Afr. Mat. DOI 10.1007/s13370-015-0381-0

A hybrid class of expansive-contractive mappings in cone b-metric spaces H. Olaoluwa1 · J. O. Olaleru1

Received: 12 July 2014 / Accepted: 13 August 2015 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015

Abstract In this research work, we generalize classical results on the existence of common fixed points of generalized expanding mappings by removing the constraint on the signs of the underlining coefficients in the inequality considered. The hybrid class obtained surprisingly regroups both classes of generalized contractive maps and classes of generalized expansive maps. The research work also provides a clear understanding of the boundary between contractive and expansive mappings in literature and is applied to product metric-type spaces. Keywords space

Expanding maps · Contractive maps · Fixed point theorem · Cone b-metric

Mathematics Subject Classification

47H10 · 54H25

1 Introduction Ever since Polish mathematician Banach proved the Banach contraction principle in a complete metric space in [4], fixed point theory and its applications have been a major attraction to researchers. Many contractive conditions have been studied in metric spaces (see, for example [21,22]). Expanding mappings have enjoyed a relatively lower popularity with the results of Wang et al. [26], Daffer and Kaneko [7], Kumar and Garg [13] among others. However, with the generalization of metric spaces to b-metric spaces [3], cone metric spaces [10,15,16,20] and recently, cone b-metric spaces [11], fixed point theorems for expanding mappings have been proved in cone metric spaces (e.g. [1,23,25]).

B

H. Olaoluwa [email protected] J. O. Olaleru [email protected]

1

Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria

123

H. Olaoluwa, J. O. Olaleru

The most recent and perhaps most general theorems on expanding mappings in cone metric spaces were initially proposed by Han and Xu [8] who considered a pair of surjective expansion mappings. [27] was an erratum of the results in [8] and a valid generalization of the results of Aage and Salunke [1]. Even though the maps considered in [27] are not necessarily continuous, the coefficients ai in the expanding conditions are taken to be positive. In this manuscript, the theorems in [27] are improved: the coefficients in the “expanding” inequalities are not constrained to be positive thus paving way to a more general class which can be, in a classical sense, contractive, expansive or neither. The number of the maps considered is extended to four with the surjectivity assumption removed, and the ambient space is generalized to cone b-metric spaces. The following definitions and results will be needed in the sequel. Definition 1.1 (See [10]) Let E be a real Banach space. A subset P of E is called a cone if and only if: (a) P is closed, non-empty and P  = {0}; (b) a, b ∈ R, a, b ≥ 0, x, y ∈ P imply that ax + by ∈ P; (c) P ∩ (−P) = {0}. Given a cone P, define a partial ordering ≤ with respect to P by x ≤ y if and only if y − x ∈ P. We shall write x  y for y − x ∈ int P, where int P stands for interior of P. Also we will use x < y to indicate that x ≤ y and x  = y. The cone P in a normed space E is called normal whenever there is a real number k > 0, such that for all x, y ∈ E, 0 ≤ x ≤ y implies x ≤ ky. The least positive number satisfying this norm inequality is called the normal constant of P. In the following, we always suppose that E is a Banach space, P is a cone in E with int (P)  = ∅ and ≤ is a partial ordering with respect to P. Definition 1.2 (See [10,11]) Let X be a non-empty set and let E be a real Banach space equipped with the partial ordering ≤ with respect to the cone P ⊂ E. Consider a mapping d : X × X −→ E and the following axioms: (c1 ) (c2 ) (c3 ) (c3 )

0 ≤ d(x, y) for all x, y ∈ X and d(x, y) = 0 if and ony 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. d(x, y) ≤ s[d(x, z) + d(z, y)] for all x, y, z ∈ X.

When (c1 ) − (c3 ) are satisfied, d is called a cone metric on X and (X, d) a cone metric space; when (c3 ) is satisfied instead of (c3 ), d is called a cone b-metric and (X, d) a cone b-metric space. Obviously, cone b-metric spaces generalize b-metric spaces and cone metric spaces. When s = 1, a cone b-metric space becomes a cone metric space. Here are some examples: Example 1.3 Let X = {1, 2, . . . , n}; E = R2 ; P = {(x, y) ∈ E : x ≥ 0, y ≤ 0}. Define d : X × X → E by   1 , −|x − y| if x  = y |x−y| d(x, y) = 0 if x = y. > 1 and If n ∈ / {2, 3}, then (X, d) is a cone b-metric space with the coefficient s = (n−1)(n−2) 2n−3 not a cone metric space since the triangle inequality fails for the points 1, 2, n. If n ∈ {2, 3}, then (X, d) is a cone metric space.

123

A hybrid class of expansive-contractive mappings in cone. . .

Example 1.4 (See [11]) Let X = l p with 0 < p < 1, where l p = {{xn } ⊂ 1  ∞ ∞ p p p. R: n=1 |x n | < ∞}. Let d : X × X → R+ be defined by d(x, y) = n=1 |x n − yn | Then (X, d) is a b-metric space. Put E = l 1 , P = {{xn } ∈ E : xn ≥ 0, ∀n ≥ 1}. Letting  ¯ ¯ is a cone b-metric space with d¯ : X × X → E be defined by d(x, y) = d(x,y) , (X, d) 2n n≥1 1

the coefficient s = 2 p > 1 but it is not a cone metric space. Definition 1.5 (See [11]) Let (X, d) be a cone b-metric space, {xn } a sequence in X and x ∈ X. We say that {xn } is • a Cauchy sequence if for every c ∈ E with 0  c, there is some k ∈ N such that, for all n, m ≥ k, d(xn , xm )  c; • a convergent sequence if for every c ∈ E with 0  c, there is some k ∈ N such that, for all n ≥ k, d(xn , x)  c. Such x is called limit of the sequence {xn }. Note that every convergent sequence in a cone b-metric space X is a Cauchy sequence. A cone b-metric space X is said to be complete if every Cauchy sequence in X is convergent in X. The following lemma will be needed in the sequel. Lemma 1.6 (See [9]) Let f , g, S and T be self mappings on a cone b-metric space X with the coefficient s ≥ 1. Suppose that f (X ) ⊂ T (X ) and g(X ) ⊂ S(X ). Define {xn } and {yn } by y2n+1 = f x2n = T x2n+1 and y2n+2 = gx2n+1 = Sx2n+2 , n ≥ 0. Suppose that there is λ ∈ [0, 1s ) such that d(yn , yn+1 ) ≤ λd(yn−1 , yn ) for each n ≥ 1. Then {yn } is Cauchy. Proof For n ∈ N, we have d(yn , yn+1 ) ≤ λd(yn−1 , yn ) ≤ λ2 d(yn−2 , yn−1 ) ≤ · · · ≤ λn d(y0 , y1 ). For any n, p ∈ N, we have: d(yn , yn+ p ) ≤ s[d(yn , yn+1 ) + d(yn+1 , yn+ p )] = sd(yn , yn+1 ) + sd(yn+1 , yn+ p ) ≤ sd(yn , yn+1 ) + s 2 [d(yn+1 , yn+2 ) + d(yn+2 , yn+ p )] = sd(yn , yn+1 ) + s 2 d(yn+1 , yn+2 ) + s 2 d(yn+2 , yn+ p ) ≤ .. . ≤ sd(yn , yn+1 ) + s 2 d(yn+1 , yn+2 ) + · · · + s p−1 d(yn+ p−2 , yn+ p−1 ) +s p−1 d(yn+ p−1 , yn+ p ) ≤ [sλn + s 2 λn+1 + s 3 λn+2 + · · · + s p−1 λn+ p−2 + s p−1 λn+ p−1 ]d(y0 , y1 ) = sλn

p−1

k=0

(sλ)k d(y0 , y1 ) = sλn

1 d(y0 , y1 ) 1 − (sλ)

1 → 0 as Given 0  c, choose τ > 0 such that c + {y ∈ P : y < τ } ⊂ P. Since sλn 1−sλ 1 n n → ∞, there is n 0 ∈ N such that sλ 1−sλ d(y0 , y1 ) ∈ {y ∈ P : y < τ } for all n > n 0 . 1 d(y0 , y1 )  c for all n > n 0 . Thus. for all n > n 0 and p ∈ N, It follows that sλn 1−sλ d(yn , yn+ p )  c and {yn } is Cauchy.  

To avoid imposing stiff conditions on a set of maps for the existence of their common fixed points, the coefficients considered in the next theorems in this manuscript are not all necessarily positive. There arise situations where the triangle inequality has to be multiplied

123

H. Olaoluwa, J. O. Olaleru

by some of these coefficients. Since multiplication by a negative real number changes the sense of an inequality, it is important to state beforehand, mathematically, the effect of multiplication on the triangle inequality. Lemma 1.7 Let (X, d) be a cone b-metric space with coefficient s ≥ 1. From the triangle inequality, for any real number a and any x, y, z ∈ X ,   1 + s2 s−1 1 − s2 s+1 ad(x, z) ≥ a+ |a| d(x, y) + a− |a| d(z, y) 2s 2s 2 2 When s = 1 (case of cone metric spaces), ad(x, z) ≥ ad(x, y) − |a|d(z, y), ∀x, y, z ∈ X , ∀a ∈ R. Proof Let a ∈ R. If a ≥ 0, then from the triangle inequality, for any x, y, z ∈ X , d(x, y) ≤ sd(x, z) + sd(z, y), that is d(x, z) ≥ 1s d(x, y) − d(z, y), and by multiplying by a ≥ 0, ad(x, z) ≥ a s d(x, y) − ad(z, y). If a ≤ 0, then from the triangle inequality, d(x, z) ≤ sd(x, y) + sd(y, z) for any x, y, z ∈ X ; multiplying by a ≤ 0, one gets ad(x, z) ≥ sad(x, y) + sad(y, z). Thus the following inequalities are obtained

ad(x, z) ≥ as d(x, y) − ad(z, y), a ≥ 0 ad(x, z) ≥ sad(x, y) + sad(y, z), a ≤ 0. Combining both inequalities, ad(x, z) ≥ ϕ(a)d(x, y) + ψ(a)d(z, y) for any a ∈ R, where

a

, a≥0 −a, a ≥ 0 ϕ(a) = s and ψ(a) = sa, a ≤ 0. sa, a ≤ 0 From a + |a| = 2

a, a ≥ 0 0, a ≤ 0

one gets  ϕ(a) = ψ(a)

1 a+|a| 1 a−|a| 2 · s + 2 · s = 2s  1 a−|a| = − a+|a| 2 + 2 · s = −2

Thus ad(x, z) ≥ [ and a ∈ R.

1+s 2 2s

a+

1−s 2 2s

and

+ +

a − |a| = 2





0, a ≥ 0 a, a ≤ 0,



s 1 s 1+s 2 1−s 2 2 a + 2s − 2 |a| = 2s a + 2s |a|   1  s s s−1 s+1 2 a + − 2 − 2 |a| = 2 a − 2 |a|

|a|]d(x, y) + [ s−1 2 a−

s+1 2 |a|]d(z,

y) for any x, y, z ∈ X  

The following definition is needed in the sequel. Definition 1.8 (See [12]) Two self mappings f and g of a set X are said to be weakly compatible if they commute at their coincidence points, i.e., f gx = g f x whenever f x = gx, x ∈ X.

2 Main results 2.1 Common fixed points of a hybrid class in cone b-metric spaces We now state our main theorem on the fixed points of an hybrid class of four mappings in cone b-metric spaces. The theorem extends and generalizes the results of Han and Xu [8] to cone b-metric spaces and to four maps.

123

A hybrid class of expansive-contractive mappings in cone. . .

Theorem 2.1 Let (X, d) be a cone b-metric space with the coefficient s ≥ 1. Suppose mappings f, g, S, T : X → X satisfy T (X ) ⊂ f (X ), S(X ) ⊂ g(X ) and d( f x, gy) ≥ a1 d(Sx, T y) + a2 d( f x, Sx) + a3 d(gy, T y) + a4 d( f x, T y) + a5 d(gy, Sx) (2.1) for all x, y ∈ X , where a1 , a2 , a3 , a4 , a5 ∈ R satisfy for all x, y ∈ X ⎧   2 1−s 2 1−s 1+s ⎪ a1 + a3 + 1+s ⎪ 2s a4 + 2s |a4 | > s 1 − a2 + 2 a4 + 2 |a4 | ≥ 0 ⎪ ⎪   2 ⎨ 1−s 2 1−s 1+s a1 + a2 + 1+s 2s a5 + 2s |a5 | > s 1 − a3 + 2 a5 + 2 |a5 | ≥ 0 ⎪ ⎪ 2sa3 + (1 + s 2 )(a1 + a4 ) + (1 − s 2 )(|a1 | + |a4 |) > 0 ⎪ ⎪ ⎩ 2sa2 + (1 + s 2 )(a1 + a5 ) + (1 − s 2 )(|a1 | + |a5 |) > 0.

(2.2)

(1) If S = T = I d X then f and g have a common fixed point. (2) If the pairs { f, S} and {g, T } are weakly compatible then they both have coincidence points. If in addition 1 < a1 + a4 + a5 , then f, g, S, T have a unique common fixed point. Proof Suppose x0 is an arbitrary point in X . Since T (X ) ⊂ f (X ) and S(X ) ⊂ g(X ), there exists sequences {xn }, {yn } in X such that for any n ≥ 0

y2n+1 = f x2n+1 = T x2n y2n+2 = gx2n+2 = Sx2n+1 . By (2.1), we have d(y2n+1 , y2n+2 ) = d( f x2n+1 , gx2n+2 ) ≥ a1 d(Sx2n+1 , T x2n+2 ) + a2 d( f x2n+1 , Sx2n+1 ) + a3 d(gx2n+1 , T x2n+2 ) +a4 d( f x2n+1 , T x2n+2 ) + a5 d(gx2n+2 , Sx2n+1 ) = a1 d(y2n+2 , y2n+3 ) + a2 d(y2n+1 , y2n+2 ) + a3 d(y2n+2 , y2n+3 ) +a4 d(y2n+1 , y2n+3 ).

(2.3)

From Lemma 1.7, we have that 

a4 d(y2n+1 , y2n+3 ) ≥

1 + s2 1 − s2 a4 + |a4 | d(y2n+3 , y2n+2 ) 2s 2s  s+1 s−1 + a4 − |a4 | d(y2n+2 , y2n+1 ). 2 2

Thus (2.3) implies that  1 + s2 1 − s2 d(y2n+1 , y2n+2 ) ≥ a1 + a3 + a4 + |a4 | d(y2n+3 , y2n+2 ) 2s 2s  s−1 s+1 a4 − |a4 | d(y2n+2 , y2n+1 ). + a2 + 2 2 Thus d(y2n+2 , y2n+3 ) ≤ δ1 d(y2n+1 , y2n+2 ), with δ1 =

1 − a2 + a1 + a3 +

1−s 1+s 2 a4 + 2 |a4 | . 2 1+s 1−s 2 2s a4 + 2s |a4 |

(2.4)

123

H. Olaoluwa, J. O. Olaleru

Similarly, d(y2n , y2n+1 ) = d( f x2n+1 , gx2n ) ≥ a1 d(Sx2n+1 , T x2n ) + a2 d( f x2n+1 , Sx2n+1 ) + a3 d(gx2n , T x3n ) +a4 d( f x2n+1 , T x2n ) + a5 d(gx2n , Sx2n+1 ) = a1 d(y2n+2 , y2n+1 ) + a2 d(y2n+1 , y2n+2 ) +a3 d(y2n , y2n+1 ) + a5 d(y2n , y2n+2 ).

(2.5)

Lemma 1.7, (2.5) implies that  1 + s2 1 − s2 d(y2n , y2n+1 ) ≥ a1 + a2 + a5 + |a5 | d(y2n+2 , y2n+1 ) 2s 2s  s−1 s+1 + a3 + a5 − |a5 | d(y2n , y2n+1 ). 2 2 Thus d(y2n+1 , y2n+2 ) ≤ δ2 d(y2n , y2n+1 ), with δ2 = From (2.2), we have that δ1 <

1 s

1 − a3 + a1 + a2 +

1−s 1+s 2 a5 + 2 |a5 | . 1+s 2 1−s 2 2s a5 + 2s |a5 |

(2.6)

and δ2 < 1s . Thus from (2.4) and (2.6),

d(yn , yn+1 ) ≤ hd(yn−1 , yn ), h = max{δ1 , δ2 } <

1 . s

(2.7)

From Lemma 1.6, we have that {yn } is a Cauchy sequence in X . Suppose g(X ) is complete. Then there exists u ∈ g(X ) say u = gv such that gx2n+2 = y2n+2 → u. In fact yn → u. Let us prove that T v = u. From (2.1) we have d( f x2n+1 , gv) ≥ a1 d(Sx2n+1 , T v) + a2 d( f x2n+1 , Sx2n+1 ) + a3 d(gv, T v) +a4 d( f x2n+1 , T v) + a5 d(gv, Sx2n+1 ) d(y2n+1 , u) ≥ a1 d(y2n+2 , T v) + a2 d(y2n+1 , y2n+2 ) + a3 d(u, T v) +a4 d(y2n+1 , T v) + a5 d(u, y2n+2 ) From Lemma 1.7 we have  2 ⎧ ⎨ a1 d(y2n+2 , T v) ≥ 1+s 2s a1 +  2 1+s ⎩ a d(y 4 2n+1 , T v) ≥ 2s a4 +

1−s 2 2s |a1 | 1−s 2 2s

 

d(u, T v) +

|a4 | d(u, T v) +

 s−1

(2.8) 

2

a1 −

s+1 2 |a1 |

2

a4 −

s+1 2 |a4 |

 s−1



d(u, y2n+2 ) d(u, y2n+1 ).

Thus (2.8) becomes  1 + s2 1 − s2 1 + s2 1 − s2 a1 + |a1 | + a3 + a4 + |a4 | d(u, T v) d(y2n+1 , u) ≥ 2s 2s 2s 2s +a2 d(y2n+1 , y2n+2 )  s−1 s+1 + a1 − |a1 | + a5 d(u, y2n+2 ) 2 2  s−1 s+1 + a4 − |a4 | d(u, y2n+1 ) 2 2

123

A hybrid class of expansive-contractive mappings in cone. . .

d(u, T v) ≤

1 1+s 2 2s a1



+

1−s 2 2s |a1 | + a3

+

1+s 2 2s a4

1−s 2 2s |a4 |

+

 1−s 1+s a4 + |a4 | d(y2n+1 , u) 2 2   1+s 1−s a1 + |a1 | − a5 d(u, y2n+2 ) − a2 d(y2n+1 , y2n+2 ) . + 2 2

×

1+

As n → ∞, d(u, T v) = 0 so u = T v. Since u = T v ∈ f (X ), there exists w ∈ X such that u = f w. Now we shall show that Sw = u. d( f w, gx2n+2 ) ≥ a1 d(Sw, T x2n ) + a2 d( f w, Sw) + a3 d(gx2n , T x2n ) +a4 d( f w, T x2n ) + a5 d(gx2n , Sw) d(u, y2n ) ≥ a1 d(Sw, y2n+1 ) + a2 d(u, Sw) + a3 d(y2n , y2n+1 ) +a4 d(u, y2n+1 ) + a5 d(y2n , Sw)

(2.9)

By Lemma 1.7 we have  2 ⎧ ⎨ a1 d(Sw, y2n+1 ) ≥ 1+s 2s a1 +  2 ⎩ a d(Sw, y ) ≥ 1+s a + 2n 5 5 2s

1−s 2 2s |a1 | 1−s 2 2s

 

d(Sw, u) +

|a5 | d(Sw, u) +

 s−1 2

a1 −

s+1 2 |a1 |

2

a5 −

s+1 2 |a5 |

 s−1

 

d(y2n+1 , u) d(y2n , u).

(2.9) becomes 

d(u, y2n ) ≥

d(Sw, u) ≤

1 + s2 1 − s2 1 + s2 1 − s2 a1 + |a1 | + a2 + a5 + |a5 | d(Sw, u) 2s 2s 2s 2s +a3 d(y2n , y2n+1 )  s−1 s+1 + a1 − |a1 | + a4 d(y2n+1 , u) 2 2  s−1 s+1 + a5 − |a5 | d(y2n , u). 2 2 1 1+s 2 2s a1



+

1−s 2 2s |a1 | + a2

+

1+s 2 2s a5

+

1−s 2 2s |a5 |

1−s 1+s × (1 + a5 + |a5 |)d(u, y2n ) 2 2   1+s 1−s a1 + |a1 | − a4 d(u, y2n+1 ) − a3 d(y2n , y2n+1 ) . + 2 2 As n → ∞, d(Sw, u) ≤ 0 hence u = Sw. If S = T = I d X , then u = gv = v and u = f w = w; thus u is a common fixed point of f and g. If { f, S} and {g, T } are both weakly compatible pairs, from u = gv = T v = f w = Sw we have

f u = f Sw = S f w = Su = w1 (say) gu = gT v = T gv = T u = w2 (say).

123

H. Olaoluwa, J. O. Olaleru

From (2.1), we have d( f u, gu) ≥ a1 d(Su, T u) + a2 d( f u, Su) + a3 d(gu, T u) + a4 d( f u, T u) + a5 d(gu, Su)(a1 + a4 + a5 − 1)d(w1 , w2 ) ≤ 0 If a1 + a4 + a5 > 1, d(w1 , w2 ) ≤ 0 so w1 = w2 . Now let us prove that u = gu. From (2.1), we have: d(u, gu) = d( f w, gu) ≥ a1 d(Sw, T u) + a2 d( f w, Sw) + a3 d(gu, T u) +a4 d( f w, T u) + a5 d(gu, Sw), i.e., (a1 + a4 + a5 − 1)d(u, gu) ≤ 0. Thus have we proved that u = gu = T u = f u = Su and so u is a common fixed point of the four maps f, g, S, T . From the expanding condition, it follows that the common fixed point is unique. We can prove the same if we suppose S(X ), T (X ) or f (X ) complete.   Note that condition (2.2) imposed on the coefficients ai in Theorem 2.1 is equivalent to the following condition: ⎧ 2 (s+1)(s 2 −s+1) ⎪ a4 − (s+1)(s2s+s−1) |a4 | > s ⎪ a1 + sa2 + a3 + 2s ⎪ ⎪ ⎪ ⎪ a1 + a2 + sa3 + (s+1)(s 2 −s+1) a5 − (s+1)(s 2 +s−1) |a5 | > s ⎪ ⎪ 2s 2s ⎪ ⎨ 2(1 − a2 ) + (1 − s)a4 + (1 + s)|a4 | ≥ 0 (2.10) ⎪ ⎪ 2(1 − a3 ) + (1 − s)a5 + (1 + s)|a5 | ≥ 0 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ 2sa3 + (1 + s )(a1 + a4 ) + (1 − s )(|a1 | + |a4 |) > 0 ⎪ ⎩ 2sa2 + (1 + s 2 )(a1 + a5 ) + (1 − s 2 )(|a1 | + |a5 |) > 0 which in a more compact way can be written as ⎧ 3 2 (s+1)(s 2 −s+1) ⎪ ai+2 − (s+1)(s2s+s−1) |ai+2 | > s ⎨ j=i a j + sai + 2s 2(1 − ai ) + (1 − s)ai+2 + (1 + s)|ai+2 | ≥ 0 ⎪ ⎩ 2sa5−i + (1 + s 2 )(a1 + ai+2 ) + (1 − s 2 )(|a1 | + |ai+2 |) > 0, i ∈ {2, 3} When s = 1 (case of cone metric spaces), (2.11) becomes ⎧ 3 ⎨ j=1 a j + ai+2 − |ai+2 | > 1 1 − ai + |ai+2 | ≥ 0 ⎩ a5−i + (a1 + ai+2 ) > 0, i ∈ {2, 3} When ai ≥ 0 for all i ≥ 2, (2.12) becomes

3 j=1 a j > 1 1 − ai + ai+2 ≥ 0, i ∈ {2, 3}

(2.11)

(2.12)

(2.13)

The following implications hold: (2.10) ⇐⇒ (2.11) ⇐ (2.12) ⇐ (2.13). When X is a cone metric space (i.e. s = 1), and if S = T , f = g, a4 = a5 = 0 in (2.1), with condition (2.13) satisfied, we obtain Theorem 2.1 in [25]. With s = 1, S = T = I d X in (2.1), and condition (2.13) satisfied, we obtain the following corollary (of Theorem 2.1), which is the properly stated second main result of Han and Xu [8]: Corollary 2.2 (Theorem 2.5 of Xu et al. [27]) Let (X, d) be a cone metric space. Suppose mappings f, g : X → X are surjective and such that d( f x, gy) ≥ a1 d(x, y)+a2 d( f x, x)+ a3 d(gy, y) + a4 d( f x, y) + a5 d(gy, x) for all x, y ∈ X , where the real numbers ai satisfy

123

A hybrid class of expansive-contractive mappings in cone. . .

ai ≥ 0 for i ≥ 2, and such that (2.13) holds, i.e., a1 + a2 + a3 > 1, a2 ≤ 1 + a4 and a3 ≤ 1 + a5 . Then f and g have a common fixed point. If in addition, f = g in Corollary 2.2, we obtain the following corollary: Corollary 2.3 (Theorem 2.1 of Xu et al. [27]) Let (X, d) be a cone metric space. Suppose the mapping f : X → X is onto and such that d( f x, f y) ≥ a1 d(x, y) + a2 d( f x, x) + a3 d( f y, y) + a4 d( f x, y) + a5 d( f y, x) for all x, y ∈ X , where the real numbers ai satisfy ai ≥ 0 for i ≥ 2, a1 + a2 + a3 > 1 and a3 ≤ 1 + a5 . Then f has a fixed point. Proof Without loss of generality, one can assume, by symmetry, that a2 = a3 and a4 = a5 . Hence, (2.13) becomes a4 ≥ 0, a1 + a2 + a3 ≥ 1 and a3 ≤ 1 + a5 .   The following example illustrates the validity of our result: Example 2.4 Let E = C 1 ([0, 1], R), P = {ϕ ∈ E : ϕ(t) ≥ 0, t ∈ [0, 1]}, X = [0, 1] and d : X × X → E defined by d(x, y) = |x − y|ψ, where ψ ∈ P, ψ(t) = t + 3, 2 2 ∀t ≥ 0. Let f, g, S, T : X → X be defined by f x = x 2 , gx = x2 , Sx = x6 and 2

x for all x ∈ X . These maps satisfy inequality (2.1) of Theorem 2.1 with a1 = 6 T (x) = 12 and a2 = a3 = a4 = a5 = −1. Every condition in Theorem 2.1 is satisfied and 0 is the unique common fixed point of f, g, S and T .

If the coefficients ai in Theorem 2.1 are made to be functions instead of constants, we obtain the following theorem whose proof is omitted due to its similarity with the proof of Theorem 2.1: Theorem 2.5 Let (X, d) be a cone b-metric space with the coefficient s ≥ 1. Suppose mappings f, g, S, T : X → X satisfy T (X ) ⊂ f (X ), S(X ) ⊂ g(X ) and d( f x, gy) ≥ a1 (x, y)d(Sx, T y) + a2 (x, y)d( f x, Sx) + a3 (x, y)d(gy, T y) +a4 (x, y)d( f x, T y) + a5 (x, y)d(gy, Sx)

(2.14)

for all x, y ∈ X , where a1 , a2 , a3 , a4 , a5 : X × X → R satisfy for all x, y ∈ X  ⎧   1+s 2 1−s 2 1+s ⎪ a +a + a + |a | (x, y) ≥ h1 1−a2 + 1−s 1 3 4 4 ⎪ 2s 2s 2 a4 + 2 |a4 | (x, y) ≥ 0 ⎪ ⎪   ⎪   2 ⎨ 1−s 2 1+s a1 +a2 + 1+s a + |a | (x, y) ≥ h1 1−a3 + 1−s 5 5 2s 2s 2 a5 + 2 |a5 | (x, y) ≥ 0 (2.15)  ⎪ ⎪ 2sa3 + (1 + s 2 )(a1 + a4 ) + (1 − s 2 )(|a1 | + |a4 |) (x, y) > 0 ⎪ ⎪ ⎪  ⎩ 2sa2 + (1 + s 2 )(a1 + a5 ) + (1 − s 2 )(|a1 | + |a5 |) (x, y) > 0 where h ∈ [0, 1s ). (1) If S = T = I d X then f and g have a common fixed point. (2) If the pairs { f, S} and {g, T } are weakly compatible then they both have coincidence points. If in addition 1 < inf x,y {a1 (x, y) + a4 (x, y) + a5 (x, y)}, then f, g, S, T have a unique common fixed point. Theorem 2.5, more general than Theorem 2.1, can be used to obtain several parallel results one of which is the following corollary:

123

H. Olaoluwa, J. O. Olaleru

Corollary 2.6 Let (X, d) be a cone b-metric space with coefficient s ≥ 1. Suppose mappings f, g, S, T : X → X satisfy T (X ) ⊂ f (X ), S(X ) ⊂ g(X ); suppose in addition that f (X ), g(X ), S(X ) or T (X ) is complete and d( f x, gy) ≥ k1 A x,y + k2 [d( f x, T y) + d(gy, Sx)], A x,y ∈ {d(Sx, T y), d( f x, Sx), d(gy, T y)}

(2.16)

> 1 and k2 > 21 . If the pairs for all x, y ∈ X , where the constants k1 , k2 satisfy k1 + k2 (1−s) s { f, S} and {g, T } are weakly compatible then f, g, S, T have a unique common fixed point. Proof If A x,y ∈ {d(Sx, T y), d( f x, Sx), d(gy, T y)}, then k1 A x,y + k2 [d( f x, T y) + d(gy, Sx)] = a1 (x, y)d(Sx, T y) + a2 (x, y)d( f x, Sx) +a3 (x, y)d(gy, T y) + a4 (x, y)d( f x, T y) +a5 (x, y)d(gy, Sx). where a1 , a2 , a3 : X × X → {0, k1 }, a4 , a5 : X × X → {k2 } are chosen such that one and only one of a1 , a2 , a3is simultaneously non  null. The conditions (2.15) of Theorem 2.5 is s(1−k1 +k2 ) sk2 satisfied for h = max sk1 +k2 , ∈ [0, 1). From (2) of Theorem 2.5, the maps have k2 a unique common fixed point under the stated conditions.  

2.2 Relating classes of contractive maps, expansive maps and hybrid maps Let f, g, S, T : X → X be self-maps of a cone metric space (X, d) such that one of f (X ), g(X ), S(X ) and T (X ) is complete and such that T (X ) ⊂ f (X ) and S(X ) ⊂ g(X ). We have: (1) a class of generalized contractive mappings in the classical sense (i.e. in literature; see [2]) if d(Sx, T y) ≤ A1 d( f x, gy) + A2 d(gy, T y) + A3 d( f x, Sx) +A4 d( f x, T y) + A5 d(gy, Sx), ∀x, y ∈ X, where the Ai satisfy Ai ≥ 0, ∀i,

A4 = A5 ,



Ai < 1

(2.17)

(2.18)

(2) a class of generalized expansive mappings in the classical sense (i.e. in literature; see [8] in the case of two maps) if (2.1) and (2.13) hold, i.e. d( f x, gy) ≤ a1 d(Sx, T y) + a2 d( f x, Sx) + a3 d(gy, T y) 3

+a4 d( f x, T y) + a5 d(gy, Sx), ∀x, y ∈ X,

and ai ≥ 0 ∀i ≥ 2, j=1 a j > 1, and 1 − al + al+2 ≥ 0, l = 2, 3. (3) an hybrid class of contractive-expansive mappings if (2.1) and (2.12) hold, i.e. d( f x, gy) ≤ a1 d(Sx, T y) + a2 d( f x, Sx) + a3 d(gy, T y) + a4 d( f x, T y) + a5 d(gy, Sx), ∀x, y ∈ X, ⎧ 3 ⎨ j=1 a j + a4,5 − |a4,5 | > 1 × 1 − ai + |ai+2 | ≥ 0 ⎩ a5−i + (a1 + ai+2 ) > 0, i = 2, 3

123

A hybrid class of expansive-contractive mappings in cone. . .

With no loss of generality,  we can assume in (2.17) and (2.18) that A1  = 0. In deed, if A1 = 0, setting M := i≥2 Ai , (2.17) and (2.18) remain true for A1 = 1−M 2  = 0. A similar argument can be used for a1 . Thus, the transformations A1 a1 = 1,

A2 a1 = −a3 ,

A3 a1 = −a2 ,

A4 a1 = −a4 ,

A5 a1 = −a5

(2.19)

are possible and transform (2.17) to (2.1) and vice-versa. Thus (2.17) ⇐⇒ (2.1). One can easily check that if (2.18) is satisfied then, under (2.19), condition (2.12) is satisfied. Thus the class of generalized contractive mappings in the classical sense [(2.17)– (2.18)] is contained in the hybrid class [(2.1), (2.12)] introduced in this paper. Thus several contractive conditions in literature (see [2,15,16] and the references therein) are generalized by Theorem 2.1. On the other hand, the class of generalized expansive mappings in the classical sense [(2.1), (2.13)] is contained in also contained in the hybrid class since (2.13) ⇒ (2.12) as mentioned earlier. Thus, several expansive conditions in literature (see [1,7,8,13,23,26] and the references therein) are also generalized by Theorem 2.1. Finally, one can prove that the generalized contractive class [(2.17)–(2.18)] coincides with the generalized expansive class [(2.1), (2.13)] if and only if A1 (or a1 )  = 0 and Ai (or ai ) = 0. In deed, considering (2.19), A1 a1 = 1 implies that a1 > 0, since A1 > 0. Since [by (2.18), (2.13)] Ai , ai ≥ 0 for all i ≥ 2, then [A2 a1 = −a3 , A3 a1 = −a2 , A4 a1 = −a4 , A5 a1 = −a5 ] ⇒ Ai = ai = 0, ∀i ≥ 2. The subclass of maps in such case satisfy T (X ) ⊂ f (X ), S(X ) ⊂ g(X ) and d(Sx, T y) ≤ A1 d( f x, gy), A1 < 1 or equivalently d( f x, gy) ≥ a1 d(Sx, T y), a1 = A11 > 1. The coincidence between the contractive class and the expansive class is precisely expressed by the fact that if { f, g, S, T } is an ordered quadruplet of maps, { f, g, S, T } is contractive if and only if {S, T, f, g} is expansive. In particular, if f is a contraction (see [4]) then { f, f, 1, 1} are contractive in the classical sense and {1, 1, f, f } are expansive; conversely, if f is a surjective expansion (see [26]) then { f, f, 1, 1} are expansive in the classical sense and {1, 1, f, f } are contractive.

3 Applications 3.1 Multipled fixed points and fixed points in product spaces The following definitions and propositions were stated by Olaoluwa and Olaleru [18] as extension of the concepts of coupled (e.g. [5,17,19]), tripled and even quadruple fixed points. Let X be a nonempty set. Define, for any vector x = (x1 , x2 , . . . , xm ) ∈ X m , the circular matrix of x by ⎛ ⎞ x1 x2 . . . xm−2 xm−1 xm ⎜ x2 x3 . . . xm−1 xm x1 ⎟ ⎜ ⎟ ⎜ x2 ⎟ t (x) := ⎜ x3 x4 . . . xm x1 ⎟. ⎜ .. .. .. .. ⎟ .. ⎝. . . . ⎠ . xm x1 . . . xm−3 xm−2 xm−1 In the sequel, ti (x) denotes the i-th line of t (x) and ti j (x) the (i, j)-th element of the matrix. Definition 3.1 [18] Let X be a nonempty set and F : X m → X and g : X → X be two mappings. An element x = (x1 , x2 , . . . , xm ) ∈ X m is said to be a coincidence point of m-order (or simply, multipled coincidence point) of F and g if

123

H. Olaoluwa, J. O. Olaleru

⎧ F(x1 , x2 , . . . , xm−1 , xm ) ⎪ ⎪ ⎪ ⎪ F(x ⎪ 2 , x3 , . . . , xm , x1 ) ⎨ F(x3 , x4 , . . . , xm , x1 , x2 ) ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎩ F(xm , x1 , x2 , . . . , xm−1 )

= g(x1 ) = g(x2 ) = g(x3 ) or equivalently F(t (x)) = g(x ), ∀i ∈ [1, m]. i i = g(xm )

If in addition, all the xi are fixed points of g, then x is said to be a common fixed point of m-order (or common multipled fixed point) of F and g. If g = I d X , then x is said to be a fixed point of m-order or multipled fixed point of F. When m = 1, 2, 3, 4, we obtain the notions of fixed points, coupled fixed points, tripled fixed points and quadruple fixed points respectively. Example 3.2 [18] Let X = R and F : X m → X be defined for all x = (x1 , x2 , . . . , xm ) by F(x) = 2x1 + x2 + x3  + · · · xm − 1. The system F(t  i (x)) = xi ∀i ∈ {1, . . . , m}, is satisfied by all x such that mj=1 x j = 1. In particular, m1 , . . . , m1 and (1, 0, . . . , 0) are both multipled fixed points of F. Definition 3.3 [18] Let X be a nonempty set and F : X m → X and g : X → X be two mappings. The mappings F and g are called (w1 ) w-compatible if g(F(x1 , x2 , . . . , xm )) = F(gx1 , gx2 , . . . , gxm ) at any coincidence point (x1 , x2 , . . . , xm ) of F and g. (w2 ) w*-compatible if g(F(x, x, . . . , x)) = F(gx, gx, . . . , gx) whenever gx = F(x, x, . . . , x). Consider the mappings F : X m → X and g : X → X and their associate mappings ˜ F : X m → X m and g˜ : X m → X m defined for all x = (x1 , x2 , . . . , xm ) ∈ X m by

˜ F(x) = (F(t1 (x)), F(t2 (x)), . . . , F(tm (x))) (3.1) g(x) ˜ = (gx1 , gx2 , . . . , gxm ) Lemma 3.4 [18] (i) An element x = (x1 , x2 , . . . , xm ) ∈ X m is a multipled fixed point of F or multipled coincidence point (or common multipled fixed point) of F and g if and only if it is a fixed point of F˜ or multipled coincidence point (or common multipled fixed point) of F˜ and g. ˜ (ii) The maps F and g are w-compatible if and only if F˜ and g˜ are w-compatible in X m . Proposition 3.5 [18] (i) If x is unique as multipled coincidence (or common multipled fixed) point of F and g, then x = ti (x) for all i ∈ {2, 3, . . . , m}, which leads to x1 = x2 = · · · = xm . (ii) If F and g are w*-compatible mappings with only one multipled coincidence point, they are also w-compatible.  Example 3.6 Let X = R and F : X m → X be defined by F(x) = 1 − m + mj=1 x j . F(ti (x)) = xi ∀i ∈ {1, . . . , m} ⇐⇒ j=i x j = m −1. The determinant of the system 0 1 1 ... 1 1 1 0 1 ... 1 1 .. .. .. .. .. .. = (−1)m−1 (m − 1)  = 0 hence the system has a unique solution, (1, . . . , 1), . . . . . . 1 1 1 ... 1 0 which is the unique multipled fixed point of F. The notions of coupled fixed points, tripled fixed points and multipled fixed points in general are relative to mappings defined on X m , with m ≥ 1. It is therefore of interest to equip such higher dimensional spaces with the same structure of cone b-metric spaces. The following subsection arises from this motivation.

123

A hybrid class of expansive-contractive mappings in cone. . .

3.2 Finite product cone b-metric spaces Definition 3.7 Let (X i , di ), i ∈ {1, 2, . . . m} be m cone b-metric spaces with respect to cones Pi and coefficients si , where Pi ⊂ E for all i and Pi ∩ (−P j ) = {0} for all i, j. The ! set Z = i=m i=1 X i together with d : Z × Z → E defined by m

di (xi , yi ), ∀x = (x1 , x2 , . . . , xm ), y = (y1 , y2 , . . . , ym ) d(x, y) = i=1

m is a cone b-metric space with respect to cone P = i=1 Pi and coefficient s = max si . Z is called product cone b-metric space. When X i = X for each i ∈ {1, 2, . . . m}, where (X, d) is a cone b-metric space with respect to cone P ⊂ E, we define the product cone b-metric space X m with respect to P by considering the cone b-metric D : X m × X m → E by m

d(xi , yi ), x = (xi )1≤i≤m , y = (yi )1≤i≤m . (3.2) D(x, y) = i=1

Convergence of sequences in a product cone b-metric space and convergence of their coordinates are equivalent as expressed in the next Proposition, easy to prove. Proposition 3.8 Let (X, d) be a cone b-metric space and (X m , D) the product cone b-metric space. ( p1 ) A sequence {xn } = {(xn1 , xn2 , . . . , xnm )} converges to x = (x 1 , x 2 , . . . , x m ) if and only if the sequences {xni } converge to x i for all i ∈ {1, 2, . . . , m}. ( p2 ) A sequence {xn } = {(xn1 , xn2 , . . . , xnm )} is a Cauchy sequence in X m if and only if the sequences {xni } are Cauchy sequences for all i ∈ {1, 2, . . . , m}. ( p3 ) (X m , D) is complete if and only if (X, d) is complete.

3.3 Consequences The following result on common multipled fixed points of a class of hybrid mappings can be obtained from the main result discussed in Sect. 2. The technique of proof is similar to the one considered in [18]. Theorem 3.9 Let (X, d) be a cone b-metric space with the coefficient s ≥ 1, f, g : X m → X and S, T : X → X be four mappings such that T (X ) ⊂ f (X m ), S(X ) ⊂ g(X m ) and m

ai d(Sxi , T u i ) + b1 d( f x, Sx1 ) + b2 d(gu, T u 1 ) d( f x, gu) ≥ i=1

+b3 d( f x, T u 1 ) + b4 d(gu, Sx1 )

(3.3)

where ai , (i = 1, 2, . . . , m), for all x = (x1 , x2 , . . . , xm ), u = (u 1 , u 2 , . . . , u m ) ∈ b1 , b2 , b3 satisfy ⎧ m 2 (s+1)(s 2 −s+1) ⎪ b3 − (s−1)(s2s−s−1) |b3 | > s i=1 ai + sb1 + b2 + ⎪ 2s ⎪ ⎪ m 2 ⎪ (s+1)(s 2 −s+1) ⎪ ⎪ b4 − (s−1)(s2s−s−1) |b4 | > s ⎪ i=1 ai + b1 + sb2 + 2s ⎪ ⎨ 2(1 − b1 ) + (1 − s)b3 + (1 + s)|b3 | ≥ 0 ⎪ ⎪ 2(1 − b2 ) + (1 − s)b4 + (1 + s)|b4 | ≥ 0 ⎪ ⎪ m m ⎪ ⎪ ⎪ 2sb2 + (1 + s 2 )( i=1 ai + b3 ) + (1 − s 2 )(| i=1 ai | + |b3 |) > 0 ⎪ ⎪ ⎩ m m 2sb1 + (1 + s 2 )( i=1 ai + b4 ) + (1 − s 2 )(| i=1 ai | + |b4 |) > 0 Xm,

123

H. Olaoluwa, J. O. Olaleru

(1) If S = T = I d X then f and g have a common multipled fixed point. (2) If the pairs { f, S} and {g, T } are weakly m compatible then they both have multipled coincidence points. If in addition 1 < i=1 ai + b3 + b4 , then f, g, S, T have a unique common multipled fixed point (u, . . . , u) ∈ X m and for every (x01 , x02 , . . . , x0m ) ∈ X m , the sequences {xn } = {(xn1 , xn2 , . . . , xnm )} ⊂ X m and {u n } = {u 1n , u 2n , . . . , u m n } defined by  i u i2n+1 := f ti x2n+1 = T x2n ∀i = 1, 2, . . . , m (3.4) i u i2n+2 := gti x2n+2 = Sx2n+1 converge both to (u, u, . . . , u). Proof From (3.3) and by simple interchanges, we have d( f tk x, gtk u) ≥

m

ai d(Sxi , T u i ) + b1 d( f tk x, Sxk ) + b2 d(gtk u, T u k ))

i=1

+b3 d( f tk x, T u k ) + b4 d(gtk u, Sxk ) for every k ∈ {1, . . . , m}. Summing the m inequalities, " m # m m m





d( f (ti x), g(ti u)) ≥ ai d(Sxi , T u i ) + b1 d( f ti x, Sxi ) i=1

i=1

+b2

m

i=1

i=1

d(gti u, T u i ) + b3

i=1 m

d( f ti x, T u i ) + b4

i=1

m

d(gti u, Sxi )

i=1

In view of (3.1) and (3.2), " m #

˜ ˜ + b2 D(gu, ˜ T˜ u) + b1 D( f˜x, Sx) D( f x, gu) ˜ ≥ ai D( Sx, ˜ T˜ u) i=1

˜ +b3 D( f˜x, T˜ u) + b4 D(gu, ˜ Sx), where f˜, g, ˜ S˜ and T˜ are defined for all x = (xi )1≤i≤m ∈ X m and u = (u i )1≤i≤m ∈ X m by ⎧ f˜(x) = ( f t1 x, f t2 x, . . . , f tm x) ⎪ ⎪ ⎨ g(x) ˜ = (gt1 x, gt2 x, . . . , gtm x) ˜ S(u) = (Su 1 , Su 2 , . . . , Su m ) ⎪ ⎪ ⎩ ˜ T (u) = (T u 1 , T u 2 , . . . , T u m ). Condition (2.1) f˜, g, ˜ S˜ and T˜ .

is satisfied for m T (X ) ⊂ f (X ) ⇒ T˜ (X m ) ⊂ f˜(X m ) We have ˜ m ) ⊂ g(X S(X ) ⊂ g(X m ) ⇒ S(X ˜ m ). m m ˜ m ) or ˜ m ), S(X If one of f (X ), g(X ), S(X ) or T (X ) is complete then f˜(X m ), g(X m m T˜ (X ) is complete in X , hence by Theorem 2.1 applied to the product cone b-metric space ˜ and {g, X m , the pairs { f˜, S} ˜ T˜ } have unique points of coincidence which are unique multiple coincidence points of { f, S} and {g, T }. (1) If S = T = I d X then S˜ = T˜ , hence by Theorem 2.1, f˜ and g˜ have a common multipled fixed point which is a multipled fixed point of f and g.

123

A hybrid class of expansive-contractive mappings in cone. . .

˜ and {g, (2) If the pairs { f, S} and {g, T } are weakly compatible then { f˜, S} ˜ T˜ } are w˜ ˜ ˜ compatible. By Theorem 2.1, f , g, ˜ S and T have a coincident point, which is multipled m ˜ T˜ coincidence point of f, g, S and T . If in addition 1 < i=1 ai +b3 +b4 , then f˜, g, ˜ S, have a unique common fixed point which is the unique common multipled fixed point of f, g, S, T . Because of the uniqueness, it is of the form (u, . . . , u), for some u ∈ X . Also, from Theorem 2.1, for any (x01 , x02 , . . . , x0m ) ∈ X m , the sequences {xn } = {(xni )1≤i≤m } and {u n } = {(u in )1≤i≤m } defined by

u 2n+1 := f˜x2n+1 = T˜ x2n (3.5) ˜ 2n+1 u 2n+2 := gx ˜ 2n+2 = Sx converge to (u, . . . u) ∈ X m and (3.5) is equivalent to  i u i2n+1 := f ti x2n+1 = T x2n i u i2n+2 := gti x2n+2 = Sx2n+1

∀i = 1, 2, . . . , m

The sequences {xn } and {u n } thus defined in (3.4) converge to (u, u, . . . , u) ∈ X m .

 

Acknowledgments The authors are grateful to the referees for their helpful suggestions contributing to the improvement of the paper.

References 1. Aage, C.T., Salunke, J.N.: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 27(6), 1101–1106 (2011) 2. Abbas, M., Rhoades, B.E., Nazir, T.: Common fixed points for four maps in cone metric spaces. Appl. Math. Comput. 216, 80–86 (2010) 3. Bakhtin, I.A.: The contraction mapping principle in almost metric spaces. Funct. Anal. Gos. Ped. Inst. Unian. 30, 26–37 (1989) 4. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922) 5. Bhashkar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal. Theory Methods Appl. 65(7), 1379–1393 (2006) 6. Ciric, L.B.: Generalized contractions and fixed-point theorems. Publ. l’Inst. Math. (Beograd) (NS) 12(26), 19–26 (1971) 7. Daffer, P.Z., Kaneko, H.: On expansive mappings. Math. Jpn. 37(1992), 733–735 (1992) 8. Han, Y., Xu, S.: Some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 3, 1–9 (2013) 9. Huang, H., Xu, S.: Correction: Fixed point theorems of contractive mappings in cone b-metric spaces and applications. Fixed Point Theory Appl. 2014, 55 (2014) 10. Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2), 1468–1476 (2007) 11. Hussain, N., Shah, M.H.: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, 1677–1684 (2011) 12. Jungck, G., Rhoades, B.E.: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 29(3), 227–238 (1998) 13. Kumar, S., Garg, S.K.: Expansion mapping theorems in metric spaces. Int. J. Contemp. Math. Sci. 4(36), 1749–1758 (2009) 14. Nashine, H.K., Kadelburg, Z., Radenovic, S.: Coupled common fixed point theorems for w*-compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 218, 5422–5432 (2012) 15. Olaleru, J.O.: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 1–10 (2009) (Article ID 657914) 16. Olaleru, J.: Common fixed points of three self-mappings in cone metric spaces. Appl. Math. E-notes 11, 41–49 (2011)

123

H. Olaoluwa, J. O. Olaleru 17. Olaoluwa, H., Olaleru, J.O., Chang, S.S.: Coupled fixed point theorems for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2013, 68 (2013) 18. Olaoluwa, H., Olaleru, J.: Multipled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2014, 43 (2014) 19. Olaoluwa, H., Olaleru, J.O.: An approach to the approximation of common coupled fixed points of contractive maps in metric-type spaces. Adv. Fixed Point Theory 5(1), 13–31 (2015) 20. Olaoluwa, H., Olaleru, J.O.: On common fixed points and multipled fixed points of contractive mappings in metric-type spaces (2015). doi:10.1016/j.jnnms.2015.06.001 21. Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977) 22. Rhoades, B.E.: Contractive definitions. In: Rassias, T.M. (ed.) Nonlinear Analysis, pp. 513–526. Word Scientific Publishing Company, NJ (1988) 23. Sahin, I., Telci, M.: A theorem on common fixed points of expansion type mappings in cone metric spaces. An. St. Univ. Ovid. Const. 18, 329–336 (2010) 24. Samet, B., Vetro, C.: Coupled fixed point, F-invariant set and fixed point of N -order. Ann. Funct. Anal. 1(2), 46–56 (2011) 25. Shatanawi, W., Awawdeh, F.: Some fixed and coincidence point theorems for expansive maps in cone metric spaces. Fixed Point Theory Appl. 2012, 19 (2012) 26. Wang, S.-Z., Li, B.-Y., Gao, Z.-M., Lseki, K.: Some fixed points of expansion mappings. Math. Jpn. 29, 631–636 (1984) 27. Xu, S., Cheng, S., Han, Y.: Correction: some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 2014, 178 (2014)

123