Nonlinear Analysis: Modelling and Control, Vol. 24, No. 5, 819–837 ISSN: 1392-5113 https://doi.org/10.15388/NA.2019.5.8 eISSN: 2335-8963 Existence of common fixed points for linear combinations of contractive maps in enhanced probabilistic metric spaces∗ Shahnaz Jafaria , Maryam Shamsa , Asier Ibeasb , Manuel De La Senc a Department of Pure Mathematics, University of Shahrekord, Shahrekord 88186-34141, Iran jafari.shahnaz@yahoo.com; maryam.shams@sci.sku.ac.ir b Department of Telecommunications and Systems Engineering, Faculty of Engineering, Universitat Autónoma de Barcelona (UAB), 08193 Bellaterra, Cerdanyola del Vallés, Barcelona, Spain asier.ibeas@uab.cat c 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 manuel.delasen@ehu.eus Received: September 23, 2018 / Revised: May 14, 2019 / Published online: September 27, 2019 Abstract. In this paper, we introduce the concept of enhanced probabilistic metric space (briefly EPM-space) as a type of probabilistic metric space. Also, we investigate the existence of fixed points for a (finite or infinite) linear combination of different types of contractive mappings in EPM-spaces. Furthermore, we investigate about the convergence of sequences (generated by a finite or infinite family of contractive mappings) to a common fixed point. The useful application of this research is the study of the stability of switched dynamic systems, where we study the conditions under which the iterative sequences generated by a (finite or infinite) linear combination of mappings (contractive or not), converge to the fixed point. Also, some examples are given to support the obtained results. In the end, a number of figures give us an overview of the examples. Keywords: enhanced probabilistic metric space, fixed point, linear combination, convergent sequence, dynamic systems. 1 Introduction In recent times, fixed point theory has become an important tool in pure and applied sciences, such as biology [4], chemistry [25], economics [10], engineering and physics, to ∗ The authors would like to express their sincere appreciation to the Shahrekord University and the Center of Excellence for Mathematics for financial support. Also, this work was partially supported by the Spanish Ministry of Economy and Competitiveness and by the European Fund of Regional Development (MINECO/FEDER) through grants RT12018-094336-B-100 and RT12018-094902-B-C22 and to the Basque Government for its support through grant IT1207-19. c 2019 Authors. Published by Vilnius University Press This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 820 S. Jafari et al. cite just a few. The Banach’s fixed point theory, widely known as the contraction principle, is an important tool in the theory of metric spaces [14, 22]. Moreover, since the location of the fixed point can be obtained by means of an iteration process, it can be implemented on a computer to find the fixed point of contraction mappings easily. Several authors have extended the Banach’s fixed point theorem in various ways. For instance, a fixed point theorem for multivalued contractive mappings was established by Nadler [28]. Rich showed that a contractive mapping from complete metric space X into the family K(X) of all nonempty compact subsets of X has a fixed point [29]. You can see similar works in [11, 30]. However, it is important to notice that some results presented in the literature as generalizations to well-known theorems are not actually real generalizations in the sense that they can be obtained from previous exisiting results, (see, for instance, [6,19,20, 31]). The concept of metric space was introduced by the French mathematician Maurice Frechet in 1906, and since then several generalizations of it have been proposed in the literature, see [1,5]. One of the important generalizations is the probabilistic metric space (PM space), which was introduced by the Austrian mathematician K. Menger in 1942 [26] by using the notion of distribution functions in place of nonnegative real numbers. Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces, which are endowed with a triangular norm [33]. A generalization of the Banach contraction principle on a complete Menger space was worked by Sehgal and Bharuch-Reid in 1972 [33]. After this initial work, the fixed point theory in probabilistic metric spaces has been developed in many works such as [13,21]. In 1984, Khan et al. introduced the concept of altering distance function [23]. A ϕ-function is the extension of altering distance function and has been worked by many authors, [24, 27, 34]. For instance, the concepts of (α, ψ)-type contractive and α-admissible mappings were introduced by Gopal et al. [17], who also established some fixed point theorems for these mappings in complete Menger spaces. After that, Shams and Jafari generalized this concept to (α, β, ψ)-contractive and α−β-admissible mappings and proved some fixed point theorems for such maps [34]. The family of contraction mappings was introduced by Ciric [12] and Taskovic [35]. The existence of a common fixed point for a finite or infinite family of self-mappings and contractive maps is worked by many authors [2, 7]. In this way, the existence of a fixed point for infinite families of self-mappings of a complete metric space satisfying some new conditions of common contractivity was studied by Allahyari et al. in [3]. Also, the study on fixed point and convergence of sequences generated by a family of k-contractions and ϕ-contractions in Menger spaces was worked by De la Sen et al. in [15]. In this paper, we introduce the concept of enhanced probabilistic metric space (briefly, EPM-space) as a type of PM-space and show that it is a real generalization of PM-spaces in the sense of [19]. Also, motivated by the definition of contractive mappings introduced in [16] and [34] in Menger PM-spaces, we investigate the existence of common fixed points for a (finite or infinite) linear combination of these contractive mappings in EPM- spaces. Furthermore, we investigate about the convergence of sequences (generated by a family of contractive mappings) to a common fixed point. Although there exist many works related to the existence of a fixed point for different types of contractive maps in Menger PM-spaces [9, 16, 17], this is the first time that this research is being investigated http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 821 on linear combinations of these contractive maps. The useful potential application of this research is the study of the stability of switched dynamic systems, where we study the conditions under which the iterative sequences generated by (finite or infinite) linear combination of mappings (contractive or not), converge to the fixed point. Also, some examples and application to switched dynamic systems are given to support the obtained results. The rest of the paper is organized as follows: in the first section, we introduce some previous and new notions employed in the article. The second section is our main result, which is related to the existence of a fixed point for a linear combination of contractive maps and about the convergence of sequences generated by these combinations. The final section contains some numerical examples that illustrate the main results. 2 Background results In this section, we bring some notions, definitions and known results, which are related to our work in order to make the paper self-contained. For more details on PM-space, we refer the reader to [18]. Also, we introduce the notion of enhanced probabilistic metric space (briefly, EPM-space) and bring some properties of this space. We denote by R the set of real numbers, Z the set of integers, Z+ = {z ∈ Z: z > 0}, Z0+ = Z+ ∪ {0} and f m is composition of f for m times. Definition 1. A distribution function is a function F : (−∞, ∞) → [0, 1], that is non- decreasing and left-continuous on R. Moreover, inf t∈R F (t) = 0 and supt∈R F (t) = 1. The set of all the distribution functions is denoted by D, and the set of those distribution functions such that F (0) = 0 is denoted by D+ . We will denote the specific Heaviside distribution function by: ( 1, t > 0, H(t) = 0, t 6 0. Definition 2. An enhanced probabilistic metric space (briefly, EPM-space) is an ordered pair (X, F ), where X is a nonempty set, and F is a mapping from X × X into D+ such that, if Fx,y denotes the value of F at the pair (x, y), the following conditions hold: (EPM1) Fx,y (t) = H(t) if and only if x = y. (EPM2) Fx,y (t) = Fy,x (t). (EPM3) If Fx,y (t) = 1 and Fy,z (s) = 1, then Fx,z (t + s) = 1 for all x, y, z ∈ X and s, t > 0. P∞ (EPM4) 1 / FP∞ (t) 6 i=1 λi / Fxi ,yi (t) for all λi ∈ [0, 1], P∞ P∞ i=1 λi xi , i=1 λi yi λ i=1 i = 1, and for all t > 0 and xi , yi ∈ X, i ∈ Z+ . We can see that an EPM-space is a particular case of a PM-space. Hence, all properties and previous results in PM-spaces (in this section) hold in EPM-space as follows. Definition 3. A binary operation T : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if the following conditions hold: Nonlinear Anal. Model. Control, 24(5):819–837 822 S. Jafari et al. (a) T is commutative and associative. (b) T is continuous. (c) T (a, 1) = a for all a ∈ [0, 1]. (d) T (a, b) 6 T (c, d) whenever a 6 c and b 6 d for a, b, c, d ∈ [0, 1]. The following are three basic continuous t-norms [32]: (i) The minimum t-norm TM defined by TM (a, b) = min{a, b}. (ii) The product t-norm Tp defined by Tp (a, b) = ab. (iii) The Lukasiewicz t-norm TL defined by TL (a, b) = max{a + b − 1, 0}. These t-norms are related in the following way: TL 6 TP 6 TM . Definition 4. A Menger enhanced probabilistic metric space (briefly, Menger EPM- space) is a triple (X, F, T ), where (X, F ) is a EPM-space, and T is a continuous t-norm such that, for all x, y, z ∈ X and s, t > 0,  Fx,y (t + s) > T Fx,z (t), Fz,y (s) . It is straightforward to check that a Menger EPM-space is also a Menger PM-space. However, the converse is not true in general as the following example shows. Example 1. Let X = R, T = Tp , and ( min{|x|,|y|}+t max{|x|,|y|}+t , x, y ∈ [0, ∞) or x, y ∈ (−∞, 0], Fx,y (t) = t t |x|+t |y|+t elsewhere for all t > 0. Clearly, (X, F, Tp ) is a Menger PM-space. We show that (X, F, Tp ) is not a EPM-space since does not satisfy (EPM4). To see P∞this, consider P∞ λi = (1/2)i , i xi = 1/((3/2) λi ) and yi = 1/λi . It is easy to see, i=1 λi =P1, i=1 λi xi = 2, P ∞ ∞ i=1 λi yi = ∞. Thus, 1/F i=1 λi xi , i=1 λi yi (t) = ∞ and i=1 λi /Fxi ,xi (t) = P∞ P∞ 3 + 3/5t. So (X, F, Tp ) is not a EPM-space. Definition 5. Let (X, F, T ) be a Menger EPM-space. Then (i) A sequence xn in X is said to be convergent to x if, for every  > 0 and 0 < λ < 1, there exists a positive integer N such that Fxn ,x () > 1 − λ whenever n > N . (ii) A sequence xn in X is called a Cauchy sequence if, for every  > 0 and λ > 0, there exists a positive integer N such that Fxn ,xm () > 1 − λ whenever n, m > N . (iii) A Menger EPM-space is said to be complete if and only if every Cauchy sequence in X converges to a point in X. (iv) A sequence xn is called G-Cauchy if limn→∞ Fxn ,xn+m (t) = 1 for each m ∈ Z+ and t > 0. (v) The space (X, F, T ) is called G-complete if every G-Cauchy sequence in X is convergent to a point in X. http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 823 According to [7], the (, λ)-topology in (X, F, T ) is introduced by the family of neighborhoods Nx of a point x ∈ X given by  Nx = Nx (, λ):  > 0, λ ∈ (0, 1) , where  Nx (, λ) = y ∈ X: Fx,y () > 1 − λ . The (, λ)-topology is a Hausdorff topology. Therefore, in this topology, a function f is continuous in x0 ∈ X if and only if f (xn ) → f (x0 ) for every sequence xn → x0 , [7]. Definition 6. (See [8].) A function φ : [0, ∞) → [0, ∞) is said to be a Φ-function if it satisfies the following conditions: (i) φ(t) = 0 if and only if t = 0. (ii) φ(t) is strictly monotone increasing and φ(t) → ∞ as t → ∞. (iii) φ(t) is left-continuous in t ∈ (0, ∞). (iv) φ(t) is continuous at t = 0. In the sequel, the class of all Φ-functions will be denoted by Φ. Also, we denote by Ψ the class of all continuous functions ψ : [0, ∞) → [0, ∞) such that ψ(0) = 0 and ψ n (an ) → 0 whenever an → 0 as n → ∞. Definition 7. (See [34].) Let (X, F, T ) be a Menger PM-space, f : X → X be a given mapping, and α, β : X × X × (0, ∞) → [0, ∞) be two functions. We say that f is α−β-admissible if (i) x, y ∈ X, for all t > 0, α(x, y, t) > 1 ⇒ α(f x, f y, t) > 1; (ii) x, y ∈ X, for all t > 0, β(x, y, t) 6 1 ⇒ β(f x, f y, t) 6 1. Now we introduce the following definition that is a particular case of Definition 7. So all properties and previous results related to α−β-admissible hold for enhanced α−β- admissible. Definition 8. Let (X, F, T ) be a Menger EPM-space, {fi }i∈N be a family of self-maps on X, and α, β : X × X × (0, ∞) → [0, ∞) be two bilinear functions. We say that {fi }i∈N is enhanced α−β-admissible if (i) x, y ∈ X, for all t > 0, α(x, y, t) > 1 ⇒ α(fi x, fj y, t) > 1, i, j ∈ Z+ ; (ii) x, y ∈ X, for all t > 0, β(x, y, t) 6 1 ⇒ β(fi x, fj y, t) 6 1, i, j ∈ Z+ . Definition 9. (See [34].) Let (X, F, T ) be a Menger EPM-space, f : X → X be a given mapping. We say that f is an (α, β, ψ)-contractive mapping if there exist two functions α, β : X × X × (0, ∞) → [0, ∞) and ψ ∈ Ψ satisfying the following inequality:     1 1 α(x, y, t) − 1 6 β(x, y, t)ψ −1 (1) Ff x,f y (ϕ(ct)) Fx,y (ϕ(t)) for all x, y ∈ X, t > 0 such that Fx,y (ϕ(t)) > 0, where c ∈ (0, 1) and ϕ ∈ Φ. Nonlinear Anal. Model. Control, 24(5):819–837 824 S. Jafari et al. Remark 1. If α(x, y, t) = β(x, y, t) = 1, then we have the following theorem in PM- space [16] that also hold in EPM-space. Theorem 1. Let (X, F, T ) be a G-complete Menger EPM-space, and f : X → X be a mapping satisfying the following inequality:   1 1 −16ψ −1 , (2) Ff x,f y (ϕ(ct)) Fx,y (ϕ(t)) where c ∈ (0, 1), ϕ ∈ φ and ψ ∈ Ψ . Then f has a fixed point, i.e., there exists a point u ∈ X such that f u = u. Remark 2. We say that the mapping f : X → X is ψ-contractive if it satisfies condi- tion (2). Theorem 2. (See [34].) Let (X, F, T ) be a G-complete Menger EPM-space, and let f : X → X be a (α, β, ψ)-contractive mapping satisfying the following conditions: (i) f is α−β-admissible. (ii) There exists x0 ∈ X such that α(x0 , f x0 , t) > 1 and β(x0 , f x0 , t) 6 1 for all t > 0. (iii) There exists a sequence {xn } in X fulfilling this condition: if α(xn , xn+1 , t) > 1 and β(xn , xn+1 , t) 6 1 for all n ∈ Z+ and for all t > 0, and xn → x as n → ∞, then α(xn , x, t) > 1 and β(xn , x, t) 6 1 for all n ∈ Z+ and for all t > 0. Then f has a fixed point, i.e., there exists a point u ∈ X such that f u = u. Theorem 3. With the same hypotheses of Theorem 2, if for all x ∈ X and for all t > 0, there exists z ∈ X such that α(x, z, t) > 1 and β(x, z, t) 6 1, then f has a unique fixed point. Remark 3. It should be noticed that the convergent point x of the sequence (xn ) in (iii) can be different from the fixed point u. Proposition 1. In order to calculate the fixed point of f in Theorems 1 and 2, we can set up the sequence xn+1 = f (xn ) with x0 ∈ X arbitrary. Thus, the convergence points of this iteration are the fixed points of f . We can see this result in the proof of Theorems 1 and 2 in [16] and [34], respectively. Lemma 1. (See [15].) The following properties hold: (i) If for all n ∈ Z+ , fn : X → X are continuous and {fn } uniformly converges to {f }, then {fnm } uniformly converge to {f m } for all m ∈ Z+ . (ii) If {fn } uniformly converges to f and fn commutes with f in X for all n ∈ Z0+ , then {fnm } uniformly converge to {f m } for all m ∈ Z+ . http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 825 3 Main results In this section, we prove that a linear combination of a finite or infinite family of (α, β, ψ)- contractive and ψ-contractive mappings have a fixed point in a Menger EPM-space. Also, we discuss about the convergence of iterative sequences generated by linear combinations of (α, β, ψ)-contractive and ψ-contractive maps. Moreover, we will show the application of this fixed point theory to switched dynamic systems. The subsequent result shows that when the family of operators {fn }n∈N are not contractive but converge to a contractive map f , then the iterated sequence generated by fn , converge to the fixed point of f . We denote by Fix(f ) = {x ∈ X: f (x) = x} the set of fixed points of f . Theorem 4. Let (X, F, T ) be a complete Menger EPM-space, and let {fn } be a sequence of operators fn : X → X such that Fix(fn ) = {x∗n } for all n ∈ Z+ , and {fn } uniformly converges to f , where f : X → X is a (α, β, ψ)-contractive map with f (x∗ ) = x∗ . Consider the sequence {xn } ⊂ X generated by the iterated scheme xn+1 = fn (xn ) for any given x1 ∈ X and n ∈ Z+ . Assume that some of the conditions below holds: (i) {fnm } uniformly converge to {f m } for all m ∈ Z+ . (ii) {fn } uniformly converges to f and either fn : X → X is continuous for all n ∈ Z0+ or {fn } commutes with f for all n ∈ Z0+ . Then {xn } → x∗ . Proof. Since {fn } uniformly converges to {f } for any given λ ∈ (0, 1) and t > 0, there exists N1 (∈ Z0+ ) = N1 (λ, t) such that Ffn xn ,fxn (ϕ(ct)) > 1 − λ, where ϕ ∈ Φ and c ∈ (0, 1). The proof is performed by contradiction. Assume that {xn } does not converge to x∗ . Then there exists some real constant λ0 ∈ (0, 1) such that for some subsequence {fnm } of {fn }, which generates the sequence xnj +1 = fnj xnj for some given x1 ∈ X for all j ∈ Z0+ , we have: 1 −1 1 − λ0 1 6 −1 limj→∞ (limm→∞ Fxnj+m ,x∗ (ϕ(ct))) 1 = −1 limj→∞ (limm→∞ Ffnj mx nj ,x ∗ (ϕ(ct))) 1 6 −1 limj→∞ (limm→∞ T (Ffnj mx m nj ,f xnj ( ϕ(ct) m m ∗ ϕ(ct) 2 ), Ff xnj ,f x ( 2 ))) 1 6 T (limj→∞ limm→∞ F mx fnj m nj ,f xnj ( ϕ(ct) m m ∗ ϕ(ct) 2 ), limm→∞ Ff xnj ,f x ( 2 )) − 1. (3) Nonlinear Anal. Model. Control, 24(5):819–837 826 S. Jafari et al. From condition (i) and according to Lemma 1, {fnmj } uniformly converge to f m . Hence, from (3) we get 1 1 −16 −1 1 − λ0 T (1, limm→∞ Ff m xnj ,f m x∗ ( ϕ(ct) 2 )) 1 = − 1. (4) limm→∞ Ff m xnj ,f m x∗ ( ϕ(ct) 2 ) Since f is (α, β, ψ)-contractive and ψ n (an ) → 0 whenever an → 0, from (4) we deduce that     1 1 α(x, y, t) − 1 6 α(x, y, t) − 1 1 − λ0 limm→∞ Ff m xnj ,f m x∗ ( ϕ(ct) 2 )   m 1 6 lim β(x, y, t)ψ ϕ( t ) − 1 = 0. (5) m→∞ Fxnj ,x∗ ( c2m ) Note that (5) yields α(x, y, t)(1/(1 − λ0 ) − 1) 6 0 implying 1 − λ0 > 1 and λ0 6 0, which is the contradiction. Hence, {xnj } → x∗ and then {xn } → x∗ . So the theorem is proved. The following theorem shows that a linear combination of a family of (α, β, ψ)- contractive maps that satisfy Theorem 2 has a common fixed point. Theorem 5. Let {fi }m i=1 be a finite family of self-maps Tm on X that is enhanced α−β- admissible satisfying all conditions of Theorem 2, i=1 Fix(f ) = {u1 , u2 , . . . , um }, and Pi m fi are continuous at the common fixed points. Let f = i=1 λi fi , where λi ∈ (0, 1) also P m and i=1 λi = 1. Then {u1 ,Tu2 , . . . , um } are also fixed points of f . m Moreover, for each u ∈ i=1 Fix{fi }, let {xin } be the sequence associated to each fi such that {xn } → u as n → ∞ and xin+1 = fi (xin ) with xi0 ∈ X arbitrary. Then the i following results hold: Pm (i) Consider {tn } to be the corresponding sequence of f such that tn = i=1 λi xin . Then there exists a sequence {wn } satisfying wn = f (tn ) such that {wn } → u as n → ∞. (ii) Consider {zn } be one of the sequences associated to f such that zn+1 = f (zn ) with z0 ∈ X arbitrary. Then {zn } → u as n → ∞. Proof. We show that u is a fixed point of f (linear combination of fi ), where u ∈ T m i=1 Fix(fi ). Afterward, we prove that the sequences {wn } and {zn } converges to u. Since u is a common fixed point of fi , we have f (u) = λ1 f1 (u) + λ2 f2 (u) + · · · + λm fm (u) = λ1 u + λ2 u + · · · + λm u = (λ1 + λ2 + · · · + λm )u = u Pm since i=1 λi = 1. Hence, f (u) = u and u is a fixed point of f . http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 827 Now, in case (i), we want to prove that the sequence {wn } converges to u as n → ∞. According to Proposition 1, we know that each sequence {xin } converges to u as n → ∞. Also, we have wn = f (tn ). Hence, lim wn = lim f (tn ) n→∞ n→∞  = lim λ1 f1 (tn ) + λ2 f2 (tn ) + · · · + λm fm (tn ) . (6) n→∞ On the other hand, m m  m ! X X  X lim tn = lim λi xin = lim λi xin = λi u = u. (7) n→∞ n→∞ n→∞ i=1 i=1 i=1 From (6), (7) and because of continuity of fi at the fixed point, we get m ! X lim wn = λ1 f1 (u) + λ2 f2 (u) + · · · + λm fm (u) = λi u = u. n→∞ i=1 This means the sequence {wn } converges to the fixed point u. Now in case (ii), we prove that the iterative sequence {zn } converges to the fixed point u. According to (EPM4)-property, we have m 1 1 X 1 = P 6 λi Ff x,f y (ϕ(ct)) F mi=1 λi fi x, Pmi=1 λi fi y (ϕ(ct)) i=1 F fi x,fi y (ϕ(ct)) m   X 1 = λi − 1 + 1. (8) i=1 F fi x,fi y (ϕ(ct)) From (8) and since each fi is (α, β, ψ)-contractive, we get   X m   1 1 α(x, y, t) −1 6 λi α(x, y, t) −1 Ff x,f y (ϕ(ct)) i=1 Ffi x,fi y (ϕ(ct)) m   X 1 6 λi β(x, y, t)ψ −1 i=1 Fx,y (ϕ(t))   1 = β(x, y, t)ψ −1 , Fx,y (ϕ(t)) this means that f is (α, β, ψ)-contractive. Also, f is α−β-admissible because m m ! m Xm X X X α(f x, f y, t) = α λi fi x, λi fi y, t = λi λj α(fi x, fj y, t) (9) i=1 i=1 i=1 j=1 since {fi }m i=1 is enhanced α−β-admissible. Hence, from (9) we get m X X m α(f x, f y, t) > λi λj > 1. i=1 j=1 Nonlinear Anal. Model. Control, 24(5):819–837 828 S. Jafari et al. Also, by the same proof we have β(f x, f y, t) 6 1. It is easy to check that f satisfy all conditions of Theorem 2. Hence, according to Proposition 1, the sequence {zn } defined by zn+1 = f (zn ), converges to the fixed point u. So the theorem is proved. We show that the linear combination of contractive maps in PM-space is not necessary contractive and Theorem 5 is not applicable in PM-space. Example 2. Consider the Menger PM-space (X, F, Tp ), where X = [−6, 6] and Fx,y (t) is the same as Example 1. Let λ1 = λ2 = 1/2, and fi : X → X, i = 1, 2, defined by f1 (x) = x and f2 (x) = −4. We define the functions α, β : X × X × (0, ∞) → [0, ∞) by ( 1, x, y ∈ [−6, 0] or x, y ∈ [0, 6], α(x, y, t) = 0 otherwise, ( 2, x, y ∈ [−6, 0] or x, y ∈ [0, 6], β(x, y, t) = 1 otherwise. Let ϕ(t) = ψ(t) = t. Then f1 is (α, β, ψ)-contractive map. To view this, suppose x, y < 0 and x < y, then α(x, y, t) = 1 and β(x, y, t) = 2. So inequality (1) holds for all c ∈ (1/2, 1). In the same way, we can show that for other cases, inequaility (1) holds, and f1 is (α, β, ψ)-contractive map. It is easy to see that f2 is (α, β, ψ)-contractive map. Also, f1 and f2 have common fixed point x = −4. We show that Theorem 5(ii) is not applicable in this case since f = f1 /2 + f2 /2 = x/2 − 2 is not (α, β, ψ)-contractive map. To see this, consider x = 6 and y = 3, then there is not any c ∈ (0, 1) such that inequality (1) holds. Hence, f is not (α, β, ψ)-contractive map. Remark 4. Notice that the previous result also holds when λi ∈ [0, 1] instead of λi ∈ T some j ∈ {1, 2, . . . , m} and λi 6= 0 for the remaining (0, 1). In this case, if λj = 0 for elements, Theorem 5 holds for λi 6=0 Fix{fi }. Pm Corollary Pm 1. In Theorem 5, if we have f = fn = i=1 λin fi with λin ∈ (0, 1) and i=1 λin = 1 for each n > 1, then the results of Theorem 5 hold as well. In fact, in equations (6) and (7) of the proof of Theorem 5, we set up λin instead of λi , we get limn→∞ λin xin as follows: Since limn→∞ xin = u, there exists k: n > k ⇒ |xin − u| 6 δ. Thus we have m X m X m X m X λin xin − u = λin xin − λin xin − u  λin u = i=1 i=1 i=1 i=1 m X 6 λin δ 6 δ. i=1 Pm This means limn→∞ i=1 λin xin = u. Hence, the results hold in the case of time-varying linear combinations of contractive maps. http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 829 The subsequent result shows that a linear combination of ψ-contractive maps that satisfy Theorem 1, has a common fixed point. Proposition 2. Let {fiT }mi=1 be a finite family of self-maps on X satisfying all con- m ditions of Theorem 1, i=1 Fix(f Pim) = {u1 , u2 , . . . , um } and also continuous Pm at the common fixed points. Let f = i=1 λin fi , where λin ∈ (0, 1) and i=1 λin = 1. Then {u1 , u2 , . . . , um } are also fixed points of f . Tm Moreover, for each u ∈ i=1 Fix{fi }, let {xin } be the sequence associated to each fi such that {xin } → u and xin+1 = fi (xin ) with xi0 ∈ X arbitrary. Then the following conditions hold: Pm (i) Consider {tn } to be the corresponding sequence of f such that tn = i=1 λin xin . Then there exists a sequence {wn } satisfying wn = f (tn ) such that {wn } con- verges to u as n → ∞. (ii) Consider {zn } be one of the sequences associated to f such that zn+1 = f (zn ) with z0 ∈ X arbitrary. Then {zn } converges to u as n → ∞. Proof. The proof of this result follows the same steps as the proof of Theorem 5. Theorem 6. Let {fi }m i=1 be a finite family of self-maps on X that satisfy all conditions of Theorem 2, and {gni }ki=1 be a finite family of sequences of continuous operators gni : X → X such that {gni } uniformly convergence to g, where g : X → X is a (α, β, ψ)- i contractive map. Also, assume that Fix(fi ) ∩ Fix(g Pm n ) = {uP 1 , u2 , . . . , um } and {fi } is k i continuousP in the common fixed points. Let f = i=1 λ i fi + i=1 µi gn , where λi , µi ∈ m Pk (0, 1) and i=1 λi + i=1 µi = 1. Then {u1 , u2 , . . . , um } are also fixed points of f . Moreover, for each u ∈ Fix(fi ) ∩ Fix(gni ), let {xin } be the sequence associated to each fi with xi0 ∈ X arbitrary, and let {yni } be the sequence associated to gni with y0i ∈ X arbitrary such that {xin } → u and {yni } → u as n → ∞ and xin+1 = fi (xin ), i i i yPn+1 = gn (yn ). Consider {tn } to be the corresponding sequence of f such that tn = m i P k i λ x i=1 i n + i=1 µi yn . Then there exists a sequence {wn } satisfying wn = f (tn ) such that {wn } converges to u as n → ∞. Proof. We show that u is a fixed point of f (linear combination of fi and gni ), where u ∈ Fix(fi ) ∩ Fix(gni ). Afterwards, we prove that the sequence {wn } converges to u. Since u is a common fixed point of fi and gni , we have m k ! X X f (u) = λ i fi + µi gni u i=1 i=1 = λ1 f1 (u) + λ2 f2 (u) + · · · + λm fm (u) + µ1 gn1 (u) + µ2 gn2 (u) + · · · + µk gnk (u) = (λ1 + λ2 + · · · + λm + µ1 + µ2 + · · · + µk )u. (10) Pm Pk Since i=1 λi + i=1 µi = 1, from (10) we get f (u) = u and u is a fixed point of f . Nonlinear Anal. Model. Control, 24(5):819–837 830 S. Jafari et al. Now we prove that the sequence {wn } converges to u. According to Proposition 1, we know that each sequence {xin } converges to u, and also the sequence {yni } converges to u by Theorem 4. We have wn = f (tn ). Hence, m X k X lim wn = lim f (tn ) = lim λi fi (tn ) + lim µi gni (tn ). (11) n→∞ n→∞ n→∞ n→∞ i=1 i=1 On the other hand, we know m k ! m k X X X X lim tn = lim λi xin + µi yni = λi (u) + µi (u) = u. (12) n→∞ n→∞ i=1 i=1 i=1 i=1 Now from (11), (12) and since the function fi and gni are continuous at the common fixed points, we get Xm Xk lim wn = λi fi (u) + µi gni (u) = u. n→∞ i=1 i=1 So the theorem is proved. Pm Pk Corollary 2. In Theorem 6, if we have f = i=1 λin fi + i=1 µin gni , where λin , µin ∈ Pm Pk (0, 1) and i=1 λin + i=1 µin = 1 for n > 1, then the results hold, too. Theorem 7. Let {fi }m i=1 be a finite family of (α, β, ψ)-contractive self-maps on X that 1 satisfy Theorem 2, and {gi }m i=1 be a family of operators from X to X. Also, assume 2 that Fix(fi ) ∩ Fix(g Pm1i ) = {u1 ,Pu2 , . . . , um } and {fi } are continuous inP the common fixed m1 +m2 m1 +m2 poins. Let f = i=1 λin fi + i=m 1 +1 λ in gi , where λ in ∈ (0, 1) and i=1 λin = 1 for n > 1. Assume that we have the following condition: (i) λin → λ∗i , i = 1, 2, . . . , m1 , and λin → 0, i = m1 +1, . . . , m1 +m2 , as n → ∞. Then {u1 , u2 , . . . , um } are also fixed points of f . Moreover, for each u ∈ Fix(fi ) ∩ Fix(gi ), let {xin } and {yni } be the sequences associated to each fi and gi , respectively, such that {xin } and {yni } converge to u as n → ∞ with xin+1 = fi (xin )P i and yn+1 = i m i gP(y i n ). Consider {t n } to be the corresponding sequence of f such that tn = λ x i=1 i n + k i i=1 µi yn . Then there exists a sequence {wn } satisfying wn = f (tn ) such that {wn } converges to u as n → ∞. Proof. We show that u is a fixed point of f : m1 mX ! X 1 +m2 m1 X mX 1 +m2 f (u) = λin fi + λin gi u = λin fi (u) + λin gi (u) i=1 i=m1 +1 i=1 i=m1 +1 mX ! 1 +m2 = λin u = u, i=1 hence, f has the fixed point u. http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 831 Now we want to show that the sequence {wn } converges to u. We have lim wn = lim f (tn ) n→∞ n→∞ m1 X mX 1 +m2 = lim λin fi (tn ) + lim λin gi (tn ). (13) n→∞ n→∞ i=1 i=m1 +1 Since limn→∞ λin = λ∗i , i = 1, 2, . . . , m1 , and limn→∞ λin = 0, i = m1 + 1, . . . , m1 + m2 , in (13), we get lim wn = lim f (tn ) n→∞ n→∞ m1 X mX 1 +m2 = lim λ∗i fi (tn ) + lim λin gi (tn ) . (14) n→∞ n→∞ i=1 i=m1 +1 | {z } 0 On the other hand, m1 mX ! X 1 +m2 lim tn = lim λin xin + λin yni , (15) n→∞ n→∞ i=1 i=m1 +1 We know limn→∞ λin = 0, i = m1 + 1, . . . , m1 + m2 , and also, according to Proposition 1, we know that each sequence {xin } converges to u. Hence, in (15), we get m1 mX ! X 1 +m2 m1 X i i ∗ lim tn = lim λin xn + lim λin yn = λi u = u. (16) n→∞ n→∞ n→∞ i=1 i=m1 +1 i=1 | {z } 0 Now from (16) and since the function fi is continuous in u, in (14) we conclude that m1 X m1 X lim wn = λ∗i fi (u) = λ∗i u = u. n→∞ i=1 i=1 This means the sequence {wn } converges to the fixed point of f . So the theorem is proved. The subsequent result will show that a linear combination of an infinite family of contractive maps in Menger EPM-space have a common fixed point. Also, we prove that the iterative sequences generated by this linear combination converges to the fixed point. Theorem 8. Let {fi }i∈N be an infinite family of self-maps on X that is enhanced α−β- admissible P∞ satisfying Theorem 2, P and ∩ Fix(fi ) = {u1 , u2 , u3 , . . . }. Consider f = ∞ i=1 λ i f i , where λi ∈ (0, 1) and i=1 λi = 1 for all i ∈ N . Then {u1 , u2 , u3 , . . . } are also fixed points of f . Nonlinear Anal. Model. Control, 24(5):819–837 832 S. Jafari et al. T∞ Moreover, for each u ∈ i=1 Fix{fi }, let {xin } be the sequence associated to each fi with xi0 ∈ X arbitrary such that {xin } → u and xin+1 = fi (xin ). Consider {zn } be one of the sequences associated to f such that zn+1 = f (zn ) with z0 ∈ X arbitrary. Then {zn } → u as n → ∞. Proof. It is easy to see that u is a fixed points of f . Now we show that the iterative sequence {zn } converges to the fixed point u. According to (EPM4)-property, we have ∞ 1 1 X 1 = P 6 λi Ff x,f y (ϕ(ct)) F i=1 λi fi x, i=1 λi fi y (ϕ(ct)) ∞ P ∞ i=1 Ffi x,fi y (ϕ(ct)) ∞   X 1 = λi − 1 + 1, (17) i=1 Ffi x,fi y (ϕ(ct)) From (17) and because each fi is (α, β, ψ)-contractive, we get   ∞   1 X 1 α(x, y, t) −1 6 λi α(x, y, t) −1 Ff x,f y (ϕ(ct)) i=1 Ffi x,fi y (ϕ(ct)) ∞   X 1 6 λi β(x, y, t)ψ −1 i=1 Fx,y (ϕ(t))   1 = β(x, y, t)ψ −1 , Fx,y (ϕ(t)) this means that f is (α, β, ψ)-contractive map. Also, by the same proof of Theorem 5, f is α−β-admissible. It is easy to check that f satisfy all conditions of Theorems 2. Thus, according to Proposition 1, the sequence zn+1 = f (zn ) converges to the fixed point u. So the theorem is proved. P∞ Corollary 3. In Theorem 8, if we have f = fn = i=1 λin fi , where λin ∈ (0, 1) and P ∞ i=1 λin = 1 for n > 1, then the results hold, too. Notice that in Theorems 5 and 8, we prove that, when {fi }m i=1 is a family of enhanced α−β-admissible self-maps, then the linear combination of {fi }m i=1 is α−β-admissible. Now an open question arises: is the linear combination of {fi }m i=1 , α−β-admissible when each element of the {fi }m i=1 family is α−β-admissible? 4 Numerical examples In this section, we consider some numerical examples obtained by simulation, which illustrate the main results discussed in the previous sections. The first, second and third examples are concerned with a finite linear combination of contractive mappings with constant and time-varying coefficients. The other examples illustrate the results previously established in Theorem 8 and Corollary 3. http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 833 Example 3. Let X = R, Fx,y (t) = t/(t + |x − y|) for all x, y ∈ X, t > 0, and the t-norm T is defined as T (a, b) = ab. Then (X, F, T ) is a G-complete Menger EPM-space. Consider the dynamic system given by zn+1 = f (zn ) with f = λ1 f1 + λ2 f2 + λ3 f3 , λ1 = 1/2, λ2 = 1/5, λ3 = 31/0 and fi : X → X, i = 1, 2, 3, by ( ( x x , x ∈ [0, 1], 1 , x ∈ [0, 1], f1 (x) = 4 f2 (x) = sin(x), f3 (x) = x52 0 otherwise, 4 4 otherwise. We define the functions α, β : X × X × (0, ∞) → [0, ∞) by ( ( 1 1, x, y ∈ [0, 1], , x, y ∈ [0, 1], α(x, y, t) = β(x, y, t) = 2 0 otherwise, 2 otherwise. Consider c = 3/4 and ϕ, ψ : [0, ∞) → [0, ∞) by ϕ(t) = ψ(t) = t. We want to prove that mappings fi and f satisfy the hypotheses of Theorem 5. It is easy to see that fi and f satisfy all conditions of Theorem 2 and x = 0 is a common fixed point of f and fi . Define the sequence {xin } such that xin+1 = fi (xin ), i = 1, 2, 3, with x10 = 1/4, x20 = π/4 and x30 = 1/2. Hence, we get (x1n ) = 1/4n , (x2n ) = (1/4n ) sinn (π/4) and (x3n ) = (1/2)(1/5n ). It is easy to see these sequences converge to zero as n → ∞. P3 Now let {tn } be the corresponding sequence of f such that tn = i=1 λi xin . We have P3 limn→∞ tn = limn→∞ i=1 λi xin = 0. On the other hand, since function fi is continu- ous in x = 0. Hence, we get 3 X lim wn = lim f (tn ) = lim λi fi (tn ) = 0. n→∞ n→∞ n→∞ i=1 Now we define the sequence (zn ) by zn+1 = f (zn ) with z0 = 1. Then we get zn = (9/20)n + (1/4n ) sinn (1/4). Hence, limn→∞ zn = 0, i.e., the sequence {zn } converges to the fixed point. For a better understanding of the above example, see Fig. 1 in which sequences {tn } converges to the fixed point u. Also, see Fig. 2 in which the sequence {wn } converges to u and Fig. 3, where the sequence {zn } converges to u, as predicted by Theorem 5. Example 4. Let X = R, Fx,y (t) = t/(t + |x − y|) for all x, y ∈ X, t > 0, and the t-norm T is defined as T (a, b) = ab. Then (X, F, T ) is a G-complete Menger EPM- space. Consider the switched dynamic system zn+1 = fn (zn ) given by fn = λ1n f1 + λ2n f2 + λ3n f3 with λ1n = 1/2, λ2n = 1/5, λ3n = 3/10. Define: λ1n+1 = λ3n , λ2n+1 = λ1n , λ3n+1 = λ2n . Let the functions fi , αi , βi and ϕ, ψ be the same as Example 3. Then Corollary 1 holds and the switched dynamic system is stable. See Fig. 4, where the iterated sequence converge to the fixed (equilibrium) point x = 0. Also, if we define: λ1n+1 = λ3n − 1, λ2n+1 = λ1n + 1/2, λ3n+1 = λ2n + 1/2 for n ∈ N , then the results hold, and the switched dynamic system is stable. This is an example of how these techniques are useful to study switched dynamic systems and prove that the stability holds independent of the switching law. Nonlinear Anal. Model. Control, 24(5):819–837 834 S. Jafari et al. Pm of thei sequence {tn Figure 1. Convergence i } Figure 2. Convergence of the sequence {wn } defined by tn = i=1 λi xn , where {xn } defined by wn = f (tn ) to the fixed point. converge to u. Figure 3. Convergence of the sequence {zn } Figure 4. Convergence of the sequence {zn } defined Pm by zn+1 = f (zn ), where f = defined Pm by zn+1 = fn (zn ), where fn = i=1 λi fi for different initial conditions. i=1 λin fi for different initial conditions. Example 5. Let X = [0, 1], Fx,y (t) = t/(t + |x − y|) for all x, y ∈ X, t > 0, and the t-norm T is defined as T (a, b) = ab. Then (X, F, T ) is a G-complete Menger EPM- space. Consider the dynamic system zn+1 = f (zn ) given by f = λ1 f1 + λ2 f2 + λ3 f3 with λ1 = 1/4, λ2 = 2/4, λ3 = 1/4 and fi : X → X by f1 (x) = x/2, f2 (x) = x and f3 (x) = x/4 for all x ∈ X. We consider c = 1/2 and ψ : [0, ∞) → [0, ∞) by ψ(t) = ϕ(t) = t. It is easy to check that each fi is ψ-contractive map and x = 0 is common fixed point for fi . Now f x = x/8 + x/2 + x/16 = 11x/16, and x = 0 is a fixed point of f . Consider z0 = 1. Then we get limn→∞ zn = limn→∞ (11/16)n = 0, i.e., the iterative sequence {zn } converges to the fixed point as predicted by Proposition 2, and the dynamic system is stable (asymptotically stable). Example 6. Let X = [0, 1], Fx,y (t) = t/(t + |x − y|) for all x, y ∈ X, t > 0, and the t-norm T is defined as T (a, b) = ab. Then (X, F, T ) is a G-complete Menger PEPM-space. ∞ Consider the switched dynamic system given by zn+1 = f (zn ) with f = i=1 λi fi (x) i 2 with {λi } = 1/2 and fi : X → X by fi (x) = x /(10i). We define ϕ, ψ : [0, ∞) → http://www.journals.vu.lt/nonlinear-analysis Existence of common fixed points for linear combinations of contractive maps 835 P∞ Figure 5. Convergence of the sequence {zn } defined by zn+1 = f (zn ), where f = i=1 λi fi for different initial conditions. [0, ∞) by ϕ(t) = ψ(t) = t and αi , βi : X × X × (0, ∞) → [0, ∞) by αi (x, y, t) = 2 and βi (x, y, t) = 1/2 for all x, y ∈ X. It is easy to see that fi and f satisfy the hypotheses P∞ 2 for c =P of Theorem [2/(5i), 1), and hence, fi have the fixed point x = 0. We have ∞ f = i=1 λi fi (x) = i=1 (1/2i )(x2 /(10i). According to Theorem 8, f has a common fixed point. In fact, x = 0 is common fixed point for fP . Now we define the sequence ∞ {zn } by zn+1 = f (zn ) with z0 = 1. Then we get zn = ( i=1 (1/2i )(1/(10i))n . Hence, limn→∞ zn = 0. i.e., the sequence {zn } converge to the fixed point. See Fig. 5, where the iterative sequence of infinite linear combination of contractive maps converges to the fixed point. Example 7. Let X = [0, 1], Fx,y (t) = t/(t + |x − y|) for all x, y ∈ X, t > 0, and the t-norm T is defined as T (a, b) = ab. Then (X, F, T ) is a G-complete Menger EPM-space. Consider the switched dynamic system given by zn+1 = fn (zn ) with fn = P ∞ i i=1 λin fi (x) with {λin } = 1/2 . Define λin+1 = λin . Let the functions fi , αi , βi and ϕ, ψ be the same as Example 3. Then Corollary 3 holds, and the switched dynamic system is stable. The fixed point theory is a useful tool in this content. 5 Conclusion In this paper, we investigate the existence of fixed points for a (finite or infinite) linear combination of (α, β, ψ)-contractive and ψ-contractive mappings. Also, we investigate about the convergence of sequences (generated by a linear combination of (α, β, ψ)- contractive and ψ-contractive mappings) to the fixed point and prove that these sequences converge to the common fixed point. Also, we study about linear combination of mappings such that some of them are contractive and some of them are not. In this case, we provide the condition that ensure that these linear combinations have common fixed points. Acknowledgment. The authors would like to thank the editor and the referees for their constructive comments and suggestions, particulary for drawing our attention to bring some examples to show that our results are real generalization. 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