Kumar and Rathee Fixed Point Theory and Applications (2015) 2015:45 DOI 10.1186/s13663-015-0290-9

0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

Open Access

Fixed point and common fixed point results in cone metric space and application to invariant approximation

Anil Kumar1,2* and Savita Rathee1

Correspondence: anilkshk84@gmail.com 1 Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 124001, India 2Department ofMathematics, Govt. College Bahu, Jhajjar, Haryana 124142, India

Abstract

In this work, the concept of almost contraction for multi-valued mappings in the setting of cone metric spaces is defined and then we establish some fixed point and common fixed point results in the set-up of cone metric spaces. As an application, some invariant approximation results are obtained. The results of this paper extend and improve the corresponding results of multi-valued mapping from metric space theory to cone metric spaces. Further our results improve the recent result of Arshad and Ahmad (Sci. World J. 2013:481601, 2013).

MSC: 46T99; 47H10; 54H25

Keywords: fixed point; common fixed point; cone metric space; multi-valued mappings; best approximation

ringer

1 Introduction

Fixed point theory has many applications in different branches of science. This theory itself is a beautiful mixture of analysis, topology, and geometry. Since the appearance of the Banach contraction mapping principle, there has been a lot of activity in this area and several well-known fixed point theorems came into existence as a generalization of that principle. Many authors generalized and extended the notion of metric spaces such as ¿-metric spaces, partial metric spaces, generalized metric spaces, complex-valued metric space etc. For a useful discussion of these generalizations of metric spaces, one may refer to [1].

In 2007, Huang and Zhang [2] introduced the concept of cone metric space as a generalization of metric space, in which they replace the set of real numbers with a real Banach space. Although they proved several fixed point theorems for contractive type mappings on a cone metric space when the underlying cone is normal. Rezapour and Hamlbarani [3] proved such fixed point theorems omitting the assumptions of normality of cone. After that, the study of fixed point theorems in cone metric spaces was followed by many others (e.g., see [4-17] and the references therein).

On the other side, Nadler [18] and Markin [19] initiated the study of fixed point theorems for multi-valued mappings and established the multi-valued version of the Banach contraction mapping principle. Since the theory of multi-valued mappings has many applications, it became a focus of research over the years. Recently, many authors worked

© 2015 Kumar and Rathee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly credited.

out results on multi-valued mappings defined on a cone metric space when the underlying cone is normal or regular (see [20-23]). In 2011, Jankovic etal. [24] showed that most of the fixed point results in the set-up of normal cone metric space can be obtained as a consequence of the corresponding results in metric spaces. In the light of this, Arshad and Ahmad [25] improved Wardowski's results by proving the same without the assumption of the normality of the cones.

Here, the concept of almost contraction for multi-valued mappings in the setting of cone metric spaces is defined and then we establish some fixed point and common fixed point results in the set-up of cone metric spaces. In this way our results extend the results of Arshad and Ahmad [25] and also improve the corresponding results of both single-valued and multi-valued mappings existing in the literature. Before starting our work we need the following well-known definitions and results.

Definition 1 Let E be a real Banach space with norm || • || and P be a subset of E. Then P is called a cone if

(1) P is nonempty, closed, and P = [0}, where 0 is the zero element of E;

(2) for any non-negative real numbers a, b and for any x,y e P, one has ax + by e P;

(3) x e P and -x e P implies x = 0.

Given a cone P c E, we define a partial ordering ^ with respect to P by x ^ y if and only if y - x e P. We shall write x -< y if x ^ y and x = y while x ^ y if and only if y - x e int P, where int P is the interior of P. A cone P is called normal if there is a number K >0 such that for all x, y e E,

0 < x < y implies ||x||<K||y||.

The least positive number K satisfying the above inequality is called the normal constant of P. In the following we suppose that E is a real Banach space and P is a cone in E with intP = $ and ^ is a partial ordering with respect to P.

Definition 2 [2] Let X be a nonempty set. Suppose the mapping d: X x X ^ E satisfies the following:

(d1) 0 < d(x, y) for all x, y e X;

(d2) d(x, y) =0 if and only if x = y;

(d3) d(x, y) = d(y, x) for all x, y e X;

(d4) d(x, y) ^ d(x, z) + d(z, y) for all x, y, z e X.

Then d is called a cone metric on X, and (X, d) is called a cone metric space.

Definition 3 [2] Let (X, d) be a cone metric space and let [xn} be a sequence in X. Then the sequence [xn} obeys the following.

(1) [xn} converges to x, if for every c e E with 0 ^ c there exists a positive integer N such that d(xn, x) ^ c, for all n > N. We denote this by limn^ oo xn = x.

(2) [xn} is said to be Cauchy if for every c e E with 0 ^ c there exists a positive integer N such that d(xn,xm) ^ c, for all n, m > N.

A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.

Lemma 4 [24] Let P be a cone in Banach space E. Then the following properties hold:

(1) If c e int P and an ^ 0, then there exists a positive integer N such that for all n > N, we have an ^ c.

(2) If a < ka, where a e P and 0 < k <1, then a = 0.

Definition 5 [25] Let (X, d) be a cone metric space and let C(X) be the family of all nonempty and closed subsets of X.Amap H: C(X) x C(X) ^ E is called an H-cone metric on C(X) induced by d if the following conditions hold:

(H1) 0 < H(A, B) for all A, B e C(X).

(H2) H(A, B) =0 if and only if A = B.

(H3) H(A, B)=H(B, A) for all A, B e C(X).

(H4) H(A, B) ^ H(A, C) + H(C, B) for all A, B, C e C(X).

(H5) If A, B e C(X), 0 < e e E with H (A, B) ^ e, then for each a e A there exists b e B such that d(a, b) < e.

Example 6 Let (X, d) be a metric space. Then the mapping Hu : C(X) x C(X) ^ R defined by

is an H-cone metric induced by d. It is also known as the usual Hausdorff metric induced by d.

It is to be noted that (C(X),H) is a complete metric space whenever (X, d) is a complete metric space.

Definition 7 Let X be a nonempty set, T : X ^ C(X) be a multi-valued mapping, and f: X ^ X. Then an element x e X is said to be

(i) a fixed point of T,if x e Tx;

(ii) a common fixed point of T and f, if x = fx e Tx;

(iii) a coincidence point of T and f, if w = fx e Tx, and w is called the point of coincidence of T and f.

We denote C(f, T) = [x e X: fx e Tx}, the set of coincidence point off and T. The set of fixed point of T and the set of common fixed point off and T is denoted by F(T) and F(f, T), respectively.

Definition 8 [26] Let X be a nonempty set, T : X ^ C(X) be a multi-valued mapping, andf: X ^ X. Thenf is called T-weakly commuting at x e X ifffx e Tfx.

2 Main result

We start this section with the following definition.

Definition 9 Let (X, d) be a cone metric space and let there exist an H-cone metric on C(X) induced by d. A map T: X ^ C(X) is said to be a multi-valued almost contraction

if there exist two constants X e (0,1) and L > 0 such that

H(Tx, Ty) < Xd(x,y) + Ld(y, u)

for all x, y e X and u e Tx.

Theorem 10 Let (X, d) be a complete cone metric space and let there exist an H-cone metric on C(X) induced by d. Suppose T : X ^ C(X) is a multi-valued almost contraction. Then T has a fixed point in X.

Proof Let x0 be an arbitrary fixed element and xi e Tx0, if x0 = xi, then x0 is fixed point of T; ifx0 = x1, then Q < d(x0,x1). As X >0, we have H(Tx0, Tx1) -< e, where e = H(Tx0, Ix1) + Xd(x0,x1). Then, by the definition of H-cone metric there exists x2 e Tx1 such that

d(x1,x2) -< e = H(Tx0, Tx1) + Xd(x0,x1).

Clearly, H(Tx1, Tx2) -< H(Tx1, Tx2) + X2d(x0,x1). Since x2 e Tx1, for e = H(Tx1, Tx2) + X2d(x0,x1), there exists x3 e Tx2 such that

d(x2,x3) -< H(Tx1, Tx2) + X2d(x0,x1).

Inthe sameway, we can findasequence {xn} in X such that xn+1 e Txn, foreachn e N U{0} and

d(xn,xn+1) -< H(Txn-1, Txn) + Xnd(x0,x1). (2.2)

Since T is a multi-valued almost contraction, in view of (2.2) we have

d(xn,xn+1) -< Xd(xn-1,xn) + Ld(xn, u) + Xnd(x0,x1) (2.3)

for each u e Txn-1. Also, as xn e Txn-1, for each n > 1 we have

d(xn,xn+1) -< Xd(xn-1,xn) + Xnd(x0,x1). (2.4)

By repeated use of (2.4), we get

d(xn,xn+1) -< Xnd(x0,x1) + nXnd(x0,x1)

(n + 1)Xnd(x0, x1).

Now, for any m > n,

d(xn, xm) ^^ d(xn, xn+1) + d(xn+1, xn+2) + *** + d(xm_1, xm)

^ (n + 1)Xnd(x0,x1) + (n + 2)Xn+1d(x0,x1) + ••• + mXm-1d(x0,x1)

^ d(x0,x1)^2(i + 1)Xi.

Let c e E be any with 0 ^ c. Choose S >0 suchthat c + Ns (0) c int P,where Ns (0) = [x e E: ||x|| < S}. Also, since the series ^0=1(n + 1)^n is convergent, there exists a natural number N such that d(x0,xi) £°0n(i + 1)^ e Ns(0), for all n > N. Thus d(x0,xi) £°0n(i + 1)^ « c, for all n > N. Hence, (2.6) implies

d (x n, x m ) c

for all m > n > N. Thus, the sequence [xn} is Cauchy. Since X is complete, there exists z e X such that limn^o d(xn, z) = 0. Now we shall show that z is a fixed point of T, that is, z e Tz. As xn+1 e Txn and

H(Txn, Tz) < H(Txn, Tz) + k"d(x0,x1),

using the definition of the H-cone metric there exists yn e Tz such that

d(xn+1,yn) < H(Txn, Tz) + knd(x0,x1). (2.7)

Since T is a multi-valued almost contraction and xn+1 e Txn, it follows from (2.7) that

d(xn+1,yn) < Xd(xn, z) +Ld(z,xn+1) + knd(x0,x1).

Then, by the triangle inequality, we get

d(z,yn) ^ d(z,xn+1) + d(xn+1,yn)

Further, since d(xn , z) ^ 0 and Xn ^ 0 as n the right-hand side of the inequality

(2.8) tends to 0 as n ^o. Now, by Lemma 4, for any c e E with 0 ^ c there exists a positive integer N1 such that d(z,yn) ^ c for all n > Nl. Thus, the sequence [yn} converges

Now we present an example in support of the proved result.

Example 11 Let X = [0,1], E = CR([0,1]) with the norm = supxeX \ф{х)\ + supxeX and consider the cone P = e E: q>(t) > 0}. Suppose ц>, ф e E are defined as

y(x)=x and ф(х) = x2n for each n > 1

Then в < ф < у and = 2, ||ф|| = 2n + 1. Given any^ > 0 we can find a positive integer n such that 2n +1 > 2K. So, ||ф| ^ K||у || for any K > 0. Thus, P is non-normal cone. Now, define d: X x X ^ E by

d(x, y) = \x - y\q>,

where у :[0,1] ^ R with у (t) = ef. Then (X, d) be a complete cone metric space. Let C(X) be the family of all nonempty and closed subsets of X and define a mapping H: C(X) x

^ d(z,xn+i) + Xd(xn,z) +Ld(z,xn+i) + Xnd(x0,xi).

to z. As yn e Tz and Tz is closed, we get z e Tz.

C(X) ^ E as

H(A, B) = Hu(A, B)y for all A, B e C(X),

where Hu is the usual Hausdorff metric induced by d(x,y) = |x -y|. Also define T: X ^ C(X) by

T (x) =

[0,§] for x e [0,±], [2, f + i] forx e (¿1].

Now we shall show that T is a multi-valued almost contraction, that is, we show that T will satisfy condition (2.1). For this, we consider the following possible cases: Case (1). If x e [0, ¿] and y e (¿,1], then condition (2.1) canbewrittenas

et < k\x -y\et + L\y - u\et

for all u e Tx = [0, f].Here, we observe that \f-(f + ¿)|< |, \x-yje (0,1], and \y- u\ > ^ for all u e [0, ¿]. Thus, the inequality (2.9) is true for any k e (0,1) and L > 10. Case (2). If x e (^,1] and y e [0, 2], then condition (2.1) takes the form

x 1 y f + 2J-2

et < k\x -y\ef + L\y - u\ef

(2.10)

for all u e Tx = [f, f + i]. In this case |(f + i)- 11 < |, |x -y| e (0,1] and |y - u|> | for all u e [|, f + i]. Thus, the inequality (2.10) is true for any k e (0,1) and L > 5. Case (3). If x,y e [0, 2], then

H (Tx, Ty) = Hu (Tx, Ty)V = Hu

= 2 \x - y\e < kd(x, y) +Ld(y, u)

for any k e [2,1) and L > 0, where u is arbitrary element of Tx. Case (4). If x,y e (2,1], then

"2 x 1 + - 2 "2 y + f 1"

_3 3 2_

H (Tx, Ty) = Hu (Tx, Ty)V

= Hu I

= f \x - y\et

< kd(x, y) +Ld(y, u)

for any k e [f, 1) and L > 0, where u is arbitrary element of Tx.

Now, from all the cases, it is concluded that the multi-valued mapping T satisfies the inequality (2.1) for X = 2 and L = 5. Hence, T is an almost multi-valued contraction that satisfies all the hypotheses of Theorem 10. Thus, the mapping T has a fixed point. Here x = 0 is such a fixed point.

Remark 12

(i) Theorem 3.1 of Arshad and Ahmad [25], Theorem 2.4 of Doric [27], and Theorem 3.1 of Wardowski [20] are direct consequences of Theorem 10.

(ii) In Example 11, forx = 2 andy = |, we get Tx = [0, 4], Ty = [|, H], therefore H(Tx, Ty) = 36et. Then it can easily be checked that there does not exist any

X e (0,1) such that the mapping T satisfies the conditions (D1), (D2), (D3), (D4) given in Definition 2.1 of Doric [27]. Hence, Theorem 2.4 of Doric [27] cannot be applied to Example 11. It is also to be noted that Theorem 3.1 of Arshad and Ahmad [25] and Theorem 3.1 of Wardowski [20] are not applicable to Example 11.

In [28] Haghi etal. proved the following lemma.

Lemma 13 LetX be a nonempty set andf: X ^ Xbe a function. Then there exists a subset E c X such thatf (E) =f (X) andf: E ^ X is one to one.

Theorem 14 Let (X, d) be a cone metric space and let there exist an H-cone metric on C(X) induced by d. Supposef: X ^ X is a self map such thatf (X) is a complete subspace ofX and T: X ^ C(X) is a multi-valued mapping with Tx c f (X) for each x e X. If there exist two constants X e (0,1) and L > 0 such that

for allx, y e X and u e Tx. Then T andf have a coincidence point in X. Moreover, ifffx = fx for each x e Cf, T), then T andf have a common fixed point in X.

Proof By Lemma 13, there exists E c X such that f (E) =f (X) andf: E ^ X is one to one. Now, we define a map g: f (E) ^ C(f (E)) by g(f (x)) = Tx. Clearly g is well defined as f is one to one. Also

for all fx,fy e f (E) and u e Tx = g(fx). Then, by Theorem 10, there exists fx0 e f (E) such thatfx0 eg(fx0) = Tx0. Thus x0 is a coincidence point off and T and hence ffx0 = fx0. Let w = fx0, therefore fw = ffx0 = fx0 e TxBy (2.11), we have

H(Tx0, Tw) < Xd(fx0,fw) + Ld(fw,fw)

H(Tx, Ty) < Xd(fx,fy) + Ld(fy, u)

H(g(fx),g(fy))= H(Tx, Ty)

< Xd(fx,fy) + Ld(fy, u)

which gives Tx0 = Tw. Therefore, w = fw e Tw, that is, w is a common fixed point off and T. □

Theorem 15 Let (X, d) be a cone metric space and let there exist an H-cone metric on C(X) induced by d. Assume K is a nonempty closed subset ofX such that for each x e K andy e K there exists z e SK such that

d(x, z) + d(x, y) = d(x, y).

Suppose T: K ^ C(X) andf: K ^ X are two non-self maps satisfying

H(Tx, Ty) < kdfx,fy) + Ldfy, u) (2.13)

for all x,y e K and u e Tx with some k e (0,1) and L > 0 such that k(1 + L)< 1. Further assume

(i) SK c fK;

(ii) (UxeKTx) n K C fK;

(iii) fx e SK ^ Tx C K;

(iv) fK is closed in X.

Then T andf have a coincidence point in X. Moreover, ifffx = fx for each x e C(f, T), then there exists a common fixed point off and T.

Proof Let x e SK. We construct two sequences {xn} in K and {yn} in fK in the following way. Since SK c fK, there exists x0 e K such that fx0 = x e SK. So, by (iii) we get Tx0 c K. Since (UxeK Tx) n K C fK, we have Tx0 c fK. Let yi e Tx0, then there exists xi e K such that y1 = fx1. Consider the element H(Tx0, Tx1) e E. If the right-hand side of (2.13) is 0 at x = x0 and y = x1, then, as fx1 e Tx0, we have d(fx0,fx1) = 0 and hence fx1 = fx0. This and fx1 e Tx0 imply fx0 e Tx0. Thus, x0 is coincidence point off and T.

Assume the right-hand side of (2.13)isnot 0. Let e e P be a fixed element such that e = 0. Since k > 0, we have H(Tx0, Tx1) -< e, where e = H(Tx0, Tx1) + ke. Then, as y1 e Tx0, by the definition of an H-cone metric there exists y2 e Tx1 such that

d(y1,y2) -< e = H(Tx0, Tx1) + ke.

If y2 e K, then from (ii), we have y2 e fK. Therefore, there exists x2 e K such that y2 = fx2. If y2 e K, then, as fx1 e K, there exists a point p e SK such that

d(fx1, p) + d(p, y2) = d(fx1, y2). (2.14)

Sincep e SK c fK, there exists a point x2 e K such thatp = fx2. Then, by (2.14)

d(fx1,fx2) + d(fx2, y2) = d(fx1, y2). (2.15)

Clearly, H(Tx1, Tx2) < H(Tx1, Tx2) + k2e. Then, again using the definition of the H-cone metric, there exists y3 e Tx2 such that

d(y2,yf) < H(Tx1, Tx2) + k2e.

If y3 e K, then, again using (ii), we have y3 e fK. So, there is a point x3 e K such that y3 = fx3. If y3 e K, then there exists a point q e SK such that

d(fx2, q) + d(q, y3) = d(fx2, yf).

(2.11)

Again, since q e SK c fK, there exists a point x3 e K such that q = fx3. Then, by (2.16),

d(fx2,fx3) + d(fx3, y3) = d(fx2, y3). (2.17)

Repeating the foregoing procedure we construct two sequences [xn} in K and [yn} in fK such that

(a) yn+\ e Txn, for each n e N U [0};

(b) d(yn,yn+i) ^ H(Txn-i, Txn) + Xne;

(c) if yn e K, then yn = fxn;

(d) if yn e K, then fxn e SK with

d(fxn-i,fxn) + d(fxn, yn) = d(fn-i, yn). (2.18)

Now we show that the sequence {/xn} is Cauchy and for this we define two sets P and Q as follows:

P = fxi e [fxn} :fx = yt\, Q = fx e [fxn}:fxi = y^.

Clearly, iffxn e Q, thenfxn-1 and_/xn+L lies in P. Now, it can be concluded that there are three possibilities.

Case 1. Iffxn e P and fxn+L e P, then fxn = yn and fxn+L = yn+L. Therefore, by using (b)

d(fxn,fxn+i) = d(yn, yn+i)

^ H(Txn-1, Txn) + Xne

< Xd(fxn-1,fxn) + Ld(fxn, u) + Xne for each u e Txn-1. (2.19) Thus, in view of (a) and (2.19), we have

d(fxn,fxn+i) < Xd(fxn-1,fxn) + Xne. (.20)

Case 2. Iffxn e P and fxn+L e Q, then fxn = yn and yn+L e K. Thus, (d) implies_/xn+L e SK with

d(yn,fxn+i) + d(fxn+i, yn+i) = d(yn, yn+i). (2.21)

Regarding (2.21) and (b), we get

d(fxn,fxn+i) < d(yn,yn+i)

^ H(Txn-1, Txn) + Xne

< Xd(fxn-1,fxn) + Ld(fxn, u) + Xne for each u e Txn-1. (2.22) Since fxn = yn e Txn-1, (2.21) gives

d(fxn,fxn+i) < Xd(fxn-1,fxn) + Xne. (.3)

Case 3. Iffxn e Q and fxn+1 e P, then fxn-1 e P, yne K and fxn+1 = yn+1. Thus, from (d) fxn e SK such that

d(yn-1,fxn) + d(fxn, yn) = d(yn-1, yn). (2.24)

By the triangle inequality, (2.24), and (b), we obtain

d(fxn,fxn+1) = d(fxn, yn+1)

^ d(fxn,yn) + d(yn,yn+1) ^ d(yn-1,yn) - d(yn-1,fxn) + d(yn,yn+1)

^ d(yn-1,yn) - d(yn-1,fxn) + H(Txn-1, Txn) + kne. (2.25)

Since yn e Txn-1, on account of (2.13), we derive from (2.25) that

d(fxmfxn+1) < d(yn-1,yn) - d(yn-1,fxn) + kd(fxn-1,fxn) + Ldfxn,yn) + kne ^ d(yn-1,yn) + Ld(fxn,yn) + kne, as k e (0,1) ^ d(yn-1,yn)+Ld(yn-1,yn)-Ld(yn-1,fxn) + kne

^ (1 + L)d(yn-1,yn) + kne. (.21)

Therefore, by using (b) and (2.13), we conclude that

d(fxn,fxn+1) < (1 + L)H(Txn-2, Txn-1) + (1 + L)kn-1e + kne

< (1 + L)kd(fxn-2,fxn-1) + (1 + L)Ld(fxn-1,yn) + (1 + L)kn-1e + kne

< (1 + L)kd(fxn-2,fxn-1) + (1 + L)kn-1e + kne. (2.27)

Now, we define

/ = maxj k, (1 + L)k}. Thus, from Case 1-Case 3, it follows that

d(fxn,fxn+1) < xZn + (xn-1 + xn)e, (2.8)

Zn e {d(/xn-1^^n), d(/xn-2^^n-1^. Now we claim that for each n >1,

d(fxn,fxn+1) < /2-1Z2 + 2n/n e, (2.29)

Z2 e {d(/x0,fx1),d(fx1,fx2)}.

We shall prove it by mathematical induction. If n = 2, then (2.28) gives d(fx2,fx3) — fZ2 + (f + f2)e

= fZ2 + f (1 + f)e — f22 Z2 + 2 • (2fe), as f < 1.

Thus, (2.29) holds for n = 2. Let (2.29) be true for 2 < n < m, then we have to show that it is also true for n = m + 1. From (2.28), for n = m + 1, we have

d(fxm+1,fxm+2) — fZm+1 + f + fm+1) e, (0)

Zm+1 e {d(fxm,fxm+i), d(fxm-1,fxm)\.

Case 1. If Zm+L = d(fxm,fxm+i), then (2.30) implies

d(fxm+1,fxm+2) — fld(fxm,fxm+i) + ( fm + f™+1)e

— ff m-1Z2 + 2mfm e) + fm + fm+1) e

1 / m-1 m \ m+1

— f2 (f 2 Z2 + 2mf2 e) +2f 2 e

(m+1)-1 , m+1

= f 2 Z2 + 2(m + 1)f 2 e. Case 2. If £m+1 = d(fxm-1,fxm), then it follows from (2.30) that

d(fxm+1,fxm+2) — fld(fxm-1,fxm) + (fm + f™+1)e

— ff + (m -1)f^e) + (fm + fim+1)e

m m+1 m+1

— f T Z2 + (m -1)f e + f ""^(1 + f)e

m m+1 m+1

— f2 Z2+ (m -1)f 2 e + f 2 (1 + f)e

m m+1 m+1

— f T Z2 + 2(m - 1)f e + 2 • 2f e

(m+1)-1 m+1

= f 2 Z2 + 2(m + 1)f e.

Therefore, in both cases, (2.29) is true for n = m + 1. Thus, by the principle of mathematical induction the inequality (2.29) holds for each n > 1. Now, by the triangle inequality and (2.29), for any m > n,we have

d(fxn,fxm ) — d(fxn,fxn+\) + d(fxn+1,fxn+2) + ••• + d(fxm-1,fxm)

r n-1 n -i

— 2 Z2 + 2nf2 ej

r-n n + 1 -1

+ 2 Z2 + 2(n + 1)f ej + •••

i- m-2 . . m-1 -i

+ [f 2 Z2 + 2(m - 1)f 2 ej

r n-1 n m-2 -i

— |f ^ + f2 + ••• + f JZ2 + Rn(f)2e, (2.31)

where Rn(/) is the remainder of the convergent series ^n/n. As / <1, by (2.31) we get

n-1 /1^2-

d(fxn,fxm) <--Z2 + Rn(/)2e ^ 0, as n ^ro. (2.32)

Thus, by Lemma 4, for any c e E with 0 ^ c there exists a positive integer N1 such that d(fxn,fxm) ^ c for all n > N1. Hence, the sequence {fxn} is Cauchy. Also, {fxn} is a sequence in K nf (K) and, being a closed subset of complete space X, Knf (K) is complete. Therefore, there exists z e K n f (K) such that fxn ^ z as n ^ro. Further, as z e f (K), there exists w e K such that z = f (w). By the construction of {fxn} there is a subsequence fxm such that

m = ym e Txm-1.

Now we shall show that w is a coincidence point of T andf, that is,fw e Tw. Asfxm e Txm-1 and

H(Txm-1, Tw) -< H(Txm-1, Tw) + km-1e,

using the definition of H-cone metric, there exists zm e Tw such that

dfxm, zm) < H(Txm-1, Tw) + km-1e. (.3)

Regarding (2.13) and fxm e Txm-1, we obtain from (2.33)

dfxm, zm) ^ kdfxm-1,fw) + Ldfw,fxm) + km e.

Then, by the triangle inequality, we get

dfw, zm) < dfwfxm) + dfxm, zm)

^ dfw,fxm) + kdfxm-1,fw) + Ldfw,fxm) + km-1e. (2.34)

Since the subsequence {fxm} converges to z = f (w) and k <1, the right-hand side of the inequality (2.34) converges to 0 as m ^ro. Therefore, in view of Lemma 4, for any c e E with 0 ^ c, we can choose a positive integer N2 such that dfw,zm) ^ c for all m > N2. Thus the sequence zm converges to f(w). As zm e Tw and Tw is closed, we get fw e Tw. Since w e Cf, T), it follows thatffw = fw. Let z = fw and sofz = ffw = fw e Tw. In view of (2.13) we have

H(Tw, Tz) < kdfw,fz) + Ldfz,fz) = 0,

which gives Tw = Tz. Therefore, z = fz e Tz, that is, z is a common fixed point off and T.

If we let f = I (identity map) in Theorem 15, we obtain the following result as an extension of Theorem 9 of [29] to a cone metric space.

Corollary 16 Let (X, d) be a cone metric space and let there exist an H-cone metric on C(X) induced by d. Assume K is a nonempty closed subset ofX such that for each xeK and ye K there exists z e SK such that

d(x, z) + d(z, y) = d(x, y).

Suppose that T: K ^ C(X) is a non-self map satisfying

H(Tx, Ty) — Xd(x,y) + Ld(y, u) (2.5)

for all x,y e K and u e Tx with some X e (0,1) and L > 0 such that X(1 + L)< 1. Further assume xe SK ^ Tx c K, then there exists zeK such that z e Tz.

Now, we present a non-trivial example which shows the generality of Corollary 16 over the corresponding existing theorems.

Example 17 Let X = [-1, o), E = CR([0,1]) with supremum norm, P = [y e E : y(t) > 0} and d : X x X ^ E defined by d(x,y) = |x - y|y, where y :[0,1] ^ R with y(t) = el. Then (X, d) is a cone metric space with a non-normal cone P. Let C(X) be the family of all nonempty and closed subsets of X and define a mapping H: C(X) x C(X) ^ E as

H(A, B) = Hu(A, B)v for all A, B e C(X),

where Hu is the usual Hausdorff metric induced by d(x,y) = |x - y|. Take K = [0,1] and define T: K ^ C(X) as given in Example 11 of [29]:

T (x) =

for x e [0,2), for x = 2, [17,9 + 9] for xe (2,1].

{9} {-1}

Here SK = [0,1}. Clearly, for each xeK and ye K there exists a point z = 0 or z = 1 e SK such that d(x, z) + d(z, y) = d(x, y). Further, as 0 e SK ^ T0 = [0} c [0,1] = K and 1 e SK ^ T1 = [lL8, 1] c K, so x e SK ^ Tx c K. Now, using a routine calculation as done in Example 11, it can easily be shown that the inequality (2.35) holds for S = 9 and L =9. Thus, all the conditions of Corollary 16 are satisfied and hence T has a fixed point in K. Here x = 1 is such a point.

Remark 18

(i) Theorem 1 of Assad and Kirk [30] is a direct consequence of Corollary 16.

(ii) In Example 17, for x = 1 and y = 2, it can be checked that the inequality

H(Tx, Ty) < Sd(x,y)

is not satisfied for any S e (0,1). Therefore, Theorem 1 of Assad and Kirk [30] is not applicable to Example 17.

3 Application to invariant approximation

Since the appearance of Meinardus' result in best approximation theory, several authors have obtained best approximation results for single-valued maps as an application of fixed point and common fixed point results. The best approximation results for multi-valued mappings was obtained by Kamran [26], Al-Thagafi and Shahzad [31], Beg et al. [32], O'Regan and Shahzad [33], and Markin and Shahzad [34]. Further, best approximation results in the setting of cone metric space were for the first time considered by Rezapour [35] (see also [36]).

In this section the best approximation results for a multi-valued mapping in the setting of cone metric spaces are obtained.

Definition 19 Let M be a nonempty subset of a cone metric space X. A point y e M is said to be a best approximation to p e X, if d(y,p) < d(z,p) for all z e M. The set of best approximations top in M is denoted by BM(p).

As an application of Theorem 14, we obtain the following theorem, which ensures the existence of a best approximation.

Theorem 20 Let M be subset of a cone metric space X, p e X, and let there exist an H-cone metric on C(M) induced by d. Suppose f: M ^ M is a single-valued mapping and T: M ^ C(M) is a multi-valued mapping such that for all x, y e BM (p) andue Tx we have

where X e (0,1) andL > 0. Also the following conditions hold:

(i) f (Bm(p))= Bm(p).

(ii) ffv =fvfor veCf, T) n Bm(p).

(iii) d(y, p) < dfx, p) for all xe BM (p) and y e Tx.

(iv) f (BM(p)) is complete. Then F f, T ) n Bm (P) = <£.

Proof First we show that T |bmp : BM(p) ^ C(BM(p)) is a multi-valued mapping. For this let x e BM (p) and« e Tx. Then, asf (BM (p)) = BM(p), we getfxe BM (p) and hence dfx, p) < d(z,p) for all zeM. Since u e Tx, by (iii) we obtain

d(u,p) < dfx,p) < d(z,p) for all z e M.

Thus, u e BM(p) and hence Tx ç BM(p) for all x e BM(p). Since Tx is closed for all x e M, therefore also Tx is closed for all x e BM(p). So, T|bmp is a multi-valued mapping from BM(p) to C(BM(p)). Moreover, Tx ç BM(p) =f (BM(p)) for each x e BM(p). Further, as f (Bm(p)) = Bm(p) andf : M ^ M, the mapping f |bm p : Bm(p) ^ Bm (p) is single-valued. Clearly

H (Tx, Ty) < Xdfx,fy) + Ldfy, u),

F f |bm(p), T Ibm (p)) = F f, T) n Bm (p).

Therefore, by applying Theorem 14 for X = Bm (p), F f, T ) n Bm (P) =

Corollary 21 LetM be subset of a cone metric space X, peX, and let there exist an H-cone metric on C(M) induced by d. Suppose T: M ^ C(M) is a multi-valued mapping such that for all x, ye BM(p) and ueTx we have

H(Tx, Ty) — Xd(x,y) + Ld(y, u), (3.2)

where X e (0,1) andL > 0. Also the following conditions hold:

(i) d(y,p) — d(x,p) for all x e BM (p) and y e Tx.

(ii) BM(p) is complete. Then F(T) n Bm(P) = 4>.

Let M be a subset of a cone metric space (X, d) and let there exist an H-cone metric on C(M) induced by d. A family F = h : A e C(M)} of functions from [0,1] into C(M) with the property hA (1) = A for each A e C(M) is said to be contractive if there exists a mapping y : (0,1) ^ (0,1) such that for all A, Be C(M) and t e (0,1), we have

H (hA (t), hB(t)) — y(t)H(A, B).

Such a family F is said to be jointly continuous if A ^ A0 in C(M) and t ^ to in (0,1) imply hA(t) ^ hA0(t0).

Theorem 22 Let M be a subset of a cone metric space (X, d), and let there exist an H-cone metric on C(M) induced by d. Suppose F = [hA : A e C(M)} is a contractive and joint continuous family, T: M ^ C(M) is a multi-valued mapping and there exists L > 0 such that

H(Tx, Ty) — d(x,y) + Ld(y, u) (3.3)

for all x, y e M and u e hTx (k), where k e (0,1) is any fixed element. IfM is compact and T is continuous, then T has a fixed point in M.

Proof For each n > 1, if we put kn = n+i, then kn is a real sequence which lies in (0,1) with kn ^ 1 as n ^ o. Now, we define Tn : M ^ C(M) by

Tnx = hTx(kn) for all x e M.

Firstly, we show that Tn is a multi-valued almost contraction on M. For this let x,y e M, as F is a contractive family, we have

H(Tnx, Tny) = Hfax(kn), hjy(kn))

— y(kn)H(Tx, Ty)

— y(kn)d(x, y) + Ly(kn)d(y, u) (.4)

for all u e hTx(kn) = Tnx. For each n > 1, define Xn = y(kn) and Ln = Ly(kn). Clearly, Xn e (0,1) and Ln > 0 for each n > 1. Therefore, for each n > 1, (3.4) implies

H(Tnx, Tny) — Xnd(x,y) + Lnd(y, u)

for all x, y e M and u e Tnx. Hence, T is a multi-valued almost contraction. Since M is compact, by Theorem 10 for each n > 1 there exists xn e M such that xn e Tnxn. Again compactness of M implies that there exists a convergent subsequence {xm} of {xn} such that xm ^ zeM as m ^ro. Since T is continuous, the family F is jointly continuous and xm e Tmxm = hTxm (km), we have z e Tz, as km ^ 1 as m ^ ro. Thus T has a fixed point in M. □

The following theorem ensures the existence of a fixed point from the set of best approximations.

Theorem 23 Let M be a subset of a cone metric space (X, d) and let there exist an H-cone metric on C(X) induced by d. Suppose T : X ^ C(X), p e X, and BM(p) is nonempty, compact, and it has a joint contractive family F = {hA : A e C(BM(p))}. If T is continuous on BM(p), (3.3) holds for all x, y e BM(p) and also d(y,p) < d(x,p) for all x e BM(p) andy e Tx. Then BM(p) n F(T) = 0.

Proof We claim Tx c BM(p) for each x e BM(p). To prove this let x e BM(p), then d(x,p) ^ d(z, p) for all zeM .If u e Tx, then by the given hypothesis

d(u, p) ^ d(x, p) ^ d(z, p) for all zeM.

Thus, u e BM(p). So T : BM(p) ^ C(BM(p)) is a multi-valued mapping and hence by applying Theorem 22 for Bm(p), it follows that Bm(p) n F(T) = 0. □

Remark 24 All results in this paper hold as well in the frame of tvs-cone metric spaces (see [16]).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the referees for their valuable comments in modifying the first version of this paper.

Received: 14 October 2014 Accepted: 12 March 2015 Published online: 27 March 2015

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