On Fixed Point Results for Matkowski Type of Mappings in G-Metric Spaces

Filomat 29:10 2015, 2301 2309 DOI 10.2298/FIL1510301G Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Fixed Point Results for Matowsi Type of Mappings in G-Metric Spaces Lj. Gajić a, M. Stojaović b a Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Serbia b Department of Mathematics, Faculty of Technical Sciences, University of Novi Sad, Serbia Abstract. The main result of this paper is a fixed point theorem for Matowsi type mapping with contractive iterate at a point in a class of G-metric spaces. Our result unifies, generalizes and complements some well nown results in metric and G-metric spaces. 1. Introduction and preliminaries In 1975 Matowsi introduced the following class of mappings: Definition 1.1. [9] Let T be a mapping on a metric space X, d. Then T is called a wea contraction if there exists a function γ from [0, to itself satisfying the following: i γ is nondecreasing, ii lim n γ n t = 0 for all t > 0, iii dtx, Ty γdx, y for all x, y X. In the same paper he proved the existence and uniqueness of a fixed point for such type of mappings. This result is significant because the concept of wea contraction of Matowsi type is independent of Meir- Keeler contraction [12], and it was generalized in different directions [10], [11], [17]. Matowsi generalized his own result proving a theorem of Segal- Guseman type [6]. Theorem 1.2. [10] Let X, d be a complete metric space, T : X X, α : [0, 5 [0, and γt = αt, t, t, 2t, 2t for t 0. Suppose that 1. α is nondecreasing with respect to each variable, 2. lim t t γt =, 3. lim t γ n t = 0, t 0, 2010 Mathematics Subject Classification. Primary 47H10 Keywords. wea contraction, fixed point, mapping with contractive iterate at a point, G-metric space Received: 20 August 2014; Accepted: 11 Novemvber 2014 Communicated by Dragana Cvetović-Ilić This wor is supported by Ministry of Science and Technological Development, Republic of Serbia Email addresses: gajic@dmi.uns.ac.rs Lj. Gajić, stojaovic@sbb.rs M. Stojaović

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2302 4. for every x X, there exists a positive integer n = nx such that for all y X dt n x, T n y αdx, y, dx, T n x, dx, T n y, dt n x, y, dt n y, y then T has a unique fixed point a X and for each x X, lim T x = a. The aim of this paper is to show that this result is valid in a more general class of spaces. On 1963. S. Gähler introduced 2-metric spaces, but other authors proved that there is no relation between two distance functions and there is no easy relationship between results obtained in the two settings. B. C. Dhage introduced a new concept of the measure of nearness between three or more objects. But topological structure of so called D-metric spaces was incorrect. Finally, Z. Mustafa and B. Sims [13] introduced correct definition of a generalized metric space as follows. Definition 1.3. [13] Let X be a nonempty set, and let G : X X X R + be a function satisfying the following properties G1 Gx, y, z = 0 if x = y = z; G2 0 < Gx, x, y, for all x, y X, with x y; G3 Gx, x, y Gx, y, z, for all x, y, z X, with z y; G4 Gx, y, z = Gx, z, y = Gy, z, x =..., symmetry in all three variables; G5 Gx, y, z Gx, a, a + Ga, y, z, for all x, y, z, a X. Then function G is called a generalized metric, abbreviated G-metric on X, and the pair X, G is called a G-metric space. Clearly these properties are satisfied when Gx, y, z is the perimeter of the triangle in with vertices x, y and z. R 2, moreover taing a in the interior of the triangle shows that G5 is the best possible. Example 1.1[13] Let X, d be an ordinary metric space, then X, d defines G-metrics on X by G s x, y, z = dx, y + dy, z + dx, z, G m x, y, z = max { dx, y, dy, z, dx, z }. Example 1.2[13] Let X = {a, b}. Define G on X X X by Ga, a, a = Gb, b, b = 0, Ga, a, b = 1, Ga, b, b = 2, and extend G to X X X by using the symmetry in the variables. Then it is clear the X, G is a G-metric space. The following useful properties of a G-metric are readily derived from the axioms. Proposition 1.4. [13] Let X, G be a G-metric space, then for any x, y, z and a from X it follows that: 1. if Gx, y, z = 0, then x = y = z, 2. Gx, y, z Gx, x, y + Gx, x, z, 3. Gx, x, y 2Gy, y, x, 4. Gx, y, z Gx, a, z + Ga, y, z, 5. Gx, y, z 2 3 Gx, y, a + Gx, a, z + Ga, y, z, 6. Gx, y, z Gx, a, a + Gy, a, a + Gz, a, a.

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2303 Definition 1.5. [13] Let X, G be a G-metric space, and let {x n } be a sequence of points of X. A point x X is said to be the limit of the sequence {x n } if lim Gx, x n, x m = 0, and one says that the sequence {x n } is G-convergent to x. n,m Proposition 1.6. [13] Let X, G be a G-metric space, then for a sequence {x n } X and a point x X the following are equivalent: 1. {x n } is G-convergent to x, 2. Gx n, x n, x 0 as n, 3. Gx n, x, x 0 as n. Definition 1.7. [13] Let X, G be a G-metric space, a sequence {x n } is called G-Cauchy if for every ɛ > 0, there is N N such that Gx n, x m, x l < ɛ, for all n, m, l N, that is, if Gx n, x m, x l 0 as n, m, l. Proposition 1.8. [13] In a G-metric space X, G, the following are equivalent: 1. the sequence {x n } is G-Cauchy, 2. for every ε > 0, there exists an n 0 N such that Gx n, x m, x m < ε, for all n, m n 0. A G-metric space X, G is G-complete or G is a complete G-metric, if every G-Cauchy sequence in X, G is G-convergent in X, G. Proposition 1.9. [13] Let X, G be a G-metric space, then the function Gx, y, z is jointly continuous in all three of its variables. Recently, Samet at all [15] and Jleli,Samet [7] observed that some fixed point theorems in context of G- metric space can be proved by simple transformation using related existing results in the setting of metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before see [14]. Very recently, Karapinar and Agarwal suggest new contraction conditions in G-metric space in a way that the techniques in [15], [7] are not applicable. In this approach [8], contraction conditions can not be expressed in two variables. So, in some cases, as it is noticed even in Jleli-Samet paper [7], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper. For more fixed point results for mappings defined in G-metric spaces, we refer the reader to [1], [2], [3], [4] [5], [14]. Definition 1.10. Let X be a nonempty set and a function δ : X X [0, satisfies the following properties: 1. δx, y = 0 if and only if x = y, 2. δx, y δx, z + δz, y for any points x, y, z X, Then the pair X, δ is called a quasi -metric space. The sequence {x n } X converges to x X, iff lim n δx n, x = lim n δx, x n = 0. 2. Main result Let α : [0, 5 [0, be a nondecreasing function with respect to each variable and let γt = αt, t, 2t, 3t, 3t for t 0. Following Matowsi [10], let Γ be the set of all functions γ : [0, [0, such that 1 lim γ n t = 0, t > 0; 2 lim t t γt =.

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2304 Simple examples of the function γ Γ are: γt = q t, q 0, 1, γt = t 1+t, γt = ln1 + t, t [0,. Remar 2.1. It is obvious that γ is nondecreasing function and that consequences of the condition lim t γ n t = 0, t 0, are: a γt < t, t > 0; b γ0 = 0. Definition 2.2. If X, G be a G-metric space, T : X X, and for every x X, there exists a positive integer n = nx such that for all y X GT nx x, T nx x, T nx y αgx, x, y, Gx, x, T nx y, Gx, T nx x, T nx x, Gy, T nx x, T nx x, Gy, y, T nx y, 2.1 then we say that T is a wea contraction in X. Lemma 2.3. If X, G is a G-metric space and T : X X is a wea contraction in X, then for every x X, the orbit {T x} is bounded. Proof. For any x X and any integer s, 0 s < nx we define the sequence u x, s = u = G x, x, Tx nx+s, = 0, 1, 2,... and the number hx, s = h = max { Gx, x, T nx x, Gx, T nx x, T nx x, Gx, x, T s x }. The property 2 of the function γ, implies that there exists a c, c > h, such that t γt > h, t > c. It is easy to see that u 0 < c. Next, we show that u j < c for all j = 0, 1, 2,.... The assumption that there exists a positive integer j such that u j c, but u i < c for i < j will lead to contradiction. Under the last assumption, G T nx x, T nx x, T j 1nx+s x h + u j 1 < 2u j, G T j 1nx+s x, T j 1nx+s x, T jnx+s x 2u j 1 + u j < 3u j. Now, since α is nondecreasing with respect to each variable and the mapping T is a wea contraction, we get u j = G x, x, T jnx+s x G x, x, T nx x + G T nx x, T nx x, T jnx+s x h + αu j, u j, u j, 2u j, 3u j h + γu j. The inequality u j γu j h contradicts to the choice of c. Hence, u j < c for all j = 0, 1,.... It completes the proof that for any fixed x X, sup Gx, x, T x = M <, meaning that the orbit {T x} is bounded. Lemma 2.4. If X, G is a G-metric space and T : X X is a wea contraction in X, then for every x 0 X, the sequence x +1 = T nx x, = 0, 1,..., is a Cauchy sequence.

Proof. It is easy to see that Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2305 x +j = T nx +j 1+...+nx x, for all, j N. Using the notation s 0 = nx +j 1 +... + nx, we get G x, x, x +j = G x, x, T s 0 x Putting we obtain = G T nx 1 x 1, T nx 1 x 1, T nx 1 T s 0 x 1. t s1 = max { Gx 1, x 1, T i x 1 : i {s 0, nx 1, s 0 + nx 1 } }, G T nx 1 x 1, T nx 1 x 1, T s 0 x 1 2ts1 + t s1 = 3t s1 and G T nx 1+s 0 x 1, T s 0 x 1, T s 0 x 1 ts1 + 2t s1 = 3t s1. The property that the mapping T is a wea contraction with respect to nondecreasing function α implies G x, x, T s 0 x αt s1, t s1, 2t s1, 3t s1, 3t s1 = γt s1. Repeating this procedure, one concludes that there exists an s j N, j = 1, 2,..., 1 such that The mapping γ is nondecreasing which yields G x j, x j, T s j x j γ Gx j 1, x j 1, T s j+1 x j 1. G x, x, x +j γ G x 0, x 0, T s x 0 γ M. Since γ Γ, we see that lim γ n M = 0. This is precisely the assertion of the lemma, that is, {x } is a Cauchy sequence. Theorem 2.5. If X, G is a complete G-metric space and T : X X is a wea contraction in X, then T has a unique fixed point a X, for every x X, lim T x = a and T na is continuous at a. Proof. By completeness of X, the Cauchy sequence {x } defined in the above lemma is convergent, i.e. lim x = a X. The proof falls naturally into six consecutive parts in which we prove that: 1. T na a = a, 2. a is a unique fixed point of T na, 3. Ta = a, 4. a is a unique fixed point of T, 5. for every x X, lim T x = a, 6. T na is continuous at a 1. Applying the same reasoning as in the last lemma, we can show that lim Gx, x, T na x = 0. It means that for every ε > 0 there exists a 1 ε N such that for 1 ε, Gx, x, T na x < 1 ε γε. 8

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2306 Since lim Gx, x, a = 0, for every ε > 0 there exists a 2 ε N such that for 2 ε, Ga, x, x < 1 ε γε 8 Next, we claim that T na a = a. Indeed, if we suppose the opposite, i.e. if we suppose that there exists an ε > 0 such that GT na a, T na a, a = ε, then for max{ 1 ε, 2 ε} ε = GT na a, T na a, a GT na a, T na a, T na x The next two relations +GT na x, T na x, x + Gx, x, a α Ga, a, x, Ga, a, T na x, Ga, T na a, T na a, Gx, T na a, T na a, Gx, x, T na x + 1 ε γε. 2 Ga, a, T na x Ga, a, x + Gx, x, T na x < 1 ε γε, 4 imply Gx, T na a, T na a Gx, a, a + Ga, T na a, T na a < 2ε, ε αε, ε, ε, 2ε, ε + 1 2 ε γε < 1 ε + γε < ε, 2 which is a contradiction, meaning that the assumption that T na a a is not correct. 2. The proof that a is the only fixed point of the mapping T na is once again by contradiction. If b X would be another fixed point of T na, then Ga, a, b = GT na a, T na a, T na b αga, a, b, Ga, a, b, 0, Gb, a, a, 0 γga, a, b < Ga, a, b, which establishes our claim that a = b. 3. The equality T a = T T na a = T na T a implies that T a = a. 4. From 2. and 3. we conclude that a is the only point from X such that T a = a. 5. Next we prove that lim T x = a, for each x X. For each x X and each s N, 0 s < na, define the sequence a = Ga, a, T na+s x, = 0, 1, 2,. If for some N, a > a 1, then a = GT na a, T na a, T na T 1na+s x α G a, a, T 1na+s x, G a, a, T na+s x, G a, T na a, T na a, G T na+s x, T na a, T na a, G T 1na+s x, T 1na+s x, T na+s x αa 1, a, 0, a 1, a + 2a 1 γa < a. By the last contradiction, we deduce that a a 1 for all N. Hence, a αa 1, a 1, a 1, a 1, 3a 1 γa 1 γ a 0.

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2307 Letting in the last relation, lim a = lim γ a 0 = 0, and consequently lim T x = a. 6. Finally, we finish the proof by showing the continuity of T na at a. We consider a sequence {y m } m X converging to a. For any m N G a, a, T na y m = GT na a, T na a, T na y m α G a, a, y m, G a, a, T na y m, G a, T na a, T na a, G y m, T na a, T na a, G y m, y m, T na y m = α G a, a, y m, G a, a, T na y m, G a, a, a, G y m, a, a, G y m, y m, T na y m α G a, a, y m, G a, a, T na y m, 0, G y m, a, a, G y m, y m, a + G a, a, T na y m. If lim m G a, a, T na y m 0, then there exists ε > 0 such that for some m1 N, G a, a, T na y m > ε for all m > m 1. On the other hand, since lim y m = a, for given ε there exist m 2 N and m 3 N such that and m > m 2 G a, a, y m < ε < G a, a, T na y m, m > m 3 G y m, y m, a < ε < G a, a, T na y m. Putting m 0 = max{m 1, m 2, m 3 }, for all m > m 0, we get G a, a, T na y m α G a, a, T na y m, G a, a, T na y m, 0, G a, a, T na y m, 2G a, a, T na y m γ G a, a, T na y m < G a, a, T na y m. Obviously, the assumption lim m G a, a, T na y m 0 induces the contradiction in the last relation. Hence, T na is continuous at a. Remar 2.6. In a symmetric G-metric space, one can put γt = αt, t, t, 2t, 2t as it is done in Matovsi paper [10], but Jleli-Samet technique can not be applied. In [7] it was shown that if X,G is a G-metric space, putting δx, y = Gx, y, y, X, δ is a quasi metric space δ is not symmetric. Simple replacement G with δ in 2.1, defines the wea contraction T in X, δ δt nx x, T nx y αδx, y, δx, T nx y, δt nx x, x, δt nx x, y, δy, T nx y. The following result is an immediate consequence of above definitions and relations. Theorem 2.7. If X, δ is a complete quasi-metric space such that δx, y 2δy, x for all x, y X and T : X X is a wea contraction in X, then T has a unique fixed point a X, for every x X, lim T x = a and T na is continuous at a. Proof. It is obvious that X, δ has the same topological structure as any G-metric space X, G where G satisfies equality δx, y = Gx, y, y. Now, repeating the proof of Theorem 2.5, we conclude that the assertion of theorem is true.

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2308 The next few corollaries are also a consequence of Theorem 2.5. Corollary 2.8. [5] Let X, G be a complete G-metric space, T : X X, γ Γ and for each x X there exists a positive integer n = nx such that G T nx x, T nx x, T nx y γ Gx, x, y, 2.2 for all y X. Then T has a unique fixed point a X. Moreover, for each x X, lim at a. T x = a and T na is continuous Corollary 2.9. [3] Let X, G be a complete G-metric space, T : X X and for each x X there exists a positive integer n = nx such that G T nx x, T nx x, T nx y q Gx, x, y for all y X and some q 0, 1. Then T has a unique fixed point a X. Moreover, for each x X, lim T na is continuous at a. T x = a and Proof. Function γt = q t, t [0,, belongs to Γ, so corollary is a consequence of Theorem 2.5. Remar 2.10. Taing γt = q t, 0 < q < 1, we obtain the fixed point result from [2] or [3], so Theorem 2.5. is also a generalization of Guseman fixed point result from [6]. Corollary 2.11. Let X, G be a complete G-metric space, T : X X, and for each x X there exists a positive integer n = nx such that G T nx x, T nx x, T nx y Gx, x, y 1 + Gx, x, y, for all y X. Then T has a unique fixed point a X. Moreover, for each x X, lim at a. Proof. Since the function γ = T x = a and T na is continuous t 1+t, t [0,, belongs to the family Γ, we can apply Theorem 2.5. If nx = 1 nx = m N, for each x X, we are going to prove that condition 3 can be omitted so, in that case we will have an improvement and another proof of Theorem 3.1 Corollary 3.2 from [16]. Proposition 2.12. Let X, G be a complete G-metric space, γ : [0, [0, be a nondecreasing function with lim γ t = 0 for t > 0. If T : X X satisfies G Tx, Tx, Ty γgx, x, y, for all x, y X, then T has a unique fixed point a X. Moreover, for each x X, lim at a. 2.3 T x = a and T na is continuous Proof. We prove, by induction, that for any x 0 = x, x X, the orbit {T x 0 } is bounded. By the properties of the function γ, lim γ Gx 0, x 0, x 1 = 0 and γt < t, t > 0, we get that there exists 0 N such that for all 0 Hence γ Gx 0, x 0, x 1 < 1 γ1. G Tx, Tx, Tx +1 < 1 γ1, 0. 2.4 For j > 0, j N, we claim that G x 0, x 0, x j 1. 2.5

Lj. Gajić, M. Stojaović / Filomat 29:10 2015, 2301 2309 2309 Indeed, for j = 0 + 1, the last relation is true by 2.4. Assuming inequality 2.5 holds for j = i, i > 0, we will prove it holds for j = i + 1. Really, letting j = i + 1, we get G x 0, x 0, x i+1 G x0, x 0, x 0 +1 + G x0 +1, x 0 +1, x i+1 Finally, for any N G x 0, x 0, x 0 +1 + γ G x0, x 0, x i 1 γ1 + γ1 = 1. G x 0, x 0, T x 0 G x0, x 0, x 0 + G x0, x 0, x G x 0, x 0, x 0 + max{1, G x0, x 0, x 1,, G x0, x 0, x 0 1 = M <. Now, the rest of the proof runs as in Theorem 2.1. References [1] M. Abbas, B.E. Rhoades, Common fixed point results for noncommuting mapping without continuity in generalized metric space, Applied Mathematics and Computation, 215 2009 262-269. [2] Lj. Gajić, Z. Lozanov-Crvenović, On mappings with contractive iterate at a point in generalized metric spaces, Fixed Point Theory and Applications. 2010, Article ID 458086, 16 pages, doi:10.1155/2010/458086. [3] LJ. Gajić, Z. Lozanov-Crvenović, A fixed point result for mappings with contractive iterate at a point in G-metric spaces, Filomat, 25 2 2011 53-58. [4] LJ. Gajić, M. Stojaović, On Ćirić generalization of mappings with a contractive iterate at a point in G-metric space, Applied Mathematics and Computation, 2019 2012 435-441. [5] LJ. Gajić, M. Stojaović, On mappings with ϕ-contractive iterate at a point on generalized metric spaces, Fixed Point Theory and Applications 2014, 2014:46 [6] L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proceedings of American Mathematical Society, 26 1970 615-618. [7] M. Jleli, B. Samet, Remars on G-metric spaces and fixed point theorems, Fixed Point Theory and Applications, 2012, 2012:210, doi:10.1186/1687-1812-2012-210. [8] E. Karapinar, R. Agarval, Further remars on G-metric spaces, Fixed Point Theory and Applications 2013, 2013:154 doi:10.1186/1687-1812-2013-154. [9] J. Matowsi, Integrable solutions of functional equations, Dis.Math., 127, Warsaw, 1975. [10] J. Matowsi, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 62 1977 344-348. [11] J. Matowsi, Fixed point theorems for contractive mappings in metric spaces, Casopis Pest. Mat., 105 1980 341-344. [12] A. Meir, E. Keeler, A theorem on contraction mappings, Journal of Mathematical Analysis and Application, 28 1969 326-329. [13] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear Convex Analysis, 72 2006 289-297. [14] Z. Mustafa, H. Obiedat, H. Awawdeh, Some fixed point theorem for mappings on complete G-metric spaces, Fixed Point Theory and Applications 2008 article ID 189870,12 pages. [15] B. Samet, C. Vetro, F. Vetro, Remars on G-Metric Spaces, International Journal of Analysis, Volume 2013 2013, Article ID 917158, 6 pages [16] W. Shatanawi, Fixed Point Theory for Contractive Mappings Satisfying Φ-Maps in G-Metric Spaces, Fixed Point Theor. Appl. 2010 Article ID 181650, 9 pages. [17] T. Suzui, C. Vetro, Three existence theorems for wea contraction of Matowsi type, International Journal of Mathematical Statistics, 6 2010 ISSN:0973-8347.