Access the full text.

Sign up today, get DeepDyve free for 14 days.

Axioms
, Volume 9 (4) – Nov 16, 2020

/lp/multidisciplinary-digital-publishing-institute/fixed-points-of-g-interpolative-iri-reich-rus-type-contractions-in-b-EXxjUuM7RX

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

- Publisher
- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2020 MDPI (Basel, Switzerland) unless otherwise stated Disclaimer The statements, opinions and data contained in the journals are solely those of the individual authors and contributors and not of the publisher and the editor(s). MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Terms and Conditions Privacy Policy
- ISSN
- 2075-1680
- DOI
- 10.3390/axioms9040132
- Publisher site
- See Article on Publisher Site

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/).

Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Nov 16, 2020

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.