De la Sen et al. SpringerPlus (2016) 5:557 DOI 10.1186/s40064-016-2057-0 RESEARCH Open Access On fixed points and convergence results of sequences generated by uniformly convergent and point‑wisely convergent sequences of operators in Menger probabilistic metric spaces Manuel De la Sen1*, Asier Ibeas2,3 and Jorge Herrera4 *Correspondence: manuel. delasen@ehu.eus Abstract 1 Institute of Research In the framework of complete probabilistic Menger metric spaces, this paper investi- and Development of Processes IIDP, Faculty gates some relevant properties of convergence of sequences built through sequences of Science and Technology, of operators which are either uniformly convergent to a strict k-contractive operator, University of the Basque for some real constant k ∈ (0, 1), or which are strictly k-contractive and point-wisely Country, PO Box 644, de Bilbao, Barrio Sarriena, convergent to a limit operator. Those properties are also reformulated for the case 48940 Leioa, Bizkaia, Spain when either the sequence of operators or its limit are strict ϕ-contractions. The defini- Full list of author information tions of strict (k and ϕ) contractions are given in the context of probabilistic metric is available at the end of the article spaces, namely in particular, for the considered probability density function. A numeri- cal illustrative example is discussed. Keywords: Strict contractions, Strict ϕ-contractions, Probabilistic metric spaces, Menger spaces, Triangular norms Background Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces. See, for instance, Pap et al. (1996), Sehgal and Bharucha-Reid (1972), Schweizer and Sklar (1960), Eldred and Veeramani (2006), De la Sen (2010a, b), Choudhury et al. (2011, 2012), De la Sen and Karapinar (2014, 2015a, b), Beg et al. (2001), Roldan et al. (2014), Jleli et al. (2014), Roldán-Lopez-de-Hierro et al. (2015), Khan et al. (1984), Choudhury and Das (2008), Gopal et al. (2014), Takahashi (1970), Shimizu and Takahashi (1996), Kaewcharoen and Panyanak (2008), Karpagam and Agrawal (2009), Suzuki (2006), Di Bari et al. (2008), Rezapour et al. (2011), Derafshpour et al. (2010), Al-Thagafi and Shahzad (2009), Karpa- gam and Agrawal (2009), Dutta et al. (2009), Chang et al. (2001), Chen et al. (2012), Chen (2012), Berinde (2007), De la Sen et al. (2015) and the wide list of references cited in those papers. In particular, fixed point theory is also a relevant tool to investigate itera- tive schemes and stability theory of continuous-time and discrete-time dynamic systems, boundedness of the trajectory solutions, stability of equilibrium points, convergence © 2016 De la Sen et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. De la Sen et al. SpringerPlus (2016) 5:557 Page 2 of 27 to stable equilibrium points and the existence oscillatory solution trajectories. See, for instance, De la Sen (1999, 2010), Berinde (2007), De la Sen and Karapinar (2014), De la Sen et al. (2010, 2015), Marchenko (2014, 2015), Delasen (1983), Istratescu (1981) and references therein. The solution of some interesting stability and solution approxima- tion problems in some integral and physical problems have been recently investigated in Eshkuvatov et al. (2015), Abdulla et al. (2015), Matinfar et al. (2015). On the other hand, fixed point theory is, in particular, receiving important research attention in the framework of probabilistic metric spaces. See, for instance, Schweizer and Sklar (1960, 1983), Pap et al. (1996), Sehgal and Bharucha-Reid (1972), Choudhury et al. (2011, 2012), De la Sen and Karapinar (2015a), Beg et al. (2001) and references therein. Also, Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm, (Pap et al. 1996; Sehgal and Bharucha-Reid 1972; Choudhury et al. 2011; De la Sen and Karapinar 2015a, b; Choudhury and Das 2008; Gopal et al. 2014) and which are very useful in the context of fixed point theory since the triangular norm plays a close role to that of the norm in normed spaces. In probabilistic metric spaces, the deterministic notion of distance is considered to be probabilistic in the sense that, given any two points x and y of a metric space, a measure of the distance between them is a probabilistic metric Fx,y(t), rather than the deterministic distance d(x, y), which is interpreted as the probability of the dis- tance between x and y being less than t (t > 0), (Sehgal and Bharucha-Reid 1972). Fixed point theorems in complete Menger spaces for probabilistic concepts of B and C-contractions can be found in Pap et al. (1996) together with a new notion of contrac- tion, referred to as (Ψ, C)-contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of C-contraction. On the other hand, cyclic contractions on subsets of complete Menger spaces were discussed in Choudhury et al. (2011, 2012), De la Sen and Karapinar (2015a). Also, some types of contractions in complete probabilistic Menger spaces have been studied through the use of the so-called altering distances. See, for instance, Khan et al. (1984), De la Sen (2010) and references therein and more recent results in Mishra et al. (2015). Some general fixed point theorems have been very recently obtained in Gopal et al. (2014) for α − ψ contractive mappings in Menger probabilistic metric spaces. Also, a parallel background literature, related to results on best proximity points and fixed points in cyclic mappings in metric and Banach spaces as well as topics related to common fixed points, is exhaus- tive. See, for instance, Eldred and Veeramani (2006), De la Sen (2010), Takahashi (1970), Shimizu and Takahashi (1996), Kaewcharoen and Panyanak (2008), Karpagam and Agrawal (2009), Suzuki (2006), Di Bari et al. (2008), Rezapour et al. (2011), Derafshpour et al. (2010), Al-Thagafi and Shahzad (2009) and (Chen et al. 2012), Chen 2012) as well as references therein. On the other hand, fuzzy metric spaces have been investigated more recently and some ad -hoc versions of fixed point theorems have been obtained in that framework. See, for instance, Roldan et al. (2014), Jleli et al. (2014), Roldán-Lopez-de- Hierro et al. (2015) and some references therein. Recent research has been also focused on some basic convergence properties of the iterates in iterative schemes. For instance, a new averaged algorithm for finding a common fixed point of a countably infinite fam- ily of generalized k-strictly pseudocontractive multi-valued mappings is studied in Chidume and Opkala (2015) and the computational errors of iterated schemes in the De la Sen et al. SpringerPlus (2016) 5:557 Page 3 of 27 set-valued case are investigated in Reich and Zaslavski (2015) and in Dinevari and Frigon (2015), in this last case under the support of graph theory. On the other hand, some variants of homopotopy methods together with Picard and Picard–Padé iterative meth- ods to solve Michaelis–Menten equation have been investigated in Vazquez-Leal et al. (2015) while an iterative scheme being strongly convergent to a common fixed point of a countable family of strictly pseudo-contractive mappings has been focused on in Cham- narnpan et al. (2013). Such a fixed point is proved to be also a solution of variational inequality problem related to quadratic minimization problems. See also Chidume et al. (2015). This paper investigates some properties of convergence of sequences being built through sequences of operators which are either uniformly convergent to a strict k-con- tractive operator, for some real constant k ∈ (0, 1), or which are strictly k-contractive and point-wisely convergent to a limit operator. The obtained properties are reformulated later on for the case when either the sequence of operators or its limit operator are strict ϕ-contractions. The appropriate definitions of strict (k and ϕ) contractions are given ad- hoc in the context of probabilistic metric spaces, namely, for the considered probability density function. Denote R+ = {z ∈ R : z > 0}, R0+ = R+ ∪ {0}, Z + = {z ∈ Z : z > 0}, Z 0+ = Z + ∪ {0}, n̄ = {1, 2, . . . , n}, and denote also by L, the set of distance distribution functions H : R → [0, 1], (Schweizer and Sklar 1983), which are non-decreasing and left continu- ous such that H(0) = 0 and supt∈R H (t) = 1. Let X be a nonempty set and let the proba- bilistic metric (or distance) F : X × X → L a symmetric mapping from X × X, where X is an abstract set, to the set of distance distribution functions L of the form H : R → [0, 1] which are functions of elements Fx,y for every (x, y) ∈ X × X . Then, the ordered pair (X, F ) is a probabilistic metric space (PM) (Pap et al. 1996; Sehgal and Bharucha-Reid 1972; Chang et al. 2001) if     1. ∀x, y ∈ X Fx,y (t) = 1; ∀t ∈ R+ ⇔ x = y 2. Fx,y (t) = Fy,x (t); ∀x, y ∈ X, ∀t ∈ R (1)    3. ∀x, y, z ∈ X; ∀t1 , t2 ∈ R+ (Fx,y (t1 ) = Fy,z (t2 ) = 1) ⇒ Fx,z (t1 + t2 ) = 1 A particular distance distribution function Fx,y ∈ L is a probabilistic metric (or dis- tance) which takes values Fx,y(t) identified with a probability distance density function H : R → [0, 1] in the set of all the distance distribution functions L. A Menger PM-space is a triplet (X, F , ∆), where (X, F ) is a PM-space which satisfies:   Fx,y (t1 + t2 ) ≥ ∆ Fx,z (t1 ), Fz,y (t2 ) ; ∀x, y, z ∈ X, ∀t1 , t2 ∈ R0+ (2) under ∆ : [0, 1] × [0, 1] → [0, 1] is a t-norm (or triangular norm) belonging to the set T of t-norms which satisfy the properties: 1. Δ(a, 1) = a 2. Δ(a, b) = Δ(b, a) 3. Δ(c, d) ≥ Δ(a, b) if c ≥ a, d ≥ b 4. ∆(∆(a, b), c) = ∆(a, ∆(b, c)) (3) De la Sen et al. SpringerPlus (2016) 5:557 Page 4 of 27 A property which follows from the above ones is Δ(a, 0) = 0 for a ∈ [0, 1]. Typical con- tinuous t-norms are the minimum t-norm defined by ΔM(a, b) = min (a, b), the product t-norm defined by ΔP(a, b) = a.b and the Lukasiewicz (or nilpotent-minimum) t-norm defined by ΔL(a, b) = max (a + b − 1, 0) which are related by the inequalities ΔL ≤ ΔP ≤ ΔM. The (probabilistic) diameter of a subset A of X is a function from R0+ to [0, 1] defined by DA (z) = supt 0 2. Fx,y (t) = Fy,x (t); (4) 3. if Fx, y(t) = 1 and Fy, z(s) = 1 then Fx, z(t + s) = 1. ••  The triplet (X, F , ∆) is a Menger space, where (X, F ) is a PM-space and ∆ : [0, 1] × [0, 1] → [0, 1] is a triangular norm, which satisfies the inequality Fx,z (t + s) ≥ ∆ Fx,y (t), Fy,z (s) ; ∀x, y, z ∈ X; ∀t, s ∈ R+.   ••  ∆M : [0, 1] × [0, 1] → [0, 1] is the minimum triangular norm defined by ∆M (a, b) = min(a, b). ••  A sequence {xn} ⊆ X in a probabilistic space (X, F ) is said to be: 1. almost surely (a.s.) convergent to a point x ∈ X, denoted by {xn} → x (a.s.), if for every ε ∈ R+ and  ∈ (0, 1), there exists some N = N (ε, ) ∈ Z 0+ such that: Fxn ,x (ε) > 1 − ; ∀n(∈ Z 0+ ) ≥ N , De la Sen et al. SpringerPlus (2016) 5:557 Page 5 of 27 2.  Probabilistically convergent (or, equivalently, it converges in probability) to a point x ∈ X, denoted by {xn} → x, if the probabilistic (X, F ) is induced by a metric space (X, d) and for every ε ∈ R+ and  ∈ (0, 1) there exists some N = N (ε, ) ∈ Z 0+ such that: Fxn ,x (d(xn , x) − ε) ≤ 1 − ; ∀n(∈ Z 0+ ) ≥ N , 3. Cauchy if for every ε ∈ R+ and  ∈ (0, 1), there exists some N = N (ε, ) ∈ Z 0+ such that Fxn ,xm (ε) > 1 − ; ∀n, m(∈ Z 0+ ) ≥ N . A good treaty on almost sure convergence and martingale approaches has been given in Stout (1974). Note that a PM-space (X, F ) is complete if every Cauchy sequence is almost surely convergent. Lemma 1 Consider a PM-space (X, F ) and a sequence {xn } ⊂ X . Then, the following properties hold: (i) {xn} → x a.s. if and only if limn→∞ Fxn+ℓ ,x (ε) = limn,m→∞ Fxn+m ,x (ε) = 1; ∀ℓ ∈ Z 0+ , ∀ε ∈ R+ . (ii) {xn} → x if and only if limn→∞ Fxn ,x (d(xn , x) − ε) = 0; ∀ε ∈ R+, where (X, F ) is induced by a metric space (X, d). (iii) If {xn} → x a.s. and, furthermore, the PM space (X, F ) is induced by a metric space (X, d) then {xn} → x. (iv) If {xn} → x a.s. then {xn} is Cauchy. The converse is true if (X, F ) is complete. (v) Assume that the PM space (X, F ) is induced by a metric space (X, d) and that the distribution function is Fx,y (t) = H (t − d(x, y)); ∀x, y ∈ X. Then, almost sure con- vergence and convergence in probability are equivalent for any sequence {xn} ⊂ X. Proof Since {xn} → x (a.s.), then, for every given ε ∈ R+ and  ∈ (0, 1), there exists some N = N (ε, ) ∈ Z 0+ such that Fxn ,x (ε) > 1 − ; ∀n(∈ Z 0+ ) ≥ N . Then, lim inf n→∞ lim inf ε→0+ Fxn ,x (ε) > 1 −  for any given  ∈ (0, 1) since Fxn ,x (ε)   is left-continuous and non-decreasing. This leads to the existence of the limit limn→∞ Fxn+j ,x (ε) = limn→∞ Fxn ,x 0+ = 1, ∀ε ∈ R+ , ∀j ∈ Z 0+. Conversely, since limn→∞   Fxn ,x (ε) = 1, there is a ball BN (x, ε, ) in the PM space (X, F ) which contains xn for all n(∈ Z 0+ ) ≥ N and some N = N(ɛ, λ) so that Fxn ,x (ε) > 1 − ; ∀n ≥ N . Property (i) has been proved. On the other hand if {xn} → x, if for every given ε ∈ R+ and  ∈ (0, 1) there exists some N = N (ε, ) ∈ Z 0+ such that Fxn ,x (d(xn , x) − ε) ≤ 1 − ; ∀n(∈ Z 0+ ) ≥ N . The condition for d(xn, x) ≤ ɛ holds trivially since Fxn ,x (0) = 0, ∀n ∈ Z 0+ so that we discuss the case d(xn, x) > ɛ. By taking λ = 1 – δ and arbitrary δ ∈ (0, 1), one gets lim supn→∞ Fxn ,x (d(xn , x) − ε) ≤ δ so that the limit limn→∞ Fxn ,x (d(xn , x) − ε) = 0 exists for any ε ∈ R+ if {xn} → x. The converse is proved closely to the proof of its counterpart in Proposition (i) by defining a ball BN (x, d(xn , x) − ε, ) in the PM space (X, F ) which contains xn for all n(∈ Z 0+ ) ≥ N . Property (ii) has been proved. De la Sen et al. SpringerPlus (2016) 5:557 Page 6 of 27 To prove Property (iii), note from Property (i) that {xn} → x a.s. if and only if limn→∞ Fxn ,x 0+ = 1 and the sequence in the metric space (X, d) satisfies d(xn, x) → 0   as n → ∞. Assume that {xn} → x fails, so that lim inf n→∞ Fxn ,x (d(xn , x) − ε) > 0 for some ε ∈ R+, and proceed by contradiction by assuming that there exists some ε ∈ R+ such that lim inf n→∞ (d(xn , x) − ε) > limn→∞ d(xn , x) = 0. But this leads to the contra- diction lim inf n→∞ Fxn ,x (−t) > 0 for some t ∈ R+. The proof of Property (iv) is given as follows. Now, if {xn} → x a.s. then for any given ε ∈ R+ and  ∈ (0, 1): lim Fxn+ℓ ,x (ε/2) = lim Fxn ,x (ε/2) = 1; ∀ℓ ∈ Z + . n→∞ n→∞ Then, by the third property of (5), one gets limn→∞ Fxn+ℓ ,xn (ε) = 1; ∀ℓ ∈ Z 0+. Assume that {xn} is not Cauchy. Thus, for some given ε ∈ R+ and  ∈ (0, 1), and any N ∈ Z 0+, there is a subsequence {nk } ⊆ Z 0+ with mk > nk ≥ N such that 1 −  ≥ limk→∞ Fxnk , xmk (ε) and then lim supk→∞ Fxnk , xmk (ε) < 1 but implies that limn→∞ Fxn+ℓ ,xn (ε) < 1, a contra- diction. Then, if limn→∞ Fxn+ℓ ,xn (ε) = 1, and by Property (i), if {xn} → x a.s., then {xn} is Cauchy. The converse is true if the metric space is complete from the definition of com- pleteness. Property (iv) has been proved. To prove Property (v), note that Fxn ,x (ε) = H(ε − d(xn , x)). If {xn} → x then limn→∞ Fxn ,x (d(xn , x) − ε) = 0, and then limn→∞ Fxn ,x (ε − d(xn , x)) = 1 for any ε ∈ R+ so that from Property (i), {xn} → x a.s.. The converse, that is,{xn} → x (a.s.) ⇒ {xn} → x if (X, F ) is induced by (X, d) is Property (iii), already proved. Thus, if (X, F ) is induced by (X, d) then {xn} → x a.s. ⇔ {xn} → x. Example 1 Consider the probabilistic metric (Z + , F ) space with probability density function  1 if (j − 1)/i < t ≤ j/i Fij (t) = ; ∀i, j ∈ Z + . 0 otherwise Then, the sequence of probability densities Fij (t) = {F11 (t), F21 (t), F22 (t), F31 (t), F32 (t),   F33 (t), . . .} defined for all i, j(≤ i) ∈ Z + and t ∈ (0, 1) does not con- verge to one as i, j → ∞, so that it does not converge almost surely  since lim n→∞F(n−1)/n,1(1) = 1, since the argument t = t(n) satisfies, t ∈ n−1 n , 1 , and limn→∞ Fj/n,(n−1)/n (t) = limj(≤n−1),n→∞ Fj/n,(n−1)/n (t) = 0; ∀j(≤ n − 1) ∈ Z + , t ∈ [0, 1). However, Fn,n+1 (t) converges in probability as n → ∞; ∀t ∈ R+ since         0 if 1 < t ≤ 1 + lim 1/n ∨ (t = 0) lim 1 − Fn,n+1 (t) = n→∞ n→∞ 1 if t ∈ (0, 1]  0 if t = 0 = 1 if t ∈ (0, 1] De la Sen et al. SpringerPlus (2016) 5:557 Page 7 of 27 Main results concerning the uniform convergence of operators The concept of (probabilistic) k-contraction follows: Definition 1 Let (X, F ) be a PM-space. Then, T : X → X is k-contractive (or a k-con- traction) if FTx,Ty (kt) ≥ Fx,y (t); ∀x, y ∈ X, ∀t ∈ R+ for some real constant k ∈ (0, 1). A wider class of contractive (in a weaker sense) operators T : X → X is that which sat- isfies FTx,Ty (t) > Fx,y (t), ∀x, y(≠x) ∈ X, ∀t ∈ R+. Proposition 1 Let (X, F ) be a PM-space. If T : X → X is a -contraction then it is also a weak contraction. The converse is not true.k Proof Note that if T : X → X is a k-contraction then   FTx,Ty (t) ≥ Fx,y k −1 t ≥ Fx,y (t) since F : R → R0+ which is non-decreasing and left-continuous. Furthermore, if x � = y and Fx,y (t1 ) < 1 for some t1 ∈ R+ then the last above inequality has to be strict for all t ∈ R+, i.e. FTx,Ty (t) > Fx,y (t); ∀t ∈ R+. Otherwise, we would have the following contra- diction for t = t1: 1 = lim FT n x,T n y (t) ≥ Fx,y k −n t = Fx,y +∞− = Fx,y (t) > Fx,y (t1 ); ∀t ∈ R+     n→∞ Since Fx,y (t1 ) > Fx,y (t1 ) is impossible. On the other hand, it always exist such a t1 ∈ R+ such that Fx,y (t1 ) < 1; ∀x, y(� = x) ∈ X . Otherwise, x = y from the first condition of (4). As a result, if T : X → X is a k-contraction and x � = y then FTx,Ty (t) > Fx,y (t); ∀x, y(�= x) ∈ X . Thus, if T : X → X is a k-contraction then it is also contractive and we have specifically proved that for any real constant k ∈ (0, 1) and any x, y(�= x) ∈ X :   FTx,Ty (t) ≥ Fx,y k −1 t ⇒ FTx,Ty (t) > Fx,y (t) To prove that the inverse is not true, it suffices to prove that for some k ∈ (0, 1) and any x, y(� = x) ∈ X , the logic implication below fails:   FTx,Ty (t) > Fx,y (t) ⇒ FTx,Ty (t) ≥ Fx,y k −1 t If the above implication were true then the contradiction 1 > 1 would follow from: 1 ≥ lim FT n+1 x,T n+1 y (t) > lim FT n x,T n y (t) ≥ lim Fx,y k −n t = Fx,y +∞− = 1.     n→∞ n→∞ n→∞   De la Sen et al. SpringerPlus (2016) 5:557 Page 8 of 27 The result below refers to the case of uniform convergence of the sequence {Tn } on X to a strict k-contractive operator on X in the framework of a complete Menger space (X, F , ∆) : Theorem 1 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators Tn : X → X , such that FTn = xn∗ , ∀n ∈ Z + , {Tn } → → {T } a.s. and T : X → X is   a (strict) -contraction with kFT = {x*}. Then, the following properties hold: limn→∞ Fxn∗ ,x∗ (t) = limn→∞ Fxn∗ ,x∗ 0+ = 1; ∀t ∈ R+ ,   (i) xn → x∗ a.s. and T m xn∗ → x∗ a.s. as n, m → ∞,  ∗ (ii) limm,n→∞ FT m xn∗ ,x∗ (t) = limn→∞ FT ℓ xn∗ ,x∗ (t) = limm,n→∞ FT m xn∗ ,x∗ 0+ = 1; ∀ℓ ∈ Z 0+ , ∀t ∈ R+ ,    m ∗ limn→∞ Fxn∗ ,x∗ (t) = limn→∞ Fxn∗ ,x∗ 0+ = 1; ∀t ∈ R+ , T xn → x∗ , ∀m ∈ Z 0+ .   Proof Fix any t ∈ R+, γ ∈ (0, 1) and choose a natural number N = N (γ , t) such that n ≥ N implies FTx,Tn x (t) ≥ 1 − γ ; ∀x ∈ X, ∀m ∈ Z 0+ ∀m ∈ Z +, since {Tn } → → {T } a.s., ∀x ∈ X and, in particular, the above inequality holds for the fixed point x*of T : X → X , where k ∈ (0, 1) is the contraction coefficient. Also, since {x*n}, {Tx*n} ⊂ X, x* = Tx* ∈ X, x*n = Tnx*n ∈ X, T : X → X is a k-contraction and F ∈ ∆F is non-decreasing and left-continuous everywhere in its definition domain, one has for all n ≥ N and all t ∈ R+ that: Fxn∗ ,x∗ (t) = FTn xn∗ ,Tx∗ (t)   ≥ ∆M FTx∗ ,Txn∗ (ηt), FTxn∗ ,Tn xn∗ ((1 − η)t) (5)     ≥ ∆M Fxn∗ ,x∗ k −1 ηt , 1 −      > min Fxn∗ ,x∗ k −1 ηt , 1 −  for any given real constant η ∈ (0, 1) since FTxn∗ ,Tn xn∗ (t) ≥ 1 − γ > 1 − ; ∀x ∈ X, ∀t ∈ R+, and any real constants λ ∈ (γ, 1), and γ ∈ (0, 1), ∀n(∈ Z 0+ ) ≥ N since {Tn } → → {T } a.s.. The following cases can occur in (5): (a) Fxn∗ ,x∗ k −1 ηt ≥ Fxn∗ ,x∗ (t) > 1 − ; ∀n(∈ Z 0+ ) ≥ N ; ∀t ∈ R+   Then, {x*n} → x* a.s. and η ∈ [k, 1) since F ∈ ∆F is non-decreasing and left-continu- ous.   (b) Fxn∗ ,x∗ k −1 ηt ≤ min Fxn∗ ,x∗ (t), 1 −  ; ∀j ∈ Z 0+   j j     for a given η ∈ (0, k], some t ∈ R+ and some subsequence xn∗j ⊆ xn∗ of fixed points     of Tnj ⊆ {Tn }. If Fxn∗ ,x∗ (t) > 1 − , one also finds that xn∗j → x∗ a.s. with η ∈ (0, k]. j The convergence of the subsequence ensures that {x*n} → x*a.s. If Fxn∗ ,x∗ (t) ≤ 1 −  and j then Fxn ,x k t1 ≤ 1 −  with t1 = k t and we deduce from (5) under the same −1  −1  ∗ ∗ j reasoning with the replacement t → t1 that either Fxn∗ ,x∗ k −1 t1 > 1 − ; ∀t1 ∈ R+   j De la Sen et al. SpringerPlus (2016) 5:557 Page 9 of 27 and {x*n} → x*a.s. or Fxn∗ ,x∗ k −1 ηt1 = Fxn∗ ,x∗ k −2 ηt ≤ 1 −  for some         j j   subsequence xn∗j ⊆ xn∗ of fixed points of Tnj ⊆ {Tn }. But, since 1 = limj→∞ limi→∞ Fxn∗ ,x∗ k ηt = limj→∞ limti →∞ Fxn∗ ,x∗ (ηti ) ≤ 1 −  would be  −i  j j a contradiction, some finite tl = k−ℓt exists such that Fxn∗ ,x∗ (tℓ ) = j Fxn∗ ,x∗ k −(ℓ+1) t > 1 − ; ∀n(∈ Z 0+ ) ≥ Nℓ , ∀t ∈ R+ and some Nℓ = Nℓ (, t) for any   j     subsequence xn∗j ⊆ xn∗ implying that limn→∞ Fxn∗ ,x∗ (t) = limn→∞ Fxn∗ ,x∗ 0+ = 1   so that {x*n} → x*a.s. On the other hand, for any given real constant γ ∈ (0, 1) and t ∈ R+ and some N = N(λ, t), one gets, since xn∗ → x∗, a.s. that   FT m xn∗ ,x∗ (t) = FT m xn∗ ,T m x∗ (t) ≥ Fxn∗ ,x∗ k −m t ≥ Fxn∗ ,x∗ (t) > 1 − ;   ∀n(∈ Z 0+ ) ≥ N ; ∀m ∈ Z 0+ and lim lim lim FT m xn∗ ,x∗ (t) = lim lim FT ℓ xn∗ ,x∗ (t) = 1; ∀ℓ ∈ Z 0+ t→0+ n→∞m→∞ t→0+ n→∞ so that, furthermore, T m xn∗ → x∗ a.s.; ∀m ∈ Z 0+ and Tmx*n → x* a.s. as n, m → ∞.   Property (i) has been proved. On the other hand, note that   FT m xn∗ ,x∗ (t) ≥ ∆M FT m xn∗ ,T m x∗ (t/2), FT m x∗ ,x∗ (t/2)   ≥ ∆M FT m xn∗ ,T m x∗ (t/2), Fx∗ ,x∗ (t/2)   ≥ ∆M FT m xn∗ ,T m x∗ (t/2), 1 ≥ Fxn∗ ,x∗ k −m t/2 ; ∀t ∈ R+ ; ∀m, n ∈ Z + (6)   and lim lim FT m xn∗ ,x∗ (t) ≥ lim lim Fxn∗ ,x∗ (t) m→∞ n→∞ m→∞ n→∞ = lim Fxn∗ ,x∗ 0+ = F +∞− = 1; ∀t ∈ R+     n→∞ since it has been already proved that xn∗ → x∗ a.s., where F(+∞−) denotes the left-   limit of F(t) as t → +∞.  On the  other hand, for any finite m ∈ Z +, one has from (6) that FT m xn∗ ,x∗ (t) ≥ Fxn∗ ,x∗ k −m t/2 = Fxn∗ ,x∗ (t1 ) for t1 = t1 (t, m) = k −m t/2. Since t ∈ R+, there exists N1 = N1 (γ , m,t) ≥ N such that FT m xn∗ ,x∗ (t)  > 1 − γ ; ∀n(∈ Z 0+ ) ≥ N1, where N = N (γ , t) = min z ∈ Z 0+ : Fxn∗ ,x∗ (t) > 1 − γ for any given t ∈ R+ and γ ∈ (0, 1). Then, FT m xn∗ ,x∗ (t) ≥ Fxn∗ ,x∗ (t) ≥ Fxn∗ ,x∗ (t1 ) > 1 − γ for t ≥ 2k m t1 Property (i) includes the convergence results {x*n} → x*a.s. and Tmx*n → x*a.s. as n, m → ∞. It is now proved that Tmx*n → x* a.s. as n → ∞, ∀m ∈ Z 0+. Proceed by con- tradiction by assuming that T m xn∗ ¬ → x∗ a.s. for  some Z + then there exists some    m ∈  real constant γ1 ∈ (0, 1)and some subsequence xnj ⊆ xn such that the following ∗ ∗ contradiction follows to xn∗ → x∗ a.s.   1 − γ1 ≥ lim sup lim Fxn∗ ,x∗ (t) ≥ lim lim Fxn∗ ,x∗ k −m t/2 = 1   t→0+ j j→∞ t→0+ j j→∞ De la Sen et al. SpringerPlus (2016) 5:557 Page 10 of 27 Then, lim FT m xn∗ ,x∗ (t) = lim FT ℓ xn∗ ,x∗ 0+ = lim FT m xn∗ ,x∗ 0+ = 1;     n→∞ n→∞ n,m→∞ ∀ℓ ∈ Z 0+ and T m xn∗ → x∗ a.s.;   ∀m ∈ Z 0+. Property (ii) has been proved.  The following auxiliary result is then used: Lemma 2 The following properties hold:     j (i) If Tn : X → X, ∀n ∈ Z + are all continuous and Tn → → T a.s., ∀j ∈ m̄ for some j  m+1  →  m+1  given m ∈ Z +, where T : X → X , then Tn → T a.s. (ii) If the self-maps Tn : X → X, ∀n ∈ Z + are continuous and {Tn } → → {T } a.s. then  m → m Tn → {T } a.s., ∀m ∈ Z +. (iii) If the self-maps Tn : X → X, ∀n ∈ Z + are strictly contractive and {Tn } → → {T } a.s. then Tnm → m } a.s., ∀m ∈ Z .   → {T +  m → m (iv) If {Tn } → → {T } a.s. and Tn commutes with T in X, ∀n ∈ Z 0+ then Tn → {T } a.s., ∀m ∈ Z +.       j+1 j j Proof Note that Tn x = Tn Tn x , ∀n ∈ Z + and, since Tn → → T a.s., ∀j ∈ m̄, then j Tjn(Tℓx) → Tj(Tℓx) as n → ∞; ∀j ∈ m̄, ∀ℓ ∈ n̄, ∀x ∈ X and Tj(Tnx) → Tj(Tx) = Tj+1x as n → ∞ a.s., ∀j ∈ m̄, ∀x ∈ X since Tn : X → X is continuous for any n ∈ Z +, that is,   FT j (T x),T j (Tx) (t) ≥ ∆M FT j (T x),T j (T x) (t/2), FT j (Tn x),T j (Tx) (t/2) n n  n n n  (7) ≥ min 1 − γ , FT j (Tn x),T j (Tx) (t/2) for any given t ∈ R+, γ ∈ (0, 1) and  ∈ (γ , 1), ∀j ∈ m − 1 ∪ {0} and all n(∈ Z 0+ ) ≥ N1 and some N1 (∈ Z 0+ ) = N1 (γ , t) such that max0≤j≤m FT j (T x),T j (T x) (t/2) ≥ 1 − γ > 1 −      n n n j since Tn → → T j a.s., ∀j ∈ m − 1 ∪ {0}. Since Tn : X → X is continuous ∀n ∈ Z 0+ there is N2 (∈ Z 0+ ) = N2 (, t) such that FT j (Tn x),T j (Tx) (t/2) > 1 −  for all n(∈ Z 0+ ) ≥ N1, ∀j ∈ Z 0+, ∀t ∈ R+ and any given real constant  ∈ (γ , 1). Then, from (7), min F j+1 x,T j+1 x (t) 0≤j≤m Tn > min(1 − , 1 − ) = 1 − ; ∀j ∈ m − 1, ∀t ∈ R+ , ∀n(∈ Z 0+ ) ≥ max(N1 , N2 ) (8) Then, min F j (t) > 1 − ; ∀t ∈ R+ , ∀n(∈ Z 0+ ) ≥ max(N1 , N2 ) j 0≤j≤m+1 Tn x,T x (9) and min0≤j≤m+1 limn→∞ limt→0+ FT j x,T j x (t) = 1. Thus, Tjn(Tnx) → Tj+1x a.s. as n → ∞,   m+1  n ∀j ∈ m̄. Then, Tnm+1 → . Property (i) has been proved. Also, if Tn : X → X ,  → T  m → m ∀n ∈ Z + are all continuous and {Tn } →→ {T } a.s., assumed it also holds that Tn → {T }  m+1  →  m+1  a.s. for some m ∈ Z + then Tn → T a.s. via complete induction from Property (i) and Property (ii) follows as well. Property (iii) follows from Property (ii) since if the self-maps Tn : X → X , ∀n ∈ Z + are continuous since they are strictly contractive. De la Sen et al. SpringerPlus (2016) 5:557 Page 11 of 27 On the other hand, if Tn commutes with T in X for all n ∈ Z 0+ then T TnTix = TnTjx = TjTnx; ∀i ≤ j , j ∈ Z 0+, ∀n ∈ Z 0+, ∀x ∈ X which used in (7) yields j−i   by using {Tn } → → {T } a.s., ∀n ∈ Z 0+ and TnTx = TTnx: min F j+1 x,T j+1 x (t) 0≤j≤m Tn   ≥ min 1 − γ , FTn (T j x),T (T j x) (t/2) = 1 − ; (10) ∀j ∈ m − 1, ∀t ∈ R+ , ∀n(∈ Z 0+ ) ≥ max(N1 , N2 ) and any given m ∈ Z 0+. Then, {Tn } → → {T } a.s. and Property (iv) is proved.  The subsequent result is concerned with the probabilistic convergence properties of the sequences {xn } ⊂ X generated by the iterated scheme xn+1 = Tn xn; ∀n ∈ Z + for any given x1 ∈ X: Theorem 2 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators Tn : X → X , such that FTn = xn∗ , ∀n ∈ Z +, {Tn } → → {T } a.s. and T : X → X is   * a (strict) k-contraction with FT = {x }. Consider the sequence {xn } ⊂ X generated by the iter- ated scheme xn+1 = Tn xn; ∀n ∈ Z + for any given x1 ∈ X. Then, the following properties hold: (i) T m xn → x∗ a.s. as m → ∞; ∀n ∈ Z 0+ and T m xn → x∗ a.s. as n, m → ∞, (ii) FT m xn+1 ,T m x0 (t) ≥ FTn xn ,Txn k −m t ; ∀n ∈ Z 0+ , ∀t ∈ R+ , ∀m ∈ Z 0+   n+1 FT m xn+1 ,T m x0 (t) ≥ FTn xn ,Txn k −m t > 1 − γ ; ∀n(∈ Z 0+ ) ≥ N1 ,   ∀t ∈ R+ , ∀m ∈ Z 0+ n+1 lim F m m 0 (t) = 1; ∀n(∈ Z 0+ ) ≥ N1 , ∀t ∈ R+ m→∞ T xn+1 ,T xn+1 lim F m m 0 (t) = 1; ∀t ∈ R+ , m,n→∞ T xn+1 ,T xn+1 where x0n+1 = Txn, for some N1 (∈ Z 0+ ) = N1 (γ , t, m), (iii) Assume that some of the conditions below holds:  m → m 1. Tn → T a.s.; ∀m ∈ Z + 2. {Tn } →→ T a.s. and either Tn : X → X is continuous for all n ∈ Z 0+ or Tn commutes with T for all n ∈ Z 0+. Then, {xn } → x∗ a.s. (iv) If (X, F , ∆M ) is a (non-necessarily complete) Menger space, each elements of the sequence {Tn } of operators Tn : X → X has a unique fixed points FTn = xn∗ , ∀n ∈ Z + , and commute everywhere in X with a (strict) k-contraction   T : X → X of unique fixed point FT = {x*}, then: lim inf Fxn ,x∗ (t) ≥ lim sup Fxn ,(Tn .....Tn−j+1 .Tn−j )x∗ t − ; ∀t ∈ R+   n→∞ j,(n−j)→∞ Proof For any given γ ∈ (0, 1) and t ∈ R+, there exists N1 (∈ Z 0+ ) = N1 (γ , t) such that FTn xn ,Txn (t) > 1 − γ since {Tn } → → {T } a.s.. On the other hand, since T : X → X be a (strict) k-contraction with a unique fixed point x* ∈ X, one gets for any given any given γ ∈ (0, 1) and some N1 (∈ Z 0+ ) = N γ , k −m t that:   FT m xn ,x∗ (t) = FT m xn ,T m x∗ (t) ≥ Fxn ,x∗ k −m t   De la Sen et al. SpringerPlus (2016) 5:557 Page 12 of 27 and then T m xn → x∗ a.s. as m → ∞; ∀n ∈ Z 0+ and T m xn → x∗ a.s. as n, m → ∞. On the other hand, if xn+1 = Tnxn; ∀n ∈ Z 0+ for any x1 ∈ X, since {Tn } → → {T } a.s. and T : X → X is k-contractive, then FT m xn+1 ,T m (Txn ) (t) = FT m Tn xn ,T m+1 xn (t) ≥ FTn xn ,Txn k −m t > 1 − γ ;   ∀n(∈ Z 0+ ) ≥ N1 , ; ∀t ∈ R+ , ∀m ∈ Z 0+ lim FT m xn+1 ,T m+1 xn (t) = lim FT m Tn xn ,T m+1 xn (t) = 1; ∀n(∈ Z 0+ ) ≥ N1 , m→∞ m→∞ ∀t ∈ R+ lim FT m Tn xn ,T m+1 xn (t) = 1; ∀t ∈ R+ m,n→∞ Properties (i) and (ii) have been proved. To prove Property (iii), proceed by contradiction by assuming that {xn }¬ → x∗ a.s.. Then, there exists some real constant γ0 ∈ (0, 1) such that the following contradiction holds for some subsequence Tnm of {Tn } which generates the sequence xnj +1 = Tnj xnj   for some given x1 ∈ X, ∀j ∈ Z 0+:     1 − γ0 ≥ lim sup lim sup Fxnj +m ,x∗ (t) = lim sup lim sup FTnm xnj ,x∗ (t) m→∞ m→∞ j j→∞ j→∞    ≥ lim sup lim sup ∆M FTnm xnj ,T m xnj (t/2), FT m xnj ,T m x∗ (t/2) m→∞ j j→∞      −m  ≥ ∆M lim inf lim sup FTnm xnj ,T m xnj (t/2) , lim Fxnj ,x∗ k t/2 j→∞ m→∞ j m→∞ = min(1, 1) = 1 (11)   from Lemma 2, since either Tnmj → → → T a.s., ∀m ∈ Z +, or {Tn } → T a.s. with Tn : X → X m being either everywhere continuous or it commutes with T : X → X ,  ∀n ∈Z 0+. Note that (7) yields the contradiction γ0 ≤ 0. Thus, {xn }¬ → x∗ a.s. is false and xnj → x∗ a.s. and then {xn } → x∗ a.s. Property (iii) has been proved. To prove Property (iv), note that for any real constant σ ∈ (0, 1): Fxn+m ,x∗ (t) = FT̄ (n,n+m)xn ,x∗ (t)   ≥ ∆M FT̄ (n,n+m)xn ,T m xn+m (σ t), FT m xn+m ,T m x∗ ((1 − σ )t)   ≥ ∆M FT̄ (n,n+m)xn ,T̄ (n,n+m)T m xn (σ t), Fxn+m ,x∗ k −m (1 − σ )t  ≥ FT̄ (n,n+m)xn ,T̄ (n,n+m)T m xn (σ t)     ≥ ∆M FT̄ (n,n+m)xn ,T̄ (n,n+m)T m x∗ σ 2 t , FT̄ (n,n+m)T m x∗ ,T̄ (n,n+m)T m xn (σ (1 − σ )t)     = ∆M FT̄ (n,n+m)xn ,T̄ (n,n+m)T m x∗ σ 2 t , FT m T̄ (n,n+m)x∗ ,T m T̄ (n,n+m)xn (σ (1 − σ )t)     ≥ ∆M Fxn+m ,T̄ (n,n+m)T m x∗ σ 2 t , FT̄ (n,n+m)x∗ ,xn+m k −m σ (1 − σ )t ;  (12) ∀j ∈ Z , ∀t ∈ R+, since the elements of {Tn } commute with T, Fxn+m ,x∗  −m 0+ k σ (1 − σ ) ≥ Fxn+m ,x∗ (t), ∀n, m ∈ Z + and limm→∞ FT̄ (n,n+m)x∗ ,xn+m k −m σ (1 − σ ) = 1,    ∀t ∈ R+,where T̄ (n, n + m) = Tn+m−1 · · · · · Tn is a composite operator with m con- secutive members of the sequence {Tn}. Thus, for any given t ∈ R+ and  ∈ (0, 1), De la Sen et al. SpringerPlus (2016) 5:557 Page 13 of 27 there exist m0 , n0 ∈ Z 0+ such that for any n(∈ Z 0+ ) ≥ n0 and m(∈ Z 0+ ) ≥ m0, FT̄ (n,n+m)x∗ ,xn+m k −m t/4 > 1 − . Then, one gets from (12), since x* = Tmx* that     Fxn+m ,x∗ (t) ≥ Fxn+m ,T̄ (n,n+m)x∗ σ 2 t ; ∀n ≥ n0 , ∀m ≥ m0 , ∀t ∈ R+   lim inf Fxn ,x∗ (t) ≥ lim sup Fxn ,T̄ (n−m,n)x∗ σ 2 t ; ∀m ≥ m0 , ∀t ∈ R+ n→∞ n→∞ for any σ ∈ (0, 1) and renaming subscripts, the above inequality becomes identical to:   Fxn ,x∗ (t) ≥ Fxn ,T̄ (j,n)x∗ σ 2 t ; ∀j(∈ Z 0+ ) ≥ n0 , m(∈ Z 0+ ) ≥ m0 , ∀t ∈ R+ (13)    2 lim inf Fxn ,x∗ (t) ≥ lim lim sup Fxn ,T̄ (n−m,n)x∗ σ t n→∞ σ →1− n→∞ = lim sup Fxn ,T̄ (n−m,n)x∗ t − ; m(∈ Z 0+ ) ≥ m0 , ∀t ∈ R+ (14)   n→∞ and Property (iv) is proved.  Main results concerning the point‑wise convergence of operators The first result of this section is the following one: Theorem 3 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators such that: 1. The operators Tn : X → X ; ∀n ∈ Z 0+ of the sequence {Tn} are all strict k-contractions for some real constant k ∈ (0, 1), 2. {Tn } → T for some T : X → X . Then, T : X → X is a strict k-contraction and xn∗ → x∗ a.s., where FTn = xn∗ ;     ∀n ∈ Z +, and FT = {x*}. Furthermore, {xn } → x∗ a.s., where xn+1 = Tnxn; ∀n ∈ Z 0+ for any given x0 ∈ X. Proof We have that FTn x,Tn y (t) ≥ Fx,y k −1 t ; ∀n ∈ Z 0+ , ∀t ∈ R+ for any x, y ∈ X , and     FTx,Ty (t) ≥ ∆M FTx,Tn x (t/2), FTn x,Ty (t/2)    ≥ ∆M FTx,Tn x (t/2), ∆M FTn x,Tn y (t/4), FTn y,Ty (t/4)      ≥ ∆M FTx,Tn x (t/2), ∆M Fx,y k −1 t/4 , FTn y,Ty (t/4) ; ∀n ∈ Z 0+ , ∀t ∈ R+ (15) Thus, since {Tn } → T , Tn : X → X are strict k-contractions, then everywhere continu- ous, and ∆M : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm, De la Sen et al. SpringerPlus (2016) 5:557 Page 14 of 27      FTx,Ty (t) ≥ ∆M lim inf FTx,Tn x (t/2), ∆M Fx,y k −1 t/4 , lim inf FTn y,Ty (t/4) n→∞ n→∞      ≥ ∆M lim inf FTx,Tn x (t/2), ∆M Fx,y k −1 t/4 , lim inf FTn y,Ty (t/4) n→∞ n→∞      −1 = ∆M 1, ∆M Fx,y k t/4 , 1     = ∆M Fx,y k −1 t/4 , 1   = Fx,y k −1 t/4 ; ∀n ∈ Z 0+ , ∀t ∈ R+ , ∀x, y ∈ X (16) so that FT n x,T n y (t) ≥ Fx,y k −n t/4 ; ∀n ∈ Z 0+ ,   ∀t ∈ R+ , ∀x, y ∈ X (17) lim FT n x,T n y (t) = 1; ∀t ∈ R+ , ∀x, y ∈ X (18) n→∞ Eq. (17), leading to (18), establishes that T : X → X is a strict k-contraction. It has to be proved that it has a unique fixed point x*. Assume on the contrary that there are two x∗ , x̄∗ (�= x∗ ) ∈ F (T ) so that Fx̄∗ ,x∗ (t) = FT n x̄∗ ,T n x∗ (t) ≥ Fx̄∗ ,x∗ k −n t ;   ∀n ∈ Z 0+ , ∀t ∈ R+ Fx̄∗ ,x∗ (t) = lim FT n x̄∗ ,T n x∗ (t) ≥ lim Fx̄∗ ,x∗ k −n t = 1;   ∀t ∈ R+ n→∞ n→∞ and then x̄∗ = x∗ by the property 1 of (1) of the PM space (X, F ).On the other hand, one gets by taking y = y(x) = Tx and zn = zn(x) = Tnx for any given x ∈ X that lim F n+1 x,T n x (t) = lim FTzn ,zn (t) = 1; ∀t ∈ R+ , ∀x ∈ X n→∞ T n→∞ and, for any given  ∈ (0, 1) and t ∈ R+, there is N0 = N0 (ε, ) ∈ Z 0+ such that Fzn+1 ,zn (t) > 1 − ; ∀n(∈ Z 0+ ) ≥ N0 , ∀t ∈ R+ so that {zn } is a Cauchy sequence which converges to some limit point z* = z*(x) ∈ X since (X, F , ∆M ) is complete. Since the fixed point x*of T : X → X is unique, any limit point z* = z*(x) ∈ X of any sequence {T n x} for any arbitrary initial point x ∈ X is the fixed point x* of T : X → X . It remains to prove that {xn } → x∗ a.s., where xn+1 = Tnxn;∀n ∈ Z 0+, for any given initial point x0 ∈ X. Note that Fxn+1 ,x∗ (t)    ≥ ∆M Fxn+1 ,Tn+1 m x n+1 (t/2), ∆M FTn+1 m x m ∗ n+1 ,T xn+1 (t/4), FT m xn+1 ∗ ,T m x∗ (t/4)      ≥ ∆M Fxn+1 ,Tn+1 m x n+1 (t/2), ∆M ∆M FTn+1m x m ∗ n+1 ,Tn+1 xn+1 (t/8), FT m xn+1 ∗ ,T m x∗ n+1 n+1 (t/8) , FT m xn+1 ∗ ,T m x∗ (t/4)     −m    −m  ≥ ∆M Fxn+1 ,Tn+1 m x n+1 (t/2), ∆M ∆M Fxn+1 ,xn+1 ∗ k t/8 , FT m xn+1∗ ,T m x∗ n+1 n+1 (t/8) , Fxn+1 ∗ ,x∗ k t/4 ; ∀n ∈ Z 0+ , ∀m ∈ Z + , ∀t ∈ R+ (19) De la Sen et al. SpringerPlus (2016) 5:557 Page 15 of 27 Note also that FT m xn+1 ∗ ,T m x∗ (t) = F  ∗  p  m−p ∗  (t) n+1 n+1 T p T m−p xn+1 ,Tn Tn+1 xn+1 =F  ∗ T T m−1 xn+1   m−1 ∗  (t) ,Tn Tn+1 xn+1 (20) for any p, n, (m ≥ p) ∈ Z 0+ , ∀t ∈ R+ and, since Tn : X → X; ∀n ∈ Z 0+ and T : X → X are all strict k-contractions, they are also continuous so that if {Tn } → T , then {Tn x} → Tx a.s. and limn→∞ FTn x,Tx (t) = 1; ∀t ∈ R+ as n → ∞ for any given    p m−p ∗ x ∈ X. Then, Tn Tn+1 xn+1 → T p T m−p xn+1 ∗ a.s. as m → ∞, ∀n ∈ Z 0+ since   Tn : X → X; ∀n ∈ Z 0+ and T : X → X are strict k-contractions. This implies from (20) that ∗ ,T m x∗ (t) = 1; lim FT m xn+1 ∀t ∈ R+ , ∀n ∈ Z 0+ m→∞ n+1 n+1 (21) On the other hand,  m  Tn xn → xn∗ a.s. as m → ∞; ∀n ∈ Z 0+ (22) since {Tn } is a sequence of strict k-contractive operators with FTn = xn∗ ; ∀n ∈ Z 0+.   Furthermore, one has:  −m   −m  lim Fxn+1 ,xn+1 ∗ k t = lim Fxn+1 ∗ ,x∗ k t/4 = 1; ∀t ∈ R+ (23) m→∞ m→∞ In a similar way to (19), we get: Fxn+1 ∗ ,x∗ (t) = FT m x∗ ,T m x∗ (t) n+1 n+1   ∗ ,x∗ (t/2), FT m x∗ ,T m x∗ (t/2) ≥ ∆M FT m xn+1 n+1 n+1 n+1   −m   ∗ ,x∗ k ≥ ∆M Fxn+1 t/2 , FT m xn+1 ∗ ,T m x∗ (t/2) ; n+1 n+1 ∀m ∈ Z 0+ , ∀t ∈ R+ (24) and taking limits as m → ∞, one gets that limn→∞ FT m xn+1 ∗ ,T m x∗ (t/2) = n+1 n+1 ∗ ,x∗ (t) = 1; ∀t ∈ R + so that x n → x a.s.. On the other hand, the use of  ∗ ∗ limn→∞ Fxn+1 (21)–(23) in (19) as well as limn→∞ Fxn+1∗ ,x∗ (t) = 1; ∀t ∈ R + got from (24) yields for all t ∈ R+, since ∆M : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm, lim inf Fxn+1 ,x∗ (t) n→∞     −m  ≥ lim inf ∆M Fxn+1 ,Tn+1 m x n+1 (t/2), ∆M ∆M Fxn+1 ,xn+1 ∗ k t/8 , n,m→∞   −m  FT m xn+1 ∗ ,T m x∗ (t/8) , Fx∗ ,x∗ k n+1 n+1 n+1 t/4    lim Fx ,x∗ k −m t/8 ,   ≥ ∆M lim Fxn+1 ,Tn+1 m x (t/2), ∆ M ∆ M n,m→∞ n+1 m→∞ n+1 n+1   −m  lim FT m xn+1∗ ,T m x∗ (t/8) , lim Fx∗ ,x∗ k n+1 n+1 n+1 t/4 m→∞ m→∞ ≥ ∆M (1, ∆M (∆M (1, 1), 1)) = 1 (25) so that ∃limn→∞ Fxn ,x∗ (t) = 1; ∀t ∈ R+ and {xn } → x∗ a.s..  De la Sen et al. SpringerPlus (2016) 5:557 Page 16 of 27 As it occurs in the deterministic counterpart, (Berinde 2007), the uniform conver- gence of a sequence of operators {Tn } can be weakened if such operators possess certain additional contractive properties. See also Istratescu (1981). In this case, it is possible to get some close properties to those proved in “Main results concerning the uniform con- vergence of operators” section for the case of uniform convergence. Firstly, two defini- tions which are then used follow below: Definition 2 (Berinde 2007) A non-decreasing function ϕ : R0+ → R0+ (i.e. ϕ(t1) ≤ ϕ (t2) if t1 ≤ t2) is said to be a comparison function if {ϕ n (t)} → 0, ∀t ∈ R+. If, furthermore, (t − ϕ(t)) → +∞ as t → + ∞ then it is said to be a strict comparison function. (t)t Example 2 Note that ϕ(t) = 1+(t)t for t ∈ R0+ with  : R0+ → R0+ being such that (t)t is non-decreasing is a strict comparison function since it is non-decreasing and (t)t ϕ n (t) = ϕ ϕ n−1 (t) = 1+n(t)t for all n ∈ Z + implying that {ϕ n (t)} → 0; ∀t ∈ R0+ as   n → ∞. Example 3 Let (X, F ) be a PM-space, let T : X → X be a mapping  on X and let −1 (t)−1 (t) Fx,y ϕ : X × X × R0+ → R0+ be defined as ϕx,y (t) =  −1  ; ∀x, y ∈ X, ∀t ∈ R0+ 1+(t) Fx,y (t)−1 leading to the n-the composite function with itself resulting to be   (t) −1 (t)−1 Fx,y n (t) = ϕx,y  −1  ; ∀t ∈ R0+ , ∀n ∈ Z which satisfies {ϕ n (t)} → 0; ∀t ∈ R0+ as 1+n(t) Fx,y (t)−1 n → ∞. Then, ϕ : X × X × R0+ → R0+ is a strict comparison function for any x, y ∈ X   −1 (t) − 1 is non-decreasing for all t ∈ R provided that λ(0) = 0 and (t) Fx,y 0+ for each pair x, y ∈ X × X . Note, in particular, that ϕx,x(t) = 0; ∀t ∈ R0+; ∀x ∈ X so that the null-   function ϕ is both non-increasing and non-decreasing. Definition 3 Let (X, F ) bea PM-space.Then, G : X → X is said to be a strict ϕ-con- −1 traction if GTx,Ty −1 (t) − 1 , ∀x, y ∈ X,∀t ∈ R for some strict comparison (t) ≤ 1 + ϕ Gx,y + function ϕ : R0+ → R0+. The next result follows: Theorem 4 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators such that: 1. The operators Tn : X → X of the sequence {Tn} are all strict ϕ-contractions, 2. {Tn } → T for some T : X → X . Then, T : X → X is a strict ϕ-contraction and xn → x∗ a.s., where  ∗ * FTn = xn ; ∀n ∈ Z + and FT = {x }. Furthermore {xn } → x a.s., where xn+1 = Tnxn;  ∗ ∗ ∀n ∈ Z 0+ for any given x0 ∈ X. De la Sen et al. SpringerPlus (2016) 5:557 Page 17 of 27   Proof We have FT−1 n x,Tn y −1 (t) − 1 ; ∀x, y ∈ X,∀t ∈ R , ∀n ∈ Z (t) − 1 ≤ ϕ Fx,y + 0+ for some strict comparison function ϕ : R0+ → R0+, since all the operators of the sequence {T} are strict nϕ-contractions, what is equivalent to 1 FTn x,Tn y (t) ≥  ; ∀x, y ∈ X, ∀t ∈ R+ , ∀n ∈ Z 0+ −1 1+ϕ Fx,y (t) − 1 and then � � FTx,Ty (t) ≥ ∆M FTx,Tn x (t/2), FTn x,Ty (t/2) � � �� ≥ ∆M FTx,Tn x (t/2), ∆M FTn x,Tn y (t/4), FTn y,Ty (t/4)    1 ≥ ∆M FTx,Tn x (t/2), ∆M  � � , FTn y,Ty (t/4); −1 1 + ϕ Fx,y (t/4) − 1 (26) ∀n ∈ Z 0+ , ∀t ∈ R+ Thus, since {Tn } → T , Tn : X → X all strict ϕ-contractions, then everywhere continu- ous, and ∆M : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm, one gets:    1 FTx,Ty (t) ≥ lim inf ∆M FTx,Tn x (t/2), ∆M  � � , FTn y,Ty (t/4) n→∞ −1 1 + ϕ Fx,y (t/4) − 1    1 ≥ ∆M lim inf FTx,Tn x (t/2), ∆M  � � , lim inf FTn y,Ty (t/4) n→∞ −1 n→∞ 1 + ϕ Fx,y (t/4) − 1    1 = ∆M 1, ∆M  � � , 1 −1 1 + ϕ Fx,y (t/4) − 1   1 = ∆M  � � , 1 −1 1 + ϕ Fx,y (t/4) − 1 1 = � �; ∀t ∈ R+ , ∀x, y ∈ X −1 1+ϕ Fx,y (t/4) − 1 (27) so that   −1 −1 FTx,Ty (t) ≤ 1 + ϕ Fx,y (t/4) − 1       FT−1 −1 n x,T n y (t) ≤ 1 + ϕ F n−1 T x,T n−1 y (t/4) − 1 ≤ · · · ≤ 1 + ϕ n Fx,y −1 t/2n+1 − 1 ; ∀n ∈ Z 0+ , ∀t ∈ R+ , ∀x, y ∈ X (28)   since limn→∞ ϕ n Fx,y −1 t/2n+1 − 1 = 0 since {ϕ n (t)} → 0; ∀t ∈ R+. Then,   −1 t/2n+1 = 1, ∀t ∈ R , ∀x, y ∈ X. Thus, one also has that:   limn→∞ Fx,y + lim FT n x,T n y (t) = 1; ∀t ∈ R+ , ∀x, y ∈ X (29) n→∞ Eq. (28), leading to (29), establishes that T : X → X is a strict ϕ-contraction. It has to be proved that it has a unique fixed pointx*. Assume on the contrary that there are two points x∗ , x̄∗ (� = x∗ ) ∈ F (T ) so that De la Sen et al. SpringerPlus (2016) 5:557 Page 18 of 27     Fx̄∗ ,x∗ (t) = FT n x̄∗ ,T n x∗ (t) = lim FT n x̄∗ ,T n x∗ (t) ≤ 1 + lim ϕ n Fx,y −1 t/2n+1 − 1 n→∞ n→∞ = 1; ∀t ∈ R+ (30) and then x̄∗ = x∗ by the property 1 of (1) of the PM space (X, F ). On the other hand, one gets from (29) by taking y = y(x) = Tx and zn = zn(x) = Tn x for any given x ∈ X that lim F n+1 x,T n x (t) = lim FTzn ,zn (t) = 1; ∀t ∈ R+ , ∀x ∈ X n→∞ T n→∞ and, for any given  ∈ (0, 1) and t ∈ R+, there is N0 = N0 (ε, ) ∈ Z 0+ such that Fzn+1 ,zn (t) > 1 − ; ∀n(∈ Z 0+ ) ≥ N0 , ∀t ∈ R+ so that {zn } is a Cauchy sequence which converges to some limit point z* = z*(x) ∈ X since (X, F , ∆M ) is complete. Since the fixed point x*of T : X → X is unique, any limit point z* = z*(x) ∈ X of any sequence {T n x} for any arbitrary initial point x ∈ X is the fixed point x* of T : X → X . It remains to prove that {xn } → x∗ a.s., where xn+1 = Tnxn; ∀n ∈ Z 0+, for any given initial point x0 ∈ X. Note that, since Tn is a strict ϕ-contraction with FTn = xn∗ ; ∀n ∈ Z 0+, we can perform the   two next replacements to a close set of inequalities to those got in (19) and (24) within the proof of Theorem 1 1 FTn+1 m x m ∗ (t) → n+1 ,Tn+1 xn+1   −1 1 + ϕm Fxn+1 ,x∗ (t) − 1 n+1  −m  1 Fxn+1 ∗ ,x∗ k t →   −1 1 + ϕ m Fx∗ ∗ (t) − 1 n+1 ,x for all t ∈ R+ and m ∈ Z 0+. Thus, one gets instead of (24), � � ∗ ,x∗ (t) = FT m x∗ ,T m x∗ (t) ≥ ∆M FT m x∗ ,x∗ (t/2), FT m x∗ ,T m x∗ (t/2) Fxn+1 n+1 n+1 n+1 n+1 n+1 n+1   1 ≥ ∆M  � � , FT m x∗ ,T m x∗ (t/2); ∀n, m ∈ Z 0+ , ∀t ∈ R+ n+1 n+1 n+1 1 + ϕ m Fx−1 ∗ ,x∗ (t/2) − 1 n+1 (31) Since Tnm → T m then limn→∞ FT m xn+1   ∗ ,T m x∗ (t/2) = 1; ∀t ∈ R + ; ∀m ∈ Z 0+ , ∀t ∈ R + n+1 n+1 and the above constraint implies that   1 1 lim inf  −1 − � �  ≥ 0; ∀m ∈ Z 0+ , ∀t ∈ R+ n→∞ Fx∗ ,x∗ (t) 1 + ϕ m F −1 ∗ (t/2) − 1 n+1 x ,x∗ n+1    Equivalently, lim supn→∞ Fx−1 −1 ∗ ,x∗ (t) − 1 − ϕ Fx∗ ,x∗ (t/2) − 1 ≤ 0; ∀t ∈ R+. Since, n+1 n+1 −1 furthermore, F (t) is non-increasing and ϕ(t) is non-decreasing, one has:    lim sup Fx−1 −1 ∗ ,x∗ (t) − 1 − ϕ Fx∗ ,x∗ (t) − 1 n→∞ n+1 n+1    ≤ lim sup Fx∗ ,x∗ (t) − 1 − ϕ Fx−1 −1 ∗ ,x∗ (t/2) − 1 ≤ 0; ∀t ∈ R+ ; n→∞ n+1 n+1 De la Sen et al. SpringerPlus (2016) 5:557 Page 19 of 27 Define ση = sup{σ ∈ R0+ : σ − ϕ(σ ) ≤ η} which is defined for any given η ∈ R̄0+ (the extended nonnegative real semi-line) from the prop- erty of strict ϕ-contractions, since ϕ is a strict comparison function, lim supt→+∞ (t − ϕ m (t)) ≤ limt→+∞ (t − ϕ(t)) = +∞; ∀n ∈ Z 0+ as t → + ∞. It turns out that limη→0 ση = 0. Now, define σn = Fx−1 ∗ ∗ (t) − 1; ∀n ∈ Z 0+ , ∀t ∈ R +. Taking into n ,x account that    lim sup Fx−1 n −1 ∗ ,x∗ (t) − 1 − ϕ Fx∗ ,x∗ (t) − 1 n ≤ 0; ∀t ∈ R+ (32) n→∞   * * One concludes that limη→0 ση = limη→0 Fx−1 ∗ ,x∗ (t) − 1 = 0; ∀t ∈ R + so that {xn} → x n a.s.. Also, one can get, instead of (19) in the proof of Theorem 1, that � � �� Fxn+1 ,x∗ (t) ≥ ∆M Fxn+1 ,Tn+1 m x n+1 (t/2), ∆M F m m ∗ Tn+1 xn+1 ,T xn+1 (t/4), F m ∗ m T xn+1 ,T x ∗ (t/4) � ≥ ∆M Fxn+1 ,Tn+1 m x n+1 (t/2),    1 ∆M ∆M FT m xn+1∗ ,T m x∗ (t/8), n+1 n+1 � � , 1 + ϕ m Fx−1 ∗ n+1 ,xn+1 (t/8) − 1  1 � � ; ∀n ∈ Z 0+ , ∀m ∈ Z + , ∀t ∈ R+ m −1 1 + ϕ Fx∗ ,x∗ (t/4) − 1 n+1   and then limn→∞ ϕ n Fx,y −1 t/2n+1 − 1 = 0 since {ϕ n (t)} → 0; ∀t ∈ R . Take limits in   + the above expression by using the continuity of the minimum triangular norm and the fact that ϕ is a strict ϕ-comparison function by using close arguments to those used to get (32). We then conclude in a similar way the validity of (21) to (25) by replacing the conditions of k-contractions by conditions of strict ϕ-contractions so that there exist the limits limn→∞ Fxn∗ ,x∗ (t)= limn→∞ Fxn ,x∗ (t) = 1; ∀t ∈ R+ and {xn } → x∗ a.s..  A close result to Theorem 4 is now got in the case when {Tn } → → T , with Tn : X → X , so that T : X → X is a strict ϕ-contraction without requesting that all the elements of the sequence {Tn} be strict ϕ-contraction. Theorem 5 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators such that: 1. {Tn } → → T such that Tn : X → X ; ∀n ∈ Z 0+ for some T : X → X which is a strict ϕ -contraction, 2. xn∗ ∈ FTn � = ∅; ∀n ∈ Z 0+ Then, {Tn} has a subsequence of strict ϕ-contractions and xn∗ → x∗ a.s., where   FTn = xn∗ ; ∀n ∈ Z + and FT = {x*}. Furthermore {xn } → x∗ a.s., where xn+1 = Tnxn;   ∀n ∈ Z 0+ for any given x0 ∈ X. De la Sen et al. SpringerPlus (2016) 5:557 Page 20 of 27   Proof We have FTx,Ty−1 −1 (t) − 1 ; ∀x, y ∈ X,∀t ∈ R , ∀n ∈ Z (t) − 1 ≤ ϕ Fx,y + 0+ for some strict comparison function ϕ : R0+ → R0+, since T : X → X is a strict ϕ-contraction, equivalently, FTx,Ty (t) ≥  1 −1 ; ∀x, y ∈ X,∀t ∈ R + , ∀n ∈ Z 0+, and then 1+ϕ Fx,y (t)−1 � � FTn x,Tn y (t) ≥ ∆M FTn x,Tx (t/2), FTx,Tn y (t/2) � � �� ≥ ∆M FTx,Tn x (t/2), ∆M FTx,Ty (t/4), FTn y,Ty (t/4)    1 ≥ ∆M FTx,Tn x (t/2), ∆M  � � , FTn y,Ty (t/4); −1 1 + ϕ Fx,y (t/4) − 1 ∀n ∈ Z 0+ , ∀t ∈ R+ . Thus, since {Tn } → → T and T : X → X is a strict ϕ-contraction, then everywhere continu- ous, and ∆M : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm, one gets:    1 lim inf FTn x,Tn y (t) ≥ lim inf ∆M FTx,Tn x (t/2), ∆M  � � , FTn y,Ty (t/4) n→∞ n→∞ −1 1 + ϕ Fx,y (t/4) − 1    1 ≥ ∆M lim inf FTx,Tn x (t/2), ∆M  � � , lim inf FTn y,Ty (t/4) n→∞ −1 n→∞ 1 + ϕ Fx,y (t/4) − 1    1 1 = ∆M 1, ∆M  � � , 1 = � �; −1 −1 1 + ϕ Fx,y (t/4) − 1 1 + ϕ Fx,y (t/4) − 1 ∀t ∈ R+ , ∀x, y ∈ X so that one gets the following recursion:   −1 lim sup lim sup FT m x,T m y (t) n n m→∞ n→∞    −1 ≤ 1 + lim sup lim sup ϕ F m−1 m−1 (t/4) − 1 m→∞ n→∞ Tn x,Tn y      m −1 m+1 ≤ · · · ≤ 1 + lim sup lim sup ϕ Fx,y t/2 −1 =1 m→∞ n→∞   since limm→∞ ϕ m Fx,y−1 t/2n+1 − 1 = 0 since {ϕ m (t)} → 0; ∀t ∈ R+. Then,   −1 t/2m+1 = 1, ∀t ∈ R , ∀x, y ∈ X, and lim n→∞ FT n x,T n y (t) = 1; ∀t ∈ R +,   limm→∞ Fx,y + ∀x, y ∈ X. So, there is a subsequence Tnn of {Tn } whose elements are strict -contrac-    ϕ tions and the elements of such a subsequence have unique fixed points xnk . Eq. (32)∗ can also be got under the conditions of this theorem so that one concludes that {x*n} → x* a.s.. The remaining of the proof is close to its counterpart in Theorem 4.  We now reformulate close results to the above ones associated with strict ϕ-contrac- tions via a dual class of contractions referred to as dual strict ϕ-contractions which oper- ate directly on contractive conditions on the probability density function instead on its inverse. For that purpose, we first introduce the concept of dual strict comparison func- tion as follows: De la Sen et al. SpringerPlus (2016) 5:557 Page 21 of 27 Definition 4 A non-increasing function ϕ : R0+ → R0+ (i.e. ϕ(t1) ≥ ϕ(t2) if t1 ≤ t2) is said to be a dual comparison function if {ϕ n (t)} → 1, ∀t ∈ R+. If, furthermore, (t − ϕ(t)) → +∞ as t → + ∞ then it is said to be a dual strict comparison function. Definition 5 Let (X, F) be a PM-space. Then, G : X → X is said to be a dual strict ϕ -contraction if GTx,Ty (t) ≥ ϕ Gx,y (t) , ∀x, y ∈ X,∀t ∈ R+ for some dual strict comparison   function ϕ : R0+ → R0+. Note that if T is any strict k-contraction for any given k ∈ (0, 1) then   FTx,Ty (t) ≥ Fx,y k −1 t ≥ Fx,y (t) = ϕ Fx,y (t) ; ∀x, y ∈ X;   ∀t ∈ R+ if ϕ ≡ 1 which is a dual strict comparison function since F is non-decreasing and left-continuous. If {ϕ n (t)} → 1, although ϕ : R0+ → R0+ be non-necessarily unity but a dual strict comparison function, then FT n+1 x,T n+1 y (t) ≥ FTx,Ty k −n t ≥ ϕ Fx,y k −n t = ϕ n Fx,y (t) ;        ∀x, y ∈ X, ∀t ∈ R+ . Since {ϕ n (t)} → 1, ∀t ∈ R+ because it is a dual strict comparison function, all the lim- its of the above chain equalize unity. So, if T : X → X is any strict k-contraction then it is also a dual strict ϕ-contraction. The converse is not true in general. Assume now that T : X → X is a dual strict ϕ-contraction for some dual strict comparison function ϕ so that FTx,Ty (t) ≥ ϕ Fx,y (t) ; ∀x, y ∈ X, ∀t ∈ R+. Since F is non-decreasing and ϕ is non-   increasing, we have for k ∈ (0, 1):      Fx,y k −1 t ≥ Fx,y (t); ϕ Fx,y k −1 t   ≤ ϕ Fx,y (t) ; ∀x, y ∈ X, ∀t ∈ R+ so that FTx,Ty (t) ≥ ϕ Fx,y (t) ≥ ϕ Fx,y k −1 t ; ∀x, y ∈ X, ∀t ∈ R+ and any given      k ∈ (0, 1). One then gets that FT n x,T n y (t) ≥ ϕ n Fx,y k −n t so that FT n x,T n y (t) → 1 as    n → ∞; ∀x, y ∈ X, ∀t ∈ R+. Then, if T : X → X is a ϕ-contraction, it is not a strict k-con- traction, in general. The next result follows: Theorem 6 Let (X, F , ∆M ) be a complete Menger space and let {Tn } be a sequence of operators such that: 1. The operators Tn : X → X; ∀n ∈ Z 0+ of the sequence {Tn} are all dual strict ϕ-con- tractions, 2. {Tn } → T for some T : X → X . Then, T : X → X is a dual strict ϕ-contraction and xn∗ → x∗ a.s., where FTn = xn∗ ;     ∀n ∈ Z + and FT = {x*}. Furthermore {xn } → x∗ a.s., where xn+1 = Tnxn; ∀n ∈ Z 0+ for any given x0 ∈ X. De la Sen et al. SpringerPlus (2016) 5:557 Page 22 of 27 Proof We have FTn x,Tn y (t) ≥ ϕ Fx,y (t) ; ∀x, y ∈ X,∀t ∈ R+ , ∀n ∈ Z 0+ for some strict   comparison function ϕ : R0+ → R0+, since all the operators of the sequence {Tn} are dual strict ϕ-contractions; ∀x, y ∈ X,∀t ∈ R+. Under close steps to those used in the proof of Theorem 4, we get instead of (28):      FT n x,T n y (t) ≥ ϕ FT n−1 x,T n−1 y (t/4) ≥ · · · ≥ ϕ n Fx,y t/2n+1 ; ∀n ∈ Z 0+ , ∀t ∈ R+ , ∀x, y ∈ X (33) and limn→∞ ϕ Fx,y t/2n n+1= 1, since {ϕ n (t)} → 1; ∀t ∈ R+. Then limn→∞    FT n x,T n y (t) = 1; ∀t ∈ R+, ∀x, y ∈ X and one can conclude in a similar way to Theorem 4 that T, which is the point-wise limit of the sequence {Tn } of dual strict ϕ-contractions, is also a dual strict ϕ-contraction. We also can prove the remaining properties of the state- ment in a close way to the proof of Theorem 4.  It turns out that a similar result to Theorem 6, under the guidelines of Theorem 5, can be directly formulated for the case when {Tn } → → T with Tn : X → X where the limit operator T : X → X is a dual strict ϕ-contraction. Numerical example This section contains some numerical examples illustrating the theoretical results previ- ously established, in particular Theorems 1, 3 and 6 from “Main results concerning the uniform convergence of operators” and “Main results concerning the point-wise con- vergence of operators” sections, respectively. For this purpose, we extend the example proposed in de la Sen and Ibeas (2015): Example 4 Consider the space X = R0+ = [0, +∞) with the probabilistic met- this example as d(x, y) = ‖x − y‖ where ‖ · ‖ stands for the Euclidean norm. Clearly, ric given by Fx,y (t) = t+d(x,y)t where d(x, y) is a deterministic metric, selected in [0, +∞), Fx,y (t), ∆M is a Menger PM-space. Moreover, consider now the iterative   scheme given by xn+1 = Tnxn; n ∈ Z 0+ with the family of nonlinear operators Tn (n ≥ 0) xn being defined by Tn xn = f (n)(1+x n) on [0, + ∞) and f (n) = 2n+3 n+1 . This family of non- linear operators is simultaneously a strict k and dual strict ϕ-contraction (according to Definitions 1 and 4, respectively) so that conditions from Theorems 1, 3 and 6 hold and they can be applied simultaneously. A numerical simulation of the iterative scheme is performed in this section to show these theoretical results. Initially, we must ensure that they are strict k and dual strict ϕ-contractions. Thus, according to Definition 1 each Tn has to satisfy FTn x,Tn y (kt) ≥ Fx,y (t), ∀x, y ∈ [0, + ∞) and ∀t ≥ 0 for some real constant 0 < k < 1. Therefore, x y x−y Tn x − Tn y = − = f (n)(1 + x) f (n)(1 + y) f (n)(1 + x)(1 + y) De la Sen et al. SpringerPlus (2016) 5:557 Page 23 of 27 so that kt ktMn FTn x,Tn y (kt) = =  ; ∀t ∈ R+ , ∀n ∈ Z 0+ kt + � � x−y ktMn + x − y Mn where Mn = f(n)(1 + x)(1 + y). On the other hand: t Fx,y (t) =  ; ∀t ∈ R+ t + x − y  so that the condition FTn x,Tn y (kt) ≥ Fx,y (t) becomes: ktMn t  ≥  ; ∀t ∈ R+ . (34) ktMn + x − y   t + x − y  qt If we denote now q = kMn and consider the function g(q) = qt+�x−y� we have:     dg(q) d qt x − yt =   = 2 ≥ 0; ∀t ∈ R+ dq dq qt + x − y   qt + x − y Therefore, g(q) is a non-decreasing function and if q ≥ 1 we have g(q) ≥ g(1). Thus, Eq. (34) holds provided that q = kMn ≥ 1, condition that can be achieved if: 1 1 k≥ ≥ inf Mn Mn Moreover, inf Mn = inf(f(n)(1 + x)(1 + y)) = inf f(n) = 2 since X = [0, +∞) and Eq. (34) holds if k ≥ 12. Therefore, we can choose 12 ≤ k < 1 and Tn is a strict k-contraction for every non-negative n. Furthermore, if we select now the dual strict comparison function ϕ(t) = 1, Tn is also a strict dual ϕ-contraction since FTn x,Tn y (kt) ≥ Fx,y (t) still holds when k = 1 (Definition 4). Therefore, Theorems 1, 3 and 6 hold and the nonlinear iterative scheme: xn xn+1 = (35) f (n)(1 + xn ) converges to the unique fixed point of T = limn→∞ Tn = limn→∞ f (n)(1+x) x x = 2(1+x) which is x* = 0. Figure 1 shows the evolution of the iterative scheme (35) for different initial conditions. It can be seen in Fig. 1 that the sequence of iterates converges to zero as predicted by Theorem 2(iii), Theorems 3 and Theorem 6. Moreover, the sequence of probability functions Fxn ,x∗ (t) = Fxn ,0 (t) converges to unity as Figs. 2, 3 and 4 show. Conclusions This paper has investigated some relevant properties of convergence of sequences built through sequences of operators which are either uniformly convergent to either a strict k-contractive operator, for some given real constant k ∈ (0, 1), or which are strictly k-contractive and point-wisely convergent to some limit operator. Those convergence De la Sen et al. SpringerPlus (2016) 5:557 Page 24 of 27 15 8 xn 6 2 0 1 2 3 4 5 6 7 8 n Fig. 1 Evolution of the sequence of iterates for different initial conditions 1 0.9 0.8 0.7 0.6 x ,0 0.5 1 F 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 t Fig. 2 Convergence of the sequence of probability functions Fxn ,x ∗ (t) to unity. Display of Fx1 ,0 (t) properties are also reformulated for the case when either the sequence of operators or its limit are strict ϕ-contractions. The limits of the built convergent sequences are either the limits of the sequence of fixed points of the corresponding sequences of operators or the fixed points of the limit operator of the sequence of operators. The definitions of strict k-contractions and ϕ-contractions are given in the context of probabilistic metric spaces for the given probability density function. A numerical illustrative example is proposed and discussed in detail. De la Sen et al. SpringerPlus (2016) 5:557 Page 25 of 27 1 0.9 0.8 0.7 0.6 x ,0 0.5 3 F 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 t Fig. 3 Convergence of the sequence of probability functions Fxn ,x ∗ (t) to unity. Display of Fx3 ,0 (t) 1 0.9 0.8 0.7 0.6 x ,0 0.5 6 F 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 t Fig. 4 Convergence of the sequence of probability functions Fxn ,x ∗ (t) to unity. Display of Fx6 ,0 (t) Authors’ contributions MS contribution consists of the theoretical developments and mathematical results with their proofs. AI and JH con- tributions consist of the statement, development of the numerical simulations of the last section and the verification, correction and testing of the organization and complete details of the whole manuscript. All authors read and approved the final manuscript. Author details 1 Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, PO Box 644, de Bilbao, Barrio Sarriena, 48940 Leioa, Bizkaia, Spain. 2 Department of Telecommunications and Systems Engineering, Faculty of Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Bellaterra, Cerdanyola del Vallés, Barcelona, Spain. 3 Present Address: Departamento de Ingeniería, Facultad de Ciencias Naturales e Ingeniería, Universidad de Bogotá Jorge Tadeo Lozano, 22 Street, No. 4‑96, Mod. 7A, Bogotá, D.C. 110311, Colombia. 4 Departamento de Ingeniería, Facultad de Ciencias Naturales e Ingeniería, Universidad de Bogotá Jorge Tadeo Lozano, 22 Street, No. 4‑96, Mod. 7A, Bogotá, D.C. 110311, Colombia. Acknowledgements The authors thank the reviewers for their useful comments and to University Jorge Tadeo Lozano by its support through Grant 644-11-14. Competing interests The authors declare that they have no competing interests. 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