Dynamical system related to primal-dual splitting projection methods
Ewa M. Bednarczuk, Raj Narayan Dhara, Krzysztof E. Rutkowski
OON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO SOMEDYNAMICAL SYSTEMS WITH LOCALLY LIPSCHITZ RIGHT-HAND SIDES
EWA M. BEDNARCZUK , , RAJ NARAYAN DHARA , , AND KRZYSZTOF E. RUTKOWSKI Abstract.
We investigate the existence, uniqueness and extendability of solutions toan autonomous dynamical system in a Hilbert space X with right-hand side which iscontinuous and bounded on a bounded and closed subset ˆ D ⊂ X and locally Lipschitzon set ˆ D \ { ¯ z } , where ¯ z is the only stationary point of the differential equation. Introduction
Let X be a Hilbert space. In this paper we investigate the existence, uniqueness and thebehaviour, as t goes to + ∞ , of an autonomous dynamical system of the form ˙ x ( t ) = F ( x ( t )) , t ≥ t > ,x ( t ) = x ∈ ˆ D \ { ¯ z } . (DS)where F : ˆ D → X is a continuous function, locally Lipschitz on ˆ D except a single point ¯ z ∈ ˆ D , and ˆ D is a closed and bounded set in X . When F ( x ) := P C ( x ) ( ¯ w ) − x , where C is a multivalued mapping, ¯ w ∈ X and P C ( x ) ( ¯ w ) is the orthogonal projection of ¯ w on C ( x ) ,the dynamical system (DS) is related to some algorithmic scheme for solving optimizationproblems.First order dynamical systems related to optimization problems have been discussed bymany authors (see e.g. [1, 3, 8, 6, 7]). In those papers, a natural assumption is thatthe vector field F is globally Lipschitz and consequently, the existence and uniqueness ofsolutions to dynamical system is guaranteed by classical results (see e.g. [9, Theorem 7.3]).A survey of existing results going beyond classical Cauchy-Picard theorem from finite toinfinite settings journey can be found in [16].Main difficulties in adapting the existence theory to autonomous ODE in infinite dimen-sional settings are due to the lack of compactness, see [18, Remark 5.1.1]. For instance,assuming the continuity of the right-hand side F is not enough to adapt Peano’s theoremto infinite dimensional spaces, [13], even in Hilbert spaces, [27].In general, [14] proved that in every infinite-dimensional Banach space there exists acontinuous field F such that there is no solution to the related (DS) whereas the globalLipschitz condition, due to Cauchy-Lipschitz-Picard-Lindeloff, of the right-hand side fieldgives the uniqueness and/or extendability of the solution, see [9, Theorem 7.3]. Someattempts to weaken the global Lipschitz condition of the right-hand side vector field havebeen done in the context of existence of solutions, see e.g. [18, Theorem 5.1.1] and [23, Mathematics Subject Classification.
Key words and phrases. autonomous ordinary differential equations, locally Lipschitz vector field, exis-tence and uniqueness of solutions, extendability of solutions, projected dynamical systems. Warsaw University of Technology, 00-662 Warszawa, Koszykowa 75, Systems Research Institute of the Polish Academy of Sciences, Newelska 6 . Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Re-public. Cardinal Stefan Wyszyński University, 01-815 Warsaw, Dewajtis 5. RND acknowledges the support of IBS PAN, Warszawa, Poland and the Czech Science Foundation,project GJ19-14413Y. a r X i v : . [ m a t h . C A ] F e b N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 2
19, 24, 15] and the references therein. It is observed that the local Lipschitz condition, givesthe local existence and uniqueness for the related initial value problems.In [10] a smooth vector field is constructed such that the respective autonomous dynamicalsystem has a bounded maximal solution which is not globally defined.In finite-dimensional settings, under the assumption of local Lipschitzness and someboundedness of the vector field, [25] showed the existence and uniqueness of trajectoryon [ t , + ∞ ) . They applied their results to investigations of projected dynamical systems.The contribution of the present investigation is as follows.- Existence and uniqueness of solutions to dynamical system (DS) with continuousright-hand side vector field F defined on a bounded and closed set ˆ D , which is locallyLipschitz on ˆ D \ ¯ z , where ¯ z is the only stationary point of F on ˆ D (Proposition 3.4for the Euler method and Proposition 3.7 for the approach via contraction mappingprinciple) (section 3).- Extendability of solutions to dynamical system (DS) satisfying the above properties,Theorem 4.1 (section 4).- Behaviour of solutions at + ∞ (section 5).- Application to projected dynamical systems (PDS) (section 6).2. Formulation of the problem
Let ¯ w, ¯ z ∈ X , X - Hilbert space with the inner product (cid:104)· | ·(cid:105) , ¯ w (cid:54) = ¯ z and associatednorm (cid:107) · (cid:107) = (cid:112) (cid:104)· | ·(cid:105) . Let D ⊂ X be a closed convex subset of X such that ¯ w, ¯ z ∈ D and (cid:104) ¯ z − x | ¯ w − x (cid:105) ≤ for all x ∈ D . (2.1)Note that condition (2.1) immediately implies that ¯ w and ¯ z are boundary points of the set D .The definition of the set D comes from the application. This is also one of the mainmotivations that we assume X to be a Hilbert space.Let r be such that (cid:107) ¯ w − ¯ z (cid:107) > r > . Throughout this note we consider another setrelated to D (see Figure 2.1): ˆ D = { x ∈ D | (cid:107) x − ¯ w (cid:107) ≥ r } . We consider the following Cauchy problem ˙ x ( t ) = F ( x ( t )) , t ≥ ,x (0) = x ∈ ˆ D \ { ¯ z } , (DS )where F : ˆ D → X is a continuous function on ˆ D and locally Lipschitz on ˆ D \ { ¯ z } andbounded on ˆ D ( (cid:107) F ( x ) (cid:107) ≤ M , M > , x ∈ ˆ D ). Moreover we assume:(A) ¯ z is the only zero point of F in ˆ D , i.e. F ( x ) = 0 iff x = ¯ z .(B) for all x ∈ ˆ D , for all h ∈ [0 , we have x + hF ( x ) ∈ ˆ D Together with assumptions (A), (B) we also consider the following assumption:(C) (cid:104) F ( x ) | ¯ w − x (cid:105) ≤ for all x ∈ ˆ D . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 3
Figure 2.1.
Illustration of considered sets.
Remark 2.1.
The motivation for considering non-convex set ˆ D comes from the followingapplication. Consider F : D → X defined as F ( x ) = P C ( x ) ( ¯ w ) , (2.2) where P C ( x ) ( ¯ w ) is the projection of ¯ w onto C ( x ) , C : D ⇒ X is a multifunction given by C ( x ) = H ( ¯ w, x ) ∩ H ( x, g ( x )) , H ( a, b ) = { h ∈ X | (cid:104) h − b | a − b (cid:105) ≤ } , a, b ∈ X and g : X → X satisfies ¯ z ∈ H ( x, g ( x )) for all x ∈ H and P .Under suitable assumption on g , the function F given by (2.2) is locally Lipschitz on D \ { ¯ w, ¯ z } (see e.g. [4] ), continuous on D \ { ¯ w } and bounded on D . Fact 2.1.
We have: x ∈ ¯ B ( ¯ w +¯ z , (cid:107) ¯ w − ¯ z (cid:107) ) if and only if (cid:104) ¯ z − x | ¯ w − x (cid:105) ≤ . (2.3) Proof.
Let o = ( ¯ w + ¯ z ) . Then (cid:107) x − o (cid:107) − (cid:107) ¯ z − ¯ w (cid:107) = (cid:107)
12 ( x − ¯ w ) + 12 ( x − ¯ z ) (cid:107) − (cid:107) ¯ z − x + x − ¯ w (cid:107) = 14 (cid:107) x − ¯ w (cid:107) + 12 (cid:104) x − ¯ w | x − ¯ z (cid:105) + 14 (cid:107) x − ¯ z (cid:107) − (cid:107) ¯ z − x (cid:107) − (cid:104) ¯ z − x | x − ¯ w (cid:105) − (cid:107) x − ¯ w (cid:107) = (cid:104) ¯ z − x | ¯ w − x (cid:105) . This shows that (cid:107) x − o (cid:107) = (cid:107) ¯ z − ¯ w (cid:107) + (cid:104) ¯ z − x | ¯ w − x (cid:105) , i.e. x ∈ ¯ B ( ¯ w +¯ z , (cid:107) ¯ w − ¯ z (cid:107) ) if andonly if (2.3) holds. (cid:3) Remark 2.2.
Fact 2.1 implies that
D ⊂ ¯ B ( ¯ w +¯ z , (cid:107) ¯ w − ¯ z (cid:107) ) , hence D is bounded. Moreover,this easily implies that (cid:107) ¯ w − ¯ z (cid:107) = d := sup x,y ∈D (cid:107) y − x (cid:107) that is the pair ¯ z , ¯ w realizes maximal distance between two points in D (the diameter of D ). Lemma 2.1.
Let x ( · ) ∈ C ([ a, b ] , ˆ D ) , [ a, b ] ⊂ R + . Then the function f ( t ) := F ( x ( t )) iscontinuous: f ( · ) ∈ C ([ a, b ] , X ) Definition 2.3.
Let T = [ t ; T ) , t < T ≤ + ∞ or T = [ t ; T ] , t < T < + ∞ . Solution of (DS) on interval T is any function x ( · ) ∈ C ( T , ˆ D ) satisfying N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 4 (1) initial condition x ( t ) = x ;(2) equation ˙ x ( t ) = F ( x ( t )) for all t ∈ T , where the differentiation is understood in thesense of strong derivative on space X , where at the boundary point of the interval T ,in the case when it belongs to T , the differentiation is understood in the one-sidedway. Definition 2.4.
A solution x ( t ) to problem (DS) on interval T = [0 , T ] (or T = [0 , T ) )is called non-extendable if there is no solution x ( · ) ∈ C ( T , ˆ D ) on any interval T of thisproblem satisfying conditions:(1) T (cid:41) T ;(2) ∀ t ∈ T , x ( t ) = x ( t ) . Remark 2.5. If x ( t ) is a solution of Cauchy problem (DS ) on interval T = [0 , T ] (or T = [0 , T ) ), then restriction of x ( t ) on any interval T = [ t , t ] ⊂ T (or T = [ t , t ) ⊂ T )is a solution of Cauchy problem (DS) on T with initial condition x = x ( t ) . Lemma 2.2. (about extendability) Let x ( t ) be defined and differentiable in a continuousway in left-sided neighbourhood of t , i.e. x ( · ) ∈ C (( t − γ, t ) , ˆ D ) (2.4) and assume that the limit x := lim t → t − ˙ x ( t ) (2.5) exists and x ∈ F ( ˆ D ) . Then(1) x ( t ) is extendable in a continuous way to function ˜ x ( · ) ∈ C (( t − γ, t ] , ˆ D ) (2) ˙˜ x (cid:96) = x (where (cid:96) denotes the left derivative of x ( · ) at t )Proof. It follows from the existence of the left-hand limit, that the derivative is bounded insome left-sided half-neighbourhood of t : ∃ ζ ∈ (0 , γ ] , ∃ L > ∀ t ∈ ( t − ζ, t ) (cid:107) ˙ x ( t ) (cid:107) ≤ L. (2.6)By the weakened formula on finite-increments we obtain Lipschitz continuity of function x ( t ) on ( t − ζ, t ) with some constant L . Therefore, for the function x ( t ) , the Cauchycondition for the existence of the left derivative at time t is satisfied, and ∃ x = lim t → t − x ( t ) . (2.7)Put ˜ x ( t ) = (cid:26) x ( t ) , t ∈ ( t − γ, t ); x t = t . It is obvious that, the function constructed in this way is continuous on ( t − ζ, t ] . Now,it is enough to show that ˙˜ x (cid:96) ( t ) = x , i.e. lim t → t − t − t ( x ( t ) − x ) = x , or lim ∆ t → t ( x − x ( t − ∆ t )) = x . To use the formula of Newton-Leibniz we introduce a function z ( t ) = (cid:26) x (cid:48) ( t ) , t ∈ ( t − γ, t ); x t = t . By (2.4) and (2.5), function z ( t ) is continuous on ( t − ζ, t ] . We cannot yet claim that ˙˜ x ( t ) = x , our aim is to prove it. N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 5
For any δ ∈ (0 , ζ ) we can rewrite formula of Newton-Leibniz as (cid:90) t − δt − ∆ t z ( t ) dt = x ( t − δ ) − x ( t − ∆ t ) . (2.8)We take the limit with δ tending to zero. Then on the one hand, x ( t − δ ) → x (see(2.7)). On the other hand, (cid:90) t − δt − ∆ t z ( t ) dt → (cid:90) t t − ∆ t z ( t ) dt, because (cid:107) (cid:90) t t − ∆ t z ( t ) dt − (cid:90) t − δt − ∆ t z ( t ) dt (cid:107) = (cid:107) (cid:90) t t − δ z ( t ) dt (cid:107) ≤ (cid:90) t t − δ (cid:107) z ( t ) (cid:107) dt ≤ δL → . Here, we used continuity of z ( t ) , estimation (2.6) and, from fact (2.5) with (2.5), estimation (cid:107) z ( t ) (cid:107) ≤ L .Taking the limit with both sides in (2.8) we obtain (cid:90) t t − ∆ t z ( t ) dt = x − x ( t − ∆ t ) . (2.9)Then from (2.8) and (2.9) we have (cid:90) tt − ∆ t − ˜ x ( t ) − ˜( t − ∆ t ) for all t ∈ [ t − ∆ t, t ] , where function inside integral is continuous. Now applying at thepoint t = t the theorem on differentiation of the integral with respect to the upper limit,we obtain ˙˜ x (cid:96) ( t ) = z ( t ) = x , as required. (cid:3) Existence and uniqueness of solutions to (DS)In this section we are concerning with the existence and the uniqueness of solutions to(DS) defined on closed intervals, namely [ t , T ] , where T > t is finite. In deriving existenceresults we present two approaches: Euler method (Section 3.1) and contraction mappingprinciple (Section 3.2). Proposition 3.1.
Assume that (C) holds. Then any solution x ( t ) of (DS ) satisfies thecondition (cid:107) x ( t ) − ¯ w (cid:107) is nondecreasing with respect to t ≥ . Proof.
Let us note that x ( t ) is continuously differentiable on [0 , + ∞ ) , therefore by (C) wehave ddt (cid:107) x ( t ) − ¯ w (cid:107) = (cid:104) ˙ x ( t ) | x ( t ) − ¯ w (cid:105) = (cid:104) F ( x ( t )) | x ( t ) − ¯ w (cid:105) ≥ . (cid:3) Now we show the uniqueness of trajectories.
Proposition 3.2.
Let t > and let x ∈ ˆ D \ { ¯ z } . Assume that assumptions (A) and(C) holds. If (DS) is solvable in a given interval [ t , T ] , then the solution is unique on thisinterval. N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 6
Proof.
Now we show the uniqueness of solutions of (DS) on [ t , T ] . Suppose that x ( · ) and x ( · ) solve (DS) on interval [ t , T ] . Let ¯ t ∈ [ t , T ] be such that ¯ t := sup { t ∈ [ t , T ] | (cid:107) x ( t ) − x ( t ) (cid:107) = 0 } . (3.1)Let us note that x ( t ) = x = x ( t ) . Consider two cases:Case 1 : x (¯ t ) = x (¯ t ) = ¯ z . Then, by Proposition 3.1, (cid:107) x ( t ) − ¯ w (cid:107) ≥ (cid:107) ¯ z − ¯ w (cid:107) and (cid:107) x ( t ) − ¯ w (cid:107) ≥ (cid:107) ¯ z − ¯ w (cid:107) for t ≥ ¯ t . However, { x ∈ H | (cid:107) x − ¯ w (cid:107) ≥ (cid:107) x − ¯ z (cid:107)} ∩ ˆ D = { ¯ z } . Therefore, by assumption (A), x ( t ) = x ( t ) = ¯ z for all t ∈ [¯ t, T ] .Case 2 : x (¯ t ) = x (¯ t ) (cid:54) = ¯ z and ¯ t < T . Then, by local Lipschitzness of F ( · ) on ˆ D\{ ¯ z } thereexists a neighbourhood of x (¯ t ) , namely U ( x (¯ t )) such that F is locally Lipschitzin U ( x (¯ t )) with some constant L x (¯ t ) , i.e., ∀ x , x ∈ U ( x (¯ t )) (cid:107) F ( x ) − F ( x ) (cid:107) ≤ L x (¯ t ) (cid:107) x − x (cid:107) . Since x and x are Lipschitz functions with constant M there exists a neighbour-hood V (¯ t ) ∩ [ t , T ] such that ∀ t ∈ V (¯ t ) ∩ [ t , T ] x ( t ) ∈ U ( x (¯ t )) ∧ x ( t ) ∈ U ( x (¯ t )) . Then for t ∈ V (¯ t ) ∩ [ t , T ] ddt (cid:18) (cid:107) x ( t ) − x ( t ) (cid:107) (cid:19) = (cid:104) ˙ x ( t ) − ˙ x ( t ) | x ( t ) − x ( t ) (cid:105) = (cid:104) F ( x ( t )) − F ( x ( t )) | x ( t ) − x ( t ) (cid:105) ≤ L x (¯ t ) (cid:107) x ( t ) − x ( t ) (cid:107) . By using Gronwall’s inequality for the function t → (cid:107) x ( t ) − x ( t ) (cid:107) we obtain that (cid:107) x ( t ) − x ( t ) (cid:107) ≤ , i.e., x ( t ) = x ( t ) for t ∈ V (¯ t ) ∩ [ t , T ] . This contradicts 3.1with ¯ t (cid:54) = T . (cid:3) Proposition 3.3. x ( t ) is a solution of (DS) on I = [ t , T ] ( T > t is arbitrary) if and onlyif satisfies the condition x ( t ) = x + (cid:90) tt F ( x ( s )) ds, ∀ t ∈ I , (3.2) where the integral is understood in the sense of Riemann and x ( t ) ∈ ˆ D , t ∈ I . Let us define B t ,T := C ([ t , t + T ] , ˆ D ) and B Rt ,x ,T := { x ∈ B t ,T | sup t ∈ [ t ,t + T ] (cid:107) x ( t ) − x (cid:107) ≤ R } . Let us note that B t ,T is a complete metric space due to the fact that ˆ D is a closed subsetof a Hilbert space X . Moreover, in the sequel we consider on ˆ D the topology induced bythe topology of the space.3.1. Euler method.
We start with the following construction of Euler trajectories.For any λ ∈ (0 , define c λn , n = 0 , , . . . as follows c λ := x , c λn +1 := c λn + λF ( c λn ) , n = 0 , , · · · . (3.3)Then, for any λ ∈ (0 , define a continuous trajectory on [ t , T ] as follows c λ ( t ) = c λn + ( t − t − nλ ) F ( c λn ) , t ∈ [ t + nλ, t + ( n + 1) λ ] n = 0 , , · · · . (3.4) N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 7
Proposition 3.4.
Let t > and let x ∈ ˆ D \ { ¯ z } . Assume that (B) hold.(1) If X is finite dimensional, then for all T > t there exists a solution of x ( t ) of (DS) on [ t , T ] in the class B t ,T ,(2) If X is infinite dimensional, then there exists R > and T > t such that thereexists a solution of x ( t ) of (DS) on [ t , T ] in the class B Rt ,x ,T ,Proof. Let us start with the initial settings.(1) In case X is finite dimensional we take any T > t . Let us note that in this case ˆ D is closed and bounded, hence compact. Since F is continuous on ˆ D , F is uniformlycontinuous, i.e. ∀ ε > ∃ δ > ∀ x , x ∈ ˆ D (cid:107) x − x (cid:107) < δ = ⇒ (cid:107) F ( x ) − F ( x ) (cid:107) < ε. (2) In case X is infinite dimensional let T = RM + t , where R is such that F ( · ) isLipschitz on B ( x , R ) . Let m λ := (cid:100) ( T − t ) λ − (cid:101) . Let us note that, by the factthat x ∈ ˆ D \ { ¯ z } and, by assumption (B), for any λ ∈ (0 , and all t ∈ [ t , T ] , c λ ( t ) ∈ ˆ D . For any λ ∈ (0 , function c λ ( · ) given by (3.4) is differentiable on [ t , T ] \ { t , t + λ, . . . , t + m λ λ } as a piecewise affine function.For all λ ∈ (0 , and any t ∈ [ t , T ] ( t = t + aλ + ˜ t , a ∈ N , ≤ ˜ t < λ ) we have (cid:107) c λ ( t ) − x (cid:107) = (cid:107) x + λ a − (cid:88) n =0 F ( c λ ( t + nλ )) + ˜ tF ( c λ ( t + aλ )) − x (cid:107)≤ λ a − (cid:88) i =0 M + ˜ tM = M ( aλ + ˜ t ) ≤ M ( T − t ) = RM M = R. Let us note that in this case F is uniformly continuous on B ( x , R ) ∩ ˆ D , i.e. ∀ ε > ∃ δ > ∀ x , x ∈ B ( x , R ) ∩ ˆ D (cid:107) x − x (cid:107) < δ = ⇒ (cid:107) F ( x ) − F ( x ) (cid:107) < ε. Now let us continue the proof in both cases 1. and 2. together. For any λ ∈ (0 , define ∆ λ ( t ) := ˙ c λ ( t ) − F ( c λ ( t )) , t + nλ < t < min { t + ( n + 1) λ, T } ,n = 0 , . . . , m λ , t = t , t + λ, . . . , t + m λ λ. (3.5)Note that for all t ∈ [ t , T ] , c λ ( t ) = x + (cid:90) tt ˙ c λ ( s ) ds = x + (cid:90) tt F ( c λ ( s )) + ∆ λ ( s ) ds, We have (cid:107) ∆ λ ( t ) (cid:107) = (cid:107) F ( c λn ) − F ( c λ ( t )) (cid:107) , t ∈ [ t + nλ, min { t + ( n + 1) λ, T } ] ,n = 0 , , . . . , m λ . Let us note that c λ ( · ) is Lipschitz continuous on [ t , T ] because it is differentiable almosteverywhere and the norm of its derivative is bounded by M . Therefore ∀ n = 0 , . . . , m λ sup t ∈ [ t + nλ, min { t +( n +1) λ,T } ] (cid:107) c λ ( t + nλ ) − c k ( t ) (cid:107) ≤ M | nλ − t | ≤ M λ
Fix any ε > and take λ ∈ (0 , such that M λ < δ . Then for all n = 0 , . . . , m λ wehave sup t ∈ [ t + nλ, min { t +( n +1) λ,T } ] (cid:107) c λ ( t + nλ ) − c λ ( t ) (cid:107) = sup t ∈ [ t + nλ, min { t +( n +1) λ,T } ] (cid:107) ( t − ( t + nλ )) F ( c λn ) (cid:107) ≤ M λ < δ,
N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 8 and consequently ∀ n = 0 , . . . , m λ sup t ∈ [ t + nλ, min { t +( n +1) λ,T } ] (cid:107) F ( c λn ) − F ( c λ ( t ) (cid:107) < ε. Hence, for all λ < δM , we have ∀ n = 0 , . . . , m λ ∀ t ∈ [ t + nλ, min { t + ( n + 1) λ, T } ] (cid:107) ∆ λ ( t ) (cid:107) < ε. Thus, (cid:107) ∆ λ ( · ) (cid:107) + ∞ → as λ → on [ t , T ] . Let { λ k } k ∈ N be a sequence in (0 , such that λ k → as k → . By the Ascoli-Arzela Theorem, there exists a uniformly convergent subsequence of { c λ k ( t ) } k ∈ N , namely { c λ ki ( t ) } i ∈ N , which converges to x ( t ) = lim i → + ∞ c λ ki ( t ) for t ∈ [ t , T ] , i.e. ∃{ λ k i } i ∈ N ∀ ε > ∃ i ∈ N ∀ i ≥ i ∀ t ∈ [ t , T ] (cid:107) x ( t ) − c λ ki ( t ) (cid:107) < ε. (3.6)Therefore, for all t ∈ [ t , T ] , c λ ki ( t ) = x + (cid:90) tt F ( c λ ki ( s )) + ∆ λ ki ( s ) ds, x ( t ) = x + (cid:90) tt F ( x ( s )) ds. By Proposition 3.3, x ( t ) is a solution of (DS) on [ t , T ] . Since c λ ki ( t ) ∈ ˆ D , i ∈ N , t ∈ [ t , T ] , by the closedness of ˆ D , we obtain that x ( t ) ∈ ˆ D , [ t , T ] . (cid:3) Corollary 3.5.
Let t ≥ , x ∈ ˆ D \ { ¯ z } be arbitrary fixed. Assume that assumptions (A),(B), (C) are satisfied. Then there exist R > , T (cid:48) > such that for all T ∈ [ t , T (cid:48) ) thereexists solution to (DS) on [ t , t + T ] and it is unique in the class B Rt ,x ,T .Proof. The proof follows from Proposition 3.2 and Proposition 3.4. (cid:3)
Contraction mapping principle for extended vector field.
We consider the follow-ing Cauchy problem ˙ x ( t ) = ˜ F ( x ( t )) , t ≥ ,x (0) = x ∈ ˆ D \ { ¯ z } , (DS )where ˜ F is such that ˜ F ( x ) = F ( x ) for all x ∈ ˆ D and ˜ F is continuous on X . Lemma 3.1. ( [12, Lemma 1.2] ) Let X, Y be Banach spaces, Ω ⊂ X closed and f : Ω → Y continuous. Then there is a continuous extension ˜ f : X → Y of f such that ˜ f ( X ) ⊂ conv f (Ω) (:=convex hull of f (Ω) ). Definition 3.6.
Let T = [ t ; T ) , t < T ≤ + ∞ or T = [ t ; T ] , t < T < + ∞ . Solution of (DS ) on interval T is any function x ( · ) ∈ C ( T , X ) satisfying(1) initial condition x ( t ) = x ;(2) equation ˙ x ( t ) = ˜ F ( x ( t )) for all t ∈ T , where the differentiation is understood inthe sense of strong derivative on space X , where at the boundary point of theinterval T , in the case when it belongs to T , the differentiation is understood in theone-sided way. N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 9
Proposition 3.7. x ( · ) is a solution of (DS ) on I = [ t , T ] ( T > t is arbitrary) if andonly if satisfies the condition x ( t ) = x + (cid:90) tt ˜ F ( x ( s )) ds, ∀ t ∈ I , (3.7) where the integral is understood in the sense of Riemann. Proposition 3.8.
Let t ≥ , x ∈ ˆ D \ { ¯ z } . Then there exists T > such that there existsa solution of (DS ) on interval [ t , t + T ] .Proof. For a given x ∈ C ([ t , t + T ]; X ) , define S [ x ] to be the function on [ t , t + T ] ,given by S [ x ]( t ) := x + (cid:90) tt ˜ F ( x ( τ )) dτ, t ∈ [ t , t + T ] , (3.8)where ˜ F is an extension of F given by Lemma 3.1. In the following, boundedness of F or ˜ F will be used as per their restrictive sense.Step 1. If x ∈ C ([ t , t + T ]; ˆ D ) then S ( x ) makes sense, since the right hand side is welldefined.Step 2. Let us prove that S [ x ]( · ) ∈ C ([ t , t + T ]; X ) for any T > and for x ∈ C ([ t , t + T ]; X ) . Assume t , t ∈ [ t , t + T ] with t < t . It is evident that S [ x ]( t ) = S [ x ]( t ) + (cid:90) t t ˜ F ( x ( τ )) dτ. (3.9)Then the continuity of S [ x ] gives us as t → t , (cid:107) S [ x ]( t ) − S [ x ]( t ) (cid:107) X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t t ˜ F ( x ( τ )) dτ (cid:13)(cid:13)(cid:13)(cid:13) X ≤ max τ ∈ [ t ,t + T ] (cid:107) ˜ F ( x ( τ )) (cid:107) X · | t − t | . Thus S : C ([ t , t + T ]; X ) −→ C ([ t , t + T ]; X ) .Step 3. Denote C := C ([ t , t + T ]; X ) . Consider the following form of a ball in C , wherewe intend to look for a fixed point. C D := (cid:26) x ( t ) ∈ C : | x − x | C ≡ max t ∈ [ t ,t + T ] (cid:107) x ( t ) − x (cid:107) X ≤ / , x ∈ ˆ D (cid:27) . Clearly, C D ( ⊆ C ) is a complete metric space with the metric induced by the normof C . Let us show that for choosing T small enough the operator S maps C D into itself and has a fixed point.We have, by Step 2, S [ x ]( · ) ∈ C , whenever x ( · ) ∈ C D . We now show that S [ x ]( · ) ∈ C D . It follows form (3.9) that | S [ x ] − x | C = max t ∈ [ t ,t + T ] (cid:107) S [ x ]( t ) − x (cid:107) X = max t ∈ [ t ,t + T ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t (cid:16) ˜ F ( x ( τ )) (cid:17) dτ (cid:13)(cid:13)(cid:13)(cid:13) X ≤ max τ ∈ [ t ,t + T ] (cid:107) ˜ F ( x ( τ )) (cid:107) X T =: cT, Therefore, for a choice of T ≤ / c , | S [ x ] − x | C ≤ / . Hence, S [ x ]( · ) ∈ C D implies S : C D → C D for every T ≤ / c .Step 4 We shall show now that a sequence { x n ( · ) } n ≥ ⊆ C D is a Cauchy sequence. Letsstart with the initial point { x } ∈ ˆ D be given and define x ( · ) := x . Denote x ( · ) := S [ x ]( · ) , and that x n +1 ( · ) := S [ x n ]( · ) , n = 1 , , · · · . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 10
Moreover, the followings hold successively. | x n +1 − x n | C = | S [ x n ] − S [ x n − ] | C ≤ cT | x n − x n − | C ≤ · · · ≤ ( cT ) n | x − x | C = ( cT ) n max t ∈ [0 ,T ] (cid:107) x ( t ) − x (cid:107) X ≤ c n T n +1 (cid:107) F ( x ) (cid:107) X Let m, n ∈ N such that m > n and cT = δ ∈ [0 , then | x m − x n | C ≤ | x m − x m − | C + | x m − − x m − | C + · · · + | x n +1 − x n | C ≤ ( δ m + δ m − + · · · + δ n +1 ) (cid:107) F ( x ) (cid:107) X c = δ n +1 (cid:107) F ( x ) (cid:107) X c m − n (cid:88) k =0 δ k ≤ δ n +1 (cid:107) F ( x ) (cid:107) X c ∞ (cid:88) k =0 δ k δ< = δ n +1 (cid:107) F ( x ) (cid:107) X c (1 − δ ) . Let ε > . Moreover, since δ ∈ [0 , , we can find a large number N ∈ N so that δ N +1 < εc (1 − δ ) / (cid:107) F ( x ) (cid:107) X . Therefore, for m, n > N ∈ N , | x m − x n | C ≤ ε. Hence, we have that the sequence { x n ( · ) } n ≥ ⊆ C D is Cauchy. Therefore, { x n ( · ) } n ≥ converges to some ¯ x ( · ) ⊆ C D , where ¯ x ( · ) satisfies ¯ x ( t ) = x + (cid:90) t ˜ F (¯ x ( τ )) dτ, ∀ t ∈ [ t , t + T ] . (3.10)By Proposition 3.7, ¯ x ( · ) is a solution of (DS ) for t ∈ [ t , t + T ] . (cid:3) Remark 3.9.
The proof of the above proposition will not work in the formulation of ˜ F defined only on set ˆ D . This comes from the fact that the operator S [ x ]( · ) : ˆ D → C ([ t , t + T ] , X ) may map a function x ( · ) outside of ˆ D for which we cannot apply Step 4. in the proof.However, the case when x ∈ int ˆ D , we have the following corollary. Corollary 3.10.
We have the following relationships between (DS) and (DS ) :(1) if x ∈ int ˆ D , then there exists a function x ( · ) ∈ C ([ t , t + T ] , ˆ D ) , which is aunique solution of (DS) and (DS ) on [ t , t + T ] for some T > ;(2) if x ∈ ∂ ˆ D and assumption (B) holds, then the solution of (DS) is unique on [ t , t + T ] for some T > and the solution of (DS ) exists on [ t , t + T ] forsome T > .Proof. The proof will follow the same lines of proof of the Proposition 3.8 up to Step 3replace ˜ F with F and then as follows.We consider the following two cases.Case 1. Suppose x ∈ ˆ D such that ρ := inf y ∈ ∂ ˆ D (cid:107) x − y (cid:107) H =: dist( x , ∂ ˆ D ) > .Case 2. Suppose x ∈ ˆ D such that ρ = 0 . Then one can follow the proof of Proposition 3.8.We show for the Case 1., we look for solution to (DS).Let us consider C D := (cid:26) x ( t ) ∈ C : | x − x | C ≡ max t ∈ [ t ,t + T ] (cid:107) x ( t ) − x (cid:107) X ≤ ρ/ , x ∈ ˆ D (cid:27) . Given the fact that x ∈ ˆ D := { x ∈ D | (cid:107) x − ¯ w (cid:107) ≥ r > } , it implies ρ := || x − ¯ w || > . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 11
Let us consider the following two possible cases at a moment for fixed r > ,(i) if ρ > r , then consider the ball B ρ ( x ) ⊂ ˆ D ;(ii) if ρ < r , then consider the ball B r − ρ ( x ) ⊂ ˆ D .Thereafter following the Step 4. of Proposition 3.8 we can show the existence of Cauchysequence in C D and so on.Step 5. Moreover, C D is a closed subset of C . Indeed, its an implication of the facts ofcontinuity of S and x n ∈ D ⇒ lim n →∞ x n =: ˆ x ∈ D, since D is closed in H. Step 6. Finally, D (cid:51) ˆ x must be a fixed point of S : C D → C D . Indeed, ˆ x = lim n →∞ x n = lim n →∞ S [ x n − ] continuity of S = S (cid:104) lim n →∞ x n − (cid:105) = S [ˆ x ] . Hence, we reach at the solution to (DS). (cid:3)
In the following example we show that the existence of solutions of (DS) is not guaranteedwithout assumption (B), however there are still solutions of (DS ) due to Proposition 3.8. Example 3.11.
Let X = R , ¯ w = ( − , , ¯ z = (1 , , ˆ D = ¯ B ((0 , , \ B (( − , , and let F : ˆ D → X be defined as F (( x , x )) = (1 − x , , ( x , x ) ∈ ˆ D Then assumption (A) and (C) is satisfied. Consider x = x (0) = (0 , − . Then there is nosolution of (DS) . By extending F ( x ) in the continuous way: ˜ F (( x , x )) = (1 − x , , ( x , x ) ∈ X , we obtain that one solution of (DS ) is x ( t ) = (1 − e − t , − . The following example shows that by considering (DS ) with assumption (B) we mayloose the uniqueness of solutions in the sense of Definition 3.6. Example 3.12.
Let X = R , ¯ w = (0 , − , ¯ z = (1 , , ˆ D = [0 , × [ − , \ B ((0 , − , and let F : ˆ D → X be defined as F (( x , x )) = (1 − x , − x ) , ( x , x ) ∈ ˆ D Then assumptions (A), (B) and (C) are satisfied. Consider x = x (0) = (0 , . By extending F ( x ) in the continuous way: ˜ F (( x , x )) = (1 − x , − x ) ( x , x ) ∈ ˆ D , (1 − x , x ) ( x , x ) ∈ Γ := { (1 − e − s , e − s + s − , s ∈ (0 , } , continuous otherwise on X . We obtain that there are more solutions than one of the system (DS ) . For example: ( x ( t ) , x ( t )) = (1 − e − t , e − t + t − , t ∈ [0 , , ( x ( t ) , x ( t )) = (1 − e − t , , t ∈ [0 , . Extendability of solutions to (DS)In this section we provide conditions for the existence and uniqueness to problem (DS)on ( t , + ∞ ) for any t ≥ , see Theorem 4.1. The proof is based on two lemmas, Lemma4.1 and Lemma 4.2 . The proposed approach follows the lines presented in Lecture 3 of thelecture notes in [2]. For more general results and examples on extendability of solutions seee.g. [17] and the references therein.Let T = [ t ; T ) , t < T ≤ + ∞ or T = [ t ; T ] , t < T < + ∞ . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 12
As a consequence of the results of Proposition 3.4 and Corollary 3.5 we have the following’non branching’ result.
Lemma 4.1.
Suppose that assumptions (A), (B) and (C) are satisfied. Let x ( t ) , x ( t ) besolutions to problem (DS) in the sense of Definition 2.3 on T , T , respectively. Then oneof these solutions is a prolongation of the other (in particular, they coincide if T = T ) .Proof. By contrary suppose that, x ( t ) (cid:54)≡ x ( t ) on T ∩ T . Consider the set T (cid:54) = := { t ∈ T ∩ T | x ( t ) (cid:54) = x ( t ) } . Let us note that t / ∈ T (cid:54) = (by initial condition of (DS)). Furthermore, the set T (cid:54) = is openin the set T ∩ T , because it is an inverse image of ( t , + ∞ ) under continuous mapping t → (cid:107) x ( t ) − x ( t ) (cid:107) defined on T ∩ T .Put T ∗ = inf T (cid:54) = . Let us note that T ∗ / ∈ T (cid:54) = (hence x ( T ∗ ) = x ( T ∗ ) ). Indeed, if T ∗ = t then, t / ∈ T (cid:54) = because x ( t ) = x ( t ) . If T ∗ > t , then T ∗ is a boundary point of T (cid:54) = , so T ∗ / ∈ T (cid:54) = since T (cid:54) = is open in T ∩ T . This means that in any right-hand side half-neighbourhood of thepoint T ∗ there exists t > T ∗ such that t ∈ T (cid:54) = (cid:40) T ∩ T , and the intersection of thisright-hand side half-neighbourhood with T (cid:54) = is nonempty.Take any α > T ∗ and t , t ∈ T (cid:54) = ∩ [ T ∗ , α ) . By Remark 2.5, functions x ( t ) , x ( t ) aresolutions to Cauchy problem (cid:26) ˙ x ( t ) = F ( x ( t )) , t > T ∗ x ( T ∗ ) = x ( T ∗ ) (4.1)on interval [ T ∗ , t ] . Since x ( t ) , x ( t ) ∈ ˆ D for all t ∈ [ T ∗ , t ] and the set ˆ D is bounded, wehave R = max i =1 , sup t ∈ [ T ∗ ,t ] (cid:107) x i ( t ) − x ( T ∗ ) (cid:107) < + ∞ . (4.2)By Corollary 3.5, there exists T (cid:48) > t , such that for any T ∈ ( t , T (cid:48) ] , solution of the Cauchyproblem (4.1) on interval [ T ∗ , T ∗ + T ] satisfying (cid:107) x ( t ) − x ( T ∗ ) (cid:107) ≤ R (4.3)is unique. Taking T = min { T (cid:48) , t − T ∗ } we come to a contradiction with Corollary 3.5,because, by (4.2) the condition (4.3) holds both for x ( t ) and x ( t ) , but the functions x ( t ) and x ( t ) are different in any right-hand side half-neighbourhood of T ∗ . (cid:3) Before proving our main result we prove the following elementary Lemma.
Lemma 4.2.
Let x ( t ) be a Lipschtiz function on ( a, b ) , a, b ∈ R with Lipschitz constant L and values in Hilbert space X . Then the limit lim t → b − x ( t ) exists.Proof. It is enough to show that x ( t ) has the Cauchy property at b − , in the sense that ∀ ε > ∃ δ > ∀ t , t ∈ ( a, b ) b − t < δ ∧ b − t < δ = ⇒ (cid:107) x ( t ) − x ( t ) (cid:107) < ε. (4.4)Since x ( t ) is Lipschitz on ( a, b ) we have ∀ t , t ∈ ( a, b ) (cid:107) x ( t ) − x ( t ) (cid:107) < L | t − t | By the right-hand side half-neigbourhood of a given t ∈ R we mean an interval in a form [ t, α ) for any α > t . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 13
Let us take any ε > and δ = ε L . Then for any t , t , < b − t < δ , < b − t < δ wehave (cid:107) x ( t ) − x ( t ) | < L | t − t | < L ( | b − t | + | b − t | ) < ε, which proves (4.4). (cid:3) Now we are ready to prove the main result of this section.
Theorem 4.1.
Suppose that assumptions (A), (B) and (C) hold. There exists a uniquesolution of (DS) on [ t , + ∞ ) .Proof. By Corollary 3.5, there exists solution of problem (DS) on some interval [ t , T ] ( T > t ) in the class B Rt ,x ,T for some R > . By Lemma 4.1, for any two solutions of ourproblem (DS ) on different intervals one is the prolongation of the other.Consider now, for any T > t , all functions from C ([ t , T ] , ˆ D ) . Among these functionsthere exist solutions of problem (DS) or not. Put T = { T > t | ∃ solution to (DS) from C ([ t , T ] , ˆ D ) } ,T = sup T . (4.5)If T = + ∞ , there exists solution ˜ x ( t ) ∈ C ([ t , + ∞ ) , ˆ D ) to problem (DS). Indeed,by taking a monotone increasing sequence T n → + ∞ and the corresponding sequence ofsolutions { x n ( t ) } , by Lemma 4.1 we get, for all n ∈ N solution x n +1 is the prolongation of x n . Hence, the function ˜ x ( t ) = (cid:26) x n ( t ) , t ∈ [ T n − , T n ) , n ≥ x ( t ) , t ∈ [ t , T ) is a solution defined on [ t , + ∞ ) . Other solutions (which do not coincide with restrictionsof ˜ x ( t ) on smaller intervals) do not exist by Lemma 4.1. In the rest of the proof we showthat this is the only possible case.Consider now T < + ∞ . Then two cases are possible:(a) T ∈ T ,(b) T / ∈ T .In case ((a)) there exists a solution x ( · ) ∈ C ([ t , T ] , ˆ D ) to problem (DS). But then, byCorollary 3.5, applied to our problem (DS) with t = T solution can be extended beyond T and both one-sided derivatives ˙ x − ( T ) and ˙ x + ( T ) exist and both equal F ( x ( T )) : left - bythe definition of solutions on [ t , T ] , right - by the definition of solution to our problem withthe beginning of the interval from T . In consequence, we get solution on larger intervaland arrive to a contradiction with the definition of T . This excludes case ((a)).In case ((b)), by the arguments analogous to the case T = + ∞ , we get the existenceand uniqueness of solutions x ( t ) of (DS) on the semi-interval [ t , T ) . Case ((b)) splits intwo subcases:(1) lim sup t → T − (cid:107) x ( t ) (cid:107) = + ∞ (i.e. solution is unbounded in any left-sided interval of T ),(2) lim sup t → T − (cid:107) x ( t ) (cid:107) < + ∞ .The subcase 1 is impossible in view of boundednes of the set ˆ D . Now we show thatthe subcase 2 is also impossible. Indeed, let the function x ( t ) be bounded on the wholehalf-interval [0 , T ) : ∃ C ≥ ∀ t ∈ [ t , T ) (cid:107) x ( t ) (cid:107) ≤ C. We have ∀ t ∈ [ t , T ) (cid:107) F ( x ( t )) (cid:107) ≤ M. N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 14
But then, from the equation (DS), it follows that the function x ( t ) is Lipschitz continuouswith a constant M on ( t , T ) , since (cid:107) ˙ x ( t ) (cid:107) ≤ M for all t ∈ ( t , T ) . Hence, by Lemma4.2, there exists the limit Y = lim t → T − x ( t ) . Let us put Y to be the value of x ( t ) at T . The obtained function Y ( t ) will be continuousfrom the left at T . Then, by Lemma 2.1, the function F ( Y ( T )) is also continuous fromthe left at T and hence lim t → T − F ( x ( t )) = lim t → T − F ( Y ( t )) = F ( Y ) . Since for t < T we have ˙ x ( t ) = F ( x ( t )) , from the last formula we get lim t → T − ˙ x ( t ) = F ( Y ) . However, by Lemma about extendability at point (Lemma 2.2), it follows that the function x ( t ) can be extended from [ t , T ) onto [ t , T ] with preservation of continuous differen-tiability (let us denote the obtained function by Y ( t ) ) and ˙ Y ( t ) = F ( Y ) and Y ( t ) is asolution on [ t , T ] . We arrive to a contradiction in the subcase 2 of case ((b)) (solutionson [ t , T ] do not exist). (cid:3) Behaviour of trajectories at + ∞ Let x ( t ) , t ∈ [ t , + ∞ ) be a solution of (DS). In this section we investigate the conver-gence properties of x ( · ) when t → + ∞ . We start with the following Lemma. Lemma 5.1.
For all x ∈ D we have (cid:107) x − ¯ z (cid:107) ≤ (cid:107) ¯ w − ¯ z (cid:107) − (cid:107) ¯ w − x (cid:107) . Proof.
This follows from (2.1): we have (cid:107) ¯ w − ¯ z (cid:107) = (cid:107) ¯ w − x (cid:107) + 2 (cid:104) ¯ w − x | x − ¯ z (cid:105) + (cid:107) x − ¯ z (cid:107) ≥ (cid:107) ¯ w − x (cid:107) + (cid:107) x − ¯ z (cid:107) , for all x ∈ D . (cid:3) With this observation we can establish the following results.
Proposition 5.1.
Suppose that assumptions (A), (B) and (C) hold. Suppose that thereexists an increasing sequence { t n } n ∈ N such that t n → + ∞ and x ( t n ) → ¯ z . Then x ( t ) → ¯ z as t → + ∞ .Proof. Let { t n } n ∈ N , t n → + ∞ be such that x ( t n ) → ¯ z . We will show that for all ε > , forevery increasing sequence { s n } n ∈ N , s n → + ∞ there exists n ∈ N such that for all n ≥ n , (cid:107) x ( s n ) − ¯ z (cid:107) ≤ ε . Take any ε > and an increasing sequence { s n } n ∈ N , s n → + ∞ .We have ddt (cid:107) x ( t ) − ¯ w (cid:107) = 2 (cid:104) F ( x ( t )) | x ( t ) − ¯ w (cid:105) ≥ , hence function (cid:107) x ( · ) − ¯ w (cid:107) is nondecreasing. Moreover, by Lemma 5.1 and convergence of x ( t n ) , for all ε (cid:48) > there exists n (cid:48) ∈ N such that for all n > n (cid:48) (cid:107) x ( t n ) − ¯ w (cid:107) ≥ (cid:107) ¯ w − ¯ z (cid:107) − ε (cid:48) . Take ε (cid:48) = ε and n such that s n ≥ t n (cid:48) .Then, by Lemma 5.1 and fact that (cid:107) x ( · ) − ¯ w (cid:107) is nondecreasing we obtain: for all n ≥ n (cid:107) x ( s n ) − ¯ z (cid:107) ≤ (cid:107) ¯ w − ¯ z (cid:107) − (cid:107) x ( s n ) − ¯ w (cid:107) ≤ (cid:107) ¯ w − ¯ z (cid:107) − (cid:107) x ( t (cid:48) n ) − ¯ w (cid:107) ≤ ε. (cid:3) Theorem 5.2.
Suppose that assumptions (A), (B) and (C) hold. Assume that either
N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 15 (a) X is finite-dimensional and lim t → + ∞ x ( t ) exists, or(b) X is infinite-dimensional and for every sequence { t n } n ∈ N , t n → + ∞ x ( t n ) (cid:42) ˜ x = ⇒ ˜ x = ¯ z, (5.1) where x ( t ) is a unique solution of (DS) .Then the trajectory x ( t ) satisfies the condition lim t → + ∞ x ( t ) = ¯ z , where convergence isunderstood in the sense of the norm of X .Proof. (a) Suppose X is finite dimensional. Then the set ˆ D is compact (closed andbounded). Take any sequence { t n } n ∈ N , t n → + ∞ . Since x ( t n ) ∈ ˆ D , there existsa subsequence { t n k } such that x ( t n k ) converges to some element ˜ x ∈ ˆ D . By thecontinuity of F ( · ) on ˆ D we have lim k → + ∞ (cid:107) F ( x ( t n k )) (cid:107) = (cid:107) F (˜ x ) (cid:107) . (5.2)Since X is finite-dimensional of dimension, say, (cid:96) , then x ( t ) := ( x ( t ) , ..., x (cid:96) ( t )) , ˙ x ( t ) = ( ˙ x ( t ) , ..., ˙ x (cid:96) ( t )) . By assumption, there exists ˜ x ∈ ˜ D such that lim t → + ∞ x ( t ) = ˜ x , ˜ x := (˜ x , ..., ˜ x (cid:96) ) ,and by the continuity of F , lim t → + ∞ ˙ x ( t ) = lim t → + ∞ F ( x ( t )) = F (˜ x ) = ( F (˜ x ) , ..., F (cid:96) (˜ x )) , (5.3)or equivalently, lim t → + ∞ ˙ x i ( t ) = F i (˜ x ) for ≤ i ≤ (cid:96). (5.4)For any ≤ i ≤ (cid:96) and any n ∈ N , by the (Lagrange) Mean Value Theorem, thereexists ξ in ∈ ( n, n + 1) such that x i ( n + 1) − x i ( n ) = ˙ x i ( ξ in ) . This shows that lim n → + ∞ ˙ x i ( ξ in ) = 0 . By (5.4) it must be lim n → + ∞ ˙ x i ( ξ in ) = lim t → + ∞ ˙ x i ( t ) = F i (˜ x ) = 0 . (5.5)Hence, F (˜ x ) = 0 , which proves that ˜ x = ¯ z .(b) By 5.1, we have ˜ x = ¯ z , i.e., x ( t n k ) converges weakly to ¯ z . By Lemma 5.1, thefollowing inequality holds for this subsequence (cid:107) x ( t n k ) − ¯ w (cid:107) + (cid:107) x ( t n k ) − ¯ z (cid:107) ≤ (cid:107) ¯ w − ¯ z (cid:107) , k = 1 , , . . . and hence lim inf k →∞ (cid:107) x ( t n k ) − ¯ w (cid:107) + lim inf k →∞ (cid:107) x ( t n k ) − ¯ z (cid:107) ≤ (cid:107) ¯ w − ¯ z (cid:107) . ( ∗ ) Since the norm is weakly lower semicontinuous, we also have (cid:107) ¯ z − ¯ w (cid:107) ≤ lim inf k →∞ (cid:107) x ( t n k ) − ¯ w (cid:107) . This and ( ∗ ) implies lim inf k →∞ (cid:107) x ( t n k ) − ¯ z (cid:107) = 0 . Consequently, there is a subsequence t n km such that lim m →∞ (cid:107) x ( t n km ) − ¯ z (cid:107) = 0 . Thus we have shown that for any sequence { t n } n ∈ N , t n → ∞ , there exists asubsequence { t n km } m ∈ N such that the above condition holds.This means that (cid:107) x ( t ) − ¯ z (cid:107) → as t → + ∞ . (cid:3) N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 16
Proposition 5.3.
Suppose that assumptions (A), (B) and (C) hold. Assume that X isfinite dimensional and that for all t ∈ [ t , + ∞ ) such that x ( t ) (cid:54) = ¯ z , we have (cid:104) F ( x ( t )) | ¯ w − x ( t ) (cid:105) < . Then lim t → + ∞ x ( t ) = ¯ z .Proof. Let g ( t ) := ddt (cid:107) x ( t ) − ¯ w (cid:107) , t ≥ t . We start by showing that there exists a sequence { t k } , t k → + ∞ such that lim k → + ∞ g ( t k ) = 0 .By contrary, suppose that there exist ε > and t (cid:48) ≥ t such that g ( t ) > ε for all t > t (cid:48) .Hence, for all t > t (cid:48) (cid:107) x ( t ) − ¯ w (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) = (cid:90) tt g ( s ) ds = (cid:90) t (cid:48) t g ( s ) ds + (cid:90) tt (cid:48) g ( s ) ds ≥ (cid:90) t (cid:48) t g ( s ) ds + (cid:90) tt (cid:48) ε ds = (cid:90) t (cid:48) t g ( s ) ds + ( t − t (cid:48) ) ε. By taking t > ε (cid:32) (cid:107) ¯ z − ¯ w (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) − (cid:90) t (cid:48) t g ( s ) ds (cid:33) + t (cid:48) we arrive to (cid:107) x ( t ) − ¯ w (cid:107) > (cid:107) ¯ z − ¯ w (cid:107) , i.e. x ( t ) / ∈ ˆ D - a contradiction. In this way we provedthat there exists a sequence { t k } k ∈ N such that t k → + ∞ and lim k → + ∞ g ( t k ) = 0 .Since X is finite-dimensional and ˆ D is closed, bounded, hence compact. There exists asubsequence of { t k } k ∈ N , namely { t k n } n ∈ N such that x ( t k n ) converges and lim n → + ∞ x ( t k n ) =˜ x ∈ ˆ D . Without loss of generality we may assume that the sequence { t k n } n ∈ N is increasing.By Fact 2.1, ddt (cid:107) x ( t ) − ¯ w (cid:107) = 2 (cid:104) F ( x ( t )) | x ( t ) − ¯ w (cid:105) ≥ for all t ≥ t . We have n → + ∞ g ( t k n ) = lim n → + ∞ (cid:104) F ( x ( t k n )) | x ( t k n ) − ¯ w (cid:105) = 2 (cid:104) F (˜ x ) | ˜ x − ¯ w (cid:105) , hence, by assumption ˜ x = ¯ z . Now the assertion follows from Proposition 5.1. (cid:3) Remark 5.4.
In infinite dimensional Hilbert space X the assertion of Proposition 5.3 remainstrue under additional assumptions. We assume that F can be extended to conv ˆ D and F : conv ˆ D → X is a weak-strong continuous on conv ˆ D , i.e., for any weakly convergentsequence ˆ D (cid:51) x n (cid:42) ¯ x we have lim n → + ∞ F ( x n ) = F (¯ x ) , where the limit is strong. Thisfollows from the fact that if v n → v and u n (cid:42) u , then (cid:104) v n | u n (cid:105) → (cid:104) v | u (cid:105) . Indeed, |(cid:104) v n | u n (cid:105) − (cid:104) v | u (cid:105)| = |(cid:104) v n − v | u n (cid:105) + (cid:104) u n | v (cid:105) − (cid:104) v | u (cid:105)|≤ (cid:107) u n (cid:107)(cid:107) v n − v (cid:107) + |(cid:104) u n | v (cid:105) − (cid:104) v | u (cid:105)| Proposition 5.5.
Suppose that assumptions (A), (B) and (C) hold. Assume that for all t ∈ [ t , + ∞ ) such that x ( t ) (cid:54) = ¯ z , we have (cid:104) F ( x ( t )) | ¯ w − x ( t ) (cid:105) < α ( t ) , where α :[ t , + ∞ ) → R − is an integrable function on any interval [ t , T ] , T > t and there exist T (cid:48) > t , √ T (cid:48) − t √ T (cid:48) > ( (cid:107) ¯ w − ¯ z (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) ) and ε ≤ − √ T (cid:48) such that sup [ t ,T (cid:48) ] α ( s ) < ε . Then lim t → + ∞ x ( t ) = ¯ z .Proof. Let us note that in the case when there exists t (cid:48) ∈ [ t , + ∞ ) such that x ( t (cid:48) ) = ¯ z ,then x ( t ) = ¯ z for all t > t (cid:48) since F ( x ( t (cid:48) )) = F (¯ z ) = 0 . N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 17
Consider now the situation that x ( t ) (cid:54) = ¯ z for any t ∈ [ t , + ∞ ) . By contradiction, supposethat x ( t ) (cid:54)→ ¯ z . Then, in view of Proposition 5.1, there exists ε > such that x ( t ) / ∈ B (¯ z, ε ) for all t ∈ [ t , + ∞ ) .We have that for all t > T (cid:48) (cid:107) x ( t ) − ¯ w (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) = (cid:90) tt dds (cid:107) x ( s ) − ¯ w (cid:107) ds = 2 (cid:90) tt (cid:104) F ( x ( s )) | x ( s ) − ¯ w (cid:105) ds ≥ − (cid:90) tt α ( s ) ds ≥ − t − t ) · sup s ∈ [ t ,t ] α ( s ) ≥ − t − t ) · ε. Thereby for such t > (cid:107) ¯ w − ¯ z (cid:107) c + t we arrive to a contradiction with x ( t ) ∈ ˆ D ⊂ D . (cid:3) Proposition 5.6.
Assume that for all ε such that < ε < (cid:107) x − ¯ z (cid:107) , inf x ∈ ˆ D\ B (¯ z,ε ) (cid:104) F ( x ) | ¯ w − x (cid:105) < . Then lim t → + ∞ x ( t ) = ¯ z .Proof. If there exists t (cid:48) ∈ [0 , + ∞ ) such that (cid:104) F ( x ( t (cid:48) )) | w − x ( t (cid:48) ) (cid:105) = 0 then we are done -in view of assumptions of the Proposition, x ( t (cid:48) ) = ¯ z , and by Lemma 5.1, Proposition 3.1, x ( t ) = ¯ z for all t ≥ t (cid:48) .Suppose that for all t ∈ [0 , + ∞ ) we have (cid:104) F ( x ( t )) | w − x ( t ) (cid:105) < . For any t > t (cid:107) ¯ z − ¯ w (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) ≥ (cid:107) x ( t ) − ¯ w (cid:107) − (cid:107) x ( t ) − ¯ w (cid:107) = (cid:90) tt dds (cid:107) x ( s ) − ¯ w (cid:107) ds = 2 (cid:90) tt (cid:104) F ( x ( s )) | x ( s ) − ¯ w (cid:105) ds ≥ − t − t ) · sup s ∈ [ t ,T ] (cid:104) F ( x ( s )) | x ( s ) − ¯ w (cid:105) = 2( t − t ) · inf s ∈ [ t ,T ] (cid:104) F ( x ( s )) | w − x ( s ) (cid:105) Therefore inf s ∈ [ t ,t ] (cid:104) F ( x ( s )) | w − x ( s ) (cid:105) → as t → + ∞ . Note that α ( s ) := (cid:104) F ( x ( s )) | w − x ( s ) (cid:105) is a continuous function on every [ t , t ] , t > t . Hence there exists an increasingsequence { t n } n ∈ R + , t n → + ∞ such that (cid:104) F ( x ( t n )) | x ( t n ) − ¯ w (cid:105) → . We claim that x ( t n ) → ¯ z .Suppose by contrary, that x ( t n ) (cid:54)→ ¯ z as n → + ∞ . Then there exists ε > and asubsequence { t n k } k ∈ N such that x ( t n k ) ∈ ˆ D \ B (¯ z, ε ) . Since inf x ∈ ˆ D\ B (¯ z,ε ) (cid:104) F ( x ) | ¯ w − x (cid:105) < we have that there exists c < such that (cid:104) F ( x ( t k n )) | ¯ w − x ( t k n ) (cid:105) < c , a contradiction to lim k → + ∞ (cid:104) F ( x ( t k n )) | ¯ w − x ( t k n ) (cid:105) = 0 .Hence x ( t n ) → ¯ z . Now the assertion follows from Proposition 5.1. (cid:3) Projective dynamical system
In this section we give an example of the system (DS). We consider projective dynamicalsystem ˙ x ( t ) = P C ( x ) ( ¯ w ) − x,x ( t ) = x ∈ ˆ D , (PDS)where C ( x ) : D ⇒ X is a multifunction such that:(A (cid:48) ) for all x ∈ D , ¯ z ∈ C ( x ) and P C ( x ) ( ¯ w ) = x iff x = ¯ z ,(C (cid:48) ) (cid:104) P C ( x ) ( ¯ w ) − x | ¯ w − x (cid:105) ≤ for all x ∈ D ,(D (cid:48) ) for all x ∈ D , C ( x ) is closed and convex. N THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 18
Condition (D (cid:48) ) ensures that the projection onto C ( x ) , x ∈ D is uniquely defined andtherefore (C (cid:48) ) is equivalent to the condition: ∀ x ∈ D ∀ h ∈ C ( x ) (cid:104) h − x | ¯ w − x (cid:105) ≤ . Let us note that in this setting ((A (cid:48) ), (C (cid:48) ), (D (cid:48) )) the assumption (B) is satisfied since forall x ∈ ˆ D and for any h ∈ [0 , x + h ( P C ( x ) ( ¯ w ) − x ) = (1 − h ) x + hP C ( x ) ( ¯ w ) ∈ D , (cid:107) x + h ( P C ( x ) ( ¯ w ) − x ) − ¯ w (cid:107) = (cid:107) x − ¯ w (cid:107) − h (cid:104) P C ( x ) ( ¯ w ) − x | ¯ w − x (cid:105) + h (cid:107) P C ( x ) ( ¯ w ) − x (cid:107) ≥ (cid:107) x − ¯ w (cid:107) ≥ r, i.e. x + h ( P C ( x ) ( ¯ w ) − x ) ∈ ˆ D .As a consequence of Theorem 4.1 we can formulate the following theorem. Theorem 6.1.
Suppose that (A (cid:48) ), (C (cid:48) ) , (D (cid:48) ) holds. Assume that P C ( x ) ( ¯ w ) is locallyLipschitz continuous on ˆ D \ { ¯ z } and continuous on ˆ D . Then the system (PDS) has aunique solution on [ t , + ∞ ) .Proof. The proof follows from Theorem 4.1. (cid:3)
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