# Global Dynamical Systems Involving Generalizedf-Projection Operators and Set-Valued Perturbation in Banach Spaces

Authors: Yun-zhi Zou; Xi Li; Nan-jing Huang; Chang-yin Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.


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## Body

## 1. Introduction

It is well known that dynamics system has long time been an interest of many researchers. This is largely due to its extremely wide applications in a huge variety of scientific fields, for instance, mechanics, optimization and control, economics, transportation, equilibrium, and so on. For details, we refer readers to references [1–10] and the references therein.In 1994, Friesz et al. [3] introduced a class of dynamics named global projective dynamics based on projection operators. Recently, Xia and Wang [7] analyzed the global asymptotic stability of the dynamical system proposed by Friesz as follows:
(1.1)dxdt=PK(x-ρN(x))-x,
where N:ℝn→ℝn is a single-valued function, ρ>0 is a constant, PKx denotes the projection of the point x on K; here K⊂ℝn is a nonempty, closed, and convex subset.Later, in 2006, Zou et al. [9] studied a class of global set-valued projected dynamical systems as follows:
(1.2)dx(t)dt∈PK(g(x(t))-ρN(x(t))-g(x(t))),fora.a.t∈[0,J],x(0)=b,
where N:ℝn→2ℝn is a set-valued function, g:ℝn→ℝn is a single-valued function, ρ>0 is a constant, PKx denotes the projection of the point x on K, b is a given point in ℝn.The concept of generalizedf-projection operator was first introduced by Wu and Huang [11] in 2006. They also proved that the generalized f-projection operator is an extension of the projection operator PK in Rn and it owns some nice properties as PK does; see [12, 13]. Some applications of generalized f-projection operator are also given in [11–13]. Very recently, Li et al. [14] studied the stability of the generalized f-projection operator with an application in Banach spaces. We would like to point out that Cojocaru [15] introduced and studied the projected dynamical systems on infinite Hilbert spaces in 2002.To explore further dynamic systems in infinite dimensional spaces in more general forms has been one of our major motivations and efforts recently, and this paper is a response to those efforts. In this paper, we introduce and study a new class of generalized dynamical systems involving generalizedf-projection operators. By using the fixed-point theorem due to Nadler [16], we prove that the equilibrium points set of this class of generalized global dynamical systems is nonempty and closed. We also show that the solutions set of the systems with set-valued perturbation is continuous with respect to the initial value. The results presented in this paper generalize many existing results in recent literatures.


## 2. Preliminaries

Let𝒳 be a Banach space and let K⊂𝒳 be a closed convex set, let N:𝒳→2𝒳 be a set-valued mapping, and let g:𝒳→𝒳 be a single-valued mapping. The normalized duality mapping J from 𝒳 to 𝒳* is defined by
(2.1)J(x)={x*∈X*:〈x,x*〉=‖x‖2=‖x*‖2},
for x∈X. For convenience, we list some properties of J(·) as follows.X is a smooth Banach space, J(·) is single valued and hemicontinuous; that is, J is continuous from the strong topology of 𝒳 to the weak* topology of 𝒳*.LetC(𝒳) denote the family of all nonempty compact subsets of 𝒳 and let ℋ(·,·) denote the Hausdorff metric on C(𝒳) defined by
(2.2)H(A,B)=max⁡⁡{sup⁡a∈Ainf⁡b∈Bd(a,b),sup⁡b∈Binf⁡a∈Ad(a,b)},∀A,B∈C(X).In this paper, we consider a new class of generalized set-valued dynamical system, that is, to find those absolutely continuous functionsx(·) from [0,h]→𝒳 such that
(2.3)dx(t)dt∈ΠKf(g(x(t))-ρN(x(t)))-g(x(t)),fora.a.t∈[0,h],x(0)=b,
where b∈𝒳, ρ>0 is a constant and f:K→R∪{+∞} is proper, convex, and lower semicontinuous and ΠKf:𝒳→2K is a generalized f-projection operator denoted by
(2.4)ΠKfx={u∈K:G(J(x),u)=inf⁡ξ∈KG(J(x),ξ)},∀x∈X.It is well known that many problems arising in the economics, physical equilibrium analysis, optimization and control, transportation equilibrium, and linear and nonlinear mathematics programming problems can be formulated as projected dynamical systems (see, e.g., [1–10, 15, 17] and the references therein). We also would like to point out that problem (2.3) includes the problems considered in Friesz et al. [3], Xia and Wang [7], and Zou et al. [9] as special cases. Therefore, it is important and interesting to study the generalized projected dynamical system (2.3).Definition 2.1.
A pointx* is said to be an equilibrium point of global dynamical system (2.3), if x* satisfies the following inclusion:
(2.5)0∈ΠKf(g(x)-ρN(x))-g(x).Definition 2.2.
A mappingN:𝒳→𝒳 is said to be
(i)α-strongly accretive if there exists some α>0 such that
(2.6)(N(x)-N(y),J(x-y))≥α∥x-y∥2,∀x,y∈K;
(ii)ξ-Lipschitz continuous if there exists a constant ξ≥0 such that
(2.7)‖N(x)-N(y)‖≤ξ‖x-y‖,∀x,y∈K.Definition 2.3.
A set-valued mappingT:𝒳→𝒳 is said to be ξ-Lipschitz continuous if there exists a constant ξ>0 such that
(2.8)H(T(x),T(y))≤ξ‖x-y‖,∀x,y∈K,
where ℋ(·,·) is the Hausdorff metric on C(𝒳).Lemma 2.4 (see [14]).
Let𝒳 be a real reflexive and strictly convex Banach space with its dual 𝒳* and let K be a nonempty closed convex subset of 𝒳. If f:K→R∪{+∞} is proper, convex, and lower semicontinuous, then ΠKf is single valued. Moreover, if 𝒳 has Kadec-Klee property, then ΠKf is continuous.Lemma 2.5 (see [18]).
Let𝒳 be a real uniformly smooth Banach space. Then 𝒳 is q-uniformly smooth if and only if there exists a constant Cq>0 such that, for all x,y∈𝒳,
(2.9)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+Cq‖y‖q.Lemma 2.6 (see [19]).
Let(𝒳,d) be a complete metric space and let T1,T2 be two set-valued contractive mappings with same contractive constants θ∈(0,1). Then
(2.10)H(F(T1),F(T2))≤11-θsup⁡x∈X⁡H(T1(x),T2(x)),
where F(T1) and F(T2) are fixed-point sets of T1 and T2, respectively.Lemma 2.7 (see [19]).
Let𝒳 be a real strictly convex, reflexive, and smooth Banach space. For any x1,x2∈𝒳, let x^1=ΠKfx1 and x^2=ΠKfx2. Then
(2.11)〈J(x1)-J(x2),x^1-x^2〉≥2M2δ(‖x^1-x^2‖2M),
where
(2.12)M=∥x^1∥2+∥x^2∥22.We say that𝒳 is 2-uniformly convex and 2-uniformly smooth Banach space if there exist k,c>0 such that
(2.13)δX(ϵ)≥kϵ2,ρX(t)≤ct2,
where
(2.14)δX(ϵ)=inf{1-∥x+y2∥:∥x∥=∥y∥=1,∥x-y∥≥ϵ},ρX(t)=sup⁡{12(‖x+y‖+‖x-y‖)-1:‖x‖=1,‖y‖≤t}.Based on Lemma2.7, we can obtain the following lemma.Lemma 2.8.
Let𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Then
(2.15)‖ΠKfx-ΠKfy‖≤64ck‖x-y‖,∀x,y∈X.Proof.
According to Lemma2.7, we have
(2.16)〈J(x)-J(y),ΠKfx-ΠKfy〉≥2M12δ(‖ΠKfx-ΠKfy‖2M1),
where
(2.17)M1=∥ΠKfx∥2+∥ΠKfy∥22.
Since δX(ϵ)≥kϵ2, (2.16) yields
(2.18)‖ΠKfx-ΠKfy‖≤2k‖J(x)-J(y)‖.
From the property of J(·), we have
(2.19)‖J(x)-J(y)‖≤2M22ρX(4‖x-y‖/M2)‖x-y‖≤32c‖x-y‖.
It follows from (2.18) and (2.19) that
(2.20)‖ΠKfx-ΠKfy‖≤64ck‖x-y‖.
This completes the proof.


## 3. Equilibrium Points Set

In this section, we prove that the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.Theorem 3.1.
Let𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Let N:𝒳→C(𝒳) be μ-Lipschitz continuous and let g:𝒳→𝒳 be α-Lipschitz continuous and β-strongly accretive. If
(3.1)1+α2-2βC2+64ck(α+ρμ)<1,
then the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.Proof.
Let(3.2)T(x)=x-g(x)+ΠKf(g(x)-ρN(x)),∀x∈K.
Since N:𝒳→C(𝒳) and ΠKf are continuous, we know that T:𝒳→C(𝒳). From Definition 2.1, it is easy to see that x* is an equilibrium point of the generalized set-valued dynamical system (2.3) if and only if x* is a fixed-point of T in 𝒳, that is:
(3.3)x*∈T(x*)=x*-g(x*)+ΠKf(g(x*)-ρN(x*)).
Thus, the equilibrium points set of (2.3) is the same as the fixed-points set of T. We first prove that F(T) is nonempty. In fact, for any x,y∈𝒳 and a1∈T(x), there exists u∈N(x) such that
(3.4)a1=x-g(x)+ΠKf(g(x)-ρu).
Since u∈N(x), and N:𝒳→C(𝒳), it follows from Nadler [16] that there exists v∈N(y) such that
(3.5)‖u-v‖≤H(N(x),N(y)).
Let
(3.6)a2=y-g(y)+ΠKf(g(y)-ρv).
Then a2∈T(y). From (3.4) to (3.6), we have
(3.7)‖a1-a2‖=‖x-y-(g(x)-g(y))+ΠKf(g(x)-ρu)-ΠKf(g(y)-ρ(v))‖≤‖x-y-(g(x)-g(y))‖+‖ΠKf(g(x)-ρu)-ΠKf(g(y)-ρ(v))‖.
Since g is α-Lipschitz continuous and β-strongly accretive,
(3.8)‖x-y-(g(x)-g(y)‖2≤‖x-y‖2-2〈g(x)-g(y),J(x-y)〉+C2‖g(x)-g(y)‖2≤(1+α2-2βC2)‖x-y‖2.
From Lemma 2.8, where ΠKf is Lipchitz continuous, we have
(3.9)‖ΠKf(g(x)-ρu)-ΠKf(g(y)-(ρv))‖≤64ck(‖g(x)-g(y)‖+ρ‖u-v‖)≤64ck(α‖x-y‖+ρ‖u-v‖).
From the selection of v and the Lipschitz continuity of N,
(3.10)‖u-v‖≤H(N(x),N(y))≤μ‖x-y‖.
In light of (3.7)–(3.10), we have
(3.11)‖a1-a2‖≤(1+α2-2βC2+64ck(α+ρμ))‖x-y‖=L‖x-y‖,
where
(3.12)L=1+α2-2βC2+64ck(α+ρμ).
Now (3.11) implies that
(3.13)d(a1,T(y))=inf⁡a2∈T(y)‖a1-a2‖≤L‖x-y‖.
Since a1∈T(x) is arbitrary, we have
(3.14)sup⁡a1∈T(x)d(a1,T(y))≤L‖x-y‖.
Similarly, we can prove
(3.15)sup⁡a2∈T(y)d(T(x),a2)≤L‖x-y‖.
From (3.14), (3.15), and the definition of the Hausdorff metric ℋ on C(𝒳), we have
(3.16)H(T(x),T(y))≤L‖x-y‖,∀x,y∈K.
Now the assumption of the theorem implies that L<1 and so T(x) is a set-valued contractive mapping. By the fixed-point theorem of Nadler [16], there exists x* such that x*∈T(x*), and thus x* is the equilibrium point of (2.3). This means that F(T) is nonempty.
Now we prove thatF(T) is closed. Let {xn}⊂F(T) with xn→x0(n→∞). Then xn∈T(xn) and (3.16) imply that
(3.17)H(T(xn),T(x0))≤L‖xn-x0‖.
Thus,
(3.18)d(x0,T(x0))≤‖x0-xn‖+d(xn,T(xn))+H(T(xn),T(x0))≤(1+L)‖xn-x0‖⟶0asn→∞.
It follows that x0∈F(T) and so F(T) are closed. This completes the proof.Remark 3.2.
Theorem3.1 is a generalization of Theorem 1 in Zou et al. [9] from ℝn to Banach space 𝒳.


## 4. Sensitivity of the Solutions Set

In this section, we study the sensitivity of the solutions set of the generalized dynamical system with set-valued perturbation for (2.3) as follows:
(4.1)dx(t)dt∈ΠKf(g(x(t))-ρN(x(t)))-g(x(t))+F(x(t)),fora.a.[t∈0,h],x(0)=b,
where g and b are the same as in (2.3), F:𝒳→2𝒳 is a set-valued mapping, and N:𝒳→𝒳 is a single-valued mapping. Let S(b) denote the set of all solutions of (4.1) on [0,h] with x(0)=b.Now we prove the following result.Theorem 4.1.
Let𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Let g:𝒳→𝒳 be α-Lipschitz continuous, let N:𝒳→𝒳 be μ-Lipschitz continuous, and let F:𝒳→C(𝒳) be a ω-Lipschitz continuous set-valued mapping with compact convex values. If
(4.2)64ck(α+ρμ)+α+ω<1,h(64ck(α+ρμ)+α+ω)<1,
then S(b) is nonempty and continuous.Proof.
Let(4.3)M(x)=ΠKf(g(x)-ρN(x))-g(x)+F(x).
Then M:𝒳→C(𝒳) is a set-valued mapping with compact convex values since F:𝒳→C(𝒳) is a set-valued mapping with compact convex values. For any x1,x2∈𝒳 and a1∈M(x1), there exists u∈F(x1) such that
(4.4)a1=ΠKf(g(x1)-ρN(x1))-g(x1)+u.
Since u∈F(x1), and F:𝒳→C(𝒳), it follows from Nadler [16] that there exists v∈F(x2) such that
(4.5)‖u-v‖≤H(F(x1),F(x2)).
Let
(4.6)a2=ΠKf(g(x2)-ρN(x2))-g(x2)+v.
Then a2∈M(x2). From (4.4) and (4.6), we have
(4.7)‖a1-a2‖=‖ΠKf(g(x1)-ρN(x1))-ΠKf(g(x2)-ρN(x2))-(g(x2)-g(x2))+u-v‖≤‖g(x1)-g(x2)‖+‖ΠKf(g(x)-ρN(x1))-ΠKf(g(y)-ρN(x2))‖+‖u-v‖.
Since g is α-Lipschitz continuous,
(4.8)‖g(x1)-g(x2)‖≤α‖x2-x2‖.
From Lemma 2.8, ΠKf is Lipschitz continuous. It follows from the continuity of N and g that
(4.9)‖ΠKf(g(x1)-ρN(x1))-ΠKf(g(x2)-ρN(x2))‖≤64ck(‖g(x1)-g(x2)‖+ρ‖N(x1)-N(x2)‖)≤64ck(α+ρμ)‖x1-x2‖.
From the selection of v and the Lipschitz continuity of F, we know
(4.10)‖u-v‖≤H(F(x1),F(x2))≤ω‖x1-x2‖.
In light of (4.7)–(4.10), we have
(4.11)‖a1-a2‖≤(α+64ck(α+ρμ)+ω)‖x1-x2‖=θ‖x-y‖,
where
(4.12)θ=α+64ck(α+ρμ)+ω.
Now (4.11) implies that
(4.13)d(a1,M(x2))=inf⁡a2∈M(x2)‖a1-a2‖≤θ‖x1-x2‖.
Since a1∈M(x1) is arbitrary, we obtain
(4.14)sup⁡a1∈M(x1)d(a1,M(x2))≤θ‖x1-x2‖.
Similarly, we can prove
(4.15)sup⁡a2∈M(x2)d(M(x1),a2)≤θ‖x1-x2‖.
From (4.13), to (4.15), and the definition of the Hausdorff metric ℋ on C(𝒳), we have
(4.16)H(M(x1),M(x2))≤θ‖x1-x2‖,∀x1,x2∈X.
Now (4.2) implies that 0<θ<1, and so M(x) is a set-valued contractive mapping. Let
(4.17)Q(x,b)={y∈C([0,h],X)∣y(t)=b+∫0t‍z(s)ds,z(s)∈M(x(s))},
where
(4.18)C([0,h],X)={f:[0,h]⟶X∣fiscontinuous}.
Since M:𝒳→C(𝒳) is a continuous set-valued mapping with compact convex values, by the Michael's selection theorem (see, e.g., Theorem 16.1 in [20]), we know that Q(x,b) is nonempty for each x and b∈𝒳. Moreover, it is easy to see that the set of fixed-points of Q(x,b) coincides with S(b). It follows from [21] or [8] that Q(x,b) is compact and convex for each x and b∈𝒳. Suppose that bm is the initial value of (4.1); that is, x(0)=bm(m=0,1,2,…) and bm→b0(m→∞). Since
(4.19)Q(x,b0)=Q(x,bm)-bm+b0,
it is obvious that Q(x,bm) converges uniformly to Q(x,b0).
Next we prove thatQ(x,bm) is a set-valued contractive mapping. For any given x1,x2∈C([0,h],𝒳), since M:𝒳→C(𝒳) is a continuous set-valued mapping with compact convex values, by the Michael's selection theorem (see, e.g., Theorem 16.1 in [20]), we know that M(x1(s)) has a continuous selection r1(s)∈M(x1(s)). Let
(4.20)c1(t)=bm+∫0t‍r1(s)ds.
Then c1∈Q(x1,bm). Since r1(s)∈M(x1(s)) is measurable and M(x2(s)) is a measurable mapping with compact values, we know that there exists a measurable selection r2(s)∈M(x2(s)) such that
(4.21)‖r1(s)-r2(s)‖≤H(M(x1(s)),M(x2(s))).
Thus, it follows from (4.16) that
(4.22)‖r1(s)-r2(s)‖≤H(M(x1(s)),M(x2(s)))≤θ‖x1-x2‖.
Let
(4.23)c2(t)=bm+∫0t‍r2(s)ds.
Then c2∈Q(x2,bm) and
(4.24)‖c1-c2‖≤∫0h‍‖r1(s)-r2(s)‖ds≤hH(M(x1(s)),M(x2(s)))≤hθ‖x1-x2‖.
Hence, we have
(4.25)d(c1,Q(x2,bm))=inf⁡c2∈Q(x2,bm)‖c1-c2‖≤hθ‖x1-x2‖.
Since c1∈Q(x1,bm) is arbitrary, we obtain
(4.26)sup⁡c1∈Q(x1,bm)⁡d(c1,Q(x2,bm))≤hθ‖x1-x2‖.
Similarly, we can prove that
(4.27)sup⁡c2∈Q(x2,bm)⁡d(Q(x1,bm),c2)≤hθ‖x1-x2‖.
From the definition of the Hausdorff metric ℋ on C(𝒳), (4.26) and (4.27) imply that
(4.28)H(Q(x1,bm),Q(x2,bm))≤hθ‖x1-x2‖,∀x1,x2∈X,m=0,1,2,….
Since hθ<1, it is easy to see that Q(x,b) has a fixed-point for each given b∈𝒳, and so S(b) is nonempty for each given b∈𝒳. Setting
(4.29)Wm(x)=Q(x,bm),m=0,1,2,…,
we know that Wm(x) are contractive mappings with the same contractive constant hθ. By Lemma 2.6 and (4.28), we have
(4.30)H(F(Wm),F(W0))≤11-hθsup⁡x∈XH(Wm(x),W0(x))⟶0.
Thus, F(Wm)→F(W0), which implies that S(bm)→S(b); that is, the solution of (4.1) is continuous with respect to the initial value of (4.1). This completes the proof.Remark 4.2.
Theorem4.1 is a generalization of Theorem 2 in Zou et al. [9] from ℝn to Banach space 𝒳.



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