Abstract
Certain common fixed point results involving four mappings satisfying generalized contractive conditions on a cone metric type space are obtained. Our results substantially improve and extend a number of known results. An example is given in support of the new results developed here. As an application, we establish the existence of a solution for an implicit integral equation. MSC: Primary 47H09; 47H10; secondary 49M05
Introduction
The Banach contraction principle is a remarkable result in the metric fixed point theory. Over the years, it has been generalized in different directions and spaces by several mathematicians (see [-] ). In , Zamfirescu Cvetković et al. [] and Hussain and Shah [] introduced the notion of a cone metric type space, which is a generalization of a cone metric space, and proved some common fixed point theorems and KKM-mappings results respectively. In this paper we consider mappings on a cone metric type space (CMTS) with a solid cone. We prove new common fixed point theorems involving four mappings satisfying a generalized contractive and a generalized nonexpansive conditions. Our results are generalization of theorems proved in [, , , , ] and some other authors. Almost all of our results are proved without the assumption of the continuity of mappings.
The paper is organized as follows. In Section  we review some definitions and wellknown results which will be needed in the sequel. In Section  and  we prove the existence of common fixed points of four mappings. We also present some corollaries which establish the fact that our results are generalization of several recent results in the literature. In Section , we present an application to integral equations to illustrate the usability of the obtained results.
Definitions and notation
Let E be a real Banach space and P a subset of E. We use the symbol  to denote the zero element of E and the symbol int P to denote the interior of P. The subset P is called a cone if and only if:
(I) P is closed, nonempty and P = {}; (II) a, b ∈ R, a, b ≥ , and x, y ∈ P such that ax + by ∈ P; (III) P ∩ (-P) = {}. For a given cone P, a partial ordering ≤ with respect to P is introduced as follows: x ≤ y if and only if y -x ∈ P; x < y means x ≤ y but x = y; if y -x ∈ int P, one writes x y; if int P = ∅, the cone P is called a solid cone.
In this paper we always suppose that E is a real Banach space, P ⊆ E is a solid cone and ≤ is partial ordering with respect to P. Definition . Let X be a nonempty set and E be a real Banach space with a cone P. A vector-valued function d : X × X → E is said to be a cone metric type function on X with the constant k ≥  if:
The pair (X, d) is called a cone metric type space (CMTS).
Remark . For k = , the above definition of CMTS reduces to that of a cone metric space introduced in [] .
Definition . Let (X, d) be a CMTS and {x n } be a sequence in X:
(I) {x n } converges to x if for every c ∈ E with  c there exists n  ∈ N such that d(x n , x) c for all n > n  . We write lim n→∞ x n = x, or x n → x as n → ∞. Let E be an ordered real Banach space with a cone P. The following properties hold (a, b, c ∈ E):
Then there exists a positive integer n  such that x n c for each n > n  .
Definition . Let F, G : X → X be mappings on a set X:
(i) If y = Fx = Gx for some x ∈ X, then x is called a coincidence point of F and G, and y is called a point of coincidence of F and G; (ii) The pair {F, G} is called weakly compatible if F and G commute at all of their coincidence points, that is,
The pair {F, G} is called occasionally weakly compatible (in brief OWC) [] if FGx = GFx for some x ∈ C(F, G).
Generalized contractive operators
First we show the existence of a unique fixed point.
Theorem . Let (X, d) be a cone metric type space with the constant k ≥  and let P be a solid cone. Let T : X → X be a mapping satisfying the contractive condition
for all x, y ∈ X and for some constant λ ∈ (,  k+k  ). If TX is complete, then T has a unique fixed point in X.
Proof For x  ∈ X, we construct a sequence {x n } in X by
We show that {x n } is a Cauchy sequence in X. For n ≥ , we have
where
Thus we have the following three cases: http://www.fixedpointtheoryandapplications.com/content/2013/1/169
and so kα <  if and only if
Now, by (C  ) and (C  ), it follows that for every c ∈ int P, there exists a positive integer n  such that d(x n , x m ) c for every m > n > n  , so {x n }is a Cauchy sequence. Since TX is a complete subspace of X, there exists p ∈ TX such that lim n→∞ x n = p = lim n→∞ Tx n . We show that p is a fixed point of T. Consider
Let  c. Then at least one of the following three cases holds: .
Thus Tp = p. Suppose that there exists q ∈ X such that Tq = q; then from (.), we have
where
Thus we have the following two cases: Now we present some new common fixed point theorems involving four mappings defined on a cone metric type space.
Theorem . Let (X, d) be a cone metric type space with the constant k ≥  and let P be a solid cone. Suppose that the self-mappings F, G, S, T : X → X are such that SX ⊂ GX and TX ⊂ FX. Let x  ∈ X, let the sequences {x n } and {z n } be defined by z n = Gx n+ = Sx n and z n+ = Fx n+ = Tx n+ for n ≥  and let O(x  ) = {z n : n ≥ }. Assume that there exists x  ∈ X such that for some constant λ ∈ (,
one of SX, TX, FX, or GX is a complete subspace of X, then {S, F} and {T, G} have a point of coincidence in X.
Proof We consider two cases of an odd integer n and of an even n.
, and from (.) there exists some
Thus we have the following three cases:
Thus we have the following three cases:
So, inequality (.) is satisfied in this case, too. Therefore, (.) is satisfied for all n ∈ N, and by iterating we get
Since k ≥ , for m > n we have
Now, by (C  ) and (C  ), it follows that for every c ∈ int P there exists a positive integer n  such that d(z n , z m ) c for every m > n > n  , so {z n }is a Cauchy sequence. Let us suppose that SX is a complete subspace of X, then there exists z ∈ SX such that lim n→∞ z n = lim n→∞ Sx n = z. Then we have
that is, for any  c, for sufficient large n, we have d(z n , z) c. Since z ∈ SX ⊂ GX, then there exists y ∈ X such that z = Gy. Let us prove that z = Ty. From (d  ) and (.), we get
Therefore we have three cases: Since TX ⊂ FX, there exists v ∈ X such that z = Fv. From (d  ) and (.), we have
Therefore we have the following three cases: Proof From Theorem ., we get that the mappings {S, F} and {T, G} have a point of coincidence in z ∈ X. Now we prove that z is a unique point of coincidence of pairs {S, F} and {T, G}. Suppose that there exists z * ∈ X such that Fv Using (C  ), we get Sz = z = Fz. Similarly, the weak compatibility of the pair {T, G} implies Tz = z = Gz. Hence Sz = Fz = Tz = Gz = z.
Remark . Theorem . is a generalization of Theorem . of [].
If we choose T = S and G = F in Theorems . and ., we get the following results for two mappings on a cone metric type space.
Corollary . Let (X, d) be a cone metric type space with the constant k ≥  and let P be a solid cone. Suppose that the self-mappings F, S : X → X are such that SX ⊂ FX. Let x  ∈ X, let the sequences {x n } and {z n } be defined by z n = Fx n+ = Sx n and z n+ = Fx n+ = Sx n+ for n ≥ , and let O(x  ) = {z n : n ≥ }. Assume that there exists x  ∈ X such that for some constant λ ∈ (,
one of SX or FX is a complete subspace of X, then S and F have a unique point of coincidence in X. Moreover, if {S, F} is a weakly compatible pair, then S and F have a common fixed point.
The following example shows that Theorem . is different from Theorem . of Cvetković et al. [] .
Example . Let X = {, , }, and let d(x, y) = |x -y| for each x, y ∈ X. Let T : X → X be given by T = , T =  and T = . Let S = T and let F = G = I. We first show that we cannot invoke Theorem . of [] to show the existence of a fixed point for T. On the contrary, assume that there exists λ ∈ (, ) such that
for each x, y ∈ X. Let x =  and y = . Then, from the above, we get  ≤ λ, a contradiction. ). Thus T has a fixed point.
Generalized nonexpansive maps
The aim of this section is to present coincidence points results for four mappings without satisfying the condition of continuity and commutation on cone metric type spaces with a non-normal cone. Proof We define sequences {x n } and {z n } as in the proof of Theorem .. We prove that {z n } is a Cauchy sequence by considering the cases when n is odd and even, respectively.
For n = l + , l ∈ N, we have
Similarly, for n = l, l ∈ N, we have
Thus,
Clearly α < . Hence, inequality (.) holds for both the cases. By the same arguments as in Theorem ., we conclude that {z n } is a Cauchy sequence. Let us suppose that SX is a complete subspace of X, then there exists z ∈ SX such that lim n→∞ z n = lim n→∞ Sx n = z. Then we have
that is, for any  c, for sufficient large n, we have d(z n , z) c. Since z ∈ SX ⊂ GX, then there exists y ∈ X such that z = Gy. We prove that z = Ty. From (d  ) and (.), we have
Since {z n } converges to z, so for each t ∈ int P, there exists n  ∈ N such that for every n > n  ,
Now, by (C  ) it follows that d(z, Ty) = , that is, Ty = z. So, we have Ty = Gy = z, that is, z is a point of coincidence of T and G.
Since TX ⊂ FX, there exists u ∈ X such that z = Fu. From (d  ) and (.), we have 
because of (C  ), we get z = z * . Therefore, z is the unique point of coincidence of S, F, G Recall that if (X, ) is a partially ordered set and f , g : X → X is such that for x, y ∈ X, x y implies fx fy, then the mapping f is said to be nondecreasing. Further, for x, y ∈ X, gx gy implies fx fy, then the mapping f is said to be g-nondecreasing [] .
The Proof Using Lemma ., there exists E ⊆ X such that g(E) = g(X) and g : E → X is one-toone. We define a mapping G : g(E) → g(E) by
for all gx ∈ g(E). As g is one-to-one on g(E) = g(X) and f (X) ⊆ g(X), so G is well defined. Thus, it follows from (.) and (.) that for all gx, gy ∈ g(X) for which g(y) g(x). Since f is a g-nondecreasing mapping, for all gy gx ∈ g(X), it implies fy fx, which gives Ggy Ggx. Thus G is a nondecreasing mapping on g(X). Also, there exists x  ∈ X such that g(x  ) f (x  ) = G(gx  ). Suppose that the assumption (a) holds. Since f is continuous, G is also continuous. Using Theorem . [] to the mapping G, it follows that G has a fixed point u ∈ g(X). Suppose that the assumption (b) holds. We conclude similarly from Theorem . [] that the mapping G has a fixed point u ∈ g(X). Finally, we prove that f and g have a coincidence point. Since u is a fixed point of G, we get u = Gu. Since u ∈ g(X), there exists a point u  ∈ X such that u = gu  . It follows that gu  = u = Gu = Ggu  = fu  . Thus, u  is a required coincidence point of f and g. This completes the proof. 
Application to the existence of solutions of integral equations


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