Open Access

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

Journal of Inequalities and Applications20122012:258

https://doi.org/10.1186/1029-242X-2012-258

Received: 5 March 2012

Accepted: 23 October 2012

Published: 5 November 2012

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.

Keywords

multifunctionfunctional equationiterationK-convexupper semi-continuity

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 ) , x S ,
(1.1)
where S is a subset of a linear space over , F : S S is a given function, λ i s ( i = 1 , , n ) are real constants, f : S S 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 x S , 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 ) , x I : = [ a , b ] ,
(1.2)
where n 2 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 x I . 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 n 3 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 ) , x I ,
(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 K Y be a convex cone, i.e., K + K K and λ K K for all λ 0 . Let Ω X be a convex set. A multifunction T : X Y is said to be K-convex on Ω if
λ T ( x ) + ( 1 λ ) T ( y ) T ( λ x + ( 1 λ ) y ) + K , x , y Ω , 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 : I cc ( 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 ) ) min F ( 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 ) ) max F ( 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 , D cc ( I ) and for an arbitrary real λ, the following properties hold:
  1. (a)

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

     
  2. (b)

    h ( λ A , λ B ) = | 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 : I cc ( I ) is increasing (resp. strictly increasing) if max F ( x 1 ) min F ( x 2 ) (resp. max F ( x 1 ) < min F ( x 2 ) ) for all x 1 , x 2 I with x 1 < x 2 . A multifunction F : I cc ( I ) is upper semi-continuous (USC) at a point x 0 I if for every open set v R 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 y λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ,
y min 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 ) ) = λ min F 1 F 2 ( x 1 ) + ( 1 λ ) 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 y λ F 2 ( x 1 ) + ( 1 λ ) F 2 ( x 2 ) ,
y max 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 max F ( b ) = b for F USIC + ( I , m , M ) ( min F ( 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 , x I . 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
λ F ( x 1 ) + ( 1 λ ) F ( x 2 ) λ F n 0 ( x 1 ) + ( 1 λ ) 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 ) ) min F ( 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 ) ) max F ( 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
L F ( x ) = 1 λ 1 ( G ( x ) i = 2 n λ i F i ( x ) ) , x I .
(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 ) = max A + max B , we have
Hence, L F ( x ) is R -convex on I. Obviously, L F ( x ) is strictly increasing and L F ( x ) > x for x int I . Similar to the proof of Theorem 4.1 in [29], by Lemma 3.4 and condition (4.1), L F ( x ) USIC ( I , m , M ) . Thus, we have proved that L F ( 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 ) ) , x I .

This completes the proof. □

We note the fact that A + B C if the sets A, B, C satisfy A = C B . 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 ) , x I .
(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 ) , x I : = [ 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.

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University
(2)
College of Mathematics and Information Science, Neijiang Normal University

References

  1. Kuczma M, Choczewski B, Ger R Encyclopedia Math. Appl. 32. In Iterative Functional Equations. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
  2. Targonski G: Topics in Iteration Theory. Vandenhoeck & Ruprecht, Göttingen; 1981.MATHGoogle Scholar
  3. Baron K, Jarczyk W: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 2001, 61: 1–48. 10.1007/s000100050159MathSciNetView ArticleMATHGoogle Scholar
  4. Zhang J, Yang L, Zhang W: Some advances on functional equations. Adv. Math. (China) 1995, 24: 385–405.MathSciNetMATHGoogle Scholar
  5. Dhombres JG: Itération linéaire d’ordre deux. Publ. Math. (Debr.) 1977, 24: 177–187.MathSciNetMATHGoogle Scholar
  6. Jarczyk W: On an equation of linear iteration. Aequ. Math. 1996, 51: 303–310. 10.1007/BF01833285MathSciNetView ArticleMATHGoogle Scholar
  7. Matkowski J, Zhang W: On linear dependence of iterates. J. Appl. Anal. 2000, 6: 149–157.MathSciNetView ArticleMATHGoogle Scholar
  8. Mukherjea A, Ratti JS: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 1983, 7: 899–908. 10.1016/0362-546X(83)90065-2MathSciNetView ArticleMATHGoogle Scholar
  9. Mukherjea A, Ratti JS: On a functional equation involving iterates of a bijection on the unit interval II. Nonlinear Anal. 1998, 31: 459–464. 10.1016/S0362-546X(96)00322-7MathSciNetView ArticleMATHGoogle Scholar
  10. Tabor J, Tabor J: On a linear iterative equation. Results Math. 1995, 27: 412–421.MathSciNetView ArticleMATHGoogle Scholar
  11. Yang D, Zhang W: Characteristic solutions of polynomial-like iterative equations. Aequ. Math. 2004, 67: 80–105. 10.1007/s00010-003-2708-4View ArticleMathSciNetMATHGoogle Scholar
  12. Malenica M:On the solutions of the functional equation ϕ ( x ) + ϕ 2 ( x ) = F ( x ) . Mat. Vesn. 1982, 6: 301–305.MathSciNetMATHGoogle Scholar
  13. Zhao L:A theorem concerning the existence and uniqueness of solutions of the functional equation λ 1 f ( x ) + λ 2 f 2 ( x ) = F ( x ) . J. Univ. Sci. Tech. 1983, 32: 21–27. in ChineseGoogle Scholar
  14. Zhang W:Discussion on the iterated equation i = 1 n λ i f i ( x ) = F ( x ) . Chin. Sci. Bull. 1987, 32: 1444–1451.MATHGoogle Scholar
  15. Mai J, Liu X:Existence, uniqueness and stability of C m solutions of iterative functional equations. Sci. China Ser. A 2000, 43: 897–913. 10.1007/BF02879796MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang W:Discussion on the differentiable solutions of the iterated equation i = 1 n λ i f i ( x ) = F ( x ) . Nonlinear Anal. 1990, 15: 387–398. 10.1016/0362-546X(90)90147-9MathSciNetView ArticleMATHGoogle Scholar
  17. Si J:Existence of locally analytic solutions of the iterated equation i = 1 n λ i f i ( x ) = F ( x ) . Acta Math. Sin. 1994, 37: 590–600. in ChineseMathSciNetMATHGoogle Scholar
  18. Trif T: Convex solutions to polynomial-like iterative equations on open intervals. Aequ. Math. 2010, 79: 315–325. 10.1007/s00010-010-0020-7MathSciNetView ArticleMATHGoogle Scholar
  19. Xu B, Zhang W: Decreasing solutions and convex solutions of the polynomial-like iterative equation. J. Math. Anal. Appl. 2007, 329: 483–497. 10.1016/j.jmaa.2006.06.087MathSciNetView ArticleMATHGoogle Scholar
  20. Zhang W, Nikodem K, Xu B: Convex solutions of polynomial-like iterative equations. J. Math. Anal. Appl. 2006, 315: 29–40. 10.1016/j.jmaa.2005.10.006MathSciNetView ArticleMATHGoogle Scholar
  21. Kulczycki M, Tabor J: Iterative functional equations in the class of Lipschitz functions. Aequ. Math. 2002, 64: 24–33. 10.1007/s00010-002-8028-2MathSciNetView ArticleMATHGoogle Scholar
  22. Tabor J, Żoldak M: Iterative equations in Banach spaces. J. Math. Anal. Appl. 2004, 299: 651–662. 10.1016/j.jmaa.2004.06.011MathSciNetView ArticleMATHGoogle Scholar
  23. Zhang W: Solutions of equivariance for a polynomial-like iterative equation. Proc. R. Soc. Edinb. A 2000, 130: 1153–1163. 10.1017/S0308210500000615View ArticleMathSciNetMATHGoogle Scholar
  24. Johansson KH, Rantzer A, Aström KJ: Fast switches in relay feedback systems. Automatica 1999, 35: 539–552. 10.1016/S0005-1098(98)00160-5View ArticleMATHGoogle Scholar
  25. Choi C, Nam D: Interpolation for partly hidden diffusion processes. Stoch. Process. Appl. 2004, 113: 199–216. 10.1016/j.spa.2004.03.014MathSciNetView ArticleMATHGoogle Scholar
  26. Lawry J: A framework for linguistic modelling. Artif. Intell. 2004, 155: 1–39. 10.1016/j.artint.2003.10.001MathSciNetView ArticleMATHGoogle Scholar
  27. Starr RM: General Equilibrium Theory. Cambridge University Press, Cambridge; 1997.View ArticleMATHGoogle Scholar
  28. Nikodem K, Zhang W: On a multivalued iterative equation. Publ. Math. 2004, 64: 427–435.MathSciNetMATHGoogle Scholar
  29. Xu B, Nikodem K, Zhang W: On a multivalued iterative equation of order n . J. Convex Anal. 2011, 18: 673–686.MathSciNetMATHGoogle Scholar
  30. Borwein J: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 1977, 13: 183–199. 10.1007/BF01584336View ArticleMathSciNetMATHGoogle Scholar
  31. Kuroiwa D, Tanaka T, Ha TXD: On cone convexity of set-valued maps. Nonlinear Anal., Theory Methods Appl. 1997, 30: 1487–1496. 10.1016/S0362-546X(97)00213-7MathSciNetView ArticleMATHGoogle Scholar
  32. Aubin JP, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990.MATHGoogle Scholar
  33. Radström H: An embedding theorem for space of convex sets. Proc. Am. Math. Soc. 1952, 3: 165–169.View ArticleGoogle Scholar
  34. Smajdor W: Local set-valued solutions of the Jensen and Pexider functional equations. Publ. Math. 1993, 43: 255–263.MathSciNetMATHGoogle Scholar

Copyright

© Gong; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.