SIMULATION TYPE FUNCTIONS AND ...

SIMULATION TYPE FUNCTIONS AND COINCIDENCE POINT RESULTS STOJAN RADENOVI‚ AND SUMIT CHANDOK In this paper, we obtain some sucient conditions for the existence and uniqueness of point of coincidence by using simulation functions in the context of metric spaces and prove some interesting results. Our results generalize the corresponding results of [6], [13], [21] and [25] in several directions. Also, we provide an example which shows that our main result is a proper generalization of the result of Jungck [American Math. Monthly 83(1976) 261-263] and L-de-Hierro et al. [J. Comput. Appl. Math 275(2015) 345-355] and of Olgun et al. [Turk. J. Math. (2016) 40:832-837]. Abstract.

1.

INTRODUCTION AND PRELIMINARIES

To begin with, we have following denitions and notations which will be used in the sequel. Denition 1.1. [4] A mapping G : [0, +∞)2 → R is called C−class function if it is continuous and satises the following conditions: (1) G (s, t) ≤ s; (2) G (s, t) = s implies that either s = 0 or t = 0, for all s, t ∈ [0, +∞). Author in [4] denote C−class function as C. For C−class functions see also [5] and [11]. In [6], authors generalized the simulation function introduced by Khojasteh, Shukla and Radenovi¢ ([21]) using the function of C−class as follows: Denition 1.2. A mapping G : [0, +∞)2 → R has a property CG , if there exists an CG ≥ 0 such that (3) G (s, t) > CG implies s > t; (4) G (t, t) ≤ CG , for all t ∈ [0, +∞). Some examples of C−class functions that have property CG are as follows: a) G (s, t) = s − t, CG = r, r ∈ [0, +∞); , CG = 0; b) G (s, t) = s − (2+t)t 1+t s r c) G (s, t) = 1+kt , k ≥ 1, CG = 1+k , r ≥ 2. For more examples of C−class functions that have property CG see [6] and [11]. Recently, Khojasteh et al. ([21]) (also see [2], [7], [9], [10], [13], [23], [24], [29], [30]) introduced new approach in the xed point theory by using the following: 2010

47H10, 54H25, 46Nxx. Point of coincidence; Common xed point; Compatible mappings; Commuting mappings; Simulation function; C-class function; Property CF . 0*Correspondence Mathematics Subject Classication.

Key words and phrases.

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Denition 1.3. A simulation function is a mapping ζ : [0, ∞)2 → R satisfying the follow-

ing: (5) ζ (t, s) < s − t for all t, s > 0; (6) if {tn } , {sn } are sequences in (0, +∞) such that limn→∞ tn = limn→∞ sn > 0, and tn < sn , then lim supn→∞ ζ (tn , sn ) < 0. Ansari et al. [6] generalized the simulation function as the following: Denition 1.4. A CG −simulation function is a mapping ζ : [0, +∞)2 → R satisfying the following: (7) ζ (t, s) < G (s, t) for all t, s > 0, where G : [0, +∞)2 → R is element of C; (8) if {tn } , {sn } are sequences in (0, +∞) such that limn→∞ tn = limn→∞ sn > 0, and tn < sn , then lim supn→∞ ζ (tn , sn ) < CG . Some examples of simulation functions and CG −simulation functions are s d) ζ (t, s) = s+1 − t for all t, s ≥ 0. e) ζ (t, s) = s − ϕ (s) − t for all t, s ≥ 0, where ϕ : [0, +∞) → [0, +∞) is lower semi continuous function and ϕ (t) = 0 if and only if t = 0. f ) Example 12, 13 and 14 from [6]. For more examples of simulation functions and CG −simulation functions see [6], [7], [9],[10], [13], [21],[23], [24],[29], [30] and [31]. Denition 1.5. Let f and g be self maps of a set X. If w = f x = gx for some x ∈ X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. The pair (f, g) of self maps is weakly compatible if they commute at their coincidence points. The following result of [1] will be needed in sequel. Proposition 1.1. Let f and g be weakly compatible self maps of a set X. If f and g have a unique point of coincidence w = f x = gx, then w is the unique common xed point of f and g. Assertions similar to the following result of Radenovi¢ et al. [26] were used (and proved) in the proofs of several xed point results in various papers. Lemma 1.2. Let (X, d) be a metric space and let {xn } be a sequence in X such that (1.1) lim d (xn , xn+1 ) = 0. n→∞

If {xn } is not a Cauchy sequence, then there exist ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that n (k) > m (k) > k and the next sequences tend to ε+ when k → +∞ : (1.2)




d xm(k) , xn(k) , d xm(k) , xn(k)+1 , d xm(k)−1 , xn(k) ,

d xm(k)−1 , xn(k)+1 , d xm(k)+1 , xn(k)+1 .

Denition 1.6. Given two self-mappings f, g : X → X and a sequence {xn }n∈N∪{0} ⊆ X, the sequence {xn } is a Picard-Jungck sequence of the pair (f, g) (based on x0 ) if yn = f xn = gxn+1 for all n ∈ N∪ {0} (see also [13], Denition 4.4.)

SIMULATION TYPE FUNCTIONS...

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3

MAIN RESULTS

Let ZG be the family of all CG −simulation functions ζ : [0, +∞)2 → R. Each simulation function as in Denition 1.3 is also a CG −simulation function as in Denition 1.4, but the converse is not true. For this claim see Example 3.3 of [13] using C−class function G (s, t) = s − t. In this section, we establish some results on the existence and uniqueness of coincidence point by using simulation functions in the framework of metric spaces. Now, we begin with the following denition:

Denition 2.1. (see [6], [13]) Let (X, d) be a metric space and f, g : X → X be selfmappings. A mapping f is called a (ZG , g) −contraction if there exists ζ ∈ ZG such that (2.1) ζ (d (f x, f y) , d (gx, gy)) ≥ CG for all x, y ∈ X with gx 6= gy. In the case, g = iX (identity mapping on X ) and CG = 0 we get Z−contraction of [21] (Denition 2.3). Now, we state our rst new and very important result for the notion of (ZG , g) −contraction. It generalize the corresponding results of [6], [13], [21] and [25] in several directions.

Theorem 2.1. Let (X, d) be a metric space, f, g : X → X be self-mappings and f be a

(ZG , g) −contraction. Suppose that there exists a Picard-Jungck sequence {xn }n∈N∪{0} of (f, g) . Also assume that, at least, one of the following conditions hold: (i) (f (X) , d) or (g (X) , d) is complete; (ii) (X, d) is complete, g is a continuous and f and g are commuting; (iii) (X, d) is complete, g is continuous and f and g are compatible. Then f and g have unique point of coincidence.

Proof. First of all we shall prove that the point of coincidence of f and g is unique (if it exists). Suppose that z1 and z2 are distinct points of coincidence of f and g. From this it follows that there exist two points v1 and v2 (v1 6= v2 ) such that f v1 = gv1 = z1 and f v2 = gv2 = z2 . Then (2.1) implies that (2.2) CG ≤ ζ (d (f v1 , f v2 ) , d (gv1 , gv2 )) = ζ (d (z1 , z2 ) , d (z1 , z2 )) < G (d (z1 , z2 ) , d (z1 , z2 )) ≤ CG ,

which is a contradiction. In order to prove that f and g have a point of coincidence, suppose that there is a sequence {yn } such that yn = f xn = gxn+1 where n ∈ N∪{0} , so-called Jungck's sequence. If yk = yk+1 for some k ∈ N ∪ {0} , then gxk+1 = yk = yk+1 = f xk+1 and f and g have a point of coincidence. Therefore, suppose that yn 6= yn+1 for all n ∈ N ∪ {0} . Substituting x = xn+1 , y = xn+2 in (2.1) we obtain that CG ≤ ζ (d (f xn+1 , f xn+2 ) , d (gxn+1 , gxn+2 )) = ζ (d (yn+1 , yn+2 ) , d (yn , yn+1 ))

(2.3) < G (d (yn , yn+1 ) , d (yn+1 , yn+2 )) . Using (3) of Denition 1.2, we have d (yn , yn+1 ) > d (yn+1 , yn+2 ) . Hence, for all n ∈ N ∪ {0} we get that d (yn+1 , yn+2 ) < d (yn , yn+1 ) . Therefore there exists D ≥ 0 such that

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limn→∞ d (yn , yn+1 ) = D ≥ 0. Suppose that D > 0. Since d (yn+1 , yn+2 ) < d (yn , yn+1 ) and both d (yn+1 , yn+2 ) and d (yn , yn+1 ) tend to D. Using (8) of Denition 1.4, we get

(2.4)

CG ≤ lim sup ζ (d (yn+1 , yn+2 ) , d (yn , yn+1 )) < CG , n→∞

which is a contradiction. Hence limn→∞ d (yn , yn+1 ) = D = 0. Further we have to prove that n 6= m implies yn 6= ym . Indeed, suppose that yn = ym for some n > m. Then we choose xn+1 = xm+1 (which is obviously possible by the denition of Jungck's sequence yn ) and hence also yn+1 = ym+1 . Then following the previous arguments, we have d (yn , yn+1 ) < d (yn−1 , yn ) < ... < d (ym , ym+1 ) = d (yn , yn+1 ) ,

which is a contradiction. Now, we have to show that {yn } is a Cauchy sequence. On the contrary, suppose that {yn } is not a Cauchy sequence. Consider x = xm(k)+1 , y = yn(k)+1 in (2.1), we obtain (2.5)







CG ≤ ζ d ym(k)+1 , yn(k)+1 , d ym(k) , yn(k) < G d ym(k) , yn(k) , d ym(k)+1 , yn(k)+1 .

Using (3) of Denition 1.2, it follows that d ym(k) , yn(k) > d ym(k)+1 , yn(k)+1 . Now, if the
sequence {yn } is not
a Cauchy sequence, then by Lemma 1.2, we have d ym(k) , yn(k) and d ym(k)+1 , yn(k)+1 tend to ε > 0, as k → ∞. Therefore, using (2.5), we

have

(2.6)




CG ≤ lim sup ζ d ym(k)+1 , yn(k)+1 , d ym(k) , yn(k) < CG , n→∞

which is a contradiction. Therefore, the Jungck's sequence {yn } is a Cauchy sequence. Suppose that (i) holds, i.e., (g (X) , d) is complete. Then there exists v ∈ X such that gxn → gv as n → ∞. We shall prove that f v = gv. It is clear that we can suppose yn 6= f v, gv for all n ∈ N ∪ {0} . Therefore, by (2.1), we have (2.7)

CG ≤ ζ (d (f xn , f v) , d (gxn , gv)) < G (d (gxn , gv) , d (f xn , f v)) .

Using (3) of Denition 1.2, we get d (f xn , f v) < d (gxn , gv) . It implies that f xn → f v as n → ∞. Hence, f v = gv is a (unique) point of coincidence of f and g. Similarly, we can prove that f v = gv is a (unique) point of coincidence of f and g , when (f (X) , d) is complete. Suppose that (ii) holds. Since (X, d) is a complete, then there exists v ∈ X such that gxn → v , when n → ∞. As g is continuous, g 2 xn → gv when n → ∞. Putting now x = gxn , y = v in (2.1) we obtain



(2.8) CG ≤ ζ d (f gxn , f v) , d g 2 xn , gv < G d g 2 xn , gv , d (f gxn , f v) . Using (3) of Denition 1.2 and f , g are commuting, we have d (f gxn , f v) = d (gf xn , f v) < d (g 2 xn , gv) . Since, gf xn = g 2 xn+1 , the result follows, i.e., f v = gv is a (unique) point of coincidence of f and g. Finally, suppose that (iii) holds. Since (X, d) is a complete, then there exists v ∈ X such that f xn → v , when n → ∞. As g is continuous, g(f xn ) → gv when n → ∞.

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Consider (2.9)

CG ≤ ζ (d (f (gxn ) , f v) , d (g (f xn ) , gv)) < G (d (g (f xn ) , gv) , d (f (gxn ) , f v)) .

Using (3) of Denition 1.2 and continuity of g , we have d (f (gxn ) , f v) < d (g (f xn ) , gv) → 0, as n → ∞. It implies that d (f (gxn ) , f v) → 0, as n → ∞. Further, as f and g are compatible, we have (2.10) d (f v, gv) ≤ d (f v, f (gxn )) + d (f (gxn ) , g (f xn )) + d (g (f xn ) , gv) → 0 + 0 + 0 = 0. Hence, the result is proved for all three cases (i), (ii) and (iii), i.e., the mappings f and g have a unique point of coincidence. 

Theorem 2.2. In addition to hypotheses of Theorem 2.1, if f and g are weakly compatible, then they have a unique common xed point in X.

Proof. Using Theorem 2.1, f and g have unique point of coincidence. Further, if f and g are weakly compatible, then according to Proposition 1.1, they have a unique common xed point.  Remark 2.1. It is obvious that inequality (2.1) implies that f is a continuous if g is a continuous. Indeed, since d (gx, gy) > 0 for all x, y ∈ X for which gx 6= gy, we have that d (f x, f y) < d (gx, gy) . Hence, the assumptions in [6] and [13] that f is continuous are superuous. Also, it is clear that (ii) implies (iii). In the sequel we introduce the following generalized (ZG , g) −contraction:

Denition 2.2. Let (X, d) be a metric space and f, g : X → X be self-mappings. A

mapping f is called a generalized (ZG , g) −contraction if there exists ζ ∈ ZG such that (2.11)    ζ d (f x, f y) , max d (gx, gy) , d (gx, f x) , d (gy, f y) ,

d (gx.f y) + d (gy, f x) 2

≥ CG

for all x, y ∈ X with gx 6= gy. In the case that g = iX (identity mapping on X ) and CG = 0 we get Z−contraction of [25] (Denition 2, Theorem 2). The next result also generalize several ones in existing literature. Since its proof is similar with the proof of Theorems 2.1-2.2, so we omitted it.

Theorem 2.3. Let (X, d) be a metric space, f, g : X → X be self-mappings and f be

a generalized (ZG , g) −contraction. Suppose that there exists a Picard-Jungck sequence {xn }n∈N∪{0} of (f, g) . Also assume that, at least, one of the following conditions hold: (i) (f (X) , d) or (g (X) , d) is complete; (ii) (X, d) is complete and f and g are continuous and commuting; (iii) (X, d) is complete and f and g are continuous and compatible. Then f and g have unique point of coincidence. Moreover, if f and g are weakly compatible, then they have a unique common xed point in X.

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In the sequel, we describe how to use our Theorem 2.1 in order to guarantee existence and uniqueness of solution of nonlinear equations. It is worth to notice that the two examples given in [13] are not suitable and support their main result. Namely, both examples support even known Jungck's Theorem [17]. On the other words, there is λ ∈ (0, 1) such that d (f x, f y) ≤ λd (gx, gy) for all x, y ∈ X = [0, +∞), f x = x + 10, gx = 109 x + πx ex + sin 2(1+x) + 1, d (x, y) = |x − y| , ([13], Example 5.11) i.e., f x = arctan (x + 2) , gx = log (x + 3) , ([13], Example 5.12.). Also, ([6], Example 20), in the same metric space, where f x = x + 2, gx = 4x + e2x is such. The following example shows that our Theorem 2.1 is a proper generalization of the corresponding results of Jungck [17], L.-de-Hierro et al. [13] and of Olgun et al. [25].

Example 2.1. Let X = [0, 1] and d : X × X → [0, +∞) be dened by d (x, y) = |x − y| . x Then (X, d) is a complete metric space. Dene f, g : X → X as f x = 2+x , gx = x2 . Then, f is not Jungck's contraction in the sense that there is λ ∈ (0, 1) such that d (f x, f y) ≤ s λd (gx, gy) for all x, y ∈ X. However, putting ζ (t, s) = s+1 − t, G (s, t) = s − t, CG = 0, we have that f is a (Z, g) −contraction with respect to ζ. Indeed, we obtain d (gx, gy) − d (f x, f y) ≥ 0 1 + d (gx, gy) 1 |x − y| y x 2 − ⇔ − 1 + 12 |x − y| x + 2 y + 2 |x − y| 2 |x − y| = − ≥ 0, 2 + |x − y| (x + 2) (y + 2)     whenever x, y ∈ X. Further, since f (X) = 0, 31 ⊆ 0, 12 = g (X) there exists a PicardJungck sequence {xn }n∈N∪{0} of (f, g) . As both (f (X) , d) or (g (X) , d) are complete, this means that all the conditions of Theorem 2.2 are satised, i.e., the mappings f and g have a coincidence point x = 0. On the other word, they have a unique common xed point, x which is the only solution of equation x+2 = x2 , x ∈ [0, 1] . ζ (d (f x, f y) , d (gx, gy)) ≥ CG = 0 ⇔

Finally, we introduce the following:

Denition 2.3. Let (X, d) be a metric space and f, g : X → X be self-mappings. A mapping f is called a (ZG , g) −quasi-contraction of ‚iri¢-Das-Naik type if there exists ζ ∈ ZG such that (2.12) ζ (d (f x, f y) , max {d (gx, gy) , d (gx, f x) , d (gy, f y) , d (gx.f y) , d (gy, f x)}) ≥ CG for all x, y ∈ X with gx 6= gy. In the case that g = iX (identity mapping on X ) and CG = 0 we get Z−quasi-contraction of ‚iri¢ type. Finally, we have the following open question: Does the following Theorem hold?

Theorem 2.4. Let f be a (ZG , g) −quasi-contraction of of ‚iri¢-Das-Naik type in a metric

space (X, d) and suppose that there exists a Picard-Jungck sequence {xn }n∈N∪{0} of (f, g) . Also assume that, at least, one of the following conditions hold:

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(i) (f (X) , d) or (g (X) , d) is complete; (ii) (X, d) is complete and f and g are continuous and commuting; (iii) (X, d) is complete and f and g are continuous and compatible. Then f and g have unique point of coincidence. Moreover, if f and g are weakly compatible, then they have a unique common xed point in X.

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Stojan Radenovi¢

Nonlinear Analysis Research Group,Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

E-mail address :

[email protected]

Sumit Chandok

School of Mathematics, Thapar University, Patiala-147004, India.

E-mail address :

[email protected]; [email protected]