V. Popa, A.-M. Patriciu ( Vasile Alecsandri Univ. Bacău, Romania)

UDC 51791 V Popa, A-M Patriciu ( Vasile Alecsandri Univ Bacău, Romania) FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS SATISFYING AN IMPLICIT RELATION OF A NEW TYPE РЕЗУЛЬТАТИ ПРО НЕРУХОМУ ТОЧКУ НА ПОВНИХ G-МЕТРИЧНИХ ПРОСТОРАХ ДЛЯ ВIДОБРАЖЕНЬ, ЩО ЗАДОВОЛЬНЯЮТЬ НЕЯВНЕ СПIВВIДНОШЕННЯ НОВОГО ТИПУ We prove some general fixed-point theorems in complete G-metric space that generalize some recent results Доведено загальнi теореми про нерухому точку у повних G-метричних просторах, що узагальнюють деякi результати, отриманi нещодавно 1 Introduction In, 4] Dhage introduced a new class of generalized metric space, named D-metric space Mustafa and Sims 7, 8] proved that most of the claims concerning the fundamental topological structures on D-metric spaces are incorrect and introduced appropriate notion of generalized metric space, named G-metric space In fact, Mustafa, Sims and other authors, 9 11] studied many fixed-point results for self mappings in G-metric spaces under certain conditions Quite recently 1], Mustafa et al obtained new results for mappings in G-metric spaces In 1, 14], Popa initiated the study of fixed points in metric spaces for mappings satisfying an implicit relation Let T be a self mapping of a metric space (X, d) We denote by Fix (T ) the set of all fixed points of T T is said to satisfy property (P ) if Fix (T ) = Fix (T n ) for each n N An interesting fact about mappings satisfying property (P ) is that they have not nontrivial periodic points Papers dealing with property (P ) are, between others,, 1 15] The purpose of this paper is to prove a general fixed-point theorem in complete G-metric space which generalize the results from 1, 10 1] for mappings satisfying a new form of implicit relation In the last part of this paper is proved a general theorem for mappings in G-metric space satisfying property (P ), which generalize some results from 1] Preliminaries Definition 1 8] Let X be a nonempty set and G: X R + be a function satisfying the following properties: (G 1 ) G(x, y, z) = 0 if x = y = z; (G ) 0 < G(x, x, y) for all x, y X with x y; (G ) G(x, x, y) G(x, y, z) for all x, y, z X with z y; (G 4 ) G(x, y, z) = G(x, z, y) = G(y, z, x) = (symmetry in all three variables); (G 5 ) G(x, y, z) G(x, a, a) + G(a, y, z) for all x, y, z, a X (rectangle inequality) Then the function G is called a G-metric and the pair (X, G) is called a G-metric space Note that if G(x, y, z) = 0 then x = y = z 8] Lemma 1 8] G(x, y, y) G(x, x, y) for all x, y X Definition 8] Let (X, G) be a metric space A sequence (x n ) in X is said to be: a) G-convergent to x X if for any ε > 0 there exists k N such that G(x, x n, x m ) < ε for all m, n k; c V POPA, A-M PATRICIU, 01 814 ISSN 107-190 Укр мат журн, 01, т 65, 6

FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS 815 b) G-Cauchy if for ε > 0, there exists k N such that for all n, m, p k, G(x n, x m, x p ) < ε that is G(x n, x m, x p ) 0 as m, n, p A G-metric space is said to be G-complete if every G-Cauchy sequence in X is G-convergent Lemma 8] Let (X, G) be a G-metric space Then, the following properties are equivalent: 1) (x n ) is G-convergent to x; ) G(x, x n, x n ) 0 as n ; ) G(x n, x, x) 0 as n Lemma 8] Let (X, G) be a G-metric space Then the following properties are equivalent: 1) The sequence (x n ) is G-Cauchy ) For every ε > 0, there exists k N such that G(x n, x m, x m ) < ε for n, m > k Definition 8] Let (X, G) and (X, G ) be two G-metric spaces and f : (X, G) (X, G ) Then, f is said to be G-continuous at x X if for ε > 0, there exists δ > 0 such that for all x, y X and G(a, x, y) < δ, then G (fa, fx, fy) < ε f is G-continuous if it is G-continuous at each a X Lemma 4 8] Let (X, G) and (X, G ) be two G-metric spaces Then, a function f : (X, G) (X, G ) is G-continuous at a point x X if and only if f is sequentially continuous, that is, whenever (x n ) is G-convergent to x we have that f(x n ) is G-convergent to fx Lemma 5 8] Let (X, G) be a G-metric space Then, the function G(x, y, z) is continuous in all three of its variables Quite recently, the following theorem is proved in 1] Theorem 1 Let (X, G) be a complete G-metric space and T : X X be a mapping which satisfies the following condition, for all x, y X G(T x, T y, T y) maxag(x, y, y), bg(x, T x, T x) + G(y, T y, T y)], bg(x, T y, T y) + G(y, T y, T y) + G(y, T x, T x)], (1) where a 0, 1) and b 0, 1 ) Then T has a unique fixed point The purpose of this paper is to prove a general fixed point theorem in G-metric space for mappings satisfying a new type of implicit relation which generalize Theorem 1 and other results from 1,, 10 1] Implicit relations Definition 1 Let F u be the set of all continuous functions F (t 1,, t 6 ): R 6 + R such that (F 1 ) F is nonincreasing in variables t 5 and t 6 ; (F ) there exists h 0, 1) such that for each u, v 0 and F (u, v, v, u, u + v, 0) 0, then u hv; (F ) F (t, t, 0, 0, t, t) > 0 t > 0 Example 1 F (t 1,, t 6 ) = t 1 maxat, b(t + t 4 ), b(t 4 + t 5 + t 6 ), where a 0, 1) and b 0, 1 ) (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u maxav, b(v + u) 0 If u > v, then u1 maxa, b] 0, a contradiction Hence u v, which implies u hv, where h = = maxa, b < 1 (F ) F (t, t, 0, 0, t, t) = t(1 maxa, b) > 0 t > 0 ISSN 107-190 Укр мат журн, 01, т 65, 6

816 V POPA, A-M PATRICIU Example F (t 1,, t 6 ) = t 1 at b(t +t 4 ) c(t 5 +t 6 ), where a, b, c 0, a+b+c < 1 and a + c < 1 (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u av b(v + u) c(u + v) 0 Then u hv, where h = a + b + c 1 b c < 1 (F ) F (t, t, 0, 0, t, t) = t1 (a + c)] > 0 t > 0 Example F (t 1,, t 6 ) = t 1 at b maxt, t 4 c maxt 5, t 6, where a, b, c 0, a + b + c < 1 (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u av b maxu, v c(u + v) 0 If u > v, then u1 (a + b + c)] 0, a contradiction Hence, u v which implies u hv, where h = a + b + c < 1 1 c (F ) F (t, t, 0, 0, t, t) = t1 (a + c)] > 0 t > 0 Example 4 F (t 1,, t 6 ) = t 1 k maxt, t,, t 6, where k 0, 1 ) (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u k(u + v) 0 which implies u hv, where h = k k 1 < 1 (F ) F (t, t, 0, 0, t, t) = t(1 k) > 0 t > 0 Example 5 F (t 1,, t 6 ) = t 1 at bt c maxt 4 +t 5, t 6, where a, b, c 0, a+b+c < < 1, a + 4c < 1 (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u av bv c(u + v) 0 Then u hv, where h = a + b + c < 1 1 c (F ) F (t, t, 0, 0, t, t) = t1 (a + 4c)] > 0 t > 0 Example 6 F (t 1,, t 6 ) = t 1 k max k 0, 1) t, t, t 4, t 4 + t 6, t 4 + t, t 5 + t 6 0, where (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u k max u, v, u, u + v, u + v 0 If u > v, then u(1 k) 0, a contradiction Hence, u v which implies u hv, where h = k < 1 (F ) F (t, t, 0, 0, t, t) = t(1 k) > 0 t > 0 Example 7 F (t 1,, t 6 ) = t 1 k max t, t, t 4, t 5 + t 6, where k 0, ) (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u k max u, v, u + v 0 If u > v, then u(1 k) 0, a contradiction Hence, u v which implies u hv, where h = k < 1 (F ) F (t, t, 0, 0, t, t) = t k max t, t = t 1 k ] > 0 t > 0 ISSN 107-190 Укр мат журн, 01, т 65, 6

FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS 817 Example 8 F (t 1,, t 6 ) = t 1 t 1(at + bt + ct 4 ) dt 5 t 6, where a, b, c 0, a + b + c < 1, a + d < 1 (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u u(av + bv + cu) 0 If u > 0, then u av bv cu 0 which implies u hv, where h = a + b < 1 If u = 0, then u hv 1 c (F ) F (t, t, 0, 0, t, t) = t 1 (a + d)] > 0 t > 0 Example 9 F (t 1,, t 6 ) = t 1 k max t, t + t 4, t 5 + t 6, where k 0, ) (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u k max v, u + v 0 If u > 0, then u(1 k) 0, a contradiction Hence u v which implies u hv, where h = k < 1 (F ) F (t, t, 0, 0, t, t) = t 1 k ] > 0 t > 0 Example 10 F (t 1,, t 6 ) = t 1 k max t, t t 4, t 5 t 6, where k 0, ) (F ) Let u, v 0 be and F (u, v, v, u, u + v, 0) = u k max v, uv 0 If u > v, then u(1 k) 0, a contradiction Hence, u v which implies u hv, where 0 h = k < 1 (F ) F (t, t, 0, 0, t, t) = t(1 k) > 0 t > 0 4 Main results Theorem 41 Let (X, G) be a G-metric space and T : (X, G) (X, G) be a mapping such that F (G(T x, T y, T y), G(x, y, y), G(x, T x, T x), G(y, T y, T y), G(x, T y, T y), G(y, T x, T x)) 0 (41) for all x, y X, where F satisfies property (F ) Then T has at most a fixed point Proof Suppose that T has two distinct fixed points u and v Then by (41) we have successively F (G(T u, T v, T v), G(u, v, v), G(u, T u, T u), G(v, T v, T v), G(u, T v, T v), G(v, T u, T u)) 0, F (G(u, v, v), G(u, v, v), 0, 0, G(u, v, v), G(v, u, u)) 0 By Lemma 1 G(v, u, u) G(u, v, v) Since F is nonincreasing in variable t 6 we obtain F (G(u, v, v), G(u, v, v), 0, 0, G(u, v, v), G(u, v, v)) 0, a contradiction of (F ) Hence u = v Theorem 41 is proved Theorem 4 Let (X, G) be a complete G-metric space and T : (X, G) (X, G) satisfying inequality (41) for all x, y X, where F F u Then T has a unique fixed point Proof Let x 0 X be an arbitrary point in X We define x n = T x n 1, n = 1,, Then by (41) we have successively F (G(T x n 1, T x n, T x n ), G(x n 1, x n, x n ), G(x n 1, T x n 1, T x n 1 ), ISSN 107-190 Укр мат журн, 01, т 65, 6

818 V POPA, A-M PATRICIU G(x n, T x n, T x n ), G(x n 1, T x n, T x n ), G(x n, T x n 1, T x n 1 )) 0, F (G(x n, x n+1, x n+1 ), G(x n 1, x n, x n ), G(x n 1, x n, x n ), G(x n, x n+1, x n+1 ), G(x n 1, x n+1, x n+1 ), 0) 0 By (G 5 ), G(x n 1, x n+1, x n+1 ) G(x n 1, x n, x n ) + G(x n, x n+1, x n+1 ) Since F is nonincreasing in variable t 5 we obtain which implies by (F ) that Then F (G(x n, x n+1, x n+1 ), G(x n 1, x n, x n ), G(x n 1, x n, x n ), G(x n, x n+1, x n+1 ), G(x n 1, x n, x n ) + G(x n, x n+1, x n+1, 0) 0 G(x n, x n+1, x n+1 ) hg(x n 1, x n, x n ) G(x n, x n+1, x n+1 ) hg(x n 1, x n, x n ) h n G(x 0, x 1, x 1 ) Moreover, for all m, n N, m > n, we have repeated use the rectangle inequality G(x n, x m, x m ) G(x n, x n+1, x n+1 ) + G(x n+1, x n+, x n+ ) + + G(x m 1, x m, x m ) (h n + h n+1 + + h m 1 )G(x 0, x 1, x 1 ) hn 1 h G(x 0, x 1, x 1 ), which implies lim n,m G(x n, x m, x m ) = 0 Hence, (x n ) is a G-Cauchy sequence Since (X, G) is G-complete, there exists u X such that lim n x n = u We prove that u = T u By (F 1 ) we have successively F (G(T x n 1, T u, T u), G(x n 1, u, u), G(x n 1, T x n 1, T x n 1 ), G(u, T u, T u), G(x n 1, T u, T u), G(u, T x n 1, T x n 1 )) 0, F (G(x n, T u, T u), G(x n 1, u, u), G(x n 1, x n, x n ), G(u, T u, T u), G(x n 1, T u, T u), G(u, x n, x n )) 0 By continuity of F and G, letting n tend to infinity, we obtain F (G(u, T u, T u), 0, 0, G(u, T u, T u), G(u, T u, T u), 0) 0 By (F ) we obtain G(u, T u, T u) = 0, hence u = T u and u is a fixed point of T By Theorem 41 u is the unique fixed point of T Theorem 4 is proved Corollary 41 Theorem 1 Proof The proof follows from Theorem 4 and Example 1 ISSN 107-190 Укр мат журн, 01, т 65, 6

FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS 819 Corollary 4 (Theorem 11]) Let (X, G) be a G-complete metric space and T : (X, G) (X, G) be a mapping satisfying the following condition: G(T x, T y, T z) αg(x, y, z) + βg(x, T x, T x) + G(y, T y, T y) + G(z, T z, T z)], (4) for all x, y, z X and 0 α + β < 1 Then T has a unique fixed point Proof By (4) for z = y we obtain G(T x, T y, T y) αg(x, y, y) + βg(x, T x, T x) + G(y, T y, T y)], for all x, y X By Theorem 4 and Example for α = a, β = b and c = 0 it follows that T has a unique fixed point Corollary 4 (Theorem 11]) Let (X, G) be a G-complete metric space and T : (X, G) (X, G) be a mapping satisfying the condition G(T x, T y, T z) αg(x, y, z) + β maxg(x, T x, T x), G(y, T y, T y), G(z, T z, T z), (4) for all x, y, z X and 0 α + β < 1 Then T has a unique fixed point Proof By (4) for z = y we obtain G(T x, T y, T y) αg(x, y, y) + β maxg(x, T x, T x), G(y, T y, T y), for all x, y X By Theorem 4 and Example for α = a, β = b and c = 0 it follows that T has a unique fixed point Corollary 44 (Theorem 1 10]) Let (X, G) be a G-complete metric space and T : (X, G) (X, G) be a mapping satisfying the condition for all x, y, z X, where k G(T x, T y, T z) k maxg(x, y, z), G(x, T x, T x), G(y, T y, T y), G(y, T z, T z), G(x, T y, T y), G(y, T z, T z), G(z, T x, T x), Proof By (44) for z = y we obtain 0, 1 ) Then T has a unique fixed point (44) G(T x, T y, T y) k maxg(x, y, y), G(x, T x, T x), G(y, T y, T y), G(x, T y, T y), G(y, T x, T x) By Theorem 4 and Example 4, T has a unique fixed point Corollary 45 Let (X, G) be a G-complete metric space and T : (X, G) (X, G) be a mapping which satisfy the following inequality for all x, y X, where k G(T x, T y, T y) k maxg(y, T y, T y) + G(x, T y, T y), G(y, T x, T x), (45) 0, 1 ) Then T has a unique fixed point Proof By Theorem 4 and Example 5 for a = b = 0 and c = k, T has a unique fixed point Remark 41 In Theorem 8 10], k 0, 1 ) ISSN 107-190 Укр мат журн, 01, т 65, 6

80 V POPA, A-M PATRICIU Corollary 46 Let (X, G) be a G-metric space and T : (X, G) (X, G) be a mapping satisfying the following inequality for all x, y, z X, G(T x, T y, T z) h max G(x, y, z), G(x, T x, T x), G(y, T y, T y), G(z, T z, T z), G(y, T x, T x) + G(y, T y, T y) + G(y, T z, T z) G(x, T x, T x) + G(y, T y, T y) + G(z, T z, T z),, (46) where k 0, 1) Then T has a unique fixed point Proof If y = z, by (46) we obtain that G(T x, T y, T y) h max G(x, y, y), G(x, T x, T x), G(y, T y, T y), G(y, T x, T x) + G(y, T y, T y), h max G(x, y, y), G(x, T x, T x), G(y, T y, T y), G(x, T x, T x) + G(y, T y, T y), for all x, y X By Theorem 4 and Example 6, T has a unique fixed point Remark 4 G(x, T x, T x) + G(y, T y, T y) G(y, T x, T x) + G(y, T y, T y), G(x, T y, T y) + G(y, T x, T x), Corollary 46 is a generalization of Theorem 6 1], where k 0, 1 ) Remark 4 By Theorem 4 and Examples 7 10 we obtain new results 5 Property (P ) in G-metric spaces Theorem 51 Under the conditions of Theorem 4, T has property (P ) Proof By Theorem 4, T has a fixed point Therefore, Fix (T n ) for each n N Fix n > 1 and assume that p Fix (T n ) We prove that p Fix (T ) Using (41) we have F (G(T n p, T n+1 p, T n+1 p), G(T n 1 p, T n p, T n p), G(T n 1 p, T n p, T n p), G(T n p, T n+1 p, T n+1 p), By rectangle inequality G(T n 1 p, T n+1 p, T n+1 p), G(T n p, T n p, T n p)) 0 G(T n 1 p, T n+1 p, T n+1 p) G(T n 1 p, T n p, T n p) + G(T n p, T n+1 p, T n+1 p) By (F 1 ) we obtain F (G(T n p, T n+1 p, T n+1 p), G(T n 1 p, T n p, T n p), G(T n 1 p, T n p, T n p), G(T n p, T n+1 p, T n+1 p), G(T n 1 p, T n p, T n p) + G(T n p, T n+1 p, T n+1 p), 0) 0 ISSN 107-190 Укр мат журн, 01, т 65, 6

FIXED-POINT RESULTS ON COMPLETE G-METRIC SPACES FOR MAPPINGS 81 By (F ) we obtain G(T n p, T n+1 p, T n+1 p) hg(t n 1 p, T n p, T n p) h n G(p, T p, T p) Since p T n p, then Therefore G(p, T p, T p) = G(T n p, T n+1 p, T n+1 p) G(p, T p, T p) h n G(p, T p, T p) which implies G(p, T p, T p) = 0, ie, p = T p and T has property (P ) Theorem 51 is proved Corollary 51 In the condition of Corollary 46, T has property (P ) Remark 51 Corollary 51 is a generalization of the results from Theorem 6 1] Corollary 5 In the condition of Corollary 44 with k 0, 1 property (P ) Remark 5 We obtain other new results from Examples 1 10 ), instead k 0, 1), T has 1 Abbas M, Nazir T, Radanović S Some periodic point results in generalized metric spaces // Appl Math and Comput 010 17 P 4094 4099 Chung R, Kasian T, Rasie A, Rhoades B E Property (P ) in G-metric spaces // Fixed Point Theory and Appl 010 Art ID 401684 1 p Dhage B C Generalized metric spaces and mappings with fixed point // Bull Calcutta Math Soc 199 84 P 9 6 4 Dhage B C Generalized metric spaces and topological structures I // An şti Univ Iaşi Ser mat 000 46, 1 P 4 5 Jeong G S More maps for which F (T ) = F (T n ) // Demonstr math 007 40, P 671 680 6 Jeong G S, Rhoades B E Maps for which F (T ) = F (T n ) // Fixed Point Theory and Appl Nova Sci Publ, 007 6 7 Mustafa Z, Sims B Some remarks concerning D-metric spaces // Int Conf Fixed Point Theory and Appl 004 P 184 198 8 Mustafa Z, Sims B A new approach to generalized metric spaces // J Nonlinear Convex Analysis 006 7 P 89 97 9 Mustafa Z, Obiedat H, Awawdeh F Some fixed point theorems for mappings on G-complete metric spaces // Fixed Point Theory and Appl 008 Article ID 189870 10 p 10 Mustafa Z, Sims B Fixed point theorems for contractive mappings in complete G-metric spaces // Fixed Point Theory and Appl 009 Article ID 917175 10 p 11 Mustafa Z, Obiedat H A fixed point theorem of Reich in G-metric spaces // Cubo A Math J 010 1 P 8 9 1 Mustafa Z, Khandagji M, Shatanawi W Fixed point results on complete G-metric spaces // Stud Sci Math hung 011 48, P 04 19 1 Popa V Fixed point theorems for implicit contractive mappings // Stud cerc St Ser Mat Univ Bacău 1997 7 P 19 1 14 Popa V Some fixed point theorems for compatible mappings satisfying implicit relations // Demonstr math 1999 P 157 16 15 Rhoades B E, Abbas M Maps satisfying contractive conditions of integral type for which F (T ) = F (T n ) // Int Pure and Appl Math 008 45, P 5 1 Received 1401, after revision 19111 ISSN 107-190 Укр мат журн, 01, т 65, 6