Convex solutions of the multi-valued iterative equation of order n
© Gong; licensee Springer 2012
Received: 5 March 2012
Accepted: 23 October 2012
Published: 5 November 2012
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.
was proved in . K-convex multifunctions, which are generalization of vector-valued convex functions, have wide applications in optimization (cf. ) and play an important role in various questions of convex analysis (cf. ). 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
A convex multifunction  may be stated as θ-convex and the convexity of a real-valued function may be stated as -convex, and concavity as -convex, where and . Let be the set of all multifunctions , where denotes the family of all nonempty closed subintervals of I.
Considering -convex multifunctions and -convex multifunctions, the following lemmas are obvious.
3 Some lemmas
Remark 3.1 If (resp. ), , then F must be single-valued on (resp. ).
Lemma 3.2 (resp. ) for (resp. ).
This completes the proof of . □
where and .
Remark 3.2 The condition for ( for ) guarantees that the iterations , , are also multifunctions.
which shows that is -convex on I.
which shows that is -convex on I. The proof is completed. □
is a closed subset of . is a closed subset of .
By Lemma 3.2, one can prove the following result.
Lemma 3.4 (resp. ) if (resp. USIC−∗ ).
The proof of Lemma 3.5 is similar to that of Lemma 3.3 in . We omit it here.
4 Convex solutions
This completes the proof. □
We have the following result.
Corollary 4.1 Under the same conditions as in Theorem 4.1, there exists a multifunction such that (4.4) holds.
For multifunctions in the other class , we have a similar result to Theorem 4.1. It can be proved similarly.
Theorem 4.2 Suppose that , () and and with . Then for arbitrary constants satisfying (4.1), Eq. (1.3) has a unique solution if condition (4.2) holds.
Corollary 4.2 Under the same conditions as in Theorem 4.2, there exists a multifunction such that (4.4) holds.
Remark 4.1 Although the assumption (or ) implies that F is single-valued on (or ), but Eq. (1.3) cannot be considered on the interval (or ) 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 -convex multifunction in . The same applies to the case of . Consequently, Eq. (1.3) has no solution in (resp. ).
Hence, if Eq. (1.3) has a convex multi-valued solution F in , 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  cannot be used. Discussing non-strictly-increasing K-convex solutions of Eq. (1.3) will be the subject of our next work.
Let and . 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 .
Remark 5.1 Example (5.1) cannot be solved by known single-valued results.
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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