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Elsevier Editorial System(tm) for Applied Mathematics and Computation Manuscript Draft Manuscript Number: AMC-D-15-02505R1 Title: A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations Article Type: Full Length Article Keywords: b-metric space; cyclic map; fixed point; integral equation. Corresponding Author: Prof. stojan radenovic, Corresponding Author's Institution: MF First Author: stojan radenovic Order of Authors: stojan radenovic; Tatjana Dosenovic; Tatjana Aleksic Lamport; Zorana Golubovic Abstract: In this paper we obtain some equivalences between cyclic contractions and noncyclic contractions in the framework of b-metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b-metric spaces established by George et al. [J. Nonlinear Funct. Anal. 2015: 5, 2015] and Nashine et al. [Filomat. 28(10): 2047-2057,2014]. Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results.

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A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations Stojan Radenović1 , Tatjana Došenović2 , Tatjana Aleksić Lampert3 , Zorana Golubovíć4 1

Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap University, Cao Lanch City, Dong Thap Province, Viet Nam 2 Faculty of Technology, University of NoviSad, Serbia 3 Faculty of Sciences, Department of Mathematics, Radoja Domanovića 12, 34 000 Kragujevac, Serbia 4 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia

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Abstract. In this paper we obtain some equivalences between cyclic contractions and noncyclic contractions in the framework of b-metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b-metric spaces established by George et al. [J. Nonlinear Funct. Anal. 2015: 5, 2015] and Nashine et al. [Filomat. 28(10): 2047-2057, 2014]. Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results. Keywords: b-metric space, cyclic map, fixed point, integral equation MSC: 47H10, 54H25.

––––––––––––––––––––––––––––––– 1. INTRODUCTION AND PRELIMINARIES Fixed point theory is one of the traditional theory in mathematics and has a large number of applications in many branches of nonlinear analysis. It is well known that the celebrated Banach contraction principle [4] is a basic result in fixed point theory, which has been extended in many different directions. One of the most interesting generalizations was given by Kirk et al. [16] in 2003 by introducing the following notion of cyclic representation. Definition 1.1 ([16]) Let A and B be nonempty subsets of a metric space (X, d) and T : A ∪ B → A ∪ B. Then T is called a cyclic map if T (A) ⊆ B and T (B) ⊆ A. The following interesting theorem for a cyclic map was given in [16]. Theorem 1.2 Let A and B be nonempty closed subsets of a complete metric space (X, d). Suppose that T : A ∪ B → A ∪ B is a cyclic map such that d (T x, T y) ≤ λd (x, y) 1

E-mail addresses: [email protected]; [email protected] (S. Radenović); [email protected] (T. Došenović); [email protected] (T. A. Lampert); [email protected] (Z. Golubović). The second, third and fourth authors are thankful to the Ministry of Education, Science and Technological Development of Serbia.

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for all x ∈ A and y ∈ B, where k ∈ [0, 1) is a constant. Then T has a unique fixed point u and u ∈ A ∩ B. It should be noticed that cyclic contractions (unlike Banach-type contractions) need not be continuous, which is an important gain of this approach. Following the work of Kirk et al., several authors proved many fixed point results for cyclic mappings, satisfying various (nonlinear) contractive conditions. For some results and observations, the reader refers to [23] and [24]. Berinde initiated the concept of almost contractions and obtained several interesting fixed point theorems (see [5]). Here we recall them, but in the context as in [25]. Definition 1.3 Let f and g be two self-mappings on a metric space (X, d) . They are said to satisfy almost generalized contractive condition, if there exist a constant δ ∈ (0, 1) and some L ≥ 0 such that for all x, y ∈ X, d (f x, gy) ≤ δM (x, y) + LN (x, y) , where

d (x, gy) + d (y, f x) M (x, y) = max d (x, y) , d (x, f x) , d (y, gy) , 2

N (x, y) = min {d (x, fx) , d (y, gy) , d (x, gy) , d (y, fx)} . Khan et al. [14] introduced the concept of altering distance function as follows. Definition 1.4 A function ϕ : [0, ∞) → [0, ∞) is called an altering distance function if the following properties hold: (1) ϕ is continuous and nondecreasing; (2) ϕ (t) = 0 if and only if t = 0. Also, there are some generalizations of usual metric spaces. One well-known generalization is b-metric space (see [3, 6]) or metric type space (for short, MTS) called by some authors (see [10, 12, 13]). The following definition is introduced in [3] and [6]. Definition 1.5 Let X be a (nonempty) set and s ≥ 1 be a given real number. A function d : X × X → [0, ∞) is called a b-metric on X if, for all x, y, z ∈ X, the following conditions hold: (b1) d (x, y) = 0 if and only if x = y; (b2) d (x, y) = d (y, x); (b3) d (x, z) ≤ s [d (x, y) + d (y, z)] (b-triangular inequality). In this case, the pair (X, d) is called a b-metric space (metric type space). Otherwise, for more definitions such as b-convergence, b-completeness, b-Cauchy sequence in b-metric spaces, we see [1-3, 6-13, 15, 17, 18, 21, 26]. Note that every metric space is a b-metric space (metric type space), but the converse is not necessarily true (see [1,2, 6-13, 15, 17, 21, 25, 26]). 2

In the sequel we use the following lemma to show the fixed point results in the framework of b-metric spaces. Lemma 1.6 ([12], Lemma 3.1) Let {yn } be a sequence in a b-metric space (X, d) with s ≥ 1 such that d (yn , yn+1 ) ≤ λd (yn−1 , yn ) for some λ ∈ [0, 1s ), and each n = 1, 2, .... Then {yn } is a b-Cauchy sequence in (X, d). Besides that Definition 1.1 authors still introduced the following more general definition: Definition 1.7 ([16]) Let X be a nonempty set. Let p be a positive integer, A1 , A2 , ..., Ap be nonempty subsets of X, Y = ∪pi=1 Ai and T : Y → Y. Then Y = ∪pi=1 Ai is said to be a cyclic representation of Y with respect to T if (i) Ai (i = 1, 2, ..., p) are nonempty closed sets; (ii) T (A1 ) ⊆ A2 , ..., T (Ap−1 ) ⊆ Ap , T (Ap ) ⊆ A1 . George et al. [7] proved the following fixed point results for cyclic contractions in the setting of b-metric spaces. Theorem 1.8 Let {Ai }pi=1 where p is a positive integer, be nonempty closed subsets of a b-complete b-metric space (X, d) with coefficient s ≥ 1 and suppose that T : ∪pi=1 Ai → ∪pi=1 Ai is a cyclical operator that satisfies the condition T (Ai ) ⊆ Ai+1 for all i ∈ {1, 2, ..., p} such that d (T x, T y) ≤ λd (x, y) for all x ∈ Ai , y ∈ Ai+1 and λ ∈ 0, 1s . Then T has a unique fixed point. Theorem 1.9 Let {Ai }pi=1 where p is a positive integer, be nonempty closed subsets of a b-complete b-metric space (X, d) with coefficient s ≥ 1 and suppose that T : ∪pi=1 Ai → ∪pi=1 Ai is a cyclical operator that satisfies the condition T (Ai ) ⊆ Ai+1 for all i ∈ {1, 2, ..., p} such that d (T x, T y) ≤ λ [d (x, T x) + d (y, T y)] 1 for all x ∈ Ai , y ∈ Ai+1 and λ ∈ 0, s+1 . Then T has a unique fixed point. Theorem 1.10 Let {Ai }pi=1 where p is a positive integer, be nonempty closed subsets of a b-complete b-metric space (X, d) with coefficient s ≥ 1 and suppose that T : ∪pi=1 Ai → ∪pi=1 Ai is a cyclical operator that satisfies the condition T (Ai ) ⊆ Ai+1 for all i ∈ {1, 2, ..., p} 3

such that d (T x, T y) ≤ λ [d (x, T y) + d (y, T x)] 1 . for all x ∈ Ai , y ∈ Ai+1 and λ ∈ 0, s+1 Then T has a unique fixed point. Theorem 1.11 Let {Ai }pi=1 where p is a positive integer, be nonempty closed subsets of a b-complete b-metric space (X, d) with coefficient s ≥ 1 and suppose that T : ∪pi=1 Ai → ∪pi=1 Ai is a cyclical operator that satisfies the condition T (Ai ) ⊆ Ai+1 for all i ∈ {1, 2, ..., p} such that d (T x, T y) ≤ λ max {d (x, y) , d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} 1 for all x ∈ Ai , y ∈ Ai+1 and λ ∈ 0, s(s+1) . Then T has a unique fixed point. Further, George et al. [7] introduced a cyclic generalized (ψ, φ)-rational contraction and proved some fixed point results in b-metric spaces. Definition 1.12 Let (X, d) be a b-complete b-metric space, A1 , A2 , ..., Ap where p is a positive integer, be nonempty closed subsets of X and Y = ∪pi=1 Ai . An operator T : Y → Y is called a cyclic generalized (ψ, φ)-rational contraction if (1) T is a cyclic operator; (2) for any x ∈ Ai , y ∈ Ai+1 (i = 1, 2, ..., p), ψ and ϕ are altering distance functions, L ≥ 0, Ap+1 = A1 , it satisfies that ψ s4 d (T x, T y) ≤ ψ (M (x, y)) − φ (M (x, y)) + Lψ (m (x, y)) where

1 + d (x, T x) d (x, T y) + d (y, T x) M (x, y) = max d (x, y) , d (x, T x) , d (y, T y) , (1.1) 1 + d (x, y) 2s and m (x, y) = min {d (x, T x) + d (y, T y) , d (x, T y) , d (y, T x)} .

(1.2)

Theorem 1.13 Let (X, d) be a b-complete b-metric space, A1 , A2 , ..., Ap where p is a positive integer, be nonempty closed subsets of X and Y = ∪pi=1 Ai . Suppose that T : Y → Y is a cyclic generalized (ψ, φ)-rational contraction. Then T has a unique fixed point in ∩pi=1 Ai . Corollary 1.14 Let (X, d) be a b-complete b-metric space and T be a mapping from X to X such that for all x, y ∈ X and L ≥ 0, ψ s4 d (T x, T y) ≤ ψ (M (x, y)) − φ (M (x, y)) + Lψ (m (x, y)) 4

where ψ ad φ are altering distance functions and M (x, y) and m(x, y) are given by (1.1) and (1.2), respectively. Then T has a unique fixed point. Further, in [8] authors introduced the notion of cyclic generalized ϕ-contraction in b-metric spaces as follows: Definition 1.15 Let (X, d) be a b-metric space with parameter s ≥ 1. Let p be a positive integer, A1 , A2 , ..., Ap be nonempty subsets of X and Y = ∪pi=1 Ai . An operator T : Y → Y is called a generalized ϕ-contraction for some continuous and nondecreasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0 and ϕ (t) < 2st for each t > 0, if (1) Y = ∪pi=1 Ai is a cyclic representation of Y with respect to T ; (2) for any (x, y) ∈ A1 × Ai+1 , i = 1, 2, ..., p (with Ap+1 = A1 ), d (T x, T y) ≤ M (x, y) + LN ϕ (x, y) , where L ≥ 0, M (x, y) = max ϕ (d (x, y)) , ϕ (d (x, T x)) , d (x, T y) + d (y, T x) d (x, T x) + d (y, T y) ϕ ,ϕ 2 2s

(1.3)

and N ϕ (x, y) = min {ϕ (d (x, y)) , ϕ (d (x, T x)) , ϕ (d (y, T y)) , ϕ (d (x, T y)) , ϕ (d (y, T x))} . (1.4) According to this notion authors proved the following result in [8]: Theorem 1.16 Let (X, d) be a b-complete b-metric space, p ∈ N, A1 , A2 , ..., Ap nonempty closed subsets of X and Y = ∪pi=1 Ai . Suppose that T : Y → Y is a cyclic generalized ϕ- contractive mapping, for some ϕ. Then T has a unique fixed point. Moreover, the fixed point of T belongs to ∩pi=1 Ai . Corollary 1.17 Let (X, d) be a b-complete b-metric space and let T : X → X satisfy the following condition: there exists ϕ ∈ Φ such that for all x, y ∈ X, d (T x, T y) ≤ M (x, y) + LN ϕ (x, y) , where L ≥ 0, M (x, y) and N ϕ (x, y) are given by (1.3) and (1.4), respectively. Then T has a unique fixed point. Remark 1.18 It is worth noticing that we can put N (x, y) = min {d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} instead of N ϕ (x, y) in (1.4). This is possible, because xn = T xn−1 (n = 1, 2, . . . ), then N ϕ (xn−1 , xn ) = min {ϕ (d (xn−1 , xn )) , ϕ (d (xn , xn+1 )) , ϕ (d (xn−1 , xn+1 )) , ϕ (d (xn , xn ))} = min {ϕ (d (xn−1 , xn )) , ϕ (d (xn , xn+1 )) , ϕ (d (xn−1 , xn+1 )) , 0} = 0 5

and N (xn−1 , xn ) = min {d (xn−1 , xn ) , d (xn , xn+1 ) , d (xn−1 , xn+1 ) , d (xn , xn )} = min {d (xn−1 , xn ) , d (xn , xn+1 ) , d (xn−1 , xn+1 ) , 0} = 0. Now that ϕ is continuous and nondecreasing, we have that N ϕ (xn , x) = min {ϕ (d (xn , xn+1 )) , ϕ (d (x, T x)) , ϕ (d (xn , T x)) , ϕ (d (x, xn+1 ))} → 0 (n → ∞) and N (xn , x) = min {d (xn , xn+1 ) , d (x, T x) , d (xn , T x) , d (x, xn+1 )} → 0 (n → ∞). 2. MAIN RESULTS In this section, we consider, generalize and improve the above results. We also prove that all cyclic results from [7, 8] are just equivalent to the respective ordinary fixed point results in the same setting. Firstly, we recall the following important assertion from [23] and [24]: If some ordinary fixed point theorem in the setting of b-complete b-metric spaces has a true cyclic-type extension, then these both theorems are equivalent. Now, we get the following: Theorem 2.1 Theorem 1.8 (that is, Theorem 3.1 from [7]) is equivalent with the following claim: (Claim 1 ) Let (X, d) be a b-complete b-metric space with s ≥ 1, and let T : X → X be a self mapping. Assume that there exists 0 ≤ λ < 1s such that for all x, y ∈ X, d (T x, T y) ≤ λd (x, y) . Then T has a unique fixed point. Theorem 2.2 Theorem 1.9 (that is, Theorem 3.2 from [7]) is equivalent with the following claim: (Claim 2 ) Let (X, d) be a b-complete b-metric space with s ≥ 1, and let T : X → X be 1 a self mapping. Assume that there exists 0 ≤ λ < s+1 such that for all x, y ∈ X, d (T x, T y) ≤ λ [d (x, T x) + d (y, T y)] . Then T has a unique fixed point. Theorem 2.3 Theorem 1.10 (that is, Theorem 3.3 from [7]) is equivalent with the following claim:

6

(Claim 3 ) Let (X, d) be a b-complete b-metric space with s ≥ 1, and let T : X → X be 1 a self mapping. Assume that there exists 0 ≤ λ < s(s+1) such that for all x, y ∈ X, d (T x, T y) ≤ λ [d (x, T y) + d (y, T x)] . Then T has a unique fixed point. Theorem 2.4 Theorem 1.11 (that is, Theorem 3.4 from [7]) is equivalent with the following claim: (Claim 4 ) Let (X, d) be a b-complete b-metric space with s ≥ 1, and let T : X → X be 1 a self mapping. Assume that there exists 0 ≤ λ < s(s+1) such that for all x, y ∈ X, d (T x, T y) ≤ λ max {d (x, y) , d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} . Then T has a unique fixed point. Proof Putting Ai = X (i = 1, 2, ..., p) in Theorem 1.8 (resp. Theorem 1.9, Theorem 1.10, Theorem 1.11), we obtain Claim 1 (resp. Claim 2, Claim 3, Claim 4), that is, Theorem 1.8 (resp. Theorem 1.9, Theorem 1.10, Theorem 1.11) implies Claim 1 (resp. Claim 2, Claim 3, Claim 4). Whereas, the converse proofs are the same as in [23] and [24] for the metric spaces and therefore we omit them. Remark 2.5 It is worth noticing that in all Theorems 1.8-1.11, based on Lemma 1.6, we are easy to see that the sequence {xn} defined by xn = T xn−1 (n = 1, 2, ...) is a b-Cauchy sequence. Hence the proofs of these facts in [7] are superfluous. Remark 2.6 We want to say that all Theorems 1.8-1.11 (that is, Theorems 3.1-3.4 from [7]) hold if λ = 0. In the sequel we still generalize, complement and improve some results from [7]. Our improvements are in several directions. First, we announce the following non-cyclic type result. Theorem 2.7 Let (X, d) be a b-complete b-metric space with s > 1, and let T : X → X be a self mapping. Suppose that there exist altering distance function ψ and constants ε > 1, L ≥ 0 such that the inequality ψ (s ε d (T x, T y)) ≤ ψ (M (x, y)) + Lψ (m (x, y))

(2.1)

holds for all x, y ∈ X, where M (x, y) and m(x, y) are given by (1.1) and (1.2), respectively. Then f has a unique fixed point. Proof For arbitrary x0 ∈ X, let us construct a Picard sequence {xn } by xn = T xn−1 for n = 1, 2, .... We shall prove that d (xn , xn+1 ) ≤ λd (xn−1 , xn ) ,

(2.2)

for all n = 1, 2, ..., where λ ∈ [0, 1s ). In fact, if xn = xn+1 for some n, there is nothing to prove. Hence, suppose that xn = xn+1 for all n = 1, 2, .... In this case, we get that ψ (s ε d (xn , xn+1 )) = ψ (s ε d (T xn−1 , T xn )) ≤ ψ (M (xn−1 , xn )) + Lψ (m (xn−1 , xn )) , 7

(2.3)

where m (xn−1 , xn ) = 0 and 1 + d (xn−1 , xn ) d (xn−1 , xn+1 ) M (xn−1 , xn ) = max d (xn−1 , xn ) , d (xn , xn+1 ) , 1 + d (xn−1 , xn ) 2s d (xn−1 , xn ) + d (xn , xn+1 ) ≤ max d (xn−1 , xn ) , d (xn , xn+1 ) , 2 = max {d (xn−1 , xn ) , d (xn , xn+1 )} . If d (xn−1 , xn ) ≤ d (xn , xn+1 ) , then (2.3) becomes ψ (s ε d (xn , xn+1 )) ≤ ψ (d (xn , xn+1 )) + Lψ (0) = ψ (d (xn , xn+1 )) . Note that the function ψ is nondecreasing, then s ε d (xn , xn+1 ) ≤ d (xn , xn+1 ) , which gives a contradiction (because s ε > 1). Consequently, we obtain that s ε d (xn , xn+1 ) ≤ d (xn−1 , xn ) , which implies (2.2), where λ = s1ε ∈ [0, 1s ). Now by Lemma 1.6, we claim that {xn } is a b-Cauchy sequence and hence there exists u ∈ X such that xn → u as n → ∞. Further, we show that condition (2.1) implies the existence and uniqueness of fixed point of T . Indeed, first of all, we have that 1 d (u, T u) ≤ d (u, xn+1 ) + d (T xn , T u) . s

(2.4)

ψ (s ε d (T xn , T u)) ≤ ψ (M (xn , u)) + Lψ (m (xn , u)) ,

(2.5)

We also have that

where m (xn , u) = min {d (xn , xn+1 ) + d (u, T u) , d (xn , T u) , d (u, xn+1 )} → 0 (n → ∞)

(2.6)

and 1 + d (xn , xn+1 ) M (xn , u) = max d (xn , u) , d (xn , xn+1 ) , d (u, T u) , 1 + d (xn , u) d (xn , T u) + d (u, xn+1 ) 2s → d (u, T u) (n → ∞). 8

(2.7)

Here (2.7) is because d (xn , T u) + d (u, xn+1 ) 2s d (xn , u) + d (u, T u) d (u, xn+1 ) ≤ + 2 2s d (u, T u) → (n → ∞). 2 Using (2.5)-(2.7), we speculate that ε ψ lim s d (T xn , T u) n→∞

= lim ψ (s ε d (T xn , T u)) n→∞ ≤ ψ lim M (xn , u) + Lψ lim m (xn , u) n→∞

n→∞

= ψ (d (u, T u)) ,

or equivalently, lim d (T xn , T u) ≤

n→∞

1 d (u, T u) . sε

(2.8)

Now from (2.4) and (2.8) we obtain that T u = u (because s ε−1 > 1). Finally, we shall the fixed point is unique. To this end, we assume that there exists another fixed point v. Then by (2.1), we have that ψ (s ε d (u, v)) = ψ (s ε d (T u, T v)) ≤ ψ (M (u, v)) + Lψ (m (u, v)) ,

(2.9)

where 1 + d (u, T u) M (u, v) = max d (u, v) , d (u, T u) , d (v, T v) , 1 + d (u, v) d (u, T v) + d (v, T u) 2s = d(u, v),

(2.10)

m (u, v) = min {d (u, T u) + d (v, T v) , d (u, T v) , d (v, T u)} = 0.

(2.11)

and

Uniting (2.9)-(2.11), we arrive at ψ (s ε d (u, v)) ≤ ψ (d (u, v)) , 9

which follows that s ε d (u, v) ≤ d (u, v) . Accordingly, d (u, v) = 0, i.e., u = v, that is to say the fixed point is unique. Now we announce two main results of this paper: Theorem 2.8 Theorems 1.13 without the function φ in his formulation where ε > 1 and Theorem 2.7 are equivalent. Proof For the details and explanations, the reader refers to [23], [24] and the proofs of Theorems 1.13 and 2.7. Theorem 2.9 Theorem 1.13 (that is, Theorem 3.8 from [7]) and Corollary 1.14 (that is, Corollary 3.9 from [7]) are equivalent. In the sequel we generalize the main result from [8]. Namely, we announce the following result: Theorem 2.10 Let (X, d) be a b-complete b-metric space with s ≥ 1, and let T : X → X be a self mapping. Suppose that there exists L ≥ 0 such that the inequality d (T x, T y) ≤

1 M (x, y) + LN (x, y) 2s

holds for all x, y ∈ X, where d (x, T y) + d (y, T x) , M (x, y) = max d (x, y) , d (x, T x) , d (y, T y) , 2s N (x, y) = min {d (x, y) , d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} .

(2.12) (2.13)

Then f has a unique fixed point. Proof See Theorem 3.11 and Corollary 3.12 from [12]. Remark 2.11 It is easy to check that Theorem 2.10 is a genuine generalization of Corollary 3.1 from [8]. Finally we have the following claim: Theorem 2.12 Theorem 1.16 ( that is, Theorem 2.2 from [8]) and Corollary 1.17 (that is, Corollary 3.1 from [8]) are equivalent. Remark 2.13 Suppose that X is a nonempty set, Y ⊆ X, T : Y → Y and Y = ∪pi=1 Ai is a cyclic representation of Y with respect to T . Then F (T ) = ∅ ⇒ ∩pi=1 Ai = ∅, where F (T ) denotes the set of fixed point of T . If a certain (non-cyclic) fixed point result is known, then the respective cyclic contractive condition implies that ∩pi=1 Ai = ∅. Indeed, in this case (∩pi=1 Ai , d) is a b-complete b-metric space and the restriction of T satisfies the given standard condition. Hence, if some ordinary fixed point theorem in the setting of b-complete b-metric spaces has a true cyclic-type extension, then these both theorems are equivalent. 10

Finally, we announce the following result in b-complete b-metric space which is a genuine generalization of Theorem 2.2 from [8]. Also, this result shows that Theorem 2.10 has a true cyclic-type extension. Theorem 2.14 Let (X, d) be a b-complete b-metric space with s ≥ 1, A1 , , A2 , ..., A p nonempty closed subsets of X and let Y = ∪pi=1 Ai . Suppose that T : Y → Y is a cyclic mapping and there exists L ≥ 0 such that the inequality d (T x, T y) ≤

1 M (x, y) + LN (x, y) 2s

(2.14)

holds for all x ∈ Ai , y ∈ Ai+1 , where M (x, y) and N(x, y) are given by (2.12) and (2.13), respectively. Then T has a unique fixed point. Proof The cyclical contractive condition (2.14) leads to ∩pi=1 Ai = ∅ immediately. The conclusion further follows by Theorem 2.10. The following examples support Theorem 2.14: Example 2.15. Let (X, d) = (R, d) where R is a set of real number and d (x, y) = (x − y)2 . Obviously, (X, d) is a complete b-metric space with, s = 2 where d is a continuous b-metric. Let p = 2, A1 = (−∞, 0], A2 = [0, +∞). We have that X = A1 ∪ A2 and A1 ∩ A2 = {0} . Consider the mapping T : X → X defined by T (x) = −

x for all x ∈ X. 5 (1 + x4 )

It is clear that T (A1 ) ⊆ A2 and T (A2 ) ⊆ A1 . Therefore X = A1 ∪ A2 is a cyclic representation of X with respect T. Let x ∈ A1 , x = 0 and y ∈ A2 , y = 0 or x ∈ A2 , x = 0 and 2 2 y ∈ A1 , y = 0 Then in first case we have (because |x| |y| ≤ x +y ) 2 d (T x, T y) = ≤ = ≤ ≤

2 2 2 |x| y 2 + ≤ x + y 5 (1 + x4 ) 5 (1 + y 4 ) 25 2 2 2 x 2 y x+ + y+ 25 3 (1 + x4 ) 25 3 (1 + y 4 ) 2 4 (d (x, T x) + d (y, T y)) ≤ max {d (x, T x) , d (y, T y)} 25 25 4 d (x, T y) + d (y, T x) max d (x, y) , d (x, T x) , d (y, T y) , 25 4 1 M (x, y) + LN (x, y) , 4

where L, M (x, y) , N (x, y) are given by (2.12) and (2.13), respectively. In second case we use that −y = |y| . The inequality d (T x, T y) ≤ 14 M (x, y) + LN (x, y) is also satisfied if one of x, y is equal to 0. In the case x = y = 0 it trivially holds. Hence, all the conditions of Theorem 2.14. are satisfied and T has a unique fixed point 0 ∈ A1 ∩ A2 . In the folowing example the b-metric d is not continuous. 11

Example 2.16. ([12, Example 2], [8, Example 4.2]) Let X = N ∪ {∞} and let d : X × X → [0, ∞) be defined by  0, 

if x = y,    1 1

x − y if one of x, y is even and the other is even or ∞, d (x, y) =  5, if one of x, y is odd and the other is odd and x = y or ∞,    2, otherwise. Then (X, d) is a b-metric space with s = 52 . Let T : X → X defined by 6x, if x ∈ N, T (x) = . ∞, if x = ∞.

If p = 2, A1 = N ∪ {∞} , A2 = {6x : x ∈ N} ∪ {∞} then X = A1 ∪ A2 and T (A1 ) ⊆ A2 and T (A2 ) ⊆ A1 . This means that X = A1 ∪ A2 is a cyclic representation of X with respect to T. Further, as in [8, Example 4.2]), but without the function ϕ, we obtain that d (T x, T y) ≤

1 1 5 M (x, y) + LN (x, y) = M (x, y) + LN (x, y) , 5 2· 2

for all x ∈ A1 , y ∈ A2 where L, M (x, y) , N (x, y) are given as in Example 2.15. All the conditions of Theorem 2.14 are satisfied and ∞ ∈ A1 ∩ A2 is a unique fixed point of T. An open question: Prove or disprove the following: Let {Ai }pi=1 be nonempty closed subsets of a b-complete b-metric space (X, d) with s > 1 and T : ∪pi=1 Ai → ∪pi=1 Ai satisfies the following conditions (where Ap+1 = A1 ): (1) T (Ai ) ⊆ Ai+1 for 1 ≤ i ≤ p; (2) there exists k ∈ [0, 1s ) such that d (T x, T y) ≤ k max {d (x, y) , d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} , for all x ∈ Ai , y ∈ Ai+1 , 1 ≤ i ≤ p. If (X, d) has a Fatou’s property (see [2]), then T has a unique fixed point. 3. APPLICATIONS TO EXISTENCE OF SOLUTIONS OF INTEGRAL EQUATIONS Motivated by Theorem 2.10, we study the existence of solutions for the following integral equation for an unknown function u: b u (t) = v (t) + λ G (t, z) f (z, u (z)) dz t ∈ [a, b] , (3.1) a

where f : [a, b] × R → R, G : [a, b] × [a, b] → [0, ∞), v : [a, b] → R are given continuous functions. 12

Let X be the set C [a, b] of real continuous functions on [a, b] and let d : X × X → R+ be given by d (u, v) = max |u (t) − v (t)|2 . a≤t≤b

(3.2)

It is easy to see that (X, d) is a b-complete b-metric space. Define a mapping T : X → X by b T u (t) := v (t) + λ G (t, z) f (z, u (z)) dz, t ∈ [a, b] , (3.3) a

then u is a solution of (3.1) if and only if it is a fixed point of T . We will prove that T has a unique fixed point under the following assumptions: (1) |λ| ≤ 1; b 1 (2) maxa≤t≤b a G2 (t, z) dz ≤ b−a ; (3) For all x, y ∈ R, the following inequality holds: |f (z, x) − f (z, y)|2 ≤

1 |x − y|2 . 2s

Theorem 3.1 Under the conditions (1)-(3), the equation (3.1) has a unique solution u ∈ X. ∗

13

Proof Using the conditions (1)-(3) and the Cauchy-Schwarz inequality, we have that d (T u1 , T u2 ) = max |T u1 (t) − T u2 (t)|2 t∈[a,b]

b

= max

v (t) + λ G (t, z) f (z, u1 (z)) dz a≤t≤b a 2 b

G (t, z) f (z, u1 (z)) dz

− v (t) + λ a

2 b

= |λ|2 max

G (t, z) (f (z, u1 (z)) − f (z, u2 (z))) dz

a≤t≤b a b b 2 2 2 G (t, z) dz · |f (z, u1 (z)) − f (z, u2 (z))| dz ≤ |λ| max a≤t≤b a a b b 2 2 2 = |λ| max G (t, z) dz · |f (z, u1 (z)) − f (z, u2 (z))| dz a≤t≤b a a b 1 1 2 2 ≤ |λ| · |u1 (z) − u2 (z)| dz b−a 2s a b 1 2 ≤ |λ| max |u1 (t) − u2 (t)|2 dz 4 (b − a) a a≤t≤b |λ|2 = max |u1 (t) − u2 (t)|2 a≤t≤b 4 |λ|2 = d (u1 , u2 ) 4 1 ≤ d (u1 , u2 ) 4 1 ≤ M (u1 , u2 ) + LN (u1 , u2 ) . 2s Hence, all conditions of Theorem 2.10 are fulfilled. This means that the mapping T has a unique fixed point u∗ ∈ X, that is, the integral equation (3.1) has a unique solution which belongs to X = C [a, b].

Acknowledgements The second, third and fourth author are thankful to the Ministry of Education, Science and Technological Development of Serbia

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References [1] Aghajani, A, Abbas, M, Roshan, J-R: Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca. 4 941-960 (2014) [2] Amini-Harandi, A: Fixed point theory for quasi-contraction maps in b-metric spaces. Fixed Point Theory. 15 (2), 351-358 (2014) [3] Bakhtin, I-A: The contraction principle in quasimetric spaces. Funct. Anal. 30, 26-37 (1989) [4] Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181 (1922) [5] Berinde, V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. 9, 43-53 (2004) [6] Czerwik, S: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5-11 (1993) [7] George, R, Reshma, K-P, Padmavati, A: Fixed point theorems for cyclic contractions in b-metric spaces. J. Nonlinear Funct. Anal. 2015, 5 (2015) [8] Nashine, H-K, Kadelburg, Z: Cyclic Generalized ϕ-contractions in b-metric spaces and an application to integral equations. Filomat. 28 (10), 2047-2057 (2014) [9] Huang, H, Radenović, S, Vujaković, J: On some recent coincidence and immediate consequences in partially ordered b-metric spaces. Fixed Point Theory Appl. 2015, 63 (2015) [10] Hussain, N, Ðorić, D, Kadelburg, Z, Radenović, S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012, 126 (2012) [11] Hussain, N, Parvaneh, V, Roshan, J-R, Kadelburg, Z: Fixed points of cyclic (ψ, ϕ, L, A, B)- contractive mappings in ordered b-metric spaces with applications. Fixed Point Theory Appl. 2013, 256 (2013) [12] Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010, Article ID 978121 (2010) [13] Khamsi, M-A, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal. 73, 3123-3129 (2010) [14] Khan, M-S, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bul. Aust. Math. Soc. 30 (1), 1-9 (1984) 15

[15] Kir, M, Kiziltunc, H: On Some well-known fixed point theorems in b-metric Spaces. Turkish J. Anal. Number Theory. 1 (1), 13-16 (2013) [16] Kirk, W-A, Srinavasan, P-S, Veeramani, P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory. 4, 79-89 (2003) [17] Parvaneh, V, Roshan, J-R, Radenović, S: Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations. Fixed Point Theory Appl. 2013, 130 (2013) [18] Parvaneh, V, Roshan, J-R, Radenović, S, Rajović, M: Some coincidence point results for generalized (ψ, ϕ)-weakly contractions in ordered b-metric spaces. to appear in Fixed Point Theory Appl. (2015) [19] Petrusel, G: Cyclic representations and periodic points. Studia Univ. Babes-Bolyai Math. 50, 107-112 (2005) [20] Rus, I-A: Cyclic representations and fixed points. Ann. Tiberiu Popovicu Semin. Funct. Equ. Approx. Convexity. 3, 171-178 (2005) [21] Radenović, S, Kadelburg, Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5(1), 38-50 (2011) [22] Radenović, S, Kadelburg, Z, Jandrlić, D, Jandrlić, A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 38(3), 625-645 (2012) [23] Radenović, S: A note on fixed point theory for cyclic weaker Meir-Keeler function in complete metric spaces, Int. J. Anal. Appl. 7 (1), 16-21 (2015) [24] Fadail, Z-M, Ahmed, A-G-B, Radenović, S, Rajović, M: Some remarks on mappings satisfying cyclical contractive conditions, to appear in Journal of Mathematical Analysis. (2015) [25] Roshan, J-R, Parvaneh, V, Shobkolaei, N, Sedghi, S, Shatanawi, W: Common fixed points of almost generalized (ψ, ϕ)s -contractive mappings in ordered b-metric spaces. Fixed Point Theory Appl. 2013, 159 (2013) [26] Roshan, J-R, Parvaneh, V, Kadelburg, Z: Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl. 7, 229-245 (2014) [27] Shatanawi, W, Postolache, M: Common fixed point results of mappings for nonlinear contraction of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013, 60 (2013)

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LaTeX Source Files Click here to download LaTeX Source Files: reviseamc.tex

*Response to Reviewers Click here to download Response to Reviewers: toeditor.pdf

1 Author’s Response to Reviewer’s Comments ”A note on some recent fixed point results for cyclic contractions in b-metric spaces and an applications to integral equations” Dear Professor T. E. Simos, Editor-in-Chief of Applied Mathematics and Computation, Many thanks for your kind attention and thank you for all your editorial work. According to the reviewer’s comments, we have revised the paper carefully (revision no.1). A letter to reviewer is attached. Best regards, Stojan Radenović ([email protected]) the corresponding author Tatjana Došenović ([email protected]) Tatjana Aleksić Lampert ([email protected]) Zorana Golubović ([email protected]) A letter to reviewer Dear reviewer, We appreciate very much your valuable suggestions and comments. According to your comments, we have revised the paper carefully. Now we address our replies and changes by point as follows. Only one suggestion is realized. In revised version 1, the authors consider the following correction according to reviewer’s report: Reviewer’s Comments: I like to see some numerical examples based on your results before sending to referees Done See Examples 2.15 and 2.16 We hope this version will be appropriate for possible publication in Applied Mathematics and Computation Stojan Radenović, on behalf of the coauthors.

*Response to Reviewers Click here to download Response to Reviewers: response toreviewers.pdf

1 Author’s Response to Reviewer’s Comments ”A note on some recent fixed point results for cyclic contractions in b-metric spaces and an applications to integral equations” Dear Professor T. E. Simos, Editor-in-Chief of Applied Mathematics and Computation, Many thanks for your kind attention and thank you for all your editorial work. According to the reviewer’s comments, we have revised the paper carefully (revision no.1). A letter to reviewer is attached. Best regards, Stojan Radenović ([email protected]) the corresponding author Tatjana Došenović ([email protected]) Tatjana Aleksić Lampert ([email protected]) Zorana Golubović ([email protected]) A letter to reviewer Dear reviewer, We appreciate very much your valuable suggestions and comments. According to your comments, we have revised the paper carefully. Now we address our replies and changes by point as follows. Only one suggestion is realized. In revised version 1, the authors consider the following correction according to reviewer’s report: Reviewer’s Comments: I like to see some numerical examples based on your results before sending to referees Done See Examples 2.15 and 2.16 We hope this version will be appropriate for possible publication in Applied Mathematics and Computation Stojan Radenović, on behalf of the coauthors.