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Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric spaces

Seong-Hoon Cho1*, Jong-Sook Bae2 and Kwang-Soo Na2

"Correspondence: shcho@hanseo.ac.kr 1 Department of Mathematics, Hanseo University, Seosan, Chungnam 356-706, South Korea Full list of author information is available at the end of the article

Abstract

In this paper, we establish a fixed-point theorem for multivalued contractive mappings in complete cone metric spaces. We generalize Caristi's fixed-point theorem to the case of multivalued mappings in complete cone metric spaces. We give examples to support our main results. Our results are extensions of the results obtained by Feng and Liu (J. Math. Anal. Appl. 317:103-112, 2006) to the case of cone metric spaces. MSC: 47H10; 54H25

Keywords: fixed point; multivalued map; cone metric space

1 Introduction

Banach's contraction principle plays an important role in several branches of mathematics. Because of its importance for mathematical theory, it has been extended in many directions (see [10,11,14,19, 21, 37, 46]); especially, the authors [36, 37, 39] generalized Banach's principle to the case of multivalued mappings. Feng and Liu gave a generalization of Nadler's fixed-point theorem. They proved the following theorem in [21].

Theorem 1.1 Let (X, d) be a complete metric space and let T: X ^ 2X be a multivalued map such that Tx is a closed subset of X for allx e X. Let II = {y e Tx: bd(x,y) < d(x, Tx)}, where b e (0,1).

If there exists a constant c e (0,1) such that for any x e X, there exists y e II satisfying d(y, Ty) < cd(x, y),

then T has a fixed point in X, i.e., there exists z e Xsuch thatz e Tz provided c < bandthe function d(x, Tx), xe X is lower semicontinuous.

Recently, in [22], the authors used the notion of a cone metric space to generalize the Banach contraction principle to the case of cone metric spaces. Since then, many authors [1-3, 7, 9,13,15,18, 22-28, 32-34, 41, 43, 44, 48] obtained fixed-point theorems in cone metric spaces. The cone Banach space was first used in [4, 6]. Since then, the authors [29, 30] obtained fixed-point results in cone Banach spaces. The authors [8] proved a Caristi-type fixed-point theorem for single valued maps in cone metric spaces. The author [5] studied the structure of cone metric spaces.

© 2012 Cho et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

ringer

Especially, the authors [16, 31, 35,42,45, 47] proved fixed point theorems for multivalued maps in cone metric spaces.

In this paper, we give a generalization of Theorem 1.1 to the case of cone metric spaces and we establish a Caristi-type fixed-point theorem for multivalued maps in cone metric spaces.

Consistent with Huang and Zhang [22], the following definitions will be needed in the sequel.

Let E be a topological vector space. A subset P of E is a cone if the following conditions are satisfied:

(1) P is nonempty, closed, and P = {0},

(2) ax + by e P, whenever x,y e P and a, b e R (a, b > 0),

(3) P n (-P) = {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 write x -< y to indicate that x ^ y but x = y.

For x, y e P, x ^ y stand for y - x e int(P), where int(P) is the interior of P.

If E is a normed space, a cone P is called normal whenever there exists a number K >0 such that for all x,y e E, 0 < x < y implies ||x|| < K||y||.

A cone P is minihedral [20] if sup[x,y} exists for all x,y e E.A cone P is strongly minihe-dral [20] if every upper bounded nonempty subset A of E, sup A exists in E. Equivalently, a cone P is strongly minihedral if every lower bounded nonempty subset A of E, inf A exists in E (see also [1, 8]).

If E is a normed space, a strongly minihedral cone P is continuous whenever, for any bounded chain [xa : a e r}, inf{||xa - inf{xa : a e r}||: a e T} = 0 and sup{||xa - sup{xa : a e r}|| : a e T} = 0.

From now on, we assume that E is a normed space, P c E is a solid cone (that is, int(P) = 0), and ^ is a partial ordering with respect to P.

Let X be a nonempty set. A mapping d: X x X ^ E is called cone metric [22] on X if the following conditions are satisfied:

(1) 0 < d(x, y) for all x, y e X and d(x, y) = 0 if and only if x = y,

(2) d(x, y) = d(y, x) for all x, y e X,

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

Let (X, d) be a cone metric space, and let [xn}c X be a sequence. Then

{xn} is convergent [22] to a point x e X (denoted by limn^ oo xn = x or xn ^ x) if for any c e int(P), there exists N such that for all n > N, d(xn,x) ^ c.

{xn} is Cauchy [22] if for any c e int(P), there exists N such that for all n, m > N, d(xn, xm) ^ c. A cone metric space (X, d) is called complete [22] if every Cauchy sequence is convergent.

Remark 1.1 (1) If limn^o d(xn,x) = 0, then limn^oxn = x. The converse is true if E is a normed space and P is a normal cone.

(2) If limn,m^ o d(xn,x„) = 0, then {xn} is a Cauchy sequence in X. If E is a normed space and P is a normal cone, then {xn} is a Cauchy sequence in X if and only if

limn,m^o d(xn, xm) = 0.

We denote by N(X) (resp. B(X), C(X), CB(X)) the set of nonempty (resp. bounded, closed, closed and bounded) subsets of a cone metric space or a metric space.

The following definitions are found in [16].

Let s(p) = {q e E:p < q} forp e E, and s(a,B) = (JbeB s(d(a, b)) for a e X and B e N(X). For A, B e B(X), we denote

Lemma 1.1 ([16]) Let (X, d) be a cone metric space, and let P c Ebea cone.

(1) Let p, q e E. Ifp < q, then s(q) c s(p).

(2) Let x e X and A e N(X). If0 e s(x, A), then x e A.

(3) Let q e P and let A, B e B(X) and a e A. Ifq e s(A, B), then q e s(a, B).

Remark 1.2 Let (X, d) be a cone metric space. If E = R and P = [0, o), then (X, d) is a metric space. Moreover, for A, B e CB(X), H(A, B) = inf s(A, B) is the Hausdorff distance induced by d.

Remark 1.3 Let (X, d) be a cone metric space. Then s({a}, {b}) = s(d(a, b)) for a, b e X.

Lemma 1.2 ([16, 40]) Ifun e E with un ^ 0, thenfor each c e int(P) there exists N such that un ^ cfor all n> N.

2 Fixed-point theorems for multivalued contractive mappings

Let (X, d) be a cone metric space, and let A e N(X).

A function h : X ^ 2P - {0} defined by h(x) = s(x, A) is called sequentially lower semi-continuous if for any c e int(P), there exists n0 e N such that h(xn) c h(x) - c for all n > n0, whenever limn^oxn = x for any sequence {xn} c X and x e X.

Let T: X ^ C(X) be a multivalued mapping. We define a function h : X ^ 2P - {0} as h(x) = s(x, Tx). For a b e (0,1], let Jxb = {y e Tx: s(x, Tx) c s(bd(x,y))}.

Theorem 2.1 Let (X, d) be a complete cone metric space and letT: X ^ C(X) be a multivalued map. If there exists a constant c e (0,1) such that for any x e X there exists y e J% satisfying

then T has a fixed point in X provided c < b and h is sequentially lower semicontinuous.

Proof Let xo e X. Then there exists xi e Jx° such that cd(xo,xi) e s(xi, Txi). For xi, there exists x2 e J^1 such that cd(xi,x2) e s(x2, Tx2). Continuing this process, we can find a sequence {xn} c X such that

s(A, B) =

cd(x, y) e s(y, Ty)

xn+i e Jx

cd(xm xn+i) e s(xn+i, Txn+i)

for all n = 0, i,...

We now show that {xn} is a Cauchy sequence in X. Since xn+2 e Jxbn+1, s(xn+1, Txn+1) C s(bd(xn+1,xn+2)).

From (2.2), we have cd(xn,xn+1) e s(bd(xn+1,xn+2)). Thus, bd(xn+1,xn+2) — cd (xn, xn+1). Hence,

d(xn+1, xn+2) — kd(xn, xn+1)

for all n = 0,1,..., where k = |. So we have

d(xn, xn+i) — kd(xn_i, xn) — k2d(xn-2, xn_i) — ••• — knd(x0, xi). For m > n,we have d(xn , xm )

d(xn, xn+1) + d(xn+1, xn+2) + ••• + d(xm_1, xm)

— (kn + kn+1 + ••• + A"-1) d(x0, x1) —--d(x0, *0.

v ' 1 - k

By Lemma 1.2, {xn} is a Cauchy sequence in X. It follows from the completeness of X that there exists z e X such that xn = z.

We now show that z e Tz. Suppose that z e Tz.

Since Tz is closed, there exists c e int(P) such that d(z,y) ^ c implies y g Tz. But since h is sequentially lower semicontinuous, there exists N such that d(xN, xN+1) ^ 2 and s(xn, Txn) C s(z, Tz) - §.

Thus, there exists y e Tz such that d(z, y)- f — d(xN, xN+1). Hence, d(z, y) — d(xN, xN+1) + 2 ^ c, which is a contradiction. □

Remark 2.1 By Remark 1.1, Theorem 2.1 generalizes Theorem 1.1 ([12, Theorem 3.1]).

Corollary 2.2 Let (X, d) be a complete cone metric space and letT: X ^ C(X) be a multivalued map. If there exists a constant c e (0,1) such that for any x e X,y e Tx

cd(x,y) e s(y, Ty)

then T has a fixed point in X provided h is sequentially lower semicontinuous.

By Lemma 1.1(3), we have the following result, which is Nadler's fixed-point theorem in the cone metric space.

Corollary 2.3 Let (X, d) be a complete cone metric space, and let T : X ^ CB(X) be a multivalued map. If there exists a constant c e (0,1), such that

cd(x,y) e s(Tx, Ty)

for all x e X, y e Tx, then T has a fixed point in X provided h is sequentially lower semi-continuous.

By Remark 1.1, we have the following corollaries.

Corollary 2.4 ([21]) Let (X, d) be a complete metric space and letT: X ^ C(X) be a multivalued map. If there exists a constant c e (0,1) such that

d(y, Ty) < cd(x, y)

for all x e X, y e Tx, then T has a fixed point in X provided h is sequentially lower semi-continuous.

Corollary 2.5 Let (X, d) be a complete metric space and letT: X ^ CB(X) be a multivalued map. If there exists a constant c e (0,1) such that

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

for all x e X, y e Tx, then T has a fixed point in X provided h is sequentially lower semi-continuous.

The following example illustrates our main theorem.

Example 2.1 Let X = {f e L1[0,1]: f (x) > 0}, E = C[0,1] and P = {f e E: f > 0 a.e.}. Define d: X x X ^ E by d(f,g)(t) = /0 f (x)-g(x) | dx, where 0 < t < 1. Then d is a complete cone metric on X. Consider a mapping T: X ^ CB(X) defined by

= df, a(f ))(t).

Since (Tf)(x) = {a(f), a(f) + 2f}, we have s(f, Tf) C s(d(f, a(f))), and hence we obtain a(f) e Jf.

Put a(f) =g. Then we have a(a(f)) = a(g) e T(a(f)) and for 0 < t < 1

(Tf )(x)= a(f), a(f ) + 2f},

where af) e X is defined by a(f)(x) = f£y(f (y) + 1) dy. Obviously, h(f) = s(f, Tf) is sequentially lower semicontinuous. For anyf e X, we can prove a(f) e j{. To see this, we compute for 0 < t < 1

df, a(f) + 2/) (t)

d(a(f), a(af )))(t) = d(a(f), a(g))(t)

I |af)(x) - a(g)(x)| dx

pt px px

/ / yf (y) + 0 dy - y(g(y) + 1) dy

r-1 r- x

/ / yif (y)-g(y)) dy dx

yf (y) -g(y)1 dydx

nyf (y)-g (y)1 dx dy

1 (t - y)yf(y)-g(y)|dy

<J0 j If (y)-g (y)| dy

j Jof (y)-g(y)| dy

= J d(f, g)(t).

Thus, we have g e J, and Jd(f,g) e s(g, Tg).

Therefore, all conditions of Theorem 1.1 are satisfied and T has a fixed pointf"(x) =

exr -1.

3 Fixed-point theorems for multivalued Caristi type mappings

Let (X, d) be a cone metric space with a preordering c.

A sequence {xn} of points in X is called c-decreasing if xn+0 c xn for all n > 0. The set S(x) = {y e X: y c x} is c-complete if every decreasing Cauchy sequence in S(x) converges in it.

A function/ : X ^ E is called lower semicontinuous from above if, for every sequence {xn} c X conversing to some point x e X and satisfying fxn+1 < fxn for all n e N,we have

fx ^ fxn.

Lemma 3.1 Let (X, d) be a cone metric space, and letT: X ^ N(X) be a multivalued mapping. Suppose that $ : X ^ E is a function and n : P ^ P is a nondecreasing, continuous, and subadditive function such that n(t) = 0 if and only ft = 0. We define a relation <n on X as follows:

y x if and only if $(x)-$ (y) e s( n(d(x, y))). (3.1)

Then <n is a partial order on X.

Proof The proof follows by using the cone metric axioms, properties (1) and (3) for the cone, and (3.1). □

Lemma 3.2 ([17]) LetP c E be a strongly minihedral and continuous cone, andlet (X, c) be a preordered set. Suppose that a mapping ^ : X ^ E satisfies the following conditions:

(1) x c y and x = y imply f (x) -< f (y);

(2) for every c-decreasingsequence {xn} C X, there exists y e X such that y c xn for all n e N;

(3) f is bounded from below.

Then, for each x e X, S(x) has a minimal element in S(x), where S(x) = {y e X: y c x}.

Theorem 3.1 Let (X, d) be a cone metric space such that P is strongly minihedral and continuous, and letT: X ^ N (X) be a multivalued mapping and $ : X ^ E be a mapping bounded from below. Suppose that, for each x e X, S(x) = {y e X: y —n x} is —n-complete, where —n is a partial ordering on X defined as (3.1). Iffor any x e X, there exists y e Tx satisfying

$(x)-$(y) e s(n(d(x,y))), then T has a fixed point in X.

Proof We define a partial ordering —n on X as (3.1).

Ifx —n y andx = y, then 0 -< d(y,x) and $(y)-$(x) e s(n(d(y,x))), and so 0 -< n(d(y,x)) — $(y) - $(x). Hence, $(x) -< $(y).

Let {xn} be a —n-decreasing sequence in X. Thenxn+1 e S(xn) for all n > 0, and {$(xn)} is bounded from below, because $ is bounded from below. Hence, {$(xn)} is bounded. Since P is strongly minihedral, u = inf $(xn) exists in E. Also, since P is continuous, inf{||$(xn) -u||: n e N} = 0. Hence, limn^TO $(xn) = u and u — $(xn) for all n > 0.

For m > n, since xm —n xn, $(xn) - $(xm) e s(n(d(xn,xm))). Hence n(d(xn,xm)) — $(xn) - $(xm) — $(xn) - u. Thus, limn,m^TO n(d(xn,xm)) = 0. Since n is continuous,

n(limn,m^TO d(xn,xm)) = 0. So limn,m^TO d(xn, xm) = 0.

Hence, {xn} is a —n-decreasing Cauchy sequence in S(xo). Since S(xn) is —n-complete and xn+1 e S(xn) for all n > 0, there exists x e S(xn) such that limn^TO xn = x. Thus, x —n xn for all n > 0.

By Lemma 3.2, S(x0) has a minimal element x in S(x0). By assumption, there exists y0 e Tx such that $(x) - $(y0) e s(n(d(x,y0))). Hence, y0 —n x. Since x is minimal element in S(x0), y0 = x. Thus, x e Tx. □

Corollary 3.2 Let (X, d) be a cone metric space such that P is strongly minihedral and continuous, and letT: X ^ N (X) be a multivalued mapping and $ : X ^ E be a mapping bounded from below. Suppose that, for each x e X, S(x) = {y e X: y —n x} is —n-complete, where —n is a partial ordering on X defined as (3.1).

Iffor any x e X and for any y e Tx,

$(x) - $(y) e s(n(d(x,y))), then there exists x0 e X such that Tx0 = {x0}.

Theorem 3.3 Let (X, d) be a complete cone metric space such that P is strongly minihedral and continuous. Suppose that T : X ^ N(X) is a multivalued mapping and $ : X ^ E is lower semicontinuous from above and boundedfrom below.

Iffor any x e X, there exists y e Tx satisfying $(x)-$(y) e s(n(d(x,y))), then T has a fixed point in X.

Proof We define a partial ordering <n on X as (3.1). It suffices to show that, for each x0 e X, S(x0) is <n-complete.

Let x0 e X be a fixed, and let {xn} be a <n-decreasing Cauchy sequence in S(x0). Then it is a <n-decreasing Cauchy sequence in X. Hence, $(xn+1) ^ $(xn) for all n e N. Since X is complete, there exists x e X such that limn^ oo xn = x. Since $ is lower semicontinuous from above, $(x) ^ limn^o $(xn). Thus, $(x) ^ $(xn) for all n e N. Since

xm ^^ n xn

m > n,we obtain

$(xn) - $(xm) e s(n(d(xn, xm))). Hence,

n(d(xn, xm)) < $(xn) - $(xm) < $M - $(x).

Letting m ^ro in above inequality, we have n(d(xn,x)) ^ $(xn) - $(x) because n and d are continuous. Hence, $(xn) - $(x) e s(n(d(xn,x))).

Thus, we have x <n xn, and so x <n xn <n x0. Hence, x e S(x0), and hence S(x0) is <n-complete. From Theorem 3.3, T has a fixed point in X. □

Corollary 3.4 Let (X, d) be a complete cone metric space such thatP is strongly minihedral and continuous. Suppose that T : X ^ N(X) is a multivalued mapping and $ : X ^ E is lower semicontinuous from above and bounded from below. Iffor any x e X and for any y e Tx,

$(x)-$(y) e s(n(d(x,y))),

then there exists x0 e X such that Tx0 = {x0}.

We now give an example to support Theorem 3.3.

Example 3.1 Let X = Lo [0,1], and let E = R1 and P = {(x,y): x,y > 0}. We define d: X x X ^ E by d(f,g) = (|f -gII«,, |f -g||p), where 1 < p < o. Then (X, d) is a complete cone metric space, and P is strongly minihedral and continuous. Let n(s) = s for all s e P.

We define a multivalued mapping T: X ^ N(X) by

Tf = jg e X: -1f (x) < g(x) < 1f (x) iff (x) > 0 and 1f (x) < g(x) < -1f (x)iff (x)< 01

and we define a mapping $ : X ^ P by

$/) = (I/ II», If II,).

Then $ is lower semicontinuous from above and bounded from below.

For any f e X, put g (*) = f (*) e Tf. Then we have n(d(f, g)) = (| If II», 2 If II,) = $(f) -$(g), and so $f) - $(g) e s(n(d(f, g))). Thus, all conditions of Theorem 3.3 are satisfied and T has a fixed pointf'(x) = 0.

Remark 3.1 Theorem 3.3 (resp. Corollary 3.4) is a generalization of Theorem 4.2 (resp. Corollary 4.3) in [21], and also results in [38, 50] to the case of cone metric spaces.

If n(t) = t in Theorem 3.3 (resp. Corollary 3.4), then we have generalizations of the results in [12, 36,49] to the case of cone metric spaces.

Competing interests

The authors declare that they have no competing Interests. Authors' contributions

Allauthors contributed equally and significantly In writing this paper. Allauthors read and approved the finalmanuscript. Author details

1 Department of Mathematics, Hanseo University, Seosan, Chungnam 356-706, South Korea. 2Department of Mathematics, Moyngji University, Yongin, 449-728, South Korea.

Acknowledgements

The authors would like to thank the referees for carefulreading and giving valuable comments. This research (S.H. Cho) was supported by the Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (No. 2011-0012118).

Received: 30 April 2012 Accepted: 2 August 2012 Published: 16 August 2012

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doi:10.1186/1687-1812-2012-133

Cite this article as: Cho et al.: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric spaces. Fixed Point Theory and Applications 2012 2012:133.