Journal of Fixed Point Theory and Applications

Selections of set-valued functions satisfyingthe general linear inclusion

Andrzej Smajdor and Joanna Szczawinska

Abstract. Applying the classical Banach fixed point theorem we provethat a set-valued function satisfying a general linear functional inclusionadmits a unique selection fulfilling the corresponding functional equa-tion. We also adopt the method of the proof for investigating the Rassiasstability of general linear equation.

Mathematics Subject Classification. 54C65, 54C60, 39B82.

Keywords. Set-valued function, selection, stability.

1. Introduction

Let (Y, ∥ · ∥) be a real normed space. We denote by n(Y ) the family of allnonempty subsets of Y and by cl(Y ), ccl(Y ), c(Y ) and cp(Y ) we denote col-lections of all closed, convex closed, compact and complete members of n(Y ),respectively.

The number

diamA := sup{∥a− b∥ : a, b ∈ A}is said to be the diameter of A ∈ n(Y ). We say that a set-valued functionF : K → n(Y ) (an “s.v. function” for abbreviation) is with bounded diameterif the function K ∋ x �→ diamF (x) ∈ R is bounded.

In [5], the authors proved that if S is a commutative semigroup withzero and (Y, ∥ · ∥) is a real Banach space, then F : S → ccl(Y ) is a subadditives.v. function; i.e.,

F (x+ y) ⊂ F (x) + F (y), x, y ∈ S,

with bounded diameter admits a unique additive selection. Popa in [11] provedthat if ∅ = K is a convex cone in a real vector space X (i.e., t1K + t2K ⊂ Kfor all t1, t2 ≥ 0) and F : K → ccl(Y ) (where (Y, ∥ · ∥) is a real Banach space)is an s.v. function with bounded diameter fulfilling the inclusion

F (αx+ βy) ⊂ αF (x) + βF (y), x, y ∈ K,

for α, β > 0, α+ β = 1, then there exists exactly one additive selection of F .

J. Fixed Point Theory Appl. 18 (2016) 133–145DOI 10.1007/s11784-015-0265-9Published online October 17, 2015© 2015 The Author(s)This article is published with open access at Springerlink.com

134 A. Smajdor and J. Szczawinska JFPTA2 A. Smajdor and J. Szczawinska

Nikodem and Popa in [9] and Piszczek in [10] proved the following the-orem.

Theorem 1.1. Let K be a convex cone in a real vector space X, (Y, ∥·∥) a realBanach space and α, β, p, q > 0. Consider an s.v. function F : K → ccl(Y )with bounded diameter fulfilling the inclusion

αF (x) + βF (y) ⊂ F (px+ qy), x, y ∈ K.

If α+β < 1, then there exists a unique selection f : K → Y of F satisfying theequation

αf(x) + βf(y) = f(px+ qy), x, y ∈ K.

If α+ β > 1, then F is single valued.

The case of p+ q = 1 was investigated by Popa in [13], Inoan and Popain [7] and recently, by Smajdor and Szczawinska in [14].

In this paper, we determine the conditions for which an s.v. functionF : K → n(Y ) satisfying the inclusion

αF (x) + βF (y) ⊂ F (px+ qy), x, y ∈ K, (1.1)

where |α| < p and diamF (x) ≤ M∥x∥, x ∈ K (for some M > 0) admits aselection satisfying the corresponding functional equation. It is easy to checkthat if F satisfies the opposite inclusion, then it is single valued. Theorem 2.1and also the method of its proof are used for the investigation of the Rassiasstability of general linear functional equation.

2. The main theorem

Let (X, ∥ · ∥) and (Y, ∥ · ∥) be real normed spaces and let K be a nonemptysubset of X. Consider an s.v. function F : K → n(Y ). A function f : K → Yis a selection of the s.v. function F if and only if f(x) ∈ F (x), x ∈ K. Let

Sel(F ) := {f : K → Y : f(x) ∈ F (x), x ∈ K}.

It is easy to check that if there exists a constantM > 0 such that diamF (x) ≤M∥x∥ for all x ∈ K, then the function

d(f, g) := sup

{∥f(x)− g(x)∥

∥x∥, x ∈ K \ {0}

}, f, g ∈ Sel(F ),

is a metric in Sel(F ). Moreover, if F (x) is complete for every x ∈ K, themetric space (Sel(F ), d) is complete. Obviously, the convergence in the space(Sel(F ), d) implies the pointwise convergence on the set K.

Theorem 2.1. Let (X, ∥·∥) and (Y, ∥·∥) be real normed spaces and let K ⊂ Xbe such that 0 ∈ K. Assume that p, q > 0 and α, β ∈ R are fixed and one ofthe following conditions holds:

(i) |α| < p and K ⊂ pK,(ii) |β| < q and K ⊂ qK.

Vol. 18 (2016) Selections of set-valued functions 135Selections of set-valued functions 3

Consider an s.v. function F : K → cp(Y ) such that 0 ∈ F (0) and

diamF (x) ≤ M∥x∥, x ∈ K,

for some positive constant M . If

αF (x) + βF (y) ⊂ F (px+ qy), x, y ∈ K, px+ qy ∈ K, (2.1)

then there exists a unique selection f : K → Y of F such that

αf(x) + βf(y) = f(px+ qy), x, y ∈ K, px+ qy ∈ K.

Proof. Assume that |α| < p and K ⊂ pK. Since diamF (0) = 0 and 0 ∈ F (0),F (0) = {0}. Putting y = 0 in (2.1), we obtain

αF

(x

p

)⊂ F (x), x ∈ K. (2.2)

Let

T (g)(x) := αg

(x

p

), x ∈ K, g ∈ Sel(G).

By (2.2), T (g) ∈ Sel(F ), g ∈ Sel(F ). Moreover, for every g1, g2 ∈ Sel(F ),

d(T (g1), T (g2)

)= |α| sup

{∥g1

(xp

)− g2

(xp

)∥

∥x∥, x ∈ K \ {0}

}

=|α|p

sup

{∥g1

(xp

)− g2

(xp

)∥

∥xp∥

, x ∈ K \ {0}

}

≤ |α|p

d(g1, g2).

Since |α| < p, the map T : Sel(F ) → Sel(F ) is contractive in the com-plete metric space (Sel(F ), d), so by the Banach theorem, it has a uniquefixed point f and limn→∞ Tn(g) = f for each g ∈ Sel(F ). Hence f : K → Yis the unique selection of the s.v. function F such that

f(x) = αf

(x

p

), x ∈ K.

Fix g ∈ Sel(F ) and x, y ∈ K such that px+ qy ∈ K. Then

x

p,y

p,px+ qy

p∈ K.

By (2.1),

αg

(x

p

)+ βg

(y

p

), g

(px+ qy

p

)∈ F

(px+ qy

p

).

Hence ����αg(x

p

)+ βg

(y

p

)− g

(px+ qy

p

)���� ≤ diamF

(px+ qy

p

)

≤ M

����px+ qy

p

���� .

136 A. Smajdor and J. Szczawinska JFPTA4 A. Smajdor and J. Szczawinska

Thus

��αT (g)(x) + βT (g)(y)− T (g)(px+ qy)�� ≤ |α|

pM∥px+ qy∥

for every x, y ∈ K such that px+ qy ∈ K. Proceeding by induction, we get

��αT n(g)(x) + βT n(g)(y)− T n(g)(px+ qy)�� ≤

(|α|p

)n

M ∥px+ qy∥

for every n ∈ N and all x, y ∈ K with px+qy ∈ K. Letting n → ∞, we obtain

αf(x) + βf(y) = f(px+ qy), x, y ∈ K, px+ qy ∈ K. �

Let X and Y be real vector spaces and K a convex cone in X. It is easyto check that if f : K → Y satisfies the equation

αf(x) + βf(y) = f(px+ qy), x, y ∈ K,

and f(0) = 0, then f is additive; i.e.,

f(x) + f(y) = f(x+ y), x, y ∈ K.

Corollary 2.2. Let p, q > 0 and |α| < p (or |β| < q). Assume that (X, ∥ · ∥)and (Y, ∥ · ∥) are normed spaces, K a convex cone in X and F : K → cp(Y )an s.v. function such that 0 ∈ F (0) and

diamF (x) ≤ M∥x∥, x ∈ K,

for some constant M > 0. If

αF (x) + βF (y) ⊂ F (px+ qy), x, y ∈ K,

then the unique selection f : K → Y of F fulfilling the equation

αf(x) + βf(y) = f(px+ qy), x, y ∈ K,

is additive.

Remark 2.3. If |α| = p and |β| = q, the above corollary is not true.

Proof. Let K = [0,+∞) and let F : K → c(R) be an s.v. function defined by

F (x) = [0, x], x ∈ K,

and diamF (x) = |x|, x ∈ K. Then

F (x+ y) = F (x) + F (y), x, y ∈ K,

and for every a ∈ [0, 1], the function f(x) = ax, x ∈ K, is an additive selectionof F . �

Vol. 18 (2016) Selections of set-valued functions 137Selections of set-valued functions 5

3. Stability results

In the first part of this section we present an application of Theorem 2.1to the investigation of the Rassias stability of the general linear functionalequation

αf(x) + βf(y) = f(px+ qy). (3.1)

For definition and more results in the Rassias stability theory see, for instance,[6, 8]. The general linear equation was considered by several authors (forexample, [1, 2, 12]). A set-valued approach can be found in [3].

Gajda showed in [4] (see also [8, Theorem 2.6]) that for ε > 0 there is afunction f : R → R satisfying the inequality

∣∣f(x) + f(y)− f(x+ y)∣∣ ≤ ε(|x|+ |y|), x, y ∈ R,

while there is no constant M ≥ 0 and no additive function f0 : R → R suchthat ∣∣f(x)− f0(x)

∣∣ ≤ M |x|, x ∈ R.We apply the method used in the proof of Theorem 2.1 to obtain the

stability result for equation (3.1). We will denote by R+ and Q+ the set ofall nonnegative real and rational numbers, respectively.

Theorem 3.1. Let K be a convex cone in a real normed space (X, ∥ · ∥),(Y, ∥ · ∥) a real Banach space and ε : R+ × R+ → R+ an R+-homogenousfunction. Assume that p, q > 0, α, β ∈ R are fixed and |α| < p. If a functionf : K → Y satisfies

∥αf(x) + βf(y)− f(px+ qy)∥ ≤ ε(∥x∥, ∥y∥), x, y ∈ K,

then there exists a unique function f0 : K → Y fulfilling the functional equa-tion

αf0(x) + βf0(y) = f0(px+ qy), x, y ∈ K,

and such that

∥f(x)− f0(x)− f(0)∥ ≤ M∥x∥, x ∈ K,

for some M > 0. Moreover,

∥f(x)− f0(x)− f(0)∥ ≤ ε(1, 0)

p− α∥x∥, x ∈ K,

and f0 is an additive function. In particular, if ε(1, 0) = 0, then

f(x) = f0(x)− f(0), x ∈ K.

Proof. Let g : K → Y be given by g(x) = f(x) − f(0), x ∈ K. It is easy tocheck that g(0) = 0 and

∥αg(x) + βg(y)− g(px+ qy)∥ ≤ ε(∥x∥, ∥y∥), x, y ∈ K. (3.2)

Hence, for all x, y ∈ K,

αg(x) + βg(y) ∈ g(px+ qy) + ε(∥x∥, ∥y∥)B, (3.3)

138 A. Smajdor and J. Szczawinska JFPTA6 A. Smajdor and J. Szczawinska

where B denotes the unit closed ball in the space Y . Setting y = 0 and re-placing x by x/p in (3.3), we obtain

αg

(x

p

)∈ g(x) + ε

(∥x∥p

, 0

)B = g(x) +

ε(1, 0)

p∥x∥B, x ∈ K.

Consider an s.v. function G : K → cl(Y ) given by

G(x) = g(x) +ε(1, 0)

p− α∥x∥B, x ∈ K.

Then

diamG(x) =2ε(1, 0)

p− α∥x∥, x ∈ K,

and

αG

(x

p

)= αg

(x

p

)+ α

ε(1, 0)

p− α

����x

p

����B

⊂ g(x) +ε(1, 0)

p

(1 +

α

p− α

)∥x∥B = G(x)

for all x ∈ K. The idea of the proof is the same as before so we only give asketch. The function T : Sel(G) → Sel(G), given by

T (h)(x) := αh

(x

p

), x ∈ K, h ∈ Sel(G),

is contraction with the constant |α|/p. By the Banach theorem, there existsa unique function f0 ∈ Sel(G) such that

f0(px) = αf0(x), x ∈ K,

and limn→∞ Tn(g) = f0. By the definition of G,

∥f0(x)− g(x)∥ ≤ ε(1, 0)

p− α∥x∥, x ∈ K.

Since g satisfies (3.2) and ε is positively homogeneous,

��αT (g)(x) + βT (g)(y)− T (g)(px+ qy)�� ≤ |α|

pε(∥x∥, ∥y∥)

for all x, y ∈ K. Proceeding by induction, we get

��αTn(g)(x) + βTn(g)(y)− Tn(g)(px+ qy)�� ≤

(|α|p

)n

ε(∥x∥, ∥y∥)

for every x, y ∈ K and n ≥ 1. It follows that

αf0(x) + βf0(y) = f0(px+ qy), x, y ∈ K.

Since f0(0) = 0, f0 is additive.For the end assume that f1 : K → Y is a solution of equation (3.1) such

that

∥f1(x)− g(x)∥ ≤ M∥x∥, x ∈ K,

Vol. 18 (2016) Selections of set-valued functions 139Selections of set-valued functions 7

for some positive M . Then

f1(0) = 0, αf1

(x

p

)= f1(x), x ∈ K

and

∥f1(x)− f0(x)∥ ≤(M +

ε(1, 0)

p− α

)∥x∥, x ∈ K.

It is easy to check that

∥f1(x)− f0(x)∥ ≤(|α|p

)n (M +

ε(1, 0)

p− α

)∥x∥, x ∈ K, n ∈ N.

Hence f1 = f0, which ends the proof. �

In the second part of this section, we apply Theorem 2.1 to obtain astability result for equation (3.1).

Theorem 3.2. Let K be a convex cone in a real normed space (X, ∥·∥), (Y, ∥·∥)a real Banach space and p, q > 0, α, β ∈ R fixed constants such that |α| < p.Consider η ≥ 0 and a function f : K → Y satisfying the inequality

∥αf(x) + βf(y)− f(px+ qy)∥ ≤ η (∥x∥+ ∥y∥), x, y ∈ K.

If there exists a constant λ > 0 such that

(1 + λα)∥x∥+ (1 + λβ)∥y∥ ≤ λ∥px+ qy∥, x, y ∈ K, (3.4)

then there exists a unique function f0 : K → Y fulfilling the functional equa-tion

αf0(x) + βf0(y) = f0(px+ qy), x, y ∈ K,

and such that

∥f(x)− f0(x)− f(0)∥ ≤ M∥x∥, x ∈ K,

for some M > 0. Moreover,

∥f(x)− f0(x)− f(0)∥ ≤ λη∥x∥, x ∈ K,

and f0 is an additive function.

Proof. Let g : K → Y be given by g(x) = f(x)−f(0), x ∈ K. As in the proofof Theorem 3.1, g(0) = 0 and

∥αg(x) + βg(y)− g(px+ qy)∥ ≤ η (∥x∥+ ∥y∥), x, y ∈ K.

In particular, for all x, y ∈ K,

αg(x) + βg(y) ∈ g(px+ qy) + η(∥x∥+ ∥y∥)B,

where B is the unit closed ball in Y . Define an s.v. function G : K → cl(Y )as follows:

G(x) = g(x) + λη∥x∥B, x ∈ K.

Then 0 ∈ G(0), diamG(x) = 2ηλ∥x∥, x ∈ K, and

αG(x) + βG(y) ⊂ G(px+ qy), x, y ∈ K.

140 A. Smajdor and J. Szczawinska JFPTA8 A. Smajdor and J. Szczawinska

Indeed, let x, y ∈ K be fixed. By condition (3.4),

αG(x) + βG(y) = αg(x) + βg(y) + η(λα∥x∥+ λβ ∥y∥

)B

⊂ g(px+ qy) + η (∥x∥+ ∥y∥)B + η(λα∥x∥+ λβ ∥y∥

)B

⊂ g(px+ qy) + ηλ∥px+ qy∥B= G(px+ qy).

By Theorem 2.1, there exists exactly one selection f0 : K → Y of thes.v. function G satisfying equation (3.1). Hence

∥f(x)− f0(x)− f(0)∥ ≤ λη∥x∥, x ∈ K.

Since f0(0) = 0, f0 must be additive. The proof that f0 is the only solutionof equation (3.1) such that

∥f(x)− f0(x)− f(0)∥ ≤ M∥x∥, x ∈ K,

for some M > 0 runs as before. �

Example. Let |α| < p and |β| < q. Every convex cone

K ⊂{(x1, . . . , xn) ∈ Rn : x1 ≥ 0 ∧ · · · ∧ xn ≥ 0 ∨ x1 ≤ 0 ∧ · · · ∧ xn ≤ 0

}

satisfies condition (3.4) with

λ ≥ max

{1

p− α,

1

q − β

}

and with the norm in Rn given by

∥(x1, . . . , xn)∥ = |x1|+ · · ·+ |xn|, (x1, . . . , xn) ∈ Rn.

4. The selection theorem

In this section we present a certain consequence of Theorem 2.1 for the caseof a compact-valued s.v. function F satisfying the inclusion (1.1) and suchthat the function x �→ diamF (x) maps bounded sets onto bounded ones.

Lemma 4.1. Let p, q > 0 and α, β ∈ R be fixed. Assume that X and Y are realvector spaces, K a convex cone in X and K a collection of nonempty subsetsof Y closed under intersections of chains. Let F : K → K be an s.v. func-tion fulfilling the inclusion (1.1) and 0 ∈ F (0). There exists a minimal s.v.function F0 : K → K such that

(i) F0(x) ⊂ F (x), x ∈ K,(ii) 0 ∈ F0(0),(iii) αF0(x) + βF0(y) ⊂ F0(px+ qy), x, y ∈ K.

Proof. Define the set F of all s.v. functions H : K → K such that

H(x) ⊂ F (x), x ∈ K,

0 ∈ H(0) and

αH(x) + βH(y) ⊂ H(px+ qy), x, y ∈ K.

Vol. 18 (2016) Selections of set-valued functions 141Selections of set-valued functions 9

The set F is nonempty, because F ∈ F , and partially ordered by the relation

H1 ≼ H2 ⇐⇒ H1(x) ⊂ H2(x), x ∈ K, H1, H2 ∈ F .

It is enough to prove that there is a minimal element in F . Let C be a chainin F . Put

H0(x) =∩

{H(x) : H ∈ C}, x ∈ K.

Since K is closed under intersections, H0(x) ∈ K, x ∈ K. Obviously, 0 ∈H0(0) and H0(x) ⊂ H(x), x ∈ K. Fix now x, y ∈ K. For every H ∈ C,

αH0(x) + βH0(y) ⊂ αH(x) + βH(y) ⊂ H(px+ qy),

hence

αH0(x) + βH0(y) ⊂ H0(px+ qy).

Therefore, the s.v. function H0 : K → K belongs to F and H0 ≼ H, H ∈ C.By the Kuratowski–Zorn lemma , there exists a minimal element F0 in theset F . �

Lemma 4.2. Let p, q > 0 and α, β ∈ R be fixed. Assume that X and Y arereal vector spaces, K a convex cone in X and K a family of nonempty subsetsof Y such that αK ⊂ K and λK ⊂ K, λ ∈ Q+. Consider an s.v. functionF : K → K. If F0 : K → K is a minimal s.v. function such that

(i) F0(x) ⊂ F (x), x ∈ K,(ii) 0 ∈ F0(0),(iii) αF0(x) + βF0(y) ⊂ F0(px+ qy), x, y ∈ K,

then F0 is superadditive and Q+-homogeneous; i.e.,

F0(x) + F0(y) ⊂ F0(x+ y), x, y ∈ K,

and F0(λx) = λF0(x), x ∈ K, λ ∈ Q+.

Proof. Putting y = 0 in (iii) and by (ii),

αF0

(x

p

)⊂ F0(x), x ∈ K.

Define

H(x) = αF0

(x

p

), x ∈ K.

Then H(x) ∈ K, H(x) ⊂ F0(x) ⊂ F (x), x ∈ K and 0 ∈ αF0(0) = H(0). Ob-serve that the s.v. function H : K → K satisfies the inclusion (iii). Indeed, letx, y ∈ K be fixed. Then

αH(x) + βH(y) = α

(αF0

(x

p

)+ βF0

(y

p

))

⊂ αF0

(px+ qy

p

)= H(px+ qy).

Thus, by the minimality of F0, F0(x) = H(x), x ∈ K; i.e.,

F0(x) = αF0

(x

p

), x ∈ K.

142 A. Smajdor and J. Szczawinska JFPTA10 A. Smajdor and J. Szczawinska

Similarly we get

F0(y) = βF0

(y

q

), y ∈ K.

Hence, by (iii),

F0(x) + F0(y) = αF0

(x

p

)+ βF0

(y

q

)⊂ F0(x+ y)

for every x, y ∈ K; i.e., F0 is superadditive. In particular,

nF0

(xn

)⊂ F0(x), x ∈ K, n ∈ N.

Fix now n ∈ N and consider the s.v. function

K ∋ x �→ nF0

(xn

)∈ K.

We have

nF0

(xn

)⊂ F0(x) ⊂ F (x), x ∈ K,

0 ∈ nF0(0) and

αnF0

(xn

)+ βnF0

( y

n

)⊂ nF0

(px+ qy

n

), x, y ∈ K.

Therefore,

nF0

(xn

)= F0(x), x ∈ K,

by the minimality of F0. Consequently,

F0

(λx

)= λF0(x), x ∈ K, λ ∈ Q+. �

Remark 4.3. Let (X, ∥·∥) be a real normed space and K ⊂ X a cone. Assumethat f : K → R is Q+-homogeneous. The function f maps bounded subsetsof K onto bounded sets if and only if there exists a positive constant M suchthat

|f(x)| ≤ M∥x∥, x ∈ K.

Proof. Assume that f : K → R is Q+-homogeneous (in particular, f(0) = 0)and maps bounded subsets of K onto bounded sets. Let

M = sup{|f(x)| : x ∈ K ∧ ∥x∥ ≤ 1}.Fix x ∈ K \ {0}. There exists a decreasing sequence (λn)n such that

limn→∞

λn = 1 and λn∥x∥ ∈ Q+, n ∈ N.

Since ����x

λn∥x∥

���� =1

λn< 1, n ∈ N,

we have ����f(

x

λn∥x∥

)���� ≤ M, n ∈ N.

Consequently |f(x)| ≤ M∥x∥.The opposite implication is obvious. �

Vol. 18 (2016) Selections of set-valued functions 143Selections of set-valued functions 11

Theorem 4.4. Let p, q > 0, α, β ∈ R and |α| < p. Assume that (X, ∥ · ∥) and(Y, ∥ · ∥) are real normed spaces, K is a convex cone in X and F : K → c(Y )an s.v. function such that 0 ∈ F (0) and

αF (x) + βF (y) ⊂ F (px+ qy), x, y ∈ K.

If K ∋ x �→ diamF (x) ∈ R maps bounded sets onto bounded sets, then thereexists a unique selection f : K → Y of F fulfilling the equation

αf(x) + βf(y) = f(px+ qy), x, y ∈ K.

The selection f is additive.

Proof. The family c(Y ) is closed under intersections of chains and

λ c(Y ) ⊂ c(Y ), λ ∈ R.

By Lemma 4.1, there exists a minimal s.v. function F0 : K → c(Y ) such thatF0(x) ⊂ F (x), x ∈ K, 0 ∈ F0(0) and

αF0(x) + βF0(y) ⊂ F0(px+ qy), x, y ∈ K.

Lemma 4.2 shows now that F0 is superadditive and Q+-homogeneous. SincediamF0(x) ≤ diamF (x), x ∈ K, the function K ∋ x �→ diamF0(x) ∈ Rmaps bounded sets onto bounded sets and it is Q+-homogeneous. Hence, byRemark 4.3, there exists a positive constant M such that

diamF0(x) ≤ M∥x∥, x ∈ K.

Consequently, by Theorem 2.1, the s.v. function F0 admits a unique selectionf0 : K → Y such that

αf0(x) + βf0(y) = f0(px+ qy), x, y ∈ K.

Since f0(0) = 0 and f0 satisfies the above equation, f0 must be additive.Assume that f1, f2 : K → Y are selections of F such that

αfi(x) + βfi(y) = fi(px+ qy), x, y ∈ K, i = 1, 2.

They are additive and

fi(x) = αfi

(x

p

), x ∈ K, i = 1, 2.

Hence, for every n ∈ N,

fi(x) = αnfi

(x

pn

), x ∈ K, i = 1, 2.

Fix x ∈ K arbitrary. Since K ∋ z �→ diamF (z) ∈ R maps bounded sets ontobounded sets, there exists a constant M such that

diamF (y) ≤ M, ∥y∥ ≤ 2∥x∥.

144 A. Smajdor and J. Szczawinska JFPTA12 A. Smajdor and J. Szczawinska

For every n ∈ N there exists a number qn ∈ (1, 2) such that qnpn ∈ Q+.

Obviously, ∥qnx∥ ≤ 2∥x∥, n ∈ N. Hence, for every n ∈ N,

∥f1(x)− f2(x)∥ = |α|n����f1

(x

pn

)− f2

(x

pn

)����

= |α|n����f1

(qnx

qnpn

)− f2

(qnx

qnpn

)����

=1

qn

(|α|p

)n ��f1(qnx)− f2(qnx)��

≤(|α|p

)n

diamF (qnx) ≤(|α|p

)n

M.

Letting n → +∞ ends the proof. �

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Vol. 18 (2016) Selections of set-valued functions 145Selections of set-valued functions 13

[14] A. Smajdor and J. Szczawinska, Selections of generalized convex set-valuedfunctions with bounded diameter. Fixed Point Theory, to appear.

Andrzej SmajdorInstitute of MathematicsPedagogical UniversityPodchorazych 2PL-30-084 KrakowPolande-mail: asmajdor@up.krakow.pl

Joanna SzczawinskaInstitute of MathematicsPedagogical UniversityPodchorazych 2PL-30-084 KrakowPolande-mail: jszczaw@up.krakow.pl

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