the triangular inequality of a metric d, we have d(Tµ,fµ) = d(Tµ,Tfxn) + d(Tfxn,fTxn) + d(fTxn,fµ), on taking limit as n → +∞ in the above inequality and using the fact that T and f are continuous, we obtain that d(Tµ,fµ) = 0. Thus, Tµ = fµ. Hence, µ is a coincidence point of T and f in X. We obtain the following consequences from Theorem 3.1 on taking zero value to α,β,γ,δ and λ as special cases. Corollary 3.2. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the self- mappings f and T on X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying the following contraction conditions (a) d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.2) for some α,γ,δ,λ ∈ [0,1) with 0 ≤ α + 2(γ + δ) + λ < 1, 212 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) (b) d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + λd(fx,fy), (3.3) where α,γ,λ ∈ [0,1) such that 0 ≤ α + 2γ + λ < 1, (c) d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.4) there exist α,δ,λ ∈ [0,1) such that 0 ≤ α + 2δ + λ < 1, (d) d(Tx,Ty) ≤ γ [d(fx,Tx) + d(fy,Ty)] + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.5) for some γ,δ,λ ∈ [0,1) with 0 ≤ 2(γ + δ) + λ < 1, for all x, y in X for which fx 6= fy are comparable. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings T and f are compatible, then T and f have a coincidence point in X. Corollary 3.3. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the map- pings f,T : X → X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying the following contraction conditions (i) d(Tx,Ty) ≤ β d(fx,Tx) d(fy,Ty) d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] +δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.6) where β,γ,δ,λ ∈ [0,1) such that 0 ≤ β + 2(γ + δ) + λ < 1, (ii) d(Tx,Ty) ≤ β d(fx,Tx) d(fy,Ty) d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + λd(fx,fy), (3.7) for some β,γ,λ ∈ [0,1) with 0 ≤ β + 2γ + λ < 1, (iii) d(Tx,Ty) ≤ β d(fx,Tx) d(fy,Ty) d(fx,fy) + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.8) there exist β,δ,λ ∈ [0,1) such that 0 ≤ β + 2δ + λ < 1, CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 213 (iv) d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + β d(fx,Tx) d(fy,Ty) d(fx,fy) + λd(fx,fy), (3.9) where α,β,λ ∈ [0,1) such that 0 ≤ α + β + λ < 1, for all x, y in X for which fx 6= fy are comparable. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings T and f are compatible, then T and f have a coincidence point in X. Corollary 3.4. Let (X,d,≤) be a complete partially ordered metric space. Suppose that T : X → X is a mapping such that for all comparable x,y ∈ X, the contraction condition(s) in Theorem 3.1 (or Corollaries 3.2 and 3.3 ) is satisfied. Assume that T satisfies the following hypotheses: (i). T is continuous, (ii). T(Tx) ≤ Tx for all x ∈ X. If there exists a point x0 ∈ X such that x0 ≤ Tx0, then T has a fixed point in X. Proof. Follow from Theorem 3.1 by taking f = IX (the identity map). We may remove the continuity criteria of T in Theorem 3.1, is still valid by assuming the following hypothesis in X: If {xn} is a non-decreasing sequence in X such that xn → x, then xn ≤ x for all n ∈ N. Theorem 3.5. Let (X,d,≤) be a complete partially ordered metric space. Suppose that T,f : X → X are two mappings such that T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying d(Tx,Ty) ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + β d(fx,Tx) d(fy,Ty) d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy), (3.10) for all x, y in X for which fx 6= fy are comparable and where α,β,γ,δ,λ ∈ [0,1) such that 0 ≤ α+ β + 2(γ + δ) + λ < 1. Assume that there exists x0 ∈ X such that fx0 ≤ Tx0 and {xn} is a non-decreasing sequence in X such that xn → x, then xn ≤ x for all n ∈ N. If f(X) is a complete subset of X, then T and f have a coincidence point in X. Further, if T and f are weakly compatible then T and f have a common fixed point in X. Moreover, the set of common fixed points of T and f are well ordered if and only if T and f have one and only one common fixed point in X. 214 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) Proof. Suppose f(X) is a complete subset of X. As we know from Theorem 3.1, the sequence {Txn} is a Cauchy sequence and hence, {fxn} is also a Cauchy sequence in (f(X),d) as fxn+1 = Txn and T(X) ⊆ f(X). Since f(X) is complete then there exists fu ∈ f(X) such that lim n→+∞ Txn = lim n→+∞ fxn = fu. (3.11) Also note that the sequences {Txn} and {fxn} are nondecreasing and from the hypothesis, we have Txn ≤ fu and fxn ≤ fu for all n ∈ N. Since T is a monotone f-nondecreasing, we get Txn ≤ Tu for all n. Letting n → +∞, we obtain fu ≤ Tu. Suppose that fu < Tu, define a sequence {un} by u0 = u and fun+1 = Tun for all n ∈ N. An argument similar to that in the proof of Theorem 3.1 yields that {fun} is a nondecreasing sequence and lim n→+∞ fun = lim n→+∞ Tun = fv for some v ∈ X. (3.12) So from the hypothesis, we have that sup n∈N fun ≤ fv and sup n∈N Tun ≤ fv. Notice that fxn ≤ fu ≤ fu1 ≤ fu2 ≤ ... ≤ fun ≤ ... ≤ fv. Now, we discuss the following two cases: Case 1: If there exists some n0 ≥ 1 with fxn0 = fun0, then we have fxn0 = fu = fun0 = fu1 = Tu, this is a contradiction to fu < Tu. Thus, fu = Tu, that is, u is a coincidence point of T and f in X. Case 2: Suppose fxn 6= fun+1 for all n. Then from condition (3.10), we have d(fxn+1,fun+1) = d(Txn,Tun) ≤ α d(fxn,Txn) [1 + d(fun,Tun)] 1 + d(fxn,fun) + β d(fxn,Txn) d(fun,Tun) d(fxn,fun) + γ [d(fxn,Txn) + d(fun,Tun)] + δ [d(fxn,Tun) + d(fun,Txn)] + λd(fxn,fun). On taking limit as n → +∞ in the above inequality and from equations (3.11) and (3.12), we get d(fu,fv) ≤ (2δ + λ) d(fu,fv) < d(fu,fv), since 2δ + λ < 1. So, we have fu = fv = fu1 = Tu, CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 215 this is again a contradiction to fu < Tu. Hence, we conclude that u is a coincidence point of T and f in X. Now, we suppose that T and f are weakly compatible. Let w be the coincidence point then Tw = Tfz = fTz = fw, since w = Tz = fz, for some z ∈ X. Now from (3.10), we have d(Tz,Tw) ≤ α d(fz,Tz) [1 + d(fw,Tw)] 1 + d(fz,fw) + β d(fz,Tz) d(fw,Tw) d(fz,fw) + γ [d(fz,Tz) + d(fw,Tw)] + δ [d(fz,Tw) + d(fw,Tz)] + λd(fz,fw) ≤ (2γ + 2δ + λ) d(Tz,Tw). As 2γ + 2δ + λ < 1, then d(Tz,Tw) = 0. Therefore, Tz = Tw = fw = w. Hence, w is a common fixed point of T and f in X. Now, suppose that the set of common fixed points of T and f is well ordered, we have to show that the common fixed point of T and f is unique. Let u and v be two common fixed points of T and f such that u 6= v, then from condition (3.10), we have d(u,v) ≤ α d(fu,Tu) [1 + d(fv,Tv)] 1 + d(fu,fv) + β d(fu,Tu) d(fv,Tv) d(fu,fv) + γ [d(fu,Tu) + d(fv,Tv)] + δ [d(fu,Tv) + d(fv,Tu)] + λd(fu,fv) ≤ (2γ + 2δ + λ) d(u,v) < d(u,v), since 2γ + 2δ + λ < 1, which is a contradiction and hence, u = v. Conversely, suppose T and f have only one common fixed point, then the set of common fixed points of T and f being a singleton is well ordered. Besides, in Corollary 3.2 and Corollary 3.3 by relaxing the continuity criteria on T and satisfying the hypotheses given in Theorem 3.5, then T and f have a coincidence point, a common fixed point and its uniqueness in X. Corollary 3.6. Let (X,d,≤) be a complete partially ordered metric space. Suppose that T : X → X is a mapping such that for all comparable x,y ∈ X, the contraction condition (3.10) is satisfied. Suppose that the following hypotheses are satisfied (i). if {xn} is a non-decreasing sequence in X with respect to ≤ such that xn → x ∈ X as n → +∞, then xn ≤ x, for all n ∈ N and (ii). T(Tx) ≤ Tx for all x ∈ X. If there exists a point x0 ∈ X such that x0 ≤ Tx0, then T has a fixed point in X. 216 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) Proof. Follow from Theorem 3.5 by taking f = IX (the identity map). Remark 3.7. (i). If α = γ = δ = 0 in Theorem 3.1 and 3.5, we obtain Theorem 2.1 and 2.3 of Chandok [25]. (ii). If f = I and α = γ = δ = 0 in Theorem 3.1 and 3.5, then we get Theorem 2.1 and 2.3 of Harjani et al. [26]. Some other consequences of the main result for the self mappings involving the integral type contractions are as follows. Let χ denote the set of all functions ϕ : [0,+∞) → [0,+∞) satisfying the following hypotheses: (a) each ϕ is Lebesgue integrable function on every compact subset of [0,+∞) and (b) for any ǫ > 0, we have ∫ ǫ 0 ϕ(t)dt > 0, for t ∈ [0,+∞). Corollary 3.8. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the map- pings T,f : X → X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying ∫ d(T x,T y) 0 ϕ(t)dt ≤ α ∫ d(fx,T x)[1+d(fy,T y)] 1+d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,T x) d(fy,T y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.13) for all x, y in X for which fx 6= fy are comparable, ϕ ∈ χ and where α,β,γ,δ,λ ∈ [0,1) such that 0 ≤ α+β + 2(γ +δ)+λ < 1. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings T and f are compatible, then T and f have a coincidence point in X. Similarly, we obtain the following results from Corollaries 3.2 and 3.3 in a complete partially ordered metric space. Corollary 3.9. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the self- mappings f,T on X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) satisfying the following contraction conditions (a) ∫ d(T x,T y) 0 ϕ(t)dt ≤ α ∫ d(fx,T x)[1+d(fy,T y)] 1+d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.14) for some α,γ,δ,λ ∈ [0,1) with 0 ≤ α + 2(γ + δ) + λ < 1, CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 217 (b) ∫ d(T x,T y) 0 ϕ(t)dt ≤ α ∫ d(fx,T x)[1+d(fy,T y)] 1+d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.15) where α,γ,λ ∈ [0,1) with 0 ≤ α + 2γ + λ < 1, (c) ∫ d(T x,T y) 0 ≤ α ∫ d(fx,T x)[1+d(fy,T y)] 1+d(fx,fy) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.16) where α,δ,λ ∈ [0,1) such that 0 ≤ α + 2δ + λ < 1, (d) ∫ d(T x,T y) 0 ≤ γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.17) there exist γ,δ,λ ∈ [0,1) such that 0 ≤ 2(γ + δ) + λ < 1, for all x, y in X for which fx 6= fy are comparable, and where ϕ ∈ χ. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings T and f are compatible, then T and f have a coincidence point in X. Corollary 3.10. Let (X,d,≤) be a complete partially ordered metric space. Suppose that the mappings f,T : X → X are continuous, T is a monotone f-nondecreasing, T(X) ⊆ f(X) and satisfying the following integral type contraction conditions: (i) ∫ d(T x,T y) 0 ϕ(t)dt ≤ β ∫ d(fx,T x) d(fy,T y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.18) for some β,γ,δ,λ ∈ [0,1) with 0 ≤ β + 2(γ + δ) + λ < 1, (ii) ∫ d(T x,T y) 0 ϕ(t)dt ≤ β ∫ d(fx,T x) d(fy,T y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,T x)+d(fy,T y) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.19) where β,γ,λ ∈ [0,1) such that 0 ≤ β + 2γ + λ < 1, 218 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) (iii) ∫ d(T x,T y) 0 ϕ(t)dt ≤ β ∫ d(fx,T x) d(fy,T y) d(fx,fy) 0 ϕ(t)dt + δ ∫ d(fx,T y)+d(fy,T x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.20) there exist β,δ,λ ∈ [0,1) such that 0 ≤ β + 2δ + λ < 1, (iv) ∫ d(T x,T y) 0 ϕ(t)dt ≤ α ∫ d(fx,T x)[1+d(fy,T y)] 1+d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,T x) d(fy,T y) d(fx,fy) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.21) where α,β,λ ∈ [0,1) with 0 ≤ α + β + λ < 1, for all x, y in X for which fx 6= fy are comparable, and where ϕ ∈ χ. If there exists a point x0 ∈ X such that fx0 ≤ Tx0 and the mappings T and f are compatible, then T and f have a coincidence point in X. Remark 3.11. If α = γ = δ = 0 in Corollary 3.8, then we obtain the Corollary 2.5 of Chandok [25]. Now, we give the examples for the main Theorem 3.1. Example 3.12. Define a metric d : X × X → [0,+∞) by d(x,y) = |x − y|, where X = [0,1] with usual order ≤. Let T and f be two self mappings on X such that Tx = x 2 2 and fx = 2x 2 1+x , then T and f have a coincidence point in X. Proof. Note that (X,d) is a complete metric space and thus, (X,d,≤) be a complete partially ordered metric space with respect to usual order ≤. Let x0 = 0 ∈ X then fx0 ≤ Tx0 and also note that T and f are continuous, T is a monotone f-nondecreasing and T(X) ⊆ f(X). Now consider the following for any x, y in X with x < y, d(Tx,Ty) = 1 2 |x2 − y2| = 1 2 (x + y)|x − y| ≤ 2(x + y + xy) (1 + x)(1 + y) |x − y| ≤ α 2x2|3 − x| [ (1 + y) + y2|3 − y| ] 4(1 + x)(1 + y) + 2|x − y|(x + y + xy) + β 4 x2y2 (x + y + xy) |x − 3||y − 3| |x − y| + γ 2 x2(1 + y)|x − 3| + y2(1 + x)|y − 3| (1 + x)(1 + y) + δ (1 + y)|4x2 − y2(1 + x)| + (1 + x)|4y2 − x2(1 + y)| 2(1 + x)(1 + y) + λ 2(x + y + xy) (1 + x)(1 + y) |x − y| CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 219 d(Tx,Ty) ≤ α x 2|x−3| 2(1+x) · 2(1+y)+y2|3−y| 2(1+y) 1 + 2|x−y|(x+y+xy) (1+x)(1+y) + β x 2|x−3| 2(1+x) · y 2|y−3| 2(1+y) 2|x − y| x+y+xy (1+x)(1+y) + γ [ x2|x − 3| 2(1 + x) + y2|y − 3| 2(1 + y) ] + δ [ ∣ ∣ ∣ ∣ x2 (1 + x) − y2 2 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ 2y2 (1 + y) − x2 2 ∣ ∣ ∣ ∣ ] + λ 2(x + y + xy) (1 + x)(1 + y) |x − y| ≤ α d(fx,Tx) [1 + d(fy,Ty)] 1 + d(fx,fy) + β d(fx,Tx) d(fy,Ty) d(fx,fy) + γ [d(fx,Tx) + d(fy,Ty)] + δ [d(fx,Ty) + d(fy,Tx)] + λd(fx,fy). Then, the contraction condition in Theorem 3.1 holds by selecting proper values of α,β,γ,δ,λ in [0,1) such that 0 ≤ α + β + 2(γ + δ) + λ < 1. Therefore, T and f have a coincidence point 0 ∈ X. Example 3.13. Define a distance function d : X × X → [0,+∞) by d(x,y) = |x − y|, where X = [0,1] with usual order ≤. Let T and f be two self mappings on X such that Tx = x3 and fx = x4, then T and f have two coincidence points 0, 1 in X with x0 = 1 4 . 4 Applications Now our aim is to give an existence theorem for a solution of the following integral equation. h(x) = ∫ M 0 µ(x,y,h(y))dy + g(x), x ∈ [0,M], (4.1) where M > 0. Let X = C[0,M] be the set of all continuous functions defined on [0,M]. Now, define d : X × X → R+ by d(u,v) = sup x∈[0,M] {|u(x) − v(x)|} then, (X,≤) is a partially ordered set. Now, we prove the following result. Theorem 4.1. Suppose the following hypotheses holds: (i) µ : [0,M] × [0,M] × R+ → R+ and g : R → R are continuous, (ii) for each x,y ∈ [0,M], we have µ ( x,y, ∫ M 0 µ(x,z,h(z))dz + g(x) ) ≤ µ(x,y,h(y)), (iii) there exists a continuous function N : [0,M] × [0,M] → [0,+∞] such that |µ(x,y,a) − µ(x,y,b)| ≤ N(x,y)|a − b| and 220 K. Kalyani & N. Seshagiri Rao CUBO 23, 2 (2021) (iv) sup x∈[0,M] ∫ M 0 N(x,y)dy ≤ γ for some γ < 1. Then, the integral equation (4.1) has a solution a ∈ C[0,M]. Proof. Define T : C[0,M] → C[0,M] by Tw(x) = ∫ M 0 µ(x,y,w(x))dx + g(x), x ∈ [0,M]. Now, we have T(Tw(x)) = ∫ M 0 µ(x,y,Tw(x))dx + g(x) = ∫ M 0 µ ( x,y, ∫ M 0 µ(x,z,w(z))dz + g(x) ) dx + g(x) ≤ ∫ M 0 µ(x,y,w(z))dz + g(x) = Tw(x) Thus, we have T(Tx) ≤ Tx for all x ∈ C[0,M]. For any x∗,y∗ ∈ C[0,M] with x ≤ y, we have d(Tx∗,Ty∗) = sup x∈[0,M] |Tx∗(x) − Ty∗(x)| = sup x∈[0,M] ∣ ∣ ∣ ∣ ∣ ∫ M 0 µ(x,y,x∗(x)) − µ(x,y,y∗(x))dx ∣ ∣ ∣ ∣ ∣ ≤ sup x∈[0,M] ∫ M 0 |µ(x,y,x∗(x)) − µ(x,y,y∗(x))| dx ≤ sup x∈[0,M] ∫ M 0 N(x,y)|x∗(x) − y∗(x)|dx ≤ sup x∈[0,M] |x∗(x) − y∗(x)| sup x∈[0,M] ∫ M 0 N(x,y)dx = d(x∗,y∗) sup x∈[0,M] ∫ M 0 N(x,y)dx ≤ γd(x∗,y∗). Moreover, {x∗n} is a nondecreasing sequence in C[0,M] such that x ∗ n→ x ∗, then x∗n ≤ x ∗ for all n ∈ N. Thus all the required hypotheses of Corollary 3.6 are satisfied. Thus, there exists a solution a ∈ C[0,M] of the integral equation (4.1). CUBO 23, 2 (2021) Coincidence point results of nonlinear contractive mappings in ... 221 References [1] I. Altun, M. Aslantas and H. Sahin, “Best proximity point results for p-proximal contractions”, Acta Math. Hungar. vol. 162, no. 2, pp. 393-402, 2020. [2] M. Aslantas, H. Sahin and D. Turkoglu, “Some Caristi type fixed point theorems”, J. Anal., pp. 1-15, 2020. [3] M. Aslantas, H. 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