@ Appl. Gen. Topol. 20, no. 1 (2019), 1-18doi:10.4995/agt.2019.7638 c© AGT, UPV, 2019 Common fixed points for generalized ψ-contractions in weak non-Archimedean fuzzy metric spaces Suthep Suantai a, Yeol Je Cho b and Jukrapong Tiamme c,1 a Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand (suthep.s@cmu.ac.th) b Department of Education and the RINS, Gyeongsang National University, Jinju 660-701, Re- public of Korea (yjcho@gnu.ac.kr) c Department of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University, Chiang Mai 50300, Thailand (jukrapong.benz@gmail.com) Communicated by M. Abbas Abstract Fixed point theory in fuzzy metric spaces plays very important role in theory of nonlinear problems in applied science. In this paper, we prove an existence result of common fixed point of four nonlinear mappings satisfying a new type of contractive condition in a generalized fuzzy metric space, called weak non-Archimedean fuzzy metric space. Our main results can be applied to solve the existence of solutions of non- linear equations in fuzzy metric spaces. Some examples supporting our main theorem are also given. Our results improve and generalize some recent results contained in Vetro (2011) [16] to generalized contractive conditions under some suitable conditions and many known results in the literature. 2010 MSC: 47H04; 47H10. Keywords: common fixed point; compatible mappings; fuzzy metric spaces; weak non-Achimedean fuzzy metric spaces; fuzzy topology; com- mon limit. 1Corresponding author. Received 03 May 2017 – Accepted 16 February 2019 http://dx.doi.org/10.4995/agt.2019.7638 S. Suantai, Y. J. Cho and J. Tiammee 1. Introduction Fixed point theory in fuzzy metric spaces plays an important role in studying theory of equations in a fuzzy metric space. It can be applied to solve the existence problems of nonlinear equations in fuzzy metric spaces (see [2]-[6], [14]). Zadeh [17] was the first who introduced the concept of fuzzy sets. Since then, the concept of fuzzy metric spaces was introduced and study by many authors in different ways [15]-[16]. In 1974, George and Veeramani [2], [3] defined a Hausdorff topology in fuzzy metric spaces which improve from Kramosil and Michalek [9]. Especially, in 2008, Mihet [12] proved a fixed point theorem for fuzzy ψ-contractive mappings in complete non-Archimedean fuzzy metric spaces. In 2011, Vetro [16] introduced a notion of a weak non-Archimedean fuzzy metric space which induces a Hausdorff topology and proved some common fixed point theorems for a pair of generalized contractive type mappings in this space. Recently, Martinez-Moreno [11] and others ([13],[6]) proved some common fixed point theorems for two contractive mappings with the CLRg- property in fuzzy metric spaces and other spaces. In this paper, we aim to prove a common fixed point result of four nonlinear mappings satisfying a new type of contractive condition and some compatible conditions in weak non-Archimedean fuzzy metric spaces. The new type of contractive condition introduced in this paper is a new concept and it is suitable for studying a common fixed point problem of four nonlinear mappings while the previous ones cannot be used to study our problem because they can be used to study a common fixed point of only two nonlinear mappings. Our main result can be applied to study existence problem of nonlinear equations in weak non-Archimedean fuzzy metric spaces. Our results extend and generalize many results in literature, especially, Vetro [16], Mihet [12] and others. Also, we give some examples to illustrate our main results. 2. Preliminaries In this section, we recall some notion and basic useful definitions. Definition 2.1 ([15]). A binary operation ∗ : [0,1] × [0,1] → [0,1] is called a t-norm if it satisfies the following conditions: (TN1) ∗ is associative and commutative; (TN2) a ∗ 1 = a for any a ∈ [0,1]; (TN3) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a,b,c,d ∈ [0,1]. If ∗ is continuous, then ∗ is called the continuous t-norm. The following are some examples of a continuous t-norm; a∗b = min{a,b}, a∗b = ab/ max{a,b,λ}, where λ ∈ (0,1), a ∗ b = max{a + b − 1,0}. Definition 2.2 ([16]). A fuzzy metric space is a triple (X,M,∗), where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × [0,∞), satisfying the following properties: (FM1) M(x,y,0) = 0 for all x,y ∈ X; c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 2 Common fixed points for generalized ψ-contractions (FM2) M(x,y,t) = 1 for all t > 0 if and only if x = y; (FM3) M(x,y,t) = M(y,x,t) for all x,y ∈ X and t > 0; (FM4) M(x,y, ·) : [0,∞) → [0,1] is left continuous for all x,y ∈ X; (FM5) M(x,z,t+s) ≥ M(x,y,t)∗M(y,z,s) for all x,y,z ∈ X and t,s > 0. From the above definition, if (FM5) is replaced by the following: (NA) M(x,z,max{t,s}) ≥ M(x,y,t) ∗ M(y,z,s) for all x,y ∈ X and t,x > 0, then (X,M,∗) is called a non-Archimedean fuzzy space. Obviously, every non- Archimedean fuzzy metric space is itself a fuzzy metric space. Definition 2.3 ([16]). A weak non-Archimedean fuzzy metric space is a triple (X,M,∗), where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × [0,∞), satisfying (FM1)-(FM4) and (WNA) M(x,z,t) ≥ max{M(x,y,t)∗M(x,z,t/2),M(x,y,t/2)∗M(y,z,t/2)} for all x,y,z ∈ X and t > 0. Remark 2.4. (1) Every non-Archimedean fuzzy metric spaces is itself a weak non-Archimedean fuzzy metric space. (2) A weak non-Archimedean fuzzy metric space is not necessarily a fuzzy metric space. Example 2.5 ([16]). Let X = [0,∞) and a ∗ b = ab for all a,b ∈ [0,1]. Define a mapping M : X2 × [0,∞) → [0,1] by: M(x,y,0) = 0, M(x,x,t) = 1 for all t > 0, M(x,y,t) = t for x 6= y and 0 < t ≤ 1, M(x,y,t) = t/2 for x 6= y and 1 < t ≤ 2, M(x,y,t) = 1 for x 6= y and t > 2. Then (X,M,∗) is a weak non-Archimedean fuzzy metric space, but it is not a fuzzy metric space. Definition 2.6 ([16]). Let (X,M,∗) be a weak non-Archimedean fuzzy metric space. We define an open ball in X by B(x,r,t) = {y ∈ X : M(x,y,t) > 1 − r} for any x ∈ X, r ∈ (0,1) and t > 0. Proposition 2.7 ([16]). Let (X,M,∗) be a weak non-Archimedean fuzzy metric space. Then we have the following: (1) Every open ball is an open set; (2) The family τ = {A ⊂ X : x ∈ A iff there exist t > 0 and r ∈ (0,1) with B(x,r,t) ⊂ A} is a topology on X; (3) Every weak non Archimedean fuzzy metric space (X,M,∗) is a Haus- dorff space. Proposition 2.8 ([16]). Let (X,M,∗) be a weak non-Archimedean fuzzy met- ric space. A sequence {xn} in a weak non-Archimedean fuzzy metric space (X,M,∗) is convergent to x ∈ X if and only if limn→∞ M(xn,x,t) = 1 for all t > 0. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 3 S. Suantai, Y. J. Cho and J. Tiammee Proposition 2.9 ([16]). Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and (xn) ⊂ X be a sequence convergent to x ∈ X. Then lim n→∞ M(y,xn, t) = M(y,x,t) for all y ∈ X and t > 0. Proposition 2.10. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space. Suppose that {xn} and {yn} are two sequences in X such that limn→∞ xn = x and limn→∞ yn = y. Then lim n→∞ M(xn,yn, t) = M(x,y,t) for all t > 0. Proof. Let {xn} and {yn} be two sequences in X such that limn→∞ xn = x and limn→∞ yn = y. Then, by the condition (WNA) and Proposition 2.9, we have M(y,xn, t) ≥ M(y,yn, t/2) ∗ M(yn,xn, t) and M(xn,yn, t) ≥ M(xn,x,t/2) ∗ M(x,yn, t). It follows that M(x,y,t) ≤ lim inf n→∞ M(xn,yn, t) ≤ lim sup n→∞ M(xn,yn, t) ≤ M(x,y,t). Therefore, limn→∞ M(xn,yn, t) = M(x,y,t) for all t > 0. This completes the proof. � Definition 2.11 ([16]). Let (X,M,∗) be a weak non-Archimedean fuzzy metric space. A sequence {xn} in X is called a Cauchy sequence if, for each ǫ ∈ (0,1) and t > 0, there exists N ∈ N such that M(xn,xm, t) > 1 − ǫ for all m,n ≥ N. A weak non-Archimedean fuzzy metric space (X,M,∗) is said to be complete if every Cauchy sequence is convergent. Definition 2.12 ([10]). An element x ∈ X is called a common fixed point of the mappings S,T,A,B : X → X if x = Sx = Tx = Ax = Bx. Definition 2.13 ([16]). The self-mappings S and T of a weak non-Archimedean fuzzy metric space (X,M,∗) are said to be compatible if lim n→∞ M(STxn,TSxn, t) = 1 for all t > 0 whenever {xn} is a sequence in X such that lim n→∞ Sxn = lim n→∞ Txn = u for some u ∈ X. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 4 Common fixed points for generalized ψ-contractions Definition 2.14 ([8]). The self-mappings S and T of a nonempty set X are said to be weak compatible if STz = TSz whenever Sz = Tz for some z ∈ X. 3. Common fixed points for compatible mappings In this section, we introduce the notion of ψ-contractions of four self-mappings and prove some common fixed points theorems for a ψ-contraction in complete non-Archimedean fuzzy metric spaces. Let ψ : [0,1] → [0,1] be a function such that (a) ψ is nondecreasing and left continuous; (b) ψ(t) > t for all t ∈ (0,1). We denote Ψ := {ψ : [0,1] → [0,1] : ψ satisfies (a) − (b)}. Lemma 3.1 ([8]). If ψ ∈ Ψ, then (1) limn→∞ ψ n(t) = 1 for all t ∈ (0,1); (2) ψ(1) = 1. Definition 3.2. Let X be a nonempty set and M be a fuzzy set on X2×[0,∞). Let A,B,S,T : X → X be four mappings. The four couple (A,B;S,T) is called ψ-contractive mappings if there exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,By,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t)}. Let A,B,S,T : X → X be four mappings such that A(X) ⊂ T(X) and B(X) ⊂ S(X). Suppose that there exist x0,x1 ∈ X such that Ax0 = Tx1 and M(x0,x1, t) > 0. Let x0 ∈ X. Since A(X) ⊂ T(X) and B(X) ⊂ S(X), there exist x0,x1 ∈ X such that Ax0 = Tx1 = y0. Also, since A(X) ⊂ T(X) and B(X) ⊂ S(X), there exists x2 ∈ X such that Bx1 = Sx2 = y1. Inductively, we can define a sequence {yn} in X such that Ax2n = Tx2n+1 = y2n, Bx2n+1 = Sx2n+2 = y2n+1(Ω) for each n ≥ 0. Lemma 3.3. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and A,B,S,T be four self-mappings of X satisfying the following conditions: (1) A(X) ⊂ T(X) and B(X) ⊂ S(X); (2) The pairs A,S and B,T are compatible; (3) One of S,T,A and B is continuous; (4) The four couple (A,B;S,T) is ψ-contractive mappings. Suppose that there exist x0,x1 ∈ X such that Ax0 = Tx1 and M(x0,x1, t) > 0. Then the sequence {yn} in X generated by (Ω) with initial point x0,x1 is a Cauchy sequence. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 5 S. Suantai, Y. J. Cho and J. Tiammee Proof. First, from the sequence {yn} in X generated by (Ω), we show that lim n→∞ M(yn,yn+1, t) = 1 for all t > 0. Assume that M(yn,yn+1, t) < 1 for all n ≥ 1. Since M(Ax0,Bx1, t) = M(y0,y1, t) > 0, we obtain M(y2,y1, t) = M(Ax2,Bx1, t) ≥ ψ(m(x2,x1, t)) = ψ(min{M(Sx2,Tx1, t),M(Ax2,Sx2, t),M(Bx1,Tx1, t)}) = ψ(min{M(y1,y0, t),M(y2,y1, t),M(y1,y0, t)}) = ψ(min{M(y1,y0, t),M(y2,y1, t)}). Suppose that M(y1,y0, t) > M(y2,y1, t) then M(y2,y1, t) ≥ ψ(M(y2,y1, t)) > M(y2,y1, t), which is a contradiction. Therefore M(y1,y0, t) ≤ M(y2,y1, t), which implies that M(y2,y1, t) ≥ ψ(M(y1,y0, t)) > 0. Again, we consider M(y2,y3, t) = M(Ax2,Bx3, t) ≥ ψ(m(x2,x3, t)) = ψ(min{M(Sx2,Tx3, t),M(Ax2,Sx2, t),M(Bx3,Tx3, t)}) = ψ(min{M(y1,y2, t),M(y2,y1, t),M(y3,y2, t)}) = ψ(min M(y2,y1, t),M(y3,y2, t)}) Suppose that M(y2,y1, t) > M(y3,y2, t) then M(y2,y3, t) ≥ ψ(M(y2,y3, t)) > M(y2,y3, t), which is a contradiction. Therefore M(y2,y1, t) ≤ M(y2,y3, t), which implies that M(y2,y3, t) ≥ ψ(M(y2,y1, t)) ≥ ψ2(M(y0,y1, t)) > 0. Therefore, for any n ∈ N we have M(yn+1,yn, t) ≥ ψn(M(y0,y1, t)) > 0. By Lemma 3.1, as n → ∞, we obtain lim n→∞ M(yn+1,yn, t) = 1. Next, we show that the sequence {yn} is a Cauchy sequence. If {yn} is not Cauchy, then there exist ǫ ∈ (0, 1 2 ) and t > 0 such that, for each k ≥ 1, there exist m(k),n(k) ∈ N such that m(k) > n(k) ≥ k and M(ym(k),yn(k), t) ≤ 1−2ǫ. By (WNA), we have 1 − 2ǫ ≥ M(ym(k),yn(k), t) ≥ M(ym(k),yn(k)+1, t) ∗ M(yn(k)+1,yn(k), t/2), c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 6 Common fixed points for generalized ψ-contractions which implies, by k → ∞, that 1 − 2ǫ ≥ lim sup k→∞ M(ym(k),yn(k)+1, t) ∗ M(yn(k)+1,yn(k), t/2) = lim sup k→∞ M(ym(k),yn(k)+1, t). Similarly, we obtain 1 − 2ǫ ≥ lim sup k→∞ M(ym(k)+1,yn(k), t), 1 − 2ǫ ≥ lim sup k→∞ M(ym(k)+1,yn(k)+1, t). So, we can assume that m(k) are odd numbers, n(k) are even numbers and M(xm(k),xn(k), t) ≤ 1 − ǫ for all k ≥ 1. Define q(k) = min{m(k) : M(ym(k),yn(k), t) ≤ 1−ǫ, m(k) is odd number}. By (WNA), we have 1 − ǫ ≥ M(yq(k),yn(k), t) ≥ M(yq(k),yq(k)−2, t/2) ∗ M(yq(k)−2,yn(k), t) ≥ M(yq(k),yq(k)−1, t/4) ∗ M(yq(k)−1,yq(k)−2, t/2) ∗ (1 − ǫ). Letting k → ∞, we obtain limk→∞ M(yq(k),yn(k), t) = 1 − ǫ. By (WNA) and the condition (4), we have M(yq(k),yn(k), t) ≥ M(yq(k),yn(k)+1, t) ∗ M(yn(k)+1,yn(k), t/2) ≥ M(yq(k)+1,yn(k)+1, t) ∗ M(yq(k)+1,yq(k), t) ∗ M(yn(k)+1,yn(k), t/2) = M(Sxq(k)+1,Txn(k)+1, t) ∗ M(yq(k)+1,yq(k), t) ∗ M(yn(k)+1,yn(k), t/2) ≥ ψ(m(xq(k)+1,xn(k)+1, t)) ∗ M(yq(k)+1,yq(k), t) ∗ M(yn(k)+1,yn(k), t/2),(*) where m(xq(k)+1,xn(k)+1, t) = min{M(Sxq(k)+1,Txn(k)+1, t),M(Axq(k)+1,Sxq(k)+1, t), M(Bxn(k)+1,Txn(k)+1, t)} = min{M(yq(k),yn(k), t),M(yq(k)+1,yq(k), t),M(yn(k)+1,yn(k), t)}. Since lim k→∞ m(xq(k)+1,xn(k)+1, t) = min{ lim k→∞ M(yq(k),yn(k), t), lim k→∞ M(yq(k)+1,yq(k), t), lim k→∞ M(yn(k)+1,yn(k), t)} = min{1 − ǫ,1,1} = 1 − ǫ By taking k → ∞ in (*), we obtain 1 − ǫ ≥ ψ(1 − ǫ) ∗ 1 ∗ 1 > 1 − ǫ, which is a contradiction. Therefore, {yn} is a Cauchy sequence. This completes the proof. � c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 7 S. Suantai, Y. J. Cho and J. Tiammee Now, we are ready to state and prove our main results. Theorem 3.4. Let (X,M,∗) be a complete weak non-Archimedean fuzzy metrci space and A,B,S,T be the self-mappings of X satisfying the following condi- tions: (1) A(X) ⊂ T(X) and B(X) ⊂ S(X); (2) The pairs A,S and B,T are compatible; (3) One of A,B,S and T is continuous; (4) The four couple (A,B;S,T) is a ψ-contractive mapping. Suppose that there exist x0,x1 ∈ X such that Ax0 = Tx1 and M(x0,x1, t) > 0. Assume that, for any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. Then A,B,S and T have a unique common fixed point. Proof. By Lemma 3.3, the sequence {yn} is a Cauchy sequence. Since X is complete, there exists z ∈ X such that limn→∞ yn = z. From Ax2n = Tx2n+1 = y2n and Bx2n+1 = Sx2n+2 = y2n+1 for all n ≥ 1, we obtain lim n→∞ Ax2n = lim n→∞ Tx2n+1 = lim n→∞ Bx2n+1 = lim n→∞ Sx2n+2 = z.(3.1) For the proof, we divide 4 Cases for the continuity of A,B,S and T . Case 1. Suppose that S is continuous. Then we have lim n→∞ SAx2n = lim n→∞ S2x2n = Sz.(3.2) Since A and S are compatible mappings, limn→∞ M(SAx2n,ASx2n, t) = 1. Thus, from M(ASx2n,Sz,t) ≥ M(ASx2n,SAx2n, t) ∗ M(SAx2n,Sz,t/2) as n → ∞, we obtain limn→∞ M(ASx2n,Sz,t) = 1, i.e. lim n→∞ ASx2n = Sz.(3.3) First, wet prove that z is a common fixed point of A and S. If Sz 6= z, then there exists t > 0 such that 0 < M(Sz,z,t) < 1. By the condition (4), we have M(ASx2n,Bx2n+1, t) ≥ ψ(m(Sx2n,x2n+1, t)), where m(Sx2n,x2n+1, t) = min{M(S2x2n,Tx2n+1, t),M(ASx2n,S2x2n, t),M(Tx2n+1,Bx2n+1, t)}. Lettinge n → ∞, by (3.1), (3.2) and (3.3), we obtain M(Sz,z,t) ≥ ψ(min{M(Sz,z,t),M(Sz,Sz,t),M(z,z,t)}) = ψ(min{M(Sz,z,t),1,1}) = ψ(M(Sz,z,t)) > M(Sz,z,t), c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 8 Common fixed points for generalized ψ-contractions which is a contradiction. Therefore, Sz = z. If Az 6= z, then there exists t > 0 such that 0 < M(Az,z,t) < 1. By the condition (4), we have M(Az,Bx2n+1, t) ≥ ψ(m(z,x2n+1, t)), where m(z,x2n+1, t) = min{M(Sz,Tx2n+1, t),M(Az,Sz,t),M(Tx2n+1,Bx2n+1, t)}. Letting n → ∞, by (3.1), (3.2) and (3.3), we obtain M(Az,z,t) ≥ ψ(min{M(Sz,z,t),M(Az,Sz,t),M(z,z,t)}) = ψ(min{M(z,z,t),M(Az,z,t),M(z,z,t)}) = ψ(min{1,M(Az,z,t),1}) = ψ(M(Az,z,t)) > M(Az,z,t), which is a contradiction and hence Az = z. Therefore, z is a common fixed point of A and S. Since A(X) ⊂ BT(X), there exists z∗ ∈ X such that z = Az = Tz∗. If z 6= Tz∗, then there exists t > 0 such that 0 < M(z,Bz∗, t) < 1. By the condition (4) and Lemma 3.1, we have M(z,Bz∗, t) = M(Az,Bz∗, t) ≥ ψ(m(z,z∗)) = ψ(M(z,Bz∗, t)) > M(z,Bz∗, t), which is a contradiction. Then z = Bz∗. Since B and T are compatible, we obtain M(Tz,Bz,t) = M(TBz∗,BTz∗t) = 1 for any t > 0, which implies Tz = Bz. Next, if z 6= Bz, then there exists t > 0 such that 0 < M(z,Bz,t) < 1. By the condition (4), we have M(z,Bz,t) = M(Az,Bz,t) ≥ ψ(m(z,z)) = ψ(M(z,Bz,t)) > M(z,Bz,t), which is a contradiction. Hence z = Tz and so z = Tz = Bz = Sz = Az. Case 2. Suppose that T is continuous. In the same way as in Case 1, we can obtain the result. Case 3. Suppose that A is continuous. Then lim n→∞ ASx2n = lim n→∞ A2x2n = Az.(3.4) Since S and A are compatible mappings, limn→∞ M(ASx2n,SAx2n, t) = 1. By (WNA), we have M(SAx2n,Az,t) ≥ M(SAx2n,ASx2n, t) ∗ M(ASx2n,Az,t/2). Letting n → ∞, we obtain limn→∞ M(SAx2n,Az,t) = 1, i.e., lim n→∞ SAx2n = Az.(3.5) If Az 6= z, then there exists t > 0 such that 0 < M(Az,z,t) < 1. By the condition (4), we have M(A2x2n,Bx2n+1, t) ≥ ψ(m(Ax2n,x2n+1, t)), c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 9 S. Suantai, Y. J. Cho and J. Tiammee where m(Ax2n,x2n+1, t) = min{M(SAx2n,Tx2n+1, t),M(A2x2n,SAx2n, t),M(Tx2n+1,Bx2n+1, t)}. Letting n → ∞, by (3.1), (3.4) and (3.5), we obtain M(Az,z,t) ≥ ψ(min{M(Az,z,t),M(Az,Az,t),M(z,z,t)}) = ψ(min{M(Az,z,t),1,1}) = ψ(M(Az,z,t)) > M(Az,z,t), which is a contradiction and hence Az = z. Since A(X) ⊂ T(X), there exists z∗ ∈ X such that z = Az = Tz∗. If z 6= Bz∗, then there exists t > 0 such that 0 < M(z,Bz∗, t) < 1. By the condition (4), we have M(z,Bz∗, t) = M(Az,Bz∗, t) ≥ ψ(m(z,z∗, t)) = ψ(M(z,Bz∗, t)) > M(z,Bz∗, t), which is a contradiction. Then z = Bz∗. Since B and T are compatible, we obtain M(Tz,Bz,t) = M(TBz∗,BTz∗t) = 1 for any t > 0, which implies Tz = Bz. Next, if z 6= Bz, then there exists t > 0 such that 0 < M(z,Bz,t) < 1. By the condition (4) and Lemma 3.1, we have M(z,Bz,t) = M(Az,Bz,t) ≥ ψ(m(z,z,t)) = ψ(M(z,Bz,t)) > M(z,Bz,t), which is a contradiction and hence z = Bz. Since B(X) ⊂ S(X), there exists z∗∗ ∈ X such that z = Bz = Sz∗∗. If z 6= Az∗∗, then there exists t > 0 such that 0 < M(z,Az∗∗, t) < 1. By the condition (4), we have M(Az∗∗,z,t) = M(Az∗∗,Bz,t) ≥ ψ(m(z∗∗,z,t)) ≥ ψ(M(Az∗∗,z,t)) > M(Az∗∗,z,t), which is a contradiction. Then z = Az∗∗. Since S and A are compatible mappings, we have M(Sz,Az,t) = M(SAz∗∗,ASz∗∗, t) = 1 for any t > 0 and so Sz = Az. Therefore, Sz = Tz = Az = Bz = z and so z is a common fixed point of A,B,S and T . Case 4. Suppose that B is continuous. In the same way as in Case 1, we can obtain the result. Now, we prove the uniqueness of the common fixed point of A,B,S and T . Assume that x,y ∈ X are two common fixed points of A,B,S and T . If x 6= y, then there exists t > 0 such that 0 < M(x,y,t) < 1. By the condition (4), we c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 10 Common fixed points for generalized ψ-contractions have M(x,y,t) = M(Ax,By,t) ≥ ψ(m(x,y,t)) = ψ(min{M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t)}) = ψ(min{M(x,y,t),M(x,x,t),M(y,y,t)}) = ψ(min{M(x,y,t),1,1}) = ψ(M(x,y,t)) > M(x,y,t), which is a contradiction. Therefore, x = y. This completes the proof. � If we put S = T = IX (the identity mapping on X) in Theorem 3.4, then we obtain the result of Vetro [16] as follows: Corollary 3.5 ([16]). Let (X,M,∗) be a complete weak non-Archimedean fuzzy metric space and A,B : X → X be two mappings. Assume that there exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,By,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(x,y,t),M(Ax,x,t),M(y,By,t)}. Suppose that, for any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. If there exists x0 ∈ X such that M(x0,Sx0, t) > 0 for all t > 0, then A and B have a unique common fixed point. If we put A = B and S = T = IX in Theorem 3.4, then we obtain the following: Corollary 3.6. Let (X,M,∗) be a complete weak non-Archimedean fuzzy met- ric space and A : X → X be a mapping. Assume that there exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,Ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(x,y,t),M(Ax,x,t),M(y,Ay,t)}. Suppose that, for any x,y ∈ X, with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. If there exists x0 ∈ X such that M(x0,Ax0, t) > 0 for all t > 0, then S has a unique fixed point. As a consequence of Corollary 3.6, by Remark 2.4, we obtain the result of Mihet [12] as follows: Corollary 3.7 ([12]). Let (X,M,∗) be a complete non-Archimedean fuzzy met- ric space and A : X → X be a mapping. Assume that there exists ψ ∈ Ψ such c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 11 S. Suantai, Y. J. Cho and J. Tiammee that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,Ay) ≥ ψ(M(x,y,t)), Suppose that, for any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. If there exists x0 ∈ X such that M(x0,Ax0, t) > 0 for all t > 0, then A has a unique fixed point. We now give an example to illustrate Theorem 3.4. Example 3.8. Let (X,M,∗) be a complete weak non-Archimedean fuzzy met- ric space, where X = [0,∞) with the t-norm defined by a ∗ b = ab for any a,b ∈ [0,1] and the fuzzy set M given by: M(x,y,0) = 0, M(x,x,t) = 1 for all t > 0, M(x,y,t) = 0 for x 6= y and 0 < t ≤ 1, M(x,y,t) = t2/4 for x 6= y and 1 < t ≤ 2, M(x,y,t) = 1 for x 6= y and t > 2. Define a function ψ : [0,1] → [0,1] by ψ(t) = √ t for all t ∈ [0,1]. Then we see that ψ ∈ Ψ. Define four mappings A,B,S,T : X → X by Ax = x, Bx = √ x, Sx = 2x, Tx = 4 √ x. Then we have the following: (1) AX = BX = SX = TX; (2) A,B,S and T are all continuous mappings; (3) The pair A,S and B,T are compatible; (4) The four couple (A,B;S,T) is a ψ-contractive mapping; (5) If we choose x0 = x1 = 0, then A0 = T0 and M(x0,x1, t) = M(0,0, t) > 0; (6) If we choose t = 3 2 , then, for any x,y ∈ X with x 6= y, we have 0 < M(x,y, 3 2 ) = 9 16 < 1. Therefore, all the conditions of Theorem 3.4 are satisfied. Also, we see that A0 = B0 = S0 = T0 and so 0 is a unique common fixed point of A,B,S and T . 4. Common fixed points for mappings with the common limit In this section, we prove some common fixed points for mappings with the common limit with respect to the value of the given mappings in weak non- Archimedean fuzzy metric spaces. Definition 4.1. Two pairs (A,S) and (B,T) of self-mappings of a weak non- Archimedean fuzzy metric space (X,M,∗) are said to have the common limit with respect to the value of the mapping S (resp., T) if there exist two sequence {xn} and {yn} of X such that lim n→∞ Axn = lim n→∞ Sxn = lim n→∞ Byn = lim n→∞ Tyn = Sz (resp., Tz) for some z ∈ X. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 12 Common fixed points for generalized ψ-contractions Theorem 4.2. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and A,B,S,T : X → X be four mappings. Suppose that the following condi- tions holds: (1) A(X) ⊂ T(X) or B(X) ⊂ S(X); (2) The pairs A,S and B,T are also weakly compatible; (3) The pairs (A,S) and (B,T) have the common limit with respect to the value of the mapping S ( or T ); (4) There exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,By,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t), M(By,Sx,t),M(Ax,Ty,t)}. (5) For any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. Then A,B,S and T have a unique common fixed point. Proof. Since the pairs (A,S) and (B,T) have the common limit with respect to the value of the mapping S, there exists two sequences {xn} and {yn} in X such that lim n→∞ Axn = lim n→∞ Sxn = lim n→∞ Byn = lim n→∞ Tyn = Sz for some z ∈ X. First, we show that Az = Sz. Suppose that Az 6= Sz. Then, by the condition (5), there exists t > 0 such that 0 < M(Az,Sz,t) < 1. By using the condition (4), we obtain M(Az,Byn, t) ≥ ψ(m(z,yn, t)),(4.1) where m(z,yn, t) = min{M(Sz,Tyn, t),M(Az,Sz,t),M(Byn,Tyn, t), M(Byn,Sz,t),M(Az,Tyn, t)}. By taking the limit as n → ∞ in (4.1), we have M(Az,Sz,t) ≥ ψ(M(Az,Sz,t)) > M(Az,Sz,t), which is a contradiction and so Az = Sz. Since A(X) ⊂ T(X), there exists v ∈ X such that Az = Tv. Next, we show that Bv = Tv. Suppose that Bv 6= Tv. Then, by the condition (5), there exists t > 0 such that 0 < M(Bv,Tv,t) < 1. By using the condition (4), we obtain M(Tv,Bv,t) = M(Az,Bv,t) ≥ ψ(m(z,v,t)),(4.2) c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 13 S. Suantai, Y. J. Cho and J. Tiammee where m(z,v,t) = min{M(Sz,Tv,t),M(Az,Sz,t),M(Bv,Tv,t), M(Bv,Sz,t),M(Az,Tv,t)}. = min{M(Tv,Tv,t),M(Az,Sz,t),M(Bv,Tv,t), M(Bv,Tv,t),M(Tv,Tv,t)} = min{1,M(Bv,Tv)}. Hence, in (4.2), we obtain M(Tv,Bv,t) ≥ ψ(M(Tv,Bv,t)) > M(Tv,Bv,t), which is a contradiction and so Tv = Bv. Therefore, we have u = Az = Sz = Bv = Tv. Since the pairs A,S and B,T are weakly compatible, Az = Sz and Bv = Tv, we have Au = AAz = ASz = SAz = Su, Bu = BTv = TBv = Tu.(4.3) Next, we show that Au = u. Suppose that Au 6= u. Then there exists t > 0 such that 0 < M(Au,u,t) < 1. By the condition (4), we obtain M(Au,u,t) = M(Au,Bv,t) ≥ ψ(m(u,v,t)),(4.4) where m(u,v,t) = min{M(Su,Tv,t),M(Au,Su,t),M(Bv,Tv,t) M(Bv,Su,t),M(Au,Tv,t)} = min{M(Au,Bv,t),M(Au,Au,t),M(Bv,Bv,t) M(Bv,Au,t),M(Au,Bv,t)} = min{1,M(Au,u,t)}. Hence, in (4.4), we obtain M(Au,u,t) ≥ ψ(M(Au,u,t)) > M(Au,u,t), which is a contradiction and so Au = u. Next, we show that Bu = u. Suppose Bu 6= u. Then there exists t > 0 such that 0 < M(Bu,u,t) < 1. By the condition (4), we obtain M(u,Bu,t) = M(Au,Bu,t) ≥ ψ(m(u,u,t)),(4.5) where m(u,u,t) = min{M(Su,Tu,t),M(Au,Su,t),M(Bu,Tu,t) M(Bu,Su,t),M(Au,Tu,t)} = min{M(Au,Bu,t),M(Au,Au,t),M(Bu,Bu,t) M(Bu,Au,t),M(Au,Bu,t)} = min{1,M(Au,Bu,t)}. Hence, in (4.5), we obtain M(Bu,u,t) ≥ ψ(M(Bu,u,t)) > M(Bu,u,t), c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 14 Common fixed points for generalized ψ-contractions which is a contradiction and so Bu = u. Therefore, we have u = Au = Bu = Su = Tu, that is, u is a common fixed point of A,B,S and T . The uniqueness of the common fixed point follows the proof of Theorem 3.4. This completes the proof. � Remark 4.3. We don’t need the completeness of a weak non-Archimedean fuzzy metric space (X,M,∗) in the proof of Theorem 4.2. As a consequence of Theorem 4.2, by putting S = T = IX, we obtain the following: Corollary 4.4. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and A,B : X → X be two mappings. Suppose that the following conditions holds: (1) The pairs (A,IX) and (B,IX) have the common limit with respect to the value of the mapping IX; (2) There exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,By,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(x,y,t),M(Ax,x,t),M(By,y,t), M(By,x,t),M(Ax,y,t)}. (3) For any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. Then A and B have a unique common fixed point. By putting A = B and S = T in Theorem 4.2, we obtain the following: Corollary 4.5. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and A,S : X → X be a mapping. Suppose that the following conditions holds: (1) A(X) ⊂ S(X); (2) A pair A,S is weakly compatible; (3) There exists a sequence {xn} in X such that lim n→∞ Axn = lim n→∞ Sxn = Sz for some z ∈ X; (4) There exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,Ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(Sx,Sy,t),M(Ax,Sx,t),M(Ay,Sy,t), M(Ay,Sx,t),M(Ax,Sy,t)}. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 15 S. Suantai, Y. J. Cho and J. Tiammee (5) For any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. Then A and S have a unique common fixed point. By putting A = B and S = T = IX in Theorem 4.2, we obtain the following: Corollary 4.6. Let (X,M,∗) be a weak non-Archimedean fuzzy metric space and A,S : X → X be two mappings. Suppose that the following conditions holds: (1) There exists a sequence {xn} in X such that lim n→∞ Axn = lim n→∞ xn = z for some z ∈ X; (2) There exists ψ ∈ Ψ such that, for all x,y ∈ X and t ∈ (0,∞) with M(x,y,t) > 0, the following condition holds: M(Ax,Ay,t) ≥ ψ(m(x,y,t)), where m(x,y,t) = min{M(x,y,t),M(Ax,x,t),M(Ay,y,t), M(Ay,x,t),M(Ax,y,t)}. (3) For any x,y ∈ X with x 6= y, there exists t > 0 such that 0 < M(x,y,t) < 1. Then A has a unique fixed point. Now, we give an example to illustrate Theorem 4.2. Example 4.7. Let (X,M,∗) be a complete weak non-Archimedean fuzzy met- ric space, where X = [0,30) with the t-norm defined by a ∗ b = ab for any a,b ∈ [0,1] and the fuzzy set M given by: M(x,y,0) = 0, M(x,x,t) = 1 for all t > 0, M(x,y,t) = 0 for x 6= y and 0 < t ≤ 1, M(x,y,t) = t2/4 for x 6= y and 1 < t ≤ 2, M(x,y,t) = 1 for x 6= y and t > 2. Then, for any x,y ∈ X with x 6= y, 0 < M(x,t, 3 2 ) = 9 16 < 1. Define a function ψ : [0,1] → [0,1] by ψ(t) = √ t for all t ∈ [0,1]. Then we see that ψ ∈ Ψ. Define four mappings A,B,S,T : X → X by Ax = { 1, if x ∈ {1} ∪ (5,30), x + 7, if x ∈ (1,5], Bx = { 1, if x ∈ {1} ∪ (5,30), x + 6, if x ∈ (1,5], Sx =      1, if x = 1, 7, if x ∈ (1,5], x+1 6 if x ∈ (5,30) Tx =      1, if x = 1, 9, if x ∈ (1,5], x − 4, if x ∈ (5,30). c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 16 Common fixed points for generalized ψ-contractions If we choose two sequences {xn} and {yn} defined by xn = yn = 5 + 1n for each n ≥ 1, then lim n→∞ Axn = lim n→∞ Sxn = lim n→∞ Byn = lim n→∞ Tyn = S1 = 1 ∈ X. This implies that the pairs (A,S) and (B,T) have the common limit with respect to the value of the mapping S. We see that A(X) = {1} ∪ (8,12], B(X) = {1} ∪ (7,11], S(X) = [1,5) ∪ {7}, T(X) = [1,26) and so A(X) ⊂ T(X), but B(X) 6⊂ S(X). It is easy to show that the mappings A,B,S,T satisfy the condition (4) in Theorem 4.2. 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