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Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces

Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces axioms Article Fixed Points of g-Interpolative Ciric–Reich–Rus-T ´ ype Contractions in b-Metric Spaces Youssef Errai * , El Miloudi Marhrani * and Mohamed Aamri Laboratory of Algebra, Analysis and Applications (L3A), Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, B.P 7955, Sidi Othmane, Casablanca 20700, Morocco; aamrimohamed82@gmail.com * Correspondence: yousseferrai1@gmail.com (Y.E.); marhrani@gmail.com (E.M.) Received: 15 October 2020; Accepted: 12 November 2020; Published: 16 November 2020 Abstract: We use interpolation to obtain a common fixed point result for a new type of Ciric–Reich–Rus-type ´ contraction mappings in metric space. We also introduce a new concept of g-interpolative Ciric–Reich–Rus-type ´ contractions in b-metric spaces, and we prove some fixed point results for such mappings. Our results extend and improve some results on the fixed point theory in the literature. We also give some examples to illustrate the given results. Keywords: fixed point; Ciric–Reich–Rus-type ´ contractions; interpolation; b-metric space MSC: 46T99; 47H10; 54H25 1. Introduction and Preliminaries Banach’s contraction principle [1] has been applied in several branches of mathematics. As a result, researching and generalizing this outcome has proven to be a research area in nonlinear analysis (see [2–6]). It is a well-known fact that a map that satisfies the Banach contraction principle is necessarily continuous. Therefore, it was natural to wonder if in a complete metric space, a discontinuous map satisfying somewhat similar contractual conditions may have a fixed point. Kannan [7] answered yes to this question by introducing a new type of contraction. The concept of the interpolation Kannan-type contraction appeared with Karapinar [8] in 2018; this concept appealed to many researchers [8–14], making them invest in various types of contractions: interpolative Ciric–Reich–Rus-type ´ contraction [9–11,13], interpolative Hardy–Rogers [15]; and they used it on various spaces: metric space, b-metric space, and the Branciari distance. In this paper, we will generalize some of the related findings to the interpolation Ciric–Reich–Rus-type ´ contraction in Theorems 1 and 2. In addition, we use a new concept of interpolative weakly contractive mapping to generalize some findings about the interpolation Kannan-type contraction in Theorem 3. Now, we recall the concept of b-metric spaces as follows: Definition 1 ([16,17]). Let X be a nonempty set and s  1 be a given real number. A function d : X X ! R is a b-metric if for all x, y, z 2 X , the following conditions are satisfied: (b ) d(x, y) = 0 if and only if x = y; (b ) d(x, y) = d(y, x); (b ) d(x, z)  s[d(x, y) + d(y, z)]. The pair (X, d) is called a b-metric space. Note that the class of b-metric spaces is larger than that of metric spaces. Axioms 2020, 9, 132; doi:10.3390/axioms9040132 www.mdpi.com/journal/axioms Axioms 2020, 9, 132 2 of 13 The notions of b-convergent and b-Cauchy sequences, as well as of b-complete b-metric spaces are defined exactly the same way as in the case of usual metric spaces (see, e.g., [18]). Definition 2 ([19,20]). Letfx g be a sequence in a b-metric space (X, d). g, h:X ! X, are self-mappings, and x 2 X. x is said to be the coincidence point of pair fg, hg if gx = hx. Definition 3 ([10,11]). Let Y be denoted as the set of all non-decreasing functions y: [0, ¥) ! [0, ¥), ¥ k such that y (t) < ¥ for each t > 0. Then: k=0 (i) y(0) = 0, (ii) y(t) < t for each t > 0. Remark 1 ([18]). In a b-metric space (X, d), the following assertions hold: 1. A b-convergent sequence has a unique limit. 2. Each b-convergent sequence is a b-Cauchy sequence. 3. In general, a b-metric is not continuous. The fact in the last remark requires the following lemma concerning the b-convergent sequences to prove our results: Lemma 1 ([19]). Let (X, d) be a b-metric space with s  1, and suppose that fx g and fy g are b-convergent n n to x, y, respectively, then we have: d(x, y)  lim inf d(x , y )  lim sup d(x , y )  s d(x, y). n n n n s n!¥ n!¥ In particular, if x = y, then we have lim d(x , y ) = 0. Moreover, for each z 2 X, we have: n!¥ n n d(x, z)  lim inf d(x , z)  lim sup d(x , z)  sd(x, z). n n n!¥ n!¥ 2. Results We denote by F the set of functions f : [0, ¥) ! [0, ¥) such that f(t) < t for every t > 0. Our main result is the following theorem: Theorem 1. Let (X, d) be a complete metric space, and T is a self-mapping on X such that: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) (1) is satisfied for all x, y 2 X n Fix(T); where Fix(T) = fa 2 XjTa = ag, a, b, g 2 (0, 1) such that a + b + g > 1, and f 2 F. If there exists x 2 X such that d(x, T x) < 1, then T has a fixed point in X. Proof. We define a sequence fx g by x = x and x = T x for all integers n, and we assume that n n 0 n+1 x 6= T x , for all n. n n We have: a b g d(x , x )  f([d(x , x )] [d(x , x )] [d(x , x )] ). (2) n n+1 n1 n n1 n n n+1 Using the fact f(t) < t for each t > 0, from (2), we obtain: a b g d(x , x ) < [d(x , x )] [d(x , x )] [d(x , x )] . n n+1 n1 n n1 n n n+1 Axioms 2020, 9, 132 3 of 13 which implies: 1g a+b [d(x , x )] < [d(x , x )] . (3) n n n+1 n1 We have d(x , x ) < 1, so that there exists a real l 2 (0, 1) such that d(x , x )  l and 0 1 0 1 d(x ,x )+1 0 1 l = . By (3), we obtain: a+b a+b 1g 1g d(x , x ) < [d(x , x )]  l . 1 2 0 1 By (3), we find: 1+e d(x , x )  d(x , x ) n n n+1 n1 a+b for all n, with e = 1 > 0. 1g Now, we prove by induction that for all n, (1+e) d(x , x )  l n+1 n where 0 < l < 1. For n = 1, this is the inequality at the bottom of page 3. The induction step is: 1+e n n+1 1+e (1+e) (1+e) d(x , x )  d(x , x )  l = l n+2 n+1 n+1 n Since (1 + e)  1 + ne by Bernoulli’s inequality and since l < 1, this implies: 1+ne n d(x , x )  l = lr n+1 for all n, where r = l < 1. This implies: 1 r n+k1 n+k2 n n n d(x , x )  l(r + r + + r ) = lr = Cr , n+k n 1 r 1r where C = l for some integer k, from which it follows that fx g forms a Cauchy sequence in 1r (X, d), and then, it converges to some z 2 X. Assume that z 6= Tz. By letting x = x and y = z in (1), we obtain: a b g d(x , Tz)  f([d(x , z)] [d(x , x )] [d(z, Tz)] ) n n n+1 n+1 a b g < [d(x , z)] [d(x , x )] [d(z, Tz)] n n n+1 for all n, which leads to d(z, Tz) = 0, which is a contradiction. Then, Tz = z. Example 1. Let X = [0, 2] be endowed with metric d : X X ! [0, ¥), defined by: 0, if x = y; d(x, y) = , if x, y 2 [0, 1] and x 6= y; 2, otherwise. Consider that the self-mapping T : X ! X is defined by: , if x 2 [0, 1]; T x = , if x 2 (1, 2]; 2 Axioms 2020, 9, 132 4 of 13 and the function f(t) = 0, 4t for all t 2 [0, ¥). For a = 0, 8, b = 0, 2, and g = 0, 25. We discus the following cases: Case 1. If x, y 2 [0, 1] or x = y for all x, y 2 [0, 2]; it is obvious. Case 2. If x, y 2 (1, 2] and x 6= y. We have: d(T x, Ty) = and: 3,5 2 2 a b g a+b+g f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f(2 ) =  . 5 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x, y 2 (1, 2]. Case 3. If x 2 [0, 1] and y 2 (1, 2] with x 6= . We have: d(T x, Ty) = and: 3,5 2 2 2 a b g a+g f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f 2 =  . 0,2 3 5.3 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x 2 [0, 1]nf g and y 2 (1, 2]. Case 4. If x 2 (1, 2] and y 2 [0, 1] with y 6= . We have: d(T x, Ty) = and: 3,5 2 2 2 a b g a+b f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f 2 =  . 0,25 3 5.3 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x 2 (1, 2] and y 2 [0, 1]nf g. Therefore, all the conditions of Theorem 1 are satisfied, and T has a fixed point, x = . Example 2. Let X = fa, q, r, sg be endowed with the metric defined by the following table of values: Axioms 2020, 9, 132 5 of 13 d(x, y) a q r s 1 10 5 a 0 3 3 3 q 0 3 2 r 3 0 5 s 2 5 0 Consider the self-mapping T on X as: a q r s T: . a a q s 2 1 For y(t) = for all t 2 [0, ¥); a = 0, 6; b = 0, 9; and g = 0, 7. 2 +1 We have: a b g d(Tu, Tv)  y([d(u, v)] [d(u, Tu)] [d(v, Tv)] ) for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. If we take y(t) = kt in Theorem (1) with k 2 (0, 1), then we have the following corollary: Corollary 1. Let (X, d) be a complete metric space, and T is a self-mapping on X such that: a b g d(T x, Ty)  k[d(x, y)] [d(x, T x)] [d(y, Ty)] is satisfied for all x, y 2 X n Fix(T); where Fix(T) = fa 2 XjTa = ag, and a, b, g, k 2 (0, 1) such that a + b + g > 1. If there exists x 2 X such that d(x, T x) < 1, then T has a fixed point in X. Example 3. It is enough to take in Example 1: f(t) = t for all t 2 [0, +¥). Example 4. Let X = fa, q, r, sg be endowed with the metric defined by the following table of values: d(x, y) a q r s a 0 0, 1 3, 1 4 q 0, 1 0 3 3, 9 r 3, 1 3 0 0, 9 s 4 3, 9 0, 9 0 Consider the self-mapping T on X as: a q r s T : . a a q s For k = ; a = 0, 7; b = 0, 1; and g = 0, 8. We have: a b g d(Tu, Tv)  k[d(u, v)] [d(u, Tu)] [d(v, Tv)] for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. Axioms 2020, 9, 132 6 of 13 Definition 4. Let (X, d, s) be a b-metric space and T, g : X ! X be self-mappings on X. We say that T is a g-interpolative Ciric–Reich–Rus-type ´ contraction, if there exists a continuous y 2 Y and a, b 2 (0, 1) such that: a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) (4) is satisfied for all x, y 2 X such that T x 6= gx, Ty 6= gy, and gx 6= gy. Theorem 2. Let (X, d, s) be a b-complete b-metric space, and T is a g-interpolative Ciric–Reich–Rus-type ´ contraction. Suppose that T X  gX such that gX is closed. Then, T and g have a coincidence point in X. Proof. Let x 2 X; since T X  gX, we can define inductively a sequence fx g such that: x = x, and gx = T x , for all integer n. 0 n n+1 If there exists n 2 f0, 1, 2, . . .g such that gx = T x , then x is a coincidence point of g and T. n n n Assume that gx 6= T x , for all n. By (4), we obtain: n n a b 1ab d(T x , T x )  y([d(gx , gx ] [d(gx , T x ] [d(gx , T x ] ) n+1 n n+1 n n+1 n+1 n n a b 1ab = y([d(T x , T x ] [d(T x , T x ] [d(T x , T x ] ) n n n n1 n+1 n1 1b b = y([d(T x , T x ] [d(T x , T x ] ). n n1 n n+1 Using the fact y(t) < t for each t > 0, 1b b d(T x , T x )  y([d(T x , T x )] [d(T x , T x )] ) n n n n+1 n1 n+1 1b b < [d(T x , T x )] [d(T x , T x )] . (5) n n1 n n+1 which implies: 1b 1b [d(T x , T x )] < [d(T x , T x )] . n+1 n n n1 Thus, d(T x , T x ) < d(T x , T x ) for all n  1. (6) n+1 n n n1 That is, the positive sequence fd(T x , T x )g is monotone decreasing, and consequently, there n+1 n exists c  0 such that lim d(T x , T x ) = c. From (6), we obtain: n!¥ n+1 n 1b b 1b b [d(T x , T x )] [d(T x , T x )]  [d(T x , T x )] [d(T x , T x )] n n n n n1 n+1 n1 n1 = d(T x , T x ). n n1 Therefore, with (5) together with the nondecreasing character of y, we get: 1b b d(T x , T x )  y([d(T x , T x )] [d(T x , T x )] ) n n n n+1 n1 n+1 y(d(T x , T x )). n n1 By repeating this argument, we get: 2 n d(T x , T x )  y(d(T x , T x ))  y (d(T x , T x ))    y (d(T x , T x )). (7) n n n+1 n1 n1 n2 1 0 Taking n ! ¥ in (7) and using the fact lim y (t) = 0 for each t > 0, we deduce that c = 0, n!¥ that is, lim d(T x , T x ) = 0. (8) n+1 n!¥ Axioms 2020, 9, 132 7 of 13 Then, fT x g is a b-Cauchy sequence. Suppose on the contrary that there exists an e > 0 and subsequences fT x g and fT x g of fT x g such that n is the smallest integer for which: m n n k k n > m > k, d(T x , T x )  e, and d(T x , T x ) < e. n m m k k n 1 k k k k Then, we have: d(gx , gx ) = d(T x , T x )  sd(T x , T x ) + sd(T x , T x ) n m n 1 m 1 n 1 m m m 1 k k k k k k k k se + sd(T x , T x ). m 1 k k Using (8) in the inequality above, we obtain: lim sup d(T x , T x ) = lim sup d(gx , gx )  se. (9) n 1 m 1 n m k k k k k!¥ k!¥ Putting x = x and y = x in (4), we have: n m k k a b 1ab e  d(T x , T x )  y([d(gx , gx )] [d(gx , T x )] [d(gx , T x )] ) n m n m n n m m k k k k k k k k a b 1ab = y([d(T x , T x )] [d(T x , T x )] [d(T x , T x )] ). (10) n 1 m 1 n 1 n m 1 m k k k k k k Taking the upper limit as k ! ¥ in (10) and using (8) and (9) and the property of y, we get: e  lim sup d(T x , T x )  y(0) = 0, n m k k k!¥ which implies that e = 0, a contradiction with e > 0. We deduce that fT x g is a b-Cauchy sequence, and consequently, fgx g is also a b-Cauchy sequence. Let z 2 X such that, lim d(T x , z) = lim d(gx , z) = 0. n+1 n!¥ n!¥ Since z 2 gX, there exists u 2 X such that z = gu. We claim that u is a coincidence point of g and T. For this, if we assume that gu 6= Tu, we obtain: a b 1ab d(T x , Tu)  y([d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] ) n n n n a b 1ab < [d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] . n n n At the limit as n ! ¥ and using Lemma 1, we get: a b 1ab d(z, Tu)  lim inf d(T x , Tu)  lim sup[d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] n n n n n!¥ n!¥ a 2 b 1ab [sd(z, gu)] [s d(z, z)] [d(gu, Tu)] = 0, which is a contradiction, which implies that: Tu = z = gu. Then, u is a coincidence point in X of T and g. Example 5. Let X = [0, +¥) and d : X X ! [0, ¥) be defined by: (x + y) , if x 6= y; d(x, y) = 0, if x = y. Axioms 2020, 9, 132 8 of 13 Then, (X, d) is a complete b-metric space. Define two self-mappings T and g on X by g(x) = x ; for all x 2 X and: 1, if x 2 [0, 2]; T x = , if x 2 (2, +¥). T is a g-interpolative Ciric–Reich–Rus-type ´ contraction for a = 0, 7, b = 0, 4, and: 3 2 89 t , if t 2 [0, ]; 20 20 y(t) = t+1 3 1 89 , if t 2 ( , +¥). 3 +1 20 For this, we discuss the following cases: Case 1. If x, y 2 [0, 2] or x = y for all x 2 [0, +¥). It is obvious. Case 2. If x, y 2 (2, +¥) and x 6= y. We have: 1 1 d(T x, Ty) = ( + )  1. x y Using the property of y, we get: 1 1 a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + ) (y + ) ) x y 2a 2(1a) y(8 .( ) )  1. Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Case 3. If x 2 [0, 2]nf1g and y 2 (2, +¥). We have: 1 3 9 2 2 d(T x, Ty) = (1 + )  ( ) = , y 2 4 and: a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + 1) (y + ) ) 9 9 2a 2b 2(1ab) y(4 .1 .( ) )  . 2 4 Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Case 4. If x 2 (2, +¥) and y 2 [0, 2]nf1g. We have: 1 9 d(T x, Ty) = (1 + )  , x 4 and: a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + ) (y + 1) ) 9 9 2a 2b 2(1ab) y(4 .( ) .1 )  . 2 4 Axioms 2020, 9, 132 9 of 13 Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Then, it is clear that g, T satisfies (4) for all u, v 2 Xnf1g. Moreover, one is a coincidence point of g and T. Example 6. Let the set X = fa, b, q, rg and a function d : X X ! [0, ¥) be defined as follows: d(x, y) a b q r a 0 1 16 b 1 0 9 q 16 9 0 49 25 1 r 0 4 4 4 By a simple calculation, one can verify that the function d is a b-metric, for s = 2. We define the self-mappings g, T on X, as: ! ! a b q r a b q r g : , T : . a r q q q r r q For a = 0, 3; b = 0, 8; and y(t) = for all t 2 [0, ¥). 1+t It is clear that g, T satisfies (4) for all u, v 2 Xnfb, rg. Moreover, b and r are two coincidence points of g and T. Definition 5. Let (X, d) is a metric space. A self-mapping T: X ! X is said to be an interpolative weakly contractive mapping if there exists a constant a 2 (0, 1) such that: a 1a a 1a z(d(T x, Ty))  z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ), (11) for all x, y 2 Xn Fix(T), where Fix(T) = fa 2 XjTa = ag, z: [0, ¥) ! [0, ¥) is a continuous monotone nondecreasing function with z(t) = 0 if and only if t = 0, j: [0, ¥) ! [0, ¥) is a lower semi-continuous function with j(t) = 0 if and only if t = 0. Theorem 3. Let (X, d) be a complete metric space. If T : X ! X is a interpolative weakly contractive mapping, then T has a fixed point. Proof. For any x 2 X, we define a sequence fx g by x = x and x = T x , n = 0, 1, 2, . . . 0 n 0 n+1 n If there exists n 2 N such that x = x , then x is clearly a fixed point in X. Otherwise, x 6= x n n n 0 n +1 n+1 0 0 0 for each n  0. Substituting x = x and y = x in (11), we obtain that: n n1 a 1a a 1a z(d(x , x ))  z([d(x , x )] [d(x , x )] ) j([d(x , x )] [d(x , x )] ) n+1 n n n+1 n1 n n n+1 n1 n a 1a z([d(x , x )] [d(x , x )] ). (12) n n+1 n1 n Using property of function z, we get: a 1a d(x , x )  [d(x , x )] [d(x , x )] . n+1 n n n+1 n1 n We derive: 1a 1a [d(x , x )]  [d(x , x )] . n+1 n n1 n Therefore: d(x , x )  d(x , x ), for all n  1. n+1 n n1 n Axioms 2020, 9, 132 10 of 13 It follows that the positive sequence fd(x , x )g is decreasing. Eventually, there exists c  0 n+1 n such that lim d(x , x ) = c. n n n+1 Taking n ! ¥ in the inequality (12), we obtain: z(c)  z(c) j(c). We deduce that c = 0. Hence: lim d(x , x ) = 0. (13) n+1 n Therefore, fx g is a Cauchy sequence. Suppose it is not. Then, there exists a real number e > 0, for any k 2 N,9m  n  k such that: k k d(x , x )  e. (14) m n k k Putting x = x and y = x in (11) and using (14), we get: n 1 m 1 k k a 1a a 1a z(e)  z(d(x , x ))  z([d(x , x )] [d(x , x )] ) j([d(x , x )] [d(x , x )] ). m n m n m n m 1 n 1 m 1 n 1 k k k k k k k k k k Letting k ! ¥ and using (13), we conclude: z(e)  z(0) j(0) = 0, which is contradiction with e > 0; thus, fx g is a Cauchy sequence; since (X, d) is complete, we obtain z 2 X such that lim d(x , z) = 0, and assuming that Tz 6= z, we have: n n a 1a a 1a z(d(x , Tz))  z([d(x , x )] [d(z, Tz)] ) j([d(x , x )] [d(z, Tz)] ) for all n. n+1 n n+1 n n+1 Letting n ! ¥, we get: a 1a a 1a z(d(z, Tz))  z([d(z, z)] [d(z, Tz)] ) j([d(z, z)] [d(z, Tz)] ) = z(0) j(0) = 0, which is a contradiction; thus, Tz = z. Example 7. Let the set X = [0, 3] and a function d : X X ! [0, ¥) be defined as follows: 0, if x = y; d(x, y) = 3, if x, y 2 [0, 1) and x 6= y; 2, otherwise. Then, (X, d) is a complete metric space. Let T: X ! X be defined as: 0, if x 2 [0, 1); T x = 1, if x 2 [1, 3]. 2 1 For z(t) = t , j(t) = t for all t 2 [0, +¥) and a = 0, 6. We discuss the following cases. Case 1. If x = y or x, y 2 (0, 1), or x, y 2 (1, 3] with x 6= y. It is obvious. Case 2. If x 2 (0, 1) and y 2 (1, 3]. We have: z(d(T x, Ty)) = z(d(0, 1)) = z(2) = 4, Axioms 2020, 9, 132 11 of 13 and: a 1a a 1a a [d(x, T x)] [d(y, Ty)] = [d(x, 0)] [d(y, 1)] = 2.( ) . Therefore: 3 3 a 1a a 1a a a z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ) = ( ) [4.( ) 1]  4 = z(2) = z(d(T x, Ty)). 2 2 Case 3. If x 2 (1, 3] and y 2 (0, 1). We have: z(d(T x, Ty)) = z(d(1, 0)) = z(2) = 4, and: a 1a a 1a a [d(x, T x)] [d(y, Ty)] = [d(x, 1)] [d(y, 0)] = 3.( ) . Therefore, 2 2 3 a 1a a 1a a a z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ) = ( ) [9.( ) ]  4 = z(2) = z(d(T x, Ty)). 3 3 2 Thus, a 1a a 1a z(d(Tu, Tv))  z([d(u, Tu)] [d(v, Tv)] ) j([d(u, Tu)] [d(v, Tv)] ), for all u, v 2 Xnf0, 1g. Then, T has two fixed points, which are zero and one. Example 8. Let X = fa, b, r, sg be endowed with the metric defined by the following table of values: d(x, y) a b r s a 0 1 4 1 b 1 0 5 2 r 4 5 0 3 s 1 2 3 0 Consider the self-mapping T on X as: a b r s T : . a s a s t t For z(t) = e 1 and j(t) = 2 1 for all t 2 [0, ¥); a =0, 3. We have: a 1a a 1a z(d(Tu, Tv))  z([d(u, Tu)] [d(v, Tv)] ) j([d(u, Tu)] [d(v, Tv)] ), for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. If z(t) = t in Theorem (3), then we have the following corollary: Axioms 2020, 9, 132 12 of 13 Corollary 2. Let (X, d) be a complete metric space and T : X ! X a self-mapping on X. If there exists a constant a 2 (0, 1) such that: a 1a a 1a d(T x, Ty)  [d(x, T x)] [d(y, Ty)] j([d(x, T x)] [d(y, Ty)] ), for all x, y 2 X and x 6= T x, y 6= Ty. j : [0, ¥) ! [0, ¥) is a lower semi-continuous function with j(t) = 0 if and only if t = 0. Then, T has a fixed point. Remark 2. In Corollary 2, if we take j(t) = (1 l)t for a constant l 2 (0, 1), then the result of Theorem [8] is obtained. Author Contributions: All authors contributed equally and significantly to the writing of this article. All authors read and approved the final manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 1922, 3, 133–181. [CrossRef] 2. Petrusel, ¸ A.; Petrusel, ¸ G. On Reich’s strict fixed point theorem for multi-valued operators in complete metric spaces. J. Nonlinear Var. Anal. 2018, 2, 103–112. 3. Suzuki, T. Edelstein’s fixed point theorem in semimetric spaces. J. Nonlinear Var. Anal. 2018, 2, 165–175. 4. Park, S. Some general fixed point theorems on topological vector spaces. Appl. Set-Valued Anal. Optim. 2019, 1, 19–28. 5. Ðoric, ´ D. Common fixed point for generalized (y, f)-weak contractions. Appl. Math. Lett. 2009, 22, 1896–1900. [CrossRef] 6. Dutta, P.; Choudhury, B.S. A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 2008, 406368. [CrossRef] 7. Kannan, R. Some results on fixed points. Bull. Cal. Math. Soc. 1968, 60, 71–76. 8. Karapinar, E. Revisiting the Kannan type contractions via interpolation. Adv. Theory Nonlinear Anal. Appl. 2018, 2, 85–87. [CrossRef] 9. Aydi, H.; Chen, C.M.; Karapınar, E. Interpolative Ciric–Reich–Rus ´ type contractions via the Branciari distance. Mathematics 2019, 7, 84. [CrossRef] 10. Aydi, H.; Karapinar, E.; Roldán López de Hierro, A.F. w-interpolative Ciric–Reich–Rus-type ´ contractions. Mathematics 2019, 7, 57. [CrossRef] 11. Debnath, P.; de La Sen, M. Fixed-points of interpolative Ciric-Reich–Rus-type ´ contractions in b-metric Spaces. Symmetry 2020, 12, 12. [CrossRef] 12. Gaba, Y.U.; Karapınar, E. A New Approach to the Interpolative Contractions. Axioms 2019, 8, 110. [CrossRef] 13. Karapinar, E.; Agarwal, R.; Aydi, H. Interpolative Reich–Rus–Ciric ´ type contractions on partial metric spaces. Mathematics 2018, 6, 256. [CrossRef] 14. Noorwali, M. Common fixed point for Kannan type contractions via interpolation. J. Math. Anal. 2018, 9, 92–94. 15. Karapınar, E.; Alqahtani, O.; Aydi, H. On interpolative Hardy–Rogers type contractions. Symmetry 2019, 11, 8. [CrossRef] 16. Bakhtin, I. The contraction mapping principle in quasimetric spaces. Func. Anal. Gos. Ped. Inst. Unianowsk 1989, 30, 26–37. 17. Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1, 5–11. 18. Boriceanu, M.; Bota, M.; Petrusel, ¸ A. Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 2010, 8, 367–377. [CrossRef] Axioms 2020, 9, 132 13 of 13 19. Aghajani, A.; Abbas, M.; Roshan, J. Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca 2014, 64, 941–960. [CrossRef] 20. Alqahtani, B.; Fulga, A.; Karapınar, E.; Özturk, A. Fisher-type fixed point results in b-metric spaces. Mathematics 2019, 7, 102. [CrossRef] Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. 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Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces

Errai, Youssef; Marhrani, El Miloudi; Aamri, Mohamed
Axioms , Volume 9 (4) – Nov 16, 2020
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axioms Article Fixed Points of g-Interpolative Ciric–Reich–Rus-T ´ ype Contractions in b-Metric Spaces Youssef Errai * , El Miloudi Marhrani * and Mohamed Aamri Laboratory of Algebra, Analysis and Applications (L3A), Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, B.P 7955, Sidi Othmane, Casablanca 20700, Morocco; aamrimohamed82@gmail.com * Correspondence: yousseferrai1@gmail.com (Y.E.); marhrani@gmail.com (E.M.) Received: 15 October 2020; Accepted: 12 November 2020; Published: 16 November 2020 Abstract: We use interpolation to obtain a common fixed point result for a new type of Ciric–Reich–Rus-type ´ contraction mappings in metric space. We also introduce a new concept of g-interpolative Ciric–Reich–Rus-type ´ contractions in b-metric spaces, and we prove some fixed point results for such mappings. Our results extend and improve some results on the fixed point theory in the literature. We also give some examples to illustrate the given results. Keywords: fixed point; Ciric–Reich–Rus-type ´ contractions; interpolation; b-metric space MSC: 46T99; 47H10; 54H25 1. Introduction and Preliminaries Banach’s contraction principle [1] has been applied in several branches of mathematics. As a result, researching and generalizing this outcome has proven to be a research area in nonlinear analysis (see [2–6]). It is a well-known fact that a map that satisfies the Banach contraction principle is necessarily continuous. Therefore, it was natural to wonder if in a complete metric space, a discontinuous map satisfying somewhat similar contractual conditions may have a fixed point. Kannan [7] answered yes to this question by introducing a new type of contraction. The concept of the interpolation Kannan-type contraction appeared with Karapinar [8] in 2018; this concept appealed to many researchers [8–14], making them invest in various types of contractions: interpolative Ciric–Reich–Rus-type ´ contraction [9–11,13], interpolative Hardy–Rogers [15]; and they used it on various spaces: metric space, b-metric space, and the Branciari distance. In this paper, we will generalize some of the related findings to the interpolation Ciric–Reich–Rus-type ´ contraction in Theorems 1 and 2. In addition, we use a new concept of interpolative weakly contractive mapping to generalize some findings about the interpolation Kannan-type contraction in Theorem 3. Now, we recall the concept of b-metric spaces as follows: Definition 1 ([16,17]). Let X be a nonempty set and s  1 be a given real number. A function d : X X ! R is a b-metric if for all x, y, z 2 X , the following conditions are satisfied: (b ) d(x, y) = 0 if and only if x = y; (b ) d(x, y) = d(y, x); (b ) d(x, z)  s[d(x, y) + d(y, z)]. The pair (X, d) is called a b-metric space. Note that the class of b-metric spaces is larger than that of metric spaces. Axioms 2020, 9, 132; doi:10.3390/axioms9040132 www.mdpi.com/journal/axioms Axioms 2020, 9, 132 2 of 13 The notions of b-convergent and b-Cauchy sequences, as well as of b-complete b-metric spaces are defined exactly the same way as in the case of usual metric spaces (see, e.g., [18]). Definition 2 ([19,20]). Letfx g be a sequence in a b-metric space (X, d). g, h:X ! X, are self-mappings, and x 2 X. x is said to be the coincidence point of pair fg, hg if gx = hx. Definition 3 ([10,11]). Let Y be denoted as the set of all non-decreasing functions y: [0, ¥) ! [0, ¥), ¥ k such that y (t) < ¥ for each t > 0. Then: k=0 (i) y(0) = 0, (ii) y(t) < t for each t > 0. Remark 1 ([18]). In a b-metric space (X, d), the following assertions hold: 1. A b-convergent sequence has a unique limit. 2. Each b-convergent sequence is a b-Cauchy sequence. 3. In general, a b-metric is not continuous. The fact in the last remark requires the following lemma concerning the b-convergent sequences to prove our results: Lemma 1 ([19]). Let (X, d) be a b-metric space with s  1, and suppose that fx g and fy g are b-convergent n n to x, y, respectively, then we have: d(x, y)  lim inf d(x , y )  lim sup d(x , y )  s d(x, y). n n n n s n!¥ n!¥ In particular, if x = y, then we have lim d(x , y ) = 0. Moreover, for each z 2 X, we have: n!¥ n n d(x, z)  lim inf d(x , z)  lim sup d(x , z)  sd(x, z). n n n!¥ n!¥ 2. Results We denote by F the set of functions f : [0, ¥) ! [0, ¥) such that f(t) < t for every t > 0. Our main result is the following theorem: Theorem 1. Let (X, d) be a complete metric space, and T is a self-mapping on X such that: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) (1) is satisfied for all x, y 2 X n Fix(T); where Fix(T) = fa 2 XjTa = ag, a, b, g 2 (0, 1) such that a + b + g > 1, and f 2 F. If there exists x 2 X such that d(x, T x) < 1, then T has a fixed point in X. Proof. We define a sequence fx g by x = x and x = T x for all integers n, and we assume that n n 0 n+1 x 6= T x , for all n. n n We have: a b g d(x , x )  f([d(x , x )] [d(x , x )] [d(x , x )] ). (2) n n+1 n1 n n1 n n n+1 Using the fact f(t) < t for each t > 0, from (2), we obtain: a b g d(x , x ) < [d(x , x )] [d(x , x )] [d(x , x )] . n n+1 n1 n n1 n n n+1 Axioms 2020, 9, 132 3 of 13 which implies: 1g a+b [d(x , x )] < [d(x , x )] . (3) n n n+1 n1 We have d(x , x ) < 1, so that there exists a real l 2 (0, 1) such that d(x , x )  l and 0 1 0 1 d(x ,x )+1 0 1 l = . By (3), we obtain: a+b a+b 1g 1g d(x , x ) < [d(x , x )]  l . 1 2 0 1 By (3), we find: 1+e d(x , x )  d(x , x ) n n n+1 n1 a+b for all n, with e = 1 > 0. 1g Now, we prove by induction that for all n, (1+e) d(x , x )  l n+1 n where 0 < l < 1. For n = 1, this is the inequality at the bottom of page 3. The induction step is: 1+e n n+1 1+e (1+e) (1+e) d(x , x )  d(x , x )  l = l n+2 n+1 n+1 n Since (1 + e)  1 + ne by Bernoulli’s inequality and since l < 1, this implies: 1+ne n d(x , x )  l = lr n+1 for all n, where r = l < 1. This implies: 1 r n+k1 n+k2 n n n d(x , x )  l(r + r + + r ) = lr = Cr , n+k n 1 r 1r where C = l for some integer k, from which it follows that fx g forms a Cauchy sequence in 1r (X, d), and then, it converges to some z 2 X. Assume that z 6= Tz. By letting x = x and y = z in (1), we obtain: a b g d(x , Tz)  f([d(x , z)] [d(x , x )] [d(z, Tz)] ) n n n+1 n+1 a b g < [d(x , z)] [d(x , x )] [d(z, Tz)] n n n+1 for all n, which leads to d(z, Tz) = 0, which is a contradiction. Then, Tz = z. Example 1. Let X = [0, 2] be endowed with metric d : X X ! [0, ¥), defined by: 0, if x = y; d(x, y) = , if x, y 2 [0, 1] and x 6= y; 2, otherwise. Consider that the self-mapping T : X ! X is defined by: , if x 2 [0, 1]; T x = , if x 2 (1, 2]; 2 Axioms 2020, 9, 132 4 of 13 and the function f(t) = 0, 4t for all t 2 [0, ¥). For a = 0, 8, b = 0, 2, and g = 0, 25. We discus the following cases: Case 1. If x, y 2 [0, 1] or x = y for all x, y 2 [0, 2]; it is obvious. Case 2. If x, y 2 (1, 2] and x 6= y. We have: d(T x, Ty) = and: 3,5 2 2 a b g a+b+g f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f(2 ) =  . 5 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x, y 2 (1, 2]. Case 3. If x 2 [0, 1] and y 2 (1, 2] with x 6= . We have: d(T x, Ty) = and: 3,5 2 2 2 a b g a+g f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f 2 =  . 0,2 3 5.3 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x 2 [0, 1]nf g and y 2 (1, 2]. Case 4. If x 2 (1, 2] and y 2 [0, 1] with y 6= . We have: d(T x, Ty) = and: 3,5 2 2 2 a b g a+b f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) = f 2 =  . 0,25 3 5.3 3 Then: a b g d(T x, Ty)  f([d(x, y)] [d(x, T x)] [d(y, Ty)] ) for all x 2 (1, 2] and y 2 [0, 1]nf g. Therefore, all the conditions of Theorem 1 are satisfied, and T has a fixed point, x = . Example 2. Let X = fa, q, r, sg be endowed with the metric defined by the following table of values: Axioms 2020, 9, 132 5 of 13 d(x, y) a q r s 1 10 5 a 0 3 3 3 q 0 3 2 r 3 0 5 s 2 5 0 Consider the self-mapping T on X as: a q r s T: . a a q s 2 1 For y(t) = for all t 2 [0, ¥); a = 0, 6; b = 0, 9; and g = 0, 7. 2 +1 We have: a b g d(Tu, Tv)  y([d(u, v)] [d(u, Tu)] [d(v, Tv)] ) for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. If we take y(t) = kt in Theorem (1) with k 2 (0, 1), then we have the following corollary: Corollary 1. Let (X, d) be a complete metric space, and T is a self-mapping on X such that: a b g d(T x, Ty)  k[d(x, y)] [d(x, T x)] [d(y, Ty)] is satisfied for all x, y 2 X n Fix(T); where Fix(T) = fa 2 XjTa = ag, and a, b, g, k 2 (0, 1) such that a + b + g > 1. If there exists x 2 X such that d(x, T x) < 1, then T has a fixed point in X. Example 3. It is enough to take in Example 1: f(t) = t for all t 2 [0, +¥). Example 4. Let X = fa, q, r, sg be endowed with the metric defined by the following table of values: d(x, y) a q r s a 0 0, 1 3, 1 4 q 0, 1 0 3 3, 9 r 3, 1 3 0 0, 9 s 4 3, 9 0, 9 0 Consider the self-mapping T on X as: a q r s T : . a a q s For k = ; a = 0, 7; b = 0, 1; and g = 0, 8. We have: a b g d(Tu, Tv)  k[d(u, v)] [d(u, Tu)] [d(v, Tv)] for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. Axioms 2020, 9, 132 6 of 13 Definition 4. Let (X, d, s) be a b-metric space and T, g : X ! X be self-mappings on X. We say that T is a g-interpolative Ciric–Reich–Rus-type ´ contraction, if there exists a continuous y 2 Y and a, b 2 (0, 1) such that: a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) (4) is satisfied for all x, y 2 X such that T x 6= gx, Ty 6= gy, and gx 6= gy. Theorem 2. Let (X, d, s) be a b-complete b-metric space, and T is a g-interpolative Ciric–Reich–Rus-type ´ contraction. Suppose that T X  gX such that gX is closed. Then, T and g have a coincidence point in X. Proof. Let x 2 X; since T X  gX, we can define inductively a sequence fx g such that: x = x, and gx = T x , for all integer n. 0 n n+1 If there exists n 2 f0, 1, 2, . . .g such that gx = T x , then x is a coincidence point of g and T. n n n Assume that gx 6= T x , for all n. By (4), we obtain: n n a b 1ab d(T x , T x )  y([d(gx , gx ] [d(gx , T x ] [d(gx , T x ] ) n+1 n n+1 n n+1 n+1 n n a b 1ab = y([d(T x , T x ] [d(T x , T x ] [d(T x , T x ] ) n n n n1 n+1 n1 1b b = y([d(T x , T x ] [d(T x , T x ] ). n n1 n n+1 Using the fact y(t) < t for each t > 0, 1b b d(T x , T x )  y([d(T x , T x )] [d(T x , T x )] ) n n n n+1 n1 n+1 1b b < [d(T x , T x )] [d(T x , T x )] . (5) n n1 n n+1 which implies: 1b 1b [d(T x , T x )] < [d(T x , T x )] . n+1 n n n1 Thus, d(T x , T x ) < d(T x , T x ) for all n  1. (6) n+1 n n n1 That is, the positive sequence fd(T x , T x )g is monotone decreasing, and consequently, there n+1 n exists c  0 such that lim d(T x , T x ) = c. From (6), we obtain: n!¥ n+1 n 1b b 1b b [d(T x , T x )] [d(T x , T x )]  [d(T x , T x )] [d(T x , T x )] n n n n n1 n+1 n1 n1 = d(T x , T x ). n n1 Therefore, with (5) together with the nondecreasing character of y, we get: 1b b d(T x , T x )  y([d(T x , T x )] [d(T x , T x )] ) n n n n+1 n1 n+1 y(d(T x , T x )). n n1 By repeating this argument, we get: 2 n d(T x , T x )  y(d(T x , T x ))  y (d(T x , T x ))    y (d(T x , T x )). (7) n n n+1 n1 n1 n2 1 0 Taking n ! ¥ in (7) and using the fact lim y (t) = 0 for each t > 0, we deduce that c = 0, n!¥ that is, lim d(T x , T x ) = 0. (8) n+1 n!¥ Axioms 2020, 9, 132 7 of 13 Then, fT x g is a b-Cauchy sequence. Suppose on the contrary that there exists an e > 0 and subsequences fT x g and fT x g of fT x g such that n is the smallest integer for which: m n n k k n > m > k, d(T x , T x )  e, and d(T x , T x ) < e. n m m k k n 1 k k k k Then, we have: d(gx , gx ) = d(T x , T x )  sd(T x , T x ) + sd(T x , T x ) n m n 1 m 1 n 1 m m m 1 k k k k k k k k se + sd(T x , T x ). m 1 k k Using (8) in the inequality above, we obtain: lim sup d(T x , T x ) = lim sup d(gx , gx )  se. (9) n 1 m 1 n m k k k k k!¥ k!¥ Putting x = x and y = x in (4), we have: n m k k a b 1ab e  d(T x , T x )  y([d(gx , gx )] [d(gx , T x )] [d(gx , T x )] ) n m n m n n m m k k k k k k k k a b 1ab = y([d(T x , T x )] [d(T x , T x )] [d(T x , T x )] ). (10) n 1 m 1 n 1 n m 1 m k k k k k k Taking the upper limit as k ! ¥ in (10) and using (8) and (9) and the property of y, we get: e  lim sup d(T x , T x )  y(0) = 0, n m k k k!¥ which implies that e = 0, a contradiction with e > 0. We deduce that fT x g is a b-Cauchy sequence, and consequently, fgx g is also a b-Cauchy sequence. Let z 2 X such that, lim d(T x , z) = lim d(gx , z) = 0. n+1 n!¥ n!¥ Since z 2 gX, there exists u 2 X such that z = gu. We claim that u is a coincidence point of g and T. For this, if we assume that gu 6= Tu, we obtain: a b 1ab d(T x , Tu)  y([d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] ) n n n n a b 1ab < [d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] . n n n At the limit as n ! ¥ and using Lemma 1, we get: a b 1ab d(z, Tu)  lim inf d(T x , Tu)  lim sup[d(gx , gu)] [d(gx , T x )] [d(gu, Tu)] n n n n n!¥ n!¥ a 2 b 1ab [sd(z, gu)] [s d(z, z)] [d(gu, Tu)] = 0, which is a contradiction, which implies that: Tu = z = gu. Then, u is a coincidence point in X of T and g. Example 5. Let X = [0, +¥) and d : X X ! [0, ¥) be defined by: (x + y) , if x 6= y; d(x, y) = 0, if x = y. Axioms 2020, 9, 132 8 of 13 Then, (X, d) is a complete b-metric space. Define two self-mappings T and g on X by g(x) = x ; for all x 2 X and: 1, if x 2 [0, 2]; T x = , if x 2 (2, +¥). T is a g-interpolative Ciric–Reich–Rus-type ´ contraction for a = 0, 7, b = 0, 4, and: 3 2 89 t , if t 2 [0, ]; 20 20 y(t) = t+1 3 1 89 , if t 2 ( , +¥). 3 +1 20 For this, we discuss the following cases: Case 1. If x, y 2 [0, 2] or x = y for all x 2 [0, +¥). It is obvious. Case 2. If x, y 2 (2, +¥) and x 6= y. We have: 1 1 d(T x, Ty) = ( + )  1. x y Using the property of y, we get: 1 1 a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + ) (y + ) ) x y 2a 2(1a) y(8 .( ) )  1. Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Case 3. If x 2 [0, 2]nf1g and y 2 (2, +¥). We have: 1 3 9 2 2 d(T x, Ty) = (1 + )  ( ) = , y 2 4 and: a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + 1) (y + ) ) 9 9 2a 2b 2(1ab) y(4 .1 .( ) )  . 2 4 Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Case 4. If x 2 (2, +¥) and y 2 [0, 2]nf1g. We have: 1 9 d(T x, Ty) = (1 + )  , x 4 and: a b 1ab 2 2 2a 2 2b 2 2(1ab) y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ) = y((x + y ) (x + ) (y + 1) ) 9 9 2a 2b 2(1ab) y(4 .( ) .1 )  . 2 4 Axioms 2020, 9, 132 9 of 13 Therefore, a b 1ab d(T x, Ty)  y([d(gx, gy)] [d(gx, T x)] [d(gy, Ty)] ). Then, it is clear that g, T satisfies (4) for all u, v 2 Xnf1g. Moreover, one is a coincidence point of g and T. Example 6. Let the set X = fa, b, q, rg and a function d : X X ! [0, ¥) be defined as follows: d(x, y) a b q r a 0 1 16 b 1 0 9 q 16 9 0 49 25 1 r 0 4 4 4 By a simple calculation, one can verify that the function d is a b-metric, for s = 2. We define the self-mappings g, T on X, as: ! ! a b q r a b q r g : , T : . a r q q q r r q For a = 0, 3; b = 0, 8; and y(t) = for all t 2 [0, ¥). 1+t It is clear that g, T satisfies (4) for all u, v 2 Xnfb, rg. Moreover, b and r are two coincidence points of g and T. Definition 5. Let (X, d) is a metric space. A self-mapping T: X ! X is said to be an interpolative weakly contractive mapping if there exists a constant a 2 (0, 1) such that: a 1a a 1a z(d(T x, Ty))  z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ), (11) for all x, y 2 Xn Fix(T), where Fix(T) = fa 2 XjTa = ag, z: [0, ¥) ! [0, ¥) is a continuous monotone nondecreasing function with z(t) = 0 if and only if t = 0, j: [0, ¥) ! [0, ¥) is a lower semi-continuous function with j(t) = 0 if and only if t = 0. Theorem 3. Let (X, d) be a complete metric space. If T : X ! X is a interpolative weakly contractive mapping, then T has a fixed point. Proof. For any x 2 X, we define a sequence fx g by x = x and x = T x , n = 0, 1, 2, . . . 0 n 0 n+1 n If there exists n 2 N such that x = x , then x is clearly a fixed point in X. Otherwise, x 6= x n n n 0 n +1 n+1 0 0 0 for each n  0. Substituting x = x and y = x in (11), we obtain that: n n1 a 1a a 1a z(d(x , x ))  z([d(x , x )] [d(x , x )] ) j([d(x , x )] [d(x , x )] ) n+1 n n n+1 n1 n n n+1 n1 n a 1a z([d(x , x )] [d(x , x )] ). (12) n n+1 n1 n Using property of function z, we get: a 1a d(x , x )  [d(x , x )] [d(x , x )] . n+1 n n n+1 n1 n We derive: 1a 1a [d(x , x )]  [d(x , x )] . n+1 n n1 n Therefore: d(x , x )  d(x , x ), for all n  1. n+1 n n1 n Axioms 2020, 9, 132 10 of 13 It follows that the positive sequence fd(x , x )g is decreasing. Eventually, there exists c  0 n+1 n such that lim d(x , x ) = c. n n n+1 Taking n ! ¥ in the inequality (12), we obtain: z(c)  z(c) j(c). We deduce that c = 0. Hence: lim d(x , x ) = 0. (13) n+1 n Therefore, fx g is a Cauchy sequence. Suppose it is not. Then, there exists a real number e > 0, for any k 2 N,9m  n  k such that: k k d(x , x )  e. (14) m n k k Putting x = x and y = x in (11) and using (14), we get: n 1 m 1 k k a 1a a 1a z(e)  z(d(x , x ))  z([d(x , x )] [d(x , x )] ) j([d(x , x )] [d(x , x )] ). m n m n m n m 1 n 1 m 1 n 1 k k k k k k k k k k Letting k ! ¥ and using (13), we conclude: z(e)  z(0) j(0) = 0, which is contradiction with e > 0; thus, fx g is a Cauchy sequence; since (X, d) is complete, we obtain z 2 X such that lim d(x , z) = 0, and assuming that Tz 6= z, we have: n n a 1a a 1a z(d(x , Tz))  z([d(x , x )] [d(z, Tz)] ) j([d(x , x )] [d(z, Tz)] ) for all n. n+1 n n+1 n n+1 Letting n ! ¥, we get: a 1a a 1a z(d(z, Tz))  z([d(z, z)] [d(z, Tz)] ) j([d(z, z)] [d(z, Tz)] ) = z(0) j(0) = 0, which is a contradiction; thus, Tz = z. Example 7. Let the set X = [0, 3] and a function d : X X ! [0, ¥) be defined as follows: 0, if x = y; d(x, y) = 3, if x, y 2 [0, 1) and x 6= y; 2, otherwise. Then, (X, d) is a complete metric space. Let T: X ! X be defined as: 0, if x 2 [0, 1); T x = 1, if x 2 [1, 3]. 2 1 For z(t) = t , j(t) = t for all t 2 [0, +¥) and a = 0, 6. We discuss the following cases. Case 1. If x = y or x, y 2 (0, 1), or x, y 2 (1, 3] with x 6= y. It is obvious. Case 2. If x 2 (0, 1) and y 2 (1, 3]. We have: z(d(T x, Ty)) = z(d(0, 1)) = z(2) = 4, Axioms 2020, 9, 132 11 of 13 and: a 1a a 1a a [d(x, T x)] [d(y, Ty)] = [d(x, 0)] [d(y, 1)] = 2.( ) . Therefore: 3 3 a 1a a 1a a a z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ) = ( ) [4.( ) 1]  4 = z(2) = z(d(T x, Ty)). 2 2 Case 3. If x 2 (1, 3] and y 2 (0, 1). We have: z(d(T x, Ty)) = z(d(1, 0)) = z(2) = 4, and: a 1a a 1a a [d(x, T x)] [d(y, Ty)] = [d(x, 1)] [d(y, 0)] = 3.( ) . Therefore, 2 2 3 a 1a a 1a a a z([d(x, T x)] [d(y, Ty)] ) j([d(x, T x)] [d(y, Ty)] ) = ( ) [9.( ) ]  4 = z(2) = z(d(T x, Ty)). 3 3 2 Thus, a 1a a 1a z(d(Tu, Tv))  z([d(u, Tu)] [d(v, Tv)] ) j([d(u, Tu)] [d(v, Tv)] ), for all u, v 2 Xnf0, 1g. Then, T has two fixed points, which are zero and one. Example 8. Let X = fa, b, r, sg be endowed with the metric defined by the following table of values: d(x, y) a b r s a 0 1 4 1 b 1 0 5 2 r 4 5 0 3 s 1 2 3 0 Consider the self-mapping T on X as: a b r s T : . a s a s t t For z(t) = e 1 and j(t) = 2 1 for all t 2 [0, ¥); a =0, 3. We have: a 1a a 1a z(d(Tu, Tv))  z([d(u, Tu)] [d(v, Tv)] ) j([d(u, Tu)] [d(v, Tv)] ), for all u, v 2 Xnfa, sg. Then, T has two fixed points, which are a and s. If z(t) = t in Theorem (3), then we have the following corollary: Axioms 2020, 9, 132 12 of 13 Corollary 2. Let (X, d) be a complete metric space and T : X ! X a self-mapping on X. If there exists a constant a 2 (0, 1) such that: a 1a a 1a d(T x, Ty)  [d(x, T x)] [d(y, Ty)] j([d(x, T x)] [d(y, Ty)] ), for all x, y 2 X and x 6= T x, y 6= Ty. j : [0, ¥) ! [0, ¥) is a lower semi-continuous function with j(t) = 0 if and only if t = 0. Then, T has a fixed point. Remark 2. In Corollary 2, if we take j(t) = (1 l)t for a constant l 2 (0, 1), then the result of Theorem [8] is obtained. Author Contributions: All authors contributed equally and significantly to the writing of this article. All authors read and approved the final manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 1922, 3, 133–181. [CrossRef] 2. Petrusel, ¸ A.; Petrusel, ¸ G. On Reich’s strict fixed point theorem for multi-valued operators in complete metric spaces. J. Nonlinear Var. Anal. 2018, 2, 103–112. 3. Suzuki, T. Edelstein’s fixed point theorem in semimetric spaces. J. Nonlinear Var. Anal. 2018, 2, 165–175. 4. Park, S. Some general fixed point theorems on topological vector spaces. Appl. Set-Valued Anal. Optim. 2019, 1, 19–28. 5. Ðoric, ´ D. Common fixed point for generalized (y, f)-weak contractions. Appl. Math. Lett. 2009, 22, 1896–1900. [CrossRef] 6. Dutta, P.; Choudhury, B.S. A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 2008, 406368. [CrossRef] 7. Kannan, R. Some results on fixed points. Bull. Cal. Math. Soc. 1968, 60, 71–76. 8. Karapinar, E. Revisiting the Kannan type contractions via interpolation. Adv. Theory Nonlinear Anal. Appl. 2018, 2, 85–87. [CrossRef] 9. Aydi, H.; Chen, C.M.; Karapınar, E. Interpolative Ciric–Reich–Rus ´ type contractions via the Branciari distance. Mathematics 2019, 7, 84. [CrossRef] 10. Aydi, H.; Karapinar, E.; Roldán López de Hierro, A.F. w-interpolative Ciric–Reich–Rus-type ´ contractions. Mathematics 2019, 7, 57. [CrossRef] 11. Debnath, P.; de La Sen, M. Fixed-points of interpolative Ciric-Reich–Rus-type ´ contractions in b-metric Spaces. Symmetry 2020, 12, 12. [CrossRef] 12. Gaba, Y.U.; Karapınar, E. A New Approach to the Interpolative Contractions. Axioms 2019, 8, 110. [CrossRef] 13. Karapinar, E.; Agarwal, R.; Aydi, H. Interpolative Reich–Rus–Ciric ´ type contractions on partial metric spaces. Mathematics 2018, 6, 256. [CrossRef] 14. Noorwali, M. Common fixed point for Kannan type contractions via interpolation. J. Math. Anal. 2018, 9, 92–94. 15. Karapınar, E.; Alqahtani, O.; Aydi, H. On interpolative Hardy–Rogers type contractions. Symmetry 2019, 11, 8. [CrossRef] 16. Bakhtin, I. The contraction mapping principle in quasimetric spaces. Func. Anal. Gos. Ped. Inst. Unianowsk 1989, 30, 26–37. 17. Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1, 5–11. 18. Boriceanu, M.; Bota, M.; Petrusel, ¸ A. Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 2010, 8, 367–377. [CrossRef] Axioms 2020, 9, 132 13 of 13 19. Aghajani, A.; Abbas, M.; Roshan, J. Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca 2014, 64, 941–960. [CrossRef] 20. Alqahtani, B.; Fulga, A.; Karapınar, E.; Özturk, A. Fisher-type fixed point results in b-metric spaces. Mathematics 2019, 7, 102. [CrossRef] Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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