Fixed point theorems for multivalued maps in cone metric spaces

Abstract

The aim of this article is to generalize a result which is obtained by Mizoguchi and Takahashi [J. Math. Anal. Appl. 141, 177-188 (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^Xbe 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

$$lim\underset{r\to t^{+}}{sup}\phi \left(r\right)<1forallt\in \left[0,\infty \right)$$

and if

$$H\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)$$

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

$$u_1\le u_2\le \cdots \le z,$$

then there exists u ∈ E such that

$$\underset{x\to \infty }{lim}∥u_n-u∥=0.$$

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

$$s\left(A,B\right)=\left({\cap }_{a\in A}s\left(a,B\right)\right)\bigcap \left({\cap }_{b\in B}s\left(b,A\right)\right).$$

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

$$∥c-a∥<\epsilon \mathsf{\text{implies}}a\in int\left(P\right).$$

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

$$lim\underset{n\to \infty }{sup}\phi \left(r_n\right)<1$$

(2.1.1)

for any decreasing sequence {r _n } in P,

and if

$$\phi \left(d\left(x,y\right)\right)d\left(x,y\right)\in s\left(Tx,Ty\right)$$

(2.1.2)

for all x,y ∈ X(x ≠ y), then T has a fixed point in X.

Proof. Let x₀ ∈ X and x₁ ∈ Tx ₀ . From (2.1.2), we have

$$\phi \left(d\left(x_0,x_1\right)\right)d\left(x_0,x_1\right)\in s\left(T{x}_0,T{x}_1\right).$$

Thus, we have by Lemma 1.2 (3), φ(d(x₀, x₁))d(x₀, x₁) ∈ s(x₁, Tx₁).

By definition, we can take x₂ ∈ Tx₁ such that φ(d(x ₀ , x₁))d(x ₀ , x₁) ∈ s (d (x₁, x₂)). So, d(x₁, x₂) ≤ φ(d(x ₀ , x₁))d(x ₀ , x₁).

Again, we have by (2.1.2), φ(d(x₁, x₂))d(x₁, x₂) ∈ s(Tx₁, Tx ₂ ). Thus, we have φ(d(x₁, x₂))d(x₁, x₂) ∈ s(x₂, Tx₂).

Thus, we can choose x ₃ ∈ Tx₂ such that φ(d(x₁, x₂))d(x₁, x₂) ∈ s(d(x₂, x₃)) and so d(x₂, x₃) ≤ φ(d(x₁, x₂))d(x₁, x₂).

Inductively, we can construct a sequence {x _n } in X such that for n = 1, 2, ...,

$$d\left(x_n,x_{n+1}\right)\le \phi \left(d\left(x_{n-1},x_n\right)\right)d\left(x_{n-1},x_n\right),x_{n+1}\in T{x}_n.$$

(2.1.3)

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

$$lim\underset{n\to \infty }{sup}\phi \left(d\left(x_n,x_{n+1}\right)\right)=r.$$

Thus, for any l ∈ (r, 1), there exists n₀ ∈ ℕ such that for all n ≥ n₀, φ(d(x_{n- 1}, x _n )) < l.

Without loss of generality, we may assume n₀ = 1. Then, we have

$$d\left(x_n,x_{n+1}\right)\le \phi \left(d\left(x_{n-1},x_n\right)\right)d\left(x_{n-1},x_n\right)<ld\left(x_{n-1},x_n\right)<l^{n}d\left(x_0,x_1\right).$$

For m > n, we have

$$d\left(x_n,x_m\right)\le \frac{l^n}{1-l}d\left(x_0,x_1\right).$$

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

$$\phi \left(d\left(x_n,u\right)\right)d\left(x_n,u\right)\in s\left(x_{n+1},Tu\right).$$

Thus, there exists v _n ∈ Tu such that

$$\phi \left(d\left(x_n,u\right)\right)d\left(x_n,u\right)\in s\left(d\left(x_{n+1},v_n\right)\right).$$

Hence, d(x_{n+ 1}, v _n ) ≤ d(x _n , u). Thus, we have

$$d\left(u,v_n\right)\le d\left(u,x_{n+1}\right)+d\left(x_{n+1},v_n\right)\le d\left(u,x_{n+1}\right)+d\left(x_n,u\right).$$

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 2.2-2.5.

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

$$\phi \left(d\left(x,y\right)\right)d\left(x,y\right)\in s\left(Tx,Ty\right)$$

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

$$kd\left(x,y\right)\in s\left(Tx,Ty\right)$$

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

$$H\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)$$

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

$$H\left(Tx,Ty\right)\le kd\left(x,y\right)$$

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 = L¹[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 $d\left(f,g\right)\left(t\right)={\int }_0^t\left|f\left(x\right)-g\left(x\right)\right|dx$, where 0 ≤ t ≤ 1. Then, d is a cone metric on X. Consider a mapping T: X → CB(X) defined by

$$\left(Tf\right)\left(x\right)={\int }_0^{x}y\left(f\left(y\right)-1\right)dy.$$

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.

$$\begin{array}{c}d\left(Tf,Tg\right)\left(t\right)\\ ={\int }_0^t|{\int }_0^{x}y\left(f\left(y\right)-1\right)dy-{\int }_0^{x}y\left(g\left(y\right)-1\right)dy|dx\\ ={\int }_0^t\left|{\int }_0^{x}y\left(f\left(y\right)-g\left(y\right)\right)dy\right|dx\\ \le {\int }_0^t{\int }_0^{x}y|f\left(y\right)-g\left(y\right)|dydx\\ ={\int }_0^t{\int }_y^{t}y|f\left(y\right)-g\left(y\right)|dxdy\\ ={\int }_0^t\left(t-y\right)y|f\left(y\right)-g\left(y\right)|dy\\ \le {\int }_0^t\frac{t^2}{4}|f\left(y\right)-g\left(y\right)|dy\\ \le \frac{1}{4}{\int }_0^t|f\left(y\right)-g\left(y\right)|dy\\ =\frac{1}{4}d\left(f,g\right)\left(t\right).\end{array}$$

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 $f*\left(x\right)=-e^{\frac{x^2}{2}}+1$.