Real-valued non compactness measures in topological vector spaces and a pplications

Abstract

A non compactness measure with values in the lattice of extended non negative real numbers $[0, +\infty ]$ is introduced in the general setting of a Hausdorff topological vector space E. This generalizes the classical Kuratowski and Hausdorff non compactness measures. In order to achieve this, we introduce the notions of basic and sufficient collections of zero neighborhoods. We then show that our measure satisfies most of the properties of the classical non compactness measures. We particularly show that if E is locally p-convex for some $0 < p \le 1$, our measure is stable by the transition to the closed p-convex hull. This allows us to obtain, as applications, generalizations of the well-known three fixed point theorems, namely Schauder, Darbo, and Sadovskii’s ones in the setting of locally p-convex spaces. As another application, we establish a quantification of Ascoli theorem in the space C(X, E) of vector-valued continuous functions on a Hausdorff completely regular space with values in a topological vector space E, giving an alternative of Ambrosetti theorem initially stated in the metric spaces setting.

1 Introduction

One of the main tools when seeking for fixed points of a self-map of a metric space X, in which no compactness is assumed, is the notion of non compactness measure. The first such a measure was introduced by Kuratowski in [12] for a bounded subset A of X. This can be thought of as a measurement of the lack of compactness of A and it is defined as follows:

$$\begin{aligned} \alpha (A) = \inf \left\{ r > 0 , \exists n \in {\mathbb {N}}, \exists A_i \subset X \text { with } \delta (A_i) \le r \text { and } A \subset \cup _{i =1}^n A_{i} \right\} , \end{aligned}$$

(1)

where $\delta (A)$ stands for the diameter of A.

A similar non compactness measure is the so-called Hausdorff (or ball) non compactness measure $\beta$ of A defined by:

$$\begin{aligned} \beta (A) = \inf \{r > 0 : \exists n \in {\mathbb {N}}, \exists x_1, \ldots x_n \in X : A \subset \cup _{i=1}^n B(x_i, r)\}, \end{aligned}$$

(2)

where B(x, r) is the ball centered at x with radius r. These two non compactness measures are equivalent in the sense that, for every bounded subset A of X,

$$\begin{aligned} \beta (A) \le \alpha (A) \le 2 \beta (A). \end{aligned}$$

In the setting of a locally convex space E, whose topology is given by a separating family ${\mathbb {P}}$ of semi-norms, Sadovskii [19] introduced extensions of $\alpha$ and $\beta$ taking their values in the set of non negative functions on ${\mathbb {P}}$, rather than non negative real numbers, as in the normed spaces case. These extensions, again denoted by $\alpha$ and $\beta$ are the functions assigning to a bounded set $A \subset E$ the mappings $\alpha (A) : P \mapsto \alpha (A)(P)$ and $P \mapsto \beta (A)(P)$ respectively, where $\alpha (A)(P)$ (resp. $\beta (A)(P)$) is defined by (1) (resp. (2)), the diameters (resp. the balls) being taken with respect to the semi-norm P.

Later, Kaniok [7] extended the definition of Sadovskii in the more general setting of a topological vector space E, using zero neighborhoods in E instead of semi-norms. Recall that, for a given such neighborhood U, a set $A \subset E$ is said to be U-small (or small of order U [16]), if $A - A \subset U$. For every $A \subset E$, the U-measures of non compactness $\alpha _{U}(A)$ and $\beta _{U}(A)$ of A are defined as:

$$\begin{aligned} \alpha _U(A)&= \inf \left\{ r>0:A \text { is covered by a finite number of } r U \text {-small sets } A_i \right\} , \end{aligned}$$

(3)

$$\begin{aligned} \beta _U(A)&= \inf \left\{ r > 0 : \exists x_1,\ldots , x_n \in E \text { such that } A \subset \cup _{i = 1}^{n} \left( x_i + r U\right) \right\} , \end{aligned}$$

(4)

with $\inf \emptyset = +\infty$. In case E is a normed space and U is the closed unit ball of E, $\alpha _U$ and $\beta _U$ are nothing but the Kuratowski and Hausdorff measures of non compactness, respectively. Thus, if ${\mathscr {U}}$ denotes a fundamental system of balanced and closed zero neighborhoods in E and ${\mathscr {F}}_{\mathscr {U}}$ is the space of all functions $\varphi : \mathscr {U} \rightarrow {\mathbb {R}}$, endowed with the pointwise ordering, then, after Kaniok, the Kuratowski (resp. the Hausdorff) measure $\alpha (A)$ (resp. $\beta (A)$) of a subset A of E is the function defined from $\mathscr {U}$ into $[0, +\infty ]$ by $\alpha (A)(U) = \alpha _U(A)$ (resp. $\beta (A)(U) = \beta _U(A)$).

The invariance under the transition to the closed convex hull ${\overline{co}}(A)$ of the subset $A \subset E$ is one of the main properties of the Kuratowski and the Hausdorff measures $\alpha$ and $\beta$. This property remains valid in the setting of locally convex spaces as shown by Sadovskii in [19, Theorem 1.2.3], and is the key of the proof of the classical Darbo and Sadovskii fixed point theorems.

In this paper, we introduce new non compactness measures, again denoted by $\alpha$ and $\beta$, generalizing to the setting of a topological vector space E the Kuratowski and the Hausdorff ones. As in the normed case, our measures take their values in $[0, +\infty ]$. We then establish some of their properties. In particular, we show that, if the space E is locally p-convex, $0 < p \le 1$, then these measures are invariant under the transition to the closed s-convex hull, $0 < s \le p$. As applications, we first extend the Schauder, the Darbo, and the Sadovskii fixed point theorems to the locally p-convex setting. Up to our knowledge, particular attention has been given recently to the study of fixed point results for multi-valued maps defined on p-convex sets (see [5] and the references therein). The usual hypothesis that a multi-valued map has non empty p-convex values does not permit to derive directly the single-valued counterpart for these results when $0< p < 1$, since in this case the singletons fail to be p-convex. This is the case, for instance, for the p-convex version of Kakutani fixed point theorem given in [5, Corollary 2.13]. Hence, an investigation of p-convex versions of the three above-mentioned fixed point theorems for single-valued maps remains of interest. A complete study of these versions has been recently achieved in [20, 21] in the particular setting of p-normed spaces. Here, we extend this study to the general setting of locally p-convex spaces for the so-called Yanyan continuous maps (Sect. 3, Theorems 4, 5 and Corollary 1). However, whether the three theorems are still valid for continuous maps remains an open question.

A further application consists of a quantification of Ascoli theorem in the space of vector-valued continuous functions on a completely regular space, via an alternative version of Ambrosetti theorem.

After investigating in Sect. 1 non compactness measures with respect to a given zero neighborhood, we define, in Sect. 2, the non compactness measures $\alpha$ and $\beta$ in a topological vector space E. In order to maintain the scalar character of these measures and to overcome some difficulties occurring when involving the whole set of zero neighborhoods in E, we introduce the notions of basic and sufficient collections of zero neighborhoods (Definition 1). Next, we show that several properties of non compactness measures extend naturally to ours.

In Sect. 3, using our measures, we obtain extensions of Schauder, Darbo and Sadovskii fixed point theorems in the context of locally p-convex spaces. Our results generalize recent fixed point theorems given by Xiao and Lu [20] and Xiao and Zhu [21] in complete p-normed spaces.

Section 4 is devoted to the quantification of Ascoli theorem using non compactness measures in C(X, E), the space of all continuous functions from a Hausdorff completely regular space X into a topological vector space E. Such a quantification was first given by Ambrosetti [1], by showing that, if X and E are two metric spaces with X compact, and D is a bounded and equicontinuous subset of C(X, E), then

$$\begin{aligned} {\widehat{\alpha }}(D) = \sup _{x \in X}\alpha (D(x)), \end{aligned}$$

(5)

where $D(x) := \{f(x) : f \in D\}$, and $\alpha$ and ${\widehat{\alpha }}$ stand for the standard Kuratowski measures of non compactness on the metric spaces E and C(X, E) respectively. Ascoli theorem is then the particular case where one of the sides of the equality (5) is zero. A general version of Ascoli theorem was given by Kelley [8, p. 234] taking X to be a Hausdorff or a regular k-space and E to be a Hausdorff uniform space. It states that a subset D of C(X, E) is compact in the topology of uniform convergence on compacta if and only if D is closed, D(x) is relatively compact for each $x \in X$ and D is equicontinuous on every compact subset of X. Here, we present a quantified version of Ascoli theorem in a different setting than Ambrosetti’s. More precisely, using our non compactness measure $\alpha$, we give an extension of Ambrosetti theorem, letting X be a Hausdorff completely regular topological space and the range space E a Hausdorff topological vector space (see Theorem 7).

2 Non compactness measures with respect to a zero neighborhood

Throughout this paper, and unless otherwise stated, E will denote a Hausdorff topological vector space over the field ${\mathbb {K}} \in \{ {\mathbb {R}}, {\mathbb {C}}\}$, and p will be a real number with $0 < p \le 1$. The set of all balanced zero neighborhoods in E is denoted by $\mathscr {V}_{0}$. Recall that $U \in \mathscr {V}_0$ is said to be shrinkable, if it is absorbing, balanced, and $r {\overline{U}} \subset \mathring{U}$, for every $0< r <1$ ; here ${\overline{U}}$ stands for the closure of U and $\mathring{U}$ for its interior. Any topological vector space admits a local base at zero consisting of shrinkable sets (see [10] or [6] for details).

Recall that a subset A of E is said to be p-convex if it satisfies $\lambda A + \mu A \subset A$ for all $\lambda , \mu \ge 0$ such that $\lambda ^p + \mu ^p = 1$. The case $p =1$ is the usual case of a convex set. Note that if A is p-convex and contains 0, then it is s-convex for every positive $s \le p$. In particular, an absolutely p-convex set (i.e., a balanced and p-convex set) is absolutely s-convex for every $0 < s \le p$. Such a result fails if $0 \notin A$. Actually, a singleton $\{x\}$, $x \in E \setminus \{0\}$, is convex but not s-convex, for any $0< s < 1$.

We will frequently use the fact that if $A \subset E$ is p-convex, then

$$\begin{aligned} \lambda A + \mu A \subset \root p \of {\lambda ^p + \mu ^p}\ A , \qquad \lambda , \mu \ge 0. \end{aligned}$$

The p-convex hull of $A \subset E$, denoted by $co_p(A)$ is the smallest p-convex subset of E containing A. This is, equivalently, the set

$$\begin{aligned} co_{p}(A) = \bigcup _{n \ge 1} \left\{ \sum _{i = 1}^n \mu _i x_i, \quad 0 \le \mu _i, \ \sum _{i=1}^n\mu _i^p =1, \quad x_i \in A \right\} . \end{aligned}$$

The set $co_1 \left( A\right)$ is simply denoted by co(A).

The topological vector space E is said to be a locally p-convex space, if E has a local base at zero consisting of p-convex sets. The topology of a locally p-convex space is always given by an upward directed family ${\mathbb {P}}$ of p-semi-norms, where a p-semi-norm on E is any non negative real-valued and subadditive functional $\Vert \ \Vert _p$ on E, such that $\Vert \lambda x\Vert _p = |\lambda |^p \Vert x\Vert _p$ for every $x \in E$, $\lambda \in {\mathbb {K}}$. If E is Hausdorff, then for every $x \ne 0$, there is some $P \in {\mathbb {P}}$ such that $P(x) \ne 0$. Whenever the family ${\mathbb {P}}$ is reduced to a singleton, one says that $(E, \Vert \ \Vert _p)$ is a p-semi-normed space. A p-normed space is a Hausdorff p-semi-normed space.

Notice that the case $p = 1$ is the usual locally convex case. Furthermore, a p-normed space is a metric vector space with the translation invariant metric $d_p(x,y) = \Vert x - y\Vert _p$, $x, y \in E$. The classical Lebesgue spaces $L_p(\mu )$ defined on a complete measure space $(\varOmega , \mathscr {M}, \mu )$, $\mu$ being a positive measure on $\mathscr {M}$, are examples of p-normed spaces, where the p-norm is given by

$$\begin{aligned} \Vert f\Vert _p = \int \limits _{\varOmega }|f(x)|^p d\mu , \quad f \in L_p(\mu ). \end{aligned}$$

If P is a continuous p-semi-norm on E, then the ball $B_P(0, s) := \{x \in E : P(x) < s\}$ is shrinkable, for every $s > 0$. Indeed, if $r < 1$ and $x \in r \overline{B_P(0,s)}$, then there exists a net $(x_i)_i \subset B_P(0,s)$ such that $rx_i$ converges to x. By continuity of P, we get $P(x) \le r^p s < s$, saying that $r \overline{B_P(0,s)} \subset B_P(0,s)$. More generally, it can be shown that every p-convex $U \in \mathscr {V}_0$ is shrinkable. From now on, denote by ${\mathscr {N}}_0$ the set of all shrinkable zero neighborhoods in E.

Throughout all the sequel, the results are presented for the Kuratowski type non compactness measure $\alpha$. Similar arguments can be used to show their analogous for the Hausdorff type one $\beta$. Therefore, if $W \in {\mathscr {V}}_0$, we will denote by $\alpha _W$ the Kuratowski type W-measure of non compactness as given by (3).

Notice that the properties 1–3 in the proposition below are mentioned in [7] without proof.

Proposition 1

Let $A,B \subset E$, $U,V \in \mathscr {V }_ 0$. Then the following assertions hold:

1. $\alpha _U$is semi-additive, i.e., $\alpha _U(A \cup B) = \max (\alpha _U (A),\alpha _{U}(B))$.

2. $\alpha _{rU}\left( sA\right) =\frac{\left| s\right| }{|r|} {\alpha }_U\left( A\right)$, for every scalars $r, s\ne 0$. In particular, if $rU\subset V\subset sU$, then $\alpha _{U}(A)=0$if and only if $\alpha _{V}(A)=0$.

3. If $V+V\subset U$, then $\alpha _{U}(A+B)\le \max (\alpha _{V}(A),\alpha _{V}(B))$.

4. If $V\subset U$, then $\alpha _{U}\le \alpha _{V}$.

If U is shrinkable, then

5. $\alpha _{U}=\alpha _{{\overline{U}}}$.

6. $\alpha _{U}(A)=\alpha _{U}({\overline{A}})$.

If U is p-convex for some $0<p\le 1$, then

7. $\alpha _{U}(A+B)\le \root p \of {\alpha _{U}(A)^{p}+{\alpha }_{U}(B)^{p}}$. In particular, if U is convex, i.e., $p=1$, then $\alpha _{U}$ is algebraically semi-additive, i.e. $\alpha _{U} (A + B) \le \alpha _U(A) + \alpha _U(B)$.

8. $\alpha _{U}$is uniformly continuous, in the sense that, for every $\varepsilon > 0$, there exists $W \in \mathscr {V}_0$such that, whenever A and B are W-close, we have either $\alpha _U(A) = \alpha _U(B) = +\infty$, or $|\alpha _U(A) - \alpha _U(B)| < \varepsilon$, where A and B are said to be W-close, if $A \subset B + W$and $B \subset A + W$.

Proof

1. It is clear that $\alpha _{U}(A\cup B)\le \max (\alpha _U(A), \alpha _{U}(B))$. For the converse, assume that $\alpha _{U}(A \cup B) < \max (\alpha _{U}(A), \alpha _U (B))$ and choose a scalar r so that $\alpha _{U}(A\cup B)<r<\max (\alpha _{U}(A),\alpha _U(B))$. Then there is a finite covering of both A and B by rU-small sets $C_{1}, \ldots , C_n$. This contradicts $r<\max (\alpha _{U}(A),\alpha _U(B))$.

2. The homogeneity with respect to A follows easily. The second part derives from the fact that $\mu \in S_{A}(U)$, if and only if, $\frac{\mu }{r} \in S_{A}(rU)$, where $S_{A}(U)$ denotes the set of all $r>0$ such that A is covered by finitely many rU-small sets. Now, if $rU\subset V\subset sU$, then $\frac{1}{s}\alpha _{U}\le \alpha _{V}\le \frac{1}{r}\alpha _{U}$. Therefore, $\alpha _{U}(A)=0$ if and only if $\alpha _{V}(A)=0$.

3. Indeed, if $\alpha _{V}(A)=+\infty$ or $\alpha _{V}(B)=+\infty$, the inequality is trivial. Now, if $r>\alpha _{V}(A)$ and $s>\alpha _{V}(B)$, then there are rV-small sets $(A_{i})_{i=1,\ldots ,n}$ and sV-small ones $(B_{j})_{j=1,\ldots ,m}$, such that $A\subset \cup _{i=1}^{n}A_{i}$ and $B\subset \cup _{J=1}^{m}B_{j}$. Therefore, $A+B\subset \cup _{\underset{j=1,\ldots ,m}{i=1,\ldots ,n}}(A_{i}+B_{j})$. But

$$\begin{aligned} (A_{i}+B_{j})-(A_{i}+B_{j})&=(A_{i}-A_{i})+(B_{j}-B_{j}) \\&\subset rV+sV \\&\subset \max (r,s)(V+V) \\&\subset \max (r,s)U. \end{aligned}$$

Passing to the infimum on r and s, we get

$$\begin{aligned} \alpha _{U}(A+B)\le \max (\alpha _{V}(A),\alpha _{V}(B))\le \alpha _{V}(A)+\alpha _{V}(B). \end{aligned}$$

4. Indeed, one has $S_{A}(V)\subset S_{A}(U)$. Therefore $\inf S_{A}(U)\le \inf S_{A}(V)$.

5. The inequality $\alpha _{U}\ge \alpha _{{\overline{U}}}$ is due to 4. above. Assume, that for some $C\subset E$, $\alpha _{U}(C)>\alpha _{{\overline{U}}}(C)$ and choose r and s so that $\alpha _{U}(C)> s> r > \alpha _{{\overline{U}}}(C)$. Then there exist $r {\overline{U}}$-small sets $C_{1},\ldots ,C_n$ covering C. But $C_{i}-C_{i} \subset r {\overline{U}} \subset sU$. Then the $C_{i}$’s are also sU-small. Hence $\alpha _{U}(C)\le s$, whereby $\alpha _{U}(C) = \alpha _{{\overline{U}}}(C)$.

6. Indeed, by $\left( 1\right)$, $\alpha _{U}(A)\le \alpha _{U}( {\overline{A}})$. Assume that $\alpha _{U}(A)<\alpha _{U}({\overline{A}})$ and choose $r>0$, with $\alpha _{U}(A)< r < \alpha _{U}({\overline{A}})$. Then there exist rU-small subsets $A_{1},\ldots ,A_{m}$ of E, such that $A\subset \cup _{i=1}^{m}A_{i}$. Since $A_{i}-A_{i}\subset rU$, we get ${\overline{A}}_i - A_{i}\subset r{\overline{U}}$. Similarly, ${\overline{A}}_{i} - {\overline{A}}_{i}\subset r{\overline{U}}$. But ${\overline{A}}\subset \cup _{i=1}^{m}\overline{A_{i}}$. Then $\alpha _{U}({\overline{A}})\le r$, a contradiction.

7. Again, if $\alpha _{U}(A)=+\infty$ or $\alpha _U(B) = + \infty$, the inequality is trivial. Now, assume $r>\alpha _{U}(A)$ and $s>\alpha _{U}(B)$. Then there are rU-small sets $(A_{i})_{i=1, \ldots , n}$ and sU-small ones $(B_{j})_{j=1,\ldots ,m}$, such that $A\subset \cup _{i=1}^{n}A_{i}$ and $B\subset \cup _{J=1}^{m}B_{j}$. Then

$$\begin{aligned} (A_{i}+B_{j})-(A_{i}+B_{j})&= (A_{i}-A_{i})+(B_{j}-B_{j}) \\&\subset rU+sU \\&\subset (r^p +s^p)^{\frac{1}{p} } U. \end{aligned}$$

It follows that : $\alpha _U(A+B) \le \root p \of {\alpha _U(A)^p + \alpha _U(B)^p}$.

8. Fix $\varepsilon > 0$, and assume first that $\alpha _U(B) = +\infty$, while $\alpha _U(A) < +\infty$. Choose then $W \in \mathscr {V}_0$, $r > \alpha _U(A)$, and $\varepsilon '> 0$ small enough so that $W + W \subset U$, $\root p \of {r^p + \varepsilon ^{\prime p}} < r +\varepsilon$, and A and B are $\varepsilon 'W$-close. Then A is covered by finitely many rU-small sets $A_1, \ldots , A_n$. For $i =1,\ldots , n$, set $B_i := B \cap (A_i + \varepsilon 'W)$. Then the sets $B_i$, $i = 1 \dots , n$, constitute a covering of B. Moreover, for every $b_1, b_2 \in B_i$, there are $a_1, a_2 \in A_i$ such that $b_1 - a_1 \in \varepsilon 'W$ and $b_2 - a_2 \in \varepsilon 'W$. But then

$$\begin{aligned} b_1 - b_2&= (b_1 - a_1) + (a_1 - a_2) + (a_2 - b_2) \\&\in \varepsilon 'W + \varepsilon 'W + rU \\&\in \varepsilon 'U + rU \\&\subset \root p \of {\varepsilon ^{\prime p }+ r^p}U \\&\subset (r + \varepsilon )U. \end{aligned}$$

It follows that $B_i$ is $(r + \varepsilon )U$-small. Hence $\alpha _U(B) \le r +\varepsilon$, contradicting our assertion on B. Therefore $\alpha _U(B)$ and $\alpha _U(A)$ are simultaneously finite or infinite. Now, by the foregoing proof, whenever $\alpha _U(A) < r$, we get $\alpha _U(B)\le r + \varepsilon$. Therefore $\alpha _U(B) \le \alpha _U(A) + \varepsilon$. Similarly, $\alpha _U(A) \le \alpha _U(B) + \varepsilon$. Thus $|\alpha _U(A) -\alpha _U(B)| \le \varepsilon$ and $\alpha _U$ is uniformly continuous. $\square$

Note that in the context of non locally convex spaces, there is no hope for non compactness measures to be invariant under the transition to the convex hull, at least for classical concrete non compactness measures. For example if for $0< p < 1$ one defines the non compactness measure of a subset A of $\ell ^p$ as in (1), the diameter $\delta (A)$ being relative to the distance defined by the p-norm of $\ell ^{p}$, then $\alpha (B_p) \le \root p \of {2}$, while $\alpha (co(B_{p}))=+\infty$, where $B_p$ is the closed unit ball of $\ell ^{p}$. For $co(B_{p})$ is the whole space $\ell ^{p}$.

Our following main result establishes the analogous of such a property in the general setting of locally p-convex spaces. As a first step, we show the property for the non compactness measure $\alpha _{U}$, when U is taken to be p-convex.

Theorem 1

If $U\in \mathscr {V}_{0}$is p-convex for some $0<p\le 1$, then $\alpha _U(co_{s}(A))=\alpha _{U}(A)$for every $A\subset E$and every $0<s\le p$.

Proof

Let $A\subset E$ and s be such that $0<s\le p$. Since $A \subset co_s(A)$, we clearly have $\alpha _U(co_s(A)) \ge \alpha _U(A)$. Assume $\alpha _U(A) < \alpha _U(co_s(A))$ and choose $r > 0$, so that $\alpha _U(co_s(A))>r>\alpha _U(A)$. Then there exist rU-small sets $A_1,\dots ,A_n$, $n\ge 1$, such that $A\subset \cup _{i=1}^{n}A_{i}$. Then, for each i, $A_i \subset A_i + rU \subset co_s(A_i) + rU$. Since $co_s(A_i) + rU$ is s-convex, it follows that $co_s(A_i) \subset co_s(A_i) + rU$. Hence $co_s(A_i) - co_s(A_i) \subset r U$. We may then (and we will do) assume each $A_i$s-convex. Now, choose from each $A_i$ some $a_i$. Then $A_i \subset a_i + rU$ and, since U is a neighborhood of 0, there exists $M > 0$, such that $a_i \in MU$ for every $i = 1, \ldots , n$. Therefore

$$\cup _{i=1}^{n}A_i \subset MU + rU \subset \root s \of {M^s + r^s}U.$$

If x belongs to $co_s(A)$, then there are $n \in {\mathbb {N}}$ and, for every $1 \le i \le n$, $\mu _i >0$ and $x_i \in A_i$, such that $\sum _i\mu _i^s = 1$ and $x = \sum _{i=1}^{n}\mu _i x_i$. For arbitrary $\delta >0$, since the set $P :=\{(\lambda _1, \dots , \lambda _n) \in {\mathbb {R}}^n, \lambda _i > 0 \text { and } \sum _i \lambda _i^s = 1\}$ is precompact, there exist $m \in {\mathbb {N}}$ and, for every $j = 1, \ldots , m$, $\mu ^{j} :=(\mu _1^j, \ldots , \mu _n^j) \in P$, such that $P\subset \cup _{j=1}^m(\mu ^j + \varDelta (\delta ))$, with $\varDelta (\delta ):=\{(\lambda _1, \ldots , \lambda _n) \in {\mathbb {R}}^n : \max _{i=1}^{n} |\lambda _i| < \delta \}$. Set, for every $j \in \{1, \ldots , m\}$, $S^j := \mu _1^j A_1 + \mu _2^j A_2 + \ldots + \mu _n^j A_n$. Then clearly $S^{j}-S^{j}\subset rU$, and there exists $j\in \{1, \ldots , m\}$, such that $\max _{i=1}^{n} |\mu _i - \mu _i^j| < \delta$. Therefore, for $y := \sum _{i=1}^n \mu _i^j x_i \in S^j$, we have

$$\begin{aligned} x - y&= \sum _{i=1}^{n}\mu _ix_i-\sum _{i=1}^{n}\mu _i^{j}x_i \\&= \sum _{i=1}^{n}(\mu _i-\mu _i^{j})x_i \\&\in \delta \root s \of {n(M^s + r^s)}U. \end{aligned}$$

Choosing $\delta$ small enough, so that $\delta \root s \of {n(M^{s} + r^{s})} \le \varepsilon$, we conclude that $co_s(A)$ is covered by the sets $S^j + \varepsilon U$. But for every $a, b \in S^j$ and $y, z \in U$, we have $(a + \varepsilon y) -(b + \varepsilon z) = (a - b) + \varepsilon (x - y)\in rU +\varepsilon \root s \of {2} U \subset r \root s \of {1 + 2\frac{\varepsilon ^s}{r^s}}U$. Since $\varepsilon$ was arbitrary, $S^j + \varepsilon U$ is $r{\overline{U}}$-small. But $\alpha _U = \alpha _{ {\overline{U}}}$, then $\alpha _U(co_s(A)) \le r$. This contradicts our assumption on r. $\square$

3 Basic and sufficient collections of zero neighborhoods and relative non compactness measures

In order to define a new non compactness measure in E, we introduce the notions of basic and sufficient collections of zero neighborhoods in a topological vector space. To do this, let us introduce an equivalence relation on $\mathscr {V}_{0}$ by saying that U is related to V, written $U \mathscr {R} V$, if and only if there exist $r, s > 0$ such that $r U \subset V \subset s U$.

Definition 1

We say that $\mathscr {C}\subset \mathscr {V}_{0}$ is a basic collection of zero neighborhoods (BCZN in short), if it contains at most one representative member from each equivalence class with respect to $\mathscr {R}$. It will be said to be sufficient (SCZN in short), if it is basic and, for every $V \in \mathscr {V}_{0}$, there exists some $U \in \mathscr {C}$ and some $r > 0$ such that $r U\subset V$.

In the normed case, if f is a continuous functional on E, $U := \{x \in E : |f(x)| < 1\}$, and V is the open unit ball of E, then $\{U\}$ is basic but not sufficient, but $\{V\}$ is sufficient.

If $(E, \tau )$ is a locally convex space, whose topology is given by an upward directed family ${\mathbb {P}}$ of semi-norms, so that no two of them are equivalent, the collection $(B_{P})_{P \in {\mathbb {P}}}$ is a SCZN, where $B_{P}$ is the open unit ball of P.

Further, if $\mathscr {W}$ is a fundamental system of zero neighborhoods in a topological vector space, then there exists an SCZN $\mathscr {C}$ consisting of $\mathscr {W}$ members.

Following an idea of [14], a subset A of E is called uniformly bounded with respect to a sufficient collection $\mathscr {C}$ of zero neighborhoods, if there exists $r > 0$ such that $A \subset rV$, for every $V \in \mathscr {C}$. In the locally convex space $C_c(X) := C_c(X, {\mathbb {K}})$, the set $B_\infty := \{f \in C(X) : \Vert f\Vert _\infty \le 1 \}$ is uniformly bounded with respect to the SCZN $\{B_K, K \in \mathscr {K}\}$, where $B_K$ is the (closed or) open unit ball of the semi-norm $P_K$.

Lemma 1

If A is an arbitrary bounded set in E, then there exists a SCZN $\mathscr {C}$with respect to which A is uniformly bounded.

Proof

Fix a SCZN $\mathscr {C}_0$. Since A is bounded, for every $V \in \mathscr {C}_0$, there exists $r_V > 0$ such that $A \subset r_V V$. Now, take $\mathscr {C }:= \{ r_V V, V \in {\mathscr {C}}_0\}$. Then $\mathscr {C}$ fulfills the required condition with $r = 1$. $\square$

We are now in a position to introduce a non compactness measure in the topological vector space E.

Definition 2

Let $\mathscr {C}$ be a SCZN in E. For every $A\subset E$, we define the non compactness measure of A with respect to $\mathscr {C}$ as:

$$\begin{aligned} \alpha _{\mathscr {C}}(A)=\sup _{U\in \mathscr {C}}\alpha _{U}(A). \end{aligned}$$

The semi-additivity of $\alpha _{\mathscr {C}}$, i.e. $\alpha _{\mathscr {C}}(A \cup B) = \max (\alpha _{\mathscr {C}}(A), \alpha _{{\mathscr {C}}}(B))$ (hence, $\alpha _{{\mathscr {C}}}$ is monotone in the sense that $A\subset B$ implies $\alpha _{\mathscr {C}}(A) \le \alpha _{\mathscr {C}}(B)$) is readily derived from Proposition 1 (1). Let us show that $\alpha _{\mathscr {C}}$ is regular, i.e. $\alpha _{\mathscr {C}}(A) = 0$ if and only if A is a precompact subset of E. If $\alpha _{\mathscr {C}}(A)=0$ and $U \in \mathscr {V}_{0}$ is arbitrary, since $\mathscr {C}$ is a SCZN, there exists $V \in \mathscr {C}$ and $r > 0$ such that $V \subset rU$. Therefore $\alpha _U(A) \le r\alpha _V(A) = 0$. Hence A is covered by a finite number of U-small sets. Since U is arbitrary, A is precompact. Conversely, if A is precompact, then $\alpha _U(A) = 0$, for every $U \in {\mathscr {V}}_0$. In particular $\alpha _U(A) = 0$ for every $U \in {\mathscr {C}}$. Hence $\alpha _{\mathscr {C}}(A) = 0$.

Some properties are shared by all the $\alpha _{\mathscr {C}}$’s. Indeed, if $\mathscr {C}$ and $\mathscr {C\,}'$ are two SCZN, then, for every subset A of E, we have:

$$\begin{aligned} \alpha _{\mathscr {C}}(A) = 0 \text { if and only if } \alpha _\mathscr {C\,}' (A)=0. \end{aligned}$$

This derives from the definition of a SCZN and Proposition 1 (2). However, there may exist a bounded set $A\subset E$ such that $\alpha _{\mathscr {C}}(A) < \alpha _\mathscr {C}'(A) = +\infty ,$ as the following example shows.

Example 1

Take in $E=C_{c}({\mathbb {R}})$, the space $C({\mathbb {R}})$ of all continuous functions from ${\mathbb {R}}$ into ${\mathbb {R}}$ endowed with the topology of uniform convergence on compact sets of ${\mathbb {R}}$, the collections $\mathscr {C}:=(B_n)_n$ and $\mathscr {C}' := (\frac{1}{n}B_n)_n$, where $B_n := \{f\in C({\mathbb {R}}), \Vert f\Vert _n := \sup _{|x| \le n} |f(x)| \le 1\}$, $n \in {\mathbb {N}}$. Then, for $A:=\{f\in C(\mathbb {R}),|f(x)|\le 1,$$x\in \mathbb {R}\}$, we have $\alpha _{\mathscr {C}}(A)=2$, while $\alpha _{\mathscr {C}'}(A) = +\infty$. Indeed, if $\alpha _{B_n}(A) < 2$ holds for some n, then also $\alpha _{B_n}(A') < 2$, where $A' :=\{f_k : k \in \mathbb {N}\}$ and $f_k$ being defined on $\mathbb {R}$ by:

$$ f_{k} (x) = \left\{ {\begin{array}{*{20}l} { - 1} \hfill & : \hfill & {x \le \frac{1}{{k + 1}}} \hfill \\ {2k(k + 1)x - (2k + 1)} \hfill & : \hfill & {\frac{1}{{k + 1}} < x \le \frac{1}{k}} \hfill \\ 1 \hfill & : \hfill & {{\text{otherwise}}{\text{.}}} \hfill \\ \end{array} } \right. $$

Choose a real number r so that $\alpha _{B_n} (A')< r < 2$. Then there exist $rB_n$-small subsets $S_1, \ldots , S_m$ of E covering $A'$. At least one of these sets, say $S_{i}$, contains infinitely many elements of $A'$. But, whenever $f_h, f_k \in S_i$ are such that $h < k$, one has

$$\begin{aligned} r \ge \left| f_k\left( \frac{1}{h+1}\right) - f_h\left( \frac{1}{h+1}\right) \right| =2. \end{aligned}$$

This contradicts $r<2$. Hence $\alpha _{B_n} (A) \ge 2$ holds for every n. Since $A \subset B_n$ and $B_n$ is convex, we get $\alpha _{B_n}(A)=2$ for every n. This shows that $\alpha _{{\mathscr {C}}} (A) = 2$. But $\alpha _{\frac{1}{n} B_n} = n \alpha _{B_n} (A)$. We conclude that $\alpha _{{\mathscr {C}}'} (A) = + \infty$.

Similar properties as in Proposition 1 are obtained directly for the non compactness measure $\alpha _{\mathscr {C}}\!\!:$

Proposition 2

Let ${\mathscr {C}}$be a SCZN in E consisting of shrinkable sets and $A, B \subset E$. Then, the following hold:

• 1. $\alpha _{\mathscr {C}}(A) = \alpha _{\mathscr {C}} ({\overline{A}})$.

• 2. $\alpha _{\mathscr {C}} (s A) =\left| s\right| {\alpha }_{\mathscr {C}}\left( A\right)$, for every scalar $s\ne 0$.

  Moreover, if $\mathscr {C}$ consists of p-convex sets for some $0<p\le 1$, then

• 3. $\alpha _{\mathscr {C}}(A+B) \le \root p \of {\alpha _{\mathscr {C}}(A)^{p} + \alpha _{\mathscr {C}} ( B)^p}$.

Some properties of subsets of E can be characterized through the non compactness measure.

Proposition 3

1. For every SCZN $\mathscr {C}$in E, a subset A of E is bounded if and only if $\alpha _{U}(A) < + \infty$for every $U \in {\mathscr {C}}$.

2. If a SCZN $\mathscr {C}$in E consists of p-convex sets for some $0 < p \le 1$and if A is $\mathscr {C}$-uniformly bounded, then $\alpha _{\mathscr {C}}(A) < + \infty$.

3. If E is locally p-convex, then A is bounded if and only if there exists a SCZN $\mathscr {C}$with $\alpha _\mathscr {C }(A) < + \infty$.

Proof

1. If A is bounded, then its balanced hull B and also $C := B - B$ are bounded. Therefore, for every $U \in \mathscr {C}$ there is $r > 0$ such that $C \subset rU$. But $A \subset B$ and $B - B \subset rU$. Therefore $\alpha _U(A) \le r < + \infty$. Conversely, for every $U \in {\mathscr {V}}_0$, there exists $V \in \mathscr {C}$ and $\rho > 0$ satisfying $V + V \subset \rho U$. But $\alpha _V(A) < + \infty$. Then, there exists $r > 0$ and rV-small subsets $A_1,\dots , A_m$ of E such that $A \subset \cup _{i = 1}^m A_i$. Fix $a_i \in A_i$ arbitrarily. Then $A \subset \{a_i, i = 1, \ldots m\} + r V$. But $\{a_i, i = 1, \ldots m\}$ is bounded. Then there exists $s > r$ so that $\{a_i, i = 1, \ldots n\} \subset s V$. It follows that $A \subset s (V+ V) \subset s\rho U$. Hence A is bounded in E.

2. If A is $\mathscr {C}$-uniformly bounded, there exists $r > 0$ such that $A \subset r U$, $U \in \mathscr {C}$. Since $rU - rU \subset r \root p \of { 2}U$ for every $U\in \mathscr {C}$, $\alpha _{U}(A)\le r \root p \of {2}$ for every $U \in \mathscr {C}$. Hence $\alpha _{\mathscr {C}}(A) \le r \root p \of {2} < + \infty$.

3. Assume that E is locally p-convex and that A is bounded. Then by Lemma  1, there exists a SCZN $\mathscr {C}$ consisting of p-convex sets such that A is $\mathscr {C}$-uniformly bounded. Hence by (2), $\alpha _{\mathscr {C}}(A) < + \infty$. For the converse, if $\alpha _{\mathscr {C} }(A) < r$, for some $r > 0$, then by (1), A is bounded. $\square$

We formulate our next main theorem stating the invariance of $\alpha _{\mathscr {C}}$ under the closed s-convex hull in a Hausdorff locally p-convex space E, $0<s\le p$. This is a consequence of Theorem 1 and Proposition 1 (6). This shows that in the case $s=p=1$, $\alpha _{\mathscr {C}}$ is a non compactness measure in the sense of Sadovskii [19] and Park [15]. This is

Theorem 2

Let ${\mathscr{C}} \subset {\mathscr{V}}_0$be a SCZN consisting of p-convex subsets in a Hausdorff topological vector space E. Then, $\alpha _{\mathscr {C}}({\overline{co}}_s A) = \alpha _{\mathscr {C}}(A)$for every $A \subset E$and every $0 < s \le p$.

We conclude this section by the following proposition which is an extension of the well-known Cantor type intersection property of non compactness measures in the setting of metric spaces. The proof in this particular setting was given by Kuratowski [12].

Proposition 4

Let E be a Hausdorff topological vector space, $\mathscr {C}$a SCZN in E consisting of shrinkable sets, and $(A_{n})_{n\ge 0}$a decreasing sequence of non empty sets in E such that $A_{n_{0}}$is complete for some $n_{0}$, and $\lim _{n\rightarrow \infty }\alpha _U (A_n) = 0$for every $U\in \mathscr {C}$. Then $\cap _{n=0}^{\infty } \overline{A_{n}}$is non empty and compact.

Proof

With no loss of generality, we may assume that $A_{0}$ is complete. Let $\varepsilon > 0$ and $U \in {\mathscr {C}}$. Since $\alpha _U(A_n) \rightarrow 0$, there exists $N \in \mathbb {N}$ such that $\alpha _U(A_n) < \varepsilon$ for every $n \ge N$. Choose a sequence $(x_n) \subset E$ such that $x_n \in A_n$ for each n. Since $\alpha _U (F) = 0$ for every finite $F\subset E$, it follows, for every $n \ge N$:

$$\begin{aligned} \alpha _U\left( ( x_n) \right)&=\alpha _U(\left\{ x_n : n \ge N\right\} ) \\&\le \alpha _{U}(A_{N})<\varepsilon . \end{aligned}$$

Since $\varepsilon >0$ and $U\in \mathscr {C}$ were arbitrary, $\alpha _{\mathscr {C}}\left( \left\{ x_{n},n\ge 0\right\} \right) =0$, that is, $\left\{ x_{n},n\ge 0\right\}$ is precompact, then relatively compact. Therefore $(x_{n})_{n}$ admits a cluster point $x\in E$. But $A_{k}$ contains all the $x_{n}$’s but a finite number. Then x belongs to the closure of $A_{k}$ for each k. Therefore $\cap _{n=0}^{\infty } \overline{A_{n}}$ is non empty. Now, the compactness of $\cap _{n=0}^{\infty }\overline{A_{n}}$ follows easily from the monotonicity, the closure invariance of $\alpha _{\mathscr {C}}$, and the fact that $\alpha _{U}\left( A_{n}\right) \rightarrow 0$ for every $U\in \mathscr {C}$. $\square$

4 Application: Schauder, Darbo and Sadovskii-type fixed point results in locally p-convex spaces

In this section, we will use our results on the introduced non compactness measure to establish the versions of the well known Darbo and Sadovskii fixed point theorems for the so-called Yanyan continuous maps in the setting of locally p-convex spaces. Similarly as with the normed case, the key idea is to come back to a compact set and to apply Schauder fixed point theorem. So, we need for this to establish a p-convex version of Schauder fixed point theorem in our setting of locally p-convex spaces (Theorem 4). This will present an extension of the following recent version established in the p-normed spaces setting.

Throughout this section, C will be a non empty subset of a Hausdorff topological vector space E.

Theorem 3

([21, Theorem 2.13]) Let $(E, \Vert \ \Vert _p)$be a complete p-normed space and C be a compact s-convex subset of E, where $0 < s \le p$. Then, every continuous map $T : C \rightarrow C$has a fixed point.

Let us say that, in a Hausdorff locally p-convex space $(E, \tau )$, a family $\mathbb {P}$ of continuous p-semi-norms on E is sufficient, if the collection $\{U_P, P \in \mathbb {P}\}$ is a SCZN, where $U_P$ denotes the open unit ball of P.

A map $T: C \rightarrow C$ will be said to be lipschitzian, if there exists a sufficient family $\mathbb {P}$ of p-semi-norms on E such that:

$$\begin{aligned} \forall P \in \mathbb {P},\quad \exists L_{P}>0,\quad P(Tx-Ty)\le L_{P}P(x-y), \quad x, y \in C. \end{aligned}$$

If $L_P <1$ for every $P \in \mathbb {P}$, then T is called a contraction on C. If there exists a sufficient family $\mathbb P$ of p-semi-norms on E, such that, for every $P \in \mathbb P$, T is continuous from $(C, \tau _P)$ into $(C, \tau _P)$, where $\tau _P$ is the topology induced on C by the single p-semi-norm P, we say that T is Yanyan-continuous. It is clear that a contraction is lipschitzian, a lipschitzian map is Yanyan-continuous, and that a Yanyan-continuous map is continuous. Moreover, in a normed space, every continuous map is Yanyan-continuous, but need not be lipschitzian. Now, if we consider again the topological linear space $C_c(\mathbb R)$ of Example 1, then the mapping T defined for every $f \in C(\mathbb R)$ by $T(f) := f^2$ is Yanyan-continuous, but not lipschitzian. Indeed, the family $(\Vert \ \Vert _n)_n$ of semi-norms defined in Example 1 is sufficient and satisfies

$$\begin{aligned} \Vert T(f) - T(g))\Vert _n \le \Vert f + g\Vert _n \Vert f - g\Vert _n, \quad f, g \in C(\mathbb R). \end{aligned}$$

Restricting ourselves to the open ball $B_n(g, r) :=\{f \in C(\mathbb R) : \Vert f - g\Vert _n < r\}$, for any $r > 0$, we get:

$$\begin{aligned} \Vert T(f) - T(g)\Vert _n \le (\Vert g\Vert _n +r) \Vert f - g\Vert _n, \end{aligned}$$

whence the Yanyan-continuity of T at g and then on $C(\mathbb R)$ since g is arbitrary. Now, if T were lipschitzian with respect to some sufficient family $\mathbb {P}$ of semi-norms defining the topology of $C(\mathbb R)$, then for every $P \in \mathbb {P}$, there would exist $M_P > 0$, such that

$$\begin{aligned} P(T(f) - T(g)) \le M_P P(f - g), \quad f, g \in C(\mathbb R). \end{aligned}$$

But for every $n \ge 1$, there exist $P_n \in \mathbb {P}$ and a positive number R(n) such that

$$\begin{aligned} \Vert f\Vert _n \le R(n)P_n(f), \quad f \in C(\mathbb {R}). \end{aligned}$$

Choose $g , f \in C(\mathbb R)$, $n \ge 1$ with $\Vert g\Vert _n \ne 0$, and $f = \lambda g$, $\lambda \ne 1$. Then, we have

$$\begin{aligned} |\lambda ^2 - 1| \Vert g^2\Vert _n = \Vert T(f) - T(g)\Vert _n \le R(n) P_n(T(f) -T(g)) \le R(n) M_{P_n} |\lambda - 1| P_n(g). \end{aligned}$$

As $\Vert g^2\Vert _n = \Vert g\Vert _n^2 \ne 0$, this means that

$$\begin{aligned} |\lambda + 1| \le \frac{R(n) M_{P_n}}{\Vert g\Vert _n^2} P_n(g). \end{aligned}$$

Since $\lambda \ne 1$ is arbitrary, this is impossible.

Notice at this point that, for every p-semi-norm P on a topological vector space E, the set $\ker P := \{ x \in E : P(x) = 0\}$ is a (closed if P is continuous) vector subspace of E.

Our next main result is a Schauder-type fixed point theorem for Yanyan-continuous maps in locally p-convex spaces.

Theorem 4

Let $(E, \tau )$be a Hausdorff locally p-convex space and $C \subset E$be a compact and s-convex subset of E, with $0 < s \le p$. Then, every Yanyan-continuous map $T : C \rightarrow C$has a fixed point.

For the proof, we need the following lemma.

Lemma 2

Let P be a continuous p-semi-norm on a Hausdorff topological vector space $(E, \tau )$, C a compact s-convex subset of E with $0 < s \le p$, and $T : C \rightarrow C$a mapping. If T is P-continuous (i.e., $\forall \varepsilon > 0$, $\exists \eta > 0$, such that $P(Tx - Ty) \le \varepsilon$for every $x, y \in C$with $P(x - y) < \eta$), then there exists $x_P \in C$, such that $x_P - T x_P \in \ker P$.

Proof

It is easily shown that the quotient space $E/ \ker P$, endowed with the p-norm $\Vert \pi (x) \Vert _p := P(x)$, is a p-normed space, where $\pi : E \rightarrow E/ \ker P$ stands for the canonical surjection. Thinking of $\pi (C)$ as a subset of the completion $\widehat{E/ \ker P}$ of $E/ \ker P$, we may assume that $E / \ker P$ is a complete p-normed space. For every $x \in C$, put ${\overline{T}}(\pi (x)) = \pi (Tx)$. It follows from the P-continuity of T that ${\overline{T}}$ is a continuous self map on $\pi (C)$. By continuity of P, $\pi$ is continuous. Therefore $\pi (C)$ is compact. But $\pi (C)$ is also s-convex. Hence, by Theorem 3, there exists $x_P \in C$ such that ${\overline{T}} (\pi (x_P)) = \pi (x_P)$, that is, $x_P - Tx_P \in \ker P$ as desired. $\square$

Proof of Theorem 4

Let $\mathbb {P}$ be a sufficient family of p-semi-norms on E with respect to which T is Yanyan-continuous on C. By Lemma 2, there exists a family $(x_{P})_{P\in \mathbb {P}}\subset C$ with $x_{P}-Tx_{P} \in \ker P$ for every $P \in \mathbb {P}$. Denote by $A_{P}$, for $P \in \mathbb {P}$, the closure of the set

$$\begin{aligned} C_{P}:=\{x_{Q}, Q\in \mathbb {P}, \text { and }P \le cQ \text { for some } c > 0\}. \end{aligned}$$

Since $\mathbb {P}$ is sufficient, the collection $(A_{P})_{P\in \mathbb {P}}$ satisfies the finite intersection property. By compactness of C, the set $A:=\cap \{A_{P},P\in \mathbb {P}\}$ contains at least an element $x\in C$. We claim that $Tx=x$. Otherwise, there would exist $P\in \mathbb {P}$ such that $P(Tx-x)>0$. Choose $\eta$ so that, $0< \eta < \frac{P(Tx - x)}{2}$ and whenever $x, y \in C$ with $P(x - y) \le \eta$, we have $P(Tx - Ty) < \frac{P(Tx - x)}{2}$. Now, choose $Q \in \mathbb {P}$, so that $P \le cQ$, for some $c>0$ and $P(x - x_Q) < \eta$. Then

$$\begin{aligned} P(Tx-x)&\le P(Tx - Tx_Q) + P(Tx_Q - x_Q)+P(x_Q-x) \\&< \frac{P(x - Tx)}{2} + c Q(Tx_Q - x_Q) + \frac{P(x - Tx)}{2} \\&= P(Tx-x). \end{aligned}$$

This is absurd. $\square$

Darbo [4] and Sadovskii [18] introduced the notions of set-contractions and condensing maps. They established their famous fixed point theorems in the setting of Banach spaces. Following them, if E is a Hausdorff locally p-convex space, we will say that a map $T : C \rightarrow C$ is a set-contraction (resp. condensing), if there is some SCZN $\mathscr {C}$ in E consisting of p-convex sets, such that:

$$\begin{aligned} \forall U \in {\mathscr {C}}, \exists 0< k_U < 1,\quad \alpha _U(T(A)) \le k_U \alpha _U(A), \quad A \subset C, \end{aligned}$$

(resp.          $\forall U \in {\mathscr {C}},\quad \alpha _{U}(T(A)) < \alpha _{U}(A), \quad A \subset C \quad \text { with }\quad \alpha _U (A) > 0$).

It is clear that a contraction on C is a set-contraction and a set-contraction on C is condensing.

Now, our next result is a Sadovskii-type fixed point theorem in the setting of locally p-convex spaces.

Theorem 5

Let $C\subset E$be a complete s-convex subset of a Hausdorff locally p-convex space E, with $0 < s \le p$. If $T:C\rightarrow C$is Yanyan-continuous and condensing, then T has a fixed point.

Proof

Let $\mathscr {C}$ be a sufficient collection of p-convex zero neighborhoods in E with respect to which T is condensing and fix $U\in \mathscr {C}$. Choose some $x_{0}\in C$ and let $\mathscr {F}$ be the family of all closed s-convex subsets A of C with $x_{0}\in A$ and $T(A)\subset A$. Note that $\mathscr {F}$ is not empty since $C\in \mathscr {F}$. Let $A_{0}=\cap _{A \in {\mathscr {F}}} A$. Then $A_{0}$ is a non empty closed s-convex subset of C, such that $T(A_{0})\subset A_{0}$. We shall show that $A_{0}$ is compact. Let $A_1 = {\overline{co}}_s (T(A_0) \cup \{x_0\})$. Since $T(A_{0})\subset A_{0}$ and $A_{0}$ is closed and s-convex, $A_{1}\subset A_{0}$. Hence, $T(A_{1})\subset T(A_{0})\subset A_{1}$. It follows that $A_{1}\in \mathscr {F}$ and therefore $A_{1}=A_{0}$. By Proposition 1 and Theorem 1, we get $\alpha _U (T(A_0)) = \alpha _U (A_0)$. Our assumption on T shows that $\alpha _U(A_0) = 0$. Since U was arbitrary, $A_0$ is compact as desired. Now, the conclusion follows from Theorem 4 applied to $T : A_0 \rightarrow A_0$. $\square$

Corollary 1

(Darbo-type fixed point theorem) Let $C\subset E$be a complete s-convex subset of a Hausdorff locally p-convex space E, with $0<s\le p$. If $T:C \rightarrow C$is Yanyan continuous and a set-contraction, then T has a fixed point.

Notice that, if T is a contraction on C then it is both Yanyan continuous and a set-contraction. In this case, the preceding corollary can be sharpened, with a standard proof as in Theorem 2.2 of [3], in the following way:

Theorem 6

(The contraction principle) Let C be a sequentially complete subset of a Hausdorff locally p-convex space E. If $T:C\rightarrow C$is a contraction on C , then T has a unique fixed point $x^{*}$in C,  and the iterative sequence $\left( T^n x \right)$converges to $x^{*}$for every $x \in C.$

The following two recent results are now particular cases of Theorem 5 and Corollary 1. The cases $s=p=1$ are the standard Darbo and Sadovskii fixed point theorems. Note that, for a p-normed space E (which is a metric space), $\alpha$ stands for the standard Kuratowski non compactness measure in E.

Corollary 2

(Sadovskii-type, [20, Theorem 4.3]) Let $(E,\Vert$·$\ \Vert _p)$be a complete p-normed space and C be a bounded, closed and s-convex subset of E, where $0 < s \le p$. Then, every continuous and $\alpha$-condensing map $T : C \rightarrow C$has a fixed point.

Proof

Take in Theorem 5$\mathscr {C=}\left\{ B_{p}\left( 0,1\right) \right\}$, where $B_p(0,1)$ stands for the closed unit ball of E, and note that it can be easily shown that $\alpha \left( A\right) =\left( \alpha _{\mathscr {C}}(A) \right) ^{p}$ for every $A \subset E$, and that T satisfies the conditions of Theorem 5. $\square$

Corollary 3

(Darbo-type, [20, Theorem 4.1]) Let $(E,\Vert \cdot \ \Vert _{p})$be a complete p-normed space and C be a bounded, closed and s-convex subset of E, where $0 < s \le p$. Then, every map $T : C \rightarrow C$which is continuous and a set-contraction has a fixed point.

We conclude this section by the following open question to which it is alluded in the introduction, on the p-convex versions of the well-known Schauder, Darbo and Sadovskii fixed point theorems in the setting of locally p-convex spaces.

Question : Can the Yanyan-continuity condition in Theorems 4, 5 and Corollary 1 be weakened to continuity ?

5 Noncompactness measure in C(X, E)

In this section, X will be a Hausdorff completely regular topological space, $\mathscr {B}$ will be the von Neumann bornology of E, i.e., the family of all bounded subsets of E, and ${\mathscr {K}}$ an upward directed collection of compact subsets of X covering the whole X. The smallest such a family is ${\mathscr {S}} := \{F \subset X \ \text {finite}\}$ and the largest is $\mathscr {G} := \{K \subset X\ \text {compact}\}$. The linear space of all continuous functions from X into E will be denoted by C(X, E). It will be endowed with the topology $\tau _{\mathscr {K}}$ of uniform convergence on the elements of ${\mathscr {K}}$. A fundamental system of zero neighborhoods for $\tau _{{\mathscr {K}}}$ is given by the sets:

$$\begin{aligned} N(K, U) := \{f \in C(X, E) : f(K) \subset U\}, \quad K \in \mathscr {K}, \quad U \in {\mathscr {N}}_0. \end{aligned}$$

The topological vector space obtained by endowing C(X, E) with the topology $\tau _{{\mathscr {K}}}$ will be denoted by $C_{\mathscr {K}}(X, E)$. In case ${\mathscr {K}} = {\mathscr {S}}$ (resp. ${\mathscr {K}} = {\mathscr {G}}$), we will rather write $C_s(X, E)$ (resp. $C_c(X, E)$).

Recall that the topology of X is generated by the uniformity induced on X by C(X, [0, 1]), the space of all continuous functions from X into [0, 1]. In other terms, the topology of X is nothing but the initial topology associated to C(X, [0, 1]).

Now, let us associate to any collection ${\mathscr {C}}$ of zero neighborhoods in E the collection

$$\begin{aligned} \widehat{{\mathscr {C}}} := \{N(K, U),\ K \in {\mathscr {K}}, \ U \in {\mathscr {C}}\} \end{aligned}$$

of zero neighborhoods in $C_{\mathscr {K}}(X, E)$.

Lemma 3

If $\mathscr {C}$is a basic (resp. sufficient) collection of zero neighborhoods in E, then so is also the collection $\widehat{\mathscr {C}}$in $C_{\mathscr {K}}(X, E)$. Moreover, if U is a closed shrinkable zero neighborhood, then N(K, U) is also closed and shrinkable in $C_{\mathscr {K}}(X, E)$.

Proof

Suppose that, for some $K, K' \in {\mathscr {K}}$ and some zero neighborhoods U and $U'$, there exist $r, s > 0$ such that $r N(K, U) \subset N(K', U') \subset s N(K, U)$. Then $K = K'$ and $U \mathscr {R }U'$, therefore, since $\mathscr {C}$ is basic $U = U'$. Indeed, if some $x \in K$ exists with $x \notin K'$, then there is a continuous function g from X into [0, 1] such that $g(x) = 1$ and g is identically 0 on $K'$. Let $a \in E$, with $a \notin sU$. Then the function $f := g \otimes a$ defined by $f(y) = g(y) a$ belongs to $N(K', U')$, but $f(x) = a \notin sU$. Hence $K = K'$. Now, assume that there exists $a \in U$, with $ra \notin U'$ and denote by g the constant function $g(y) = a$. Then $g \in N(K, U)$, but $rg(K') = ra \notin U'$. This contradicts our assumption. Similarly, we show that $U' \subset sU$. Hence $U = U'$, since $\mathscr {C}$ is basic.

Now, assume ${\mathscr {C}}$ is sufficient. Then, by the foregoing proof, $\widehat{{\mathscr {C}}}$ is basic. Moreover, let W be a zero neighborhood in $C_{\mathscr {K}}(X, E)$. Then there exist a compact set $K \in {\mathscr {K}}$ and a zero neighborhood V in E such that $N(K, V) \subset W$. But there exist $r > 0$ and $U \in {\mathscr {C}}$, such that $rU \subset V$. This leads to $rN(K, U) \subset W$.

Assume now that $U \in {\mathscr {V}}_0$ is closed and shrinkable, and $K \in {\mathscr {K}}$. Since the evaluation $\delta _x : f \mapsto f(x)$ is continuous from $C_{\mathscr {K}}(X, E)$ into E, $N(K, U) = \cap \{\delta _x^{-1}(U), \quad x \in K\}$ is a closed zero neighborhood in $C_{\mathscr {K}}(X, E)$. Furthermore, for arbitrary $1> r > 0$, we have $r {\overline{U}} = {r U \subset \mathring{U}}$. Therefore, $r \overline{N(K, U)} = r N(K,U) = N(K, rU) \subset N(K,{\mathring{U}}) \subset N(K,{\mathring{U}})$. Hence N(K, U) is shrinkable. $\square$

In the following, we let ${\mathscr {C}}$ denote a SCZN in E. With no loss of generality, since every topological vector space admits a fundamental system of zero neighborhoods consisting of closed shrinkable and balanced sets, we may assume that $\mathscr {C}$ consists also of such sets. Therefore, by Lemma 3, the associated collection $\widehat{{\mathscr {C}}}$ consists also of closed shrinkable and balanced sets.

Let ${\widehat{\alpha }}_\mathscr {C}$ be the non compactness measure in the space $C_{\mathscr {K}}(X, E)$ defined by the SCZN $\widehat{\mathscr {C}}$. This is

$$\begin{aligned} {\widehat{\alpha }}_{\mathscr {C}}(D) = \sup _{\begin{array}{c} K \in \mathscr {K} \\ U \in {\mathscr {C}} \end{array}} {\widehat{\alpha }}_{N(K,U)}(D), \end{aligned}$$

where ${\widehat{\alpha }}_{N(K, U)}$ stands for the ${N(K, U)}\text{-measure of non compactness}$ in the space $C_{\mathscr {K}}(X, E)$.

Next, we address the question whether the equality

$$\begin{aligned} {\widehat{\alpha }}_{{\mathscr {C}}} (D) = \sup _{K \in {\mathscr {K}}} \sup _{x \in K} \alpha _{\mathscr {C}} (D(x)) \end{aligned}$$

holds for a subset D of C(X, E). Here $D(x) := \{f(x) : f \in D\}$.

Definition 3

A subset $D \subset C(X, E)$ is said to be equicontinuous on a subset Y of X, if it is equicontinuous at each point of Y. It is said to be uniformly equicontinuous on Y, if for every $U \in \mathscr {N}_0$, there exist $\mu > 0$ and finitely many functions $g_1,..., g_n \in C(X,[0,1])$ such that, whenever $\max _{i = 1}^n |g_i(x) - g_i(y)| < \mu$, with $x, y \in Y$, we have $f(x) - f(y) \in U$ for every $f \in D$.

Proposition 5

If $D \subset C(X, E)$is equicontinuous on a compact $K \subset X$, then D is uniformly equicontinuous on K.

Proof

Let V be an arbitrary zero neighborhood in E and choose $U\in \mathscr {N}_0$ such that $U + U \subset V$. Since D is equicontinuous at every point of K, for every $t \in K$, there exists an open neighborhood $\varOmega _t$ of t in X such that:

$$\begin{aligned} f(t) - f(s) \in U, \quad f \in D, \quad s \in \varOmega _t. \end{aligned}$$

(6)

By the complete regularity of X, there exists a function $g_t \in C(X, [0,1])$ such that $g_t(t) = 1$, and $g_t$ vanishes identically outside of $\varOmega _t$. By compactness of K, there are $n \in \mathbb N$ and $t_1, t_2, \ldots , t_n \in K$, so that $K \subset \cup _{i=1}^n \{g_{t_i} > \frac{3}{4}\}$. Put $\eta = \frac{1}{4}$ and $g_i = g_{t_i}$. If $x, y \in K$ satisfy $\max (|g_i(x) - g_i(y)|,\ i = 1,\ldots ,n) < \eta$, then there is $i_0 \in \{1, \ldots ,n\}$, such that $g_{i_0}(x) > \frac{3}{4}$. Since $|g_{i_0}(x) - g_{i_0}(y)| < \frac{1}{4}$, it follows $g_{i_0}(y) > \frac{1}{2}$, so that both x and y belong to $\varOmega _{t_{i_0}}$. Therefore, using (6), for every $f \in D$ we get

$$\begin{aligned} f(x) - f(y) = f(x) - f(t_{i_0}) + f(t_{i_0}) - f(y) \in U + U \subset V. \end{aligned}$$

Hence, since x, y were arbitrary in K, D is uniformly equicontinuous on K. $\square$

The following corollary is an immediate consequence of Proposition 5.

Corollary 4

If a subset D of C(X, E) is equicontinuous on X, then it is uniformly equicontinuous on every compact $K \subset X$.

Now, we are in a position to establish an alternative of Ambrosetti theorem in the general setting of topological vector spaces. This is our next main result:

Theorem 7

Let D be a subset of C(X, E). If D is equicontinuous on X, then the following equality holds:

$$\begin{aligned} {\widehat{\alpha }}_{{\mathscr {C}}}(D) = \sup _{K \in {\mathscr {K}}} \sup _{x \in K} \alpha _{{\mathscr {C}}}(D(x)). \end{aligned}$$

In order to prove this result, we first give the following lemma:

Lemma 4

Let K be a compact subset of X, U a closed zero neighborhood in E, and D a subset of C(X, E). If D is equicontinuous on K, then the following equality holds:

$$\begin{aligned} {\widehat{\alpha }}_{N(K, U)}(D) = \sup \{\alpha _U (D(x)), \quad x \in K\}. \end{aligned}$$

Proof

If ${\widehat{\alpha }}_{N(K, U)}(D) = +\infty$, then obviously ${\widehat{\alpha }}_{N(K, U)}(D) \ge \sup \{\alpha _U (D(x)),\ x \in K\}$. Now, assume ${\widehat{\alpha }}_{N(K, U)}(D) < +\infty$ and consider an arbitrary $\eta > {\widehat{\alpha }}_{N(K, U)}(D)$. Then, there exist $\eta N(K,U)$-small subsets $G_1,\ldots , G_m$ of C(X, E) such that $D \subset \cup _{i = 1}^m G_i$. Therefore, for $x \in K$, $D(x) \subset \cup _{i=1}^m G_i(x)$ and $G_i(x) - G_i(x) \subset \eta U$. Hence $\alpha _U(D(x)) \le \eta$. It follows that $\alpha _U(D(x)) \le \widehat{\alpha }_{N(K,U)}(D)$, for every $x \in K$.

For the converse, it is clear that if $\sup \{\alpha _U(D(x)), x \in K\} = +\infty$, then $\widehat{\alpha }_{N(K, U)}(D) = \sup \{\alpha _U((D(x)), x \in K\}$. Assume now that $\sup \{\alpha _U(D(x)), x \in K\} < +\infty$ and consider arbitrary $\eta > \sup \{\alpha _U(D(x))$, $x \in K\}$ and zero neighborhoods $V,W\in \mathscr {N}_0$, such that $V + V \subset W$. By Proposition 5, D is uniformly equicontinuous on K. Therefore there exist $\mu > 0$ and a finite subset F of C(X, [0, 1]), such that:

$$\begin{aligned} (\forall x, y \in K), (\max _{g \in F} |g(x) - g(y)| < \mu ) \Longrightarrow (f(x) - f(y) \in V, f \in D). \end{aligned}$$

(7)

Now, for arbitrary $x \in K$, put $K_x := \{t \in X : |g(x) - g(t)| < \mu , g \in F \}$. This is an open subset of X containing x. Then the collection $(K_x)_{x \in K}$ is an open covering of K. By compactness, there are $n \in \mathbb {N}$ and $x_1, x_2, \ldots , x_n \in K$, such that $K \subset \cup _{i = 1}^n K_i$, with $K_i := K_{x_i}$. By the very definition of $K_i$ and (7), we have

$$\begin{aligned} \left( x \in K_i\right) \Longrightarrow \left( f(x) - f(x_i) \in V, \quad f \in D\right) . \end{aligned}$$

(8)

Now, for every $x \in K$, there is some $i \in \{1, \ldots , n\}$ such that $x \in K_i$. Therefore, due to (8), $D(x) \subset D(x_i) + V$. It follows that $D(K) \subset \cup _{i = 1}^n D(x_i) + V$. As $\eta > \alpha _U(D(x_i))$, $i = 1 \ldots , n$, there exist, $m \in \mathbb {N}$ and $\eta U$-small subsets $E_1, \ldots , E_m$ of E, such that $\cup _{i = 1}^n D(x_i) \subset \cup _{j = 1}^m E_j$. Let M be the set of all mappings from the set $\{1, 2, \dots , n\}$ into the set $\{1, 2, \ldots , m\}$. Then M is a finite set and, clearly, $D \subset \cup _{\mu \in M} D_\mu$, where $D_\mu := \{f \in D : f(x_i) \in E_{\mu (i)}, i = 1, \ldots , n\}$. Moreover, the sets $D_\mu$ are $\eta N(K, U)$-small. Indeed, if $\mu \in M$, $f, g \in D_\mu$ and $x \in K$ are given, then there exists $i \in \{1, \ldots , n\}$ such that $x \in K_i$. By (8), $f(x) - f(x_i) \in V$ and $g(x) - g(x_i) \in V$. Since $E_{\mu (i)} - E_{\mu (i)} \subset \eta U$, we get $f(x_i) - g(x_i) \in \eta U$. It follows that:

$$\begin{aligned} f(x) - g(x)=\, & {} f(x) - f(x_i) + f(x_i) - g(x_i) + g(x_i) \\&- g(x) \in V + \eta U + V \subset W + \eta U. \end{aligned}$$

Since this holds for $x \in K$ and every $W \in \mathscr {N}_0$, we get $f(x) - g(x) \in \overline{\eta U} = \eta {\overline{U}} = \eta U$, for U is closed. Hence $D_{\mu (i)} - D_{\mu (i)} \subset N(K, \eta U) = \eta N(K, U)$, saying that $D_\mu$ is $\eta N(K, U)$-small. It follows that $\eta \ge \widehat{\alpha }_{N(K, U)}(D)$, achieving the proof. $\square$

Proof of Theorem 7

By definition, $\displaystyle \widehat{\alpha }_{{\mathscr {C}}}(D) = \sup _{\begin{array}{c} K \in {\mathscr {K}} \\ U \in {\mathscr {C}} \end{array}} \widehat{\alpha }_{N(K, U)}(D)$. Since D is equicontinuous on X, it is uniformly equicontinuous on each compact $K \subset X$. Then, Lemma 4 yields

$$\begin{aligned} \widehat{\alpha }_{N(K,U)}(D)=\sup \{\alpha _{U}(D(x)), \quad x \in K\}. \end{aligned}$$

Then

$$\begin{aligned} \widehat{\alpha }_{{\mathscr {C}}}(D)&= \sup _{\begin{array}{c} K \in {\mathscr {K}} \\ U\in {\mathscr {C}} \end{array}} \sup _{x \in K} \alpha _U(D(x)) \\&= \sup _{K \in \mathscr {K}} \sup _{x \in K} \alpha _\mathscr {C}(D(x)), \end{aligned}$$

as claimed. $\square$

Since, in any topological vector space, a subset is precompact if and only its non compactness measure is zero, a first immediate corollary of Theorem 7 is the following:

Corollary 5

An equicontinuous subset D of C(X, E) is precompact in $C_{\mathscr {K}}(X, E)$if and only if D(x) is precompact in E, for every $x \in X$.

Notice here that, according to Corollary 5, an equicontinuous subset D of C(X, E) is precompact in $C_s(X, E)$ if and only if it is so in $C_c(X, E)$.

Actually, Corollary 5 can be improved, whenever X is a $k_{\mathscr {K}}$-space and E is a topological vector space. At this point, recall that the completely regular space X is said to be a $k_{\mathscr {K}}$-space, if a function f defined on X, with values in $\mathbb R$ (or equivalently in any completely regular space), is continuous, provided its restriction to any $K \in {\mathscr {K}}$ is relatively continuous. Whenever ${\mathscr {K}} ={\mathscr {G}}$, we get a so-called $k_\mathbb R$-space (see [11] for further details on such spaces).

Let us denote by ${\mathscr {L}}_{pc}(C_{\mathscr {K}}(X, E), E)$ the topological vector space $\mathscr {L} (C_{\mathscr {K}}(X, E), E)$ of all linear continuous mappings defined on C(X, E) with values in E, endowed with the topology $\tau _{pc}$ of uniform convergence on the precompact subsets of $C_{\mathscr {K}}(X, E)$. A fundamental system of zero neighborhoods for $\tau _{pc}$ is given by the sets of the form

$$\begin{aligned} N(D, U) := \{T \in {\mathscr {L}}(C_{\mathscr {K}}(X, E), E) : T(D) \subset U\}, \end{aligned}$$

D running over the set of precompact subsets of $C_{\mathscr {K}}(X, E)$ and U over ${\mathscr {N}}_0$. By $\varDelta$ we will mean the evaluation map defined from X into ${\mathscr {L}}(C_\mathscr {K}(X, E), E)$ by $\varDelta (x) = \delta _x$.

The following lemma can be deduced from Proposition 16 of [13] (see also Lemma 2.1 of [9], and [17] in the case E is locally convex). For the sake of completeness, we include a proof of it.

Lemma 5

Let X be a Hausdorff completely regular space and E a Hausdorff topological vector space. Then the evaluation map $\varDelta : X \rightarrow {\mathscr {L}}_{pc}(C_{\mathscr {K}}(X, E), E)$is continuous if and only if every precompact subset of $C_{\mathscr {K}}(X, E)$is equicontinuous.

In particular, if X is a $k_{\mathscr {K}}$-space, every precompact subset of $C_{\mathscr {K}}(X, E)$is equicontinuous on X.

Proof

Necessity: Let D be a $\tau _{{\mathscr {K}}}$-precompact subset of C(X, E), $x_0$ an element of X, and U a zero neighborhood in E. Since $\varDelta$ is continuous at $x_0$, there exists a neighborhood $\varOmega _0$ of $x_0$, such that $\varDelta (x) - \varDelta (x_0) \in N(D, U)$, $x\in \varOmega _0$. This means that $f(x) - f(x_0) \in U$ for every $f \in D$ and $x\in \varOmega _0$, showing that D is equicontinuous at $x_0$. Since $x_0$ is arbitrary, D is equicontinuous on X.

Sufficiency: Choose arbitrarily $x_0 \in X$, $U \in \mathscr {N}_0$, and a $\tau _{\mathscr {K}}$-precompact set $D \subset C(X, E)$. By assumption D is equicontinuous. Therefore there exists a neighborhood $\varOmega _0$ of $x_0$, such that $f(x) - f(x_0) \in U$, for every $f \in D$ and $x \in \varOmega _0$. This is $\varDelta (x) - \varDelta (x_0) \in N(D, U)$, showing that $\varDelta$ is continuous at $x_0$.

Now, choose arbitrary $K \in {\mathscr {K}}$, $x_0 \in K$, and consider N(D, U), a zero neighborhood in $\mathscr {L}_{pc}(C_{\mathscr {K}}(X, E), E)$, where D is a precompact set in $C_{\mathscr {K}}(X, E)$ and $U \in {\mathscr {V}}_0$. Consider $V \in {\mathscr {V}}_0$, such that $V + V + V \subset U$. Then, for the zero neighborhood N(K, V) in $C_{\mathscr {K}}(X, E)$, there exist, $m \in \mathbb N$ and $f_1, \ldots , f_m \in D$, such that $D \subset \underset{1 \le j \le m}{\cup }(f_j + N(K, V))$. This means:

$$\begin{aligned} \forall f \in D,\ \exists j \in \{1,\dots ,m\} : f(x) - f_j(x) \in V, \quad x \in K. \end{aligned}$$

(9)

Since $\{f_1, \ldots , f_m\}$ is equicontinuous, there exists a neighborhood $\varOmega _0$ of $x_0$, such that:

$$\begin{aligned} f_j(x) - f_j(x_0) \in V, \quad j = 1, \ldots , m, \quad x \in \varOmega _0. \end{aligned}$$

(10)

It derives from (9) and (10) that, for every $f \in D$ there is some $j \in \{1, \ldots , m\}$ such that for every $x \in \varOmega _0 \cap K$, we have:

$$\begin{aligned} \varDelta (x)(f) - \varDelta (x_0)(f)&= f(x) - f(x_0) \\&= f(x) - f_j(x) + f_j(x) - f_j(x_0) + f_j(x_0) - f(x_0) \\&\in V + V + V \subset U. \end{aligned}$$

This shows that the restriction to K of $\varDelta$ is continuous at $x_0$ then on K, since $x_0$ was arbitrary. As X is a $k_{\mathscr {K}}$-space, $\varDelta$ is continuous on X. We conclude by the first part of the lemma. $\square$

Corollary 6

Let X be a Hausdorff completely regular $k_{{\mathscr {K}}}$-space, E a Hausdorff topological vector space, and D a subset of C(X, E). Then D is precompact in $C_{\mathscr {K}}(X, E)$if and only if it is equicontinuous on X and D(x) is precompact in E for every $x \in X$.

Proof

Since D is precompact in $C_{\mathscr {K}}(X, E)$, it is equicontinuous on X by Lemma 5. Hence, by Theorem 7, $\alpha _{{\mathscr {C}}}(D(x)) = 0$, for every $x \in X$. Therefore D(x) is precompact for every $x\in X$, whence the necessity. The sufficiency derives immediately again from Theorem 7. $\square$