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Convex solutions of the multivalued iterative equation of order n
Journal of Inequalities and Applications volume 2012, Article number: 258 (2012)
Abstract
A multivalued iterative functional equation of order n is considered. A result on the existence and uniqueness of Kconvex 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 polynomiallike iterative equation
where S is a subset of a linear space over ℝ, F:S\to S is a given function, {\lambda}_{i}s (i=1,\dots ,n) are real constants, f:S\to S is the unknown function, and {f}^{i} is the i th iterate of f, i.e., {f}^{i}(x)=f({f}^{i1}(x)) and {f}^{0}(x)=x for all x\in 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 singlevalued solutions for Eq. (1.1); for example, in [5–11] for the case of linear F, [12, 13] for n=2, [14] for general n, [15, 16] for smoothness, [17] for analyticity, [18–20] for convexity, [21–23] in highdimensional spaces. However, a multifunction (called multivalued function or setvalued 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 multivalued solutions for Eq. (1.1), i.e., the equation
where n\ge 2 is an integer, {\lambda}_{i}s (i=1,\dots ,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
and {F}^{0}(x):\equiv \{x\} for all x\in I. In 2004, Nikodem and Zhang [28] discussed Eq. (1.2) for n=2 with an increasing upper semicontinuous (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 {\lambda}_{1}, {\lambda}_{2} are both constants such that {\lambda}_{1}>{\lambda}_{2}\ge 0 and {\lambda}_{1}+{\lambda}_{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\ge 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 multivalued solutions for a modified form of the equation
was proved in [29]. Kconvex multifunctions, which are generalization of vectorvalued 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 multivalued solutions for the iterative equation (1.2). In this note, we study the convexity of multivalued solutions for Eq. (1.2). We prove the existence and uniqueness of Kconvex solutions in some class of multifunctions for Eq. (1.3).
2 Kconvex multifunctions
As in [30], let X and Y be linear spaces and K\subset Y be a convex cone, i.e., K+K\subset K and \lambda K\subset K for all \lambda \ge 0. Let \mathrm{\Omega}\subset X be a convex set. A multifunction T:X\to Y is said to be Kconvex on Ω if
A convex multifunction [32] may be stated as θconvex and the convexity of a realvalued function may be stated as {\mathbb{R}}^{+}convex, and concavity as {\mathbb{R}}^{}convex, where {\mathbb{R}}^{+}:=[0,+\mathrm{\infty}) and {\mathbb{R}}^{}:=(\mathrm{\infty},0]. Let \mathcal{F}(I) be the set of all multifunctions F:I\to \mathit{cc}(I), where \mathit{cc}(I) denotes the family of all nonempty closed subintervals of I.
Considering {\mathbb{R}}^{+}convex multifunctions and {\mathbb{R}}^{}convex multifunctions, the following lemmas are obvious.
Lemma 2.1 Let F(x)\in \mathcal{F}(I). Then the multifunction F(x) is {\mathbb{R}}^{+}convex on I if and only if
Lemma 2.2 Let F(x)\in \mathcal{F}(I). Then the multifunction F(x) is {\mathbb{R}}^{}convex on I if and only if
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\in \mathit{cc}(I) and for an arbitrary real λ, the following properties hold:

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

(b)
h(\lambda A,\lambda B)=\lambda h(A,B),

(c)
h(A+C,B+D)\le h(A,B)+h(C,D),
where
As defined in [[32], Definition 3.5.1], a multifunction F:I\to \mathit{cc}(I) is increasing (resp. strictly increasing) if maxF({x}_{1})\le minF({x}_{2}) (resp. maxF({x}_{1})<minF({x}_{2})) for all {x}_{1},{x}_{2}\in I with {x}_{1}<{x}_{2}. A multifunction F:I\to \mathit{cc}(I) is upper semicontinuous (USC) at a point {x}_{0}\in I if for every open set v\subset \mathbb{R} with F({x}_{0})\subset V, there exists a neighborhood {U}_{{x}_{0}} of {x}_{0} such that F(x)\subset V for every x\in {U}_{{x}_{0}}. F is USC on I if it is USC at every point in I. For convenience, let
and
Remark 3.1 If F\in {USIC}^{+}(I) (resp. {USIC}^{}(I)), I=[a,b], then F must be singlevalued on [a,b) (resp. (a,b]).
Lemma 3.2 {F}_{1}\circ {F}_{2}\in {USIC}^{+}(I) (resp. {USIC}^{}(I)) for {F}_{1},{F}_{2}\in {USIC}^{+}(I) (resp. {USIC}^{}(I)).
Proof By Lemma 2.2 in [29], we only need to prove that {F}_{1}\circ {F}_{2} is {\mathbb{R}}^{+}convex on I (resp. {\mathbb{R}}^{}convex on I). We first prove that {F}_{1}\circ {F}_{2} is {\mathbb{R}}^{+}convex on I for {F}_{1},{F}_{2}\in {USIC}^{+}(I). By Lemma 2.1, the fact that {F}_{2} is {\mathbb{R}}^{+}convex on I implies that
Hence, for all y\in \lambda {F}_{2}({x}_{1})+(1\lambda ){F}_{2}({x}_{2}),
holds. Note that {F}_{1} is strictly increasing. Consequently,
So
By
we have
because {F}_{1} is {\mathbb{R}}^{+}convex. Hence, by (3.1)
{F}_{1}\circ {F}_{2}\in {USIC}^{+}(I) is proved.
Next, we prove {F}_{1}\circ {F}_{2} is {\mathbb{R}}^{}convex on I for {F}_{1},{F}_{2}\in {USIC}^{}(I). By Lemma 2.2, the fact that {F}_{2} is {\mathbb{R}}^{}convex on I implies that
Hence, for all y\in \lambda {F}_{2}({x}_{1})+(1\lambda ){F}_{2}({x}_{2}),
holds. Note that {F}_{1} is strictly increasing. Consequently,
So
By
it follows that
because {F}_{1} is {\mathbb{R}}^{}convex. Hence, by (3.2)
This completes the proof of {F}_{1}\circ {F}_{2}\in {USIC}^{}(I). □
Define
where I=[a,b] and M>m>0.
Remark 3.2 The condition maxF(b)=b for F\in {USIC}^{+}(I,m,M) (minF(a)=a for F\in {USIC}^{}(I,m,M)) guarantees that the iterations {F}^{n}, n=2,3,\dots , are also multifunctions.
Lemma 3.3 {USIC}^{+}(I,m,M) and {USIC}^{}(I,m,M) are complete metric spaces equipped with the distance
Proof By Lemma 3.1 in [29], we only need to prove that if \{{F}_{n}\}\subset {USIC}^{\sigma}(I,m,M) such that {lim}_{n\to \mathrm{\infty}}{F}_{n}=F(x) in USI(I,m,M), i.e.,
then F(x) is {\mathbb{R}}^{\sigma}convex on I, where \sigma =+ or \sigma =. We first prove the case of {USIC}^{+}(I,m,M). By (3.3), we have {lim}_{n\to \mathrm{\infty}}h({F}_{n}(x),F(x))=0, \mathrm{\forall}x\in I. Hence,
Note that by Lemma 3.1,
and
Hence,
By (3.4) and (3.5), we have for every \epsilon >0, there exists {n}_{0}\in \mathbb{N} such that
and
\mathrm{\forall}{x}_{1},{x}_{2}\in I, \lambda \in [0,1]. Consequently,
because {F}_{{n}_{0}}(x) is {\mathbb{R}}^{+}convex on I. Hence,
which shows that F(x) is {\mathbb{R}}^{+}convex on I.
Next we prove the case of \sigma =. By (3.6) and (3.7), we have for every \epsilon >0,
because {F}_{{n}_{0}}(x) is {\mathbb{R}}^{}convex on I. Hence,
which shows that F(x) is {\mathbb{R}}^{}convex on I. The proof is completed. □
Define
{USIC}_{\ast}^{+}(I,m,M) is a closed subset of {USIC}^{+}(I,m,M). {USIC}^{\ast}(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}\in {USIC}_{\ast}^{+}(I,{m}^{i},{M}^{i}) (resp. {USIC}^{\ast}(I,{m}^{i},{M}^{i})) if F\in {USIC}_{\ast}^{+}(I,m,M) (resp. USIC^{−∗} (I,m,M)).
Lemma 3.5 If {F}_{1},{F}_{2}\in {USIC}_{\ast}^{+}(I,m,M) (resp. {USIC}^{\ast}(I,m,M)), then
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 {\lambda}_{1}>0, {\lambda}_{i}\le 0 (i=2,\dots ,n) and {\sum}_{i=1}^{n}{\lambda}_{i}=1 and G\in {USIC}^{\ast}(I,{m}_{0},{M}_{0}) with {M}_{0}>{m}_{0}>0. Then for arbitrary constants M>m>0 satisfying
Eq. (1.3) has a unique solution F\in {USIC}^{\ast}(I,m,M) if
Proof Define the mapping L:{USIC}^{\ast}(I,m,M)\to \mathcal{F}(I) by
By Lemma 3.2, {F}^{i}(x), i=2,\dots ,n are strictly increasing {\mathbb{R}}^{}convex on I because F(x) is strictly increasing {\mathbb{R}}^{}convex. Since G(x) is {\mathbb{R}}^{}convex on I and max(A+B)=maxA+maxB, we have
Hence, LF(x) is {\mathbb{R}}^{}convex on I. Obviously, LF(x) is strictly increasing and LF(x)>x for x\in intI. Similar to the proof of Theorem 4.1 in [29], by Lemma 3.4 and condition (4.1), LF(x)\in {USIC}^{\ast}(I,m,M). Thus, we have proved that LF(x) is a selfmapping on {USIC}^{\ast}(I,m,M). By Lemma 3.5 and condition (4.2), L is a contraction map. By Lemma 3.3, {USIC}^{\ast}(I,m,M) is a complete metric space. Using Banach’s fixed point principle, L has a unique fixed point F in {USIC}^{\ast}(I,m,M), i.e.,
This completes the proof. □
We note the fact that A+B\supset C if the sets A, B, C satisfy A=CB. Hence, every solution F of Eq. (1.3) satisfies
We have the following result.
Corollary 4.1 Under the same conditions as in Theorem 4.1, there exists a multifunction F\in {USIC}^{\ast} (I,m,M) such that (4.4) holds.
For multifunctions in the other class {USIC}_{\ast}^{+}(I,m,M), we have a similar result to Theorem 4.1. It can be proved similarly.
Theorem 4.2 Suppose that {\lambda}_{1}>0, {\lambda}_{i}\le 0 (i=2,\dots ,n) and {\sum}_{i=1}^{n}{\lambda}_{i}=1 and G\in {USIC}_{\ast}^{+}(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\in {USIC}_{\ast}^{+}(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\in {USIC}_{\ast}^{+} (I,m,M) such that (4.4) holds.
Remark 4.1 Although the assumption F\in {USIC}^{\ast}(I) (or {USIC}_{\ast}^{+}(I)) implies that F is singlevalued on [a,b) (or (a,b]), but Eq. (1.3) cannot be considered on the interval [a,b) (or (a,b]) as a singlevalued case and the point b (or a) as a multivalued 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 {\mathbb{R}}^{+}convex multifunction in {USIC}^{+\ast}(I,m,M). The same applies to the case of {USIC}_{\ast}^{}(I,m,M). Consequently, Eq. (1.3) has no solution in {USIC}^{+\ast}(I,m,M) (resp. {USIC}_{\ast}^{} (I,m,M)).
Remark 4.3 By Theorem 4.1 and Theorem 4.2, we actually only prove the existence and uniqueness of Kconvex (K={\mathbb{R}}^{+} and K={\mathbb{R}}^{}, i.e., K is not a nontrivial convex cone) multivalued solutions for Eq. (1.3). In fact, there is no convex multivalued (i.e., \{0\}convex multivalued) solutions for Eq. (1.3) in the multifunction class USI(I). Since F(x) is a convex multivalued function on I if and only if
Hence, if Eq. (1.3) has a convex multivalued 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 Kconvex solutions, i.e., strictly increasing Kconvex solutions of Eq. (1.3). It is very difficult to discuss Kconvex solutions of Eq. (1.3) which are not strictly increasing because the method in [29] cannot be used. Discussing nonstrictlyincreasing Kconvex 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
where n=3, {\lambda}_{1}=\frac{5}{4}, {\lambda}_{2}=0, {\lambda}_{3}=\frac{1}{4} and
Clearly, G\in {USIC}^{\ast}(I,{m}_{0},{M}_{0}), where
Let m=\frac{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\in {USIC}^{\ast}(I,m,M).
Remark 5.1 Example (5.1) cannot be solved by known singlevalued 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 multivalued iterative equation of order n. J Inequal Appl 2012, 258 (2012). https://doi.org/10.1186/1029242X2012258
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DOI: https://doi.org/10.1186/1029242X2012258