@ Appl. Gen. Topol. 18, no. 1 (2017), 91-105 doi:10.4995/agt.2017.6322 c© AGT, UPV, 2017 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance Chirasak Mongkolkehaa, Yeol Je Chob,∗ and Poom Kumamc,d,∗ a Department of Mathematics Statistics and Computer Sciences, Faculty of Liberal Arts and Sci- ence, Kasetsart University, Kamphaeng-Saen Campus, Nakhonpathom 73140, Thailand (faascsm@ku.ac.th) b Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea, Center for General Education, China Medical University Taichung, 40402, Tai- wan (yjcho@gnu.ac.kr) c KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand. (poom.kum@kmutt.ac.th) d KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Communicated by S. Romaguera Abstract The purpose of this article is to prove some fixed point theorems for simulation functions in complete b−metric spaces with partially ordered by using wt-distance which introduced by Hussain et al. [12]. Also, we give some examples to illustrate our main results. 2010 MSC: 47H09; 47H10; 54H25. Keywords: Fixed point; simulation function; b-metric space; wt-distance; w-distance; generalized distance. Received 02 July 2016 – Accepted 06 December 2016 http://dx.doi.org/10.4995/agt.2017.6322 C. Mongkolkeha, Y. J. Cho and P. Kumam 1. Introduction Since Banach’s fixed point theorem (or Banach’s contraction principle) proved by Banach [4] in 1922, many authors have extended, improved and generalized in several ways. In 2015, Khojasteh et al. [15] introduced the notion of a simulation func- tion to generalize Banach’s contraction principle. Recently, Roldán-López-de- Hierroet et al. [18] modified the notion of a simulation function and showed the existence and uniqueness of coincidence points of two nonlinear mappings using the concept of a simulation function. On the other hand, in 1989, Bakhtin [3] (see also Czerwik [8]) introduced the concept of a b-metric space (or a space of metric type) and proved some fixed point theorems for some contractive mappings in b-metric spaces which are generalizations of Banach’s contraction principle in metric spaces. In 1996, Kada et al. [14] introduced some generalized metric, which is called the w-distance and gave some examples of w-distance and, using the w-distance, they also improved Caristi’s fixed point theorem, Ekeland’s variational principle and the nonconvex minimization theorem of Takahashi [20]. Later, Shioji et al. [19] studied the relationship between weakly contractive mappings and weakly Kannan mappings under the conditions, the w-distance and the symmetric w- distance. In 2012, Imdad and Rouzkard [13] proved some fixed point theorems in a complete metric space equipped with a partial ordering via the w-distance. Recently, Hussain et al. [12] introduced the concept of the wt-distance in generalized b-metric spaces, which is a generalization of the w-distance, and also proved some fixed point theorems in a partially ordered b-metric space by using the wt-distance. Also, Abdou et al. [1] proved some common fixed point theorems in Menger probabilistic metric type spaces by using the wt-distance. In this paper, we consider some simulation functions to show the existence of fixed points of some nonlinear mappings in complete b-metric spaces via the wt-distance. Furthermore, we also give some examples to illustrate the main results. Our result improve, extend and generalize several results given by some authors in literatures. 2. Preliminaries and generalized distances Now, we give some definitions and their examples Definition 2.1. Let (X,≤) be a partially ordered set.The elements x,y ∈ X are said to be comparable with respect to the order ≤ if either x ≤ y or y ≤ x. Let us denote X≤ by the subset of X ×X defined by X≤ = {(x,y) ∈ X ×X : x ≤ y or y ≤ x}. Definition 2.2. Let (X,≤) be a partially ordered set and f : X → X be a self-mapping of X. We say that (1) f is inverse increasing if, for all x,y ∈ X, f(x) ≤ f(y) implies x ≤ y; (2) f is nondecreasing if, for all x,y ∈ X, x ≤ y implies f(x) ≤ f(y). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 92 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance Definition 2.3. Let (X,≤) be a partially ordered set and T : X → X be a self-mapping of X. Then (1) F(T) = {x ∈ X : T(x) = x}, i.e., F(T) denotes the set of all fixed points of T ; (2) T is called a Picard operator (briefly, PO) if there exists x∗ ∈ X such that F(T) = {x∗} and {Tn(x)} converges to x∗ for all x ∈ X; (3) T is said to be orbitally U-continuous for any U ⊂ X × X if, for any x ∈ X, Tni (x) → a ∈ X as i → ∞ and (Tni (x),a) ∈ U for any i ∈ N imply that Tni+1(x) → Ta ∈ X as i →∞; (4) T is said to be orbitally continuous on X if x ∈ X and Tni (x) → a ∈ X as i →∞ imply that Tni+1(x) → T(a) ∈ X as i →∞. Definition 2.4. Let (X,d) be a metric space. A function p : X ×X → [0,∞) is said to be the w-distance on X if the following are satisfied: (1) p(x,z) ≤ p(x,y) + p(y,z) for all x,y,z ∈ X; (2) for any x ∈ X, p(x, ·) : X → [0,∞) is lower semi-continuous (i.e., if x ∈ X and yn → y ∈ X, then p(x,y) ≤ lim infn→∞p(x,yn); (3) for any ε > 0, there exists δ > 0 such that p(z,x) ≤ δ and p(z,y) ≤ δ imply d(x,y) ≤ ε. Let X be a metric space with a metric d. A w-distance p on X is said to be symmetric if p(x,y) = p(y,x) for all x,y ∈ X. Obviously, every metric is the w-distance, but not conversely. Next, we recall some examples in [21] to show that the w-distance is a generalized metric. Example 2.5. Let (X,d) be a metric space. A function p : X ×X → [0,∞) defined by p(x,y) = c for all x,y ∈ X is a w-distance on X, where c is a positive real number. But p is not a metric since p(x,x) = c 6= 0 for any x ∈ X. Example 2.6. Let (X,‖·‖) be a normed linear space. A function p : X×X → [0,∞) defined by p(x,y) = ‖x‖ + ‖y‖ for all x,y ∈ X is a w-distance on X. Example 2.7. Let F be a bounded and closed subset of a metric spaces X. Assume that F contain at least two points and c is a constant with c ≥ δ(F), where δ(F) is the diameter of F. Then a function p : X ×X → [0,∞) defined by p(x,y) = { d(x,y), if x,y ∈ F, c, if x /∈ F or y /∈ F, is a w-distance on X. Definition 2.8. Let X be a nonempty set and s ≥ 1 be a given real number. A functional D : X ×X → [0,∞) is called a b-metric if, for all x,y,z ∈ X, the following conditions are satisfied: (1) D(x,y) = 0 if and only if x = y; (2) D(x,y) = D(y,x); (3) D(x,z) ≤ s[D(x,y) + D(y,z)]. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 93 C. Mongkolkeha, Y. J. Cho and P. Kumam A pair (X,D) is called a b-metric space with coefficient s. In Definition 2.8, every metric space is a b-metric space with s = 1 and hence the class of b-metric spaces is larger than the class of metric spaces. Some examples of b-metric spaces are given by Berinde [5], Czerwik [9], Heinonen [11] and, further, some examples to show that every b-metric space is a real generalization of metric spaces are as follows: Example 2.9. The set R of real numbers together with the functional D : R×R → [0,∞) defined by D(x,y) := |x−y|2 for all x,y ∈ R is a b-metric space with coefficient s = 2. However, we know that D is not a metric on X since the ordinary triangle inequality is not satisfied. Indeed, D(3, 5) > D(3, 4) + D(4, 5). In 2014, Hussain et al. [12] introduced the concept of the wt-distance as follow: Definition 2.10. Let (X,D) be a b-metric space with constant K ≥ 1. A function P : X × X → [0,∞) is called the wt-distance on X if the following are satisfied: (1) P(x,z) ≤ K(P(x,y) + P(y,z)) for all x,y,z ∈ X; (2) for any x ∈ X, P(x, ·) : X → [0,∞) is K-lower semi-continuous (i.e., if x ∈ X and yn → y ∈ X, then P(x,y) ≤ lim infn→∞KP(x,yn); (3) for any ε > 0, there exists δ > 0 such that P(z,x) ≤ δ and P(z,y) ≤ δ imply D(x,y) ≤ ε. Example 2.11 ([12]). Let (X,D) be a b-metric space. Then the metric D is a wt-distance on X. Example 2.12 ([12]). Let X = R and D1 = (x−y)2. A function P : X×X → [0,∞) defined by P(x,y) = ‖x‖2 +‖y‖2 for all x,y ∈ X is a wt-distance on X. Example 2.13 ([12]). Let X = R and D1 = (x−y)2. A function P : X×X → [0,∞) defined by P(x,y) = ‖y‖2 for all x,y ∈ X is a wt-distance on X. The following two lemmas are crucial for our resuts. Lemma 2.14 ([12]). Let (X,D) be a b-metric space with constant K ≥ 1 and P be a wt-distance on X. Let {xn}, {yn} be two sequences in X and {αn}, {βn} two sequences in [0,∞) converging to zero. Then the following conditions hold: for all x,y,z ∈ X, (1) if P(xn,y) ≤ αn and P(xn,z) ≤ βn for all n ∈ N, then y = z. In particular, if P(x,y) = 0 and P(x,z) = 0, then y = z; (2) if P(xn,yn) ≤ αn and P(xn,z) ≤ βn for all n ∈ N, then {yn} converges to z; c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 94 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance (3) if P(xn,xm) ≤ αn for all n,m ∈ N with m > n, then {xn} is a Cauchy sequence; (4) P(y,xn) ≤ αn for all n ∈ N, then {xn} is a Cauchy sequence. 3. The classes of simulation functions In 2015, Khojasteh et al. [15] introduced the notion of a simulation function which generalizes the Banach contraction as follow: Definition 3.1 ([15]). A simulation function is a mapping ζ : [0,∞)×[0,∞) → R satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(t,s) < s− t for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that limn→∞ tn = limn→∞sn > 0, then lim sup n→∞ ζ(tn,sn) < 0. Now, we recall some examples of the simulation function given by Khojasteh et al. [15]. Example 3.2. Let ζi : [0,∞) × [0,∞) → R for i = 1, 2, 3 be defined by (1) ζ1(t,s) = ψ(s) −φ(t) for all t,s ∈ [0,∞), where φ,ψ : [0,∞) → [0,∞) are two continuous functions such that ψ(t) = φ(t) = 0 if and only if t = 0 and ψ(t) < t ≤ φ(t) for all t > 0; (2) ζ2(t,s) = s− f(t,s) g(t,s) t for all t,s ∈ [0,∞), where f,g : [0,∞)× [0,∞) → (0,∞) are two continuous functions with respect to each variable such that f(t,s) > g(t,s) for all t,s > 0. (3) ζ3(t,s) = s−ϕ(s) − t for all t,s ∈ [0,∞), where ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 if and only if t = 0 Then ζi for i = 1, 2, 3 are a simulation function. Recently, Roldán-López-de-Hierro et al. [18] modified the notion of a simu- lation function as follow: Definition 3.3 ([18]). A simulation function is a mapping â ζ : [0,∞) × [0,∞) → R satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(t,s) < s− t for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that limn→∞ tn = limn→∞sn > 0 and tn < sn for all n ∈ N, then lim sup n→∞ ζ(tn,sn) < 0. Note that the classes of all simulation functions ζ : [0,∞) × [0,∞) → R denote by Z and every simulation function in the original sense of Khojasteh et al. [15] is also a simulation function in the sense of Roldán-López-de-Hierroet et al. [18], but the converse is not true as in the following example. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 95 C. Mongkolkeha, Y. J. Cho and P. Kumam Example 3.4 ([18]). Let k ∈ R be such that k < 1 and let ζ ∈ Z be the function defined by ζ(t,s) = { 2s− 2t, if s < t, ks− t, otherwise. Then ζ is a simulation function in the sense of Definition 3.3, but ζ does not satisfy the condition (ζ3) of Definition 3.1. Definition 3.5. Let (X,d) is a complete metric space. A mapping T : X → X is called Z-contraction if there exists ζ ∈Z such that (3.1) ζ(d(Tx,Ty),d(x,y)) ≥ 0 for all x,y ∈ X. Remark 3.6. If we take ζ(t,s) = λs − t for all s,t ≥ 0, where λ ∈ [0, 1) in Definition 3.5, then the Z-contraction become to the Banach contraction. 4. Fixed point theorems for simulation functions In this section, we consider the concept of a simulation function and show the existence of a fixed point for such mapping in complete b-metric spaces via the wt-distance. First, we improve the notion of a simulation function for our considerations as follow: Definition 4.1. Let K be a given real number such that K ≥ 1. A simulation function is a mapping ζ : [0,∞)×[0,∞) → R satisfying the following conditions: (ζ1) ζ(0, 0) = 0; (ζ2) ζ(Kt,s) < s−Kt for all s,t > 0; (ζ3) if {tn} and {sn} are two sequences in (0,∞) such that lim supn→∞Ktn = lim supn→∞sn > 0 and tn < sn for all n ∈ N, then lim sup n→∞ ζ(Ktn,sn) < 0. Example 4.2. Let λ,K ∈ R be such that λ < 1 and K ≥ 1. Define the mapping â ζ : [0,∞) × [0,∞) → R by ζ(Kt,s) =   s−Kt, if s < t, λs−Kt Ks + 1 , otherwise. Clearly, ζ verifies (ζ1), and ζ satisfies (ζ2). Indeed, s,t > 0,   0 < s < t ⇒ ζ(Kt,s) = s−Kt, 0 < t < s, ⇒ ζ(Kt,s) = λs−Kt Ks + 1 < s−Kt Ks + 1 < s−Kt. Next, we will show that ζ satisfies (ζ3). If {tn} and {sn} are sequences in (0,∞) such that lim supn→∞Ktn = lim supn→∞sn > 0 and tn < sn for all n ∈ N. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 96 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance then lim sup n→∞ ζ(Ktn,sn) = lim sup n→∞ ( λsn −Ktn Ksn + 1 ) < lim sup n→∞ ( sn −Ktn Ktn + 1 ) < lim sup n→∞ ( sn −Ktn Ktn ) < lim sup n→∞ ( sn Ktn − Ktn Ktn ) ≤ lim sup n→∞ ( sn Ktn ) − lim inf n→∞ (1) ≤ 1 − 1 = 0. Then ζ is a simulation function in the sense of Definition 4.1, but ζ does not satisfy the condition (ζ3) of Definition 3.1. Indeed, if we take K = 1, tn = 2 √ 2 and sn = 2 √ 2 − 1 n , for all n ∈ N. Then, sn < tn lim sup n→∞ ζ(tn,sn) = lim sup n→∞ ( 2 √ 2 − 1 n − 2 √ 2 ) = lim sup n→∞ ( − 1 n ) = 0. Theorem 4.3. Let (X,≤) be a partially ordered set, (X,D) be a complete b−metric space with constant K ≥ 1 and P be a wt-distance on X. Suppose that T : X → X is a nondecreasing mapping satisfying the following conditions: (i) there exists ζ ∈Z such that (4.1) ζ(KP(Tx,T2x),P(x,Tx)) ≥ 0 for all (x,Tx) ∈ X≤; (ii) for all x ∈ X with (x,Tx) ∈ X≤, inf{P(x,y) + P(x,Tx)} > 0 for all y ∈ X with y 6= Ty; (iii) there exists x0 ∈ X such that (x0,Tx0) ∈ X≤. Then T has a fixed point in X. Moreover, if Tx = x, then P(x,x) = 0. Proof. If Tx0 = x0, then we are done. Suppose that the conclusion is not true. Then there exists x0 ∈ X such that (x0,Tx0) ∈ X≤. Since T is nondecreasing, we have (Tx0,T 2x0) ∈ X≤. Continuing this process, we obtain (Tnx0,Tmx0) ∈ X≤ for all n,m ∈ N. Now, we claim that (4.2) lim n→∞ P(Tnx0,T n+1x0) = 0. By the assumption (i) and the property of ζ, we observe that (4.3) 0 ≤ ζ(KP(Tnx0,Tn+1x0),P(Tn−1x0,Tnx0)) ≤ P(Tn−1x0,Tnx0) −KP(Tnx0,Tn+1x0) for all n ∈ N. Since K ≥ 1 and using (4.3), we get (4.4) P(Tnx0,T n+1x0) ≤ KP(Tnx0,Tn+1x0) ≤ P(Tn−1x0,Tnx0). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 97 C. Mongkolkeha, Y. J. Cho and P. Kumam This mean that the sequence {P(Tnx0,Tn+1x0)} is a decreasing sequence of nonnegative real numbers and so it is convergent to some r ≥ 0. Suppose that r > 0. Case I. If K > 1, letting n → ∞ in (4.4), we get r ≤ Kr ≤ r which is a contradiction. Case II. If K = 1, putting tn = P(T n+1x0,T n+2x0) and sn = P(T nx0,T n+1x0), the sequences {Ktn} and {sn} have the same positive limit. Also, the sequences {Ktn} and {sn} have the same positive limit superior and verify that tn < sn for all n ∈ N. By the condition (ζ3) of definition 4.1 we have lim sup n→∞ ζ(KP(Tn+1x0,T n+2x0),P(T nx0,T n+1x0)) = lim sup n→∞ ζ(Ktn,sn) < 0, which is a contradiction. Therefore r = 0, that is, the claim (4.3) holds. Next, we show that (4.5) lim m,n→∞ P(Tnx0,T mx0) = 0. Suppose that this is not true. Then we can find ε0 > 0 with the sequences {mk}, {nk} such that, for any mk > nk such that (4.6) P(Tnkx0,T mkx0) > ε0 for all k ∈{1, 2, 3, · · ·}. We can assume that mk is a minimum index such that (4.6) holds. Then we also have (4.7) P(Tnkx0,T mk−1x0) ≤ ε0. Hence we have ε0 < P(T nkx0,T mkx0) ≤ K[P(Tnkx0,Tmk−1x0) + P(Tmk−1x0,Tmkx0)] < Kε0 + KP(T mk−1x0,T mkx0). Taking limit superior as k → ∞ in the above inequality and using (4.2), we have (4.8) ε0 < lim sup k→∞ P(Tnkx0,T mkx0) ≤ Kε0. Now, we claim that lim sup n→∞ P(Tnk+1x0,T mk+1x0) < ε0. If lim sup k→∞ P(Tnk+1x0,T mk+1x0) ≥ ε0, then there exists {kr} and δ > 0 such that (4.9) lim sup r→∞ P(Tnkr +1x0,T mkr +1x0) = δ ≥ ε0. By the assumption (i) and the property of ζ, we have (4.10) 0 ≤ ζ(KP(Tnkr +1x0,Tmkr +1x0),P(Tnkr x0,Tmkr x0)) ≤ P(Tnkr x0,Tmkr x0) −KP(Tnkr +1x0,Tmkr +1x0). Hence, (4.11) KP(Tnkr +1x0,T mkr +1x0) ≤ P(Tnkr x0,Tmkr x0), c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 98 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance it follows from (4.8), (4.9) and (4.11), we get that Kδ = lim sup r→∞ KP(Tnkr +1x0,T mkr +1x0) ≤ lim sup r→∞ P(Tnkr x0,T mkr x0) ≤ Kε0 ≤ Kδ. Therefore the sequence {Ktkr := KP(Tnkr +1x0,Tmkr +1x0)} and {skr := P(Tnkr x0,Tmkr x0)} have the same positive limit superior and ver- ify that tkr < skr for all r ∈ N. By the property (ζ3), we conclude that 0 ≤ lim sup r→∞ ζ(KP(Tnkr +1x0,T mkr +1x0),P(T nkr x0,T mkr x0)) = lim sup r→∞ ζ(Ktkr,skr ) < 0, which is a contradiction and hence (4.5) hold. It follows from Lemma 2.14 (iii) that {Tnx0} is a Cauchy sequence. Since X is a complete b−metric space, the sequence {Tnx0} converges to some element z ∈ X. From the fact that limm,n→∞P(T nx0,T mx0) = 0, for each ε > 0, there exists Nε ∈ N such that n > Nε implies P(TNεx0,T nx0) < ε. Since P(x, ·) is K-lower semi-continuous and the sequence {Tnx0} converges to z, we have (4.12) P(TNεx0,z) ≤ lim inf n→∞ KP(TNεx0,T nx0) ≤ Kε. Setting ε = 1 k2 and Nε = nk, by (4.12), we have (4.13) lim k→∞ P(Tnkx0,z) = 0. Now, we prove that z is a fixed point of T. Suppose that Tz 6= z. Since (Tnkx0,T nk+1x0) ∈ X≤ for each n ∈ N, using the assumption (ii), (4.2) and (4.13), we have 0 < inf{P(Tnkx0,z) + P(Tnkx0,Tnk+1x0)}→ 0 as n →∞, which is a contradiction. Therefore, Tz = z. If Tx = x, we distinguish two cases. case I If K = 1, then 0 ≤ ζ(P(Tx,T2x),P(x,Tx)) = ζ(P(x,x),P(x,x)) ≤ P(x,x) −P(x,x) = 0. Hence ζ(P(Tx,T2x),P(x,Tx)) = 0 and so, by (ζ1), we obtain P(x,x) = 0. case II If K > 1, then 0 ≤ ζ(KP(Tx,T2x),P(x,Tx)) = ζ(KP(x,x),P(x,x)) ≤ P(x,x) −KP(x,x) = (1 −K)P(x,x), it follow that P(x,x) ≤ 0 and thus we must have P(x,x) = 0. This completes the proof. � Now, we give an example to illustrate Theorem 4.3. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 99 C. Mongkolkeha, Y. J. Cho and P. Kumam Example 4.4. Let X = [0, 1] and D(x,y) = (x−y)2 with the wt-distance P on X defined by P(x,y) = |y|2. We consider the following set: X≤ = { (x,y) ∈ X ×X : x = y or x,y ∈{0}∪{ 1 2n : n ≥ 1} } with the usual ordering. Let T : X → X be a mapping defined by T(x) =   12n+1 , if x = 1 2n , n ≥ 1, 0, otherwise. for all x ∈ X. Obviously, T is nondecreasing. Also, T satisfies the condition (ii). Indeed, for any n ∈ N, we have 1 2n 6= T( 1 2n ). Moreover, for each n ∈ N, we have inf { P ( 1 2m , 1 2n ) + P ( 1 2m , 1 2m − 1 22m+1 ) : m ∈ N } = 1 22n > 0. Let ζ : [0,∞) × [0,∞) → R define by ζ(t,s) = s−Kt 1 + Ks for all s,t ∈ [0,∞). Similarly, in Example 4.2, the function define as above is simulation function in the sense of Definition 4.1. Now, we show that T satisfies the condition (i). Let given x = 1 2n with ( 1 2n ,T( 1 2n )) ∈ X≤. Then we have ζ(2P(Tx,T2x),P(x,Tx)) = ζ(2P( 1 2n+1 , 1 2n+2 ),P( 1 2n , 1 2n+1 )) = ζ(2 1 22n+4 , 1 22n+2 ) = 1 22n+2 − 2 · 1 22n+4 1 + 2 · 1 22n+2 = 22n+3 − 22n+2 (22n+2)(22n+3) · 22n+1 22n+1 + 1 = 22n+2(2 − 1) (22n+4)(22n+1 + 1) = 22n+2 (22n+4)(22n+1 + 1) > 0. Therefore, all the hypothesis of Theorem 4.3 are satisfied and, further, x = 0 is a fixed point of T . c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 100 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance Corollary 4.5. Let (X,≤) be a partially ordered set and (X,D) be a complete metric type space with constant K ≥ 1 and P be a wt-distance on X. Suppose that T : X → X is a nondecreasing mapping satisfying the following conditions: (i) there exists α ∈ [0, 1 K ) such that P(Tx,T2x) ≤ αP(x,Tx) for all x ≤ Tx; (ii) for all x ∈ X with x ≤ Tx, inf{P(x,y) + P(x,Tx)} > 0 for all y ∈ X with y 6= Ty; (iii) there exists x0 ∈ X such that x0 ≤ Tx0. Then T has a fixed point in X. Theorem 4.6. Let (X,≤) be a partially ordered set and (X,D) be a complete b-metric space with constant K ≥ 1 and P be a wt-distance on X. Suppose that T : X → X is a nondecreasing mapping and there exists ζ ∈Z such that ζ(KP(Tx,T2x),P(x,Tx)) ≥ 0 for all (x,Tx) ∈ X≤. Assume that one of the following conditions holds: (i) for all x ∈ X with (x,Tx) ∈ X≤, inf{P(x,y) + P(x,Tx)} > 0 for all y ∈ X with y 6= Ty; (ii) if both {xn} and {Txn} converge to z, then z = Tz; (iii) T is continuous on X. If there exists x0 ∈ X such that (x0,Tx0) ∈ X≤, then T has a fixed point in X. Moreover, if Tx = x, then P(x,x) = 0. Proof. In the case of T satisfying the condition (i), the conclusion was proved in Theorem 4.3. Let us prove that (ii) =⇒ (i). Suppose that the condition (ii) holds. Let y ∈ X with y 6= Ty such that inf{P(x,y) + P(x,Tx) : (x,Tx) ∈ X≤} = 0. Then we can find a sequence {zn} such that (zn,Tzn) ∈ X≤ and inf{P(zn,y) + P(zn,Tzn)} = 0. So we have lim n→∞ P(zn,y) = lim n→∞ P(zn,Tzn) = 0. Again, by Lemma 2.14, we have limn→∞Tzn = y. Moreover, limn→∞T 2zn = y. In fact, since (4.14) 0 ≤ ζ(KP(Tzn,T2zn),P(zn,Tzn)) ≤ P(zn,Tzn) −KP(Tzn,T2zn), it follow from (4.14) and K ≥ 1, we get that lim n→∞ P(Tzn,T 2zn) ≤ lim n→∞ KP(Tzn,T 2zn) ≤ lim n→∞ P(zn,Tzn) = 0. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 101 C. Mongkolkeha, Y. J. Cho and P. Kumam Letting xn = Tzn, the sequences {xn} and {Txn} converge to y. Hence, by the assumption (ii), y = Ty and so (ii) =⇒ (i). Obviously, (iii) =⇒ (ii). This completes the proof. � Now, we prove new theorems by replacing some conditions in Theorem 4.3 with other conditions. Theorem 4.7. Let (X,≤) be a partially ordered set and (X,D) be a complete b-metric space with constant K ≥ 1 and P be a wt-distance on X. Suppose that T : X → X is a nondecreasing satisfying the following conditions: (i) there exists ζ ∈Z such that ζ(KP(Tx,T2x),P(x,Tx)) ≥ 0 for all (x,Tx) ∈ X≤; (ii) there exists x0 ∈ X such that (x0,Tx0) ∈ X≤, (iii) either T is orbitally continuous at x0 or (iv) T is orbitally X≤-continuous and there exists a subsequence {Tnkx0} of {Tnx0} converges to some element x? ∈ X such that (Tnkx0,x?) ∈ X≤ for any k ∈ N. Then T has a fixed point in X. Moreover if Tx = x, then P(x,x) = 0. Proof. If Tx0 = x0, then we are done. Suppose that the conclusion is not true. Then there exists x0 ∈ X such that (x0,Tx0) ∈ X≤. Since T is monotone, we have (Tx0,T 2x0) ∈ X≤. Continuing this process, we have a sequence {Tnx0} such that (Tnx0,T mx0) ∈ X≤ for any n,m ∈ N. As in the same argument in Theorem 4.3, we can see that (4.15) lim n→∞ P(Tnx0,T n+1x0) = 0. Moreover, (4.16) lim m,n→∞ P(Tnx0,T mx0) = 0. and {Tnx0} is a Cauchy sequence converges to some element z ∈ X. Next, we prove that z is a fixed point of T. If the condition (iii) holds, then Tn+1x0 → Tz. By P(x, ·) is K-lower semi-continuous and (4.16), we have (4.17) P(T nx0,z) ≤ lim inf m→∞ KP(Tnx0,T mx0) ≤ α ′ n (say) and (4.18) P(T nx0,Tz) ≤ lim inf m→∞ KP(Tnx0,T m+1x0) ≤ β ′ n, (say) where the sequences {α ′ n := αn K } and {β ′ n := βn K } which converges to 0. By Lemma 2.14 (i), we conclude that z = Tz. Suppose that the condition (iv) hold. From the fact that {Tnkx0} → z as k → ∞, (Tnkx0,z) ∈ X≤ and T is orbitally X≤-continuous, it follows that {Tnk+1x0}→ Tz as k →∞. Similarly, since P(x, ·) is K-lower semi-continuous c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 102 Fixed point theorems for simulation functions in b-metric spaces via the wt-distance as above, we conclude that z = Tz and the remaining part of the proof follow from the proof of Theorem 4.3. � Corollary 4.8. Let (X,≤) be a partially ordered set and (X,D) be a complete metric space and p be a w-distance on X. Suppose that T : X → X is a nondecreasing satisfying the following conditions: (i) there exists ζ ∈Z such that ζ(p(Tx,T2x),p(x,Tx)) ≥ 0 for all (x,Tx) ∈ X≤; (ii) there exists x0 ∈ X such that (x0,Tx0) ∈ X≤, (iii) either T is orbitally continuous at x0 or (iv) T is orbitally X≤-continuous and there exists a subsequence {Tnkx0} of {Tnx0} converges to some element x? ∈ X such that (Tnkx0,x?) ∈ X≤ for any k ∈ N. Then T has a fixed point in X. Moreover if Tx = x, then p(x,x) = 0. Corollary 4.9. Let (X,≤) be a partially ordered set and (X,D) be a complete b-metric space with constant K ≥ 1 and P be a wt-distance on X. Suppose that T : X → X is a nondecreasing satisfying the following conditions: (i) there exists λ ∈ [0, 1 K ) such that P(Tx,T2x) ≤ λP(x,Tx) for all (x,Tx) ∈ X≤; (ii) there exists x0 ∈ X such that (x0,Tx0) ∈ X≤, (iii) either T is orbitally continuous at x0 or (iv) T is orbitally X≤-continuous and there exists a subsequence {Tnkx0} of {Tnx0} converges to some element x? ∈ X such that (Tnkx0,x?) ∈ X≤ for any k ∈ N. Then T has a fixed point in X. Moreover, if Tx = x, then P(x,x) = 0. Example 4.10. Let X = [0, 1] and D(x,y) = (x−y)2 with the wt-distance P on X defined by P(x,y) = |y|2. We consider the following set: X≤ = { (x,y) ∈ X ×X : x = y or x,y ∈{0}∪{ 1 n : n ≥ 1} } , where ≤ is the usual ordering. Let T : X → X be a mapping define by T(x) =   x2, if x = 1 n , n ≥ 2, x 2 , otherwise. Then T is a nondecreasing mapping. Also, x = 0 is an element in X such that 0 ≤ T(0) = 0 and so (0,T(0)) ∈ X≤. Hence T satisfies the condition (ii). Next, we show that T satisfies the condition (i) of Theorem 4.7 with the simulation function in given in Example 4.4. If x 6= 1 n for all n ≥ 2, then c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 103 C. Mongkolkeha, Y. J. Cho and P. Kumam (x,T(x)) ∈ X≤ and it is easy to see that T satisfies the condition (i). If x = 1 n for all n ≥ 2, then ( 1 n ,T 1 n ) ∈ X≤. Further, we have ζ(2P(Tx,T2x),P(x,Tx)) = ζ ( 2P ( 1 n2 , 1 n4 ) ,P ( 1 n , 1 n2 )) = ζ ( 2 ( 1 n4 )2 , ( 1 n2 )2) = ( 1 n2 )2 − 2 ( 1 n4 )2 1 + 2 · ( 1 n2 )2 = n8 − 2n4 n12 · n4 n4 + 2 = n8 − 2n4 n8(n4 + 2) = n4 − 2 n4(n4 + 2) > 0. Hence T satisfies the condition (i). Furthermore, for each x ∈ X, Tni (x) → 0 ∈ X as i → ∞, and also Tni+1(x) → T(0) ∈ X as i → ∞. Hence all the conditions of Theorem 4.7 are satisfied. Furthermore, x = 0 is fixed points of T. Acknowledgements. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Sci- ence for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT. The first author was supported by Thailand Research Fund (Grant No. TRG5880221) and Kasetsart University Research and Development In- stitute (KURDI). Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). The authors are also grateful to the referee by several useful suggestions that have improved the first version of the paper. References [1] A. N. Abdou, Y. J. Cho and R. Saadati, Distance type and common fixed point theorems in Menger probabilistic metric type spaces, Appl. Math. Comput. 265 (2015), 1145–1154. [2] A. D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. 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