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Existence of Generalized Nash Equilibrium in n-Person Noncooperative Games under Incomplete Preference.

1. Introduction and Preliminaries

As a kind of strategy combinations, Nash equilibrium is closely bound up with many important mathematical problems, and many problems in economy and engineering technology can also be described as a Nash equilibrium problem. Recently, the existence of the Nash equilibrium of non-cooperative games has been studied [1-4]. In [1], the existence of uncertainty for generalized Nash equilibrium is proved by introducing the uncertainty to study generalized games. Using maximization theorem, the author presented the existence of Nash equilibrium in generalized games, and in these results, the strategy set is noncompact and has infinite players [2]. In [3] the existence theorem of generalized Nash equilibrium in games is given where strategy space has abstract convex structure. Assuming strategy set is a H space, the equilibrium existence theorems have been given in [4]. In the above studies, either the partial order relation on policy sets is required to satisfy, or every total order subset in policy sets must have an upper bound or certain convexity condition.

For a long time, the preference of rational decision-makers on management and economics should satisfy the completeness. But in practice, they show the indecision on many major issues. Since the preference without completeness is a kind of more general order structure, it can make preference relation and the partial order relation unified completely. The research of preference without completeness is started from the von Neumann and Morgenstern in [5]. Aumman and Bewley have made the classic study in [6,7]; Schmeidler has studied the existence of economic equilibrium with infinite number of institutions under incomplete preference in [8].

In noncooperative games, the policy set composed of player's selection strategies is a set which cannot meet the needs of completeness. If preference does not meet the completeness, Pareto optimality is meaningless and the traditional method of partial order will lose effectiveness without the antisymmetry axiom inevitably. Therefore, it is consistent with the realistic decision-making environment to study the existence of generalized Nash equilibrium of noncooperative game, but the study of this part has seldom been seen in the past literature research.

In this article, based on the equivalence class set which corresponds to the elements of incomplete preference set being a partial order set, the problem under incomplete preference is translated into the problem with partial order. This method overcomes the difficulties which are brought about by the elements in the set without the completeness. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. The fixed point theorems provide a new way for breaking through the limitations. The existence of generalized Nash equilibrium is strictly proven in the n-person noncooperative games under incomplete preference.

Here some concepts and theorems are given, which are related to incomplete preference.

Let E be a nonempty set. An ordering relation [less than or equal to] on E may satisfy the following axiom:

Reflexive: x [less than or equal to] x, for any x [member of] E;

Symmetry: If x [less than or equal to] y and y [less than or equal to] x, then x = y, for any x, y [member of] E;

Transitive: If x [less than or equal to] y and y [less than or equal to] z, then x < z, for any x, y, z [member of] E;

Complete: If x [less than or equal to] y, y < x, for any x, y [member of] E, there is at least one inequality to be established.

Definition 1 (see [9]). Let E be a nonempty set. An order relation [less than or equal to] defined among certain elements of E is said to be partial order if the order relation satisfies reflexive, transitivity, and antisymmetry axioms. Then (E [less than or equal to]) is called a poset.

Definition 2 (see [10]). Let E be a nonempty set. An order relation defined among certain elements of E is said to be incomplete preference order if the order relation satisfies reflexive and transitivity axioms, which is denoted by [less than or equal to]. If completeness axiom is still satisfied for incomplete preference order, the order relation is said to be preference order, which is still denoted by [less than or equal to]. Then (E [less than or equal to]) is called an incomplete preference set.

Definition 3 (see [11]). Let E be an incomplete preference set. For any x, y [member of] E, we say that x, y are indifference, which is denoted by x ~ y, whenever both x [less than or equal to] y and y [less than or equal to] x hold.

Remark 4. x ~ y x = y, but x = y [??] x ~ y.

Remark 5. The indifference relation ~ the equivalence relation.

Definition 6 (see [11]). Let E be an incomplete preference set. If for any complete preference subset of E, there is denumerable set {[x.sub.n]} [subset] M such that if x [member of] M, x [not equal to] sup M, there is [mathematical expression not reproducible], then E is said to be pseudo separable in incomplete preference.

Let E be an incomplete preference set, and Q is a subset in E. The order relation [less than or equal to] in quotient set [OMEGA]/ ~ is elicited by the incomplete preference relation [less than or equal to] in E. Let [x] = [y [member of] [omega] | x ~ y}, and [x] is an equivalence class set in [OMEGA].

Definition 7 (see [11]). For any [x], [y] [member of] [OMEGA]/ if there are u e [x], v e [y] such that u [less than or equal to] v, we write [x] [less than or equal to] [y].

Lemma 8 (see [11]). Let E be an incomplete preference set, and [OMEGA] is a subset in E. The order relation [less than or equal to] in quotient set [OMEGA]/ ~ which is elicited by the incomplete preference relation [less than or equal to] in E is a partial order. Then the quotient set [OMEGA]/ ~ is a poset.

Lemma 9 (see [11]). If [OMEGA] is incomplete preference pseudo separable, then [OMEGA]/ ~ is incomplete preference pseudo separable.

Lemma 10 (see [12] (Zorn Lemma)). Let E be a nonempty partial ordered set. If every total ordered subset in E has an upper bound in E, then there is a maximal element in E.

2. Existence Theorems for Fixed Point on Incomplete Preference Sets

Partial order method is discussed and applied greatly in mathematics, and the conclusion on the partial order is becoming a very complete system [13-23]. But few scholars study the fixed point and extreme value theorems on incomplete preference set.

Definition 11. Let (E, [[less than or equal to].sup.E]), (U, [[less than or equal to].sup.U]) be incomplete preference sets, and let T : E [right arrow] [2.sup.U] be an order-increasing set-valued mapping. T is said to be order-increasing upward, if x [[less than or equal to].sup.E] y in E, for any u [member of] T(x); there is v [member of] T(y) such that u [[less than or equal to].sup.U] v; T is said to be order-increasing downward, if x [[less than or equal to].sup.E] y in E, for any v [member of] T(y); there is u [member of] T(x) such that u [[less than or equal to].sup.U] v. If T is both order-increasing upward and order-increasing downward, T is said to be order-increasing.

Definition 12. Let (E, [less than or equal to]) be incomplete preference set, and let T : E [right arrow] [2.sup.E \[PHI] be an order-increasing set-valued mapping. An element x [member of] E is called a generalized fixed point of T, if there are [x.sup.*] [member of] E, u [member of] [Tx.sup.*] such that [x.sup.*] ~ u.

Let (E, [less than or equal to]) be incomplete preference set, and let T : E [right arrow] [2.sup.u] be an order-increasing set-valued mapping. The following notation will be used in Theorem 13:

ST (x) = [x [member of] E | x [less than or equal to] u, y [member of] T (x)}. (1)

Theorem 13. Let (E, [less than or equal to]) be an incomplete preference pseudo separable set, and let T : E [right arrow] [2.sup.E] be an order-increasing set-valued mapping. If T satisfies the following conditions:

([A.sub.1]) Every increasing sequence in ST(x) has an upper bound in ST(x)

([A.sub.2]) There is a [x.sub.0] [member of] E with [x.sub.0] < u, for some u [member of] [Tx.sub.0] then T has a generalized fixed point; that is, there are [x.sup.*] [member of] E, u [member of] [Tx.sup.*] such that [x.sup.*] ~ u.

Proof. Let [omega] = {x [member of] E | x [less than or equal to] u, y [member of] T(x)}. From the condition ([A.sub.2]), it implies that [OMEGA] is a nonempty set in E. Take an arbitrary total ordered subset M [subset] [OMEGA]. Since M is also an arbitrary total ordered subset of incomplete preference pseudo separable set (E, [less than or equal to]), there is denumerable set {[x.sub.n]} [subset] M such that if x [member of] M, x [not equal to] sup M, there is [mathematical expression not reproducible].

Let

[z.sub.1] = [x.sub.1],

[z.sub.n] = max {[x.sub.n], [z.sub.n-1]}, n = 2,3 .... (2)

[z.sub.n] [subset] M [subset] [OMEGA](x).

Since M is an arbitrary total ordered subset, {[z.sub.n]} is well defined. So

[z.sub.1] [less than or equal to] [z.sub.2] [less than or equal to] ... [z.sub.n] [less than or equal to].... (3)

For any x [member of] M, x [not equal to] sup M, by the condition ([A.sub.1]), there is a point [z.sup.*] [member of] [OMEGA] such that [z.sub.n] [less than or equal to] [z.sup.*], and since (E, [less than or equal to]) is an incomplete preference pseudo separable set, there is a [mathematical expression not reproducible]. By the definition of {[z.sub.n]}, we get

[mathematical expression not reproducible]. (4)

That is, [z.sup.*] [member of] [OMEGA] is an upper bound of total ordered subset M.

Let [x] = {y [member of] [OMEGA] | x ~ y}; then [x] is equivalence class set in [OMEGA]. Assuming that [OMEGA]/ ~ = {[x], x [member of] [OMEGA]} is a quotient set corresponding to the equivalence relation ~, then applying Lemmas 8 and 9, we get that the order relation [less than or equal to] in quotient set [OMEGA]/ ~ which is elicited by the incomplete preference relation [less than or equal to] in [OMEGA] is a partial order.

Take an arbitrary total ordered subset N [subset] [OMEGA]/ ~, next, to show that the set N has an upper bound in [OMEGA]/ ~.

Let

W = [union] [x] [subset] [OMEGA]. (5)

It is easy to know that W is total ordered subset in [OMEGA]. In fact, for any x, y [member of] W, there are [x], [y] [member of] N. Since N is total ordered subset in [OMEGA]/ ~, we can get that either [x] [less than or equal to] [y] or [y] [less than or equal to] [x] is valid. According to the relation between partial order and incomplete preference, we have that either x [less than or equal to] y or y [less than or equal to] x is valid. So W is total ordered subset in [OMEGA].

For any [z] [member of] N, by the definition of W, we get z [member of] W. Since every total ordered subset in [OMEGA] has an upper bound, there is [x.sub.0] [member of] W such that z [less than or equal to] [x.sub.0]. So [z] [less than or equal to] [[x.sub.0]]; that is, every total ordered subset in [OMEGA]/ ~ has an upper bound. Then applying Zorn lemma, we have that there is a maximal element [[x.sup.*]] in [OMEGA]/ ~.By the definition of [OMEGA]/ ~, we have that [x.sup.*] is the maximal element in [OMEGA].

Since [x.sup.*] is the maximal element in Q, there is u e T(x*) such that [x.sup.*] <u. Supposing that u [>>] [x.sup.*], the monotonicity of T together with u [member of] T([x.sup.*]), [x.sup.*] < u implies that there is v [member of] T(u) such that u [less than or equal to] v. It means u [member of] [OMEGA], and it is contradictory with which [x.sup.*] is the maximal element in Q. So u < [x.sup.*] is proved. Hence there are [x.sup.*] [member of] E, u [member of] [Tx.sup.*] such that [x.sup.*] ~ u; that is, T has a generalized fixed point.

Corollary 14. Let (E, [less than or equal to]) be an incomplete preference pseudo separable set, and let T : E [right arrow] [2.sup.E]\[PHI] be an order-increasing set-valued mapping. If T satisfies the following conditions:

([A.sub.1]) Every increasing sequence in ST(x) has an upper bound in ST(x)

(A2) There is [x.sub.0] [member of] E with [x.sub.0] [less than or equal to] u, for some u [member of] [Tx.sub.0]

([A.sub.3]) If x ~ y, x, y [member of] E, then Tx = Ty then T has a fixed point; that is, there are [x.sup.*] [member of] E, u [member of] [Tx.sup.*] such that [x.sup.*] [member of] [Tx.sup.*].

3. Existence of Nash Equilibrium Points in Generalized Games under Incomplete Preferences

The incomplete preference we present in the paper is more general order relation than the preference in the field of economic management and coincident with the reality of economic phenomenon. It can be applied to the existence of generalized Nash equilibrium of noncooperative game theory.

Definition 15. Let n be a positive integer greater than 1. An n-person noncooperative game consists of the following elements:

(1) The set of n players is denoted by I = {i = 1, 2, 3, ..., n};

(2) For any i [member of] I, let St be the strategy set of player i and ([S.sub.i], [[less than or equal to].sub.i]) be an incomplete preference pseudo separable set; denote S = [S.sub.1] x [S.sub.2] x ... x [S.sub.n];

(3) Let [P.sub.i] S [right arrow] U, i = 1, 2, 3, ..., n be the payoff function for player i; denote P = {[P.sub.1], [P.sub.2], ..., [P.sub.n]}.

The game is denoted by [GAMMA] = (N, S, P, U).

Every player in the n-person noncooperative game independently chooses his own strategy [x.sub.i] [member of] [S.sub.i], i = 1, 2, 3, ..., n, to maximize his payoff function [P.sub.i]([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] U. For any x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] S, denote

[x.sub.-i] = ([x.sub.1], [x.sub.2], ..., [x.sub.i-1], [x.sub.i+1], ..., [x.sub.n]) [member of] S,

[S.sub.-i] = [S.sub.1] x [S.sub.2] x ... x [S.sub.i-1] x [S.sub.i+1] x ... x [S.sub.n]. (6)

Then [x.sub.-i] [member of] [S.sub.-i], and x can be written as x = ([x.sub.i], [x.sub.-i]).

Definition 16. Let [gamma] = (N, S, P, U) be an n-person noncooperative game. The strategy [mathematical expression not reproducible] is said to be a generalized Nash equilibrium in the noncooperative game [GAMMA] = (N, S, P, U) under the incomplete preference, if there is strategy [mathematical expression not reproducible], for every i = 1, 2, 3, ..., n; the following order inequality holds

[mathematical expression not reproducible]. (7)

Lemma 17. Let [mathematical expression not reproducible] be an incomplete preference pseudo separable set. S = [S.sub.1] x [S.sub.2] x ... x [S.sub.n] is a coordinate ordering set composed of [S.sub.2], ..., [S.sub.n], for any x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] S, the order relation [[less than or equal to].sup.s] in S induced by the partial order [mathematical expression not reproducible], denoted as the following:

[mathematical expression not reproducible]. (8)

Then (S, [[less than or equal to].sup.s]) is an incomplete preference pseudo separable set.

Proof. First we show that (S, <s) is an incomplete preference set. Since [mathematical expression not reproducible] is incomplete preference set, for any x = [mathematical expression not reproducible]. So x [[less than or equal to].sup.s] x. Hence the order relation [[less than or equal to].sup.s] satisfies reflexive axiom.

For any x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]), and z = ([z.sub.1], [z.sub.2], ..., [z.sub.n]) [member of] S, which satisfy x [z.sub.1] y, y [z.sub.1] z [member of] S. By Definition 15, we have [mathematical expression not reproducible] is incomplete preference set, we have [mathematical expression not reproducible], is incomplete preference set.

Next we prove that the incomplete preference set (S, [[less than or equal to].sup.s]) is pseudo separable, for any [mathematical expression not reproducible] is pseudo separable,

Let [M.sub.i] be an arbitrary total ordered subset of set [S.sub.i; then there is denumerable set {[x.sup.n.sub.i]} c Mt such that if [x.sub.i] [member of] [M.sub.i], [x.sub.i] [not equal to] sup [M.sub.i], there is [mathematical expression not reproducible].

Define the following

M = [M.sub.1] x [M.sub.2] x ... x [M.sub.n]. (9)

Let [mathematical expression not reproducible]; then by the definition of M, we have [x.sup.n] [member of] M. This together with [mathematical expression not reproducible]. Hence the incomplete preference set (S, [[less than or equal to].sup.s]) is pseudo separable.

Theorem 18. Let [gamma] = (N, S, P, U) be an n-person noncooperative game. Suppose that, for any x e S, the payoff function [P.sub.i], i = 1, 2, 3, ..., n, satisfies the following conditions:

([G.sub.1]) Every total ordered subset in [P.sub.i]([S.sub.i], [x.sub.-i]) has an upper bound in [P.sub.i]([S.sub.i], [x.sub.-1]);

([G.sub.2]) Every increasing sequence in the inverse image {[z.sub.i] [member of] [S.sub.i] : [P.sub.i]([S.sub.i], [x.sub.-1]) is a maximal element of [P.sub.i]([S.sub.i], [x.sub.-i])} has an upper bound;

([G.sub.3]) For any x, y [member of] S, x [[less than or equal to].sup.s] y, if there is [z.sub.i] [member of] [S.sub.i] with [P.sub.i]([S.sub.i], [x.sub.-1]) to be a maximal element of [P.sub.i]([S.sub.i], [x.sub.-i]), then there is [mathematical expression not reproducible] is a maximal element of [P.sub.i]([S.sub.i], [x.sub.-1]);

([G.sub.4]) If there are p, q [member of] S such that p [[less than or equal to].sup.s]q and [P.sub.i]([q.sub.i], [p.sub.-i]) is a maximal element of [P.sub.i]([S.sub.i], [p.sub.-i]);

(G5) If p [??] q, p, q [member of] S such that [P.sub.i]([p.sub.i], [x.sub.-i]) = [P.sub.i]([q.sub.i], [x.sub.-i]). Then there is a generalized Nash equilibrium in the n-person noncooperative game [gamma] = (N, S, P, U).

Proof. Since [mathematical expression not reproducible] is an incomplete preference pseudo separable set, for every i = 1, 2, 3, ..., n, then, from Lemma 17, (S, [[less than or equal to].sup.s]) is also an incomplete preference pseudo separable set equipped with the product order [[less than or equal to].sup.s].

For every fixed i = 1, 2, 3, ..., n, define a set-valued mapping [mathematical expression not reproducible] as the following:

[T.sub.i](x) = {[z.sub.i] [member of] [S.sub.i] :

[P.sub.i]([p.sub.i], [x.sub.-1]) is a maximal element of [P.sub.i]([p.sub.i], [x.sub.-i])}, (10)

for all x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] S.

From assumption [G.sub.1] of this theorem, for every fixed element x e S, every total ordered subset in [P.sub.i]([p.sub.i], [x.sub.-i]) has an upper bound in (U, <u). Then applying Zorn Lemma, the set [P.sub.i]([p.sub.i], [x.sub.-i])} has a maximal element. Therefore, [T.sub.i](x) is a nonempty subset of [S.sub.i]. Then we define

T (x) = [T.sub.1] (x) x [T.sub.2] (x) x ... x [T.sub.n] (x), x [member of] S. (11)

For any arbitrary x [member of] S, with respect to the set T(x), we write

ST (x) = {x [member of] S : x [[less than or equal to].sup.s] z, z [member of] T (x)}. (12)

For every i = 1, 2, 3, ..., n, we have

[mathematical expression not reproducible]. (13)

Now we prove that the operator T satisfies the conditions in Theorem 18. Firstly, we will show that the operator T is order-increasing. For any given x [[less than or equal to].sup.s] y in S, and for any z = ([z.sub.1], [z.sub.2], ..., [z.sub.n]) [member of] T(x), for every i = 1, 2, ..., n, we have [z.sub.1] [member of] [T.sub.i](x); that is, [P.sub.i]([z.sub.i], [x.sub.-i]) is a maximal element of [P.sub.i]([S.sub.i], [x.sub.-i]). Then from hypothesis [G.sub.3] of this theorem, there is [mathematical expression not reproducible] is a maximal element of [P.sub.i]([S.sub.i], [x.sub.-i]); that is, [[omega].sub.i] [member of] [T.sub.i](y). Let [omega] = ([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]). We obtain that z [[less than or equal to].sup.s] [omega] and [omega] [member of] D(y). Hence the operator T is order-increasing.

From assumption [G.sub.2] of this theorem, every increasing sequence in [T.sub.i](x) has an upper bound. Then we can similarly show that every increasing sequence in [ST.sub.i](x) has an upper bound. In fact, take an arbitrary total ordered subset M [subset] ST(x). Since M is also a total ordered subset of incomplete preference pseudo separable set (E, [greater than or equal to]), there is denumerable set {[x.sub.n]} [subset] M such that for any x [member of] M, x [not equal to] sup M, there are [mathematical expression not reproducible].

From the above discussion, we have that there is e(x) [member of] [mathematical expression not reproducible] such that u(x) <se(x). Thus we obtain a mapping e : ST(x) [right arrow] T(x) satisfying the following order inequality: x [less than or equal to] u(x) [[less than or equal to].sup.s] e(x), u(x) [member of] T(x) with e(x) [member of] T(x). For any an increasing sequence {[x.sub.n]} in ST(x), we have that there is [z.sup.*] [member of] [OMEGA]. such that [x.sup.n] [[less than or equal to].sup.s] [z.sup.*]. By the definition of {[z.sub.n]}, we get

[mathematical expression not reproducible]. (14)

So we have that {[x.sub.n]} is an increasing sequence; then {e([x.sub.n])} is an increasing sequence in T(x). since every increasing sequence in T(x) has an upper bound in T(x), we have that there is [z.sup.*] [member of] T(x) such that e([x.sub.n]) [[less than or equal to].sup.s] [z.sup.*]. Since [x.sub.n] [[less than or equal to].sup.s] e([x.sub.n]), we have [x.sub.n] [[less than or equal to].sup.s] [z.sup.*]. Hence every increasing sequence in ST(x) has an upper bound in ST(x).

Then applying Theorem 13, it implies that the operator T has a generalized point; that is, there are [x.sup.*] [member of] E, u [member of] [Tx.sup.*] such that [x.sup.*] [??] u. Since the operator T is order-increasing, there are u [member of] T([x.sup.*]) and v eT(u) such that u [[less than or equal to].sup.s] v.

From assumption [G.sub.5] of this theorem, if [x.sup.*] [??] u, we can get

[P.sub.i]([x.sup.*.sub.i], [x.sub.-i]) = [P.sub.i]([u.sub.i], [x.sub.-i]). (15)

Since u e T([x.sup.*]), we have that [P.sub.i]([u.sub.i], [x.sup.*.sub.-i]) is a maximal element of [P.sub.i]([S.sub.i], [x.sup.*.sub.-i]). This together with [P.sub.i]([x.sup.*.sub.-i], [x.sub.-i]) = [P.sub.i]([u.sub.i], [x.sub.-i]) implies that [P.sub.i]([x.sup.*.sub.i], [x.sup.*.sub.-i]) is a maximal element of [P.sub.i]([S.sub.i], [x.sup.*.sub.-i]). It implies that for any [x.sup.*] = ([x.sup.*.sub.1], [x.sup.*.sub.2], ..., [x.sup.*.sub.n]) [member of] S, i = 1, 2, 3, ..., n, the following inequality is established

[mathematical expression not reproducible]. (16)

This shows that [x.sup.*] = ([x.sup.*.sub.1], [x.sup.*.sub.2], ..., [x.sup.*.sub.n]) [member of] S is a generalized Nash equilibrium in the n-person noncooperative game [GAMMA] = (N, S, P, U).

4. Conclusion

Incomplete preference is more general order relation than complete preference in the field of economic management, because restriction on order relation is eased. So it is more consistent with the reality of economic management phenomenons. The generalized game model under the incomplete preference can play an important application in economic management problems. Generalized game plays an important role to prove existence of general equilibrium.

But many economic problems ultimately come down to nonlinear problems which are denoted by the utility function without preference in an order infinite dimensional space. The traditional general game model cannot deal with the problems such as the utility function without preference, incomplete preference, order infinite dimension space, or nonlinear problem. Now since there are no ready-made methods to deal with the problems, new research methods must be sought. The generalized game model under the incomplete preference, which is proposed in this paper, can deal with the problems. But in this paper, the research is limited to the existence of the equilibrium. The stability of equilibrium and new game model which are more close to the reality are our next step research direction.

https://doi.org/10.1155/2018/3737253

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The author was supported by the National Social Science Fund of China (13CJL006), National Science Foundation for Post-doctoral Scientists of China (2014M551264), and Research Fund for the Doctoral Program of Changshu Institute of Technology (XE1509).

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Xingchang Li (iD) (1,2)

(1) School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou 215500, China

(2) Institute of Business & Economic Research, Harbin University of Commerce, Harbin 150028, China

Correspondence should be addressed to Xingchang Li; lxctsq@163.com

Received 18 April 2018; Accepted 25 September 2018; Published 9 October 2018

Academic Editor: Xinguang Zhang
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Title Annotation:Research Article
Author:Li, Xingchang
Publication:Journal of Function Spaces
Geographic Code:9CHIN
Date:Jan 1, 2018
Words:5191
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