Existence and convergence of best proximity points for generalized pseudo-contractive and Lipschitzian mappings via an Ishikawa-type iterative scheme

Abstract

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa’s iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result.

1 Introduction

Assume that M and N are nonempty subsets of a metric space $(X,d)$. If $M \cap N = \emptyset$, then the mapping f from M to N does not have a solution for the fixed-point equation $f(\eta ) = \eta$. When the fixed-point equation does not possess a solution, then it is attempted to determine an approximate solution η such that the error $d(\eta , f\eta )$ is minimum. In this situation, the best proximity-point theorems guarantee the existence and uniqueness of such an optimization for the fixed-point equations. Naturally, the best proximity point for the nonself-mappings is defined as follows:

Definition 1.1

Let M, N be two nonempty and disjoint subsets of a metric space $(X,d)$. A mapping $\Gamma : M \rightarrow N$ is said to have a best proximity point if there exist $\eta ^{*} \in M$ such that $d(\eta ^{*},\Gamma \eta ^{*}) = d(M,N)$.

Many researchers have proved the existence results on the best proximity points for various kinds of contractions. For such results, one may refer to [2, 4, 6–8, 12, 13, 15–18]. Recently, researchers have shown an interest in approximating the best proximity points through well-known iterative processes that may be seen in [1, 3, 9–11, 14, 19, 20].

On the other hand, numerous research articles have been published on the convergence of fixed points for the class of self- and nonself-contractive-type mappings in metric spaces, Hilbert spaces, and several classes of Banach spaces. For further exploration of this topic, we refer to the monograph [5] and the references cited therein.

A fundamental result in metric fixed-point theory is the following theorem, which uses the Picard iteration method.

Theorem 1.2

[5] Let $(X, d)$ be a complete metric space and $\Gamma : X \rightarrow X$ be a contraction, that is an operator satisfying

$$ d(\Gamma \eta , \Gamma \omega ) \leq a d(\eta ,\omega ),\quad \textit{for any } \eta , \omega \in X, $$

with $a \in [0,1)$ fixed. Then, Γ has a unique fixed point.

One of the effective methods for approaching the fixed point of a mapping $\Gamma : X \rightarrow X$ is the Ishikawa iteration scheme, starting with any $\eta _{0} \in X$ and for $n \geq 0$ defined by

$$ \eta _{n+1} = (1 - \gamma _{n})\eta _{n} + \gamma _{n} \Gamma \bigl((1 - \delta _{n})\eta _{n} + \delta _{n} \Gamma \eta _{n}\bigr), $$

where $\gamma _{n}, \delta _{n} \in [0, 1]$. In this direction, we state the following theorem on the iterative approximation of a fixed point that was proved by Ishikawa [11], for Lipschitzian pseudo-contractive mapping.

Theorem 1.3

[11] Let K be a convex and compact subset of a Hilbert space H and let $\Gamma : K \rightarrow K$ be Lipschitzian pseudo-contractive and let $\eta _{1} \in K$. Then, the Ishikawa iteration $\{\eta _{n}\}$, defined by

$$ \eta _{n+1} = (1-\gamma _{n}) \eta _{n} + \gamma _{n} \Gamma \bigl[(1- \delta _{n}) \eta _{n} + \delta _{n} \Gamma \eta _{n}\bigr], $$

where $\{\gamma _{n}\}$, $\{\delta _{n}\}$ are sequences of positive numbers satisfying

$$ (i) \quad 0 \leq \gamma _{n} \leq \delta _{n} \leq 1, \quad n \geq 1 ;\qquad (ii)\quad \lim_{n \rightarrow \infty} \delta _{n} = 0 ; \qquad (iii)\quad \sum_{n=1}^{ \infty} \gamma _{n} \delta _{n} = \infty , $$

converges strongly to a fixed point of Γ.

The next result gives sufficient conditions to obtain a fixed point without assuming the Lipschitzian condition.

Theorem 1.4

[5] Let K be a closed, bounded, and convex subset of a real uniformly convex Banach space H. Let $\Gamma : K \rightarrow K$ a strongly pseudo-contractive that has at least a fixed point $\eta ^{*}$. Let $\eta _{1} \in K$, then the Ishikawa iteration $\{\eta _{n}\}$, defined by

$$ \eta _{n+1} = (1-\gamma _{n}) \eta _{n} + \gamma _{n} \Gamma \bigl[(1- \delta _{n}) \eta _{n} + \delta _{n} \Gamma \eta _{n}\bigr], $$

where $\{\gamma _{n}\}$, $\{\delta _{n}\}$ are sequences of positive numbers satisfying

$$ \begin{gathered} (i) \quad 0 \leq \gamma _{n}, \delta _{n} < 1, \quad n \geq 1 ; \qquad (ii)\quad \lim_{n \rightarrow \infty} \gamma _{n} = 0, \qquad \lim_{n \rightarrow \infty} \delta _{n} = 0 ; \\ (iii)\quad \sum_{n=1}^{\infty} \gamma _{n} = \infty , \end{gathered} $$

converges strongly to a fixed point of Γ.

Motivated by Theorems 1.3 and 1.4, a natural question arises: how can one construct the Ishikawa iteration for nonself-mappings that approximate the best proximity point of such mappings? In this context, we will initiate the construction of the Ishikawa iteration process for nonself-mappings and investigate the convergence results for the best proximity point.

Before presenting the iterative approximation for the best proximity point, let us establish the existence of a best proximity point. To do so, we will recall some basic notions and definitions:

Let M and N be two subsets of a Hilbert space H with inner product $\langle \cdot , \cdot \rangle$ and norm $\Vert \cdot \Vert$:

$$\begin{aligned} & dist(M,N) = d(M,N) = inf\bigl\{ \Vert \eta -\omega \Vert : \eta \in M, \omega \in N \bigr\} ; \\ & P_{M}(\eta ) = \bigl\{ \omega \in M : \Vert \eta -\omega \Vert = d(\eta ,M) \bigr\} ; \\ & M_{0} = \bigl\{ \eta \in M: \bigl\Vert \eta -\omega ' \bigr\Vert = d(M,N) \text{ for some } \omega '\in N \bigr\} ; \\ & N_{0} = \bigl\{ \omega \in N: \bigl\Vert \eta '- \omega \bigr\Vert = d(M,N) \text{ for some } \eta ' \in M \bigr\} . \end{aligned}$$

In [13], Kirk et al. proved the following lemma that guarantees the nonemptiness of $M_{0}$ and $N_{0}$.

Lemma 1.5

Let X be a reflexive Banach space and M be a nonempty, closed, bounded, and convex subset of X, and N be a nonempty, closed, and convex subset of X. Then, $M_{0}$ and $N_{0}$ are nonempty and satisfy $P_{N}(M_{0}) \subseteq N_{0}$, $P_{M}(N_{0}) \subseteq M_{0}$.

Definition 1.6

Let H be a Hilbert space with inner product $\langle \cdot , \cdot \rangle$ and norm $\Vert \cdot \Vert$. An operator $\Gamma : H \rightarrow H$ is said to be Lipschitzian if there exists a constant $s > 0$ such that, for all η, ω in H,

$$\begin{aligned} \Vert \Gamma \eta - \Gamma \omega \Vert \leq s \Vert \eta - \omega \Vert . \end{aligned}$$

Definition 1.7

Let H be a Hilbert space with inner product $\langle \cdot , \cdot \rangle$ and norm $\Vert \cdot \Vert$. An operator $\Gamma : H \rightarrow H$ is said to be a generalized pseudo-contraction if there exists a constant $r > 0$ such that, for all η, ω in H,

$$\begin{aligned} \Vert \Gamma \eta - \Gamma \omega \Vert ^{2} \leq r^{2} \Vert \eta - \omega \Vert ^{2} + \bigl\Vert \Gamma \eta - \Gamma \omega - r(\eta - \omega ) \bigr\Vert ^{2}. \end{aligned}$$

(1)

Remark 1.8

• 1. The condition (1), is equivalent to $\langle \Gamma \eta - \Gamma \omega , \eta - \omega \rangle \leq r \Vert \eta - \omega \Vert ^{2}$.

• 2. If $r = 1$, then a generalized pseudo-contraction reduces to a pseudo-contraction.

Definition 1.9

Let H be a Banach space with norm $\Vert \cdot \Vert$. An operator $\Gamma : H \rightarrow H$ is said to be strongly pseudo-contraction if there exists a constant $t > 1$ such that

$$\begin{aligned} \Vert \eta - \omega \Vert \leq \bigl\Vert (1+c) (\eta - \omega ) - ct ( \Gamma \eta - \Gamma \omega ) \bigr\Vert \end{aligned}$$

holds for all η, ω in H and $c>0$.

In this work, we begin by providing a set of sufficient conditions for the existence of a best proximity point for nonself-Lipschitzian, generalized pseudo-contractive mappings. Subsequently, we construct the Ishikawa iteration for nonself-mappings and establish convergence results for the best proximity point of Lipschitzian pseudo-contractive nonself-mappings. To support our main result, we present an illustrative example.

Furthermore, we delve into the convergence of the best proximity point for strongly pseudo-contractive mappings without imposing the Lipschitzian condition. This discussion expands the scope of our findings and highlights the applicability of our results in a broader class of mappings.

2 Main results

Let us prove the existence result of the best proximity point for nonself-generalized pseudo-contractive and Lipschitzian mapping in the Hilbert space settings.

Theorem 2.1

Let M, N be two closed and convex subsets of a real Hilbert space H assume M to be bounded. Let $\Gamma : M \rightarrow N$ be a generalized, pseudo-contractive, and Lipschitzian mapping with corresponding constants r and s such that $0 < r < 1$, $s>1$. If $\Gamma (M_{0}) \subseteq N_{0}$, then Γ has a unique best proximity point.

Proof

Let $\lambda \in (0,1)$ satisfying, $0 < \lambda < \frac{2(1-r)}{(1-2r + s^{2})}$. We consider a projection operator on $M_{0}$, that is, $P_{M_{0}} : \Gamma (M_{0}) \rightarrow M_{0}$. Also, we define an averaged operator $F : M_{0} \rightarrow M_{0}$, associated with $P_{M_{0}} \Gamma$,

$$\begin{aligned} F(\eta ) = (1 - \lambda ) \eta + \lambda P_{M_{0}} \Gamma \eta , \quad \text{for } \eta \in M_{0}. \end{aligned}$$

(2)

Since Γ is generalized, pseudo-contractive, and Lipschitzian, we have

$$\begin{aligned} \Vert F\eta - F\omega \Vert ^{2} =& \bigl\Vert (1 - \lambda ) \eta + \lambda P_{M_{0}} \Gamma \eta - (1 - \lambda ) \omega - \lambda P_{M_{0}} \Gamma \omega \bigr\Vert ^{2} \\ =& \bigl\Vert (1-\lambda ) (\eta -\omega ) + \lambda (P_{M_{0}} \Gamma \eta - P_{M_{0}} \Gamma \omega ) \bigr\Vert ^{2} \\ =& (1-\lambda )^{2} \Vert \eta -\omega \Vert ^{2} + 2 \lambda (1- \lambda ) \langle P_{M_{0}} \Gamma \eta - P_{M_{0}} \Gamma \omega , \eta - \omega \rangle \\ &+ \lambda ^{2} \Vert P_{M_{0}} \Gamma \eta - P_{M_{0}} \Gamma \omega \Vert ^{2}. \end{aligned}$$

(3)

Let us assume $u = \Gamma \eta - P_{M_{0}} \Gamma \eta$ and $v = \Gamma \omega - P_{M_{0}} \Gamma \omega$. Now, we claim that $u =v$. Suppose $u \neq v$, then by the strict convexity of H, we have

$$\begin{aligned} \biggl\Vert \frac{\Gamma \eta + \Gamma \omega }{2} - \frac{P_{M_{0}} \Gamma \eta + P_{M_{0}} \Gamma \omega }{2} \biggr\Vert =& \biggl\Vert \frac{u + v}{2} \biggr\Vert \\ < & max \bigl\{ \Vert u \Vert , \Vert v \Vert \bigr\} \\ =& d (M,N), \end{aligned}$$

which is a contradiction. Therefore, $u = v$. This implies that, $\Gamma \eta - \Gamma \omega = P_{M_{0}} \Gamma \eta - P_{M_{0}} \Gamma \omega$. Therefore, from (3), we obtain

$$\begin{aligned} \Vert F\eta - F\omega \Vert ^{2} =& (1-\lambda )^{2} \Vert \eta - \omega \Vert ^{2} + 2 \lambda (1-\lambda ) \langle \Gamma \eta - \Gamma \omega , \eta - \omega \rangle + \lambda ^{2} \Vert \Gamma \eta - \Gamma \omega \Vert ^{2} \\ \leq & (1-\lambda )^{2} \Vert \eta -\omega \Vert ^{2} + 2 \lambda (1- \lambda ) r \Vert \eta -\omega \Vert ^{2} + \lambda ^{2} s^{2} \Vert \eta - \omega \Vert ^{2} \\ =& \bigl((1-\lambda )^{2} + 2 \lambda (1-\lambda ) r + \lambda ^{2} s^{2} \bigr) \Vert \eta - \omega \Vert ^{2}. \end{aligned}$$

Then, $\Vert F\eta - F\omega \Vert \leq ((1-\lambda )^{2} + 2 \lambda (1- \lambda ) r + \lambda ^{2} s^{2} )^{1/2} \Vert \eta - \omega \Vert$.

Now, from $0 < \lambda < \frac{2(1-r)}{(1-2r + s^{2})}$, we obtain

$$\begin{aligned}& \lambda ^{2} \bigl(1-2r + s^{2}\bigr) < 2\lambda (1-r) = 2 \lambda (1-r) +1 - 1, \\& 1 + \lambda ^{2} - 2 \lambda + 2 \lambda r - 2 \lambda ^{2} r + \lambda ^{2} s^{2} < 1, \\& \bigl((1-\lambda )^{2} + 2 \lambda (1-\lambda ) r + \lambda ^{2} s^{2} \bigr) ^{1/2} < 1. \end{aligned}$$

This implies that F is contraction. By Theorem 1.2, F has a unique fixed point $p^{*} \in M_{0}$. Then, $P_{M_{0}} \Gamma p^{*} = p^{*}$. This implies that $d(p^{*}, \Gamma p^{*}) = d(M,N)$. □

Remark 2.2

• 1. If $0< s<1$, then Γ is a contraction nonself-mapping and the result follows from [15].

• 2. If $s=1$, then Γ is a nonexpansive nonself-mapping and the result follows from [18].

Now, we define a construction of Ishikawa iteration for the case of nonself-mapping:

Let M, N be two convex subsets of a Hilbert space H. Let us define $\Gamma : M \rightarrow N$ and assume $\Gamma (M_{0}) \subseteq N_{0}$. Consider the projective operator $P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}$. Let $\eta _{1} \in M_{0}$, then the Ishikawa iteration $\{\eta _{n}\}$, is defined by

$$\begin{aligned} \eta _{n+1} = (1-\gamma _{n}) \eta _{n} + \gamma _{n} P_{M_{0}} \Gamma \bigl[(1-\delta _{n}) \eta _{n} + \delta _{n} P_{M_{0}} \Gamma \eta _{n}\bigr], \quad n = 1,2,3,\ldots, \end{aligned}$$

(4)

where $\gamma _{n}, \delta _{n} \in [0,1]$.

Next, we extend the convergence result of Theorem 1.3, for the case of nonself-mappings, by using the proposed Ishikawa iteration for nonself-mappings.

Theorem 2.3

Let M, N be two closed and convex subsets of a Hilbert space H and assume M to be compact. Let $\Gamma : M \rightarrow N$ be a pseudo-contractive and Lipschitzian mapping with $\Gamma (M_{0}) \subseteq N_{0}$. Let $\eta _{1} \in M_{0}$, then the Ishikawa iteration $\{\eta _{n}\}$, defined in (4), with $\{\gamma _{n}\}$, $\{\delta _{n}\}$ are sequences of positive numbers satisfying

$$ (i) \quad 0 \leq \gamma _{n} \leq \delta _{n} \leq 1, \quad n \geq 1 ; \qquad (ii)\quad \lim_{n \rightarrow \infty} \delta _{n} = 0 ; \qquad (iii)\quad \sum_{n=1}^{ \infty} \gamma _{n} \delta _{n} = \infty , $$

converges strongly to a best proximity point of Γ.

Proof

First, we prove that the mapping $P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}$ is pseudo-contractive. It is enough to show that $\langle P_{M_{0}}\Gamma \eta - P_{M_{0}}\Gamma \omega , \eta - \omega \rangle \leq \Vert \eta - \omega \Vert ^{2}$, for all $\eta , \omega \in M_{0}$. Now, we assume $x = \Gamma \eta - P_{M_{0}} \Gamma \eta$ and $y = \Gamma \omega - P_{M_{0}} \Gamma \omega$. Now, we claim that $x =y$. Suppose $x \neq y$, then by the strict convexity of H, we have

$$\begin{aligned} \biggl\Vert \frac{\Gamma \eta + \Gamma \omega }{2} - \frac{P_{M_{0}} \Gamma \eta + P_{M_{0}} \Gamma \omega }{2} \biggr\Vert =& \biggl\Vert \frac{x + y}{2} \biggr\Vert \\ < & max \bigl\{ \Vert x \Vert , \Vert y \Vert \bigr\} \\ =& d (M,N), \end{aligned}$$

which is a contradiction. Therefore, $x = y$. This implies that, $\Gamma \eta - \Gamma \omega = P_{M_{0}} \Gamma \eta - P_{M_{0}} \Gamma \omega$. Since Γ is pseudo-contractive, we obtain

$$ \langle P_{M_{0}}\Gamma \eta - P_{M_{0}}\Gamma \omega , \eta - \omega \rangle = \langle \Gamma \eta - \Gamma \omega , \eta - \omega \rangle \leq \Vert \eta - \omega \Vert ^{2}. $$

Now, using that Γ is a Lipschitzian mapping, there exist $s>0$, we obtain

$$ \Vert P_{M_{0}}\Gamma \eta - P_{M_{0}}\Gamma \omega \Vert = \Vert \Gamma \eta - \Gamma \omega \Vert \leq s \Vert \eta - \omega \Vert , $$

which implies that the mapping $P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}$ is a Lipschitzian operator. Moreover, $M_{0}$ satisfies all the requirements of Theorem 1.3. This implies that the sequence $\{\eta _{n} \}$ converges to a fixed point $p^{*}$ of $P_{M_{0}}\Gamma$. Then, $P_{M_{0}} \Gamma p^{*} = p^{*}$. This implies that $d(p^{*}, \Gamma p^{*}) = d(M,N)$, that is, $p^{*}$ is a best proximity point of Γ. This completes the proof. □

The following example illustrates Theorem 2.2.

Example 2.4

Let $H = \mathbb{R}^{2}$ be a Hilbert space with the Euclidean inner product and norm. Assume $M = \{(0,\eta ) : 1/2 \leq \eta \leq 2\}$, $N = \{(1,\eta ) : 1/2 \leq \eta \leq 2 \}$. Clearly, $M_{0} = M$, $N_{0} = N$. Now, we define $\Gamma : M \rightarrow N$ by $\Gamma (0,\eta ) = (1, 1/\eta )$. Then, one can easily verify that Γ is pseudo-contractive and Lipschitzian. Assume $\eta _{0} = 0.5$, $\gamma _{n} = \delta _{n} = \frac{1}{\sqrt{n}}$ for all $n \geq 0$. Then,

$$\begin{aligned} (0, \eta _{n+1}) =& (1-\gamma _{n}) (0, \eta _{n}) + \gamma _{n} P_{M_{0}} \Gamma \bigl[(1-\delta _{n}) (0, \eta _{n}) + \delta _{n} P_{M_{0}} \Gamma (0, \eta _{n})\bigr] \\ =& \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) (0, \eta _{n}) + \frac{1}{\sqrt{n}} P_{M_{0}} \Gamma \biggl[ \biggl( \frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) (0, \eta _{n}) + \frac{1}{\sqrt{n}} P_{M_{0}} \biggl(1, \frac{1}{\eta _{n}} \biggr)\biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} P_{M_{0}} \Gamma \biggl[ \biggl(0, \biggl( \frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} \biggl(0, \frac{1}{\eta _{n}} \biggr)\biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} P_{M_{0}} \Gamma \biggl[ \biggl(0, \biggl( \frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} + \frac{1}{\sqrt{n} \eta _{n}} \biggr)\biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} P_{M_{0}} \Gamma \biggl[ \biggl(0, \frac{(\sqrt{n} - 1)\eta ^{2}_{n} +1}{\sqrt{n} \eta _{n}} \biggr)\biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} P_{M_{0}} \Gamma \biggl[ \biggl(0, \frac{(\sqrt{n} - 1)\eta ^{2}_{n} +1}{\sqrt{n} \eta _{n}} \biggr)\biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \frac{1}{\sqrt{n}} P_{M_{0}} \biggl[ \biggl(1, \frac{\sqrt{n} \eta _{n}}{(\sqrt{n} - 1)\eta ^{2}_{n} +1} \biggr) \biggr] \\ =& \biggl(0, \biggl(\frac{\sqrt{n} - 1}{\sqrt{n}} \biggr) \eta _{n} \biggr) + \biggl(0, \frac{\eta _{n}}{(\sqrt{n} - 1)\eta ^{2}_{n} +1} \biggr) \\ =& \biggl(0, \frac{\sqrt{n} - 1}{\sqrt{n}} \eta _{n} + \frac{\eta _{n}}{\sqrt{n} \eta _{n}^{2} - \eta _{n}^{2} + 1} \biggr). \end{aligned}$$

As $n \rightarrow \infty$, the Ishikawa iteration $(0, \eta _{n+1}) \rightarrow (0,1)$, in particular, at $(0, \eta _{118}) =(0, 1)$, reaches the best proximity point of Γ. This result is achieved by simple Matlab coding.

Finally, we approximate the best proximity point for strongly pseudo-contractive nonself-mappings without Lipschitzian. This is an extended version of Theorem 1.4, for the case of nonself-mappings.

Theorem 2.5

Let M, N be two closed, bounded, and convex subsets of a real uniformly convex Banach space H. Let $\Gamma : M \rightarrow N$ be a strongly pseudo-contractive that has at least a best proximity point $\eta ^{*}$ and assume that $\Gamma (M_{0}) \subseteq N_{0}$. Let $\eta _{1} \in M_{0}$, then the Ishikawa iteration $\{\eta _{n}\}$, defined in (4), with $\{\gamma _{n}\}$, $\{\delta _{n}\}$ being sequences of positive numbers satisfying

$$ \begin{gathered} (i) \quad 0 \leq \gamma _{n}, \delta _{n} < 1, \quad n \geq 1 ; \qquad (ii)\quad \lim_{n \rightarrow \infty} \gamma _{n} = 0, \qquad \lim_{n \rightarrow \infty} \delta _{n} = 0 ; \\ (iii) \quad \sum_{n=1}^{\infty} \gamma _{n} = \infty , \end{gathered} $$

converges strongly to a best proximity point of Γ.

Proof

One can easily verify that the mapping $P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}$ is strongly pseudo-contractive and the result follows by Theorem 1.4. □