Common Fixed Point Results in Complex Valued Metric Spaces with (E.A) and (CLR) Properties

Transcription

1 Advances in Analysis, Vol., No. 4, October Common Fixed Point Results in Complex Valued Metric Spaces with (E.A) (CLR) Properties Mian Bahadur Zada 1, Muhammad Sarwar 1* Pa Sumati Kumari 1 Department of Mathematics University of Malak, Chakdara Dir(L), Khyber PakhtunKhwa, Pakistan. sarwarswati@gmail.com, mbz.math@gmail.com Department of Mathematics, National Institute of Technology, Andhra Pradesh, AP, India mumy141414@gmail.com Abstract In this article, we establish common fixed point theorems for two four self-mappings in the context of complex valued metric spaces. The derived results generalize extend some well known results in the literature. Examples are also given to demonstrate the validity of our results. Keywords: Complex valued metric spaces, coincident point, fixed point, weakly compatible mappings, (E.A) property, (CLR) property. 1 Introduction Preliminaries The most well-known fixed point result is Banach contraction principle [5], which is widely used result in fixed point theory. This principle has been studied generalized in different spaces such as, rectangular metric spaces, semi metric spaces, pseudo metric spaces, Quasi metric spaces, D-metric spaces cone metric spaces etc. various fixed point theorems were developed. Recently, Azam et al.[4] introduced the notion of complex valued metric space established fixed point theorems for mappings satisfying a rational type inequality. Verma Pathak [16] introduced the concept of (E.A) (CLR) properties in complex valued metric space to prove some common fixed point theorems for two pairs of weakly compatible mappings. Mohanta Maitra [10] defined the concept of ϕ-mapping proved a common fixed point theorem for a ϕ-pair in complex valued metric spaces. Subsequently, a number of fixed point theorems have been established some applications have been discussed (see, for instance, [], [], [9], [11], [1], [14], [15]) In this article, using the concept (E.A) (CLR) properties we establish common fixed point theorems for weakly compatible mappings in the frame work of complex valued metric space. We recall some basic definitions results that we need in our discussion which can be found in [4]. Let C be the set of complex numbers z 1, z C. Define a partial order on C as follows: z 1 z if only if Re(z 1 ) Re(z ) Im(z 1 ) Im(z ). Consequently, one can say that z 1 z if one of the following conditions is satisfied: 1. Re(z 1 ) = Re(z ), Im(z 1 ) < Im(z );. Re(z 1 ) < Re(z ), Im(z 1 ) = Im(z );. Re(z 1 ) < Re(z ), Im(z 1 ) < Im(z ); 4. Re(z 1 ) = Re(z ), Im(z 1 ) = Im(z ). In particular, we will write z 1 z if z 1 z one of (1),() (4) is satisfied we will write z 1 z if only () is satisfied. Note that i. a, b R a b az bz, for all z C; ii. 0 z 1 z z 1 < z ; iii. z 1 z z z z 1 z. Azam et al.[4] defined the complex-valued metric space (X, d) in the following way: Copyright 017 Isaac Scientific Publishing

2 48 Advances in Analysis, Vol., No. 4, October 017 Definition 1. Let X be a nonempty set d : X X C be the mapping satisfying the following axioms: 1. 0 d(x, y), for all x, y X d(x, y) = 0 if only if x = y;. d(x, y) = d(y, x), for all x, y X;. d(x, y) d(x, z) + d(z, y), for all x, y, z X. Then the pair (X, d) is called a complex valued metric space. Example 1.1. [7] Let X = C. Define the mapping d : X X C by d(x, y) = e ιm x y, where x, y X 0 m π. Then (X, d) is a complex valued metric space. Definition. [4] Let {x n } be a sequence in complex valued metric (X, d) x X. Then x is called the limit of {x n } if for every c C, with 0 c there is n 0 N such that d(x n, x) c for all n > n 0 we write lim n x n = x. Lemma 1. [4] Any sequence {x n } in complex valued metric space (X, d) converges to x if only if d(x n, x) 0 as n. Definition. [1] Let S T be self-maps of a non empty set X. Then 1. x X is a fixed point of T if T x = x.. x X is a coincidence point of S T if Sx = T x.. x X is a common fixed point of S T if Sx = T x = x. Jungck [8] initiated the concept of weakly compatible maps in ordinary metric spaces while Bhatt et al.[6] defined this notion in the complex valued metric space in the following way: Definition 4. Let X be a complex valued metric space. Then a pair of self-mappings S, T : X X are weakly compatible if they commute at their coincidence points, that is if there exists a point x X such that ST x = T Sx whenever Sx = T x. Aamri Moutawakil [1] introduced property (E.A) in ordinary metric spaces which generalized the notion of non-compatible mappings while Verma Pathak [16] initiated this concept in complex valued metric space in the following way: Definition 5. Let T, S : X X be two self-maps on a complex valued metric space (X, d). Then the pair (T, S) is said to satisfy property (E.A), if there exists a sequence {x n } in X such that lim T x n = lim Sx n = x for some x X. The notion of (CLR) property in ordinary metric spaces has been introduced by Sintunavarat Kumam [1], in a similarly mode Verma Pathak [16] defined this notion in a complex valued metric space in the following way: Definition 6. Let T, S : X X be two self-maps on a complex-valued metric space (X, d). Then T S are said to satisfy the common limit in the range of S property, denoted by (CLR S ) if there exists a sequence {x n } in X such that lim T x n = lim Sx n = Sx for some x X. Mohanta Maitra [10] introduced the concept of ϕ-mappings in complex valued matric space as: Definition 7. Let P = {z C : Re(z) 0 Im(z) 0}. A nondecreasing mapping ϕ : P P is called a ϕ-mapping if a. ϕ(0) = 0 0 ϕ(z) z for z P \ {0}, Copyright 017 Isaac Scientific Publishing

3 Advances in Analysis, Vol., No. 4, October b. ϕ(z) z for every z 0, c. lim n ϕn (z) = 0 for every z P \ {0}. Definition 8. [10] Let (X, d) be a complex valued metric space. The mappings T, S : X X are called ϕ-pair if there exists a ϕ-mapping satisfying d(t x, T y) ϕ(d(sx, Sy)), for all x, y X Theorem 1.1. [10] Let (X, d) be a complex valued metric space. Suppose that the mappings T, S : X X satisfy d(t x, T y) ϕ(d(sx, Sy)), for all x, y X, where ϕ is ϕ-mapping. If T (X) S(X), T S are weakly compatible T (X) or S(X) is complete, then T S have a unique common fixed point in X. Main Results Throughout this section C + = {z C : z 0} Ψ be the class of all lower semi-continuous functions ψ : C + C + such that ψ(0) = 0 ψ(z) z for every z 0. Theorem.1. Let (X, d) be a complex valued metric space K, L : X X be two self-mappings C 1. the pair (K, L) satisfies (CLR L ) property; C. for each x, y X, d(kx, Ky) ψ(d(lx, Ly)), where ψ Ψ. Then K L have a coincident point in X. Moreover, if the pair (K, L) is weakly compatible, then K L have a unique common fixed point in X. Proof. Assume that the pair (K, L) satisfies (CLR L ) property, then there exists a sequence {x n } in X such that lim Kx n = lim Lx n = Lu for some u X. (1) n + n + Now, we show that Ku = Lu, for this putting x = x n y = z in condition (C ) of Theorem.1, we have d(kx n, Ku) ψ(d(lx n, Lu)), taking lower limit as n +, it follows that using equation (1), one can write Thus lim d(kx n, Ku) lim ψ(d(lx n, Lu)) n + n + ( ) = ψ lim d(lx n, Lu), n + d(lu, Ku) ψ(d(lu, Lu)) = ψ(0) = 0. Ku = Lu = z(say). () Since the pair (K, L) is weakly compatible, so that Ku = Lu implies that LKu = KLu using equation (), we get Lz = Kz. () That is z is a coincident point of K L in X. Copyright 017 Isaac Scientific Publishing

4 50 Advances in Analysis, Vol., No. 4, October 017 Next, we show that z is a common fixed point of K L in X. For this, we show that Kz = z. If Kz z, then on using condition (C ) of Theorem.1 with x = u y = z, we get using equations () (), we have d(ku, Kz) ψ(d(lu, Lz)), d(z, Kz) ψ(d(z, Kz)) d(z, Kz), which is not possible, thus Kz = z hence from equation (), we get Kz = Lz = z. (4) That is z is a common fixed point of K L in X. Uniqueness: Assume that z z be another fixed point of K L, i.e. Kz = Lz = z. Then on using condition (C ) of Theorem., we have d(z, z ) = d(kz, Kz ) ψ(d(lz, Lz )) = ψ(d(z, z )) d(z, z ), which is a contradiction, thus z = z. Hence z is a unique common fixed point of K L in X. If we replace (CLR) property by (E.A) property in Theorem.1 we get the following result. Theorem.. Let (X, d) be a complex valued metric space K, L : X X be two self-mappings C 1. the pair (K, L) satisfies (E.A) property; C. for each x, y X, d(kx, Ky) ψ(d(lx, Ly)), where ψ Ψ. If L(X) is closed subspace of X, then K L have a coincident point in X. Moreover, if the pair (K, L) is weakly compatible, then K L have a unique common fixed point in X. Proof. The proof easily follows from Theorem.1, so we omit it. Theorem.. Let (X, d) be a complex valued metric space K, L, M, N : X X be four self-mappings (C 1 ) either the pair (K, M) satisfies (CLR K ) property or (L, N) satisfies (CLR L ) property; (C ) for each x, y X, d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), where ψ Ψ. If K(X) N(X) L(X) M(X), then each pair (K, M) (L, N) have coincident point in X. Moreover, if the pairs (K, M) (L, N) are weakly compatible, then K, L, M N have a unique common fixed point in X. Proof. Suppose that the pair (K, M) satisfies (CLR K ) property, then there exists a sequence {x n } in X such that lim Kx n = lim Mx n = Kx for some x X. (5) n + n + Since K(X) N(X), so there exists u X such that Kx = Nu thus from (5), we get lim Kx n = lim Mx n = Kx = Nu. (6) n + n + Copyright 017 Isaac Scientific Publishing

5 Advances in Analysis, Vol., No. 4, October We claim that Lu = Nu. If not, then on putting x = x n y = u in condition (C ) of Theorem., we get d(kx n, Lu) 1 ψ(d(kx n, Nu) + d(lu, Mx n )), taking lower limit as n +, we have lim d(kx n, Lu) 1 ( ) n + ψ lim d(kx n, Nu) + lim d(lu, Mx n), n + n + using equation (6), one can write d(nu, Lu) 1 ψ(d(lu, Nu)) 1 d(lu, Nu), which is not possible, thus Lu = Nu. But L(X) M(X), so there exists v X such that Lu = Mv. Therefore Lu = Nu = Mv = Kx. (7) Next, we show that Kv = Mv. Let Kv Mv, then on using condition (C ) of Theorem. with x = v y = u, from equation (7) it follows that d(kv, Lu) 1 ψ(d(kv, Nu) + d(lu, Mv)), d(kv, Mv) 1 ψ(d(kv, Mv)) 1 d(kv, Mv), which is contradiction, thus Kv = Mv. From equation (7), we get Kv = Lu = Nu = Mv = z(say). (8) Now, using the weak compatibility of the pairs (K, M), (L, N) equation (8), we have Kv = Mv KMv = MKv Kz = Mz. (9) Nu = Lu LNu = NLu Lz = Nz. (10) That is z is coincident point of each pair (K, M) (L, N) in X. In the next steps we prove that z is a common fixed point of K, L, M N in X. For this, we have to show that Kz = z. If Kz z, then on putting x = z y = u in condition (C ) of Theorem., one can get using equations (8) (9), we have d(kz, Lu) 1 ψ(d(kz, Nu) + d(lu, Mz)), d(kz, z) 1 ψ(d(kz, z) + d(z, Kz)) = 1 ψ(d(kz, z)) d(kz, z), which is not possible. Thus Kz = z hence from equation (9), we can write Kz = Mz = z. (11) Copyright 017 Isaac Scientific Publishing

6 5 Advances in Analysis, Vol., No. 4, October 017 Similarly, putting x = v, y = z in condition (C ) of Theorem. using equations (8) (10), we get Lz = Nz = z. (1) By combining equations (11) (1), one can write Kz = Lz = Nz = Mz = z. (1) That is z is a common fixed point of K, L, M N in X. Similarly, if we assume that the pair (L, N) satisfies (CLR L ) property. Then again we can show that z is a common fixed point of K, L, M N in X. Uniqueness: For the uniqueness of a common fixed point, we suppose that z z be another fixed point of K, L, M N, i.e. Kz = Lz = Mz = Nz = z. Then on using condition (C ) of Theorem., we get d(z, z ) = d(kz, Lz ) 1 ψ(d(kz, Nz ) + d(lz, Mz)) = 1 ψ(d(z, z)) d(z, z), which is contradiction, thus z = z. Hence z is a unique common fixed point of K, L, M N in X. From above theorem one can easily derive the following corollaries. Corollary.1. Let (X, d) be a complex valued metric space L, M, N : X X be three self-mappings C 1. either the pair (L, M) satisfies (CLR L ) property or (L, N) satisfies (CLR L ) property; C. for each x, y X, d(lx, Ly) 1 ψ(d(lx, Ny) + d(ly, Mx)), where ψ Ψ. If L(X) N(X) L(X) M(X), then each pair (L, M) (L, N) have coincident point in X. Moreover, if the pairs (L, M) (L, N) are weakly compatible, then L, M N have a unique common fixed point in X. Corollary.. Let (X, d) be a complex valued metric space K, M : X X be two self-mappings C 1. the pair (K, M) satisfies (CLR K ) property; C. for each x, y X, d(kx, Ky) 1 ψ(d(kx, My) + d(ky, Mx)), where ψ Ψ. Then K M have a coincident point in X. Moreover, if the pair (K, M) is weakly compatible, then K M have a unique common fixed point in X. If we replace (CLR) property by (E.A) property in Theorem., we get the following result. Theorem.4. Let (X, d) be a complex valued metric space K, L, M, N : X X be four self-mappings C 1. one of the pairs (K, M) (L, N) satisfies (E.A) property; C. for each x, y X, d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), where ψ Ψ. If K(X) N(X) L(X) M(X) such that one of M(X) N(X) is closed subspace of X, then each pair (K, M) (L, N) have a coincidence point in X. Moreover, if the pairs (K, M) (L, N) are weakly compatible, then K, L, M N have a unique common fixed point in X. Proof. The proof easily follows from Theorem., so we omit it. Similarly to Theorem. we can derive the corollary from Theorem.4. Copyright 017 Isaac Scientific Publishing

7 Advances in Analysis, Vol., No. 4, October Examples In this section, we give two examples to illustrate our obtained results. Example.1. Let X = (, 1) be a complex valued metric space with metric d : X X C defined by d(x, y) = e ιm x y, where x, y X 0 m π. Define self-maps K, L, M N on X by: { x+1 Kx = if x ( 1, 0.5) 0.5 if x (, 1] [0.5, 1) ; { x 1 Lx = if x ( 1, 0.5) 0.5 if x (, 1] [0.5, 1) ; 10 if x ( 1, 0.5) Mx = 0.5 if x (, 1] {0.5} ; 1 x if x (0.5, 1) 0 if x ( 1, 0.5) Nx = 0.5 if x (, 1] {0.5} (1 x) if x (0.5, 1). Then K(X) = (0, 0.5], L(X) = ( 1, 0.5) {0.5}, M(X) = (0, 0.5] {10}, N(X) = [0, 1). Also, define ψ : C + C + by ψ(z) = z. First we check condition (C 1 ) of Theorem., for this let {x n } = { n+1 n } n 1 be a sequence in X. Then ( ) n + 1 lim Kx n = lim K ( ) n + 1 lim Mx n = lim M n that is there exists a sequence {x n } in X such that n = lim 0.5 = 0.5, n ( = lim 1 n + 1 ) = 0.5, n n lim Kx n = lim Mx n = 0.5 = Kx for all x [0.5, 1). = lim Kx n = lim Mx n = 0.5 = Kx for some x X. Hence (K, M) satisfies (CLR K ) property. Next, to check condition (C ) of Theorem., we discuss the following cases: Case 1. let x, y ( 1, 0.5), then Now, Kx = x + 1 d(kx, Ly) = e ιm x + 1, Ly = y 1, Mx = 10, Ny = 0. y 1 = e ιm x y e ιm, ( 1 ψ(d(kx, Ny) + d(ly, Mx)) =1 ψ e ιm x e ιm y 1 = 1 ( { }) ψ e ιm x y 1 41 e ιm, 4 ) 10 Copyright 017 Isaac Scientific Publishing

8 54 Advances in Analysis, Vol., No. 4, October 017 so that d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y ( 1, 0.5). Case. Let x, y (, 1] {0.5}, then Kx = Mx = Ly = Ny = 0.5 hence d(kx, Ly) = 0 = 1 ψ(d(kx, Ny) + d(ly, Mx)). Case. Let x, y (0.5, 1), then Kx = Ly = 0.5, Mx = 1 x Ny = (1 y). Now d(kx, Ly) = 0, so that 1 ψ(d(kx, Ny) + d(ly, Mx)) =1ψ(e ιm Kx Ny + e ιm Ly Mx ) = 1 (e ιm ) ψ y + x 0.5, d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y (0.5, 1) Therefore from the above three cases it follows that d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y X. Also K(X) N(X) L(X) M(X), the pairs (K, M) (L, N) are weakly compatible. Hence by Theorem (.), 0.5 is a unique common fixed point of K, L, M N. Example.. Let X = (0, ] {4} be a complex valued metric space with metric d : X X C defined by d(x, y) = e ιm x y, where x, y X 0 m π. Define self-maps K, L, M N on X by: Kx = { 4 if x (0, 1) 1 if x [1, ] { 1 if x (0, 1) Lx = ; 1 if x [1, ] 4 if x (0, 1) Mx = 1 if x = 1 ; x 1 if x (1, ] Nx = Also, define ψ : C + C + by ψ(z) = z. Then { K(X) = 1, 4 }, L(X) = { 1, 1 if x (0, 1) 1 if x = 1 x+1 if x (1, ]. Now, Let {x n } = { 1 n } n 1 be a sequence in X. Then ( lim Kx n = lim K 1 ) n ; } [, M(X) = (0, 1] {4}, N(X) =, 4 ]. = lim n 1 = 1, ( lim Mx n = lim M 1 ) ( = lim 1 1 ) = 1. n n n Copyright 017 Isaac Scientific Publishing

9 Advances in Analysis, Vol., No. 4, October Thus there exists a sequence {x n } in X such that lim Kx n = lim Mx n = 1 X. Hence (K, M) satisfies (E.A) property. To check condition (C ) of Theorem.4, we distinguish the following cases: Case 1. let x, y (0, 1), then Kx = 4, Ny =, Ly = 1 Mx = 4. Now d(kx, Ly) = 5 e ιm, 6 Thus ( 1 ψ(d(kx, Ny) + d(ly, Mx)) =1 ψ e ιm 4 ) + e ιm 1 4 = 1 ( ) 5 ψ e ιm = 1 ( ) 5 e ιm = 5 e ιm e ιm =d(kx, Ly). d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y (0, 1), Case. let x = y = 1, then Kx = Mx = Ly = Ny = 1 hence d(kx, Ly) = 0 = 1 ψ(d(kx, Ny) + d(ly, Mx)), Case. let x, y (1, ], then Kx = Ly = 1, Mx = x 1 Ny = y+1 Now d(kx, Ly) = 0, so that ( { 1 ψ(d(kx, Ny) + d(ly, Mx)) =1 ψ e ιm 1 y x 1 }) = 1 ( { }) e ιm y + x, d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y (1, ]. Therefore from the above three cases it follows that d(kx, Ly) 1 ψ(d(kx, Ny) + d(ly, Mx)), for all x, y X. Also K(X) N(X) L(X) M(X) such that N(X) is closed subspace of X the pairs (K, M) (L, N) are weakly compatible. Hence by Theorem.4, we can say that 1 is a unique common fixed point of K, L, M N.. 4 Conclusions The derived results extend generalize some well known results of the existing literature in the frame work of compex valued metric spaces. In particular Theorems..1 generalize Theorem 1.1. We observe that (E.A) property requires the condition of closedness of the subspace, while (CLR) property never requires any condition of closedness (or completeness) of the subspace. As we note that in Example.1 neither M(X) nor N(X) is closed subspaces of X, so Theorem.4 is not applicable. Copyright 017 Isaac Scientific Publishing

10 56 Advances in Analysis, Vol., No. 4, October 017 References 1. M. Aamri D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 70 (1), (00), J. Ahmad, A. Azam S. Saejung, Common fixed point results for contractive mappings in complex valued metric spaces, Fixed Point Theor. Appl., 014, (014) 67.. H. Aydi M. Abbas, Tripled coincidence fixed point results in partial metric spaces, Appl. Gen. Topol. 1 (), (01), A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. optim., (), (011), S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux equations integrales, Fund Math.,, (19), S. Bhatt, S. Chaukiyal R.C. Dimri, A common fixed point theorem for weakly compatible maps in complex-valued metric spaces, Int. J. Math. Sci. Appl., 1 (), (011), S. Chok D. Kumar, Some common fixed point results for rational type contraction mappings in complex valued metric spaces, J. Oper., 01, Article ID 81707, 6 pages. 8. G. Jungck, Common fixed points for non-continuous non-self mappings on a non-numeric spaces, Far East J Math Sci., 4 (), (1996), R. Kamal, R. Chugh, S. L. Singh, S. N. Mishrac, New common fixed point theorems for multivalued maps, Appl. Gen. Topol., 15 (), (014), S. K. Mohanta R. Maitra, Common fixed points for ϕ-pairs in complex valued metric spaces, Int. J. Math. Comput. Res., 1, (01), J. R. Roshan, V. Parvaneh I. Altun, Some coincidence point results in ordered b-metric spaces applications in a system of integral equations, Appl. Math. Comput., 6, (014), M. Sarwar M. Bahadur Zada, Common fixed point theorems for six self-maps satsifying common (E.A) common (CLR) properties in complex valued metric spaces, Elect. J. Math. Anal. Appl., (1), (015), W. Sintunavarat, P. Kumam, Common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space, J. Appl. Math., Vol.011, Article ID 67958, 14 pages. 14. W. Sintunavarat, P. Kumam, Generalized common fixed point theorems in complex valued metric spaces applications, J. Inequal. Appl., 01 (01) N. Wairojjana, W. Sintunavarat P. Kumam, Common tripled fixed point theorems for w-compatible mappings along with the CLR g property in abstract metric spaces, J. Inequa. Appl., 014, (014) R. K. Verma H. K. Pathak, Common fixed point theorems using property (E.A) in complex valued metric spaces, Thai J. Math., 11 (), (01), Copyright 017 Isaac Scientific Publishing