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Fixed point theorems for multivalued maps in cone metric spaces
Fixed Point Theory and Applicationsvolume 2011, Article number: 87 (2011)
Abstract
The aim of this article is to generalize a result which is obtained by Mizoguchi and Takahashi [J. Math. Anal. Appl. 141, 177188 (1989)] to the case of cone metric spaces.
MSC: 47H10; 54H25.
1 Introduction
Banach contraction principle is widely recognized as the source of metric fixed point theory. Also, this principle plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions for nonlinear equations, systems of linear equations and linear integral equations and to prove the convergence of algorithms in computational mathematics.
Because of its importance for mathematical theory, Banach contraction principle has been extended in many direction (see [1–8]). Especially, the generalizations to multivalued case are immense too (see [6, 9, 10]).
Mizoguchi and Takahashi proved the following theorem in [9].
Theorem 1.1. Let (X,d) be a complete metric space and let T: X → 2^{X}be a multivalued map such that Tx is a closed bounded subset of X for all x ∈ X. If there exists a function φ: (0, ∞) → [0,1) such that
and if
for all x,y ∈ X(x ≠ y), then T has a fixed point in X.
Recently, in [10], the authors introduced a cone metric space which is a generalization of a metric space. They generalized Banach contraction principle for cone metric spaces. Since then, in [11–23], the authors obtained fixed point theorems in cone metric spaces. And the authors [24, 25] obtained fixed point results in cone Banach spaces.
The authors [26–28] proved fixed point theorems for multivalued maps in cone metric spaces.
In this article, we extend the Hausdorff distance to cone metric spaces, and generalize Theorem 1.1 to the case of cone metric spaces.
Consistent with Huang and Zhang [17], the following definitions will be needed in the sequel.
Let E be a real Banach space. A subset P of E is a cone if the following conditions are satisfied:

(1)
P is nonempty closed and P ≠ {θ},

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

(3)
P ∩ (P) = {θ}.
Given a cone P ⊂ E, we define a partial ordering ≤ with respect to P by x ≤ y if and only if y  x ∈ P. We write x < y to indicate that x ≤ y but x ≠ y.
For x,y ∈ P, x ≪ y stand for y  x ∈ int(P), where int(P) is the interior of P. A cone P is called normal if there exists a number K > 1 such that for all x,y ∈ E, x ≤ K y whenever θ ≤ x ≤ y.
A cone P is called regular if every increasing sequence which is bounded from above is convergent. That is, if {u_{ n }} is a sequence such that for some z ∈ E
then there exists u ∈ E such that
Equivalently, a cone P is regular if and only if every decreasing sequence which is bounded from below is convergent.
It has been mentioned [17] that every regular cone is normal (see also [22]).
From now on, we assume that E is a Banach space, P is a cone in E with $int\left(P\right)\ne \varnothing $ and ≤ is a partial ordering with respect to P.
Let X be a nonempty set. A mapping d: X × X → E is called cone metric [17] on X if the following conditions are satisfied:

(1)
θ ≤ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y,

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

(3)
d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.
A sequence {x_{ n }} in a cone metric space (X, d) converges [17] to a point x ∈ X (denoted by lim_{n →∞}x_{ n }= x or x_{ n }→ x) if for any c ∈ int(P), there exists N such that for all n > N, d(x_{ n }, x) ≪ c. A sequence {x_{ n }} in a cone metric space (X, d) is Cauchy [17] if for any c ∈ int(P), there exists N such that for all n,m > N, d(x_{ n }, x_{ m }) ≪ c. A cone metric space (X,d) is called complete [17] if every Cauchy sequence is convergent.
Lemma 1.1. [17] Let (X, d) be a cone metric space and P be a normal cone, and let {x_{ n }} be a sequence in X and x,y ∈ X. Then, we have that

(1)
lim_{n →∞}x_{ n }= x if and only if lim_{n →∞}d(x_{ n }, x) = θ,

(2)
{x_{ n }} is Cauchy if and only if lim_{n , m →∞}d(x_{ n }, x_{ m }) = θ,

(3)
if lim_{n →∞}x_{ n }= x and lim_{n →∞}x_{ n }= y, then x = y.
We denote by N(X)(resp. B(X), CB(X)) the set of nonempty(resp. bounded, sequentially closed and bounded) subset of a metric space or a cone metric space.
Let (X, d) be a cone metric space.
From now on, we denote s(p) = {q ∈ E: p ≤ q} for p ∈ E, and s(a, B) = ∪_{b ∈ B}s(d(a, b)) for a ∈ X and B ∈ N(X).
For A,B ∈ B(X), we denote
Lemma 1.2. Let (X, d) be a cone metric space, and let P be a cone in Banach space E.

(1)
Let p,q ∈ E. If p ≤ q, then s(q) ⊂ s(p).

(2)
Let x ∈ X and A ∈ N(X). If θ ∈ s(x, A), then x ∈ A.

(3)
Let q ∈ P and let A, B ∈ B(X) and a ∈ A. If q ∈ s(A, B), then q ∈ s(a, B).
Remark 1.1. Let (X,d) be a cone metric space. If E = ℝ and P = [0,∞), then (X,d) is a metric space. Moreover, for A, B ∈ CB(X), H(A, B) = inf s(A, B) is the Hausforff distance induced by d.
Remark 1.2. Let (X, d) be a cone metric space. Then, s({a}, {b}) = s(d(a, b)) for a,b ∈ X.
Lemma 1.3. If u_{ n }∈ E with u_{ n }→ θ, then for each c ∈ int(P) there exists N such that u_{ n }≪ c for all n > N.
Proof. Let c ∈ int(P). There exists ϵ > 0 such that
Since u_{ n } → 0, there exists N such that u_{ n } < ϵ for all n > N. Thus, we have c  (c  u_{ n }) < ϵ and so c  u_{ n }∈ int(P) for all n > N. Therefore, u_{ n }≪ c for all n > N.
2 Fixed point theorems
Theorem 2.1. Let (X, d) be a complete cone metric space with normal cone P and let T: X → CB(X) be a multivalued map. If there exists a function φ: P → [0,1) such that
for any decreasing sequence {r_{ n }} in P,
and if
for all x,y ∈ X(x ≠ y), then T has a fixed point in X.
Proof. Let x_{0} ∈ X and x_{1} ∈ Tx_{ 0 }. From (2.1.2), we have
Thus, we have by Lemma 1.2 (3), φ(d(x_{0}, x_{1}))d(x_{0}, x_{1}) ∈ s(x_{1}, Tx_{1}).
By definition, we can take x_{2} ∈ Tx_{1} such that φ(d(x_{ 0 }, x_{1}))d(x_{ 0 }, x_{1}) ∈ s (d (x_{1}, x_{2})). So, d(x_{1}, x_{2}) ≤ φ(d(x_{ 0 }, x_{1}))d(x_{ 0 }, x_{1}).
Again, we have by (2.1.2), φ(d(x_{1}, x_{2}))d(x_{1}, x_{2}) ∈ s(Tx_{1}, Tx_{ 2 }). Thus, we have φ(d(x_{1}, x_{2}))d(x_{1}, x_{2}) ∈ s(x_{2}, Tx_{2}).
Thus, we can choose x_{ 3 }∈ Tx_{2} such that φ(d(x_{1}, x_{2}))d(x_{1}, x_{2}) ∈ s(d(x_{2}, x_{3})) and so d(x_{2}, x_{3}) ≤ φ(d(x_{1}, x_{2}))d(x_{1}, x_{2}).
Inductively, we can construct a sequence {x_{ n }} in X such that for n = 1, 2, ...,
If x_{ n }= x_{n+ 1}for some n ∈ ℕ, then T has a fixed point.
We may assume that x_{ n }≠ x_{n+ 1}for all n ∈ ℕ. From (2.1.3), {d(x_{ n }, x_{n +1})} is a decreasing sequence in P. From (2.1.1), there exists r ∈ (0,1) such that
Thus, for any l ∈ (r, 1), there exists n_{0} ∈ ℕ such that for all n ≥ n_{0}, φ(d(x_{n 1}, x_{ n })) < l.
Without loss of generality, we may assume n_{0} = 1. Then, we have
For m > n, we have
By Lemma 1.3, {x_{ n }} is a Cauchy sequence in X. It follows from the completeness of X that there exists u ∈ X such that lim_{n →∞}x_{ n }= u.
We now show that u ∈ Tu.
From (2.1.2), we have φ(d(x_{ n }, u))d(x_{ n }, u) ∈ s(Tx_{ n }, Tu) for n ∈ ℕ. By Lemma 1.2 (3), we obtain
Thus, there exists v_{ n }∈ Tu such that
Hence, d(x_{n+ 1}, v_{ n }) ≤ d(x_{ n }, u). Thus, we have
By letting n → ∞ in above inequality and by Lemma 1.1, we have lim_{n →∞}d(u, v_{ n }) = 0. Again, by Lemma 1.1, lim_{n →∞}v_{ n }= u. Since Tu is closed, u ∈ Tu.
Remark 2.1. (1) By Remark 1.1, Theorem 2.1 generalize Theorem 1.1 [Theorem 5, 13].

(2)
The authors [26, 28] obtained fixed point theorems for multivalued maps T defined on cone metric spaces (X, d) under assumption that the function I(x) = inf_{x ∈ Tx}d(x,y) is lower semicontinuous, and the author [27] obtained a fixed point theorem for multivalued maps T under assumptions that the function I(x), x ∈ X is lower semicontinuous and a dynamic process is given.

(3)
In [26–28], the authors do not use the concept of the Hausdorff metric on cone metric spaces, and their results cannot be applied directly to obtain the following corollaries 2.22.5.
Collorary 2.2. Let (X, d) be a complete cone metric space with normal cone P and let T: X → CB(X) be a multivalued map. If there exists a monotone increasing function φ: P → [0,1) such that
for all x,y ∈ X(x ≠ y), then T has a fixed point in X.
The following result is Nadler multivalued contraction fixed point theorem in cone metric space.
Collorary 2.3. Let (X, d) be a complete cone metric space with normal cone P and let T: X → CB(X) be a multivalued map. If there exists a constant k ∈ [0, 1) such that
for all x,y ∈ X, then T has a fixed point in X.
By Remark 1.1, we have the following corollaries.
Collorary 2.4. [29] Let (X, d) be a complete metric space and let T: X → CB(X) be a multivalued map. If there exists a monotone increasing function φ: (0, ∞) → [0, 1) such that
for all x,y ∈ X(x ≠ y), then T has a fixed point in X.
Collorary 2.5. [6] Let (X,d) be a complete metric space and let T: X → CB(X) be a multivalued map. If there exists a constant k ∈ [0, 1) such that
for all x,y ∈ X, then T has a fixed point in X.
The following example illustrates our main theorem.
Example 2.1. Let X = L^{1}[0, 1], E = C[0,1] and P = {f ∈ E: f ≥ 0}. Then, P is a normal cone with normal constant K = 1. Define d: X × X → E by $d\left(f,g\right)\left(t\right)={\int}_{0}^{t}\leftf\left(x\right)g\left(x\right)\rightdx$, where 0 ≤ t ≤ 1. Then, d is a cone metric on X. Consider a mapping T: X → CB(X) defined by
Let $\phi \left(t\right)=\frac{1}{2}$ for all t ∈ P. Obviously, condition (2.1.1) is satisfied.
We show that condition (2.1.2) is satisfied.
Consider the following inequality.
Thus, we have $\frac{1}{4}d\left(f,g\right)\in s\left(d\left(Tf,Tg\right)\right)=s\left(Tf,Tg\right)$. Hence, $\phi \left(d\left(f,g\right)\right)d\left(f,g\right)=\frac{1}{2}d\left(f,g\right)\in s\left(Tf,Tg\right)$.
Therefore, all conditions of Theorem 2.1 are satisfied and T has a fixed point $f*\left(x\right)={e}^{\frac{{x}^{2}}{2}}+1$.
References
 1.
Agarawl RP, O'Regan DO, Shahzad N: Fixed point theorems for generalized contractive maps of MeiKeeler type. Math. Nachr 2004, 276: 3–12. 10.1002/mana.200310208
 2.
Aubin JP, Siegel J: Fixed point and stationary points of dissipative multivalued maps. Proc Amer Math Soc 1980, 78: 391–398. 10.1090/S00029939198005533821
 3.
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int J Math Math Sci 2002, 29: 531–536. 10.1155/S0161171202007524
 4.
Covitz H, Nadler SB Jr: Multivalued contraction mappings in generalized metric spaces. Israel J Math 1970, 8: 5–11. 10.1007/BF02771543
 5.
Feng Y, Liu S: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings. J Math Anal Appl 2006, 317: 103–112. 10.1016/j.jmaa.2005.12.004
 6.
Nadler SB Jr: Multivalued contraction mappings. Pacific J Math 1969, 30: 475–478.
 7.
Vijayaraju P, Rhoades BE, Mohanraj R: A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. Int J Math Math Sci 2005, 15: 2359–2364.
 8.
Wang T: Fixed point theorems and fixed point stability for multivalued mappings on metric spaces. J Nanjing Univ Math Baq 1989, 6: 16–23.
 9.
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188. 10.1016/0022247X(89)90214X
 10.
Reich S: Some problems and results in fixed point theory. Contemp Math 1983, 21: 179–187.
 11.
Alnafei SH, Radenović S, Shahzad N: Fixed point theorems for mappings with convex diminishing diameters on cone metric spaces. Appl Math Lett 2011, 24: 2162–2166. 10.1016/j.aml.2011.06.019
 12.
Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl 2008, 341: 416–420. 10.1016/j.jmaa.2007.09.070
 13.
Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Appl Math Lett 2008.
 14.
Abdeljawad T, Karapinar E: A gap in the paper "A note on cone metric fixed point theory and its equivalence". Gazi Univ J Sci 2011,24(2):233–234. [Nonlinear Anal. 72(5), 2259–2261 (2010)]
 15.
Cho SH, Bae JS: Common fixed point theorems for mappings satisfying property ( E.A ) on cone metric spaces. Math Comput Modelling 2011, 53: 945–951. 10.1016/j.mcm.2010.11.002
 16.
Choudhury BS, Metiya N: Fixed points of weak contractions in cone metric spaces. Nonlinear Anal 2009.
 17.
Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
 18.
Ilić D, Rakočević V: Common fixed points for nmaps on cone metric spaces. J Math Anal Appl 2008, 341: 876–882. 10.1016/j.jmaa.2007.10.065
 19.
Ilić D, Rakočević V: Quasicontraction on cone metric spaces. Appl Math Lett 2008.
 20.
Karapinar E: Some nonunique fixed point theorems of Ciric type on cone metric spaces. Abstr Appl Anal 2010., 14: Article ID 123094, (2010)
 21.
Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl 2010,59(12):3656–3668. 10.1016/j.camwa.2010.03.062
 22.
Rezapour Sh, Hamlbarani R: Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J Math Anal Appl 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049
 23.
Yang SK, Bae JS, Cho SH: Coincidence and common fixed and periodic point theorems in cone metric spaces. Comput Math Appl 2011, 61: 170–177. 10.1016/j.camwa.2010.10.031
 24.
Karapinar E: Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl 2009., 9: 2009, Article ID 609281.
 25.
Karapinar E, Trkoglu DA: Best approximations for a couple in cone Banach spaces. Fixed Point Theory Appl 2010., 9: 2010 Article ID 784578
 26.
Kadelburg Z, Radenovič S: Some results on setvalued contractions in abstract metric spaces. Comput Math Appl 2011, 62: 342–350. 10.1016/j.camwa.2011.05.015
 27.
Klim D, Wardowski D: Dynamic processes and fixed points of setvalued nonlinear contractions in cone metric spaces. Nonlinear Anal 2009, 71: 5170–5175. 10.1016/j.na.2009.04.001
 28.
Wardowski D: Endpoints and fixed points of setvalued contractions in cone metric spaces. Nonlinear Anal 2009, 71: 512–516. 10.1016/j.na.2008.10.089
 29.
Daffer PZ, Kaneko H: Fixed points of generalized contractive multivalued mappings. J Math Anal Appl 1995, 192: 655–666. 10.1006/jmaa.1995.1194
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110012118). The authors express their gratitude to the referees for useful remarks and suggestions.
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Keywords
 fixed point
 multivalued map
 cone metric space