φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers

Joshi et al., Cogent Mathematics 2016, 3: 1162484 PURE MATHEMATICS RESEARCH ARTICLE φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers Received: 25 October 2015 Accepted: 26 February 2016 First Published: 07 March 2016 *Corresponding author: Deepak Singh, Department of Applied Sciences, NITTTR, Under Ministry of HRD, Government of India, Bhopal 462002, India E-mail: dk.singh1002@gmail.com Reviewing editor: Prasanna K. Sahoo, University of Louisville, USA Additional information is available at the end of the article Vishal Joshi 1, Naval Singh 2 and Deepak Singh 3 * Abstract: The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via φ-contractive multivalued mappings involving pointdependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested. Subjects: Advanced Mathematics; Analysis-Mathematics; Mathematics & Statistics; Pure Mathematics; Science Keywords: common fixed point; complex valued metric spaces; complete complex valued metric spaces; Cauchy sequence; multivalued mappings AMS Subject Classifications: 47H10; 54H25; 30L99 Deepak Singh ABOUT THE AUTHORS Vishal Joshi, having more than 16 years of teaching experience, is working as an assistant professor in the Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India. He has qualified several national level examinations and tests. His research interests are fixed point theory, topology, and functional analysis. Deepak Singh received his MSc and PhD degrees in Mathematics from the Barkatullah Vishwavidyalaya, Bhopal, Madhya Pradesh, India in 1993 and 2004, respectively. Currently, he is an associate professor at Department of Applied Sciences, National Institute of Technical Teachers Training and Research, Bhopal, Madhya Pradesh, India. He is associated as referees and reviewer for many journals of international repute. He has also delivered contributed /invited talks in many international conferences held in European and Asian countries. His current research interests include optimization, nonlinear functional analysis, fixed point theory and its applications. He has published 40 research papers in the journals of international repute. PUBLIC INTEREST STATEMENT In recent times, the notion of complex valued metric spaces is one of the developing areas in mathematical analysis. The fixed point results concerning rational contractive conditions cannot be extended in cone metric spaces, whereas in complex valued metric spaces, one can find the fixed point of mappings via rational contractive conditions. The results in this space can be utilized to find the solution of Urysohn Integral Equations., Boundary value problems and system of algebraic equations. This paper is devoted to a package of multivalued mappings with control functions, which may be useful for the researcher to find the solution for future problems specially mentioned above. 2016 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. Page 1 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 1. Introduction In 2011, Azam, Fisher, and Khan 2011 introduced the notion of complex valued metric spaces and established some fixed point results for a pair of mappings for contraction condition satisfying a rational expression. This idea is intended to define rational expressions which are not meaningful in cone metric spaces and thus many such results of analysis cannot be generalized to cone metric spaces but to complex valued metric spaces. After the establishment of complex valued metric spaces, Rouzkard and Imdad 2012 established some common fixed point theorems satisfying certain rational expressions in this spaces to generalize the result of Azam et al. 2011. Subsequently, Sintunavarat and Kumam 2012 obtained common fixed point results by replacing the constant of contractive condition to control functions. Sitthikul and Seajung 2012 established some fixed point results by generalizing the contractive conditions in the context of complex valued metric spaces. Recently, Sintunavarat, Cho, and Kumam 2013 introduced the notion of C-Cauchy sequence and C-completeness in complex valued metric spaces and applied it to obtain the common solution of Urysohn integral equations. Very recently, Singh, Singh, Badal, and Joshi in press established certain fixed point theorems which generalized numerous preceding results in the setting of complex valued metric spaces. Ahmad, Klin-Eam, and Azam 2013 established the existence of common fixed point for multivalued mappings under generalized contractive condition in complex valued metric spaces. Afterward Azam, Ahmad, and Kumam 2013 and then Kutbi, Ahmad, Azam, and Al-Rawashdeh 2014 improved the contractive condition of the result of Ahmad et al. 2013 and proved some common fixed point results for multivalued mappings in complex valued metric space. In what follows, we recall some notations and definitions that will be used in our subsequent discussion. Let C be the set of complex numbers and z 1, z 2 C. Define a partial order on C as follows: z 1 z 2 if and only if Rez 1 Rez 2 and Imz 1 Imz 2. It follows that if one of the followings conditions is satisfied. C1 Rez 1 =Rez 2 and Imz 1 =Imz 2 ; C2 Rez 1 < Rez 2 and Imz 1 =Imz 2 ; C3 Rez 1 =Rez 2 and Imz 1 < Imz 2 ; C4 Rez 1 < Rez 2 and Imz 1 < Imz 2. In particular, we will write z 1 z 2 if z 1 z 2 and one of C2, C3, and C4 is satisfied and z 1 z 2 if only C4 is satisfied. Definition 1.1 Azam et al., 2011 Let X be a non empty set. A mapping d : X X C is called a complex valued metric on X if the following conditions are satisfied: CM1 0 dx, y for all x, y X and dx, y =0 x = y; CM2 dx, y =dy, x for all x, y X; CM3 dx, y dx, z+dz, y for all x, y, z X. In this case, we say that X, d is a complex valued metric space. Page 2 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 Example 1.2 Let X = C be a set of complex number. Define d : C C C by dz 1, z 2 = x 1 x 2 + i y 1 y 2, where z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2. Then C, d is a complex valued metric space. Example 1.3 Let X = C. Define a mapping d : X X C by dz 1, z 2 =e ik z 1 z 2, where k [0, π 2 ]. Then X, d is a complex valued metric space. Definition 1.4 Azam et al., 2011 Suppose that X, d is a complex valued metric space. 1 We say that a sequence {x n } is a Cauchy sequence if for every 0 c C there exists an integer N such that dx n, x m c for all n, m N. 2 We say that {x n } converges to an element x X if for every 0 c C there exists an integer N d such that dx n, x c for all n N. In this case, we write x n x. 3 We say that X, d is complete if every Cauchy sequence in X converges to a point in X. Lemma 1.5 Azam et al., 2011 Let X, d be a complex valued metric space and let {x n } be a sequence in X. Then {x n } converges to x if and only if dx n, x 0 as n. Lemma 1.6 Azam et al., 2011 Let X, d be a complex valued metric space and let {x n } be a sequence in X. Then {x n } is a Cauchy sequence if and only if dx n, x n+m 0 as n. Ahmad et al. 2013 introduced the notion of multivalued mappings as follows. Let X, d be a complex valued metric space. Denote the family of nonempty, closed, and bounded subsets of a complex valued metric space by CBX. From now on, denote sz 1 ={z 2 C : z 1 z 2 } for z 1 C, and sa, B = sda, b = C : da, b z} for a X and B CBX. b B b B{z For A, B CBX, denote sa, B = sa, B sb, A. a A Definition 1.7 Ahmad et al. 2013 Let X, d be a complex valued metric space. Let T : X CBX be a multivalued map. For x X and A CBX, define W x A ={dx, a : a A}. b B Thus, for x, y X, W x Ty={dx, u : u Ty}. Definition 1.8 Ahmad et al. 2013 Let X, d be a complex valued metric space. A subset A of X is called bounded from below if there exists some z X such that z a for all a A. Definition 1.9 Ahmad et al. 2013 Let X, d be a complex valued metric space. A multivalued mapping F : X 2 C is called bounded from below if for each x X there exists z x C such that z x u for all u Fx. Page 3 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 Definition 1.10 Ahmad et al. 2013 Let X, d be a complex valued metric space. The multivalued mapping T : X CBX is said to have the lower bound property l.b property on X, d, if the for any x X, the multivalued mapping F x : X 2 C defined by F x Ty=W x Ty, is bounded from below. That is, for x, y X, there exists an element l x Ty C such that l x Ty u, for all u W x Ty, where l x Ty is called a lower bound of T associated with x, y. Definition 1.11 Ahmad et al. 2013 Let X, d be a complex valued metric space. The multivalued mapping T : X CBX is said to have the greatest lower bound property g.l.b property on X, d if a greatest lower bound of W x Ty exists in C for all x, y X. Denote dx, Ty by the g.l.b of W x Ty. That is, dx, Ty=inf {dx, u : u Ty}. Definition 1.12 Let Ψ be a family of non-decreasing functions, φ : C C such that φ0 =0 and φt t, when 0 t. 2. Main result We start this section with the following observation. Proposition 2.1 Let X, d be a complex valued metric space and S, T : X CBX. Let x 0 X and defined the sequence {x n } by x 2n+1 Sx 2n, x 2n+2 Tx 2n+1, n = 0, 1, 2, 2.1 Assume that there exists a mapping λ : X [0, 1 such that λu λx and λv λx, x X, u Sx, v Tx. Then λx 2n λx 0 and λx 2n+1 λx 1. Proof Let x X and n = 0, 1, 2,. Then we have λx 2n =λu 1 λx 2n 2, for u 1 Sx 2n 1 = λu 2 λx 2n 2, for u 2 Sx 2n 2 λx 0 i.e. λx 2n λx 0. Similarly, we have λx 2n+1 λx 1. The subsequent example illustrates the preceding proposition. Example 2.2 Let X ={1, 1, 1, 1, 1, }. Define d : X X C as dx, y =i x y then clearly X, d is a 2 3 4 5 complex valued metric space. Also define multivalued mappings S and T by [ ] 1 1 1 S = 0, = T, n = 0, 1, 2, 3, n + 1 n + 2 n + 1 Page 4 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 Choosing sequence {x n } as x n = 1 n+1, n = 0, 1, 2, 3, Then x 0 = 1 X. Clearlyn+1 Sx 2n and x 2n+2 Tx 2n+1. Consider a mapping λ : X [0, 1 by λx = x, for all x X. 6 Undoubtedly λu λx and λv λx, x X, u Sx, v Tx for all x, y X Consider λx 2n = that is λx 2n λx 0, n = 0, 1, 2. Also consider λx 2n+1 = 1 62n + 1 1 6 = λx 0, 1 62n + 2 1 2 6 = λx 1, that is λx 2n+1 λx 1, n = 0, 1, 2, x X. Thus Proposition 2.1 is verified. Our main theorem runs as follows. Theorem 2.3 Let X, d be a complete complex valued metric space and S, T : X CBX be multivalued mapping with g.l.b. property. Then there exist mappings λ, η, δ, ξ, ν, μ, γ : X [0, 1 such that, i λu λx, ηu ηx, δu δx, ξu ξx, νu νx, μu μx and γu γx, for all u Sx and x X; ii λv λx, ηv ηx, δv δx, ξv ξx, νv νx, μv μx and γv γx, for all v Tx and x X; iii λx +ηx +δx +2ξx +νx +μx +γx < 1, x X; iv dx, Sxdx, Ty φ λxdx, y+ηxdx, Sx+δxdy, Ty+ξx 1 + dx, y dy, Sxdy, Ty dx, Sxdy, Ty +νx + μx + γx 1 + dx, y 1 + dx, y for all x, y X and φ Ψ. dy, Sxdx, Ty 1 + dx, y ssx, Ty, 2.2 Then S and T have a common fixed point. Proof Let x 0 be an arbitrary point in X and x 1 Sx 0. From 2.2 with x = x 0 and y = x 1, we get φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx, Sx dx, Tx 1 0 1 1 ssx 0. This yields that dx 0 dx 1 + γx 0 dx 1 dx 0 Page 5 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx, Sx dx, Tx 1 0 1 1 ssx 0. x Sx 0 dx 0 dx 1 + γx 0 dx 1 dx 0 This implies that φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx 1 dx 1 sx, dx 0 dx 1 + γx 0 dx 1 dx 0 for all x Sx 0. Now since x 1 Sx 0, one can have φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx 1 dx 1 dx 0 dx 1 + γx 0 dx 1 dx 0 sx 1, φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx, Sx dx, Tx 1 0 1 1 + μx 0 dx, Sx dx, Tx 0 0 1 1 + γx 0 dx, Sx dx, Tx 1 0 0 1 sdx 1, x. x Tx 1 2.3 So there exists some x 2 Tx 1, such that φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx, Sx dx, Tx 0 0 0 1 + νx 0 dx 1 dx 1 sdx 1. Therefore dx 0 dx 1 dx 1 φ λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 + ξx 0 dx 0 dx 0 dx 0 dx 1 + νx 0 dx 1 dx 1 + γx 0 dx 1 dx 0 λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 + ξx 0 dx 0 dx 0 dx 0 dx 1 + νx 0 dx 1 dx 1 + γx 0 dx, Sx dx, Tx 1 0 0 1. + γx 0 dx 1 dx 0 Utilizing the greatest lower bound property g.l.b. property of S and T, we obtain Page 6 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 dx 1 λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx 0 dx 0 + νx 0 dx 1 dx 1 dx 0 dx 1 + γx 0 dx 1 dx 0 = λx 0 dx 0 +ηx 0 dx 0 +δx 0 dx 1 +ξx 0 dx 0 dx 0 dx, x dx, x 0 1 1 2. 2.4 So that dx 1 λx 0 dx 0 + ηx 0 dx 0 + δx 0 dx 1 Inductively, using Proposition 2.1, we can construct a sequence x n in X such that for n = 0, 1, 2, for x 2n+1 Sx 2n and x 2n+2 Tx 2n+1, with α = λx 0 +ηx 0 +ξx 0 1 δx 0 ξx 0 μx 0 < 1. Next for m > n, we get Thus we have dx + ξx 0 dx 0 0 + μx dx, x dx 0 0 1 2 λx 0 dx 0 + ηx 0 dx 0 + δx 0 dx 1 + ξx 0 dx 0 dx 1 λx 0 dx 0 + ηx 0 dx 0 + δx 0 dx 1 + ξx 0 dx 0 +dx 1 dx 1 dx 1 λx 0 +ηx 0 +ξx 0 1 δx 0 ξx 0 μx 0 dx 0 = α dx 0, where α = λx 0 +ηx 0 +ξx 0 1 δx 0 ξx 0 μx 0 < 1. dx n, x n+1 α n dx 0, dx n, x m dx n, x n+1 + dx n+1, x n+2 +... + dx m 1, x m α n + α n+1 +... + α m 1 dx 0 [ ] α n dx 1 α 0. [ ] α n dx n, x m dx 1 α 0. Which on making m, n, yields dx n, x m 0. This reflects that {x n } is a Cauchy sequence in X. Since X is complete then there exists p X such that x n p as n. Now, we show that p Tp and p Sp. From 2.2, with x = x and y = p, we have φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp + μx, p dx, Sx dp, Tp + γx, p dp, Sx dx, Tp, p ssx, Tp. Page 7 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 This implies that φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp + μx, p dx, Sx dp, Tp + γx, p dp, Sx dx, Tp, p sx, Tp x Sx φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp, p sx, Tp, for all x Sx. Since x +1 Sx, we have There exists some p k Tp, such that Thus, we have + μx dx, Sx dp, Tp, p + γx dp, Sx dx, Tp, p φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp, p + μx dx, Sx dp, Tp + γx, p dp, Sx dx, Tp, p sx +1, Tp φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp + μx, p dx, Sx dp, Tp + γx, p dp, Sx dx, Tp, p sdx +1, p 1. p 1 Tp φ λx dx, p+ηx dx, Sx +δx dp, Tp+ξx dx, Sx dx, Tp, p + νx dp, Sx dp, Tp, p sdx +1, p k. + μx dx, Sx dp, Tp + γx, p dp, Sx dx, Tp, p dx +1, p k φ λx dx, p+ηx dx, Sx +δx dp, Tp + ξx dx, Sx dx, Tp, p + μx dx, Sx dp, Tp, p + νx dp, Sx dp, Tp, p + γx dp, Sx dx, Tp, p λx dx, p+ηx dx, Sx +δx dp, Tp + ξx dx, Sx dx, Tp, p + μx dx, Sx dp, Tp, p + νx dp, Sx dp, Tp, p + γx dp, Sx dx, Tp., p Utilizing Proposition 2.1 and also using greatest lower bound property of S and T, we have Page 8 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 dx +1, p k λx 0 dx, p+ηx 0 dx k+1 +δx 0 dp, p k + ξx 0 dx k+1 dx, p k, p dx k+1 dp, p k, p + νx 0 dpk+1 dp, p k, p + γx 0 dp, x dx, p +1 k., p 2.5 We have by triangular inequality dp, p k dpk+1 +dx +1, p k. Thus, one can obtain dp, p k dpk+1 +λx 0 dx, p+ηx 0 dx k+1 +δx 0 dp, p k So that + ξx 0 dx k+1 dx, p k, p dx k+1 dp, p k, p + νx 0 dpk+1 dp, p k, p + γx 0 dp, x dx, p +1 k., p dp, p k dpk+1 + λx 0 dx, p + ηx 0 dx k+1 + δx 0 dp, p k + ξx 0 dx k+1 dx, p k, p dx k+1 dp, p k, p Which on letting k, reduces to 1 δx0 dp, p k 0 or dp, p k 0 as k. + νx 0 dpk+1 dp, p k, p + γx 0 dp, x dx, p +1 k., p By Lemma 1.5, we have lim p = p. n k Since Tp is closed then p Tp. Similarly, one can get p Sp. Thus p Tp Sp. Therefore S and T have a common fixed point. Subsequent result is an easy consequence of Theorem 2.3. Corollary 2.4 Let X, d be a complete complex valued metric space and S, T : X CBX be multivalued mapping with g.l.b. property. Then there exist mappings λ 1, η 1, δ 1, ξ 1, ν 1, μ 1, γ 1 : X [0, 1 such that, x X, i λ 1 u λ 1 x, η 1 u η 1 x, δ 1 u δ 1 x, ξ 1 u ξ 1 x, ν 1 u ν 1 x, μ 1 u μ 1 x and γ 1 u γ 1 x, for all u Sx; ii λ 1 v λ 1 x, η 1 v η 1 x, δ 1 v δ 1 x, ξ 1 v ξ x, ν 1 v ν 1 x, μ 1 v μ 1 x and γ 1 v γ 1 x, for all v Tx; iii λ 1 x+η 1 x+δ 1 x+2ξ 1 x+ν 1 x+μ 1 x+γ 1 x < 1, Page 9 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 iv dx, Sxdx, Ty λ 1 xdx, y+η 1 xdx, Sx+δ 1 xdy, Ty+ξ 1 x 1 + dx, y dy, Sxdy, Ty dx, Sxdy, Ty dy, Sxdx, Ty + ν 1 x + μ 1 + dx, y 1 x + γ 1 + dx, y 1 x ssx, Ty, 1 + dx, y for all x, y X. Then S and T have a common fixed point. Proof Proof is immediate on choosing φt =kt, where k 0, 1 in Theorem 2.3 with λ 1 x =kλx, η 1 x =kηx, δ 1 x =kδx, ξx =kξx, ν 1 x =kνx, μ 1 x =kμx, γ 1 x =kγx. Now, consider the following corollary. Corollary 2.5 Let X, d be a complete complex valued metric space and S, T : X CBX be multivalued mapping with g.l.b. property such that λ 1 dx, y+η 1 dx, Sx+δ 1 dy, Ty+ξ 1 dx, Sxdx, Ty 1 + dx, y + ν 1 dy, Sxdy, Ty 1 + dx, y + μ 1 dx, Sxdy, Ty 1 + dx, y + γ 1 dy, Sxdx, Ty 1 + dx, y ssx, Ty, for all x, y X and λ 1, η 1, δ 1, ξ 1, ν 1, μ 1, and γ 1 are non negative real numbers with λ 1 + η 1 + δ 1 + 2ξ 1 + ν 1 + μ 1 + γ 1 < 1. Then S and T have a common fixed point. Proof Proof can be obtain easily by restricting the point-dependent coefficient to constants i.e. by setting λ 1 x =λ 1, η 1 x =η 1, δ 1 x =δ 1, ξ 1 x =ξ 1, ν 1 x =ν 1, μ 1 x =μ 1 and γ 1 x =γ 1 in Corollary 2.4 with λ 1, η 1, δ 1, ξ 1, ν 1 μ 1, γ 1 0 such that λ 1 + η 1 + δ 1 + 2ξ 1 + ν 1 + μ 1 + γ 1 < 1. Remark 2.6 1 If we set η 1 = δ 1 = ξ 1 = ν 1 = 0 in Corollary 2.5, we will get the Theorem 9 of Ahmad et al. 2013. 2 If we choose λ 1 = γ 1 = ξ 1 = ν 1 = 0 in Corollary 2.5, then Theorem 15 of Ahmad et al. 2013 is obtained. 3 Setting η 1 = δ 1 = 0 in Corollary 2.5, one can obtain the Theorem 9 of Kutbi et al. 2014. Consequently all the corollaries corresponding to these results are immediate from our results. Remark 2.7 Let X, d be a complex valued metric space. If C = R, then X, d is a metric space. Furthermore, for S, T CBX, HS, T = inf ss, T is the Hausdorff metric induced by d. Utilizing aforesaid Remark 2.7, we have the following corollary from Theorem 2.3. Corollary 2.8 Let X, d be a complete metric space and let S, T : X CBX be multivalued mapping with g.l.b. property. Then there exist mappings λ, η, δ, ξ, ν, μ, γ : X [0, 1 such that, i λu λx, ηu ηx, δu δx, ξu ξx, νu νx, μu μx and γu γx, for all u Sx and x X; ii λv λx, ηv ηx, δv δx, ξv ξx, νv νx, μv μx and γv γx, for all v Tx and x X; iii λx +ηx +δx +2ξx +νx +μx +γx < 1, x X; Page 10 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 iv dx, Sxdx, Ty HSx, Ty φ λxdx, y+ηxdx, Sx+δxdy, Ty+ξx 1 + dx, y dy, Sxdy, Ty dx, Sxdy, Ty + νx + μx + γx 1 + dx, y 1 + dx, y dy, Sxdx, Ty 1 + dx, y, 2.6 for all x, y X and φ Ψ. Then S and T have a common fixed point. Remark 2.9 By choosing point-dependent control functions λ, η, δ, ξ, ν, μ, γ, function φ and mappings S and T suitably in Theorem 2.3, Corollaries 2.4, 2.5 and 2.8, one can deduce a multitude of results from the existing literature which includes the celebrated Banach fixed point theorem for multivalued mappings in complex valued complete metric spaces as well as in complete metric spaces. Following example substantiates the validity of hypothesis of our main Theorem 2.3. Example 2.10 Let X =[0, 1]. Define d : X X C as follows: dx, y = x y e i π 6. Then X, d is a complete complex valued metric space. Consider the mappings S, T : X CBX such that Sx =[0, x ] and Ty =[0, y ] for all x, y X. 4 4 Next, we define the functions λ, η, δ, ξ, ν, μ, γ : X [0, 1 by λx = x+1, ηx = x, δx = x, ξx = x, νx = x, μx = x x and γx =. 3 20 10 50 25 30 25 Undoubtedly i λu λx, ηu ηx, δu δx, ξu ξx, νu νx, μu μx and γu γx, for all u Sx =[0, x ] and x [0, 1; 4 ii λv λx, ηv ηx, δv δx, ξv ξx, νv νx, μv μx and γv γx, for all v Tx =[0, x ] and x [0, 1; 4 iii λx +ηx +δx +2ξx +νx +μx +γx < 1, x [0, 1. Also define the function φ Ψ by φt = 3t 4. Calculating the various functions involving in our contractive condition. dx, y = x y e i π 6, dx, Sx = x x π 4 ei 6, dy, Ty= y y π 4 ei 6, dy, Sx = y x π 4 ei 6, dx, Ty= x y π 4 ei 6, ssx, Ty=s x 4 y π 4 ei 6. Now consider, Page 11 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 3 dx, Sx dx, Ty λx dx, y + ηx dx, Sx + δx dy, Ty + ξx 4 1 + dx, y dy, Sx dy, Ty dx, Sx dy, Ty + νx + μx + γx 1 + dx, y 1 + dx, y dy, Sx dx, Ty 1 + dx, y Clearly, for λx = x+1, ηx = x, δx = x, ξx = x, νx = x, μx = x x and γx = and for all 3 20 10 50 25 30 25 x, y [0, 1], we have, x 4 y 4 3 [ x + 1 x x y + x x x 4 3 20 4 + y y x 10 4 + x x x y 4 4 50 1 + dx, y x y x + y y 4 4 x x x + y y 4 4 x y x + x y ] 4 4, 25 1 + dx, y 30 1 + dx, y 25 1 + dx, y 2.7 Since, one can easily calculate that x 4 y 4 3 x + 1 x y for all x X. 4 3 And all the remaining terms on the right hand side of Inequality 2.7 are non-negative for all x X. Consequently, one can obtain dx, Sxdx, Ty φ λxdx, y+ηxdx, Sx+δxdy, Ty+ξx 1 + dx, y dy, Sxdy, Ty dx, Sxdy, Ty + νx + μx + γx 1 + dx, y 1 + dx, y dy, Sxdx, Ty 1 + dx, y ssx, Ty. Hence, all the conditions of Theorem 2.3 are satisfied and x = 0 remains fixed under mappings S and T. 3. Slip-ups in some recent papers and their remedies The motivation of this section is to point out some slip-ups in the examples of some recent papers Ahmad et al. 2013 and Kutbi et al. 2014 in complex valued metric spaces. In above-mentioned papers, the authors claimed that the function d : X X C defined by dx, y = x y e iθ, 3.1 where θ = tan 1 y and x, y X =[0, 1], is a complex valued metric which is not a reality. x Unfortunately the function dx, y described by the Equation 3.1 is not a complex valued metric in its present form. It is neither symmetric nor enjoys the triangular inequality which amounts to say that the other calculations in the examples are incorrect so that these examples do not illustrate the concerned theorems as claimed by the authors. Now following examples are furnished which substantiate our viewpoints. To substantiate the claim, consider x, y X =[0, 1] and define a function d : X X C as dx, y = x y e iθ, where θ = tan 1 y. Then x dx, y = x y e i tan 1 y x 3.2 and dy, x = y x e i tan 1 x y, 3.3 so that Page 12 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 dx, y dy, x as e i tan 1 y x e i tan 1 x y, x, y X and for x y. Triangular inequality for complex valued metric spaces runs as follows: dx, y dx, z+dz, y, for all x, y, z X. If we invoke the defined function d in above inequality, then the partial ordering due to Azam et al. 2011 does not hold good always. and For example, if we choose x = 0, y = 1, z = 1 2, then dx, y = 0 1 e i tan 1 1 0 = e i tan 1 = e i π 2 = i, dz, y = 1 2 1 e i tan 1 2 = 0.2236 + 0.4472i dx, z = 1 2 Thus = i 2. tan 1 ei dx, y dx, z+dz, y, for all x, y, z X. and similarly for some other values of x, y and z, we can show that triangular inequality is not satisfied by d. Thus X, d is not a complex valued metric space. Rectification: In order to overcome the aforementioned drawbacks, we propose suitable but different rectifications for both the papers separately. Firstly, in Example 14 of Ahmad et al. 2013 if we define d : X X C by dx, y = x y e i. π 4, x, y X =[0, 1]. Then X, d is a complex valued metric space. If we consider the same mappings as in Example 14 of Ahmad et al. 2013 with the substitution θ = π, then this example demonstrates the validity of 4 the hypothesis of Theorem 9 of Ahmad et al. 2013. Finally, Example 24 of Kutbi et al. 2014 can be repaired as follows: Example 3.1 Let X =[0, 1]. Define d c : X X C by d c ξ, η = ξ η e i. π 5, ξ, η X. Then d c ξ, η is a complex valued metric space. Carrying out routine calculation on the lines of Example 24 of Kutbi et al. 2014 under the restriction θ = π, one can demonstrate Theorem 9 of 5 Kutbi et al. 2014. Notice that ξ = 0 is a common fixed point of S and F. Page 13 of 14

Joshi et al., Cogent Mathematics 2016, 3: 1162484 Remark 3.2 From the preceding discussions, we infer that in order to hold the validity of aforesaid results mentioned in Ahmad et al. 2013 and Kutbi et al. 2014, we can take many θ s ranging in [0, π 4 ]. Acknowledgements The authors are grateful to the learned referees for their accurate reading and their helpful suggestions. Funding The authors received no direct funding for this research. Author details Vishal Joshi 1 E-mail: joshinvishal76@gmail.com Naval Singh 2 E-mail: drsinghnaval12@gmail.com Deepak Singh 3 E-mail: dk.singh1002@gmail.com 1 Department of Mathematics, Jabalpur Engineering College, Jabalpur, Madhya Pradesh, India. 2 Government Science and Commerce College, Benazeer, Bhopal, Madhya Pradesh, India. 3 Department of Applied Sciences, NITTTR, Under Ministry of HRD, Government of India, Bhopal 462002, India. Citation information Cite this article as: φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers, Vishal Joshi, Naval Singh & Deepak Singh, Cogent Mathematics 2016, 3: 1162484. References Ahmad, J., Klin-Eam, C., & Azam, A. 2013. Common fixed points for multivalued mappings in complex valued metric spaces with applications. Abstract and Applied Analysis, 2013, 12, Article ID 854965. Azam, A., Fisher, B., & Khan, M. 2011. Common fixed point theorems in complex valued metric spaces. Numerical Functional Analysis and Optimization, 32, 243 253. Azam, A., Ahmad, J., & Kumam, P. 2013. Common fixed point theorems for multi-valued mappings in complex valued metric spaces. Journal of Inequality and Applications, 2013, Article 578, 12 p. Kutbi, M. A., Ahmad, J., Azam, A., & Al-Rawashdeh, A. S. 2014. Generalized common fixed point results via greatest lower bound property. Journal of Applied Mathematics, 2014, Article ID 265865. Rouzkard, F., & Imdad, M. 2012. Some common fixed point theorems on complex valued metric spaces. Computers & Mathematics with Applications, 64, 1866 1874. Singh, N., Singh, D., Badal, A., & Joshi, V. in press. Fixed point theorems in complex valued metric space. Journal of the Egyptian Mathematical Society. Sintunavarat, W., & Kumam, P. 2012. Generalized common fixed point theorems in complex valued metric spaces and applications. Journal of Inequalities and Applications, 2012, Article ID 84. Sintunavarat, W., Cho, Y. J., & Kumam, P. 2013. Urysohn integral equations approach by common fixed points in complex valued metric spaces. Advances in Difference Equations, 2013, Article 49, 14 p. Sitthikul, K., & Saejung, S. 2012. Some fixed point theorems in complex valued metric space. Fixed Point Theory and Applications, 2012, Article ID 189. 2016 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics ISSN: 2331-1835 is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at www.cogentoa.com Page 14 of 14