Dislocated Cone Metric Space over Banach Algebra and α-quasi Contraction Mappings of Perov Type

Transcription

1 Dislocated Cone Metric Space over Banach Algebra and α-quasi Contraction Mappings of Perov Type Reny George 1,2, R. Rajagopalan 1, Hossam A. Nabwey 1,3 and Stojan Radenović 4,5 1 Department of Mathematics, College of Science, Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia 2 Permanent Affiliation : Department of Mathematics and Computer Science, St. Thomas College, Bhilai, Chhattisgarh, India 3 Permanent Affiliation : Department of Basic Engineering Sciences, Faculty of Engineering, Menofia University, Menofia, Egypt 5 Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam 4 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Abstract Dislocated cone metric space over Banach algebra is introduced as generalization of a cone metric space over Banach algebra as well as a dislocated metric space. Fixed point theorems for Perov type α-quasi contraction mapping, Kannan type contraction as well as Chatterjee type contraction mappings are proved in a dislocated cone metric space over Banach algebra. Proper examples are provided to establish the validity of our claims. Keywords 1 Fixed points, Cone metric space, Dislocated cone metric space, α-admissible function. MSC : Primary 47H10, Secondary 54H25. Acknowledgment 1 This project is supported by Deanship of Scientific research at Prince Sattam bin Abdulaziz University, Al kharj, Kingdom of Saudi Arabia, under International Project Grant No. 2016/01/6714. *Correspondence : 1 Introduction 1 Stojan Radenović : stojan.radenovic@tdt.edu.vn Generalising the concept of a cone metric space, Liu and Xu in [5] introduced cone metric space over Banach algebra (in short CM S BA) and proved contraction principles in such space. They replaced the usual real contraction constant with a vector constant and scalar multiplication with vector multiplication in their results and also furnished proper examples to show that their results were different from those in cone metric space and metric space. While studying the applications of topology in logic programming semantics, Hitzler and Seda[2] introduced dislocated metric space as a generalisation of metric space and discussed the associated topologies. Later George and Khan introduced dislocated fuzzy metric space [10] and then various fixed point results were proved in dislocated spaces. For some details refer to [11]. On the other hand, Perov [8] generalized the Banach contraction principle by replacing the contractive factor with a matrix convergent to zero. Cvetkovic and Rakocevic [1] introduced Perov-type quasi-contractive mapping replacing contractive factor with bounded linear operator with spectral radius less than one and obtained some interesting fixed point results in the setup of cone metric spaces.

2 2 In this work we have introduced the concept of dislocated cone metric space over a Banach algebra (in short dcms BA) as a generalisation of CMS BA as well as dislocated metric space and proved fixed point theorems for Perov type α-quasi contraction mapping in dcms BA and CMS BA. Simple examples are given to illustrate the validity and superiority of our results. 2 Preliminaries A linear space A over K {R, C} is an algebra if for each ordered pair of elements x, y A, a unique product xy A is defined such that for all x, y, z A and scalar α: (i)(xy)z = x(yz); (iia)x(y + z) = xy + xz (iib)(x + y)z = xz + yz (iii) α(xy) = (αx)y = x(αy) A Banach algebra is a Banach space A over K {R, C} such that for all x, y A xy x y. For a given cone P A and x, y A, we say that x y if and only if y x P. Note that is a partial order relation defined on A. For more details on basic concepts of Banach algebra, solid cone, unit element e, zero element θ, invertible elements in Banach algebra etc. the reader may refer to [4, 5]. In what follows A will always denote a Banach algebra, P a solid cone in A and e the unit element of A. Definition 2.1 A sequence p n in a solid cone P of a Banach space is a c-sequence if for each c θ, there exists n 0 N such that p n c for all n n 0. Lemma 2.2 [3] For x A, lim n xn 1 n exists and the spectral radius r(x) satisfies r(x) = lim n xn 1 n = inf x n 1 n. If r(x) < λ, then λe x is invertible in A, moreover, where λ is a complex constant. (λe x) 1 = n=0 x n λ n+1, Lemma 2.3 [12] Let x A. If the spectral radius r(x) of x is less than 1 i.e., r(x) = lim n x n 1 n = infn 1 x n 1 n < 1 (2.1) then (e x) is invertible. Actually (e x) 1 = x i (2.2) i=0

3 Lemma 2.4 [12] Let a, b A. If a commutes with b, then r(a + b) r(a) + r(b), Lemma 2.5 [4] Let E be a Banach space. (i) If a, b, c E and a b c, then a c. (ii) If θ a c for each c θ, then a = θ r(ab) r(a)r(b). Lemma 2.6 [12] Let {u n } be a sequence in A with {u n } θ(n ). Then {u n } is a c-sequence. Lemma 2.7 [4] Let {u n } be a c-sequence in P. If β P is an arbitrarily given vector, then {βu n } is a c-sequence. Lemma 2.8 [3] Let α A and r(α) < 1 then {α n } is a c-sequence. 3 Remark 2.9 For more on c-sequences see [3, 4]. Definition 2.10 Let X be any non empty set and T : X X and α : X X [0, ) be mappings. Then i) T is an α-admissible mapping iff α(x, y) 1 implies α(t x, T y) 1, x, y X. ii) T is an α-dominated mapping iff α(x, y) 1 implies α(x, T x) 1, x, y X. 3 Main Results In this section first we introduce the definition of a dislocated cone metric space over Banach algebra ( in short dcms-ba)and furnish examples to show that this concept is more general than that of CMS BA. We then define convergence and cauchy sequence in a dcms-ba and then prove fixed point results in this space. Definition 3.1 Let χ be a non empty set and d lc : χ χ A be such that for all x, y, z χ, : (dcm1) θ d lc (x, y) and d lc (x, y) = θ implies x = y (dcm2) d lc (x, y) = d lc (y, x) (dcm3) d lc (x, y) d lc (x, z) + d lc (z, y) Then d lc is called a dislocated cone metric on χ and (χ, d lc ) is called a dislocated cone metric space over Banach algebra (in short dcbms BA). Note that every metric space and CMS BA is a dcms BA but the converse is not necessarily true. Inspiring from [4, 5, 9] we furnish the following examples which will establish our claim. Example 3.2 Let A = {a = (a i,j ) 3 3 : a i,j R, 1 i, j 3}, a = 1 i,j 3 a i,j, P = {a A : a i,j 0, 1 i, j 3} be a cone in A. Let χ = R + {0},. Let d lc : χ χ A be given by x + y x + y x + y d lc (x, y) = 2x + 2y 2x + 2y 2x + 2y 3x + 3y 3x + 3y 3x + 3y Then (χ, d lc ) is a dcms BA over A but not a CMS BA over Banach algebra A.

4 Example 3.3 Let χ = R and let A = C R 2(χ). For α = (f, g) and β = (u, v) in A, we define α.β = (f.u, g.v) and α = max( f, g ) where f = sup x χ f(x). Then A is a Banach algebra with unit e = (1, 1), zero elemant θ = (0, 0)and P = {(f, g) A : f(t) 0, g(t) 0, t χ} a non normal cone in A. Consider d lc : χ χ A given by d lc (x, y)(t) = ( x y (1 + t 2 )+ x (t 2 )+ y (t 4 ), x y (1 + t 2 )) Clearly (χ, d lc ) is a dcms BA over A but not a CMS BA over Banach algebra A. For any a χ, the open sphere with centre a and radius λ > θ is given by B λ (a) = {b χ : d lc (a, b) < λ}. Let U = {Y χ : x Y, r > θ, such that B r (x) Z}. Then U defines the dislocated cone metric topology for the dcms BA (χ, d lc ). 4 Definition 3.4 Let (χ, d lc ) be a dcms BA over A, p χ and {p n } be a sequence in χ. i) {p n } converges to p if for each c A, with θ c, there exist n 0 N such that d lc (p n, p) c for all n n 0. We write it as Lim n p n = p. ii) {p n } is a Cauchy sequence if and only if for each c A, with θ c, there exist n 0 N such that d lc (p n, p m ) c for all n, m n 0. iii) (χ, d lc ) is a complete dcms if and only if every Cauchy sequence in (χ, d lc ) is convergent. Proposition 3.5 Let (χ, d lc ) be a dcms BA over A, P be a soloid cone and {p n } be a sequence in χ. If {p n } converges to p χ then i) d lc (p n, p) is a c-sequence. ii) d lc (p n, p n+r ) is a c-sequence. Proof : Follows from definitions 2.1,3.1 and 3.4(i). In [13] Samet et al introduced the concept of α-admissible mappings and proved fixed point theorems for alpha-psi contractive type mappings which paved way for proving new and existing results in fixed point theory.as in [13] and others we give the following definitions : Definition 3.6 Let (χ, d lc ) be a dcms BA, T : X X and α : X X [0, ) be mappings. Then i) T is an α-admissible mapping iff α(x, y) 1 implies α(t x, T y) 1, x, y X. ii) T is an α-dominated mapping iff α(x, y) 1 implies α(x, T x) 1, x, y X. iii) α is a triangular function iff α(x, y) 1, α(y, z) 1 implies α(x, z) 1, x, y, z X. iv) (χ, d lc ) is α-regular iff for any sequence {x p } in χ with α(x p, x p+1 ) 1 and x p x as p, then α(x p, x ) 1

5 5 However for proving the uniqueness of the fixed point, different hypothesis were used by different authors. In the sequel Popescu [7] considered the following condition : (K) For all x y χ there exists w χ such that α(x, w) 1, α(y, w) 1 and α(w, T w) 1. We now introduce the following definitions : Definition 3.7 Let (χ, d lc ) be a dcms BA, T : X X and α : X X [0, ) be mappings. Then (i) T is an α-identical function iff α(t x, T x) 1 for all x χ. (ii) T is weak semi α-admissible iff α(x, y) 1 implies α(x, T 2 y) 1 for any x, y χ. (iii) T satisfies condition (G) iff α(x, T x) 1 and α(y, T y) 1 implies α(x, y) 1 or α(t x, T y) 1 for any x, y χ. (iv) T satisfies condition (G ) iff for all x y χ with α(x, T x) 1 and α(y, T y) 1 there exists w χ such that α(x, w) 1, α(y, w) 1, α(w, w) 1 and α(w, T w) 1. Example 3.8 Let χ = [0, ], T x = x 2 for all x X. Let { 1 if x, y [0, 1] or x = y α(x, y) = 0 otherwise Then T is an α-identical function and T satisfies condition (G) and (G )but T does not satisfy condition (K) and T is not α-dominated. Example 3.9 Let χ = [ n, n] for some n N, T x = x for all x χ. Let { 1 if x, y [ n, 0] or x, y (0, n] α(x, y) = 0 otherwise Then T is an α-identical function and T satisfies condition (G) and (G )but T is not α-dominated and does not satisfy condition (K). Example 3.10 Let A = [ n, 0], B = [0, n] and χ = A B for some n N. Let T x = x for all x χ and { 1 if (x, y) {A B, B A} α(x, y) = 0 otherwise Then α is not triangular and T is not α-identical but T is weak semi α-admissible and α-dominted. T does not satisfy condition (G) but satisfies condition (G ). Example 3.11 Let A = [ n, 0), B = (0, n] and χ = A {0} B for some n N. Let T x = x2 n x χ and { 1 if (x, y) {A A, B B} α(x, y) = 0 otherwise for all Then α is triangular and T is α-identical but T is not weak semi α-admissible and not α-dominted. T satisfy conditions (G) and (G ) but does not satisfy condition (K).

6 6 Lemma 3.12 Let X be a non empty set and T : X X and α : X X [0, ) be mappings. Let {x n } be the Picard sequence starting with x 0. If α(x 0, x 0 ) 1 and α(x 0, T x 0 ) 1, and if α is a triangular function and T is α-admissible then for all n 1 and 0 p q n, α(x p, x q ) 1. Proof : For proof we will make use of principle of mathematical induction. As α(x 0, x 0 ) 1, α(x 0, x 1 ) 1 and T is α-admissible, α(x 1, x 1 ) = α(t x 0, T x 0 ) 1 and so the result holds good for n = 1. Again by α-admissibility of T we get α(x 1, x 2 ) 1, α(x 2, x 2 ) 1 and then since α is triangular we get α(x 0, x 2 ) 1. Thus the result holds good for n = 2. Suppose the result is true for n = r, i.e. α(x p, x q ) 1 for all 0 p q r. We will show that it is true for n = r + 1. Its enough to consider the case α(x p, x r+1 ), 0 p r + 1. By induction hypothesis and α admissibility of T we have α(x p, x r+1 ) 1 for all 1 p r + 1. Since α(x 0, x 1 ) 1, by α admissibility of T and triangularity of function α we get α(x 0, x r+1 ) 1 and thus the result is true for n = r + 1. Hence by principle of mathematical induction the result is true for all n. Lemma 3.13 Let X be a non empty set and T : X X and α : X X [0, ) be mappings. Let {x n } be the Picard sequence starting with x 0 such that α(x 0, x 0 ) 1 and α(x 0, T x 0 ) 1. If T is α-admissible and weak semi α-admissible, then for all n 1 and 0 p q n, α(x p, x q ) 1. Proof : As α(x 0, x 0 ) 1, α(x 0, x 1 ) 1 and T is α-admissible, α(x 1, x 1 ) = α(t x 0, T x 0 ) 1 and so the result holds good for n = 1. Again by α-admissibility of T we get α(x 1, x 2 ) 1, α(x 2, x 2 ) 1. Since T is weak semi α-admissible and α(x 0, x 0 ) 1 we get α(x 0, x 2 ) 1. Thus the result holds good for n = 2. Suppose the result is true for n = r, i.e. α(x p, x q ) 1 for all 0 p q r. We will show that it is true for n = r + 1. Its enough to consider the case α(x p, x r+1 ), 0 p r + 1. By induction hypothesis and α admissibility of T we have α(x p, x r+1 ) 1 for all 1 p r + 1. If r is even, then using α(x 0, x 1 ) 1 and repeated use of weak semi α admissibility of T we get α we get α(x 0, x r+1 ) 1. If r is odd then using α(x 0, x 0 ) 1 and repeated use of weak semi α admissibility of T we get α we get α(x 0, x r+1 ) 1. Thus the result is true for n = r + 1. Hence by principle of mathematical induction the result is true for all n. Definition 3.14 Let (χ, d lc ) be a dcms BA, T : χ χ and α : X X [0, ) be mappings. Then T is a Perov type α-quasi contraction mapping iff there exist µ P such that 0 r(µ) < 1, and for all u, v χ with α(u, v) 1 d lc (T u, T v) µ.ϕ(u, v) (3.1) where ϕ(u, v) {d lc (u, v), d lc (u, T u), d lc (v, T v), d lc (u, T v), d lc (v, T u). Lemma 3.15 Let T be an α-admissible Perov type α-quasi contraction mapping in a dcms BA (χ, d lc ) where α : X X [0, ), x p be the iterative sequence defined by x p+1 = T x p for some arbitrary x 0 χ and all p N such that α(x 0, x 0 ) 1 and α(x 0, T x 0 ) 1. If α is a triangular function or T is weak semi α-admissible, then for all n 1, p, q N, we have α(x p, x q ) 1 for 0 p q n and for all 1 p q n d lc (x p, x q ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) (3.2) Proof : Let d lc (x p, x p+1 ) = d p and d lc (x p, x p ) = d p,p. For proof we will make use pf principle of mathematical induction. Note that by lemma 3.12 or lemma 3.13 as the case may be α(x p, x q ) 1 for

7 7 all 0 p q. For n = 1,p = q = 1, since α(x 0, x 0 ) 1, α(x 0, x 1 ) 1 and α(x 1, x 1 ) 1, from 3.1 we have d lc (x 1, x 1 ) = d lc (T x 0, T x 0 ) µ{d lc (x 0, x 0 ), d lc (x 0, x 1 ), d lc (x 0, x 1 ), d lc (x 0, x 1 ), d lc (x 0, x 1 ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) and thus the result holds good. Now suppose 3.2 is true for n = r, i.e. d lc (x p, x q ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )), for 1 p, q r (3.3) We will show that 3.2 is true for n = r + 1. Its enough to consider the case 1 p r + 1 and q = r + 1. Note that µ(e µ) 1 = µ µ i µ i = (e µ) 1. (3.4) and Since α(x p 1, x r ) 1, from 3.1 we have i=0 e + µ(e µ) 1 = e + i=0 µ i = (e µ) 1. (3.5) i=1 d lc (x p, x r+1 ) = d lc (T x p 1, T x r ) µ{d lc (x p 1, x r ), d lc (x p 1, x p ), d lc (x r, x r+1 ), d lc (x p 1, x r+1 ), d lc (x r, x p ) We will analyze each term on the right hand side of above inequality : (i) d lc (x p, x r+1 ) µd lc (x p 1, x r ) Case i(a) : p = 1. d lc (x 1, x r+1 ) µd lc (x 0, x r ) µ(d lc (x 0, x 1 ) + d lc (x 1, x r )) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) ( by 3.3 and 3.5 ) Case i(b) : 2 p r. By 3.3 and 3.4 we get d lc (x p, x r+1 ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) Case i(c) : p = r + 1. In this case d lc (x p, x r+1 ) µd lc (x r, x r ) and the result follows from 3.3 and 3.4 (ii) d lc (x p, x r+1 ) µd lc (x p 1, x p ) Case ii(a) : p = 1. d lc (x 1, x r+1 ) µd lc (x 0, x 1 ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) Case ii(b) : 2 p r. The result follows from 3.3. Case ii(c) : p = r + 1. d lc (x p, x r+1 ) µd lc (x r, x r+1 ) = µd lc (T x r 1, T x r ) µ 2 {d lc (x r 1, x r ), d lc (x r 1, x r ), d lc (x r, x r+1 ), d lc (x r 1, x r+1 ), d lc (x r, x r )

8 8 Case ii(c 1) : d lc (x p, x r+1 ) µd lc (x r, x r+1 ) µ 2 d lc (x r 1, x r ) By 3.3 and 3.4 we get µ 2 d lc (x r 1, x r ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) Case ii(c 2) : d lc (x p, x r+1 ) µd lc (x r, x r+1 ) µ 2 d lc (x r, x r+1 ). Now, µd lc (x r, x r+1 ) µ 2 d lc (x r, x r+1 ) implies µ(e µ)d lc (x r, x r+1 ) θ. Note that r(µ) < 1 and so (e µ) is invertble and (e µ) 1 > e. Therefore we get d lc (x r, x r+1 ) θ, i.e d lc (x p, x r+1 ) θ µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )). Case ii(c 3) : d lc (x p, x r+1 ) µd lc (x r, x r+1 ) µ 2 d lc (x r 1, x r+1 ) Again by 3.1 d lc (x p, x r+1 ) µd lc (x r, x r+1 ) µ 2 {d lc (x r 2, x r ), d lc (x r 2, x r 1 ), d lc (x r, x r+1 ), d lc (x r 2, x r+1 ), d lc (x r, x r 1 ) Continuing this process we will at most arrive at the following : (i) d lc (x p, x r+1 ) µ k d lc (x p, x q ) for some 1 p, q r, 2 k r and the result follows from which by 3.3 and 3.4. (ii) d lc (x p, x r+1 ) µ k d lc (x r, x r+1 ), and the result follows by proceeding as in Case ii(c-2). (iii) d lc (x p, x r+1 ) µ r d lc (x p, x r+1 ) which implies (e µ)d lc (x p, x r+1 ) θ and by the same argument as in Case ii(c-2) the result follows. Case ii(c 4) : d lc (x p, x r+1 ) µd lc (x r, x r+1 ) µ 2 d lc (x r, x r ). The result follows from 3.3 and 3.4. (iii) d lc (x p, x r+1 ) µd lc (x r, x r+1 ) = µd lc (T x r 1, T x r ). The result follows proceeding as in case ii(c). (iv) d lc (x p, x r+1 ) µd lc (x r, x p ). The result follows from 3.3 and 3.4. (v) d lc (x p, x r+1 ) µd lc (x p 1, x r+1 ) Using 3.1 and continuing in a similar manner as above either we will get the desired result or we get d lc (x p, x r+1 ) µd lc (x p 1, x r+1 ) µ 2 d lc (x p 2, x r+1 ) µ p 1 d lc (x 1, x r+1 ) Now d lc (x 1, x r+1 ) µ{d lc (x 0, x r ), d lc (x 0, x 1 ), d lc (x r, x r+1 ), d lc (x 0, x r+1 ), d lc (x r, x 1 ) If d lc (x 1, x r+1 ) µ{d lc (x 0, x r ) or d lc (x 0, x 1 ) or d lc (x r, x r+1 ) or d lc (x r, x 1 ) then the result follows by proceeding as in case i(a) or ii(a) or ii(c) or be 3.3 and 3.4. If d lc (x 1, x r+1 ) µd lc (x 0, x r+1 ) then d lc (x 1, x r+1 ) µd lc (x 0, x r+1 ) µ(d lc (x 0, x 1 ) + d lc (x 1, x r+1 ) µ(e µ) 1 d lc (x 0, x 1 ) µ(e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) Thus 3.2 is true for n = r + 1 and hence by principle of mathematical induction its true for all n. Theorem 3.16 Let (χ, d lc ) be a complete dcms BA, T : χ χ and α : X X [0, ) be mappings such that

9 9 (i) T is a be a Perov type α-quasi contraction mapping. (ii) α is a triangular function or T is weak semi α-admissible. (iii) T is α-admissible. (iv) there exist x 0 χ such that α(x 0, x 0 ) 1 and α(x 0, T x 0 ) 1 (v) (χ, d lc ) is α-regular Then T has a fixed point. Proof : Consider the iterative sequence defined by x p+1 = T x p for all p N. Let d lc (x p, x p+1 ) = d p and d lc (x p, x p ) = d p,p. Note that d p,p 2d p 1 and d p,p 2d p+1. We will show that {x p } is a Cauchy sequence. For 1 < p < q, let Γ p,q = {d lc (x i, x j ) : p i j q}. Then using 3.1 and by the same argument in the proof of Lemma 12 in [liu and xiu], we can find u 1 Γ p 1,q, u 2 Γ p 2,q, u p 1 Γ 1,q satisfying d lc (x p, x q ) µu 1 µ 2 u 2 µ p 1 u p 1 By Lemma 3.15, u p 1 µ(e µ) 1 (d lc (x 0, x 1 )+d lc (x 0, x 0 )). Therefore d lc (x p, x q ) µ p (e µ) 1 (d lc (x 0, x 1 )+ d lc (x 0, x 0 )). Since r(µ) < 1, by Lemma 2.8 and 2.7 we see that µ p (e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) is a c-sequence. Now let θ c be arbitrary in A. By definition 2.1, there exists a natural number p 0, such that µ p (e µ) 1 (d lc (x 0, x 1 ) + d lc (x 0, x 0 )) c for all p p 0. Thus we get d lc (x p, x q ) c for all p p 0. Therefore {x p } is a Cauchy sequence and by completeness of (χ, d lc ) there exist x χ such that lim n x p = x. By proposition 3.5 and lemma 2.6, d lc (x p, x ) θ, d lc (x p 1, x p ) θ and d lc (x p 1, x ) θ. Since (χ, d lc ) is α-regular, α(x p 1, x ) 1 and so by 3.1 d lc (x, T x ) d lc (x, x p ) + d lc (x p, T x ) = d lc (x, x p ) + d lc (T x p 1, T x ) d lc (x, x p ) + µ{d lc (x p 1, x ), d lc (x p 1, x p ), d lc (x, T x ), d lc (x p 1, T x ), d lc (x, x p )} d lc (x, x p ) + µ{d lc (x p 1, x ), d lc (x p 1, x p ), d lc (x, T x ), d lc (x p 1, x p ) + d lc (x p, T x ), d lc (x, x p )} d lc (x, x p ) + µ{d lc (x p 1, x ), d lc (x p 1, x p ), θ, (e µ) 1 d lc (x p 1, x p ), d lc (x, x p )} By proposition 3.5 and lemma 2.6, d lc (x p, x ) θ and d lc (x p 1, x p ) θ. Thus d lc (x, T x ) θ and so T x = x. In Theorem 3.16 condition (v) can be replaced with another condition as in Popescu [7]. the following : We have Theorem 3.17 Let (χ, d lc ) be a complete dcms BA, T : χ χ and α : X X [0, ) be mappings such that (i) T is a Perov type α-quasi contraction mapping. (ii) α is a triangular function or T is weak semi α-admissible.. (iii) T is α-admissible. (iv) there exist x 0 χ such that α(x 0, x 0 ) 1 and α(x 0, T x 0 ) 1 (v) If {x p } is a sequence in χ such that α(x p, x p+1 ) 1for all p and x p u χ as p, then there exists a subsequence {x p(k) } of {x p } such that α(x p(k), u) 1 for all k.

10 10 Then T has a fixed point. Proof : Proceeding as in the proof of Theorem 3.16, the Picard sequence {x p } starting with x 0 converges to x χ. By (v) there exists a subsequence {x p(k) } of {x p } such that α(x p(k), u) 1 for all k. Thus we have d lc (x, T x ) d lc (x, x p(k) ) + d lc (x p(k), T x ) = d lc (x, x p(k) ) + d lc (T x p(k) 1, T x ) d lc (x, x p(k) ) + µ{d lc (x p(k) 1, x ), d lc (x p(k) 1, x p(k) ), d lc (x, T x ), d lc (x p(k) 1, T x ), d lc (x, x p (k))} d lc (x, x p(k) ) + µ{d lc (x p(k) 1, x ), d lc (x p(k) 1, x p ), d lc (x, T x ), d lc (x p(k) 1, x p(k) ) + d lc (x p(k), T x ), d lc ( d lc (x, x p(k) ) + µ{d lc (x p(k) 1, x ), d lc (x p(k) 1, x p(k) ), θ, (e µ) 1 d lc (x p(k) 1, x p(k) ), d lc (x, x p(k) )} By proposition 3.5 and lemma 2.6, d lc (x p(k), x ) θ and d lc (x p(k) 1, x p(k) ) θ. Thus d lc (x, T x ) θ and so T x = x. Theorem 3.18 Let (χ, d lc ), T and α be as in Theorem Suppose all conditions of Theorem 3.16 or Theorem 3.17 are satisfied. If T is an α-identical function or if T is α-dominated, then T has a fixed point x χ and d lc (x, x ) = θ. Further if T satisfies condition (G) then the fixed point is unique. Proof : As in the proof of Theorem 3.16 or Theorem 3.17 we see that T has a fixed point x χ. If T is α- identical then α(x, x ) = α(t x, T x ) 1. If T is α-dominated then α(x, x ) = α(x, T x ) 1. Then from 3.1 we have d lc (x, x ) = d lc (T x, T x ) µ{d lc (x, x ), d lc (x, x ), d lc (x, x ), d lc (x, x ), d lc (x, x )} = µd lc (x, x ). Hence d lc (x, x ) = θ. Now suppose y is another fixed point of T. Then as above α(y, y ) 1 and d lc (y, y ) = θ. Since T satisfies condition (G) we have α(x, y ) 1 and then by 3.1 d lc (x, y ) = d lc (T x, T y ) Thus d lc (x, y ) θ and so x = y. µ{d lc (x, y ), d lc (x, x ), d lc (y, y ), d lc (x, y ), d lc (y, x )} Theorem 3.19 Let (χ, d lc ), T and α be as in Theorem Suppose all conditions of Theorem 3.16 or Theorem 3.17 are satisfied. If T is an α-identical function or if T is α-dominated, then T has a fixed point x χ and d lc (x, x ) = θ. Further if T satisfies condition (G ) then the fixed point is unique. Proof : As in the proof of Theorem 3.18 we see that T has a fixed point x χ and d lc (x, x ) = θ and if y is another fixed point of T then α(y, y ) 1 and d lc (y, y ) = θ. Since T satisfies condition (G ) there exists w χ such that α(x, w) 1, α(y, w) 1, α(w, w) 1 and α(w, T w) 1. By Theorem 3.16 the sequence {T n w} will converge to a fixed point say w of T. Since T is α-admissible we get α(x, T n w) 1 and α(y, T n w) 1 and then by 3.1 we have d lc (x, T n+1 w) = d lc (T x, T T n w) µ{d lc (x, T n w), d lc (x, x ), d lc (T n w, T n+1 w), d lc (x, T n w), d lc (T n w, x )} Then as n, using Proposition 3.5 and Lemma 2.6 we get d lc (x, w ) θ and so x = w. Similarly we can show that y = w. Therefore x = y. Remark 3.20 In Theorem 3.18 and 3.19 we can replace the requirement of condition (G) or condition (G ) with that of condition (K). But as in examples 3.8 and 3.9 there exists functions α and T such that T is α-identical and T satisfy condition (G) and condition (G ) but does not satisfy condition (K). Hence our approach is new and justifiable.

11 11 Since every CMS BA is a dcms BA and since in a Cone metric space (χ, d c ), d c (x, y) = θ for all x, y χ, we give the following generalised results which is easily deducted from our main results. Theorem 3.21 Let (χ, d c ) be a complete CMS BA, T : χ χ and α : X X [0, ) be mappings such that (i) T is a Perov type α-quasi contraction mapping. (ii) α is a triangular function. (iii) T is α-admissible. (iv) there exist x 0 χ such that and α(x 0, T x 0 ) 1 (v) (χ, d lc ) is α-regular Then T has a fixed point. Theorem 3.22 Let (χ, d c ) be a complete CMS BA, T : χ χ and α : X X [0, ) be mappings such that (i) T is a Perov type α-quasi contraction mapping. (ii) α is a triangular function. (iii) T is α-admissible. (iv) there exist x 0 χ such that and α(x 0, T x 0 ) 1 (v) If {x p } is a sequence in χ such that α(x p, x p+1 ) 1for all p and x p u χ as p, then there exists a subsequence {x p(k) } of {x p } such that α(x p(k), u) 1 for all k. Then T has a fixed point. Theorem 3.23 Let (χ, d c ), T and α be as in Theorem Suppose all conditions of Theorem 3.21 or Theorem 3.22 are satisfied. If T is α-identical or α-dominated function then T has a fixed point x χ. Further if T satisfies condition (G) then the fixed point is unique. Theorem 3.24 Let (χ, d c ), T and α be as in Theorem Suppose all conditions of Theorem 3.21 or Theorem 3.22 are satisfied. If T is α-identical or α-dominated function then T has a fixed point x χ. Further if T satisfies condition (G ) then the fixed point is unique. Theorem 3.25 Let (χ, d c ) be a complete CMS BA, T : χ χ, A and B be non empty subsets of χ such that χ = A B and T (A) T (B) and T (B) T (A). If there exist µ P such that 0 r(µ) < 1, and d lc (T u, T v) µ.ϕ(u, v) (3.6) for all u A, v B and ϕ(u, v) {d lc (u, v), d lc (u, T u), d lc (v, T v), d lc (u, T v), d lc (v, T u), Then T has a unique fixed point in A B. Proof : Let { 1 if (x, y) {A B, B A} α(x, y) = 0 otherwise Then T is α-admissible, weak semi α-admissible and α-dominated function and satisfies condition (G ). Hence by Theorem 3.24, T has a unique fixed point x in χ. Since T is α-dominated, α(x, x ) = α(x, T x ) 1. This is possible iff x A B.

12 12 Corollary 3.26 Let (χ, d lc ) be a complete dcms BA and T : χ χ a mappings. If there exist µ P such that 0 r(µ) < 1, and d lc (T u, T v) µ.ϕ(u, v) (3.7) for all u, v χ and ϕ(u, v) {d lc (u, v), d lc (u, T u), d lc (v, T v), d lc (u, T v), d lc (v, T u), Then T has a unique fixed point. Proof : The proof easily follows from Theorems 3.21 and 3.23 or Theorems 3.22 and 3.24, by taking α(x, y) = 1 for all x, y χ. Corollary 3.27 [Theorem 9, [6]] Let (χ, d c ) be a complete CMS BA and T : χ χ a mappings. If there exist µ P such that 0 r(µ) < 1, and d c (T u, T v) µ.ϕ(u, v) (3.8) for all u, v χ and ϕ(u, v) {d c (u, v), d c (u, T u), d c (v, T v), d c (u, T v), d c (v, T u), Then T has a unique fixed point. Proof : Since every CMS BA is a dcms BA, the proof follows from Corollary Example 3.28 Let χ = [0, ) and A be as in Example 3.9 and d lc (x, y)(t) = ( x y (1 + t 2 ), x y (1 + t 2 )). Let T : χ χ be given by { log(1 + x T x = 2 ) if x [0, 1] log(1 + 2x) otherwise and { 1 (x, y) [0, 1] or x = y α(x, y) = 0 otherwise Then α is a triangular function, T is α-admissible, α-identical and satisfy condition (G). Also for all α(x, y) 1 we see that d lc (T x, T y) q.d lc (x, y) where q = 1 2. Thus T satisfies all conditions of Theorems 3.18 and 3.19 but does not satisfy conditions of Corollary Further 0 is a unique common fixed point of T. Theorem 3.29 Let (χ, d lc ) be a complete dcms BA and T : χ χ be a mapping. Let α : X X [0, ) be a mapping satisfying conditions (iii), (iv) and (v) of Theorem 3.16 or If there exist λ, µ, ν P, such that λ commutes with µ + 3ν, µ + ν commutes with µ + 3ν, r(λ + µ + ν) + r(µ + 3ν) < 1 and for all x, y χ with α(x, y) 1 d lc (T x, T y) λd lc (x, y) + µ(d lc (x, T x) + d(y, T y)) + ν(d lc (x, T y) + d(y, T x)) (3.9) then T has a fixed point. Further if T is an α-identical function or if T is α-dominated, then T has a fixed point x χ and d lc (x, x ) = θ. Moreover if T satisfies condition (G) or (G ) then the fixed point is unique. Proof : Consider the iterative sequence defined by x p+1 = T x p for all p N. Let d lc (x p, x p+1 ) = d p and d lc (x p, x p ) = d p,p. Note that d p,p 2d p 1 and d p,p 2d p+1. By Lemma 3.12 or 3.13 α(x p, x p+1 ) 1 and therefore using 3.9 we have d p+1 = d lcb (x p+1, x p+2 ) = d lc (T x p, T x p+1 ) λd lc (x p, x p+1 ) + µ(d lc (x p, T x p ) + d lc (x p+1, T x p+1 )) + ν(d lc (x p, T x p+1 ) + d lc (x p+1, T x p )) = λd p + µ(d p + d p+1 ) + ν(d p + d p+1 + 2d p )

13 13 i.e or d p+1 β.d p where β = (λ + µ + ν)(e µ 3ν) 1 d p+1 β p+1 d p. Since λ commutes with µ + 3ν, µ + ν commutes with µ + 3ν, simple calculations shows that (λ + µ + ν)(e µ 3ν) 1 = (e µ 3ν) 1 (λ + µ + ν and using Lemma 2.4 again by simple calculations we have r(β) = r(λ + µ + ν)(e µ 3ν) 1 ) < 1. Therefore e β is invertible and (e β) 1 = i=0 (β)i. Hence d lc (x p, x p+n ) d p + d p+1 + d p+n 1 β p {e + β + β 2 + }d 0 = β p (e β) 1 d 0 Also using Lemma 2.8and Lemma 2.7, β p (e β) 1 d 0 is a c-sequence. Thus by definition 2.1 for any c A with θ c, there is N 1 N satisfying n > N 1 implies d lcb (x p, x p+n ) {β p (e β) 1 d 0 c. (3.10) Thus {x p } is a Cauchy sequence and since (χ, d lc ) is complete we have u χ such that lim x p = u. (3.11) n By proposition 3.5 and lemma 2.6, d lc (x p, u) θ, d lc (x p 1, x p ) θ and d lc (x p 1, u) θ. Since (χ, d lc ) is α-regular, α(x p 1, u) 1 and so by 3.1 Since d p d q whenever p q there exist k N such that d lc (u, T u) {d k, d k+1,...}. Then for any p > k d lc (u, T u) (d lc (u, x p ) + d lc (x p, T u)) = (d lc (u, x p ) + d lc (T x p 1, T u)) (d lc (u, x p ) + λd lc (x p 1, u) + µ(d lc (x p 1, x p ) + d lc (u, T u)) + ν(d lc (x p 1, T u) + d lc (u, x p )) i.e. d lc (u, T u) ν(e µ ν) 1 (d lc (u, x p ) + (e µ ν) 1 (λ + ν)d lc (x p 1, u) + (e µ ν) 1 µ(d lc (x p 1, x p ) + d lcb (u, T u)) By Lemma 2.6, ν(e µ ν) 1 d lc (u, x p ) θ, (e µ ν) 1 (λ + ν)d lc (u, x p 1 ) θand (e µ ν) 1 µ(d lc (x p 1, x p ) + d lc (u, T u)) θ. Hence d lc (u, T u) θ. Thus T u = u. If T is an α-identical function or if T is α-dominated then proceeding as in the proof of Theorem 3.18 and using 3.9, we get d lc (u, u) = θ. Now suppose there exists u such that T u = u. Then as above d lc (u, u ) = θ. If T satisfies condition (G) we have α(u, u ) 1 and then by 3.9 d lc (u, u ) = d lc (T u, T u ) (λ + 2ν)d lc (u, u ) (λ + 2ν) 2 d lc (u, u ) (λ + 2ν) n d lc (u, u ). As above by lemma 2.8and Lemma 2.7, (λ + 2ν) n d lc (u, u ) is a c-sequence and so by Lemma 2.6 (λ + 2ν) n d lc (u, u ) θ as n. Thus u = u. If T satisfies condition (G ) then there exists w χ such that α(u, w) 1, α(u, w) 1, α(w, w) 1

14 14 and α(w, T w) 1. Then by replacing x 0 with w in condition (iv) and proceeding as above, the sequence {T n w} will converge to a fixed point say w of T and d lc (w, w ) = θ. Since T is α-admissible we get α(u, T n w) 1 and α(u, T n w) 1 and then by 3.9 we have d lc (u, T n+1 w) = d lc (T u, T T n w) λd lc (u, T n w) + µ{d lc (u, u) + d lc (T n w, T n+1 w)} + ν{d lc (u, T n w) + d lc (T n w, u)} Then as n, using Proposition 3.5 and Lemma 2.6 we get d lc (u, w ) θ and so u = w. Similarly we can show that u = w. Therefore u = u. Theorem 3.30 Let (χ, d lc ) be a complete dcms BA and T : χ χ be a mapping. If there exist λ P such that 0 r(2λ) < 1, and d lc (T x, T y) λ[d lc (x, T x) + d lc (y, T y)] (3.12) for all x, y χ with α(x, y) 1 then T has a unique fixed point. Proof: Note that 3.12 implies 3.9. Hence the reult follows from Theorem Theorem 3.31 Let (χ, d lc ) be a complete dcms BA and T : χ χ be a mapping. If there exist λ P such that 0 r(4λ) < 1, and d lc (T x, T y) λ[d lc (x, T y) + d lc (y, T s)] (3.13) for all x, y χ with α(x, y) 1 then T has a unique fixed point. Proof: Note that 3.13 implies 3.9. Hence the reult follows from Theorem Conclusion In this paper we have introduced the concept of dislocated cone metric space over Banach algebra and proved some generalised fixed point theorems in such a space. Some new properties of mappings such as α-identical mappings, semi α-admissible mappings, mappings satisfying condition (G) and condition (G ) are also introduced. Our work is a generalisation of some work already done in metric space and cone metric space over Banach algebra. There is further scope for extending and generalising various fixed point theorems in the setting of a dislocated cone metric space over Banach algebra Competing interests The authors declare that they have no competing interests Author s contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Acknowledgement This project is supported by Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al kharj, Kingdom of Saudi Arabia, under International Project Grant No. 2016/01/6714. The authors are Thankful to the learned reviewers for their valuable suggestions which helped in bringing this paper in its present form.

15 15 References [1] M. Cvetkovic and V. Rakocevic, Quasi-contraction of Perov type, Appl. Math. Comput., 235 (2014), [2] P. Hitzler and A. K. Seda, Dislocated topologies, Proc. of the slovakian conference in applied mathematics (2000), Bratislava. [3] H.P Huang and S. Radenović,Common fixed point theorems of generalised Lipschitz mappings in cone b-metric space and applications, J. Nonlinear Sci. Appl., 8(2015), [4] H.P Huang and S. Radenović, Some fixed point results of generalised Lipchitz mappings on cone b-metric spaces over Banach algebras, J. Comput. Anal. Appl., 20(2016), [5] H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 320 (2013), [6] H. Liu, S. Xu, Fixed point theorems of quasi contractions on cone metric space with Banach algebra, Abst. Appl. Anal., Article ID ,(2013), 5 pages. [7] Ovidiu Popescu, Some new fixed point theorems for α-geraghty contraction type maps in metric spaces, Fixed Point Theory Appl., 190 (2014), [8] A.I. Perov, On cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uravn., 2 (1964), [9] Reny George, Hossam A Nabwey, K.P Reshma and R.Rajagopalan, Generalised cone b-metric spaces and contraction principles, Mat. Vesnik, 67(4)(2015), [10] Reny George and M.S Khan, Dislocated fuzzy metric spaces and associated topologies, Proceedings of the 8th Joint conference on information sciences, (2005), [11] Reny George, R Rajagopalan, S Vinayagam, Cyclic contractions and fixed points in dislocated metric spaces, Int. J. Math. Anal 7 (9) (2013), [12] W. Rudin, Functional Anal., McGraw-Hill, New York, NY, USA, 2nd edition, [13] B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α-ψcontractive mappings. Nonlinear Anal. 75, (2012)