algebra-valued modular metric spaces and related ...

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces 1

J HR-operator pairs in C ∗ -algebra-valued modular metric spaces and related fixed point results via C∗ -class functions

Bahman Moeinia , Arsalan Hojat Ansarib and Choonkil Parkc∗ a b

Department of Mathematics, Hidaj Branch, Islamic Azad University, Hidaj, Iran

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

c

Research Institute for Natural Sciences Hanyang University Seoul 04763 Korea

Abstract In this paper, we define J HR-operator pairs in a C ∗ -algebra-valued modular metric space. Next, by using C∗ -class functions, some common fixed point theorems are established for such mappings. Also, to support and elaborate of our results some examples and an application is provided for a system of nonlinear integral equations.

1

Introduction

In 2008, Chistyakov [2] introduced the notion of modular metric spaces generated by F -modular and developed the theory of this space. In 2010 Chistyakov [3] defined the notion of modular on an arbitrary set and develop the theory of metric spaces generated by modular such that called the modular metric spaces. Recently, Mongkolkeha et al. [15, 16] and Parya et al. [17] have introduced some notions and established some fixed point results in modular metric spaces. In [12], Ma et al. introduced the concept of C ∗ -algebra-valued metric spaces. The main idea consists in using the set of all positive elements of a unital C ∗ algebra instead of the set of real numbers. This line of research was continued in [8,9,11,20,21], where several other fixed point results were obtained in the framework of C ∗ -algebra valued metric, as well as (more general) C ∗ -algebra-valued b-metric spaces. ∗

Corresponding author E-mail:[email protected] E-mail:[email protected] E-mail:[email protected] 2010 Mathematics Subject Classification: 47H10, 46L07, 46A80. Keywords: C ∗ -algebra-valued modular metric space, C∗ -class function, J HR-operator pair, common fixed point, weakly compatible, integral equation.

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Moeini and Hojat Ansari

Recently, Shateri [10] introduced the concept of C ∗ -algebra-valued modular space which is a generalization of a modular space and next proved some fixed point theorems for self-mappings with contractive or expansive conditions on such spaces. This author showed that if Xρ be a complete C ∗ -algebra-valued modular space and T : Xρ → Xρ be a contractive mapping on Xρ , i.e., there exists an a ∈ A with kak < 1 and α, β ∈ R+ with α > β such that ρ(α(T x − T y))  a∗ ρ(β(x − y))a, (∀x, y ∈ Xρ ). Then T has a unique fixed point in Xρ . In this paper, by using some ideas of [13, 14] a new type of non-commuting mappings as J HR-operator pairs are defined on a C ∗ -algebra-valued modular metric space. Then via C∗ -class functions some common fixed point results are established for such mappings with contractive conditions. Also, some examples to elaborate and illustrate of our results are constructed. Finally, an application is given that shows existence and uniqueness of solution for a type of system of nonlinear integral equations.

2

Basic notions

Let X be a non empty set, λ ∈ (0, ∞) and due to the disparity of the arguments, function ω : (0, ∞) × X × X → [0, ∞] will be written as ωλ (x, y) = ω(λ, x, y) for all λ > 0 and x, y ∈ X. Definition 2.1. [2] Let X be a non empty set. a function ω : (0, ∞) × X × X → [0, ∞] is said to be a modular metric on X if it satisfies the following three axioms: (i) given x, y ∈ X, ωλ (x, y) = 0 for all λ > 0 if and only if x = y; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0 and x, y ∈ X; (iii) ωλ+µ (x, y) ≤ ωλ (x, z) + ωµ (z, y) for all λ > 0 and x, y, z ∈ X, and (X, ω) is called a modular metric space. Recall that a Banach algebra A (over the field C of complex numbers) is said to be a C ∗ -algebra if there is an involution ∗ in A (i.e., a mapping ∗ : A → A satisfying a∗∗ = a for each a ∈ A) such that, for all a, b ∈ A and λ, µ ∈ C, the following holds: ¯ ∗+µ (i) (λa + µb)∗ = λa ¯b∗ ; (ii) (ab)∗ = b∗ a∗ ; (iii) ka∗ ak = kak2 .

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces 3

Note that, form (iii), it easy follows that kak = ka∗ k for each a ∈ A. Moreover, the pair (A, ∗) is called a unital ∗-algebra if A contains the identity element 1A . A positive element of A is an element a ∈ A such that a∗ = a and its spectrum σ(a) ⊂ R+ , where σ(a) = {λ ∈ R : λ1A − a is noninvertible}. The set of all positive elements will be denoted by A+ . Such elements allow us to define a partial ordering ’’ on the elements of A. That is, b  a if and only if b − a ∈ A+ . If a ∈ A is positive, then we write a  θ, where θ is the zero element of A. Each positive element a of a C ∗ -algebra A has a unique positive square root. From now on, by A we mean a unital C ∗ -algebra with identity element 1A . Further, 1 A+ = {a ∈ A : a  θ} and (a∗ a) 2 = |a|. Lemma 2.2. [4] Suppose that A is a unital C ∗ -algebra with a unit 1A . (1) For any x ∈ A+ , we have x  1A ⇔ kxk ≤ 1. (2) If a ∈ A+ with kak < 12 , then 1A − a is invertible and ka(1A − a)−1 k < 1. (3) Suppose that a, b ∈ A with a, b  θ and ab = ba, then ab  θ. (4) By A0 we denote the set {a ∈ A : ab = ba, ∀b ∈ A}. Let a ∈ A0 if b, c ∈ A with b  c  θ, and 1A − a ∈ A0 is an invertible operator, then (1A − a)−1 b  (1A − a)−1 c. Notice that in a C ∗ -algebra, if θ  a, b, one cannot   thatθ  ab. For  conclude 1 −2 3 2 ∗ , ,b= example, consider the C -algebra M2 (C) and set a = −2 4 2 3   −1 2 . Clearly a, b ∈ M2 (C)+ , while ab is not. then ab = −4 8 In the following we define a new type of modular metric space that is called C ∗ -algebra-valued modular metric space. Definition 2.3. Let X be a non empty set. a function ω : (0, ∞) × X × X → A+ is said to be a C ∗ -algebra-valued modular metric (briefly, C ∗ .m.m) on X if it satisfies the following three axioms: (i) given x, y ∈ X, ωλ (x, y) = θ for all λ > 0 if and only if x = y; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0 and x, y ∈ X; (iii) ωλ+µ (x, y)  ωλ (x, z) + ωµ (z, y) for all λ, µ > 0 and x, y, z ∈ X. The truple (X, A, ω) is called a C ∗ .m.m space.

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If instead of (i), we have the condition (i0 ) ωλ (x, x) = θ for all λ > 0 and x ∈ X, then ω is said to be a C ∗ -algebra-valued pseudo modular metric (briefly, C ∗ .p.m.m) on X and if ω satisfies (i0 ), (iii) and (i00 ) given x, y ∈ X, if there exists a number λ > 0, possibly depending on x and y, such that ωλ (x, y) = θ, then x = y, then ω is called a C ∗ -algebra-valued strict modular metric (briefly, C ∗ .s.m.m) on X. A C ∗ .m.m (or C ∗ .p.m.m, C ∗ .s.m.m) ω on X is said to be convex if, instead of (iii), we replace the following condition: (iv) ωλ+µ (x, y) 

λ λ+µ ωλ (x, z)

+

µ λ+µ ωµ (z, y)

for all λ, µ > 0 and x, y, z ∈ X.

Clearly, if ω is a C ∗ .s.m.m, then ω is a C ∗ .m.m, which in turn implies ω is a C ∗ .p.m.m on X, and similar implications hold for convex ω. The essential property of a C ∗ .m.m ω on a set X is a following given x, y ∈ X, the function 0 < λ → ωλ (x, y) ∈ A is non increasing on (0, ∞). In fact, if 0 < µ < λ, then we have ωλ (x, y)  ωλ−µ (x, x) + ωµ (x, y) = ωµ (x, y).

(2.1)

It follows that at each point λ > 0 the right limit ωλ+0 (x, y) := limε→+0 ωλ+ε (x, y) and the left limit ωλ−0 (x, y) := limε→+0 ωλ−ε (x, y) exist in A and the following two inequalities hold: ωλ+0 (x, y)  ωλ (x, y)  ωλ−0 (x, y). (2.2) It can be check that if x0 ∈ X, the set Xω = {x ∈ X : lim ωλ (x, x0 ) = θ}, λ→∞

C ∗ -algebra-valued

is a metric space, called a C ∗ -algebra-valued modular space, 0 whose dω : Xω × Xω → A is given by d0ω = inf{λ > 0 : kωλ (x, y)k ≤ λ} for all x, y ∈ Xω . Moreover, if ω is convex, the set Xω is equal to Xω∗ = {x ∈ X : ∃ λ = λ(x) > 0 such that kωλ (x, x0 )k < ∞}, and d∗ω : Xω∗ × Xω∗ → A is given by d∗ω = inf{λ > 0 : kωλ (x, y)k ≤ 1} for all x, y ∈ Xω∗ . It is easy to see that if X is a real linear space, ρ : X → A and ωλ (x, y) = ρ(

x−y ) for all λ > 0 and x, y ∈ X, λ

(2.3)

then ρ is C ∗ -algebra valued modular (convex C ∗ -algebra-valued modular) on X if and only if ω is C ∗ .m.m (convex C ∗ .m.m, respectively) on X. On the other hand, if ω satisfy the following two conditions:

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces 5

(i) ωλ (µx, 0) = ω λ (x, 0) for all λ, µ > 0 and x ∈ X; µ

(ii) ωλ (x + z, y + z) = ωλ (x, y) for all λ > 0 and x, y, z ∈ X. If we set ρ(x) = ω1 (x, 0) with (2.3) holds, where x ∈ X, then (a) Xρ = Xω is a linear subspace of X and the functional kxkρ = d0ω (x, 0), x ∈ Xρ is a F -norm on Xρ ; (b) If ω is convex, Xρ∗ ≡ Xω∗ = Xρ is a linear subspace of X and the functional kxkρ = d∗ω (x, 0), x ∈ Xρ∗ is a norm on Xρ∗ . Similar assertions hold if replace C ∗ .m.m by C ∗ .p.m.m. If ω is C ∗ .m.m in X, we called the set Xω is C ∗ .m.m space. By the idea of property in C ∗ -algebra-valued metric spaces and C ∗ -algebravalued modular spaces, we defined the following: Definition 2.4. Let Xω be a C ∗ .m.m space. (1) The sequence (xn )n∈N in Xω is said to be ω-convergent to x ∈ Xω with respect to A if ωλ (xn , x) → θ as n → ∞ for all λ > 0. (2) The sequence (xn )n∈N in Xω is said to be ω-Cauchy with respect to A if ωλ (xm , xn ) → θ as m, n → ∞ for all λ > 0. (3) A mapping f is said to be ω-continuous with respect to A in subset D of Xω if for every sequence {xn }n∈N ⊆ D such that ωλ (xn , z) → θ as n → ∞ for all λ > 0, then ωλ (f xn , f z) → θ as n → ∞ for all λ > 0. (4) A subset D of Xω is said to be ω-closed with respect to A if the limit of the ω-convergent sequence of D always belong to D. (5) Xω is said to be ω-complete if any ω-Cauchy sequence with respect to A is ω-convergent. (6) A subset D of Xω is said to be ω-bounded with respect to A if for all λ > 0 δω (D) = sup{kωλ (x, y)k; x, y ∈ D} < ∞, where, δω (D) denotes the diameter of D in the C ∗ .m.m space. In the following we give some nontrivial examples of C ∗ .m.m spaces.

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Moeini and Hojat Ansari

Example 2.5. Let A = M2 (R), of all 2 × 2 matrices with the usual operation of addition, scalar multiplication, and matrix multiplication. Define norm on A by P 1 2 2 2 , and ∗ : A → A, given by A∗ = A, for all A ∈ A, defines a |a | kAk = ij i,j=1 convolution on A. Thus A becomes a C ∗ -algebra. For     a11 a12 b11 b12 A= ,B = ∈ A = M2 (R), a21 a22 b21 b22 we denote A  B if and only if (aij − bij ) ≤ 0, for all i, j = 1, 2. Set X = { c1n : n = 1, 2, · · · } where 0 < c < 1 and define ω : (0, ∞) × X × X → A by ωλ (x, y) = diag(k

x−y x−y k, k k) λ λ

for all x, y ∈ X and λ > 0. It is easy to check that ω satisfies all the conditions of Definition 2.3. So, (X, A, ω) is a C ∗ .m.m space. Example 2.6. Let X = R and A = M2 (R). Define ω : (0, ∞) × X × X → A+ by x−y ωλ (x, y) = diag(| x−y λ |, α| λ |)

for all x, y ∈ X, α ≥ 0 and λ > 0. Then ω is a C ∗ .m.m and by the completeness of R, Xω is a ω-complete C ∗ .m.m space. Example 2.7. Let X = L∞ (E) and H = L2 (E), where E is a lebesgue measurable set. By B(H) we denote the set of bounded linear operator on Hilbert space H. Clearly, B(H) is a C ∗ -algebra with the usual operator norm. Define ω : (0, ∞) × X × X → B(H)+ by ωλ (f, g) = π| f −g | ,

(∀f, g ∈ X).

λ

Where πh : H → H is the multiplication operator defined by πh (φ) = h.φ, for φ ∈ H. Then ω is a C ∗ .m.m and (Xω , B(H), ω) is a ω-complete C ∗ .m.m space. It suffices to verify the completeness of Xω . For this, let {fn } be a ω-Cauchy sequence with respect to B(H), that is for an arbitrary ε > 0, there is N ∈ N such that for all m, n ≥ N , kωλ (fm , fn )k = kπ| fm −fn | k = k fm λ−fn k∞ ≤ ε, λ

so {fn } is a Cauchy sequence in Banach space X. Hence, there is a function f ∈ X and N1 ∈ N such that k fnλ−f k∞ ≤ ε, (n ≥ N1 ).

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces 7

It implies that kωλ (fn , f )k = kπ| fn −f | k = k fnλ−f k∞ ≤ ε, (n ≥ N1 ). λ

Consequently, the sequence {fn } is a ω-convergent sequence in Xω and so Xω is a ω-complete C ∗ .m.m space. Definition 2.8. Let Xω be a C ∗ .m.m space. Let f, g self-mappings of Xω . a point x in Xω is called a coincidence point of f and g iff f x = gx. We shall call w = f x = gx a point of coincidence of f and g. Let C(f, S) and P C(f, S) denote the set of coincidence points and points of coincidence, respectively, of the pair (f, S). Definition 2.9. Let Xω be a C ∗ .m.m space. Two self-mappings f and g of Xω are said to be compatible iff limn→∞ ωλ (f Sxn , Sf xn ) = θ, whenever {xn } is a sequence in Xω such that limn→∞ f xn = limn→∞ Sxn = z for some z ∈ Xω . Definition 2.10. Let Xω be a C ∗ .m.m space. Two self-mappings f and g of Xω are said to be subcompatible iff there exists a sequence {xn } in Xω such that limn→∞ f xn = limn→∞ Sxn = z, for some z ∈ Xω and which satisfies lim ωλ (f Sxn , Sf xn ) = θ.

n→∞

Definition 2.11. Let Xω be a C ∗ .m.m space. Two maps f and g of Xω are said to be weakly compatible if they commute at coincidence points. Definition 2.12. Let Xω be a C ∗ .m.m space. Two self-mappings f and g of Xω are occasionally weakly compatible (owc) iff there is a point x in Xω which is a coincidence point of f and g at which f and g commute. Lemma 2.13. Let Xω be a C ∗ .m.m space and f, g owc self-mappings of Xω . If f and g have a unique point of coincidence, w = f x = gx, then w is a unique common fixed point of f and g. Proof. Since f and g are owc, then there exists a u ∈ Xω such that f u = gu = w0 ⇒ f gu = gf u, so, f w0 = gw0 . By uniqueness of point of coincidence, we have f w0 = gw0 = w0 , so w0 is a common fixed point of f and g. Now, if there is another point z ∈ Xω such that f z = gz = z, then z is a point of coincidence of f and g, thus z = w0 and this completes the proof. Lemma 2.14. Let {yn } be a sequence in Xω such that limn→∞ ωλ (yn , yn+1 ) = θ for each λ > 0. If {yn } is not a ω-Cauchy sequence in Xω , then there exist 0 > 0, λ0 > 0, and two sequences {mi } and {ni } of positive integers such that

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Moeini and Hojat Ansari

(i) mi > ni + 1 and ni → ∞ as i → ∞, (ii) kω2λ0 (ymi , yni )k > 0 and kω2λ0 (ymi −1 , yni )k ≤ 0 , i = 1, 2, 3, · · · . Proof. If {yn } is not a ω-Cauchy sequence in Xω , then there exist 0 > 0, λ0 > 0 such that for each positive integers i, there exist positive integers mi , ni with mi > ni , such that kω2λ0 (ymi , yni )k > 0 ,

(∗).

For i = 1, 2, ..., let mi be the least positive integer exceeding ni satisfying (∗), that is for i = 1, 2, ..., kω2λ0 (ymi , yni )k > 0 , kω2λ0 (ymi −1 , yni )k ≤ 0 . and since limi→∞ ωλ (yni , yni +1 ) = θ for all λ > 0, so kω2λ0 (yni , yni +1 )k ≤ 0 , thus mi > ni + 1 and ni → ∞ as i → ∞.

3

J H-operator pairs class in a C ∗ .m.m space

In 1976, Jungck [7] initiated a study of common fixed points of commuting mappings. On the other hand in 1982, Sessa [19] initiated the tradition of improving commutativity in fixed point theorems by introducing the notion of weakly commuting maps in metric spaces. After this, Jungck [6] gave the concept of weakly compatible mappings. In 2010, Pathak and Hussain [18] introduced classes of non-commuting self-mappings satisfying generalized I-contraction or I-nonexpansive type condition and established the existence of common fixed point results and applications for these mappings that called them P-operator pairs. In 2011, Hussain et al. [5] extended two classes of non-commuting self-mappings, these classes contain the occasionally weakly compatible as J H-operator pairs and weakly biased self-mappings. Here, we define concept of J H-operator pairs of [5] and J HR-operator pairs of [13] in a C ∗ .m.m space.

3.1

J H-operator pairs

Definition 3.1. Let Xω be a C ∗ .m.m space and f, S : Xω → Xω are two mappings. Then (f, S) is called a J H-operator pair if there exists a point w = f x = Sx in P C(f, S) such that kωλ (x, w)k ≤ δω (P C(f, S)), for all λ > 0.

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces 9

Example 3.2. Let X = R, A = M2 (R) with norm kAk = A ∈ A and ω : (0, ∞) × X × X → A+ defined by

P

2 2 i,j=1 |aij |

1 2

for all

x−y ωλ (x, y) = diag(| x−y λ |, α| λ |)

for all x, y ∈ X, α ≥ 0 and λ > 0. Then by Example 2.6, (Xω , A, ω) is a ω-complete C ∗ .m.m space. Now define self-mappings f and S on Xω as follows:  3  2 x if x 6= 0, x if x 6= 0, f (x) = S(x) = 1 1 if x = 0, if x = 0. 2 2 So, C(f, S) = {0, 1} and P C(f, S) = { 21 , 1}. For w = 1 ∈ P C(f, S) and x = 1, we   0 0 have f (1) = S(1) = 1, then for all λ > 0, α ≥ 0, ωλ (x, w) = θ = and 0 0 δω (P C(f, S)) = sup{kωλ (x, y)k; x, y ∈ P C(f, S)} 1 α = sup{kdiag( 2λ , 2λ )k} ≥ 0.

Thus for all λ > 0, 0 = kωλ (x, w)k ≤ δω (P C(f, S)). Therefore, (f, S) is a J H-operator pair. Obviously, f and S are occasionally weakly compatible but not weakly compatible.

3.2

J HR-operator pairs

Definition 3.3. Let Xω be a C ∗ .m.m space and f, S : Xω → Xω are two mappings. Then (f, S) is called a J HR-operator pair if P C(f, S) 6= ∅ and there exists a sequence {xn } in Xω such that limn→∞ f xn = limn→∞ Sxn = z ∈ Xω and which satisfies lim kωλ (xn , z)k ≤ δω (P C(f, S)), for all λ > 0, n→∞

Proposition 3.4. Let Xω be a C ∗ .m.m space and f, S : Xω → Xω are two mappings. If (f, S) be a J H-operator pair, then (f, S) is a J HR-operator pair. Proof. By hypothesis there exists a point z = f p = Sp ∈ P C(f, S) such that kωλ (z, p)k ≤ δω (P C(f, S)), for all λ > 0. If we define sequence {xn } in Xω by xn = p for n = 1, 2, 3, · · · . Then, limn→∞ f xn = limn→∞ Sxn = z and lim kωλ (xn , z)k ≤ δω (P C(f, S)) for all λ > 0.

n→∞

Therefore, (f, S) is a J HR-operator pair.

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Example 3.5. Let (X, A, ω) be as in Example 4.4 and define self-mappings f and S on Xω with f x = x3 and Sx = 2 − x. So, C(f, S) = {1}, P C(f, S) = {1} and δω (P C(f, S)) = 0. Define sequence {xn } in Xω by xn = n+1 n , for n = 1, 2, 3, · · · . Then limn→∞ f xn = limn→∞ Sxn = z = 1. Since for all λ > 0, 0 = lim kωλ (xn , z)k ≤ δω (P C(f, S)) = 0. n→∞

Therefore, the pair (f, S) is a J HR-operator pair. Obviously this pair is also J Hoperator pair. In the following, we shall show that the class of the J H-operator pairs is the proper subclass of J HR-operator pairs. Remark 3.6. The inverse of the Proposition 3.4 is not true in general. For this, Let (X, A, ω) be as in Example 4.4 and define self-mappings f and S on Xω by   xcos(x) if 0 < x ≤ π2 xsin(x) if 0 < x ≤ π2 , f (x) = S(x) = 1 1 elsewere, elsewere. 2 4 π We have C(f, S) = { π4 }, P C(f, S) = { 4√ } and δω (P C(f, S)) = 0. Since f ( π4 ) = 2 π S( π4 ) = 4√ , for all λ > 0, we have 2

π π kωλ ( , √ )k > 0 = δω (P C(f, S)), 4 4 2 thus, (f, S) is not a J H-operator pair. Now, if we define sequence {xn } by xn = 1 1 1 n+1 , n = 1, 2, 3, · · · , then limn→∞ f xn = limn→∞ n+1 sin( n+1 ) = limn→∞ Sxn = 1 1 limn→∞ n+1 cos( n+1 ) = z = 0 ∈ Xω and lim kωλ (xn , z)k = lim kωλ (

n→∞

n→∞

1 , 0)k = 0 ≤ 0 = δω (P C(f, S)), n+1

for all λ > 0. Therefore, the pair (f, S) is a J HR-operator pair. Remark 3.7. Suppose f and S be ω-continuous self-mappings on C ∗ .m.m space (X, A, ω), if (f, S) is a J HR-operator pair and P C(f, S) is a singleton set, then f and S are subcompatible mappings. For this, there exists a sequence {xn } in Xω such that limn→∞ f xn = limn→∞ Sxn = z for some z ∈ Xω and which satisfies limn→∞ kωλ (xn , z)k ≤ δω (P C(f, S)) = 0, for all λ > 0. So, limn→∞ xn = z and f z = Sz = z. Hence, lim ωλ (f Sxn , Sf xn ) = θ, n→∞

that is f and S are subcompatible.

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces11

Note that, class of subcompatible mappings and class of J HR-operator pairs contain properly occasionally weakly compatible mappings but, this classes are two different concept in general. Let X = [1, ∞) and A, ω defined as in Example 2.6. Suppose f and S as follows:  3  2  x + x if 1 < x < 2, x > 2,  x + 1 if 1 < x < 2, x > 2, 2 if x = 1, 2 if x = 1, fx = Sx =   4 if x = 2, 4 if x = 2. Then, (f, S) is a J HR-operator pair but, these mappings are not subcompatible. Also, it can be shown that there exist two mappings that are subcompatible but not J HR-operator pair. We leave simple example of the assertion to the reader.

4

Main results

In 2017, Ansari et al. [1] introduced the concept of complex C-class functions as follows: Definition 4.1. Suppose S = {z ∈ C : z  0}, then a continuous function F : S 2 → C is called a complex C-class function if for any s, t ∈ S, the following conditions hold: (1) F (s, t)  s; (2) F (s, t) = s implies that either s = 0 or t = 0. An extra condition on F that F (0, 0) = 0 could be imposed in some cases if required. For examples of these functions see [1]. In this section, we define a concept of C∗ -class function. The main idea consists in using the set of elements of a unital C ∗ -algebra instead of the set of complex numbers. Definition 4.2. (C∗ -class function) Suppose A is a unital C ∗ -algebra, then a continuous function F : A+ × A+ → A is called C∗ -class function if for any A, B ∈ A+ , the following conditions hold: (1) F (A, B)  A; (2) F (A, B) = A implies that either A = θ or B = θ. An extra condition on F that F (θ, θ) = θ could be imposed in some cases if required. The letter C∗ will denote the class of all C∗ -class functions. Remark 4.3. The class C∗ includes the set of complex C-class functions. It is suffitient to take A = C in Definition 4.2. The following examples show that the class C∗ is nonempty:

12

Moeini and Hojat Ansari

Example 4.4. Let A = M2 (R), of all 2 × 2 matrices with the usual operation of addition, scalar multiplication, and matrix multiplication. Define norm on A by P 1 2 2 2 , and ∗ : A → A, given by A∗ = A, for all A ∈ A, defines a |a | kAk = ij i,j=1 convolution on A. Thus A becomes a C ∗ -algebra. For     a11 a12 b11 b12 A= ,B = ∈ A = M2 (R), a21 a22 b21 b22 we denote A  B if and only if (aij − bij ) ≤ 0, for all i, j = 1, 2. (1) Define F∗ : A+ × A+ → A by        a a12 b11 b12 a11 − b11 a12 − b12 11 , F∗ = a21 a22 b21 b22 a21 − b21 a22 − b22 for all ai,j , bi,j ∈ R+ , (i, j ∈ {1, 2}). Then F∗ is a C∗ -class function. (2) Define F∗ : A+ × A+ → A by        a a11 a12 b11 b12 a12 11 =m , F∗ b21 b22 a21 a22 a21 a22 for all ai,j , bi,j ∈ R+ , (i, j ∈ {1, 2}), where, m ∈ (0, 1). Then F∗ is a C∗ -class function. Example 4.5. Let X = L∞ (E), H = L2 (E) and B(H) are defined as Examole ??. Define F∗ : B(H)+ × B(H)+ → B(H) by F∗ (U, V ) = U − ϕ(U ), where ϕ : B(H)+ → B(H)+ is is a continuous function such that ϕ(U ) = θ if and only if U = θ (θ = 0B(H) ). Then F∗ is a C∗ -class function. Let Φu denote the class of the functions ϕ : A+ → A+ which satisfy the following conditions: (ϕ1 ) ϕ is continuous and non-decreasing; (ϕ2 ) ϕ(T )  θ, T  θ and ϕ(θ)  θ. Let Ψ be a set of all continuous functions ψ : A+ → A+ satisfying the following conditions: (ψ1 ) ψ is continuous and non-decreasing; (ψ2 ) ψ(T ) = θ if and only if T = θ.

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces13

Definition 4.6. A tripled (ψ, ϕ, F∗ ) where ψ ∈ Ψ, ϕ ∈ Φu and F∗ ∈ C∗ is said to be monotone if for any A, B ∈ A+ A  B =⇒ F∗ (ψ(A), ϕ(A))  F∗ (ψ(B), ϕ(B)). and (ψ, ϕ, F∗ ) is said to be strictly monotone if for any A, B ∈ A+ A ≺ B =⇒ F∗ (ψ(A), ϕ(A)) ≺ F∗ (ψ(B), ϕ(B)). We now give detailed proofs of main results of this paper. Let (Xω , A, ω) is a C ∗ .m.m space. Here, we define a sequence as follows: Let f, S, H and T be self-mappings of Xω such that for each x, y ∈ Xω and λ > 0, f (Xω ) ⊂ T (Xω ) and H(Xω ) ⊂ S(Xω ),

(4.4)

and for each x, y ∈ Xω and λ > 0,   ψ(kωλ (f x, Hy)k1A )  F∗ ψ(M (x, y)), ϕ(M (x, y)) ,

(4.5)

for which, ψ ∈ Ψ, ϕ ∈ Φu and F∗ ∈ C∗ such that (ψ, ϕ, F∗ ) is strictly monotone and M (x, y) = (kak2 kω2λ (Sx, T y)k + kbk2 kω2λ (f x, T y)k + kck2 kω2λ (Hy, Sx)k)1A , where a, b, c ∈ A with 0 < kak2 + 2kbk2 + 2kck2 < 1 and kωλ (f x, Hy)k < ∞. Then, for arbitrary point x0 ∈ Xω , by (4.4), there exists a point x1 such that f x0 = T x1 and for this point x1 , we can chose a point x2 ∈ Xω such that Hx1 = Sx2 and so on. Inductively, one can define a sequence {yn } in Xω , such that y2n = T x2n+1 = f x2n and y2n+1 = Sx2n+2 = Hx2n+1 ,

(4.6)

for n = 1, 2, 3, · · · . Now, we give the main theorem as follows: Theorem 4.7. Let (Xω , A, ω) is a ω-complete C ∗ .m.m space. Suppose f, H, S and T are self-mappings of Xω , such that satisfying the conditions (4.4) and (4.5). In additional, if S is ω-continuous and the pairs (f, S), (H, T ) are J HR-operator pair and weakly compatible, respectively, then f , S, H and T have a unique common fixed point in Xω . Proof. We prove the the theorem in three steps. Step1. We show that if ωλn = ωλ (yn , yn+1 ) for all λ > 0, then limn→∞ ωλn = θ,

14

Moeini and Hojat Ansari

where {yn } is defined by (4.6). The inequality (4.5) implies ψ(kω λ (y2n , y2n+1 )k1A ) = ψ(kωλ (f x2n , Hx2n+1  )k1A )  F∗ ψ(M (x2n , x2n+1 )), ϕ(M (x2n , x2n+1 ))  = F∗ ψ((kak2 kω2λ (y2n−1 , y2n )k + kbk2 kω2λ (y2n , y2n )k + kck2 kω2λ (y2n+1 , y2n−1 )k)1A ),  ϕ((kak2 kω2λ (y2n−1 , y2n )k + kbk2 kω2λ (y2n , y2n )k + kck2 kω2λ (y2n+1 , y2n−1 )k)1A )   F∗ ψ((kak2 kωλ (y2n−1 , y2n )k + kck2 kω2λ (y2n+1 , y2n−1 )k)1A ),  ϕ((kak2 kωλ (y2n−1 , y2n )k + kck2 kω2λ (y2n+1 , y2n−1 )k)1A )   F∗ ψ((kak2 kωλ (y2n−1 , y2n )k + kc2 k(kωλ (y2n−1 , y2n )k + kωλ (y2n , y2n+1 )k))1A ),  ϕ((kak2 kωλ (y2n−1 , y2n )k + kc2 k(kωλ (y2n−1 , y2n )k + kωλ (y2n , y2n+1 )k))1A ) or  ψ(kωλ2n k1A )  F∗ ψ((kak2 kωλ2n−1 k + kck2 kωλ2n−1 k + kck2 kωλ2n k)1A ),  ϕ((kak2 kωλ2n−1 k + kck2 kωλ2n−1 k + kck2 kωλ2n k)1A ) . If ωλ2n  ωλ2n−1 , then   ψ(kωλ2n k1A )  F∗ ψ((kak2 + 2kck2 )kωλ2n k1A ), ϕ((kak2 + 2kck2 )kωλ2n k1A )   ≺ F∗ ψ(kωλ2n k1A ), ϕ(kωλ2n k1A )  ψ(kωλ2n k1A ). Thus ψ(kωλ2n k1A ) ≺ ψ(kωλ2n k1A ), which is a contradiction. Hence, ωλ2n  ωλ2n−1 .

(4.7)

Similarly, taking x = x2n+2 and y = x2n+1 in (4.5), we get ψ(kω λ (y2n+2 , y2n+1 )k1A )  F∗ ψ((kak2 kω2λ (y2n+1 , y2n )k + kbk2 kω2λ (y2n+2 , y2n )k + kck2 kω2λ (y2n+1 , y2n+1 )k)1A ),  ϕ((kak2 kω2λ (y2n+1 , y2n )k + kbk2 kω2λ (y2n+2 , y2n )k + kck2 kω2λ (y2n+1 , y2n+1 )k)1A )   F∗ ψ((kak2 kωλ (y2n+1 , y2n )k + kbk2 kω2λ (y2n+2 , y2n )k)1A ),  ϕ((kak2 kωλ (y2n+1 , y2n )k + kbk2 kω2λ (y2n+2 , y2n )k)1A )   F∗ ψ((kak2 kωλ (y2n+1 , y2n )k + kbk2 (kωλ (y2n , y2n+1 )k + kωλ (y2n+1 , y2n+2 )k)1A ),  ϕ((kak2 kωλ (y2n+1 , y2n )k + kbk2 (kωλ (y2n , y2n+1 )k + kωλ (y2n+1 , y2n+2 )k)1A ) ,

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces15

or  ψ(kωλ2n+1 k1A )  F∗ ψ((kak2 kωλ2n k + kbk2 kωλ2n k + kbk2 kωλ2n+1 k)1A ),  2 2 2 ϕ((kak kωλ2n k + kbk kωλ2n k + kbk kωλ2n+1 k)1A ) . If ωλ2n+1  ωλ2n , then   ψ(kωλ2n+1 k1A )  F∗ ψ((kak2 + 2kbk2 )kωλ2n+1 k1A ), ϕ((kak2 + 2kbk2 )kωλ2n+1 k1A )   ≺ F∗ ψ(kωλ2n+1 k1A ), ϕ(kωλ2n+1 k1A )  ψ(kωλ2n+1 k1A ). Thus ψ(kωλ2n+1 k1A ) ≺ ψ(kωλ2n+1 k1A ), which is a contradiction. Hence, ωλ2n+1  ωλ2n .

(4.8)

From equations (4.7) and (4.8), ωλn+1  ωλn . Therefore, {kωλn k1A }n∈N is a nonincreasing sequence in A+ . Now, again by (4.5) for x = x2 and y = x1 , we have ψ(kωλ1 k1A ) = ψ(kωλ (y2 , y1 )k1A )   F∗ ψ((kak2 kω2λ (y1 , y0 )k + kbk2 kω2λ (y2 , y0 )k + kck2 kω2λ (y1 , y1 )k)1A ),  ϕ((kak2 kω2λ (y1 , y0 )k + kbk2 kω2λ (y2 , y0 )k + kck2 kω2λ (y1 , y1 )k)1A )   F∗ ψ((kak2 kωλ (y1 , y0 )k + kbk2 kω2λ (y2 , y0 )k)1A ),  ϕ((kak2 kωλ (y1 , y0 )k + kbk2 kω2λ (y2 , y0 )k)1A )   F∗ ψ((kak2 kωλ (y0 , y1 )k + kbk2 (kωλ (y0 , y1 )k + kωλ (y1 , y2 ))k)1A ),  ϕ((kak2 kωλ (y0 , y1 )k + kbk2 (kωλ (y0 , y1 )k + kωλ (y1 , y2 ))k)1A )  = F∗ ψ((kak2 kωλ0 k + kbk2 kωλ0 k + kbk2 kωλ1 k)1A ),  ϕ((kak2 kωλ0 k + kbk2 kωλ0 k + kbk2 kωλ1 k)1A )    F∗ ψ((kak2 + 2kbk2 )kωλ0 k1A ), ϕ((kak2 + 2kbk2 )kωλ0 k1A )  ψ((kak2 + 2kbk2 )kωλ0 k1A )  ψ((kak2 + 2kbk2 + 2kck2 )kωλ0 k1A )

16

Moeini and Hojat Ansari

so, kωλ1 k1A  (kak2 + 2kbk2 + 2kck2 )kωλ0 k1A . In general, we have kωλn k1A  (kak2 + 2kbk2 + 2kck2 )n kωλ0 k1A , which implies that, limn→∞ kωλn k1A  limn→∞ (kak2 + 2kbk2 + 2kck2 )n kωλ0 k1A = θ. Therefore, limn→∞ ωλn = θ. This completes the step1. Step2. We show that the sequence {yn } defined by (4.6) is a ω-Cauchy sequence in Xω . By step1, limn→∞ ωλn = limn→∞ ωλ (yn , yn+1 ) = θ, so from Lemma 2.14 if {yn } is not a ω-Cauchy sequence in Xω , there exist 0 > 0, λ0 > 0, and two sequences {mi } and {ni } of positive integers such that (A) mi > ni + 1 and ni → ∞ as i → ∞, (B) kω2λ0 (ymi , yni )k > 0 and kω2λ0 (ymi −1 , yni )k ≤ 0 , i = 1, 2, 3, · · · . We have 0 < kω2λ0 (ymi , yni )k ≤ kωλ0 (ymi , ymi −1 )k + kωλ0 (ymi −1 , yni )k ≤ kωλ0 (ymi , ymi −1 )k + 0 .

(4.9)

As i → ∞ in (4.9), we get limi→∞ kω2λ0 (ymi , yni )k = 0 .

(4.10)

On the other hand, we have 0 < kω2λ0 (ymi , yni )k ≤ kωλ0 (ymi , yni +1 )k + kωλ0 (yni +1 , yni )k.

(4.11)

Now consider kωλ0 (ymi , yni +1 )k in (4.11) and assume that both mi and ni are even. Then by (4.5), we have ψ(kω λ0 (ymi , yni +1 )k1A ) = ψ(kωλ0 (f xmi , Hxni +1 )k1A )  F∗ ψ((kak2 kω2λ0 (Sxmi , T xni +1 )k + kbk2 kω2λ0 (f xmi , T xni +1 )k +kck2 kω2λ0 (Hxni +1 , Sxmi )k)1A ), 2 ϕ((kak2 kω2λ0 (Sxmi , T xni +1 )k + kbk  kω2λ0 (f xmi , T xni +1 )k +kck2 kω2λ0 (Hxni +1 , Sxmi )k)1A )  = F∗ ψ((kak2 kω2λ0 (ymi −1 , yni )k + kbk2 kω2λ0 (ymi , yni +1 )k +kck2 kω2λ0 (yni +1 , ymi −1 )k)1A ), 2 ϕ((kak2 kω2λ0 (ymi −1 , yni )k + kbk  kω2λ0 (ymi , yni +1 )k +kck2 kω2λ0 (yni +1 , ymi −1 )k)1A ) ,

(4.12)

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces17

so,

ψ(kωλ0 (ymi , yni +1 )k1A )   F∗ ψ((kak2 kω2λ0 (ymi −1 , yni )k + kbk2 kω2λ0 (ymi , yni +1 )k + kck2 kω2λ0 (yni +1 , ymi −1 )k)1A ), ϕ((kak2 kω2λ0 (ymi −1 , yni )k + kbk2 kω2λ0 (ymi , yni +1 )k  + kck2 kω2λ0 (yni +1 , ymi −1 )k)1A ) ,

(4.13)

By (B), (4.10), (4.11) and letting i → ∞ in equation (4.13), we get

ψ(0 1A )  F∗ ψ((kak2 + kbk2 + kck2 )0 1A ), ϕ(ψ((kak2 + kbk2 + kck2 )0 1A )  ≺ F∗ ψ(0 1A ), ϕ((0 1A )  ψ(0 1A ),



(4.14)

hence, ψ(0 1A ) ≺ ψ(0 1A ), which is a contradiction. Therefore, the sequence {yn } defined by (4.6) is a ω-Cauchy sequence, and this completes the step2. Step3. We show that f, S, H and T have a unique common fixed point. By hypothesis P C(f, S) 6= ∅, hence there exists a point x in Xω such that f x = Sx = w, from step2, {yn } is a ω-Cauchy sequence in Xω . Since Xω is complete, so there exists a point z ∈ Xω such that limn→∞ ωλ (yn , z) = θ for all λ > 0 and

lim ωλ (f x2n , z) = lim ωλ (T x2n+1 , z) = lim ωλ (Sx2n , z) = lim ωλ (Hx2n+1 , z) = θ,

n→∞

n→∞

n→∞

n→∞

for all λ > 0. From (4.5) we get

ψ(kωλ (f x, Hx2n+1 )k1A )   F∗ ψ((kak2 kω2λ (Sx, T x2n+1 )k + kbk2 kω2λ (f x, T x2n+1 )k + kck2 kω2λ (Hx2n+1 , Sx)k)1A ),  ϕ((kak2 kω2λ (Sx, T x2n+1 )k + kbk2 kω2λ (f x, T x2n+1 )k + kck2 kω2λ (Hx2n+1 , Sx)k)1A ) . (4.15)

18

Moeini and Hojat Ansari

Letting n → ∞, we have ψ(kωλ (w, z)k1A )   F∗ ψ((kak2 kω2λ (w, z)k + kbk2 kω2λ (w, z)k + kck2 kω2λ (z, w)k)1A ),  ϕ((kak2 kω2λ (w, z)k + kbk2 kω2λ (w, z)k + kck2 ω2λ (z, w)k)1A )   F∗ ψ((kak2 kωλ (w, z)k + kbk2 kωλ (w, z)k + kck2 kωλ (z, w)k)1A ),  ϕ((kak2 kωλ (w, z)k + kbk2 kωλ (w, z)k + kck2 ωλ (z, w)k)1A )    F∗ ψ(kωλ (z, w)k)1A ), ϕ(kωλ (z, w)k)1A )  ψ(kωλ (z, w)k)1A )

(4.16)

so, by definition of C∗ -class functions, ψ(kωλ (w, z)k1A ) = θ or ϕ(kωλ (w, z)k1A ) = θ, which implies that w = z. Moreover, if there is another point x0 in Xω for which f x0 = Sx0 = w0 , then by using (4.5), we get w0 = z and w = w0 = z is a unique point of coincidence of f and S. So, δω (P C(f, S)) = 0. Since (f, S) is J HR-operator pair, there exists a sequence {dn } in Xω such that limn→∞ f dn = limn→∞ Sdn = d ∈ Xω and limn→∞ kωλ (dn , d)k ≤ δω (P C(f, S)) = 0 for all λ > 0. Therefore, limn→∞ kωλ (dn , d)k = 0 for all λ > 0, hence limn→∞ dn = d. Now, we show that d = z. By using (4.5) for x = dn and y = x2n+1 , we conclude that  ψ(kωλ (f dn , Hx2n+1 )k1A )  F∗ ψ((kak2 kω2λ (Sdn , T x2n+1 )k + kbk2 kω2λ (f dn , T x2n+1 )k +kck2 kω2λ (Hx2n+1 , Sdn )k)1A ), 2 ϕ((kak2 kω2λ (Sdn , T x2n+1 )k + kbk  kω2λ (f dn , T x2n+1 )k

+kck2 kω2λ (Hx2n+1 , Sdn )k)1A )   F∗ ψ((kak2 kωλ (Sdn , T x2n+1 )k + kbk2 kωλ (f dn , T x2n+1 )k +kck2 kωλ (Hx2n+1 , Sdn )k)1A ), 2 ϕ((kak2 kωλ (Sdn , T x2n+1 )k + kbk  kωλ (f dn , T x2n+1 )k

+kck2 kωλ (Hx2n+1 , Sdn )k)1A ) ,

(4.17) for all λ > 0. Letting n → ∞, we get ψ(kω λ (d, z)k1A )  F∗ ψ((kak2 kωλ (d, z)k + kbk2 kωλ (d, z)k + kck2 kωλ (z, d)k)1A ),  ϕ((kak2 kωλ (d, z)k + kbk2 kωλ (d, z)k + kck2 kωλ (z, d)k)1A )    F∗ ψ(kωλ (z, d)k)1A ), ϕ(kωλ (z, d)k)1A )  ψ(kωλ (d, z)k1A ),

(4.18)

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces19

so, ψ(kωλ (d, z)k1A ) = θ or ϕ(kωλ (d, z)k1A ) = θ, which implies that d = z. Since S is ω-continuous and limn→∞ Sdn = d, so, Sz = z. By putting x = z and y = x2n+1 in (4.5), we conclude that ψ(kω λ (f z, Hx2n+1 )k1A )  F∗ ψ((kak2 kω2λ (Sz, T x2n+1 )k + kbk2 kω2λ (f z, T x2n+1 )k +kck2 kω2λ (Hx2n+1 , Sz)k)1A ), 2 ϕ((kak2 kω2λ (Sz, T x2n+1 )k + kbk  kω2λ (f z, T x2n+1 )k

+kck2 kω2λ (Hx2n+1 , Sz)k)1A )   F∗ ψ((kak2 kωλ (Sz, T x2n+1 )k + kbk2 kωλ (f z, T x2n+1 )k

(4.19)

+kck2 kωλ (Hx2n+1 , Sz)k)1A ), 2 ϕ((kak2 kωλ (Sz, T x2n+1 )k + kbk  kωλ (f z, T x2n+1 )k +kck2 kωλ (Hx2n+1 , Sz)k)1A ) ,

for all λ > 0. Letting n → ∞, we have ψ(kω λ (f z, z)k1A )   F∗ ψ(kbk2 kωλ (f z, z)k1A ), ϕ(kbk2 kωλ (f z, z)k1A )    F∗ ψ(kωλ (f z, z)k1A ), ϕ(kωλ (f z, z)k1A )

(4.20)

 ψ(kωλ (f z, z)k1A ), so ψ(kωλ (f z, z)k1A ) = θ or ϕ(kωλ (f z, z)k1A ) = θ which implies that f z = z. Since f (Xω ) ⊂ T (Xω ), there exists a point u ∈ Xω such that T u = f z = z. Using the inequality (4.5), we conclude that ψ(kω λ (z, Hu)k1A ) = ψ(kωλ (f z, Hu)k1A )  F∗ ψ((kak2 kω2λ (Sz, T u)k + kbk2 kω2λ (f z, T u)k +kck2 kω2λ (Hu, Sz)k)1A ), 2 ϕ((kak2 kω2λ (Sz, T u)k + kbk  kω2λ (f z, T u)k

+kck2 kω2λ (Hu, Sz)k)1A )    F∗ ψ(kck2 kω2λ (Hu, z)k1A ), ϕ(kck2 kω2λ (Hu, z)k)1A )    F∗ ψ(kωλ (Hu, z)k1A ), ϕ(ωλ (Hu, z)k)1A )

(4.21)

 ψ(kωλ (Hu, z)k1A ), so ψ(kωλ (Hu, z)k1A ) = θ or ϕ(kωλ (Hu, z)k1A ) = θ, hence, Hu = z. Since the pair (H, T ) is weakly compatible and Hu = T u = z, we have Hz = HT u = T Hu = T z.

20

Moeini and Hojat Ansari

Again from inequality (4.5), we conclude that ψ(kω λ (z, Hz)k1A ) = ψ(kωλ (f z, Hz)k1A )  F∗ ψ((kak2 kω2λ (Sz, T z)k + kbk2 kω2λ (f z, T z)k +kck2 kω2λ (Hz, Sz)k)1A ),ϕ((kak2 kω2λ (Sz, T z)k + kbk2 kω2λ (f z, T z)k +kck2 kω2λ (Hz, Sz)k)1A )   F∗ ψ((kak2 kωλ (Sz, T z)k + kbk2 kω2λ (f z, T z)k +kck2 kωλ (Hz, Sz)k)1A ),ϕ((kak2 kωλ (Sz, T z)k +kck2 kωλ (Hz, Sz)k)1A )

+

(4.22) kbk2 kωλ (f z, T z)k

   F∗ ψ(kωλ (z, Hz)k1A ), ϕ(kωλ (z, Hz)k1A )  ψ(kωλ (z, Hz)k1A ), which means that Hz = z. So, f z = Sz = Hz = T z = z, that is z is a common fixed point of f, S, H and T . The uniqueness of the common fixed point z, again follows from inequality (4.5), and this completes the proof. Corollary 4.8. Let (Xω , A, ω) is a ω-complete C ∗ .m.m space. Suppose f, H, S and T are self-mappings of Xω , such that satisfying (4.5) and   ψ(kωλ (f x, Hy)k1A )  F∗ ψ(kak2 kω2λ (Sx, T y)k1A ), ϕ(kak2 kω2λ (Sx, T y)k1A ) , (4.23) for each x, y ∈ Xω and λ > 0, for which, ψ ∈ Ψ, ϕ ∈ Φu , F∗ ∈ C∗ such that (ψ, ϕ, F∗ ) is strictly monotone, a ∈ A with 0 < kak2 < 1 and kωλ (f x, Hy)k < ∞. In additional, if S is ω-continuous and the pairs (f, S), (H, T ) are J HR-operator pair and weakly compatible, respectively, then f , S, H and T have a unique common fixed point in Xω . Example 4.9. Let (X, A, ω) is ω-complete C ∗ .m.m space defined as in Example 3.2. Define f, H, S, T : Xω → Xω by  2x if x ∈ (−∞, 2),  3 2 if x = 2, f x = Hx = 2, Sx = 4 − x, T x =  0 if x ∈ (2, ∞). Suppose, 

ψ : A+ → A+ ψ(A) = 2A,



ϕ : A+ → A+ ϕ(A) = A,



F∗ : A+ × A+ → A F∗ (A, B) = A − B.

Then, (ψ, ϕ, F∗ ) is strictly monotone. For all x, y ∈ Xω = R and λ > 0 , we have 

 0

0 0=

= kωλ (f x, Hy)k < ∞. 0 0

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces21

For every a, b, c ∈ A with 0 < kak2 + 2kbk2 + 2kck2 < 1, we get     0 0 = ψ(kωλ (f x, Hy)k1A )  F∗ ψ(M (x, y)), ϕ(M (x, y)) , 0 0 for all x, y ∈ Xω and λ > 0. S is ω-continuous. Also f (Xω ) = {2} ⊆ T (Xω ) = R, H(Xω ) = {2} ⊆ S(Xω ) = R, C(f, S) = P C(f, S) = C(H, T ) = P C(H, T ) = {2}. To see that (f, S) is J HRoperator pair, suppose {xn } is a sequence in Xω defined by xn = 2n+1 n , n = 1, 2, · · · . Then, limn→∞ f xn = limn→∞ Sxn = z = 2 and lim kωλ (xn , z)k = lim kdiag(|

n→∞

n→∞

2n+1 n

−2

λ

|, α|

2n+1 n

λ

−2

|)k = 0 ≤ δω (P C(f, S)) = 0,

for all λ > 0 and α ≥ 0. Consequently, the pair (f, S) is J HR-operator pair. Now, since C(H, T ) = {2} and HT 2 = T H2 = 2, hence, the pair (H, T ) is weakly compatible. So all conditions of Theorem 4.7 are satisfied and x = 2 is a unique common fixed point of f, H, S and T .

5

Application

Remind that if for all λ > 0 and x, y ∈ L∞ (E), define ω : (0, ∞)×L∞ (E)×L∞ (E) → B(H)+ by ωλ (x, y) = π| x−y | , λ

where, πh : H → H be defined as in Example 2.7, then (L∞ (E)ω , B(H), ω) is a ω-complete C ∗ .m.m space. Let E be a Lebesgue measurable set, X = L∞ (E) and H = L2 (E) be the Hilbert space. Consider the following system of nonlinear integral equations: Z x(t) = w(t) + ki (t, x(t)) + µ n(t, s)hj (s, x(s))ds, (5.24) E

L∞ (E)ω

for all t ∈ E, where w ∈ is known, ki (t, x(t)), n(t, s), hj (s, x(s)), i, j = 1, 2 and i 6= j are real or complex valued functions that are measurable both in t and s on E and µ is real or complex number, and assume the following conditions: (a) sups∈E

R E

|n(t, s)|dt = M1 < +∞,

(b) k1 (s, x) is ω-continuous in s and x. ki (s, x(s)) ∈ L∞ (E)ω for all x ∈ L∞ (E)ω , and there exists L1 > 1 such that for all s ∈ E, |k1 (s, x(s)) − k2 (s, y(s))| √ ≥ L1 |x(s) − y(s)| for all x, y ∈ L∞ (E)ω , 2

22

Moeini and Hojat Ansari

(c) for i = 1, 2, hi (s, x(s)) ∈ L∞ (E)ω for all x ∈ L∞ (E)ω , and there exists L2 > 0 such that for all s ∈ E, |h1 (s, x(s)) − h2 (s, y(s))| ≤ L2 |x(s) − y(s)| for all x, y ∈ L∞ (E)ω , (d) for i = 1, 2, Bi is a nonempty set consists of xpi ∈ L∞ (E)ω , such that Z ki (t, xpi (t)) = xpi (t) − w(t) − λ n(t, s)hi (s, xpi (s))ds = zpi , E

and for any xp2 ∈ B2 , this implies that R k2 (t, xp2 (t)) −w(t) − λ E n(t, s)h2 (s, Rxp2 (s))ds = k2 (t, xp2 (t) − w(t) − λ E n(t, s)h2 (s, k2 (s, xp2 (s))ds), (e) there exists a sequence {xn (t)} in L∞ (E)ω , such that  limn→∞ k1 (t, xn (t)) = limn→∞ xn (t) − w(t)  R∞ −λ a n(t, s)h1 (s, xn (s))ds = τ1 (t) ∈ L∞ (E)ω , and which satisfies lim kωλ (xn , τ1 )k ≤ k

n→∞

k1 (t, xq1 ) − k1 (t, xr1 ) k∞ λ

for all λ > 0 and some xq1 , xr1 ∈ B1 , Theorem 5.1. With the assumptions (a)-(e), the system of nonlinear integral equations (5.24) has a unique solution x∗ in L∞ (E)ω for each real or complex number µ 2 M1 with 1+|µ|L < 1. L1 Proof. Define Z f x(t) = x(t) − w(t) − µ

n(t, s)h1 (s, x(s))ds, E

Z Hx(t) = x(t) − w(t) − µ

n(t, s)h2 (s, x(s))ds, E

Sx(t) = k1 (t, x(t)), T x(t) = k2 (t, x(t)). Set a = 2kck2 =

q

1+|µ|M1 L2 .1B(H) , L1 1+|µ|M1 L2 < 1. L1

b = c = 0B(H) , then a ∈ B(H)+ and 0 < kak2 + 2kbk2 +

Define 

ψ : B(H)+ → B(H)+ ψ(B) = 12 B,



ϕ : B(H)+ → B(H)+ ϕ(B) = 41 B,

(

F∗ : B(H)+ × B(H)+ → B(H) F∗ (A, B) = √12 A.

J HR-operator pairs and C∗ -class functions in C ∗ -algebra-valued modular metric spaces23

Then, (ψ, ϕ, F∗ ) is strictly monotone. For any h ∈ H, we have ψ(kωλ (f x, Hy)k1A ) = = supkhk=1

R h

 supkhk=1

R h



1 2λ

E

E

supkhk=1

R

1 2

supkhk=1 (π| Sx−T y | h, h) 1A λ



− y) + µ

R

i n(t, s)(h (s, y(s) − h (s, x(s))ds h(t)h(t)dt 1A 2 1 E



− y) + µ

R

i n(t, s)(h (s, y(s) − h (s, x(s))ds |h(t)|2 dt 1A 2 1 E

1 2λ (x 1 2λ (x

h i 2 dt kx − yk + |µ|M L kx − yk |h(t)| ∞ 1 2 ∞ 1A E

1 L2  ( 1+|µ|M )kx − yk∞ 1A 2λ



1+|µ|M1 L2 2 (t,y(t)) √1 ( )k k1 (t,x(t))−k k∞ 2L1 λ 2

=

1+|µ|M1 L2 √1 . 1 ( )kωλ (Sx, T y)k L1 2 2

=

√1 . 1 kak2 kωλ (Sx, T y)k 2 2

1A

1A

1A

  = F∗ ψ(kak2 kωλ (Sx, T y)k 1A ), ϕ(kak2 kωλ (Sx, T y)k 1A ) Then,   ψ(kωλ (f x, Hy)k1A )  F∗ ψ(kak2 kωλ (Sx, T y)k 1A ), ϕ(kak2 kωλ (Sx, T y)k 1A ) , for all x, y ∈ L∞ (E)ω and λ > 0. By condition (b), S is ω-continuous. By (d), B1 = C(f, S) 6= ∅, B2 = C(f, S) 6= ∅ and Hxp2 = T xp2 =⇒ T (Hxp2 ) = H(T xp2 ), so (H, T ) is a weakly compatible. Also by conditions (d) and (e), P C(f, S) 6= ∅ and there exists a sequence {xn (t)} in L∞ (E)ω such that limn→∞ f xn (t) = limn→∞ Sxn (t) = τ1 ∈ L∞ (E)ω and which satisfies limn→∞ kωλ (xn , τ1 )k ≤ k

k1 (t,xq1 )−k1 (t,xr1 ) k∞ λ

for all λ > 0 and some xq1 , xr1 ∈ B1

= kωλ (Sxq1 , Sxr1 )k for all λ > 0 and some xq1 , xr1 ∈ B1 = kωλ (f xq1 , f xr1 )k for all λ > 0 and some xq1 , xr1 ∈ B1 ≤ supx,y∈P C(f,S) kωλ (x, y)k for all λ > 0 = δω (P C(f, S)).

24

Moeini and Hojat Ansari

Then, (f, S) isa J HR-operator pair. So, by Corollary 4.8, there exists a unique common fixed point x∗ ∈ L∞ (E)ω such that x∗ = f x∗ = Sx∗ = Hx∗ = T x∗ , which proves the existence of unique solution of (5.24) in L∞ (E)ω . This completes the proof.

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