in 2013 studied the existence and global bifurcationproblems for periodic solutions to a class of differential variational inequalities in finite dimensional spaces by using the topologi
Trang 1NGUYEN THI VAN ANH
BEHAVIOUR OF SOLUTIONS TO DIFFERENTIAL VARIATIONAL INEQUALITIES
Speciality: Integral and Differential EquationsCode: 9 46 01 03
SUMMARY OF DOCTORAL THESIS IN MATHEMATICS
Hanoi - 2019
Trang 2Scientific Advisor: Assoc.Prof PhD Tran Dinh Ke
Referee 1: Prof Dr Sci Nguyen Minh Tri, Vietnam Academy of Science and Technology, Institute of Mathematics.
Referee 2: Assoc.Prof Dr Nguyen Xuan Thao, Hanoi University of Science and Technology.
Referee 3: Assoc.Prof Dr Nguyen Sinh Bay, Vietnam University of Commerce.
The thesis shall be defended before the University level Thesis Assessment Council
at on
The thesis can be found in the National Library and the Library of Hanoi National
University of Education.
Trang 31 Motivation and outline
Quantitative theory of ordinary differential equation (ODE) is one of thebasic theories of mathematics which has existed for centuries and gives us manymodels describing mechanics of motion in nature and engineering In the sev-eral decades by the end of the 20th century, ODE has generalized as differentialalgebraic equation (DAE) and then it has been extensively studied Therefore,various problems of engineering and science such as in flexible mechanics, elec-trical circuit design and chemical process control, etc have been mathematicalmodeling based on the DAE However, as evidenced by the growing literaturethat has surfaced in recent years on multi-rigid-body dynamics with frictionalcontacts and on hybrid engineering systems, ODE and DAE are vastly inade-quate to deal with many naturally occurring engineering problems that containinequalities (for modeling unilateral constraints) Hence, in order to research thedifferential model with unilateral constraints, that satisfies the demands men-tioned above, mathematicians need to investigate a larger problem:Differential variational inequalities, which includes differential complementarity problem
The notion of differential variational inequality was firstly used by Aubin andCellina in 1984 In their book the authors considered the problem:
∀t ≥ 0, x(t) ∈ K,supy∈Khx0(t) − f (x(t)), x(t) − yi = 0,x(0) = x0,
Then the solvability of (1) can be studied by the topological tools of multivaluedanalysis After this work, the theory of DVIs was considered and expanded inthe work of Avgerinous and Papageorgiou in 1997 Moreover, Avgerinous andPapageorgiou studied the periodic solutions to the DVI of the form
(
−x0(t) ∈ NK(t)(x(t)) + F (t, x(t)), a.e t ∈ [0, b],x(0) = x(b)
where NK(t)(x(t)) denotes the normal cone of the convex closed set K(t) at thepoint x(t)
However, DVIs were first systematically studied by Pang and Stewart in 2008.Differential variational inequality is the problem to find an absolutely continuous
Trang 4function x : [0, T ] → Rn and an integrable function u : [0, T ] → Rm such that foralmost all t ∈ [0, T ], one has
hv − u, φ(u)i ≥ 0, ∀v ∈ K
Then u(t) solves variational inequality (3) with φ = F (t, x(t), ·) Therefore, theproperties of the solution mapping (t, x) ⇒ SOL(K, F (t, x(t), ·)) will play a keyrole in our consideration We convert the problem to a differential inclusion
x0(t) = f (t, x(t), SOL(K, F (x(t), ·))),The general boundary condition is considered by
Γ(x(0), x(T )) = 0, (5)
In the stated form, the problem is a two-point boundary-value problem (BVP) inthe sense that, linked by the abstract function Γ, both the initial state x(0) andthe terminal state x(T ) are unknown variables to be computed; in particular, theformer variable x(0) is not completely given The initial-value (IVP) version ofthe problem corresponds to the special case where Γ(x, y) ≡ x − x0, with x0 beinggiven
One of the important problem generated by DVIs is differential tarity problem (DCP), where K is a cone C
complemen-x0(t) = f (t, x(t), u(t)),
C 3 u(t) ⊥ F (t, x(t), u(t)) ∈ C∗
In turn, a proper specialization of the latter DCP yields the linear tarity system, which is studied extensively in many previous papers In thepaper of Pand and Stewart, DVIs comes from many applications including linearcomplementarity systems, differential complementarity problems, and variationalinequalities of evolution
complemen-After the work of Pang and Stewart, more and more scholars are attracted toboost the development of theory and applications for (DVIs) For instance, Chen
- Wang (2014), Liu et al in 2013 studied the existence and global bifurcationproblems for periodic solutions to a class of differential variational inequalities
in finite dimensional spaces by using the topological methods from the theory ofmultivalued maps and some versions of the method of guiding functions, Gwin-ner in 2013 obtained a stability result of a new class of differential variationalinequalities by using the monotonicity method and the technique of the Mosco
Trang 5and dynamic decision processes.
On the other hand, many applied problems of DVIs in engineering, operationsresearch, economical dynamics, and physical sciences, etc., are more preciselydescribed by partial differential equations Based on this motivation, recently,Liu–Zeng–Motreanu in 2016 and Liu et al in 2017 proved the existence of so-lutions for a class of differential mixed variational inequalities in Banach spacesthrough applying the theory of semigroups, the Filippov implicit function lemmaand fixed point theorems for condensing set-valued operators However, untilnow, only one reference, Liu et al , considered a differential hemivariational in-equality in Banach spaces which is constituted by a nonlinear evolution equationand a hemivariational inequality of elliptic type rather than of parabolic type.Also, in the paper, the authors required that the constraint set K is bounded,the nonlinear function u → f (t, x, u) maps convex subsets of K to convex setsand the C0-semigroup eAt is compact
The thesis concerns one of the important problem related to the dynamic tem linked to variational inequalities That is, we study the behaviour of solution
sys-of differential variational inequalities when time tends to infinity The recent sults of DVI in finite spaces have investigated in some works, including Liu (2013),Loi (2015) There are many open questions concerning to study the DVIs, such
re-as the stability in the sense of Lyapunov, the exsitence of decay solution andperiodic solution, the existence of global attractors of m-semiflow generated bydifferential variational inequalities In addition, DVI in Banach space is also at-tractive and has some open problem The main difficulty in research the infinitesystem is the fact that we do not what exactly the solvability and properties ofsolutions of linked variational inequalities If the solution map of variational in-equality is not suiable regularity, it is very hard to study the behaviour of solution
of DVI via diffential inclusion theory
2 Purpose, objects and scope of the thesis
2.1 Purpose: The thesis focus on studying qualtitive behavior of solutions todifferential variational inequalities in finite dimensional space and infinite dimen-sional space
2.2 Objects In this thesis, we consider three problems as follows:
1) Establish sufficient conditions ensuring the existence of mila solutions of multi
- valued dynamical systems generated by differential variational inequalities,and then prove the existence of global attractors
2) The existence of decay solutions for a class differential variational inequalities
2.2 Objects: In the thesis, we consider two types of nonautonomous semilineardifferential inclusions in Banach spaces:
∗ The first type: Differential variational inequalities in finite dimensionalspace;
∗ The second one: Differential variational inequalities of parabolic-elliptictype in infinite spaces;
Trang 6∗ The thirst one: Differential variational inequalities of parabolic-parabolictype in infinite spaces;
2.3 Scope: The scope of the thesis is defined by the following contents
• Content 1: Study the solvability of differential variational inequalities;
• Content 2: Study the existence of decay solutions for these differentialvariational inequalities;
• Content 3: Study the existence global attractors for differential variationalinequalities
3 Research Methods
The thesis use the tools of multivalued analysis, fixed point theorem, parameter semigroup theory to study the contents Moreover, we use some specialtechnique to get our purpose:
one-◦ To prove the existence of solutions to differential variational inequalities, weemploy the semigroup theory (see MNC’s estimates (see Bothe (1998) orKamenskii et al (2001))
◦ To prove the existence of solutions to differential variational inequalities: weuse fixed point theory for condensing multimaps (see Kamenskii et al (2001))
◦ In order to do research on the existence of global attractor for multi-valueddynamical systems, we use the frame work proposed by Melnik and Valero(1998) In which, we estimated measure of noncompactness to get the asymp-totically compactness of the process generated by differential inclusions
4 Structure and Results
Together with the Introduction, Conclusion, Author’s works related to the sis that have been published and References, the thesis includes four chapters:Chapter 1 is devoted to present some preliminaries In Chapter 2, we present thesolvability and the existence of global attractor for a class of differential varia-tional inequality in finite dimensional space Chapter 3 presents a sufficient con-dition ensuring the existence of attractor for differential variational of parabolic-elliptic type in Banach space Chapter 4 presents a sufficient condition ensuringthe existence of attractor for differential variational of parabolic-parabolic type
the-in the-infthe-inite dimensional spaces
Trang 7Chapter 1 PRELIMINARIES
In this chapter, we present some preliminaries including: some functionalspaces; measure of noncompactness; multi-valued calculus and fixed point prin-ciples; global attractor for multi-valued autonomous dynamical systems, someauxiliary results related to some inequalities and theorems
1.1 ONE PARAMETER SEMIGROUP
In this section, we present the basic knowledge about semigroup theory, cluding linear and nonlinear semigroup
in-1.1.1 Linear semigroup
1.1.2 Nonlinear semigroup
1.2 MEASURE OF COMPACTNESS (MNC) AND MNC ESTIMATES
In this section, we recall some notions and facts related to measure of pactness (MNC) and Hausdorff MNC, followed by some MNC estimate which isnecessary for the next chapters
noncom-1.3 MULTIVALUED CALCULUS AND SOME FIXED POINT THEOREMS 1.3.1 Multivalued calculus
In this subsection, we present some definitions and results in multivalued culus, including concept of a selector and the existence of a selection function
cal-1.3.2 Condensing map and some fixed point theorems
In this section, we recall some notions of set-valued analysis and condensingmap, then introduce some fixed point theorem for multivalued maps
1.4 GLOBAL ATTRACTORS FOR MULTIVALUED SEMIFLOWS
In this section, we present some definitions and results on global attractors formultivalued semiflows developed by Menik and Valero (1998) and the frame workfor the existence of a compact global attractor for m-semiflows generated by adifferential inclusion
1.5 SOME AUXILIARY RESULTS
In this section, we recall some notions and facts related to wellknown ties, consist of Gronwall inequality, Hanalay inequality and some theorems such asMazur Lemma, Arzela Ascoli theorem In addition, we hightlight some essentialfunctional spaces which are used in this thesis
Trang 8inequali-1.5.1 Some auxiliary inequalites
1.5.2 Some auxiliary theorems
1.5.3 Some functional spaces
Trang 9Chapter 2 DIFFERENTIAL VARIATIONAL INEQUALITY IN FINITE SPACE
In this chapter, we study behaviour of solutions of differential variationalinequalities in finite space with delay Our purpose was to give the sufficientcoditions to ensure the existence of solution and the stability of DVIs Thus, theexistence of decay solution and a global attractor for m-semiflow generated byDVIs were proved
The content of this chapter is written based on the paper [1] in the author’sworks related to the thesis that has been published
(H3) The function F : Rn → Rm is continuous and there is a positive number ηF such
that kF (v)k ≤ ηF for all v ∈ Rn.
(H4) G : K → Rm is a continuous function such that
1) G is monotone on K, i.e.
hu − v, G(u) − G(v)i ≥ 0, ∀u, v ∈ K;
Trang 102) there exists v0 ∈ K such that
lim
v∈K,kvk→∞
hv − v0, G(v)i
kvk2 > 0
(H5) h : Rn → Rn is continuous such that there are positive constants ηh, ζh verifying
kh(u)k ≤ ηhkuk + ζh, ∀u ∈ Rn
We have the following definition of integral solution to (2.1)-(2.2)
Definition 2.1 A pair of functions (x, u), where x : [−τ, T ] → Rn is continuousand u : [0, T → K] is integrable, called a solution of (2.1) − (2.3) iff the followingequalities hold
Lemma 2.1 Suppose that (H4) holds Then, for every z ∈ Rm, the solution setSOL(K, z + G(·)) of 2.4 is nonempty, convex and compact Moreover, there exists
a number ηG > 0 such that
By above setting, the differential variational inequality (2.1)-(2.3) is converted
to the differential inclusion as follows
x0(t) ∈ Ax(t) + Φ(x(t), xt), t ∈ J, (2.7)x(t) = ϕ(t), t ∈ [−τ, 0] (2.8)
Denote
PΦ(x) = {f ∈ L1(J ;Rn) : f (t) ∈ Φ(x(t), xt)}, with x ∈ C (2.9)Thanks to Lemma 2.1, we have
kΦ(v, w)k ≤ ηG(1 + ηF)[ηB(kvk + kwkCτ) + ζB] + ηhkvk + ζh
Trang 11of Noncompactness- Kamenskii et al) Thus, Φ has a Castaing represent and itimplies PΦ(x) 6= ∅ for every x ∈ C.
Let y ∈ CT and ϕ ∈ Cτ, we define the function y[ϕ] ∈ C by
y[ϕ](t) =
(
y(t), if t ∈ [0, T ],ϕ(t), if t ∈ [−τ, 0]
We consider the Cauchy operator
well-Lemma 2.3 The operator W defined by (2.10) is compact
Lemma 2.4 Let (H1)-(H5) hold Then the solution map F is compact and has
Trang 12Denote Π : BC([−τ, ∞];Rn)×L1loc(R+;R) → BC([−τ, ∞];Rn) defined by Π(x, u) :=x.
Theorem 2.2 Assume that (H1*), (H2*), (H3)-(H4) and (H5*) take place and there exists γ > 0 such that
We obtain the results related to the regularity of G(t, ·) by the following lemma.Lemma 2.6 Let the hypotheses (H1)-(H5) hold Then G(t, ·) is a compactmultimap for each t > τ
Lemma 2.7 Let the hypotheses (H1)-(H5) hold Then G(t, ·) is upper tinuous for each t ≥ 0
semicon-Lemma 2.8 Let (H1*) and (H2)-(H5) hold Then the m-semiflow G admits
an absorbing set, provided that
2ηBηG(1 + ηF) + ηh < a
Theorem 2.3 Let (H1*) and (H2)-(H5) hold Then the m-semiflow G ated by (2.1)-(2.3) admits a compact global attractor provided that
gener-2ηBηG(1 + ηF) + ηh < a
Trang 13varia-3.1 Setting problem
Let (X, k · k) be a Banach space and (U, k · kU) be another reflexive Banachspace with the dual U∗, we consider the following problem:
x0(t) − Ax(t) ∈ F (x(t), u(t)), x(t) ∈ X, t ≥ 0, (3.1)B(u(t)) + ∂φ(u(t)) 3 g(x(t), u(t)), u(t) ∈ U, t ≥ 0, (3.2)
where x is the state function with values in X, u is a control function takingvalues in U , φ : U → R is a proper, convex, lower semicontinuous function withthe subdifferential ∂φ ⊂ U × U∗
By PF we will denote the set of Bochner integrable selections of F (·, ·), thatmeans
PF : C(J ; X) × L1(J ; U ) → P(L1(J ; X)),
PF(x, u) = {f ∈ L1(J ; X) : f (t) ∈ F (x(t), u(t)) for a.e t ∈ J } (3.4)
We mention here the definition of mild solution of the problem (3.1)− (3.3).Definition 3.1 A pair of continuous functions (x, u), where x : [0, T ] → X,
u : [0, T ] → U , is a mild solution of (3.1) − (3.3) iff there exists a selection
3.2 Solvability
We consider the problem (3.1)-(3.3) with the following assumptions
(A) A is a closed linear operator generating a C0−semigroup (S(t))t≥0.
(F) F : X × U → Pc(X) is u.s.c with weakly compact and convex values and