Volume 2007, Article ID 56161,12pages doi:10.1155/2007/56161
Research Article
Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application
Ke-Qing Wu, Nan-Jing Huang, and Jen-Chih Yao
Received 7 June 2006; Revised 3 November 2006; Accepted 18 December 2006 Recommended by H. Bevan Thompson
By using the iterative technique and Nadler’s theorem, we construct a new iterative al- gorithm for solving a system of nonlinear inclusions in Banach spaces. We prove some new existence results of solutions for the system of nonlinear inclusions and discuss the convergence of the sequences generated by the algorithm. As an application, we show the existence of solution for a system of functional equations arising in dynamic program- ming of multistage decision processes.
Copyright © 2007 Ke-Qing Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is well known that the iterative technique is a very important method for dealing with many nonlinear problems (see, e.g., [1–4]). Let Ebe a real Banach space, let X be a nonempty subset ofE, and letA,B:X×X→Ebe two nonlinear mappings. Chang and Guo [5] introduced and studied the following nonlinear problem in Banach spaces:
A(u,u)=u, B(u,u)=u, (1.1)
which has been used to study many kinds of differential and integral equations in Ba- nach spaces. IfA=B, then problem (1.1) reduces to the problem considered by Guo and Lakshmikantham [1].
On the other hand, Huang et al. [6] introduced and studied the problem of finding u∈X,x∈Su, andy∈Tusuch that
A(y,x)=u, (1.2)
whereA:X×X→Xis a nonlinear mapping andS,T:X→2X are two set-valued map- pings. They constructed an iterative algorithm for solving this problem and gave an ap- plication to the problem of the general Bellman functional equation arising in dynamic programming.
LetA,B:X×X→Ebe two nonlinear mappings, letg:X→Ebe a nonlinear mapping, and letS,T:X→2X be two set-valued mappings. Motivated by above works, in this pa- per, we study the following system of nonlinear inclusions problem of finding u∈X, x∈Su, andy∈Tusuch that
A(y,x)=gu, B(x,y)=gu. (1.3) It is easy to see that the problem (1.3) is equivalent to the following problem: findu∈X such that
gu∈ATu,Su, gu∈BSu,Tu, (1.4) which was considered by Huang and Fang [7] wheng is an identity mapping. It is well known that problem (1.3) includes a number of variational inequalities (inclusions) and equilibrium problems as special cases (see, e.g, [8–10] and the references therein).
By using the iterative technique and Nadler’s theorem [11], we construct a new al- gorithm for solving the system of nonlinear inclusions problem (1.3) in Banach spaces.
We prove the existence of solution for the system of nonlinear inclusions problem (1.3) and the convergence of the sequences generated by the algorithm. As an application, we discuss the existence of solution for a system of functional equations arising in dynamic programming of multistage decision processes.
2. Preliminaries
LetPbe a cone inEand let “≤” be a partial order induced by the coneP, that is,x≤yif and only ify−x∈P. Recall that the conePis said to be normal if there exists a constant NP>0 such thatθ≤u≤vimplies thatu ≤NPv, whereθdenotes the zero element ofE.
A mapping A:E×E→Eis said to be mixed monotone if for all u1,u2,v1,v2∈E, u1≤u2andv1≤v2imply thatA(u1,v2)≤A(u2,v1).
We denote by CB(X) the family of all nonempty closed bounded subsets ofX. A set- valued mappingF:X→CB(X) is said to beH-Lipschitz continuous if there exists a con- stantλ >0 such that
HFx,F y≤λx−y, ∀x,y∈X, (2.1) whereH(·,·) denotes the Hausdorffmetric on CB(X), that is, for anyA,B∈CB(X),
H(A,B)=max
sup
x∈A
inf
y∈B
d(x,y), sup
y∈B
inf
x∈A
d(x,y)
. (2.2)
Definition 2.1. LetS,T:E→Ebe two single-valued mappings. A single-valued mapping A:E×E→Eis said to be (S,T)-mixed monotone if, for allu1,u2,v1,v2∈E,
u1≤u2, v1≤v2 imply thatASu1,Tv2
≤ASu2,Tv1
. (2.3)
Remark 2.2. It is easy to see that, ifS=T=I (I is the identity mapping), then (S,T)- mixed monotonicity ofAis equivalent to the mixed monotonicity ofA. The following example shows that the (S,T)-mixed monotone mapping is a proper generalization of the mixed monotone mapping.
Example 2.3. LetR=(−∞, +∞), letA:R×R→RandS,T:R→Rbe defined by
A(x,y)=xy, S(x)=x, T(x)= −x (2.4)
for allx,y∈R. Then it is easy to see thatAis an (S,T)-mixed monotone mapping. How- ever,Ais not a mixed monotone.
Definition 2.4. LetS,T:E→2Ebe two multivalued mappings. A single-valued mapping A:E×E→Eis said to be (S,T)-mixed monotone if, for allu1,u2,v1,v2∈E,u1≤u2and v1≤v2imply that
Ax1,y2
≤Ax2,y1
, ∀x1∈Su1,x2∈Su2, y1∈Tv1, y2∈Tv2. (2.5) Definition 2.5. If{xn} ⊂Esatisfiesx1≤x2≤ ··· ≤xn≤ ···orx1≥x2≥ ··· ≥xn≥ ···, then{xn}is said to be a monotone sequence.
Definition 2.6. LetD⊂E. A mappingg:D→Eis said to satisfy condition (C) if, for any sequence{xn} ⊂Dsatisfying{g(xn)}that is monotone,g(xn)→g(x) implies thatxn→x.
Remark 2.7. Ifg is reversible andg−1is continuous, then it is easy to see thatgsatisfies condition (C).
3. Iterative algorithm
In this section, by using Nadler’s theorem [11], we construct a new iterative algorithm for solving the system of nonlinear inclusions problem (1.3).
Letu0,v0∈E,u0< v0 (i.e.,u0≤v0 andu0=v0) and letD=[u0,v0]= {u∈E:u0≤ u≤v0}be an order interval inE. LetS,T:D→CB(D) andg:D→Esuch thatg(D)=E andgu0≤gv0. Suppose thatA:D×D→Eis an (T,S)-mixed monotone mapping and B:D×D→Eis a (S,T)-mixed monotone mapping satisfying the following conditions:
(i) for anyu,v∈D,u≤vimplies that
B(x,y)≤A(y,x), ∀x∈Su, y∈Tv; (3.1) (ii) there exist two constantsa,b∈[0, 1) such that
gu0+agv0−gu0
≤Bx0,y0
, Ay0,x0
≤gv0−bgv0−gu0
(3.2)
for allx0∈Su0andy0∈Tv0;
(iii) foru,v∈D,gu≤gvimplies thatu≤v.
Foru0andv0, we takex0∈Su0andy0∈Tv0. By virtue ofg(D)=E, there existu1,v1∈ Dsuch that
gu1=Bx0,y0
−agv0−gu0
, gv1=Ay0,x0
+bgv0−gu0
. (3.3)
It follows from (ii) that
gu0≤gu1, gv1≤gv0. (3.4)
By condition (i), we have
gv1=Ay0,x0
+bgv0−gu0
≥Bx0,y0
+bgv0−gu0
=gu1+ (a+b)gv0−gu0
≥gu1.
(3.5)
Therefore,gu0≤gu1≤gv1≤gv0. From condition (iii), we know thatu0≤u1≤v1≤v0. Now, by Nadler’s theorem [11], there existx1∈Su1andy1∈Tv1such that
x1−x0≤(1 + 1)HSu1,Su0
, y1−y0≤(1 + 1)HTv1,Tv0
. (3.6) In virtue ofg(D)=E, there existu2,v2∈Dsuch that
gu2=Bx1,y1
−agv1−gu1
, gv2=Ay1,x1
+bgv1−gu1
. (3.7) SinceBis (S,T)-mixed monotone andAis (T,S)-mixed monotone,
gu1=Bx0,y0
−agv0−gu0
≤Bx1,y1
−agv1−gu1
=gu2, gv2=Ay1,x1
+bv1−u1
≤Ay0,x0
+bgv0−gu0
=gv1. (3.8) It follows from condition (i) that
gu2=Bx1,y1
−agv1−gu1
≤Ay1,x1
−agv1−gu1
=gv2−(a+b)gv1−gu1
≤gv2.
(3.9)
Therefore,
gu0≤gu1≤gu2≤gv2≤gv1≤gv0. (3.10) So
u0≤u1≤u2≤v2≤v1≤v0. (3.11) By induction, we can get an iterative algorithm for solving the system of nonlinear inclu- sions problem (1.3) as follows.
Algorithm 3.1. Letu0,v0∈E,u0< v0, letD=[u0,v0]= {u∈E:u0≤u≤v0}be an order interval inE. LetS,T:D→CB(D) andg:D→Ewithg(D)=Eandgu0≤gv0. Suppose thatA:D×D→Eis an (T,S)-mixed monotone mapping andB:D×D→Eis (S,T)- mixed monotone mapping satisfying conditions (i)–(iii). Takingx0∈Su0andy0∈Tv0, we can get iterative sequences{un},{vn},{xn}, and{yn}as follows:
gun+1=Bxn,yn−agvn−gun, gvn+1=Ayn,xn
+bgvn−gun , xn+1∈Sun+1, xn+1−xn≤
1 + 1
n+ 1
HSun+1,Sun , yn+1∈Tvn+1, yn+1−yn≤
1 + 1 n+ 1
HTvn+1,Tvn,
(3.12)
gu0≤gu1≤gu2≤ ··· ≤gun≤ ··· ≤gvn≤ ··· ≤gv2≤gv1≤gv0, (3.13) u0≤u1≤u2≤ ··· ≤un≤ ··· ≤vn≤ ··· ≤v2≤v1≤v0 (3.14) for alln=0, 1, 2,. . . .
Remark 3.2. FromAlgorithm 3.1, we can get some new algorithms for solving some spe- cial cases of problem (1.3).
4. Existence and convergence
In this section, we will prove the existence of solutions for the system of nonlinear inclu- sions problem (1.3) and the convergence of sequences generated byAlgorithm 3.1.
Theorem 4.1. LetEbe a real Banach space, P⊂Ea normal cone inE,u0,v0∈Ewith u0< v0, andD=[u0,v0]. Letg:D→Ebe a mapping such thatg(D)=E,gu0≤gv0, andg satisfies condition (C). Suppose thatS,T:D→CB(D) are twoH-Lipschitz continuous map- pings with Lipschitz constantsα >0 andγ >0, respectively,A:D×D→Eis a (T,S)-mixed monotone mapping andB:D×D→Eis an (S,T)-mixed monotone mapping. Assume that conditions (i)–(iii) are satisfied and
(iv) there exists a constantβ∈[0, 1) witha+b+β <1 such that, for anyu,v∈D,u≤v implies that
A(y,x)−B(x,y)≤β(gv−gu) (4.1) for allx∈Su,y∈Tv.
Then there existu∗∈D,x∗∈Su∗, andy∗∈Tu∗such that gu∗=Ay∗,x∗, gu∗=Bx∗,y∗,
un−→u∗, vn−→u∗, xn−→x∗, yn−→y∗ (n−→ ∞). (4.2)
Proof. It follows from (3.12), (3.13), (3.14), and condition (iv) that
θ≤gvn−gun=Ayn−1,xn−1
−Bxn−1,yn−1
+ (a+b)gvn−1−gun−1
≤βgvn−1−gun−1
+ (a+b)gvn−1−gun−1
=(a+b+β)gvn−1−gun−1
≤ ··· ≤(a+b+β)ngv0−gu0
(4.3)
for alln=1, 2,. . . .Since the conePis normal, we have
gvn−gun≤NP(a+b+β)ngv0−gu0. (4.4) Thus, the conditiona+b+β∈[0, 1) implies that
gvn−gun−→0 (n−→ ∞). (4.5)
Now we prove that{gun}is a Cauchy sequence. In fact, for anyn,m∈N, ifn≤m, then it follows from (3.14) that
gvn−gun−
gum−gun=gvn−gum∈P (4.6) and sogum−gun≤gvn−gun. SincePis a normal cone, we conclude that
gum−gun≤NPgvn−gun. (4.7) Similarly, ifn > m, we havegun−gum≤gvm−gumand so
gun−gum≤NPgvm−gum. (4.8) It follows from (4.7) and (4.8) that
gun−gum≤NPmax gvn−gun,gvm−gum (4.9) for alln,m∈N. From (4.5) and (4.9), we know that{gun}is a Cauchy sequence inE.
Let gun→k∗∈Easn→ ∞. Since g(D)=E, there existsu∗∈D such thatgu∗=k∗. Now (4.5) implies thatgvn→gu∗asn→ ∞. Sincegsatisfies condition (C), we know that un→u∗andvn→u∗asn→ ∞. Now the closedness ofPimplies thatgun≤gu∗≤gvn for alln=1, 2,. . . .It follows from condition (iii) thatun≤u∗≤vnfor alln=1, 2,. . . .By (3.12) and theH-Lipschitz continuity of mappingsSandT, we have
xn+1−xn≤
1 + 1 n+ 1
HSun+1,Sun
≤
1 + 1 n+ 1
·αun+1−un, yn+1−yn≤
1 + 1
n+ 1
HTvn+1,Tvn≤
1 + 1 n+ 1
·γvn+1−vn.
(4.10)
Thus,{xn}and{yn}are both Cauchy sequences inD. Let
nlim→∞xn=x∗, lim
n→∞yn=y∗. (4.11)
Next, we prove thatx∗∈Su∗andy∗∈Tu∗. In fact, dx∗,Su∗=inf x∗−ω:ω∈Su∗
≤x∗−xn+dxn,Su∗≤x∗−xn+HSun,Su∗ (4.12) and sod(x∗,Su∗)=0. It follows thatx∗∈Su∗. Similarly, we havey∗∈Tu∗.
We now prove thatgu∗=A(y∗,x∗) and gu∗=B(x∗,y∗). Sinceun≤u∗≤vn,B is (S,T)-mixed monotone andAis (T,S)-mixed monotone, it follows from (i) that
gun+1=Bxn,yn
−agvn−gun
≤Bx∗,y∗−agvn−gun
≤Ay∗,x∗+bgvn−gun
−(a+b)gvn−gun
≤Ayn,xn
+bgvn−gun
−(a+b)gvn−gun
≤gvn+1.
(4.13)
Therefore,gu∗=A(y∗,x∗)=B(x∗,y∗). This completes the proof.
Theorem 4.2. LetEbe a real Banach space, P⊂Ea normal cone inE,u0,v0∈Ewith u0< v0, andD=[u0,v0]. Letg:D→Ebe a mapping such thatg(D)=E,gu0≤gv0, and g satisfies condition (C). Suppose thatS,T:D→CB(D) are twoH-Lipschitz continuous mappings with Lipschitz constantsα >0 andγ >0, respectively,A:D×D→Eis an (T,S)- mixed monotone mapping, andB:D×D→Eis a (S,T)-mixed monotone mapping. Assume that conditions (i)–(iii) are satisfied and
(iv) for anyu,v∈D,u≤vimplies that
A(y,x)−B(x,y)≤L(gv−gu) (4.14) for allx∈Su,y∈Tv, whereL:E→Eis a bounded linear mapping with a spectral radiusr(L)=β <1 anda+b+β <1.
Then there existu∗∈D,x∗∈Su∗, andy∗∈Tu∗such that gu∗=Ay∗,x∗, gu∗=Bx∗,y∗,
un−→u∗, vn−→u∗, xn−→x∗, yn−→y∗ (n−→ ∞). (4.15) Proof. It follows from (3.12), (3.13), (3.14), and condition (iv)that
θ≤gvn−gun=Ayn−1,xn−1
−Bxn−1,yn−1
+ (a+b)gvn−1−gun−1
≤Lgvn−1−gun−1
+ (a+b)gvn−1−gun−1
≤
L+ (a+b)Igvn−1−gun−1
=Jgvn−1−gun−1
(4.16)
for alln=1, 2,. . ., whereJ=L+ (a+b)IandIis the identity mapping. By induction, we conclude that
θ≤gvn−gun≤Jngv0−gu0
(4.17)
for alln=1, 2,. . . .Sincer(L)=β <1, from [12, Example 10.3(b) and Theorem 10.3(b)]
by Rudin, we have
nlim→∞Jn1/n=r(J)≤a+b+β <1. (4.18)
This implies that there existsn0∈Nsuch that
Jn≤(a+b+β)n, ∀n≥n0. (4.19) SincePis a normal cone anda+b+β <1, it follows from (4.17) and (4.19) thatgvn− gun →0 asn→ ∞. The rest argument is similar to the corresponding part of the proof inTheorem 4.1and we omit it. This completes the proof.
IfS=TinTheorem 4.1, we have the following result.
Corollary 4.3. LetEbe a real Banach space,P⊂Ea normal cone inE,u0,v0∈Ewith u0< v0, andD=[u0,v0]. Letg:D→Ebe a mapping such thatg(D)=E,gu0≤gv0, andg satisfies (iii) and condition (C). Suppose thatS:D→CB(D) isH-Lipschitz continuous with Lipschitz constantα >0, andA,B:D×D→Eare both (S,S)-mixed monotone mappings such that
(B1) for anyu,v∈D,u≤vimplies that
B(x,y)≤A(y,x), ∀x∈Su, y∈Sv; (4.20) (B2) for allu,v∈D,u≤v, there existsβ∈[0, 1) such that
A(y,x)−B(x,y)≤β(gv−gu); (4.21) for allx∈Su,y∈Sv;
(B3) there area,b∈[0, 1) witha+b+β <1 such that gu0+agv0−gu0
≤Bu0,v0
, Av0,u0
≤gv0−bgv0−gu0
. (4.22)
Then there existu∗∈Dandx∗,y∗∈Su∗such that gu∗=Bx∗,y∗=A(y∗,x∗), lim
n→∞un=lim
n→∞vn=u∗, (4.23) where
gun+1=Bun,vn
−agvn−gun
, gvn+1=Avn,un
+bgvn−gun
(4.24)
for alln=1, 2,. . . .
IfS=IinCorollary 4.3, we have the following result.
Corollary 4.4. LetEbe a real Banach space,P⊂Ea normal cone inE,u0,v0∈E,u0< v0, andD=[u0,v0]. Letg:D→Ebe a mapping such thatg(D)=E,gu0≤gv0, andgsatisfies (iii) and condition (C). Suppose thatA,B:D×D→Eare both mixed monotone and satisfy the following conditions:
(C1) there existsβ∈[0, 1) such that
A(v,u)−B(u,v)≤β(gv−gu) (4.25) for allu,v∈Dwithu≤v;
(C2) for allu,v∈D,u≤vimplies that
B(u,v)≤A(v,u); (4.26)
(C3) there area,b∈[0, 1) witha+b+β <1 such that gu0+agv0−gu0
≤Bu0,v0
, Av0,u0
≤gv0−bgv0−gu0
. (4.27)
Then there existsu∗∈Dsuch that
gu∗=Au∗,u∗=Bu∗,u∗, lim
n→∞un=lim
n→∞vn=u∗, (4.28) where
gun+1=Bun,vn−agvn−gun, gvn+1=Avn,un+bgvn−gun (4.29) for alln=1, 2,. . . .
5. An application
Dynamic programming, because of its wide applicability, has evoked much interest among people of various discipline. See, for example, [13–17] and the references therein.
LetY andZ be two Banach spaces,G⊂Y a state space,Δ⊂Za decision space, and R=(−∞, +∞). We denote byB(G) the set of all bounded real-valued functional defined onG. Definef =supx∈G|f(x)|. Then (B(G), · ) is a Banach space. Let
P= f ∈B(G) : f(x)≥0,∀x∈G. (5.1) Obviously,Pis a normal cone. In this section, we consider a system of functional equa- tions as follows.
Find a bounded functional f :G→Rsuch that f1∈S f(x), f2∈T f(x), g f(x)=sup
y∈Δ
ϕ(x,y) +F1
x,y,f1
W(x,y),f2
W(x,y), g f(x)=sup
y∈Δ
ϕ(x,y) +F2
x,y,f2
W(x,y),f1
W(x,y)
(5.2)
for allx∈G, whereW:G×Δ→G,ϕ:G×Δ→R,F1,F2:G×Δ×R×R→R,S,T: B(G)→2B(G), andg:B(G)→B(G).
As an application ofTheorem 4.1, we have the following result concerned with the existence of solution for the system of functional equations problem (5.2).
Theorem 5.1. Suppose that (1)ϕ,F1, andF2are bounded;
(2) there exist two bounded functionalsu0,v0:G→Rwithu0=v0,u0(x)≤v0(x) for allx∈G, and suppose thatS,T:D=[u0,v0]→CB(D) areH-Lipschitz continuous with Lipschitz constantsα >0 andγ >0, respectively;
(3)g:D→B(G) satisfiesg(D)=B(G),gu0≤gv0, and
(a) for any{un} ⊂Dwith{gun}being monotone,u∈D, ifgun→gu, thenun→ u;
(b) for any u,v∈D, ifu(x)≤v(x), for allx∈G, thengu(x)≤gv(x), for all x∈G;
(4) there exists a constantβ∈[0, 1) such that, for anyu,v∈D, ifu(x)≤v(x) for all x∈G, then
F1
x,y,ωW(x,y),zW(x,y)−F2
x,y,zW(x,y),ωW(x,y)
≤βgv(x)−gu0(x) (5.3)
for allz∈Su,ω∈Tv,x∈G, andy∈Δ;
(5) for anyu,v∈Dwithu(x)≤v(x) for allx∈G, F2
x,y,zW(x,y),ωW(x,y)≤F1
x,y,ωW(x,y),zW(x,y) (5.4)
for allz∈Su,ω∈Tv,x∈G, andy∈Δ;
(6) for anyz∈Su0,ω∈Tv0,x∈G, andy∈Δ, gu0(x) +agv0(x)−gu0(x)≤F2
x,y,zW(x,y),ωW(x,y), F1
x,y,ωW(x,y),zW(x,y)≤gv0(x)−bgv0(x)−gu0(x), (5.5)
wherea,b∈[0, 1) witha+b+β <1;
(7) for anyu1,u2,v1,v2∈D, ifu1(x)≤u2(x) andv1≤v2(x) for allx∈G, then F2
x,y,y1
W(x,y),x2
W(x,y)≤F2
x,y,y2
W(x,y),x1
W(x,y), F1(x,y,x1
W(x,y),y2
W(x,y)≤F1
x,y,x2
W(x,y),y1
W(x,y) (5.6)
for allx1∈Su,x2∈Su2,y1∈Tv1,y2∈Tv2,x∈G, andy∈Δ.
Then there existu∗∈D,z∗∈Su∗, andω∗∈Tu∗such that gu∗=sup
y∈Δ ϕ(x,y) +F1
x,y,ω∗W(x,y),z∗W(x,y), gu∗=sup
y∈Δ ϕ(x,y) +F2
x,y,z∗W(x,y),ω∗Wx,y) (5.7)
for allx∈G.
Proof. For anyu,v∈D, we define the mappingsA,Bas follows:
A(u,v)(x)=sup
y∈Δ
ωx,y) +F1
x,y,uW(x,y),vWx,y), B(u,v)(x)=sup
y∈Δ
ωx,y) +F2
x,y,uW(x,y,vW(x,y) (5.8)
for allx∈G. From (1.1) and (4.7), we know thatA,B:D×D→B(G) are (T,S)-mixed monotone and (S,T)-mixed monotone, respectively. By assumptions (1.3)–(4.5), it is easy to check thatA,BandS,Tsatisfy all the conditions ofTheorem 4.1. Thus,Theorem 4.1implies that there existu∗∈D,z∗∈Su∗, andω∗∈Tu∗such thatgu∗=A(ω∗,z∗)= B(z∗,ω∗), that is,
gu∗=sup
y∈Δ ϕ(x,y) +F1
x,y,ω∗W(x,y),z∗W(x,y),
gu∗=sup
y∈Δ ϕ(x,y) +F2
x,y,z∗W(x,y),ω∗W(x,y)
(5.9)
for allx∈G. This completes the proof.
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Ke-Qing Wu: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Email address:[email protected]
Nan-Jing Huang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Email address:[email protected]
Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
Email address:[email protected]
