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Convex solutions of the multi-valued iterative equation of order n

Abstract

A multi-valued iterative functional equation of order n is considered. A result on the existence and uniqueness of K-convex solutions in some class of multifunctions is presented.

MSC:39B12, 37E05, 54C60.

1 Introduction

As indicated in the books [1, 2] and the surveys [3, 4], the polynomial-like iterative equation

λ 1 f(x)+ λ 2 f 2 (x)++ λ n f n (x)=F(x),xS,
(1.1)

where S is a subset of a linear space over , F:SS is a given function, λ i s (i=1,,n) are real constants, f:SS is the unknown function, and f i is the i th iterate of f, i.e., f i (x)=f( f i 1 (x)) and f 0 (x)=x for all xS, is one of the important forms of a functional equation since the problem of iterative roots and the problem of invariant curves can be reduced to the kind of equations. Many works have been contributed to studying single-valued solutions for Eq. (1.1); for example, in [511] for the case of linear F, [12, 13] for n=2, [14] for general n, [15, 16] for smoothness, [17] for analyticity, [1820] for convexity, [2123] in high-dimensional spaces. However, a multifunction (called multi-valued function or set-valued map sometimes) is an important class of mappings often used in control theory [24], stochastics [25], artificial intelligence [26], and economics [27]. Hence, it gets more interesting to study multi-valued solutions for Eq. (1.1), i.e., the equation

λ 1 F(x)+ λ 2 F 2 (x)++ λ n F n (x)=G(x),xI:=[a,b],
(1.2)

where n2 is an integer, λ i s (i=1,,n) are real constants, G is a given multifunction, and F is an unknown multifunction. Here the i th iterate F i of the multifunction F is defined recursively as

F i (x):= { F ( y ) : y F i 1 ( x ) }

and F 0 (x):{x} for all xI. In 2004, Nikodem and Zhang [28] discussed Eq. (1.2) for n=2 with an increasing upper semi-continuous (USC) multifunction G on I=[a,b] and proved the existence and uniqueness of USC solutions under the assumption that G has fixed points a and b and λ 1 , λ 2 are both constants such that λ 1 > λ 2 0 and λ 1 + λ 2 =1. As pointed out in [29], the generalization to USC multifunctions for Eq. (1.1) is rather difficult even if n=2. Hence, discussing Eq. (1.2) for n3 evokes great interest, but the greatest difficulty is that the multifunction has no Lipschitz condition. In 2011, this difficulty was overcome by introducing the class of unblended multifunctions, the existence of USC multi-valued solutions for a modified form of the equation

λ 1 F(x)=G(x) λ 2 F 2 (x) λ n F n (x),xI,
(1.3)

was proved in [29]. K-convex multifunctions, which are generalization of vector-valued convex functions, have wide applications in optimization (cf. [30]) and play an important role in various questions of convex analysis (cf. [31]). However, up to now, there are no results on convexity of multi-valued solutions for the iterative equation (1.2). In this note, we study the convexity of multi-valued solutions for Eq. (1.2). We prove the existence and uniqueness of K-convex solutions in some class of multifunctions for Eq. (1.3).

2 K-convex multifunctions

As in [30], let X and Y be linear spaces and KY be a convex cone, i.e., K+KK and λKK for all λ0. Let ΩX be a convex set. A multifunction T:XY is said to be K-convex on Ω if

λT(x)+(1λ)T(y)T ( λ x + ( 1 λ ) y ) +K,x,yΩ,λ[0,1].

A convex multifunction [32] may be stated as θ-convex and the convexity of a real-valued function may be stated as R + -convex, and concavity as R -convex, where R + :=[0,+) and R :=(,0]. Let F(I) be the set of all multifunctions F:Icc(I), where cc(I) denotes the family of all nonempty closed subintervals of I.

Considering R + -convex multifunctions and R -convex multifunctions, the following lemmas are obvious.

Lemma 2.1 Let F(x)F(I). Then the multifunction F(x) is R + -convex on I if and only if

min ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) minF ( λ x 1 + ( 1 λ ) x 2 ) , x 1 , x 2 I,λ[0,1].
(2.1)

Lemma 2.2 Let F(x)F(I). Then the multifunction F(x) is R -convex on I if and only if

max ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) maxF ( λ x 1 + ( 1 λ ) x 2 ) , x 1 , x 2 I,λ[0,1].
(2.2)

3 Some lemmas

In order to prove our main results, we give the following useful property (cf. [33, 34]).

Lemma 3.1 For A,B,C,Dcc(I) and for an arbitrary real λ, the following properties hold:

  1. (a)

    h(A+C,B+C)=h(A,B),

  2. (b)

    h(λA,λB)=|λ|h(A,B),

  3. (c)

    h(A+C,B+D)h(A,B)+h(C,D),

where

h(A,B)=max { sup { d ( x , B ) : x A } , sup { d ( y , A ) : y B } } .

As defined in [[32], Definition 3.5.1], a multifunction F:Icc(I) is increasing (resp. strictly increasing) if maxF( x 1 )minF( x 2 ) (resp. maxF( x 1 )<minF( x 2 )) for all x 1 , x 2 I with x 1 < x 2 . A multifunction F:Icc(I) is upper semi-continuous (USC) at a point x 0 I if for every open set vR with F( x 0 )V, there exists a neighborhood U x 0 of x 0 such that F(x)V for every x U x 0 . F is USC on I if it is USC at every point in I. For convenience, let

USIC + (I):= { F F ( I ) : F  is USC, strictly increasing and  R + -convex on  I }

and

USIC (I):= { F F ( I ) : F  is USC, strictly increasing and  R -convex on  I } .

Remark 3.1 If F USIC + (I) (resp. USIC (I)), I=[a,b], then F must be single-valued on [a,b) (resp. (a,b]).

Lemma 3.2 F 1 F 2 USIC + (I) (resp. USIC (I)) for F 1 , F 2 USIC + (I) (resp. USIC (I)).

Proof By Lemma 2.2 in [29], we only need to prove that F 1 F 2 is R + -convex on I (resp. R -convex on I). We first prove that F 1 F 2 is R + -convex on I for F 1 , F 2 USIC + (I). By Lemma 2.1, the fact that F 2 is R + -convex on I implies that

min ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) min F 2 ( λ x 1 + ( 1 λ ) x 2 ) , x 1 , x 2 I,λ[0,1].

Hence, for all yλ F 2 ( x 1 )+(1λ) F 2 ( x 2 ),

ymin F 2 ( λ x 1 + ( 1 λ ) x 2 )

holds. Note that F 1 is strictly increasing. Consequently,

min F 1 ( y ) min F 1 ( min F 2 ( λ x 1 + ( 1 λ ) x 2 ) ) = min F 1 F 2 ( λ x 1 + ( 1 λ ) x 2 ) .

So

min F 1 ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) = min { F 1 ( y ) : y λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) } min F 1 F 2 ( λ x 1 + ( 1 λ ) x 2 ) .
(3.1)

By

min ( λ F 1 F 2 ( x 1 ) + ( 1 λ ) F 1 F 2 ( x 2 ) ) =λmin F 1 F 2 ( x 1 )+(1λ)min F 1 F 2 ( x 2 ),

we have

min ( λ F 1 F 2 ( x 1 ) + ( 1 λ ) F 1 F 2 ( x 2 ) ) min F 1 ( λ min F 2 ( x 1 ) + ( 1 λ ) min F 2 ( x 2 ) ) = min F 1 ( min ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) ) = min F 1 ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) )

because F 1 is R + -convex. Hence, by (3.1)

F 1 F 2 USIC + (I) is proved.

Next, we prove F 1 F 2 is R -convex on I for F 1 , F 2 USIC (I). By Lemma 2.2, the fact that F 2 is R -convex on I implies that

max ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) max F 2 ( λ x 1 + ( 1 λ ) x 2 ) , x 1 , x 2 I,λ[0,1].

Hence, for all yλ F 2 ( x 1 )+(1λ) F 2 ( x 2 ),

ymax F 2 ( λ x 1 + ( 1 λ ) x 2 )

holds. Note that F 1 is strictly increasing. Consequently,

max F 1 ( y ) max F 1 ( max F 2 ( λ x 1 + ( 1 λ ) x 2 ) ) = max F 1 F 2 ( λ x 1 + ( 1 λ ) x 2 ) .

So

max F 1 ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) = max { F 1 ( y ) : y λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) } max F 1 F 2 ( λ x 1 + ( 1 λ ) x 2 ) .
(3.2)

By

it follows that

max ( λ F 1 F 2 ( x 1 ) + ( 1 λ ) F 1 F 2 ( x 2 ) ) max F 1 ( λ max F 2 ( x 1 ) + ( 1 λ ) max F 2 ( x 2 ) ) = max F 1 ( max ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ) ) = max F 1 ( λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) )

because F 1 is R -convex. Hence, by (3.2)

This completes the proof of F 1 F 2 USIC (I). □

Define

where I=[a,b] and M>m>0.

Remark 3.2 The condition maxF(b)=b for F USIC + (I,m,M) (minF(a)=a for F USIC (I,m,M)) guarantees that the iterations F n , n=2,3, , are also multifunctions.

Lemma 3.3 USIC + (I,m,M) and USIC (I,m,M) are complete metric spaces equipped with the distance

D( F 1 , F 2 ):=sup { h ( F 1 ( x ) , F 2 ( x ) ) : x I } .

Proof By Lemma 3.1 in [29], we only need to prove that if { F n } USIC σ (I,m,M) such that lim n F n =F(x) in USI(I,m,M), i.e.,

lim n D( F n ,F)=0,
(3.3)

then F(x) is R σ -convex on I, where σ=+ or σ=. We first prove the case of USIC + (I,m,M). By (3.3), we have lim n h( F n (x),F(x))=0, xI. Hence,

lim n h ( F n ( λ x 1 + ( 1 λ ) x 2 ) , F ( λ x 1 + ( 1 λ ) x 2 ) ) =0, x 1 , x 2 I,λ[0,1].
(3.4)

Note that by Lemma 3.1,

lim n h ( λ F n ( x 1 ) , λ F ( x 1 ) ) =0, x 1 I,λ[0,1]

and

lim n h ( ( 1 λ ) F n ( x 2 ) , ( 1 λ ) F ( x 2 ) ) =0, x 2 I,λ[0,1].

Hence,

(3.5)

By (3.4) and (3.5), we have for every ε>0, there exists n 0 N such that

F n 0 ( λ x 1 + ( 1 λ ) x 2 ) F ( λ x 1 + ( 1 λ ) x 2 ) + ( ε 2 , ε 2 )
(3.6)

and

λF( x 1 )+(1λ)F( x 2 )λ F n 0 ( x 1 )+(1λ) F n 0 ( x 2 )+ ( ε 2 , ε 2 ) ,
(3.7)

x 1 , x 2 I, λ[0,1]. Consequently,

min ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) min ( λ F n 0 ( x 1 ) + ( 1 λ ) F n 0 ( x 2 ) ) ε 2 min F n 0 ( λ x 1 + ( 1 λ ) x 2 ) ε 2 min F ( λ x 1 + ( 1 λ ) x 2 ) ε

because F n 0 (x) is R + -convex on I. Hence,

min ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) minF ( λ x 1 + ( 1 λ ) x 2 ) ,

which shows that F(x) is R + -convex on I.

Next we prove the case of σ=. By (3.6) and (3.7), we have for every ε>0,

max ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) max ( λ F n 0 ( x 1 ) + ( 1 λ ) F n 0 ( x 2 ) ) + ε 2 max F n 0 ( λ x 1 + ( 1 λ ) x 2 ) + ε 2 max F ( λ x 1 + ( 1 λ ) x 2 ) + ε

because F n 0 (x) is R -convex on I. Hence,

max ( λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) ) maxF ( λ x 1 + ( 1 λ ) x 2 ) ,

which shows that F(x) is R -convex on I. The proof is completed. □

Define

USIC + (I,m,M) is a closed subset of USIC + (I,m,M). USIC (I,m,M) is a closed subset of USIC (I,m,M).

By Lemma 3.2, one can prove the following result.

Lemma 3.4 F i USIC + (I, m i , M i ) (resp. USIC (I, m i , M i )) if F USIC + (I,m,M) (resp. USIC (I,m,M)).

Lemma 3.5 If F 1 , F 2 USIC + (I,m,M) (resp. USIC (I,m,M)), then

D ( F 1 i , F 2 i ) ( j = 0 i 1 M j ) D( F 1 , F 2 ).

The proof of Lemma 3.5 is similar to that of Lemma 3.3 in [29]. We omit it here.

4 Convex solutions

Theorem 4.1 Suppose that λ 1 >0, λ i 0 (i=2,,n) and i = 1 n λ i =1 and G USIC (I, m 0 , M 0 ) with M 0 > m 0 >0. Then for arbitrary constants M>m>0 satisfying

m m 0 + i = 2 n | λ i | m i λ 1 ,M M 0 + i = 2 n | λ i | M i λ 1 ,
(4.1)

Eq. (1.3) has a unique solution F USIC (I,m,M) if

d:= 1 λ 1 i = 2 n | λ i | j = 0 i 1 M j <1.
(4.2)

Proof Define the mapping L: USIC (I,m,M)F(I) by

LF(x)= 1 λ 1 ( G ( x ) i = 2 n λ i F i ( x ) ) ,xI.
(4.3)

By Lemma 3.2, F i (x), i=2,,n are strictly increasing R -convex on I because F(x) is strictly increasing R -convex. Since G(x) is R -convex on I and max(A+B)=maxA+maxB, we have

Hence, LF(x) is R -convex on I. Obviously, LF(x) is strictly increasing and LF(x)>x for xintI. Similar to the proof of Theorem 4.1 in [29], by Lemma 3.4 and condition (4.1), LF(x) USIC (I,m,M). Thus, we have proved that LF(x) is a self-mapping on USIC (I,m,M). By Lemma 3.5 and condition (4.2), L is a contraction map. By Lemma 3.3, USIC (I,m,M) is a complete metric space. Using Banach’s fixed point principle, L has a unique fixed point F in USIC (I,m,M), i.e.,

F(x)= 1 λ 1 ( G ( x ) i = 2 n λ i F i ( x ) ) ,xI.

This completes the proof. □

We note the fact that A+BC if the sets A, B, C satisfy A=CB. Hence, every solution F of Eq. (1.3) satisfies

λ 1 F(x)+ λ 2 F 2 (x)++ λ n F n (x)G(x),xI.
(4.4)

We have the following result.

Corollary 4.1 Under the same conditions as in Theorem  4.1, there exists a multifunction F USIC (I,m,M) such that (4.4) holds.

For multifunctions in the other class USIC + (I,m,M), we have a similar result to Theorem 4.1. It can be proved similarly.

Theorem 4.2 Suppose that λ 1 >0, λ i 0 (i=2,,n) and i = 1 n λ i =1 and G USIC + (I, m 0 , M 0 ) with M 0 > m 0 >0. Then for arbitrary constants M>m>0 satisfying (4.1), Eq. (1.3) has a unique solution F USIC + (I,m,M) if condition (4.2) holds.

Corollary 4.2 Under the same conditions as in Theorem  4.2, there exists a multifunction F USIC + (I,m,M) such that (4.4) holds.

Remark 4.1 Although the assumption F USIC (I) (or USIC + (I)) implies that F is single-valued on [a,b) (or (a,b]), but Eq. (1.3) cannot be considered on the interval [a,b) (or (a,b]) as a single-valued case and the point b (or a) as a multi-valued case, respectively, because there is no meaning at the point b (or a).

Remark 4.2 By Remark 3.1, there is no strictly increasing R + -convex multifunction in USIC + (I,m,M). The same applies to the case of USIC (I,m,M). Consequently, Eq. (1.3) has no solution in USIC + (I,m,M) (resp. USIC (I,m,M)).

Remark 4.3 By Theorem 4.1 and Theorem 4.2, we actually only prove the existence and uniqueness of K-convex (K= R + and K= R , i.e., K is not a nontrivial convex cone) multi-valued solutions for Eq. (1.3). In fact, there is no convex multi-valued (i.e., {0}-convex multi-valued) solutions for Eq. (1.3) in the multifunction class USI(I). Since F(x) is a convex multi-valued function on I if and only if

min λ F ( x ) + min ( 1 λ ) F ( y ) min F ( λ x + ( 1 λ ) y )  and max λ F ( x ) + max ( 1 λ ) F ( y ) max F ( λ x + ( 1 λ ) y ) , x , y I , λ [ 0 , 1 ] .
(4.5)

Hence, if Eq. (1.3) has a convex multi-valued solution F in USI(I), then F must be strictly increasing on I, which is contradictory to (4.5).

Remark 4.4 We point out that we actually only have proved a special class of K-convex solutions, i.e., strictly increasing K-convex solutions of Eq. (1.3). It is very difficult to discuss K-convex solutions of Eq. (1.3) which are not strictly increasing because the method in [29] cannot be used. Discussing non-strictly-increasing K-convex solutions of Eq. (1.3) will be the subject of our next work.

5 Examples

We give an example to illustrate the applications of Theorem 4.1. Consider the equation

5 4 F(x)=G(x)+ 1 4 F 3 (x),xI:=[0,1],
(5.1)

where n=3, λ 1 = 5 4 , λ 2 =0, λ 3 = 1 4 and

G(x)={ [ 0 , 2 3 ] , x = 0 , 5 x + 4 3 , x ( 0 , 1 ] .
(5.2)

Clearly, G USIC (I, m 0 , M 0 ), where

m 0 = 5 18 , M 0 = 5 12 .

Let m= 1 5 and M=1. It is easy to check that both (4.1) and (4.2) hold. Thus, by Theorem 4.1, Eq. (5.1) has a unique solution F USIC (I,m,M).

Remark 5.1 Example (5.1) cannot be solved by known single-valued results.

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Acknowledgements

The author is most grateful to the Editor for the careful reading of the manuscript and anonymous referees for valuable suggestions that helped in significantly improving an earlier version of this paper. This work was supported by Key Project of Sichuan Provincial Department of Education (12ZA086) (China).

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Gong, X. Convex solutions of the multi-valued iterative equation of order n. J Inequal Appl 2012, 258 (2012). https://doi.org/10.1186/1029-242X-2012-258

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Keywords

  • multifunction
  • functional equation
  • iteration
  • K-convex
  • upper semi-continuity