Fixed Point Theory and Applications

Fixed Point Theory and Applications Some remarks on Perov type theorems --Manuscript Draft-Manuscript Number:

FPTA-D-15-00296

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Some remarks on Perov type theorems

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the Foundation of the Research Item of Strong Department of Engineering Innovation of Hanshan Normal University (2013)

Abstract:

In this paper, we consider, discuss, complement, improve and enrich some results on Perov type theorems. By using the method of $c$sequences we get much shorter proofs then ones in the papers with the Perov type theorem. Among other results we prove that Perov type theorem (in the setting of generalized metric space) is equivalent to the classical Banach fixed point theorem. Furthermore, we give an example to show that the fixed point results of quasicontraction of Perov type presented in the setting of normal and solid cone metric spaces are meaningful in the study of nonlinear integral equations.

Corresponding Author:

Stojan Radenovic

Prof. Shaoyuan Xu

VIET NAM Corresponding Author Secondary Information: Corresponding Author's Institution: Corresponding Author's Secondary Institution: First Author:

Shaoyuan Xu, PhD

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Shaoyuan Xu, PhD Stojan Radenovic

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Zoran Kadelburg [email protected] He is an expert in this field. Zhilong Li [email protected] He is an expert in this field.

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Some remarks on Perov type theorems Shaoyuan Xu School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China Stojan Radenovi´c ∗ Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap University, Cao Lanch City, Dong Thap Province, Viet Nam Abstract: In this paper, we consider, discuss, complement, improve and enrich some results on Perov type theorems. By using the method of c-sequences we get much shorter proofs then ones in the papers with the Perov type theorem. Among other results we prove that Perov type theorem (in the setting of generalized metric space) is equivalent to the classical Banach fixed point theorem. Furthermore, we give an example to show that the fixed point results of quasi-contraction of Perov type presented in the setting of normal and solid cone metric spaces are meaningful in the study of nonlinear integral equations. AMS Mathematics Subject Classification 2010 : 47H10 54H25 Keywords: cone metric space; Perov’s operator; property (P); quasi-contraction; spectral radius; solid; normal and non-solid; fixed point theorem

1

Introduction and preliminaries

There are a number of generalizations of Banach contraction principle. One such generalization is given by Perov [26], [27]. Perov proved the following theorem (also see [30]). Theorem 1.1. (Perov type theorem) Let (X, d) be a complete generalized metric space, ∗

Corresponding author: Stojan Radenovi´c. E-mail: [email protected]; [email protected]; [email protected] (S. Radenovi´c)

1

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f : X → X and A ∈ Mn,n (R+ ) is a matrix convergent to zero, such that d (f (x) , f (y)) ≼ A · d (x, y) , x, y ∈ X.

(1.1)

Then: (i) f has a unique fixed point x∗ ∈ X; (ii) the sequence of successive approximations xn = f (xn−1 ) , n ∈ N converges to x∗ for all x0 ∈ X; (iii) d (xn , x∗ ) ≼ An (In − A)−1 (d (x0 , x1 )) , n ∈ N; (iv) if g : X → X satisfies the condition d (f (x) , g (y)) ≼ c for all x ∈ X and some c ∈ Rn , then by considering the sequence yn = g n (x0 ) , n ∈ N, one has d (yn , x∗ ) ≼ (In − A)−1 (c) + An (In − A)−1 (d (x0 , x1 )) , n ∈ N.

(1.2)

Obviously, in previous theorem (X, d) is actually a cone metric space (see, [14], for instance) over the normal solid cone K = {(x1 , x2 , ..., xn ) : xi ≥ 0, i = 1, 2, ..., n} in the Banach space Rn . Otherwise, for more details on cone metric spaces, the reader refers to ([1]-[3], [9], [10], [12]-[15], [17]-[19], [21]-[24], [28], [30]-[32], [33]). In this paper, among other results we shall prove that Perov type theorem is equivalent with the classical Banach fixed point theorem (see Theorem 2.8 in the subsequent section). Also, it is worth noticing that authors in [9], [10], [17], [18] and [33] generalized Theorem 1.1. In fact, the authors of [9] introduced the quasi-contraction of Perov type and proved the corresponding fixed point result, as follows (see Definition 3.1 and Theorem 3.1 in [9]), or Definition 7, Theorem 9, Lemmas 10 and 12 in [24]). Definition 1.2. Let (X, d) be a cone metric space. A map f : X → X such that for some bounded linear operator A ∈ B (E) , ρ (A) < 1 and for all x, y ∈ X, there exists u (x, y) ∈ {d (x, y) , d (x, f (x)) , d (y, f (y)) , d (x, f (y)) , d (y, f (x))} , such that d (f (x) , f (y)) ≼ A (u (x, y)) ,

(1.3)

is called a quasi-contraction of Perov type, where ρ (A) denotes the spectral radius of A. Theorem 1.3. Let (X, d) be a complete cone metric space with a solid cone K. If a mapping f : X → X is a quasi-contraction of Perov type with A (K) ⊆ K, then f has a 2

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unique fixed point x∗ ∈ X and for any x ∈ X, the iterative sequence {f n (x)}n∈N converges to the fixed point of f. Remark 1.4. Also, let us notice that an operator A as in Theorem 1.3 is automatically continuous, and ρ (A) is well-defined. Indeed, by Propositional 19.1 in [11], every solid cone K is generating, i.e., K − K = E, so by Theorem 2.32 in [4], every linear positive operator from E to E is then continuous (also, see [18], Theorem 7). As usual, let F (f ) the set of fixed points of the mapping f : X → X. It is sad that the map f : X → X has the property P ([2], [9], [20], [22], [28]) if F (f ) = F (f n ) for each n ∈ N, i.e., if it has no periodic points. As the corollary of Theorem 3.1, authors in [9] obtained the following. Corollary 1.5. Let (X, d) be a complete cone metric space and K is a solid cone. If f : X → X is a quasi-contraction of Perov type with A (K) ⊆ K and ∥A∥ < 21 , then f has the property P. Remark 1.6. It is worth noticing that Theorem 1.3 and Corollary 1.5 are also true if the cone K is normal and non-solid. For convenience, we present the results, as follows. Theorem 1.7. Let (X, d) be a complete cone metric space with a normal and non-solid cone K. If a mapping f : X → X is a quasi-contraction of Perov type with A (K) ⊆ K, then f has a unique fixed point x∗ ∈ X and for any x ∈ X, the iterative sequence {f n (x)}n∈N converges to the fixed point of f. Proof. Since f : X → X is a quasi-contraction of Perov type with A (K) ⊆ K, we obtain d (f n (x) , f (x∗ )) ≼ A (u) , where { ( ) ( ) ( ) } u ∈ d f n−1 (x) , x∗ , d f n−1 (x) , f n (x) , d f n−1 (x) , f (x∗ ) , d (x∗ , f (x∗ )) , d (x∗ , f n (x)) . If u = d (f n−1 (x) , x∗ ) or u = d (f n−1 (x) , f n (x)) or u = d (x∗ , f n (x)) we have in all three cases (because u → θ as n → ∞) that d (x∗ , f (x∗ )) ≼ A (θ) = θ, 3

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that is f (x∗ ) = x∗ . If u = d (f n−1 (x) , f (x∗ )) or u = d (x∗ , f (x∗ )) then in both cases we obtain that d (x∗ , f (x∗ )) ≼ A (d (x∗ , f (x∗ ))) , from which it follows that d (x∗ , f (x∗ )) ≼ (I − A)−1 (θ) = θ. Hence, in all cases we have that x∗ is a fixed point of f.  Corollary 1.8. Let (X, d) be a complete cone metric space and K be a normal and non-solid cone. If f : X → X is a quasi-contraction of Perov type with A (K) ⊆ K and ∥A∥ < 12 , then f has the property P. Proof. From the proofs of the conditions (1) and (2), Theorem 3.1 in [9] follows that they also hold if the cone is normal and non-solid. Therefore, Corollary 1.5 from [9] holds if the cone is normal and non-solid. Remark 1.9. Also, it is not hard to check (see Theorem 1.13 below) that Theorem 3.2 (Perov’s Theorem) from [9] holds if the cone is normal and non-solid, which is presented as follows. Theorem 1.10. Let (X, d) be a complete cone metric space with a normal (respectively, solid or non-solid) cone K, d : X ×X → E and f : X → X a mapping contraction of Perov type. That is, for all x, y ∈ X d (f (x) , f (y)) ≼ Ad (x, y) ,

(1.4)

where A ∈ B(E), A (K) ⊆ K, then f has a unique fixed point x∗ ∈ X and for any x ∈ X, the iterative sequence {f n (x)}n∈N converges to the fixed point of f. Now let us recall Theorem 3.3 from [9], as follows. Theorem A Let (X, d) be a complete cone metric space and P is a normal cone with the normal constantK. If f : X → X is a quasi-contraction of Perov and K ∥A∥ < 1, 4

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then f has a unique fixed point x∗ ∈ X and for any x ∈ X, the iterative sequence {f n (x)}n∈N converges to the fixed point of f. Remark 1.11. It is obviously seen that Theorem 3.3 from [9] above follows immediately according to Theorem 1.7. ´ c fixed point Theorem 1.12.Theorem 3.3 from [9] is equivalent to the classical Ciri´ theorem. Proof. Since f : X → X is a quasi-contraction, it follows that ∥d (f x, f y) ∥ ≤ K∥A∥ max {∥d (x, y) ∥, ∥d (x, f x) ∥, ∥d (y, f y) ∥, ∥d (x, f y) ∥, ∥d (y, f x)} ∥, where ∥d (x, y) ∥ = D(x, y) is now a metric on X, which shows that f : X → X is a ´ c. By the the classical Ciri´ ´ c fixed point theorem, we quasi-contraction in the sense of Ciri´ ´ c fixed point theorem have the result of Theorem A. This means that the the classical Ciri´ implies Theorem 3.3 from [9]. Conversely, in Theorem A, put E = R, P = [0, +∞), ∥.∥ = |.| , A = λI with λ ∈ ´ c contrac[0, 1), ≼=≤, i.e., x ≼ y means y − x ≥ 0. Then the condition (1.3) becomes Ciri´ tion defined on a usual metric space (X, d). Theorem 1.13.Theorem 1.10 is equivalent to the classical Banach fixed point theorem. Proof. Indeed, in Theorem 1.10 put E = R, P = [0, +∞), ∥.∥ = |.| , A = λI with λ ∈ [0, 1), ≼=≤, i.e., x ≼ y means y − x ≥ 0. Then the condition (1.4) becomes Banach contraction defined on a usual metric space (X, d). So, we obtain that the classical Banach fixed point theorem holds. Conversely, we shall show that the classical Banach fixed point theorem implies Theorem 1.10. From (1.4) it follows that for all n ∈ N d (f n (x) , f n (y)) ≼ An d (x, y) .

(1.5)

Since An → θ in the Banach space B (E) of all bounded linear operators on E, there is n0 ∈ N such that ∥An0 ∥B(E) = λ < 1. Further, (1.5) implies that ∥d (f n0 (x) , f n0 (y))∥E ≤ ∥An0 ∥B(E) ∥d (x, y)∥E = λ ∥d (x, y)∥E . 5

(1.6)

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So, by the classical Banach fixed point theorem there is a unique x∗ ∈ X such that f n0 (x∗ ) = x∗ . Hence, x∗ is a unique fixed point of f in X.  Remark 1.14.The condition (1.1) implies the continuity of the mapping f. In fact, the generalized metric space is the cone metric space over normal solid cone. Hence, in it cone metric d is continuous as well as the Sandwich theorem holds. Therefore, d

d

from xn → x ⇒ f xn → f x (see [14]).  The following result concerning c-sequence is very useful in the fixed point theory of cone metric spaces (see [23], Proposition 4.1) and ([18], Proposition 5.4). Definition 1.15. Let E be a topological vector space with a solid cone K and let {un } ⊆ K be sequence. We say that {un } is a c-sequence if for every c ∈ intK there exists n0 ∈ N such that un ≪ c for all n > n0 (see [12], [23], [33]). Proposition 1.16. Let K be a solid cone in a topological vector space E and {un } be a sequence in K. Then the following conditions are equivalent: (i) {un } is a c-sequence, (ii) For each c ≫ θ there exists n0 ∈ N such that un ≺ c for n > n0 . (iii) For each c ≫ θ there exists n0 ∈ N such that un ≼ c for n > n0 . (v) There exists c ≫ θ such that for any λ ∈ (0, 1) , there exists n0 ∈ N such that un ≼ λc for all n > n0 . (vi) There exists a sequence {vn } such that vn ≫ θ for any n ∈ N, vn → θ and for any n ∈ N, there exists n0 ∈ N such that um ≼ vn for each m > n0 . Finally, following the works of Berinde ([7],[8]), authors in [10] proved the next result. Theorem 1.17. Let (X, d) be a complete cone metric space, d : X × X → E, f : X → X, A ∈ B (E) , with ρ (A) < 1 and A (K) ⊆ K, B ∈ B (E) with B (K) ⊆ K, such that d (f (x) , f (y)) ≼ A (d (x, y)) + B (d (x, f (y))) , x, y ∈ X. Then

6

(1.7)

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(i) f : X → X has a fixed point in X and for any x0 ∈ X the sequence xn = f (xn−1 ) , n ∈ N converges to a fixed point of f. (ii) If, under the above conditions, B ∈ B (E) and ρ (A + B) < 1,

(1.8)

or d (f (x) , f (y)) ≼ A (d (x, y)) + B (d (x, f n0 (x))) , x, y ∈ X, for some n0 ∈ N,

(1.9)

then f has a unique fixed point. Remark 1.18. According to Remark 1.4. A ∈ B (E) and B ∈ B (E) are superfluous. Also, previous theorem is true if the cone K is normal and non-solid. Otherwise, from the proof of Theorem 1.17 it follows that authors in [10] assume that the cone K is only solid.

2

Fixed point results

In this section we generalize, complement and improve the previous results in (solid; normal non-sold) cone metric spaces over Banach algebra. First of all, we introduce gquasi-contraction of Perov type in this setting. Definition 2.1. Let (X, d) be a cone metric space over Banach algebra A, and f, g : X → X. Then, f is called a g-quasi-contraction of Perov type if for some bounded linear operator A ∈ B (E) , ρ (A) < 1 and for all x, y ∈ X, there exists u (x, y) ∈ {d (g (x) , g (y)) , d (g (x) , f (x)) , d (g (y) , f (y)) , d (g (x) , f (y)) , d (g (y) , f (x))} , such that d (f (x) , f (y)) ≼ Au (x, y) ,

(2.1)

where ρ (A) denotes the spectral radius of A. The following result generalizes Theorem 9 from [24] as well as Theorem 3.1 from [9](see also Theorem 3.2 from [34]). Theorem 2.2. Let (X, d) be a cone metric space over Banach algebra A, K be a solid (respectively, normal and non-solid) cone and let f, g : X → X be such that f (X) ⊆ g (X) 7

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and one of f (X) or g (X) is complete. If f is a g-quasi-contraction of Perov type, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point. Remark 2.3. That f is a g-quasi-contraction of Perov type in Theorem 2.2. means that A ∈ K with ρ (A) < 1 (for all details see [24] and [33]). In the rest of the paper, we choose x0 ∈ X and denote yn = f xn = gxn+1 for all n ∈ N. The proof of the following two lemmas is the same as Lemmas 10 and 12 in [24]. Lemma 2.4. Assume that the hypotheses in Theorem 2.2 are satisfied. Then, for each n ≥ 1, and for i, j such that 1 ≤ i, j ≤ n, one has d (yi , yj ) ≼ A (I − A)−1 d (y0 , y1 ) .

(2.2)

Lemma 2.5. Assume that the hypotheses in Theorem 2.2 are satisfied. Then, {yn } is a Cauchy sequence. Now we finish the remaining part of the proof of Theorem 2.2. Proof. By Lemma 2.5 and the completeness of f (X) or g (X) , there is x∗ ∈ X such that yn = f (xn ) = g (xn+1 ) → g (x∗ ) . We will prove that f (x∗ ) = g (x∗ ) . For this, we have d (f (x∗ ) , g (x∗ )) ≼ d (f (x∗ ) , f (xn )) + d (yn , g (x∗ )) ≼ Au (x∗ , xn ) + d (yn , g (x∗ )) , where

u (x∗ , xn ) ∈ {d (g (x∗ ) , g (xn )) , d (g (x∗ ) , f (x∗ )) , d (f (xn ) , f (xn )) , d (g (x∗ ) , f (xn )) , d (g (xn ) , f (x∗ ))} . First, let the cone K be solid. Then, if u = (g (x∗ ) , g (xn )) or u = d (f (xn ) , f (xn )) or u = d (g (x∗ ) , f (xn ))(u and Au are c-sequences in all three cases), we get d (f (x∗ ) , g (x∗ )) ≼ Au + d (yn , g (x∗ )) = cn , 8

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where cn = Au + d (yn , g (x∗ )) is c-sequence. Hence, f (x∗ ) = g (x∗ ) . if u = d (g (x∗ ) , f (x∗ )) or u = d (g (xn ) , f (x∗ )) ≼ d (g (xn ) , g (x∗ )) + d (g (x∗ ) , f (x∗ )) , we get d (f (x∗ ) , g (x∗ )) ≼ Ad (g (x∗ ) , f (x∗ )) + d (yn , g (x∗ )) , i.e., d (f (x∗ ) , g (x∗ )) ≼ (I − A)−1 d (yn , g (x∗ )) = dn , where dn is c-sequence, or d (f (x∗ ) , g (x∗ )) ≼ A (d (g (xn ) , g (x∗ )) + d (g (x∗ ) , f (x∗ ))) + d (yn , g (x∗ )) = Ad (g (x∗ ) , f (x∗ )) + Ad (g (xn ) , g (x∗ )) + d (yn , g (x∗ )) , that is, d (f (x∗ ) , g (x∗ )) ≼ (I − A)−1 en , where en = Ad (g (xn ) , g (x∗ )) + d (yn , g (x∗ )) is c-sequence. Since, (I − A)−1 en is new c-sequence, we have again that f (x∗ ) = g (x∗ ) . Second, let the cone K be normal and non-solid. Then, the proof is similar to that in Remark 1.6.  Remark 2.6. By using the method with c-sequences (see also [33]) the proofs of our results are shorter then the ones in [9], page 717 and in [34], page 5–9. In the next result we improve the Corollary 3.1 from [9], that is, we announce the following. Proposition 2.7. Let (X, d) be a complete cone metric space over Banach algebra A, K be a solid (respectively, normal and non-solid) cone. If f : X → X is a quasicontraction of Perov type ( A ∈ K, ρ (A) < 1), then f has the property P. Proof. From [9], page 716, line 9- it follows that for m = n + 1 we easily obtain ( ) d f n (x) , f n+1 (x) ≼ 2An (I − A)−1 d (f (x) , x) , for arbitrary x ∈ X. Hence, if u ∈ F (f n ) we get d (u, f (u)) ≼ 2An (I − A)−1 d (f (u) , u) → θ (as n → ∞), 9

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since ρ (A) < 1 implies ∥An ∥ → 0(n → ∞). Hence in both cases for the cone K (either solid or normal and non-solid) we obtain that u ∈ F (f ) . This means that the quasi-contraction f has a the property P.  Finally, we announce the following unexpected result. Theorem 2.8. Theorem 1.1 (i.e., Perov type theorem) is equivalent to the classical Banach fixed point theorem. Proof. Indeed, putting in Theorem 1.1 n = 1 we obtain that the classical Banach fixed point theorem holds. Conversely, we shall show that the classical Banach fixed point theorem implies Perov type Theorem. From (1.1) it follows that for all n ∈ N d (f n (x) , f n (y)) ≼ An · d (x, y) .

(2.3)

Since An → θ in the Banach space B (Rn ) of all bounded linear operators on Rn , there is n0 ∈ N such that ∥An0 ∥B(Rn ) = λ < 1. Further, (2.3) implies that ∥d (f n0 (x) , f n0 (y))∥Rn ≤ ∥An0 ∥B(Rn ) ∥d (x, y)∥Rn = λ ∥d (x, y)∥Rn .

(2.4)

So, by the classical Banach fixed point theorem there is a unique x∗ ∈ X such that f n0 (x∗ ) = x∗ . Hence, x∗ is a unique fixed point of f in X.  The following result can be seen from [35] when it is only considered in the setting of normal and solid cone metric space over Banach algebra A. Theorem 2.9. Let (X, d) be a cone metric space over Banach algebra A, K be a normal (respectively, solid or non-solid) cone in A, f : X → X and A ∈ K with ρ (A) < 1. If for all x, y ∈ X d (f (x) , f (y)) ≼ Ad (x, y) , (2.5) then f has a unique fixed point x∗ ∈ X. Proof. According to the previous result we easily see Theorem 2.2 (by setting g = IX ) implies Theorem 2.9. 

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Theorem 2.10. Theorem 2.9 is equivalent to the classical Banach fixed point theorem and thus also equivalent to Perov type theorem by Theorem 2.8. Proof. In fact, in Theorem 2.9, putting A = R, K = [0, ∞), ∥.∥ = |.| , A = λ ∈ [0, 1), ≼=≤ the condition (2.5) becomes Banach contraction defined on a usual metric space (X, d) . Therefore, f has a unique fixed point x∗ ∈ X. This means that Theorem 2.9 implies the classical Banach fixed point theorem. Conversely, the result follows if we take A ∈ K instead of A ∈ Mn,n (R+ ) and take the norm ∥.∥A instead of the norm ∥.∥Rn in the previous theorem. Remark 2.11. Previous Theorem generalizes Theorems 6.1 and 6.2 from [17] and Theorem 2.1 from [35].

3

Examples In this section, we will present an example to show that one of the results obtained

in the setting of normal and non-solid cone metric spaces has meaningful applications in nonlinear integral applications. Example 3.1. Let X = L [0, 1] denote the set of all generalized real-valued Lebesgue integral functions on [0, 1]. Let A = L [0, 1]. Consider the following nonlinear integral equation ∫ 1 F (t, f (s)) ds = f (t), (3.1) 0

where F satisfies: (a) F : [0, 1] × R → R is a generalized real-valued Lebesgue integral function where R = [−∞, +∞] denoting the set of all generalized real numbers; ∫1 (b) there exists a function G(x) with 0 < 0 G(x) dx < 1 such that for a.e. x ∈ [0, 1], and a.e. y1 , y2 ∈ R, one has |F (x, y1 ) − F (x, y2 )| ≤ G(x)|y1 − y2 |. Theorem 3.2. The equation (3.1) has a unique non-negative solution in L [0, 1].

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∫1 Proof. Define a norm on A by ∥f ∥1 = 0 f (t) dt for f ∈ A. Let P = {f ∈ L [0, 1] | f = f (t) ≥ 0, for a.e. t ∈ [0, 1]}. Then P is a normal non-solid cone of the real Banach algebra A with the operations as (f + g)(t) = f (t) + g(t), (cf )(t) = cf (t), (f g)(t) = f (t)g(t), for all f = f (t), g = g(t) ∈ A and c ∈ R. Moreover, A owns the unit element e = e(t) = ∫1 1 for a.e. t ∈ [0, 1]. Let T be a self map of X defined by T f (t) = 0 F (t, f (s)) ds. It is easy shown that (X, d) is a wrtn-complete cone metric space over A (∫ Banach algebra ) 1 with the norm ∥·∥1 where the cone metric is defined by d(f, g)(t) := et 0 |f (t) − g(t)| dt . Now let us check that T : X → X is a generalized Banach contraction with the generalized ∫1 Lipschitz coefficient L = 0 G(x) dx satisfying the spectral radius ρ(L) < 1. In fact, by (b) together with the fact that ∥f g∥1 ≤ ∥f ∥1 ∥g∥1 we see (∫ 1 ) x d(T f (x), T g(x)) = e |T f (x) − T g(x)| dx 0 (∫ 1 ∫ 1 ) ( ) x = e F (x, f (t)) − F (x, g(t)) dt dx 0 0 (∫ 1 ∫ 1 ) ) ( x ≤ e F (x, f (t)) − F (x, g(t)) dt dx 0 0 (∫ 1 ) x ≤ e L f (t) − g(t) dt 0 (∫ 1 ) x ≤ Le f (t) − g(t) dt 0 ( ) = Ld f (x), g(x) , where the spectral radius ρ(L) of L satisfies ρ(L) ≤ ∥L∥ = L ∈ (0, 1). Therefore, it follows from Theorem 2.9 that the equation (3.1) has a unique non-negative solution in L [0, 1]. Remark 3.3. Example 3.1 shows that the fixed point results concerning mappings of Perov type presented in this paper in the setting of normal non-solid cone metric spaces are meaningful in the study of nonlinear integral equations.

Competing interests The authors declare that they have no competing interests. 12

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Authors’ contributions The authors contribute equally and significantly in writing this paper. All the authors read and approve the final manuscript.

Authors details 1

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China. 2 Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap University, Cao Lanch City, Dong Thap Province, Viet Nam. Email address: [email protected]; [email protected] (S. Xu); fi[email protected]; [email protected]; [email protected] (S. Radenovi´c).

Acknowledgments The research is partially supported by the foundation of the research item of Strong Department of Engineering Innovation of Hanshan Normal University, China (2013).

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