# On the Blended Solutions of Polynomial-Like Iterative Equation with Multivalued Functions.

1. Introduction

The polynomial-like iterative equation

[[lambda].sub.1]f(x)+[[lambda].sub.2][f.sup.2](x) + ... + [[lambda].sub.n][f.sup.n](x) = F(x), x [member of] X, (1)

is an interesting form of functional equations, where X is a topological vector space, F: X [right arrow] X is a given function, and [f.sup.i], i = 1,2,3, ..., n, stands for the i-th iteration of the unknown function f : X [right arrow] X, i.e., [f.sup.i](x) = f([f.sup.i-1](x)) and [f.sup.0](x) = X. Since iteration is an important problem in many mathematical subjects, such as dynamical systems and numerical computation, and many fields of natural science, in recent years some attentions have been paid to (1) and its generalizations [1-7]. On the other hand, multivalued function is an important class of mappings, which has been extensively employed in control theory , stochastics , artificial intelligence , and economics . Many nice results [12-16] were obtained on functional equations with multivalued functions. It is important to investigate (1) with multivalued functions, i.e., the equation

[[lambda].sub.1](g) + [[lambda].sub.2][g.sup.2](x) + ... + [[lambda].sub.n][g.sup.n] x) = G(x), x [member of] I := [a, b], (2)

where n [greater than or equal to] 2 is an integer, [[lambda].sub.1], [[lambda].sub.2], ... [[lambda].sub.n] are real constants, G : I [right arrow] cc(I) is a given multivalued function, cc(I) denotes the family of all nonempty convex compact subsets of I, and g : I [right arrow] cc(I) is an unknown multivalued function; the i-th iteration [g.sup.i] of the multivalued function g is defined inductively as

[g.sup.i](x) := [union]{g(y):y [member of] [g.sup.i-1](x)} (3)

and [g.sup.0](x) :[equivalent to] x for all x [member of] I. In 2004, Nikodem and Zhang  first studied (2) for n = 2 with an increasing upper semicontinuous (USC for short) multivalued function G on I = [a, b]. The result on the existence and uniqueness of USC solutions is given. As indicated in , the upper semicontinuity for multivalued functions is much weaker than the continuity for functions; the method used for continuous solutions [6, 7] is improved substantially in order to obtain USC solutions. In 2011, Xu, Nikodem, and Zhang  considered the general case n [greater than or equal to] 3 of the equation

[[lambda].sub.1]F(x) = G(x) - [[lambda].sub.2][F.sup.2](x)- ... -[[lambda].sub.n][F.sup.n](x), [for all]x [member of] I, (4)

which is a modified version of (2), for the piecewise Lipschitzian multivalued functions defined in . USC unblended multivalued solutions of this equation were given in the inclusion sense. As defined in , a multivalued function F is said to be unblended if it satisfies [[zeta].sub.k+1] [member of] F([[zeta].sub.k]) on a sequence S := {[[zeta].sup.[infinity].sub.k[greater than or equal to]0] in I; that is, [[zeta].sub.k+1] [member of] [min F([[zeta].sub.k]), maxF([[zeta].sub.k])] for each k; otherwise, F is said to be blended on S. Clearly, the blended requirement for multivalued functions is much weaker than the unblended requirement for multivalued functions. It is an interesting object to study the blended multivalued solutions of (2) in the inclusion sense. So far, we find no results on the USC blended multivalued solutions of (2) in the inclusion sense. In this paper, we investigate the existence of the USC blended multivalued solutions of (2) in the inclusion sense.

2. Preliminaries

The family cc(I) endowed with the Hausdorff distance is defined by

h (A; B) = max {sup {d (a; B) : a [member of] A}; sup {d(b;A) :b [member of] B}}; (5)

where d(a;B) = inf{[absolute value of a-b] : b [member of] B} is a complete metric space (cf., e.g., , Cor. 4.3.12).

A multivalued function F : I [right arrow] cc(I) is increasing (resp., strictly increasing) if for every x,y [member of] I with x < y, we have max F(x) [less than or equal to] min F(y) (resp., max F(x) < min F(y)) (cf. , Def. 3.5.1). A multivalued function F : I [right arrow] cc(I) is USC at a point [x.sub.0] [member of] I if for every open set V [subset] R with F([x.sub.0]) [subset] V there exists a neighborhood [mathematical expression not reproducible] of [x.sub.0] such that F(x) [subset] V for every x [member of] [mathematical expression not reproducible]. F is USC on I if it is USC at every point in I. For convenience, let

USI (I) := {F [member of] F(I) : F is USC and strictly increasing}, (6)

where F(I) is the set of all multivalued functions F : I [right arrow] cc(I). Some useful properties are summarized in the following Lemma (cf. [4,15,19]).

Lemma 1. For A, B,C,D [member of] cc(I) and for an arbitrary real [lambda], the following properties hold:

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

(ii) h(XA,XB) = [absolute value of [lambda]]h(A,B),

(iii) h(A + C,B + D) [less than or equal to] h(A, B) + h(C, D).

Lemma 2. If [F.sub.1],[F.sub.2] [member of] USI(I), then [F.sub.1] [omicron] [F.sub.2] [member of] USI(I).

As indicated in , if a function F [member of] USI (I) is not single-valued, there exists at least a point [xi] [member of] I such that the cardinal of the set F([xi]) is more than 1; i.e., card F([xi]) [greater than or equal to] 2. Actually, F([xi]) is a nontrivial interval because F([xi]) [member of] cc(I). Since F is strictly increasing, there exist two small open intervals [V.sup.-.sub.[delta]] := {x [member of] I | [xi] - [delta] < x < [xi]} and [V.sup.+.sub.[delta]] := {x [member of] I | [xi] < x < [xi] + [delta]} such that F is single-valued in both of them and satisfies min F([xi]) > F(x), [for all]x [member of] [V.sup.-.sub.[delta]], and max F([xi]) < F(x), [for all]x [member of] [V.sup.+.sub.[delta]]. We call [xi] a jump-point of F or a jump simply. For every F [member of] USI(I), let J(F) denote the set of all jumps of F. We easily see that each F [member of] USI(I) has at most countably infinite many jumps, i.e., the cardinal card J(F) [less than or equal to] N. In fact, for each [xi] [member of] J(F), the set F([xi]) is a nontrivial compact subinterval of I. By the strict monotonicity, {F([xi]) : [xi] [member of] J(F)} is a set of disjoint nonempty compact subintervals of I. Choose a rational number r([xi]) [member of] F([xi]) for each [xi] [member of] J(F). Then card J(F) [less than or equal to] card Q = N. According to above the argument, F [member of] USI(I) was divided into two cases in : one is unblended multivalued functions; the other is blended ones.

Since functions in USI(I) are strictly increasing, it suffices to discuss multivalued functions F in USI(I) which satisfy either min F(x) > x for all x [member of] int I or max F(x) < x for all x [member of] int I. Let [USI.sup.*](I) and [USI.sub.*](I) denote the two classes of multivalued functions, respectively.

Lemma 3. Suppose that F [member of] [USI.sup.*](I) (resp., [member of] [USI.sub.*](I)) is unblended on the strictly increasing (resp., decreasing) sequence S = [{[[zeta].sub.k]}.sup.[infinity].sub.k[greater than or equal to]0]. If S [member of] J(F) and satisfies that [[zeta].sub.0] = a and [lim.sub.k[right arrow][infinity]][[zeta].sub.k] = b (resp., [[zeta].sub.0] = b and [lim.sub.k[right arrow][infinity]][[zeta].sub.k] = a), then for each integer i [greater than or equal to] 1, (i)J([F.sup.i]) [subset] S, and (ii) [F.sup.i](([[zeta].sub.k],[[zeta].sub.k+1])) [subset] (([[zeta].sub.k+i],[[zeta].sub.k+1+i])(resp., [F.sup.i]((([[zeta].sub.k+1],[[zeta].sub.k])) [subset](([[zeta].sub.k+1+i],[[zeta].sub.k+i])), [for all]k [greater than or equal to] 0.

A function F [member of] USI(I) is said to be piecewiseLipschitzian on I with the sequence [LAMBDA] and constants M > m >0 if for each k [greater than or equal to] 0,

(C1) J(F) [subset] [LAMBDA],

(C2) m([x.sub.2] - [x.sub.1]) [less than or equal to] F([x.sub.2]) - F([x.sub.1]) [less than or equal to] M([x.sub.2] - [x.sub.1]) [for all][x.sub.1],[x.sub.2] [member of] ([[eta].sub.k], [[eta].sub.k+1]) with [x.sub.1] < [x.sub.2],

(C3) maxF([[eta].sub.k+1]) - min F([[eta].sub.k]) [less than or equal to] M([[xi].sub.k+1] - [[eta].sub.k]) [for all][[eta].sub.k], [[eta].sub.k+1] [member of] [LAMBDA],

where [LAMBDA] is a strictly monotonic sequence in I such that int(7) [LAMBDA] is a union of disjoint open intervals; i.e.,

[mathematical expression not reproducible] (7)

where each [[eta].sub.k] is either an element of [LAMBDA] or an endpoint of I. For a strictly increasing sequence S = [{[[zeta].sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] in I such that [[zeta].sub.0] = a and [lim.sub.k[right arrow][infinity]][[zeta].sub.k] = b, define

[USI.sup.*.sub.u] := {F [member of] [USI.sub.*](I) : F is unblended on S and J (F) [subset] S}. (8)

Similarly, for a strictly decreasing sequence S = [{[[zeta].sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] in I such that [[zeta].sub.0] = b and [lim.sub.k[right arrow][infinity]][[zeta].sub.k] = a define

[USI.sub.u*] := {F [member of] [USI.sub.*](I) : F is unblended on S and J (F) [subset] S}. (9)

Lemma 4. USI(I,[LAMBDA],m,M) is a complete metric space equipped with the distance D, defined by D([F.sub.1],[F.sub.2]) := sup{h([F.sub.1](x),[F.sub.2](x)) : x [member of] I}, [for all][F.sub.1],[F.sub.2] [member of] [PHI](I).

Define

[mathematical expression not reproducible] (10)

Clearly, [USI.sup.*] (I, S, m, M), USI, (I, S, m, M), [USI.sup.*.sub.u] (I, S, m, M), and [USI.sub.u*] (I, S, m, M) are all closed subsets of USI(I, S, m, M).

Lemma 5. [F.sup.i] [member of] [USI.sup.*](I,S,[m.sup.i],[M.sup.i]) (resp., [USI.sub.*](I,S,[m.sup.i],[M.sup.i])) if F [member of] [USI.sup.*.sub.u] (I, S, m, M) (resp., [USI.sub.u*], (I, S, m, M)).

Lemma 6. If either [F.sub.1],[F.sub.2] [member of] [USI.sup.*.sub.u](I,S,m, M) or [F.sub.1],[F.sub.2] [member of] [USI.sub.u*] (I, S, m, M), then

D([F.sup.i.sub.1],[F.sup.i.sub.2]) [less than or equal to] ([i-1.summation over (j=0)][M.sup.j])D([F.sub.1],[F.sub.2]). (11)

3. Main Results

Theorem 7. Suppose that [[lambda].sub.1] >0, [[lambda].sub.i] [less than or equal to] 0 (i = 2,..,n), and [[summation].sup.n.sub.i=1][[lambda].sub.i] = 1 and that sequences S = [{[[zeta].sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] and [S.sup.*] = [{[[zeta].sup.*.sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] are strictly increasing sequences in I such that S [subset] [S.sup.*], [[zeta].sub.0] = a, [lim.sub.k[right arrow][infinity]][[zeta].sup.*.sub.k] = b, and card of ([[zeta].sub.k],[[zeta].sub.k+1][intersection]([S.sup.*]\S) [less than or equal to] 1 and that G [member of] [USI.sup.*](I,S,[m.sub.0],[M.sub.0]), where [M.sub.0] > [m.sub.0] > 0, such that [[summation].sup.n.sub.i=1][[lambda].sub.i][[zeta].sup.*.sub.k+i] [member of] G([[zeta].sup.*.sub.k]), [for all]k [greater than or equal to] 0; then, for arbitrary constants M > m > 0 such that

[mathematical expression not reproducible] (12)

(4) has a unique solution F [member of] [USI.sup.*.sub.u](I, [S.sup.*], m, M), which is a solution of(2) in the inclusion sense and is blended in S.

Proof. Sets [USI.sup.*] (I, S, [m.sub.0], [M.sub.0]) and [USI.sup.*.sub.u] (I, [S.sup.*], m, M) are well defined since the sequences S = [{[[zeta].sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] and [S.sup.*] = [{[[zeta].sup.*.sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] are strictly increasing sequences in I such that S [subset] [S.sup.*], [[zeta].sub.0] = a, [lim.sub.k[right arrow][infinity]] = b, and card of ([[zeta].sub.k], [[zeta].sub.k+1]) [intersection]([S.sup.*]\S) [less than or equal to] 1. Define operator L : [USI.sup.*.sub.u](I, S.sup.*, m, M) [right arrow] F(I) by

L(F)(x) = 1/[[lambda].sub.1](G(x) - [n.summation over (i=2)][F.sup.i](x)), [for all]x [member of] I, (13)

where F [member of] [USI.sup.*.sub.u](I, [S.sup.*], m, M). We can deduce L(F) [member of] [USI.sup.*](I) from hypotheses [[lambda].sub.1] > 0, [[lambda].sub.i] [less than or equal to] 0 (i = 2,..,n), [[summation].sup.n.sub.i=1][[lambda].sub.i] = 1, G [member of] [USI.sup.*] (I, S, [m.sub.0],[M.sub.0]), and F [member of][USI.sup.*.sub.u] (I, [S.sup.*], m, M). Since F [member of] [USI.sup.*.sub.u] (I, [S.sup.*], m, M) and [[summation].sup.n.sub.i=1][[lambda].sub.i][[zeta].sup.*.sub.k+i] [member of] G([[zeta] .sup.*.sub.k]), [for all]k [greater than or equal to] 0, we have

[mathematical expression not reproducible] (14)

which implies

L(F) [member of] [USI.sup.*.sub.u](I, [S.sup.*]). (15)

By G in [USI.sup.*](I,S,[m.sub.0],[M.sub.0]) and card of ([[zeta].sub.k], [[zeta].sub.k+1])[intersection]([S.sup.*]\S) [less than or equal to] 1, we can obtain that G [member of] [USI.sup.*](I, [S.sup.*],[m.sub.0],[M.sub.0]). In fact, Since G [member of] [USI.sup.*](I,S,[m.sub.0],[M.sub.0]), S [subset] [S.sup.*], and card of ([[zeta].sub.k], [[zeta].sub.k+1])[intersection]([S.sup.*]\ S) [less than or equal to] 1,k [greater than or equal to] 0, then J(G) [subset] S [subset] [S.sup.*], m([x.sub.2] - [x.sub.1]) [less than or equal to] G([x.sub.2])-G([x.sub.1]) [less than or equal to] M([x.sub.2] - [x.sub.1]), [for all][x.sub.1],[x.sub.2] [member of] ([[zeta].sup.*.sub.k],[[zeta].sup.*.sub.k+1]) with [x.sub.1] < [x.sub.2], and max G([[zeta].sup.*.sub.k+1]) - min G([[zeta].sup.*.sub.k]) [less than or equal to] M([[zeta].sup.*.sub.k+1] - [[zeta].sup.*.sub.k]), [for all][[zeta].sup.*.sub.k], [[zeta].sup.*.sub.k+1] [member of] [S.sup.*]; i.e., G [member of] [USI.sup.*](I, [S.sup.*], [m.sub.0], [M.sub.0]). From G in [USI.sup.*](I, [S.sup.*], [m.sub.0], [M.sub.0]), Lemma 5, and (13), we can infer that

L(F) [member of] USI (I, [S.sup.*], 1/[[lambda].sub.1]([m.sub.0]+[n.summation over (i=2)][absolute value of [[lambda].sub.i]][m.sup.i]), 1/[[lambda].sub.1]([M.sub.0]+[n.summation over (i=2)][absolute value of [[lambda].sub.i]][M.sup.i])), (16)

which together with (15) yields

[mathematical expression not reproducible]. (17)

Furthermore, for [F.sub.1], [F.sub.2] [member of] [USI.sup.*.sub.u](I, [S.sup.*], [m.sub.0], [M.sub.0]), by (13) and Lemma 6 we have

[mathematical expression not reproducible] (18)

In view of Lemma 6,

[mathematical expression not reproducible]. (19)

We know that operator L is contract since [rho] < 1. Making use of Banach's fixed point Theorem, L has a unique fixed point F in [USI.sup.*.sub.u] (I, [S.sup.*], m, M) such that

F(x) = 1/[[lambda].sub.1](G(x) - [n.summation over (i=2)][F.sup.i](x)), [for all]x [member of] I, (20)

and since F(x), G(x), and [[summation].sup.n.sub.i=2][F.sup.i](x) are three sets, according to sets operation relations we can get

[n.summation over (i=1)][[lambda].sub.i][F.sup.i](x) [subset] G(x), [for all]x [member of] I, (21)

which implies F is a unique solution of (2) in the inclusion sense. In the following, we shall prove that F is blended in S. By reduction to absurd, we suppose F is unblended in S. Since F is unblended in [S.sup.*], for [zeta] [member of] [S.sup.*] \ S, there exists some interval [mathematical expression not reproducible] such that [mathematical expression not reproducible], which implies [mathematical expression not reproducible] because F is unblended. Therefore, [mathematical expression not reproducible], which contradicts to the condition that F is strictly increasing on [S.sup.*]; thus F is blended in S.

Theorem 8. Suppose that [[lambda].sub.1] > 0, [[lambda].sub.i] [less than or equal to] 0 (i = 2,..,n), and [[summation].sup.n.sub.i=1] [[lambda].sub.i] =1 and that sequences S = [{[zeta]}.sup.[infinity].sub.k[greater than or equal to]0] and [S.sup.*] = [{[[zeta].sup.*.sub.k]}.sup.[infinity].sub.k[greater than or equal to]0] are strictly decreasing sequences in I such that S [subset] [S.sup.*], [[zeta].sub.0] = b, [lim.sub.k[right arrow][infinity]][[zeta].sup.*.sub.k] = a, and card of ([[zeta].sub.k+1],[[zeta].sub.k])[intersection]([S.sup.*] \ S) [less than or equal to] 1 and that G [member of] [USI.sup.*](I,S,[m.sub.0],[M.sub.0]), where [M.sub.0] > [m.sub.0] > 0 such that [[summation].sup.n.sub.i=1][[lambda].sub.i][[zeta].sup.*.sub.k+i] [member of] G([[zeta]*.sub.k]), [for all]k [greater than or equal to] 0; then, for arbitrary constants M > m > 0 satisfying (12), (4) has a unique solution F [member of] [USI.sub.u*](I, [S.sup.*], m, M), which is a solution of (2) in the inclusion sense and is blended in S.

The proof is similar to Theorem 7, so we omit it.

Remark 9. Although we strengthen the condition of Theorem 4.1 in  in Theorem 7, more general solutions of (2) are given.

4. Example

In order to show the rationality conditions in Theorems, we consider the equation

[[lambda].sub.1]F(x) + [[lambda].sub.2][F.sup.2](x) = G(x), [for all]x [member of] I := [0,1], (22)

where [[lambda].sub.1] = 3/2, [[lambda].sub.2] = -1/2,

[mathematical expression not reproducible] (23)

and, obviously, G [member of] [USI.sup.*](I, S, [m.sub.0], [M.sub.0]), where

[m.sub.0] = min{3/4, 5/8} = 5/8, [M.sub.0] = {3/4, 5/8, [11/16-1/18]/1/2} = 9/8, (24)

S = [{[[zeta].sub.k]}.sup.[infinity].sub.k=0] with [[zeta].sub.k] = 1 - [2.sup.-k], k [greater than or equal to] 0, for [S.sup.*] = [{[[zeta].sup.*.sub.k]}.sup.[infinity].sub.k=0] with [[zeta].sup.*.sub.0] = 0, [[zeta].sup.*.sub.1] = 1/4, [[zeta].sup.*.sub.2] = 1/2, [[zeta].sup.*.sub.k] = 1 - [2.sup.-k+1], k [greater than or equal to] 2; one can check

[2.summation over (i=1)][[lambda].sub.i][[zeta].sup.*.sub.k+i] [member of] G([[zeta]*.sub.k]) [for all]k [greater than or equal to] 0. (25)

Taking m = 1/2, M = 3/2, we can see that conditions (12) hold. Thus, (2) has a solution F [member of] [USI.sup.*.sub.u](I, [S.sup.*], m, M) by Theorem (12).

https://doi.org/10.1155/2018/9862794

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the general item of Lingnan Normal University [grant ZL1505], KSP of Lingnan Normal University [number 1171518004], NSFC [grant number 11701476], the Fundamental Research Funds for the Central Universities [grant number 2682018CX63], and the young project of Zhanjiang Preschool Education College Scientific Research [grant number ZJYZQN201717].

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Jinghua Liu, (1) Hongjuan Duan (iD), (2) and Zhiheng Yu (iD), (3)

(1) School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

(2) Department of Information Science, Zhanjiang Preschool Education College, Zhanjiang, Guangdong 524037, China

(3) School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Correspondence should be addressed to Zhiheng Yu; mathyuzhiheng@163.com

Received 11 June 2018; Accepted 17 September 2018; Published 3 October 2018

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