() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 2, 2012 pp. 151-166 Common fixed points for generalized (ψ,φ)-weak contractions in ordered cone metric spaces Hemant Kumar Nashine and Hassen Aydi Abstract The purpose of this paper is to establish coincidence point and common fixed point results for four maps satisfying generalized (ψ,φ)-weak con- tractions in partially ordered cone metric spaces. Also, some illustrative examples are presented. 2010 MSC: 54H25, 47H10. Keywords: Coincidence point, common fixed point, weakly contractive con- dition, dominating map, dominated map, ordered set, cone met- ric space. 1. Introduction One of the simplest and useful results in the fixed point theory is the Banach– Caccioppoli contraction mapping principle. In the last years, this principal has been generalized in many directions to generalized structures as cone metrics, partial metric spaces and quasi-metric spaces has received a lot of attention. Fixed point theory in K-metric and K-normed spaces was developed by Perov et al. [24], Mukhamadijev and Stetsenko [16], Vandergraft [33]. For more details on fixed point theory in K-metric and K-normed spaces, we refer the reader to fine survey paper of Zabrejko [34]. The main idea was to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric. In 2007, Huang and Zhang [13] reintroduced such spaces under the name of cone metric spaces and reintroduced definition of convergent and Cauchy sequences in the terms of interior points of the underlying cone. They also proved some fixed point theorems in such spaces in the same work. After that, fixed point points in K-metric spaces have been the subject of intensive research (see, e.g., 152 H. K. Nashine and H. Aydi [1, 3, 7, 11, 13, 14, 15, 16, 23, 25, 30]). The main motivation for such research is a point raised by Agarwal [4], that the domain of existence of a solution to a system of first-order differential equations may be increased by considering generalized distances. Recently, Wei-Shih Du [12] used the scalarization function and investigated the equivalence of vectorial versions of fixed point theorems in K-metric spaces and scalar versions of fixed point theorems in metric spaces. He showed that many of the fixed point theorems for mappings satisfying contractive conditions of a linear type in K-metric spaces can be considered as the corollaries of corresponding theorems in metric spaces. Nevertheless, the fixed point theory in K-metric spaces proceeds to be actual, since the method of scalarization cannot be applied for a wide class of mappings satisfying contractive conditions more general than contractive conditions of a linear type. On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. One of results in this direction was given by Ran and Reurings [26] who presented its applications to matrix equations. Subsequently, Nieto and Rodŕıguez-López [22] extended the result of Ran and Reurings for nondecreasing mappings and applied it to obtain a unique solution for a first order ordinary differential equation with periodic boundary condi- tions. Thereafter, many authors obtained many fixed point theorems in ordered metric spaces. For more details, see [5, 6, 8, 10, 17, 19, 20, 21, 22, 27, 29, 31] and the references cited therein. In this paper, an attempt has been made to derive some common fixed point theorems for four maps involving generalized (ψ,φ)-weak contractions in or- dered cone metric spaces. The presented theorems generalize, extend and im- prove some recent fixed point results in K-metric spaces. 2. Preliminaries In what follows, we recall some notations and definitions that will be utilized in our subsequent discussion. Let E be always a Banach space. Definition 2.1. A non-empty subset K of E is called a cone if and only if (i) K = K, K 6= 0E where K is the closure of K, (ii) a,b ∈ R, a,b ≥ 0, x,y ∈ K ⇒ ax + by ∈ K, (iii) K ∩ (−K) = {0E}. A cone K defines a partial ordering ≤E in E by x ≤E y if and only if y − x ∈ K. We shall write x m > N. We say that {xn} converges to x ∈ X if for every c ∈ E with 0E ≪ c, there exists N ∈ N such that d(xn,x) ≪ c for all n > N. In this case, we denote xn → x as n → ∞. A cone metric space (X,d) is said to be complete if every Cauchy sequence in X is convergent in X. Definition 2.4. Let f : E → E be a given mapping. We say that f is a monotone non-decreasing mapping with respect to ≤E if for every x,y ∈ E, x ≤E y implies fx ≤E fy. Definition 2.5 ([9]). Let ψ : K → K be a given function. (i) We say that ψ is strongly monotone increasing if for x,y ∈ K, we have x ≤E y ⇐⇒ ψ(x) ≤E ψ(y). (ii) ψ is said to be continuous at x0 ∈ K if for any sequence {xn} in K, we have ‖xn − x0‖E → 0 =⇒ ‖ψ(xn) − ψ(x0)‖E → 0. Definition 2.6. Let (X,d) be a cone metric space and f,g : X → X. If w = fx = gx, for some x ∈ X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. If w = x, then x is a common fixed point of f and g. The pair {f,g} is said to be compatible if and only if lim n→+∞ d(fgxn,gfxn) = 0, whenever {xn} is a sequence in X such that lim n→+∞ fxn = lim n→+∞ gxn = t for some t ∈ X. Definition 2.7 ([2]). Let f and g be two self-maps defined on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point. Definition 2.8. Let X be a nonempty set. Then (X,d,�) is called an ordered cone metric space if and only if (i) (X,d) is a metric space, (ii) (X,�) is a partial order. Definition 2.9. Let (X,�) be a partial ordered set. Then x,y ∈ X are called comparable if x � y or y � x holds. Definition 2.10 ([2]). Let (X,�) be a partially ordered set. A mapping f is called dominating if x � fx for each x in X. 154 H. K. Nashine and H. Aydi Example 2.11 ([2]). Let X = [0,1] be endowed with usual ordering and f : X → X be defined by fx = n √ x. Since x ≤ x13 = fx for all x ∈ X. Therefore f is a dominating map. Definition 2.12 ([18]). Let (X,�) be a partially ordered set. A mapping f is called dominated if fx � x for each x in X. Example 2.13 ([18]). Let X = [0,1] be endowed with usual ordering and f : X → X be defined by fx = xn for all n ≥ 1. Since fx = xn ≤ x for all x ∈ X. Therefore f is a dominated map. 3. Common fixed point results First, let Ψ be the set of functions ψ : K → K such that (i) ψ is continuous; (ii) ψ(t) = 0E if and only if t = 0E; (iii) ψ is strongly monotone increasing. Also, let Φ be the set of functions φ : int(K)∪{0E} → int(K)∪{0E} such that (i’) φ is continuous; (ii’) φ(t) = 0E if and only if t = 0E; (iii’) φ(t) ≪E t for all t ∈ int(K); (iv’) either φ(t) ≤E d(x,y) or d(x,y) ≤E φ(t) for t ∈ int(K) ∪ {0E} and x,y ∈ X. The following Lemma will be useful later. Lemma 3.1. [30]. Let E be a Banach space, {an}, {bn} and {cn} are sequences in E such that bn → b ∈ E, cn → c ∈ E as n → +∞. Suppose also that an ∈ {bn,cn} for all n ∈ N. Then there exists a subsequence {an(p)} of {an} such that an(p) → a ∈ {b,c} as p → +∞. Our first result is the following. Theorem 3.2. Let (X,d,�) be an ordered complete cone metric space over a solid cone K. Let T,S,I,J : X → X be given mappings satisfying for every pair (x,y) ∈ X × X such that x and y are comparable, ψ(d(Sx,Ty)) ≤E ψ(Θ(x,y)) − φ(Θ(x,y)),(3.1) where Θ(x,y) ∈ {d(Ix,Jy), 1 2 [d(Ix,Sx)+d(Jy,Ty)], 1 2 [d(Ix,Ty)+d(Jy,Sx)]}, ψ ∈ Ψ and φ ∈ Φ. Suppose that (i) TX ⊆ IX and SX ⊆ JX; (ii) I and J are dominating maps and S and T are dominated maps; (iii) If for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. Also, assume either (a) {S,I} are compatible, S or I is continuous and {T,J} are weakly com- patible or Common fixed points in ordered cone metric spaces 155 (b) {T,J} are compatible, T or J is continuous and {S,I} are weakly com- patible. Then S,T,I and J have a common fixed point. Proof. Let x0 be an arbitrary point in X. Since TX ⊆ IX and SX ⊆ JX, we can define the sequences {xn} and {yn} in X by (3.2) y2n−1 = Sx2n−2 = Jx2n−1, y2n = Tx2n−1 = Ix2n, ∀n ∈ N. By given assumptions x2n+1 � Jx2n+1 = Sx2n � x2n and x2n � Ix2n = Tx2n−1 � x2n−1. Thus, for all n ≥ 0, we have (3.3) xn+1 � xn. Putting x = x2n+1 and y = x2n, from (3.3) and the considered contraction (3.1), we have ψ(d(y2n+1,y2n+2)) = ψ(d(Sx2n,Tx2n+1))(3.4) ≤E ψ(Θ(x2n,x2n+1)) − φ(Θ(x2n,x2n+1)) ≤E ψ(Θ(x2n,x2n+1)). The function ψ is strongly increasing, so we get that (3.5) d(y2n+1,y2n+2) ≤E Θ(x2n,x2n+1). Note that Θ(x2n,x2n+1) ∈ {d(Ix2n,Jx2n+1), 1 2 [d(Ix2n,Sx2n) + d(Jx2n+1,Tx2n+1)], 1 2 [d(Ix2n,Tx2n+1) + d(Sx2n,Jx2n+1)]} = {d(y2n,y2n+1), 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], 1 2 [d(y2n,y2n+2) + d(y2n+1,y2n+1)]} = {d(y2n,y2n+1), 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], 1 2 d(y2n,y2n+2)}. If Θ(x2n,x2n+1) = d(y2n,y2n+1), (3.5) becomes d(y2n+1,y2n+2) ≤E d(y2n,y2n+1). If Θ(x2n,x2n+1) = 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], then (3.5) becomes d(y2n+1,y2n+2) ≤E 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)], so d(y2n+1,y2n+2) ≤E d(y2n,y2n+1). If Θ(x2n,x2n+1) = 1 2 d(y2n,y2n+2), by (3.4) and a triangular inequality, we find that d(y2n+1,y2n+2) ≤E 1 2 d(y2n,y2n+2) ≤E 1 2 d(y2n,y2n+1) + 1 2 d(y2n+1,y2n+2), so d(y2n+1,y2n+2) ≤E d(y2n,y2n+1). In all cases, we obtained that (3.6) d(y2n+1,y2n+2) ≤E Θ(x2n,x2n+1) ≤E d(y2n,y2n+1). 156 H. K. Nashine and H. Aydi Similarly, we have (3.7) d(y2n+1,y2n) ≤E Θ(x2n,x2n−1) ≤E d(y2n,y2n−1). By (3.6) and (3.7), we get that (3.8) d(yn+1,yn) ≤E d(yn,yn−1) for all n ≥ 1. It follows that the sequence {d(yn,yn+1)} is monotone non-increasing. Since K is a regular cone and 0E ≤E d(yn,yn+1) for all n ≥ 0, there exists r ≥E 0E such that d(yn,yn+1) → r as n → +∞. By (3.6) and (3.7), we have lim n→+∞ Θ(x2n,x2n+1) = lim n→+∞ Θ(x2n,x2n−1) = r. Now, letting n → +∞ in (3.4) and using the continuity property of ψ and φ, we get ψ(r) ≤ ψ(r) − φ(r), which yields that φ(r) = 0E. Since φ(t) = 0E ⇐⇒ t = 0E, then r = 0E. Therefore, (3.9) lim n→+∞ d(yn,yn+1) = 0. Now, we will show that {yn} is a Cauchy sequence in the cone metric space (X,d). We proceed by negation and suppose that {y2n} is not a Cauchy se- quence. Then, there exists ε > 0 for which we can find two sequences of positive integers {m(i)} and {n(i)} such that for all positive integer i, (3.10) n(i) > m(i) > i, d(y2m(i),y2n(i)) ≥E ε, d(y2m(i),y2n(i)−2) 0. If lim i→+∞ Θ(x2n(i),x2m(i)−1) = ε, then using similar arguments, we obtain that ψ(ε) ≤E ψ(ε) − φ(ε), so φ(ε) = 0E, which is a contradiction. Thus {y2n} is a Cauchy sequence in X, so {yn} is also a Cauchy sequence in X. Finally, we shall prove existence of a common fixed point of the four map- pings I,J,S and T . Since X is complete, there exists a point z in X, such that {y2n} converges to z. Therefore, (3.17) y2n+1 = Jx2n+1 = Sx2n → z as n → ∞ and (3.18) y2n+2 = Ix2n+2 = Tx2n+1 → z as n → ∞. 158 H. K. Nashine and H. Aydi Assume that (a) holds. Suppose that I is continuous. Since the pair {S,I} is compatible, we have (3.19) lim n→∞ SIx2n+2 = lim n→∞ ISx2n+2 = Iz. Also, Ix2n+2 = Tx2n+1 � x2n+1. Now, by (3.1) (3.20) ψ(d(SIx2n+2,Tx2n+1)) ≤E ψ(Θ(Ix2n+2,x2n+1)) − φ(Θ(Ix2n+2,x2n+1)), where Θ(Ix2n+2,x2n+1)) ∈ {d(IIx2n+2,Jx2n+1), 1 2 [d(IIx2n+2,SIx2n+2)+ d(Jx2n+1,Tx2n+1)], 1 2 [d(IIx2n+2,Tx2n+1) + d(SIx2n+2,Jx2n+1)]}. By (3.9), (3.17), (3.18) and (3.19), we get that lim n→∞ d(IIx2n+2,Jx2n+1) = lim n→∞ 1 2 [d(IIx2n+2,Tx2n+1)+d(SIx2n+2,Jx2n+1)] = d(Iz,z), lim n→∞ 1 2 [d(IIx2n+2,SIx2n+2) + d(Jx2n+1,Tx2n+1)] = 0E. By Lemma 3.1, there exists a subsequence of {Θ(Ix2n+2,x2n+1)} still denoted Θ(Ix2n+2,x2n+1) such that from the above limits (3.21) lim n→+∞ Θ(Ix2n+2,x2n+1) ∈ {0E,d(Iz,z)}. If lim n→+∞ Θ(Ix2n+2,x2n+1) = 0E, then then letting n → +∞ in (3.20) and using the fact that lim n→∞ d(SIx2n+2,Tx2n+1) = d(Iz,z), and the continuities of ψ and φ, we obtain ψ(d(Iz,z)) ≤E ψ(0E) − φ(0E), so ψ(d(Iz,z)) = 0E, which yields that d(Iz,z) = 0E, so Iz = z. If lim n→+∞ Θ(Ix2n+2,x2n+1) = d(Iz,z), using the similar arguments we get that ψ(d(Iz,z)) − ψ(d(Iz,z)) − φ(d(Iz,z)), so similarly, Iz = z. In each case, we obtained (3.22) Iz = z. Now, Tx2n+1 � x2n+1 and Tx2n+1 → z as n → ∞, so by assumption [(iii)] we have z � x2n+1. From (3.1), (3.23) ψ(d(Sz,Tx2n+1)) ≤E ψ(d(Θ(z,x2n+1))) − φ(d(Θ(z,x2n+1))), Common fixed points in ordered cone metric spaces 159 where Θ(z,x2n+1) ∈ {d(Iz,Jx2n+1), 1 2 [d(Iz,Sz) + d(Jx2n+1,Tx2n+1)], 1 2 [d(Iz,Tx2n+1) + d(Sz,Jx2n+1)]} = {d(z,Jx2n+1), 1 2 [d(z,Sz) + d(Jx2n+1,Tx2n+1)], 1 2 [d(z,Tx2n+1) + d(Sz,Jx2n+1)]}. By (3.9), (3.17), (3.18) and (3.19), we get that lim n→∞ 1 2 [d(z,Sz)+d(Jx2n+1,Tx2n+1)] = 1 2 d(z,Sz) = lim n→∞ 1 2 [d(IIx2n+2,Tx2n+1) +d(SIx2n+2,Jx2n+1)]}, lim n→∞ d(z,Jx2n+1) = 0E. By Lemma 3.1, there exists a subsequence of {Θ(z,x2n+1)} still denoted Θ(Ix2n+2,x2n+1) such that from the above limits (3.24) lim n→+∞ Θ(Ix2n+2,x2n+1) ∈ {0E, 1 2 d(Sz,z)}. If lim n→+∞ Θ(Ix2n+2,x2n+1) = 0E, then then letting n → +∞ in (3.24) and using the fact that lim n→∞ d(Sz,Tx2n+1) = d(Sz,z), and the continuities of ψ and φ, we obtain ψ(d(Sz,z)) ≤E ψ(0E) − φ(0E), so ψ(d(Iz,z)) = 0E, which yields that Sz = z. If lim n→+∞ Θ(Ix2n+2,x2n+1) = 1 2 d(Sz,z) and using the similar arguments, we get that ψ(d(Sz,z)) ≤E ψ( 1 2 d(Sz,z)) − φ(1 2 d(Sz,z)) ≤E ψ( 1 2 d(Sz,z)), so d(Sz,z) ≤E 12d(Sz,z), which holds unless d(Sz,z) = 0E, so (3.25) Sz = z. Since S(X) ⊆ J(X), there exists a point w ∈ X such that Sz = Jw. Suppose that Tw 6= Jw. Since w � Jw = Sz � z implies w � z. From (3.1), we obtain (3.26) ψ(d(Jw,Tw)) = ψ(d(Sz,Tw)) ≤E ψ(Θ(z,w)) − φ(Θ(z,w)), where Θ(z,w) ∈ {d(Iz,Jw), 1 2 [d(Iz,Sz) + d(Jw,Tw)], 1 2 [d(Iz,Tw) + d(Sz,Jw)]} = {0E, 1 2 d(Jw,Tw)}. 160 H. K. Nashine and H. Aydi If Θ(z,w) = 0E, we easily deduce from (3.26) that d(Jw,Tw) = 0E. If Θ(z,w) = d(Jw,Tw), similarly we get that d(Jw,Tw) = 0E. Thus, we obtained (3.27) Jw = Tw. Since T and J are weakly compatible, Tz = TSz = TJw = JTw = JSz = Jz. Thus, z is a coincidence point of T and J. Now, since Sx2n � x2n and Sx2n → z as n → ∞, so by assumption [(iii)], z � x2n. Then, from (3.1) (3.28) ψ(d(Sx2n,Tz)) ≤E ψ(Θ(x2n,z)) − φ(Θ(x2n,z)), where Θ(x2n,z) ∈ {d(Ix2n,Jz), 1 2 [d(Ix2n,Sx2n) + d(Jz,Tz)], 1 2 [d(Ix2n+1,Tz) + d(Sx2n,Jz)]} = {d(Ix2n,Tz), 1 2 d(Ix2n,Sx2n), 1 2 [d(Ix2n+1,Tz) + d(Sx2n,Tz)]} We have lim n→∞ d(Ix2n,Tz) = lim n→∞ 1 2 [d(Ix2n+1,Tz) + d(Sx2n,Tz)] = d(z,Tz), and lim n→∞ d(Ix2n,Sx2n) = 0, lim n→∞ d(Sx2n,Tz) = d(z,Tz). By Lemma 3.1, there exists a subsequence of {Θ(x2n,z))} still denoted Θ(x2n,z) such that from the above limits (3.29) lim n→+∞ Θ(x2n,z) ∈ {0E,d(z,Tz)}. Similarly, letting n → ∞ in (3.28) and having in mind (3.29), we get that (3.30) z = Tz. Therefore Sz = Tz = Iz = Jz = z, so z is a common fixed point of I, J, S and T . The proof is similar when S is continuous. Similarly, the result follows when (b) holds. � Now, it is easy to state a corollary of Theorem 3.2 involving a contraction of integral type. Corollary 3.3. Let T,S,I and J satisfy the conditions of Theorem 3.2, except that condition (3.1) is replaced by the following: there exists a positive Lebesgue integrable function u on R+ such that ∫ ε 0 u(t)dt > 0 for each ε > 0 and that (3.31) ∫ ψ(d(Sx,Ty)) 0 u(t)dt ≤ ∫ ψ(Θ(x,y)) 0 u(t)dt − ∫ φ(Θ(x,y)) 0 u(t)dt. Then, S,T,I and J have a common fixed point. Common fixed points in ordered cone metric spaces 161 Corollary 3.4. Let (X,d,�) be an ordered complete cone metric space over a solid cone K. Let T,S,I : X → X be given mappings satisfying for every pair (x,y) ∈ X × X such that x and y are comparable, ψ(d(Sx,Ty)) ≤E ψ(Θ1(x,y)) − φ(Θ1(x,y)),(3.32) where Θ1(x,y) ∈ {d(Ix,Iy), 12[d(Ix,Sx)+d(Iy,Ty)], 1 2 [d(Ix,Ty)+d(Iy,Sx)]}, ψ ∈ Ψ and φ ∈ Φ. Suppose that (i) TX ⊆ IX and SX ⊆ IX; (ii) I is a dominating map and S and T are dominated maps; (iii) If for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. Also, assume either (a) {S,I} are compatible, S or I is continuous and {T,I} are weakly com- patible or (b) {T,I} are compatible, T or I is continuous and {S,I} are weakly com- patible, then S,T and I have a common fixed point. Proof. It follows by taking I = J in Theorem 3.2. � Corollary 3.5. Let (X,d,�) be an ordered complete cone metric space over a solid cone K. Let S,I : X → X be given mappings satisfying for every pair (x,y) ∈ X × X such that x and y are comparable, ψ(d(Sx,Sy)) ≤E ψ(Θ2(x,y)) − φ(Θ2(x,y)),(3.33) where Θ2(x,y) ∈ {d(Ix,Iy), 12[d(Ix,Sx)+d(Iy,Sy)], 1 2 [d(Ix,Sy)+d(Iy,Sx)]}, ψ ∈ Ψ and φ ∈ Φ. Suppose that (i) SX ⊆ IX; (ii) I is a dominating map and S is dominated maps; (iii) If for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. Also, assume {S,I} are compatible and S or I is continuous, then S and I have a common fixed point. Proof. It follows by taking S = T in Corollary 3.4. � Corollary 3.6. Let (X,d,�) be an ordered complete cone metric space over a solid cone K. Let T,S : X → X be given mappings satisfying for every pair (x,y) ∈ X × X such that x and y are comparable, ψ(d(Sx,Ty)) ≤E ψ(Θ3(x,y)) − φ(Θ3(x,y)),(3.34) where Θ3(x,y) ∈ {d(x,y), 12[d(x,Sx)+d(y,Ty)], 1 2 [d(x,Ty)+d(y,Sx)]}, ψ ∈ Ψ and φ ∈ Φ. Suppose that (i) S and T are dominated maps; (ii) If for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. 162 H. K. Nashine and H. Aydi Also, assume either S or T is continuous, then S and T have a common fixed point. Proof. It follows by taking I = IdX, the identity on X, in Corollary 3.4. � Corollary 3.7. Let (X,d,�) be an ordered complete cone metric space over a solid cone K. Let T,S,I,J : X → X be given mappings satisfying for every pair (x,y) ∈ X × X such that x and y are comparable, d(Sx,Ty) ≤E Θ(x,y) − φ(Θ(x,y)), where Θ(x,y) ∈ {d(Ix,Jy), 1 2 [d(Ix,Sx) + d(Jy,Ty)], 1 2 [d(Ix,Ty) + d(Jy,Sx)]} and φ ∈ Φ. Suppose that (i) TX ⊆ IX and SX ⊆ JX; (ii) I and J are dominating maps and S and T are dominated maps; (iii) If for a nondecreasing sequence {xn} with yn � xn for all n and yn → u implies that u � xn. Also, assume either (a) {S,I} are compatible, S or I is continuous and {T,J} are weakly com- patible or (b) {T,J} are compatible, T or J is continuous and {S,I} are weakly com- patible, then S,T,I and J have a common fixed point. Proof. It suffices to take ψ(t) = t in Theorem 3.2. � Remark 3.8. Theorem 3.2 extends Theorem 2.1 of Shatanawi and Samet [32] to cone metric spaces. Now, we state the following illustrative examples. Example 3.9 (The case of a non-normal cone). Let X = [0, 1 4 ] be equipped with the usual order. Take E = C1 R ([0,1]) and K = {ϕ ∈ E, ϕ(t) ≥ 0, t ∈ [0,1]}. Define d : X × X → E by d(x,y)(t) = |x − y|ϕ where ϕ ∈ K is a fixed function, for example ϕ(t) = et. Then, (X,d) is a complete cone metric space with a nonnormal solid cone. Also, define S,T,I,J : X → X by Sx = Tx = x2 and Ix = Jx = x. For all comparable x,y ∈ X, we have d(Sx,Ty)(t) = d(Sx,Sy)(t) = |x2−y2|et = |x−y||x+y|et ≤ 1 2 |x−y|et = 1 2 d(Ix,Jy)(t), that is, (3.1) holds for ψ(t) = t and φ(t) = 1 2 t. On the other hand, x ≤ Ix = Jx and Sx = Tx ≤ x for all x ∈ X. Also, SX = TX ⊆ IX = JX and the pairs {S,I} = {T,J} are compatible. All hypotheses of Theorem 3.2 are verified and x = 0 is a common fixed point of S,T,I and J. Example 3.10. (The case of a normal cone). Let X = [0,∞] be equipped with the usual order. Take E = R2 and K = {(x,y), x ≥ 0, y ≥ 0}. Define Common fixed points in ordered cone metric spaces 163 d : X × X → E by d(x,y) = (|x − y|,α|x − y|) where α ≥ 0 a constant. Then, (X,d) is a complete cone metric space with a normal solid cone. Also, define S,T,I,J : X → X by Sx = Tx = ax and Ix = Jx = bx where 0 < a < 1 and b > 1. For all comparable x,y ∈ X, we have d(Sx,Ty) = d(Sx,Sy) = (a|x−y|,aα|x−y|) = (a b b|x−y|, a b bα|x−y|) = a b d(Ix,Jy), that is, (3.1) holds for ψ(t) = t and φ(t) = (1 − a b )t. Also, it is clear that all other hypotheses of Theorem 3.2 are verified and x = 0 is a common fixed point of S,T,I and J. The following example (which is inspired by [18]) demonstrates the validity of Theorem 3.2. Example 3.11 (The case of a non-normal cone). Let X = [0,1] be equipped with the usual order. Take E = C1 R ([0,1]) and K = {ϕ ∈ E, ϕ(t) ≥ 0, t ∈ [0,1]}. Define d : X × X → E by d(x,y)(t) = |x − y|ϕ where ϕ ∈ K is a fixed function, for example ϕ(t) = et. Then, (X,d) is a complete cone metric space with a nonnormal solid cone. Define the self maps I, J, S and T on X by S(x) = { 0, if x ≤ 1 3 1 2 (x − 1 3 ), if x ∈ (1 3 ,1] , Tx = { 0, if x ≤ 1 3 1 3 , if x ∈ (1 3 ,1] , J(x) =    0, if x = 0 x, if x ∈ (0, 1 3 ] 1, if x ∈ (1 3 ,1] , Ix =    0, if x = 0 1 3 , if x ∈ (0, 1 3 ] 1, if x ∈ (1 3 ,1] . Then I and J are dominating maps and S and T are dominated maps with S(X) ⊆ J(X) and T(X) ⊆ I(X),i.e. S is dominated map T is dominated map I is dominating map J is dominating map for each x in X Sx ≤ x Tx ≤ x x ≤ Ix x ≤ Jx x = 0 S (0) = 0 T (0) = 0 0 = I(0) 0 = J(0) x ∈ (0, 1 3 ] Sx = 0 < x Tx = 0 < x x ≤ 1 3 = I(x) x = J(x) x ∈ ( 1 3 , 1] Sx = 1 2 (x − 1 3 ) < x Tx = 1 3 < x x ≤ 1 = I(x) x ≤ 1 = J(x) Also, {S,I} are compatible, S is continuous and {T,J} are weakly compat- ible. Define ψ : K → K and φ : int(K) ∪ {0E} → int(K) ∪ {0E} by ψ(t) = t and φ(t) = 1 2 t. The inequality (3.1) holds for all comparable x,y ∈ X. Without loss of gener- ality, take x ≤ y. We consider the following cases: (i) If x = y = 0, then d(S0,T0)(t) = 0 and (3.1) is satisfied. (ii) For x = 0 and y ∈ (0, 1 3 ], then again d(Sx,Ty)(t) = 0 and (3.1) is satisfied. 164 H. K. Nashine and H. Aydi (iii) For x = 0 and y ∈ (1 3 ,1], d(Sx,Ty)(t) = 1 3 e t < 1 2 e t = 1 2 d(Ix,Jy)(t). (iv) For x,y ∈ (0, 1 3 ], then d(Sx,Ty) = 0 and hence (3.1) is satisfied. (v) For x = (0, 1 3 ] and y ∈ (1 3 ,1], d(Sx,Ty)(t) = 1 3 et < 1 2 et = 1 2 d(Ix,Jy)(t). (vi) For x,y ∈ (1 3 ,1], d(Sx,Ty)(t) = 1 2 (1 − x)et ≤ 1 3 et ≤ 1 2 d(Jy,Ty)(t). All hypotheses of Theorem 3.2 are verified and x = 0 is a common fixed point of S,T,I and J. Acknowledgements. The authors are grateful to the editor and referee for their valuable remarks for improving this paper. References [1] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416–420. [2] M. Abbas, T. Nazir and S. Radenović, Common fixed point of four maps in partially ordered metric spaces, Appl. Math. Lett. 24 (2011), 1520–1526. [3] M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009), 511–515. [4] R. P. Agarwal, Contraction and approximate contraction with an application to multi- point boundary value problems, J. Comput. Appl. Appl. Math. 9 (1983), 315–325. [5] R. P. Agarwal, M. A. El-Gebeily and D. O’Regan, Generalized contractions in partially ordered metric spaces, Applicable Anal. 87 (2008), 109–116. [6] I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and appli- cation, Fixed Point Theory Appl. 2010(2010) Article ID 621492, 17 pages. [7] H. Aydi, H. K. Nashine, B. Samet and H. Yazidi, Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations, Nonlinear Anal. 74, no. 17 (2011), 6814–6825. [8] I. Beg and A.R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71 (2009), 3699–3704. [9] B. S. Choudhury and N. Metiya, The point of coincidence and common fixed point for a pair of mappings in cone metric spaces, Comput. Math. Appl. 60 (2010), 1686–1695. [10] Lj. B. Ćirić, N. Cakić, M. Rajović and J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 131294, 11 pages. [11] Lj. B. Ćirić, B. Samet, N. Cakić and B. Damjanović, Coincidence and fixed point the- orems for generalized (ψ,φ)-weak nonlinear contraction in ordered K-metric spaces, Comput. Math. Appl. 62 (2011), 3305–3316. [12] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259–2261. [13] L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468–1476. Common fixed points in ordered cone metric spaces 165 [14] D. Ilić and V. Rakoèević, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341 (2008), 876–882. [15] E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59, no. 12 (2010), 3656–3668. [16] E.M. Mukhamadiev and V.J. Stetsenko, Fixed point principle in generalized metric space, Izvestija AN Tadzh. SSR, fiz.-mat. igeol.-chem. nauki. 10 (4) (1969), 8-19 (in Russian). [17] H.K. Nashine and I. Altun, Fixed point theorems for generalized weakly contractive condition in ordered metric spaces, Fixed Point Theory Appl. 2011 (2011), Article ID 132367, 20 pages. [18] H. K. Nashine and M. Abbas, Common fixed point point of mappings satisfying implicit contractive conditions in TVS-valued ordered cone metric spaces, preprint. [19] H. K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ,ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2011), 2201–2209. [20] H. K. Nashine, B. Samet and C. Vetro, Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces, Math. Comput. Modelling, to appear (doi:10.1016/j.mcm.2011.03.014). [21] H .K. Nashine and W. Shatanawi, Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces, Comput. Math. Appl. 62 (2011), 1984–1993. [22] J. J. Nieto and R. Rodŕıguez-López, Contractive mapping theorems in partially ordered sets and applications to ordianry differential equations, Order 22 (2005), 223–239. [23] J. O. Olaleru, Some generalizations of fixed point theorems in cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 657914, 10 pages. [24] A. I. Perov, The Cauchy problem for systems of ordinary differential equations, in: Approximate Methods of Solving Differential Equations, Kiev, Naukova Dumka, 1964, pp. 115–134 (in Russian). [25] A. I. Perov and A.V. Kibenko, An approach to studying boundary value problems, Izvestija AN SSSR, Ser. Math. 30, no. 2 (1966), 249–264 (in Russian). [26] A. C. M. Ran and M. C. B. Reurings, A fixed point thm in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435–1443. [27] D. O’regan and A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008), 241–1252. [28] Sh. Rezapour and R. Hamlbarani, Some notes on the paper: cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008), 719–724. [29] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508–4517. [30] B. Samet, Common fixed point theorems involving two pairs of weakly compatible map- pings in K-metric spaces, Appl. Math. Lett. 24 (2011), 1245–1250. [31] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl. 60 (2010), 2508–2515. [32] W. Shatanawi and B. Samet, On (ψ,φ)-weakly contractive condition in partially ordered metric spaces, Comput. Math. Appl. 62 (2011), 3204–3214 [33] J. S. Vandergraft, Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432. [34] P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), 825–859. (Received December 2011 – Accepted August 2012) 166 H. K. Nashine and H. Aydi H. Aydi (hassen.aydi@isima.rnu.tn) Université de Sousse, Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Route GP1-4011, H. Sousse, Tunisie. H. K. Nashine (drhknashine@gmail.com,hemantnashine@rediffmail.com) Department of Mathematics, Disha Institute of Management and Technol- ogy, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur- 492101(Chhattisgarh), India. Common fixed points for generalized (,)-weak contractions in ordered cone metric spaces. By H. K. Nashine and H. Aydi