NEW ZEALAND JOU RN AL OF MATHEMATICS Volume 28 (1999), 155-170

EQU ICONTIN U ITY AND HOLDER EQU ICONTIN U ITY OF

FAMILIES OF GEN ERALIZED CONVEX M APPIN GS

W o l f g a n g W . B r e c k n e r a n d T ib e r iu T r if

(Received July 1997)

Abstract. As a natural continuation of the expository article [23], which was

focused on linear versions of the functional analytical uniform boundedness

principle, the present paper discusses, unifies and improves some nonlinear

generalizations of this principle that were obtained under the influence of con­

vex analysis and convex optimization. They refer to families consisting of

so-called rationally s-convex mappings with respect to a wedge and reveal

several characterizations of the equicontinuity at a point and on a set, respec­

tively, for such families. In particular it is proved that equicontinuity at a

point is equivalent to the local Holder equicontinuity at that point.

1. Intoduction

Let M be a nonempty subset of a topological space X, let Y be a topological

linear space, and let T be a family of mappings from M into Y. If xq is a point in

M , then T is said to be:

(i) equicontinuous at xq if for each neighbourhood W of the origin of Y there

exists a neighbourhood U of Xq such that

{f{x) — f(xo) | x G U n M} C W for every / G

(ii) bounded at Xq if the set {f(xo) | / G J7} is bounded, i.e. for each neighbourhood

W of the origin of Y there exists a number a G ]0, oo[ such that

{ /(* „ ) | / G F ) C aW.

If T is equicontinuous (respectively bounded) at each point of M , then T is

called equicontinuous (respectively pointwise bounded) on M.The main purpose of numerous investigations made in functional analysis has

been to find conditions ensuring the equicontinuity of a pointwise bounded family

of mappings. From the results obtained, we merely draw the reader’s attention to

the following theorem.

Theorem 1.1. Let X be a topological linear space of the second category, let Y be a topological linear space, and let T be a family of continuous linear mappings from X into Y which is pointwise bounded on X . Then T is equicontinuous on X.

This theorem evolved from a result established by S. Banach and H. Steinhaus

[1] for normed linear spaces, that today is commonly known as the principle of

uniform boundedness. This principle reads as follows.

1991 AMS Mathematics Subject Classification: Primary 26B25; Secondary 46N10.

156 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Theorem 1.2. Let X be a Banach space, let Y be a normed linear space, and let

J7 be a family of continuous linear mappings from X into Y such that

sup{|| f(x) || | / G T j < oo for all x G X.

Then holds

sup{|| / || | / G T] < oo.

The principle of uniform boundedness was one of the first basic results in func­

tional analysis. It and its later generalizations have been found to be very useful

not only in functional analysis, but also in the applications of functional analysis.

A good survey on the evolution of the uniform boundedness principle was given by

C. Swartz [23].

The aim of the present paper is to discuss and to improve some generalizations

of Theorem 1.1 involving families of nonlinear mappings between topological lin­

ear spaces. Thus this paper can be viewed as a natural completion of Swartz’s

expository article, which focused on linear versions of the uniform boundedness

principle.

The process of establishing nonlinear versions of the Theorems 1.1 and 1.2 was

deeply influenced by convex analysis and convex optimization. First, it was re­

marked (see, for instance, [8, pp. 134-135]) that Theorem 1.2 is a simple conse­

quence of the fact that any lower semicontinuous convex function on a Banach

space is continuous. Next, P. Kosmol [11] generalized the Theorems 1.1 and 1.2 by

proving the following result.

Theorem 1.3. Let M be a nonempty open convex subset of a topological linear space X , and let T be a pointwise bounded family of continuous convex functions

f : M —* M. Then the following assertions are true:

1° If X is of the second category, then T is equicontinuous on M .

2° If X is a Banach space, then T is locally Lipschitz equicontinuous on M.

Furthermore, P. Kosmol gave applications of this theorem to the stability of

optimization problems.

Soon after paper [11] appeared, in [14] and [12], respectively, the two asser­

tions of Theorem 1.3 were separately generalized for pointwise bounded families

of convex mappings taking values in an ordered topological linear space or in an

ordered normed linear space, respectively. It should be emphasized that in [12] the

mappings belonging to the investigated family were convex with respect to wedges

that could be distinct.

Leaving the restrictive framework of normed linear spaces, M. Jouak and

L. Thibault [10] investigated the local Lipschitz equicontinuity of families of convex

mappings defined on a nonempty open convex subset of a topological linear space

and taking values in a topological linear space ordered by a normal wedge. They

revealed that for such families the local Lipschitz equicontinuity and the equiconti­

nuity are equivalent at any point. By taking into account this result and assuming

that the domain of the mappings lies in a topological linear space of the second

category, they also proved the local Lipschitz equicontinuity of a pointwise bounded

family of continuous convex mappings.

EQUICONTINUITY AND HOLDER EQUICONTINUITY 157

But convex analysis dealt with not only convex functions, but also the more

general Jensen convex functions (see, for instance, F.A. Valentine [24], D.S. Mitri-

novic [17], A.W. Roberts and D.E. Varberg [19]). On the other hand, functional

analytical papers by M. Landsberg [16] (on topological linear spaces that are not

locally convex) and by S. Rolewicz [20], [21] (on locally bounded topological linear

spaces) led to a generalization of the semi-norms (respectively norms) called s-

semi-norms (respectively s-norms) (see also G. Kothe [15, pp. 164-166], W. Ruess

[22]). These investigations determined W.W. Breckner [2] to introduce two types

of generalized convex functions called rationally s-convex functions and s-convex

functions, respectively. From these two classes of functions the one consisting of

the rationally s-convex functions is more general.

In [3] it was shown that Kosmol’s results concerning the equicontinuity of families

of convex functions given in [11 ] remain valid for families of rationally s-convex

functions. Moreover, in [4] it was proved that also the results from [14] can be

extended to rationally s-convex mappings taking values in an ordered topological

linear space, and so a very general version of Theorem 1.1 was obtained.

To our knowledge, attempts to generalize the above-mentioned results concerning

the Lipschitz equicontinuity of families of convex functions (or convex mappings)

to families of rationally s-convex functions (or rationally s-convex mappings) have

not been done so far. This gap will be filled up by the present paper. Following

W.W. Breckner [5], who characterized the continuity of a rationally s-convex func­

tion by Holder continuity, we shall prove a similar result referring to the equicon­

tinuity of families of rationally s-convex mappings. Unlike all the previously cited

papers, for the first time here will be considered families of mappings that are nei­

ther rationally s-convex with respect to the same number s, nor with respect to

the same wedge. Since for such mixed families of rationally s-convex mappings a

generalization of Theorem 1.1 is missing, we shall derive it also here.

Summing up we can note that the results stated in the present paper collect

into a unitary theory all what was obtained under the influence of convex analysis

concerning generalizations of Theorem 1.1 to families of nonlinear mappings.

2. Preliminaries

To make our paper self-contained we recall in this section some definitions and

point out some auxiliary results that we shall need in our investigations.

Throughout our paper all linear or topological linear spaces that will occur are

over the same field K of real or complex numbers. If X is a linear space, then ox denotes its zero-element.

Given both a linear space X and a function p : X —> R, the sets B(p, r) and

B(p, r) are defined for each r e R by

B(p, r) := {x G X \ p(x) < r} and B(p, r) := {x €E X \ p(x) < r},

respectively. It is easily seen that the following proposition involving these sets is

true.

158 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Proposition 2.1. Let X be a linear space, let p\ : X —* [0, oo[ and p2 : X —> M be positively homogeneous functions, and let r\,r2 G ]0, oo[ be such that B(pi,r\) C

B(p2,r2). Then holds

p2{x) < —pi(x) for all x G X. r i

If M is an absorbing subset of a linear space X , then the function pm '• X —» R

defined by

Pm (x ) := inf{a G ]0, oo[ | x G aM}

is called the Minkowski function (or gauge) of M. It can be easily shown that pm

is positively homogeneous and satisfies M C B(pm, 1)- Besides, if M is not only

absorbing, but also balanced, then B(pm? 1) C M holds.

A nonempty family V of real-valued functions that are defined on a linear space

X is said to be a pseudonorm on X if for each p G V the following conditions are

satisfied:

(PNi) p(x) > 0 for all i G l ;

(PN2) p(ax) = \a\p(x) for all a G K and all x G X;

(PN3) there exists a p* G V such that

p(xi + x2) < p*{xi) +p*(x2) for all xi,x2 eX .

A pseudonorm V on X is said to be directed if for each pair p\,p2 G V there

exists a p G P such that

m ax{p i(x ),p2 (^)} < pix) for all x G l

Proposition 2.2 ([5]). If V is a directed pseudonorm on a linear space X , then

there exists a unique linear topology T(V) on X for which

{V G 2X | 3p G V 3r G ]0, 0 0[ : V = B(p, r)}

is a neighbourhood-base at ox-

Given a pseudonorm V on a linear space X , the linear topology T(V) on X ,

whose existence and uniqueness are assured by Proposition 2.2, is called the topology

generated by V. In view of a result by D.H. Hyers [9, Theorem 9] the topology of

each topological linear space can be generated by a directed pseudonorm.

Now let X and Y be topological linear spaces. Further, let M be a nonempty

subset of X, let T be a nonempty family of mappings from M into Y, and let

v : T —* M be a function such that v(F) C ]0,1]. Then the family T is said to be:

(i) locally v-equicontinuous at a point Xo G M if for each neighbourhood W of oy there exist a neighbourhood V of ox, a number a G ]0, 00[ and a neighbourhood

U of xq such that

Pw(f(xi) ~ f(x2)) < a[pv {xi - x 2)]I/(/) (2.1)

for all / G T and all xi, x2 G U fl M ;

(ii) locally u-equicontinuous on M if it is locally ^-equicontinuous at each point

of M.

EQUICONTINUITY AND HOLDER EQUICONTINUITY 159

In the special case when v(f) = 1 for each / G T and T is locally I'-equicontinuous

at a point xq G M (respectively on M), then T is said to be locally Lipschitz

equicontinuous at xo (respectively on M).Since in most cases the topology of a topological linear space is introduced by

means of a directed pseudonorm, it is desirable to have characterizations of the

local z/-equicontinuity of a family of mappings in terms of pseudonorms. Such a

characterization is established in the next proposition.

Proposition 2.3. Let X and Y be linear spaces endowed with the topology gen­erated by a directed pseudonorm V and Q, respectively. Further, let M be a nonempty subset of X , let J7 be a nonempty family of mappings from M into Y, and let u : T —> M be a function such that v{T) C ]0,1]. Then T is locally u-equicontinuous at a point xo & M if and only if for each q G Q there exist p G ? ,

a G ]0, oo[ and a neighbourhood U of x q such that

q{f{x i) - fix 2)) < a\p{x 1 - s2) r (/) (2-2)

for all f G T and all x i,x2 G U r I M .

Proof. Necessity. Let q be in Q. Since W := B(q, 1) is a neighbourhood of oy

and T is locally ^-equicontinuous at x q , there exist a neighbourhood V of ox, a

number a\ G ]0, 00[ and a neighbourhood U of xq such that

Pw{f{xi) - f{x2)) < a1[pv (x1- x 2)]u{f) (2.3)

for all / G T and all x\, x2 G U D M.Taking into account that W is balanced, it results that B{pw, 1) Q B(q, 1). In

virtue of Proposition 2.1 it follows that

q{y) < pw(y) for all y G Y.

Consequently, we conclude from (2.3) that

<l(f{x\)-f{x2)) < ai[pv (xi - x2)]^f) (2.4)

for all / G T and all x\, x2 G U fl M.Next we choose p G V and r G ]0, 00[ such that B(p,r) C V. Then B(p,r) C

B(py, 1) holds. By applying Proposition 2.1 we get

P v{x ) < -p(x) for all x G X. (2.5)r

Now we set a := a i( l + 1/r). Then it follows from (2.4) and (2.5) that (2.2) holds

for all / G T and all x\, x2 G U fl M.

Sufficiency. Let W be a neighbourhood of oy. Then there exist q G Q and

r G ]0, 00[ such that B(q, r) C W . Now let p G V, a\ G ]0,0 0[ and a neighbourhood

U of x q be chosen so that

q(f(x 1) - f(x2)) < ai[p(xi - x2)]^f) (2.6)

holds for all / G T and all x\,x2 G U fl M.From B(q,r) C W we conclude that B(q,r) C B{pw, 1)- Then it follows by

Proposition 2.1 that

pw(y) < -q{y) for all y G y.r

160 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Consequently, we conclude from (2.6) that

(2.7)

for all f E T and all x\, x2 E U fl M.Next we set V B(p, 1). Then V is a balanced neighbourhood of ox satisfying

B(pv, 1) Q B(p, 1). By applying Proposition 2.1 we conclude that p(x) < pv(x) for all x E X. In virtue of this result, (2.7) implies (2.1) for all f E T and all

x i , x 2 E U (1 M, where a := a\/r. Hence T is locally i'-equicontinuous at xo . □

Remark 2.4. Proposition 2.3 reveals that the local z/-equicontinuity of a family

of mappings is a natural generalization of the local s-Holder-continuity introduced

by W.W. Breckner [5] for individual real-valued functions defined on subsets of

topological linear spaces. The idea to introduce the local ^-equicontinuity of a

family of mappings by using the Minkowski function originated from [10], where

this function was used to define the Lipschitz equicontinuity of a family of mappings

in arbitrary topological linear spaces.

Let X and Y be linear spaces, and let s be a number belonging to ]0,1]. If M is

a nonempty convex subset of X and K is a subset of Y , then a mapping f : M —> Y is said to be s-convex (respectively rationally s-convex) with respect to K if for

all a E ]0,1[ (respectively all rational a E ]0,1[) and all x i, x2 E M the following

relation holds:

Obviously, each s-convex mapping with respect to K is rationally s-convex with

respect to K. But the converse of this property is not true.

Usually 1-convex (respectively rationally 1-convex) mappings with respect to

K are simply called convex (respectively rationally convex) with respect to K. Rationally convex mappings are also called Jensen convex, midconvex or midpoint convex. Any additive mapping / : X —> Y is rationally convex with respect to

{oy} (see, for instance, [6 , p. 76, Corollary 5.1.2]).

In what follows we deal with mappings that are rationally s-convex with respect

to a wedge. As usual, by a wedge we understand a nonempty subset K of a linear

space satisfying

Proposition 2.5. Let M be a nonempty convex subset of a linear space X , let

K be a wedge in a linear space Y , and let f : M —► Y be a rationally s-convex mapping with respect to K, where s E ]0,1]. If x q E M and x E M — x q , then

(1 - a)sf(xi) +asf(x2) E f ( { l -a ) x i+ax2)+K.

aK + bK C K for all a, b E [0, oof.

as[f(x0+x) - Q(s)f(x0)] E f(x0 + ax) - f(x0) + K

for every rational number a E [0,1], where

(2.8)

Proof. Let a be any rational number belonging to [0,1]. Since

-as6(s)f(x0) E [(1 - a)s - l]/(x0) + K

EQUICONTINUITY AND HOLDER EQUICONTINUITY 161

and

(1 — a)xo + a(xo + x) = xo+ax,

we have

as[f(x0 + x) - 9(s)f(xo)] G asf(x0 + x) + [(1 - a)s - l]/(x0) + K

= (1 - a)sf(x0) + asf(xo + x) - f(x0) + K C /(rc0 + ax) - f(x0) + K.

□A wedge K in a topological linear space V is said to be normal if for each

neighbourhood W of oy there exists a neighbourhood W\ of oy such that

- K ) n (Wi +K )CW .

Obviously, {oy} is a normal wedge.

Normal wedges play an important role in the theory of ordered topological linear

spaces (see, for instance, [18], [6]). In our investigations regarding the equiconti­

nuity of families of mixed rationally s-convex mappings we need a similar concept

for families of wedges. In normed linear spaces such a concept has been introduced

by P. Kosmol [12] (see also [13, p. 232, Definition 3]). It can be generalized to

arbitrary topological linear spaces as follows.

A family K of wedges in a topological linear space Y is said to be uniformly

normal if for each neighbourhood W of oy there exists a neighbourhood Wi of oy

such that

(Wi - K) n (Wi + K) C W for all K G K.

This definition implies that any wedge belonging to a uniformly normal family of

wedges is normal. Besides, if K is a normal wedge, then {—K,K} is a uniformly

normal family of wedges.

3. Characterizations of the Equicontinuity at a Point

After the preparations in Section 2 we can now state our main results revealing

several characterizations of the equicontinuity at a point in the case of families of

mixed rationally s-convex mappings. In order to simplify the formulations of these

results we shall use the following assumptions:

(Ai) X and Y are topological linear spaces;

(A2) I is a nonempty set;

(A3) K, [Ki)ieI is a uniformly normal family of wedges in Y ;

(A4) (8 i)i£i is a family of numbers in ]0, 1];

(A5) M is a nonempty convex subset of X and T {fi)iei is a family consisting

of mappings f i i M —*Y that are rationally s^-convex with respect to K*;

(As) limtx^o tSi = 0 uniformly with respect to i G /;

(A7) v : T —» M is defined by v(fi) := Si.

162 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Theorem 3.1. If the assumptions (Ai)-(A7) are satisfied and xo is an interior

point of M, then the following assertions are equivalent:

1° T is both bounded and equicontinuous at xq.

2° For each neighbourhood W of oy there exist a neighbourhood U of xq and a

number a G ]0, oo[ such that

{fi(x) | x G U fl M} C aW for all i G I.

3° T is both bounded and locally u-equicontinuous at xq.

Proof. 1° => 2° Let W be any neighbourhood of oy. Choose a neighbourhood

W\ of oy such that W\ + Wi C W . Since T is bounded at xo, there exists a number

a G ]0, oo [ such that

{/<(*o) I * G 1} C aW i.

Now, taking into consideration that T is equicontinuous at xo, we conclude that

there is a neighbourhood U of xq such that

{fi(x) — fi(xo) | x G U fl M} C aW\ for all i G I.

Consequently, we have

ifi{x) I X G u n M} C {fi{x) - fi(x0) I x G U n M ] + fi(x0)

C a{W\ + W\) C aW

for all i G I.

2° =>- 3° Obviously, assertion 2° implies that T is bounded at a?o- Thus it

merely remained to be proved that T is locally ^-equicontinuous at xo.

Let W be any neighbourhood of oy. Since K, is uniformly normal, there exists a

balanced neighbourhood W\ of oy such that

[ W i - K ^ n i W i + K ^ C W for all i G I. (3.1)

Next choose a balanced neighbourhood of oy such that W2+W2 Q W\. Accord­

ing to assertion 2° we can find a neighbourhood XJ\ of Xq and a number a G ]0, oo[

such that

{fi(x) | x G U\ D M } C aW2 for all i G I. (3.2)

Since xq is an interior point of M , we can select a balanced neighbourhood V of

ox such that xq + V + V Q Ui D M. Then U := xq + V is a neighbourhood of xo enclosed in U\ fl M. We claim that

Pw{fi(xi) ~ fi(x2)) < a[pv (xi - x2)]Si (3.3)

for al l i e / and all x\, x2 G U.To prove this assertion, we fix an i G I and x\,x2 G U. For brevity we set

s := Si. Farther we select any rational number r satisfying r > pv{x\ — x2). Then

£(xi — x2) G V, and therefore the point

x := xi + - (xi - x2) r

lies in U\ DM. Taking into account that

VX\ = x2 + —— ( x-x2),

r + 1

EQUICONTINUITY AND HOLDER EQUICONTINUITY 163

Proposition 2.5 yields

( t + t ) “ 0(s)fi(x2)] e fi(x 1) - fi(x2) + Ki,

where 9(s) is the number defined by (2.8). Consequently, we have

-^ 7 [fi{x 1) - fi(x2)] € 7-7 -p- [/<(«) - 0(s)fi(x2)] ~ K i . (3.4) ars a[r + 1JS

But, in view of (3.2) we note that

fi(x) - 0{s)fi{x2) e aW2 - aO(s)W2 C a{r + 1)S{W2 + W2) C a(r + l ) s^i-

Therefore (3.4) allows us to conclude that

A lA(*i) - f < M 1 e W'i - (3-5)ars

Similarly it is seen that

- ^ [ / i(* 2)- / i(x i)] € W i- i f i . (3.6)ars

By (3.5), (3.6) and (3.1) it follows that

A |/i(»i) - /<(*a)l e W - Ki) n (w, + ifj) c w,ars

whence

Pw(fi(xi) ~ fi(x2)) < ars.

Since r was an arbitrary rational number satisfying the inequality r > pv(x\ — x2),

the inequality (3.3) must be valid. Hence T is locally i/-equicontinuous at x q .

3° =4> 1° Let W be any neighbourhood of oy. Choose a balanced neighbourhood

W\ of oy such that W\ C W. According to assertion 3° we can find a neighbourhood

V of ox, a number a e ]0, 00[ and a neighbourhood U\ of xq such that

PWi {fi{x 1) - fi{x2)) < a[pv (x 1 - x2)]Si

for all i e I and all xi, x2 e U\ fl M. This inequality implies

PWi (fi(x) - fi(xo)) < a[pv (x - x0)]Si for all x eUxDM (3.7)

and all i e I. Let b e ]0,00[ be such that

bSi < - for all i e I a

and set U := (a?o + bV) D U\ D M. Then U is a neighbourhood of Xq satisfying

{fi(x) - fi{x0) | x e U} C W for every i e l . (3.8)

Indeed, let i be any element of I. For each x e U we have x — xq e bV, whence

\pv{.x - x0)}Si < bSi < i .

Therefore (3.7) yields

ifi(x) - fi(x 0) \xeU}C B(pWl, 1) C ^ C l ^ .

Consequently, (3.8) holds as claimed. This means that T is equicontinuous at

X q . □

164 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Remark 3.2. It should be emphasized that in Theorem 3.1 the assumption (Ae)

was needed only in the proof of the implication 3° =$■ 1°.

Corollary 3.3. If the assumptions (Ai)-(A6) are satisfied and xq is an interior point of M at which T is bounded, then the following assertions are equivalent:

10 F is equicontinuous at xo■

2° For each neighbourhood W of oy there exists a neighbourhood U of xq such

that

{fi(x) - fi(xo) | x E U fl M} C W - Ki for every i € I.

3° For each neighbourhood W of oy there exist a neighbourhood U of Xq and a

number a E ]0, oo[ such that

{fi(x) — fi(xo) | x e U fl M} C aW — Ki for every i G I.

4° For each neighbourhood W of oy there exist a neighbourhood U of xq and a

number a E ]0, oof such that

{fi(x) — fi(xo) | x G U fl M} C aW for every i € I.

Proof. 1° =>• 2° => 3° Obvious.

3° =$> 4° Let W be any neighbourhood of oy. Since K, is uniformly normal,

there exists a neighbourhood W\ of oy satisfying (3.1). Let be a balanced

neighbourhood of oy such that W2 + W2 Q W\. By assertion 3° there are a

neighbourhood U\ of xq and a number ai G ]0, oo[ such that

{fi(x) — fi(xo) | x G U\ fl M} C aiW2 — Ki for each i 6 I. (3.9)

Besides, by the boundedness of T at Xo there exists a number a2 G ]0, oo[ such that

{(2Si - 2)fi(x0) | i e / } C a2W2. (3.10)

Next we choose a balanced neighbourhood V of ox such that xq + V C U\ fl M.

After that we set U := xo + V and a := ai + a2. Then we have

ifi{x) - fi(x0) \ x e U} C aW for every i e l . (3.11)

To see this, let i e l and x € U be arbitrarily fixed. For short we set s := Sf.

Since V is balanced, the point X\ 2xo — x lies in U. Taking into account that

xo = \ x + ^ xi, we conclude that

^ ^ fi(xi) e f iM + K i ,

whence

fi(x) ~ f iixo) ^ (2s - 2)/i(x0) - [/<(xi) - fi(x0)\ + Ki.

According to (3.9) and (3.10) it follows that

- LfiW - /<(*o)] e — w2 - — W2 + Ki + Ki C W2 + W2 + Ki C Wl + Ki. a a a

But, in view of (3.9) we also have

- [/*(*) - /i(zo)] € — w2 - Ki C W2 - Ki C Wl - Ki. a a

EQUICONTINUITY AND HOLDER EQUICONTINUITY 165

Consequently, in virtue of (3.1) it follows that

- [fi(x) - fi{xo)] € W, i.e. fi(x) - fi(x0) G aW. a

Hence (3.11) holds as claimed. In other words, assertion 4° is true.

4° => 1° Let W be any neighbourhood of oy. Next choose a balanced neigh­

bourhood W\ of oy satisfying W\ + W\ C W . By assertion 4° there exist a neigh­

bourhood U of xo and a number a\ G ]0, oo[ such that

{fi(x) — fi(xo) | x G U H M} C a\W\ for every i G I.

On the other hand, the boundedness of T at Xo furnishes a number a2 G ]0, oo[

such that

{fi{xo) | i G / } C a2W i.

Consequently, we have

{fi(x ) | x G U fl M } C {fi{x) - fi{x0) | x G U n M } + fi{x0)

C + a2W i C (&1 + fl2) (W i "I- W i) ^ (®i "t- cl2)W

for every i G I. By applying the implication 2° => 1° from Theorem 3.1 we conclude

that T is equicontinuous at x q . □

Under additional assumptions concerning the space Y that occured in Theo­

rem 3.1 we can give further characterizations of the equicontinuity at a point as

consequences of this theorem. But before deriving such results we have to give a

definition.

A family T of mappings from a nonempty subset M of a topological space X

into a topological linear space Y is said to be locally equibounded at a point xq G M if there is a neighbourhood U of xo such that

( J { /(x ) | x G U fl M }

fer

is bounded. Clearly, in the special case when Y = R, then T is locally equibounded

at x0 if and only if there exist a neighbourhood U of xo and a number a G ]0, oo[

such that

{/(x) | x G U n M} C [-a, a] for every / G T.

Corollary 3.4. Suppose that the assumptions (Ai)-(Ae) are satisfied and that Y is locally bounded. Then T is both bounded and equicontinuous at an interior point

xo of M if and only if T is locally equibounded at x$.

Proof. Necessity. Let W\ be a bounded neighbourhood of oy. In virtue of the

implication 1° =£> 2° in Theorem 3.1 there exist a neighbourhood U of xo and a

number ai G ]0, oo[ such that

{fi(x) | x G U fl M } C a\W\ for all i G I .

Now let W be any neighbourhood of oy. If a2 G ]0, oof is chosen such that

Wi C a2W , then

{fi{x) \ x e U n M } C a ia 2W for all i G I .

Consequently, T is locally equibounded at xo-

Sufficiency. Obvious, because of the implication 2° =>■ 1° in Theorem 3.1. □

166 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Corollary 3.5. Suppose that the assumptions (Ai)-(Ag) are satisfied and that Y is locally convex. If x$ is an interior point of M , then the following assertions are

equivalent:

1° T is both bounded and equicontinuous at xq.

2° Given any positively homogeneous function f : Y —> R which is continuous at

oy, the family (/ o fi)i£i is locally equibounded at xq.

3° Given any continuous sublinear function f : Y —> R, the family (f o fiji^ i is

locally equibounded at xo.

4° Given any continuous semi-norm f : Y —> R, the family (/ o fi)i^i is locally

equibounded at x$.

Proof. 1° =>■ 2° Let / : Y —> R be any positively homogeneous function which

is continuous at oy. Then there is a neighbourhood W of oy such that {f(x)\xE W} C [—1,1]. By applying the implication 1° =4- 2° of Theorem 3.1 we conclude

that there exist a neighbourhood U of xq and a number a G ]0, oo[ such that

{fi(x) | x G U fl M} C aW for all i G I.

Consequently, we have

| x G U fl M} C [—a, a] for all i G I.

This means that (/ o / j) ie/ is locally equibounded at xq.

2° 3° =>• 4° Obvious.

4° =>■ 1° Let W be any neighbourhood of oy. Since Y is locally convex, we can

find a balanced convex neighbourhood W\ of oy such that W\ C W . Taking into

account that pw1 is a semi-norm, we conclude in virtue of assertion 4° that there

are a neighbourhood U of xq and a number a G ]0, oo[ such that

{,pwi (fi{x)) | x G U n M} C [0, a[ for every i G /,

whence

l~ f i ( x )\xeUn m | C B(pwi , 1) for every i G I.

But, B(pw1,1) Q W\ C W, and thus we have

{fi{x) | x G U fl M} C aW for every i G I.

By the implication 2° => 1° of Theorem 3.1 it follows that T is both bounded and

equicontinuous at x q . □

4. Characterizations of the Equicontinuity on a Set

Theorem 4.1. Suppose that the assumptions (Ai)-(Ag) are satisfied, that M is open and that T is pointwise bounded on M. Then the following assertions are

equivalent:

1° T is equicontinuous on M.

2° T is equicontinuous at some point of M.

EQUICONTINUITY AND HOLDER EQUICONTINUITY 167

3° For each neighbourhood W of oy there exist a nonempty open subset N of M

and a number a G ]0, oo[ such that

{fi{x) | x G N} C aW for every i G I.

Proof. 1° => 2° Trivial.

2° 3° Follows from Theorem 3.1.

3° =>• 1° Let xq be any point in M. We shall prove that T is equicontinuous at

x0.

Let W be any neighbourhood of oy. We take a balanced neighbourhood W\ of

oy such that W\ + W\ + C W. After that we apply assertion 3° and conclude

that there exist a nonempty open subset N of M and a number a\ G ]0, oo[ such

that

{fi{x) | x G N} C a\W\ for every z G /. (4.1)

Choose a point xi G N. Since

lim71—► OO

X q + - ( x 0 - X i )n

xq,

there exists a positive integer n such that

x2 : = x q + - (:r0 - £ i ) n

belongs to M . Since T is bounded at Xo and x2, we can choose a number a2 G ]0, oo[

such that

{ fi(x o) | i G 1} U {fi{x2) | i e 1} Q a2W\. (4.2)

By the convexity of M, we conclude that

U := -^— x2 + - !—-N (4.3)n + 1 n + 1 v ’

is a subset of M. On the other hand, the representation

U = xo + — (N - xi) n + 1

reveals that U is a neighbourhood of xo- We claim that

{fi(x ) - f i(x 0) | x G U} C (ai + 2a2)W - Ki for every i G I. (4.4)

To prove (4.4) we fix any i G I and any x G U. For short we set s := s*.

According to (4.3), x must be of the form

n 1x = — — x2 + — — x3

n + 1 n+ 1

for a suitable X3 G N. Consequently we have

( ^ T i ) fi{x^ + ( ^ T l ) e + K"

168 WOLFGANG W. BRECKNER AND TIBERIU TRIF

Taking account of (4.2) and (4.1) it follows that

f,(x) - f,(xa) e / .( i '2) + ( ^ T f ) f i f a ) - fi(xo) - Ki

C f - 4 T') oalVi + f ^ - ) a1W1- a 2W1- K i \n + I J \n + 1 /

C (ai + 2a2){W1 + WX + Wi) - K{ C (ax + 2a2)W - K{.

Hence (4.4) holds as claimed.

By applying the implication 3° =>• 1° of Corollary 3.3 it results that T is equicon­

tinuous at xo- □

Corollary 4.2. Suppose that the assumptions (Ai)-(Ay) are satisfied, that M is open and that T is pointwise bounded on M. Then T is locally is-equicontinuous on M if and only if T is locally v-equicontinuous at some point of M.

Proof. Apply the Theorems 3.1 and 4.1. □

Theorem 4.3. Suppose that the assumptions (Ai)-(Ae) are satisfied, that X is of

the second category, that M is open and that T is pointwise bounded on M . Then T is equicontinuous on M if and only if each mapping fi (i G I) is continuous on

M.

Proof. Necessity. Obvious.

Sufficiency. Let W be any neighbourhood of oy. Take a closed neighbourhood

W\ of oy satisfying W\ C W and then set

Mn := O G M | fi(x) G nW{] iei

for each positive integer n. Notice that, for every positive integer n and every ? G /, the set {x e M \ fi(x) G nWi} is closed in the induced topology on M , since it is

the inverse image of the closed set nW\ under the continuous mapping f t. Hence

each set Mn (n G N) is also closed in the induced topology on M.

On the other hand, by a well-known result from topology (see, for instance,

A. Csaszar [7, p. 386, (9.1.11)]) the set M is of the second category in the induced

topology. Taking into account that the pointwise boundedness of T implies the

equalityOO

M = ( J M n,n = 1

we conclude that there is a positive integer n such that Mn has interior points

in the induced topology on M . Thus there exist a point xo G M and an open

neighbourhood U of xq such that U fl M C Mn. So it follows that

{fi(x) \xeUn M } C nW for every i G I.

By applying the implication 3° => 1° of Theorem 4.1 we conclude that T is equicon­

tinuous on M . □

EQUICONTINUITY AND HOLDER EQUICONTINUITY 169

Corollary 4.4. Suppose that the assumptions (Ai)-(Ae) are satisfied, that X is

a Hausdorff space of finite dimension, that M is open and that T is pointwisebounded on M. Then T is equicontinuous on M.

Proof. Apply Corollary 5.2.3 given in [6, p. 82] and Theorem 4.3. □

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Wolfgang W. Breckner

Universitatea Babe§-Bolyai

Fac. de Matematica §i Informatica

Str. Kogalniceanu Nr. 1

R0-3400 Cluj-Napoca

ROMANIA

breckner@math.ubbcluj .ro

Tiberiu Trif

Universitatea Babe§-Bolyai

Fac. de Matematica §i Informatica

Str. Kogalniceanu Nr. 1

RO-34CIO Cluj-Napoca

ROMANIA

ttrif@math.ubbcluj .ro