Nonconvex integro-differential sweeping process with applications
NNonconvex integro-differential sweeping process withapplications.
Abderrahim Bouach ∗ Tahar Haddad † Lionel Thibault ‡ February 25, 2021
Abstract
In this paper, we analyze and discuss the well-posedness of a new variant of theso-called sweeping process, introduced by J.J. Moreau in the early 70’s [18] withmotivation in plasticity theory. In this variant, the normal cone to the (mildly non-convex) prox-regular moving set C ( t ), supposed to have an absolutely continuousvariation, is perturbed by a sum of a Carath´eodory mapping and an integral forcingterm. The integrand of the forcing term depends on two time-variables, that is, westudy a general integro-differential sweeping process of Volterra type. By setting up anappropriate semi-discretization method combined with a new Gronwall-like inequality(differential inequality), we show that the integro-differential sweeping process has oneand only one absolutely continuous solution. We also establish the continuity of thesolution with respect to the initial value. The results of the paper are applied to thestudy of nonlinear integro-differential complementarity systems which are combinationof Volterra integro-differential equations with nonlinear complementarity constraints.Another application is concerned with non-regular electrical circuits containing time-varying capacitors and nonsmooth electronic device like diodes. Both applicationsrepresent an additional novelty of our paper. Keywords
Moreau’s sweeping process, Volterra integro-differential equation, Differentialcomplementarity systems, Gronwall’s inequality, Prox-regular sets, Differential inclusions.
AMS subject classifications
Contents ∗ Laboratoire LMPEA, Facult´e des Sciences Exactes et Informatique, Universit´e Mohammed SeddikBenyahia, Jijel, B.P. 98, Jijel 18000, Alg´erie ( [email protected] ). † Laboratoire LMPEA, Facult´e des Sciences Exactes et Informatique, Universit´e Mohammed SeddikBenyahia, Jijel, B.P. 98, Jijel 18000, Alg´erie ( [email protected] ). ‡ Universit´e de Montpellier, Institut Montpelli´erain Alexander Grothendieck 34095 Montpellier CEDEX5 France ( [email protected] ). a r X i v : . [ m a t h . O C ] F e b In the seventies, sweeping processes are introduced and deeply studied by J. J. Moreauthrough a series of papers, in particular [18, 19]. There, it is shown that such processesplay an important role in elasto-plasticity, quasi-statics, dynamics, especially in mechanics[18]. Roughly speaking, a point is swept by a moving closed convex set C ( t ) in a Hilbertspace H , which can be formulated in the form of differential inclusion as follows (cid:40) − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) a.e. t ∈ [ T , T ] x ( T ) = x ∈ C ( T ) , (1)where N C ( t ) ( · ) denotes the normal cone of C ( t ) in the sense of convex analysis. When thesystems are perturbed, it is natural to study the following variant (cid:40) − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) a.e. t ∈ [ T , T ] x ( T ) = x ∈ C ( T ) (2)where f : [ T , T ] × H → H is a Carath´eodory mapping.Actually, diverse approaches for existence of solutions of (1) and (2) are available inthe literature: Catching-up method (see, e.g., [19]), regularization procedure (see, e.g.,[18, 20]), reduction to unconstrained differential inclusion (see, e.g., [26]). For the firstand second methods, existence and uniqueness of solutions follow, to some extent, fromthe classical Gronwall inequality and the basic relation ddt (cid:107) x ( t ) (cid:107) = 2 (cid:104) ˙ x ( t ) , x ( t ) (cid:105) , (3)(whenever meaningful), applied to two solutions or suitable approximate solutions x , x of (2), by means of the monotonicity of N C ( t ) ( · ) ( hypomonotonicity when C ( t ) is prox-regular). Those features and the Lipschitz property of the forcing term f with respectto the state variable are employed to obtain that the distance between x ( t ) and x ( t ) isnonincreasing with respect to time t . This reasoning allows in general the construction ofa Cauchy sequence of approximate solutions, converging to a solution.Several extensions of the sweeping process in diverse ways (well-posedness and optimalcontrol) have been studied in the literature (see, e.g., [1], [2], [5], [8], [16], [26], [27] andreferences therein).The present paper aims to study the following new variant of the sweeping process( P f ,f ) − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds a.e. t ∈ [ T , T ] x ( T ) = x ∈ C ( T ) , (4)where N C ( t ) ( · ) denotes the Clarke normal cone to the subset C ( t ) of H .We will have to use the following assumptions:( H ) For each t ∈ [ T , T ], C ( t ) is a nonempty closed subset of H which is r -prox-regular,for some r ∈ (0 , + ∞ ], and has an absolutely continuous variation, in the sense thatthere is some absolutely continuous function υ : [ T , T ] −→ R such that C ( t ) ⊂ C ( s ) + | υ ( s ) − υ ( t ) | B H , ∀ s, t ∈ [ T , T ] , where B H denotes the closed unit ball of H centered at the origin.( H ) f : [ T , T ] × H −→ H is (Lebesgue) measurable in time (i.e., f ( · , x ) is measurablefor each x ∈ H ), and such that( H , ) there exist non-negative functions β ( · ) ∈ L ([ T , T ] , R ) such that (cid:107) f ( t, x ) (cid:107) ≤ β ( t )(1 + (cid:107) x (cid:107) ) , for all t ∈ [ T , T ] and for any x ∈ (cid:91) t ∈ [ T ,T ] C ( t );( H , ) for each real η > L η ( · ) ∈ L ([ T , T ] , R )such that for any t ∈ [ T , T ] and for any ( x, y ) ∈ B [0 , η ] × B [0 , η ], (cid:107) f ( t, x ) − f ( t, y ) (cid:107) ≤ L η ( t ) (cid:107) x − y (cid:107) , where B [0 , η ] denotes the closed ball centered at the origin with radius η .( H ) f : Q ∆ × H −→ H is a measurable mapping such that( H , ) there exists a non-negative function β ( · , · ) ∈ L ( Q ∆ , R ) such that (cid:107) f ( t, s, x ) (cid:107) ≤ β ( t, s )(1+ (cid:107) x (cid:107) ) , for all ( t, s ) ∈ Q ∆ and for any x ∈ (cid:91) t ∈ [ T ,T ] C ( t );( H , ) for each real η > L η ( · ) ∈ L ([ T , T ] , R )such that for all ( t, s ) ∈ Q ∆ and for any ( x, y ) ∈ B [0 , η ] × B [0 , η ], (cid:107) f ( t, s, x ) − f ( t, s, y ) (cid:107) ≤ L η ( t ) (cid:107) x − y (cid:107) . Above L ([ T , T ] , R ) (resp. L ( Q ∆ , R )) stands for the space of Lebesgue integrable func-tions on [ T , T ] (resp. Q ∆ ), where Q ∆ := { ( t, s ) ∈ [ T , T ] × [ T , T ] : s ≤ t } . We called the differential inclusion (4) as integro-differential sweeping process becausethe integral of the state and the velocity are defined in the dynamical system. Onecan interpret (4) as follows: as long as x ( t ) is in the interior of the set C ( t ), we get N C ( t ) ( x ( t )) = 0 and (4) reduces to a Volterra integro-differential equation − ˙ x ( t ) = f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds a.e. t ∈ [ T , T ] x ( T ) = x (5)(for at least a small period of time) to satisfy the constraint x ( t ) ∈ C ( t ), until x ( t )hits the boundary of the set C ( t ). At this moment, if the vector field − ( f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds ) is pointed outside of the set C ( t ), then any component of this vectorfield in the direction normal to C ( t ) at x ( t ) must be annihilated to maintain the mo-tion of x within the constraint set. So, the system (4) can be considered as a Volterraintegro-differential equation (5) under control term u ( t ) ∈ N C ( t ) ( · ) which guarantees thatthe trajectory x ( t ) always belongs to the desired set C ( t ) for all t ∈ [ T , T ].The well-posedness of the classical perturbed sweeping process (2), i.e., P f , ( f ≡ P ,f ( f ≡ t (cid:90) f ( s, x ( s )) ds , i.e., for the following problem( P ,f ) − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + t (cid:90) f ( s, x ( s )) ds a.e. t ∈ [0 , T ] x (0) = x ∈ C (0) . It is also worth mentioning that Colombo and Kozaily say in their paper [9]: ” ofcourse, existence and uniqueness to ( P ,f ) is not surprising ”. Indeed, we observe thatwith the above integral t (cid:90) f ( s, x ( s )) ds , the integro-differential sweeping process ( P f ,f )is equivalent to − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + y ( t ) , ˙ y ( t ) = f ( t, x ( t )) , x (0) = x , y (0) = 0 , and so − ˙ X ( t ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) − ˙ x ( t ) − ˙ y ( t ) (cid:19) ∈ N C ( t ) × H X ( t ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) x ( t ) y ( t ) (cid:19) + f ( t,X ( t )) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) f ( t, x ( t )) + y ( t ) − f ( t, x ( t )) (cid:19) , which is a special case of the classical perturbed sweeping process (2); see, e.g., [19, 21]for the situation of unbounded moving sets.So, in [9], the aim of the authors was not the well-posedness. Their motivation for studying( P ,f ) was designing a smoother method of penalization, the motivation of which comesfrom applications to deriving necessary optimality conditions for optimal control problemswith sweeping processes.Now, if the integral involving f depends on two time -variables as in (4), the reductionof ( P f ,f ) there to the perturbed sweeping process (2) cannot be applied.To the best of our knowledge, for the problem under consideration in the case of thefunction f depending on two time -variables, that is, in the case of a general integro-differential sweeping process of Volterra type ( P f ,f ), a well-posedness result, includingthe existence, uniqueness, and stability of the solution, has not been obtained up to thepresent time.In the present paper, we obtain results on the existence and uniqueness of a solutionto the Volterra sweeping process ( P f ,f ) in a Hilbert space. This is done with the help ofa new Gronwall-like inequality (see Section 3) and of a new scheme corresponding to theexistence of absolutely continuous solutions for the quasi-autonomous sweeping processes − ˙ x n ( t ) ∈ N C ( t ) ( x n ( t )) + f ( t, x n ( t k )) + k − (cid:88) j =0 t j +1 (cid:90) t j f ( t, s, x n ( t j )) ds + t (cid:90) t k f ( t, s, x n ( t k )) ds a.e. t ∈ [ t k , t k +1 ] ,x n ( T ) = x ∈ C ( T ) , where T = t < t < ... < t n = T is a discretization of the interval [ T , T ].The outline of the paper is as follows. In Section ?? , we recall some preliminary resultsthat we use throughout. In Section 3, we prove a new Gronwall-like inequality ( differentialinequality). Then, in Section 4, we present our main existence, uniqueness, and stabilityresult. In Section 5 we use those results in the study of nonlinear integro-differentialcomplementarity systems. This is realized by transforming such systems into integro-differential sweeping processes of the form (4) where the moving set C ( t ) is described by afinite number of inequalities. We also provide sufficient verifiable conditions ensuring theabsolute continuity of the moving set. Finally, in Section 6, we give a second applicationof our results to non-regular electrical circuits containing time-varying capacitors andnonsmooth electronic device like diodes. Both applications represent an additional noveltyof our paper. Throughout H is a real Hilbert space endowed with the inner product (cid:104)· , ·(cid:105) and associatednorm (cid:107) · (cid:107) . As usual, we will denote by B H or B the closed unit ball of H and by B ( x, δ )(resp. B [ x, δ ]) the open (resp. closed) ball around x ∈ H with radius δ >
0. For anonempty subset S of H the associated distance function is denoted by d S , that is, d S ( x ) := inf y ∈ S (cid:107) x − y (cid:107) for all x ∈ H. By C ([ T , T ] , H ) we denote the space of continuous mappings from [ T , T ] into H equippedwith the supremum norm (cid:107)·(cid:107) ∞ , where we recall that −∞ < T < T < + ∞ . As usual R willdenote the set of real numbers, R + the set of non-negative reals, that is, R + := [0 , + ∞ ),and N the set positive integers. In various cases, it will be convenient to use the notationwrite I := [ T , T ]The Clarke tangent cone of S at x ∈ S , denoted by T C ( S ; x ), is the set of h ∈ H suchthat, for every sequence ( x n ) n ∈ N of S with x n −→ x as n −→ ∞ and for every sequence( t n ) n ∈ N of positive reals with t n −→ n −→ ∞ , there exists a sequence ( h n ) n ∈ N of H with h n −→ h as n −→ ∞ satisfying x n + t n h n ∈ S for all n ∈ N . This set is obviously acone containing zero and it is known to be closed and convex. The polar cone of T C ( S ; x )is the Clarke normal cone N C ( S ; x ) of S at x , that is N CS ( x ) := { υ ∈ H : (cid:104) υ, h (cid:105) ≤ , ∀ h ∈ T C ( S ; x ) } . If x / ∈ S , by convention T C ( S ; x ) and N C ( S ; x ) are empty.For a function f : H → R ∪ { + ∞} , The Clarke subdifferential ∂ C f ( x ) of f at x ( see[23] ) with f ( x ) < + ∞ is defined by ∂ C f ( x ) := { υ ∈ H : ( υ, − ∈ N C ( epi f ; ( x, f ( x ))) } , where H × R is endowed with the usual product structure and epi f is the epigraph of f , that is, epi f := { ( x, r ) ∈ H × R : f ( x ) ≤ r } . If f is not finite at x , we see that ∂ C f ( x ) = ∅ . In addition to the latter definition, thereis another link between the Clarke normal cone and the Clarke subdifferential, given by ∂ C ψ S ( x ) = N C ( S ; x ), where ψ S denotes the indicator function of the subset S of H , i.e. ψ ( y ) = 0 if y ∈ S and ψ ( y ) = + ∞ if y / ∈ S .A vector v ∈ H is a proximal subgradient of f at a point x with f ( x ) < + ∞ (see, e.g.,[12, 17, 24]) if there exist some reals σ ≥ δ > (cid:104) v, y − x (cid:105) ≤ f ( y ) − f ( x ) + σ (cid:107) y − x (cid:107) for all y ∈ B ( x, δ ) . The set ∂ P f ( x ) of all proximal subgradients of f at x is the proximal subdifferential of f at x . Of course, ∂ P f ( x ) = ∅ if f ( x ) = + ∞ . Note that one always has ∂ P f ( x ) ⊂ ∂ C f ( x ) .Taking the proximal subdifferential ∂ P ψ S ( x ) of the indicator function ψ S we obtain the proximal normal cone N PS ( x ) of S at x . So, a vector v ∈ H is a proximal normal vector of S at x ∈ S if and only if there are reals σ ≥ δ > (cid:104) v, y − x (cid:105) ≤ σ (cid:107) y − x (cid:107) for all y ∈ S ∩ B ( x, δ ) . (6)So, we always have the inclusion N PS ( x ) ⊂ N CS ( x ) for all x ∈ S .The proximal normal cone can be described in the following geometrical way (see, e.g.,[12]) N PS ( x ) = { v ∈ H : ∃ r > x ∈ Proj S ( x + rv ) } , (7)where Proj S ( u ) := { y ∈ S : d S ( u ) = (cid:107) u − y (cid:107)} . The proximal normal cone is also connected with the distance function to S through theequalities (see, e.g., [12]) ∂ P d S ( x ) = N PS ( x ) ∩ B H and N PS ( x ) = R + ∂ P d S ( x ) , where R + := [0 , + ∞ [.In many cases one has to require in (7) that the constant r be uniform for all theunit proximal normal vectors of S . The sets which satisfy that property are known as(uniformly) prox-regular sets. Given r ∈ ]0 , + ∞ ] the closed subset S is (uniformly) r - prox-regular (see [22]) (called also r -positively reached (see [15]) or r - proximally smooth (see [11])provided that, for every x ∈ S and every unit vector v ∈ N PS ( x ) one has x ∈ Proj S ( x + rv ) . The latter is equivalent to S ∩ B H ( x + rv, r ) = ∅ or equivalently (cid:10) v, x (cid:48) − x (cid:11) ≤ r (cid:13)(cid:13) x (cid:48) − x (cid:13)(cid:13) , for all x (cid:48) ∈ S . Of course, in the latter inequality, r = 0 for r = + ∞ (as usual). It is worthpointing out that for r = + ∞ , the uniform r -prox-regularity of the closed set S amountsto its convexity. Definition 2.1
For a given r ∈ (0 , ∞ ] , a subset S of the Hilbert space H is uniformly r-prox-regular, or r -prox-regular for short, if and only if for all x ∈ S and all (cid:54) = ς ∈ N PS ( x ) one has (cid:104) ς (cid:107) ς (cid:107) , y − x (cid:105) ≤ (cid:107) y − x (cid:107) r , ∀ y ∈ S. The following propositions summarize some important consequences of prox-regularityneeded in the paper. For the proof of these results, we refer the reader to [22, 10].
Proposition 2.2
Let S be a nonempty closed set in H which is uniformly r-prox-regularfor some r ∈ [0 , + ∞ ] . Then for any x i ∈ S , ς i ∈ N PS ( x i ) with i = 1 , one has : (cid:104) ς − ς , x − x (cid:105) ≥ − (cid:18) (cid:107) ς (cid:107) + (cid:107) ς (cid:107) r (cid:19) (cid:107) x − x (cid:107) . Proposition 2.3
Let S be a nonempty closed subset in H and let r ∈ (0 , ∞ ] . If the subset S is uniformly r-prox-regular then the following hold:(a) The proximal and Clarke normal cones of S coincide.(b) for all x ∈ H with d S ( x ) < r , Proj S ( x ) is nonempty and is a singleton set .(c) the Clarke and the proximal subdifferentials of d S coincide at all points x ∈ H with d S ( x ) < r . The assertion (a) in Proposition 2.3 leads us to put N S ( x ) := N CS ( x ) = N PS ( x ) when-ever the set S is r -prox-regular. Proposition 2.4
Let S be a nonempty closed subset in H and let r ∈ (0 , ∞ ] . If the subset S is uniformly r-prox-regular then the following hold:(a) For any x ∈ S and any ς ∈ ∂ P d S ( x ) one has for any y ∈ H such that d S ( y ) < r (cid:104) ς, y − x (cid:105) ≤ r (cid:107) y − x (cid:107) + d S ( y ) . (b) For any x ∈ H with d S ( x ) < r , the proximal subdifferential ∂ P d S ( x ) is a nonemptyclosed convex subset in H . We start this section with the following continuous Gronwall’s inequality [25].
Lemma 3.1 (Gronwall’s inequality)
Let
T > T be given reals and a ( · ) , b ( · ) ∈ L ([ T , T ]; R ) with b ( t ) ≥ for almost all t ∈ [ T , T ] . Let the absolutely continuous function w :[ T , T ] −→ R + satisfy (1 − α ) w (cid:48) ( t ) ≤ a ( t ) w ( t ) + b ( t ) w α ( t ) , a.e. t ∈ [ T , T ] , where ≤ α < . Then for all t ∈ [ T , T ] , one has w − α ( t ) ≤ w − α ( T ) exp( (cid:90) tT a ( τ ) dτ ) + (cid:90) tT exp( (cid:90) ts a ( τ ) dτ ) b ( s ) ds. We will need the following lemma which is a straightforward consequence of Gronwall’slemma.
Lemma 3.2
Let ρ : [ T , T ] −→ R be a nonnegative absolutely continuous function and let b , b , a : [ T , T ] −→ R + be non-negative Lebesgue integrable functions. Assume that ˙ ρ ( t ) ≤ a ( t ) + b ( t ) ρ ( t ) + b ( t ) t (cid:90) T ρ ( s ) ds, a.e. t ∈ [ T , T ] . (8) Then for all t ∈ [ T , T ] , one has ρ ( t ) ≤ ρ ( T ) exp (cid:18) t (cid:90) T ( b ( τ ) + 1) dτ (cid:19) + t (cid:90) T a ( s ) exp (cid:18) t (cid:90) s ( b ( τ ) + 1) dτ (cid:19) ds, where b ( t ) := max { b ( t ) , b ( t ) } , a.e. t ∈ [ T , T ] . Proof . Put b ( t ) = max { b ( t ) , b ( t ) } , a.e. t ∈ [ T , T ]. Setting z ( t ) = t (cid:82) T ρ ( s ) ds ⇒ ˙ z ( t ) = ρ ( t ), ¨ z ( t ) = ˙ ρ ( t ) . Then from (8) we see that¨ z ( t ) ≤ a ( t ) + b ( t ) ˙ z ( t ) + b ( t ) z ( t ) ≤ a ( t ) + b ( t ) w ( t ) , where w ( t ) = ˙ z ( t ) + z ( t ), for all t ∈ [ T , T ]. Hence, for a.e. t ∈ [ T , T ]˙ w ( t ) = ¨ z ( t ) + ˙ z ( t ) and ˙ w ( t ) ≤ a ( t ) + ( b ( t ) + 1) w ( t ) . Applying the Gronwall Lemma 3.1 with w , one obtains for all t ∈ [ T , T ] w ( t ) ≤ w ( T ) exp (cid:0) t (cid:90) T ( b ( τ ) + 1) dτ (cid:1) + t (cid:90) T a ( s ) exp (cid:0) t (cid:90) s ( b ( τ ) + 1) dτ (cid:1) ds, which gives ρ ( t ) ≤ ˙ z ( t ) + z ( t ) = w ( t ) ≤ ρ ( T ) exp (cid:18) t (cid:90) T ( b ( τ ) + 1) dτ (cid:19) + t (cid:90) T a ( s ) exp (cid:18) t (cid:90) s ( b ( τ ) + 1) dτ (cid:19) ds. We establish now the following new Gronwall-like lemma.
Lemma 3.3 (Gronwall-like differential inequality)
Let ρ : [ T , T ] −→ R be a non-negative absolutely continuous function and let K , K , ε : [ T , T ] −→ R + be non-negativeLebesgue integrable functions. Suppose for some (cid:15) > ρ ( t ) ≤ ε ( t ) + (cid:15) + K ( t ) ρ ( t ) + K ( t ) (cid:112) ρ ( t ) t (cid:90) T (cid:112) ρ ( s ) ds, a.e. t ∈ [ T , T ] . (9)0 Then for all t ∈ [ T , T ] , one has (cid:112) ρ ( t ) ≤ (cid:112) ρ ( T ) + (cid:15) exp (cid:18) t (cid:90) T ( K ( s ) + 1) ds (cid:19) + √ (cid:15) t (cid:90) T exp (cid:18) t (cid:90) s ( K ( τ ) + 1) dτ (cid:19) ds + 2 (cid:18)(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) t (cid:90) T ε ( s ) ds + (cid:15) − √ (cid:15) exp (cid:18) t (cid:90) T ( K ( τ ) + 1) dτ (cid:19)(cid:19) + 2 t (cid:90) T ( K ( s ) + 1) exp (cid:18) t (cid:90) s ( K ( τ ) + 1) dτ (cid:19)(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) s (cid:90) T ε ( τ ) dτ + (cid:15) ds, where K ( t ) := max (cid:26) K ( t )2 , K ( t )2 (cid:27) , a.e. t ∈ [ T , T ] . Proof . Set λ ( t ) = (cid:115) t (cid:82) T ε ( s ) ds + (cid:15) and z ε ( t ) = (cid:112) ρ ( t ) + λ ( t ) for all t ∈ [ T , T ] .From (9) we have for a.e. t ∈ [ T , T ]˙ ρ ( t ) ≤ ε ( t ) + (cid:15) + K ( t )( ρ ( t ) + λ ( t )) + K ( t ) (cid:112) ρ ( t ) + λ ( t ) t (cid:90) T (cid:112) ρ ( s ) + λ ( s ) ds (10)and ˙ z ε ( t ) = ˙ ρ ( t ) + 2 ˙ λ ( t ) λ ( t )2 (cid:112) ρ ( t ) + λ ( t ) = ˙ ρ ( t ) + ε ( t )2 z ε ( t ) , or equivalently ˙ ρ ( t ) = 2 z ε ( t ) ˙ z ε ( t ) − ε ( t ) , hence from (10)2 z ε ( t ) ˙ z ε ( t ) ≤ ε ( t ) + (cid:15) + K ( t ) z ε ( t ) + K ( t ) z ε ( t ) t (cid:90) T z ε ( s ) ds. Therefore,˙ z ε ( t ) ≤ ε ( t ) z ε ( t ) + (cid:15) z ε ( t ) + K ( t )2 z ε ( t ) + K ( t )2 t (cid:90) T z ε ( s ) ds ≤ λ ( t ) + √ (cid:15) K ( t ) z ε ( t ) + K ( t ) t (cid:90) T z ε ( s ) ds . Since λ ( t ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) t (cid:90) T ε ( s ) ds + (cid:15) ≤ (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ρ ( t ) + t (cid:90) T ε ( s ) ds + (cid:15) = (cid:112) ρ ( t ) + λ ( t ) = z ε ( t ) , z ε ( t ) ≤ λ ( t ) , or equivalently ε ( t ) z ε ( t ) ≤ ε ( t ) λ ( t ) . Also we have ˙ λ ( t ) = ε ( t )2 λ ( t ) . Then ε ( t ) z ε ( t ) ≤ λ ( t ), and √ (cid:15) ≤ (cid:115) (cid:15) + t (cid:82) T ε ( s ) ds = λ ( t ) ≤ z ε ( t ),hence (cid:15) z ε ( t ) ≤ √ (cid:15) K ( t ) := max (cid:26) K ( t )2 , K ( t )2 (cid:27) and applying the GronwallLemma 3.2 with z ε , one obtains for all t ∈ [ T , T ] z ε ( t ) ≤ z ε ( T ) exp (cid:0) t (cid:90) T ( K ( τ ) + 1) dτ (cid:1) + t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) (2 ˙ λ ( s ) + √ (cid:15) ds = (cid:112) ρ ( T ) + (cid:15) exp( t (cid:90) T ( K ( s ) + 1) ds ) + t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) (2 ˙ λ ( s ) + √ (cid:15) ds or equivalently z ε ( t ) ≤ (cid:112) ρ ( T ) + (cid:15) exp (cid:0) t (cid:90) T ( K ( s ) + 1) ds (cid:1) + 2 t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) ˙ λ ( s ) ds + √ (cid:15) t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) ds. On the other hand, from integration by parts, we note that t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) ˙ λ ( s ) ds = [exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) λ ( s )] tT + t (cid:90) T ( K ( s ) + 1) exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) λ ( s ) ds = λ ( t ) − exp (cid:0) t (cid:90) T ( K ( τ ) + 1) dτ (cid:1) √ (cid:15) + t (cid:90) T ( K ( s ) + 1) exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) λ ( s ) ds, which combined with what precedes gives z ε ( t ) ≤ (cid:112) ρ ( T ) + (cid:15) exp (cid:0) t (cid:90) T ( K ( s ) + 1) ds (cid:1) + √ (cid:15) t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) ds + 2 λ ( t ) − (cid:0) t (cid:90) T ( K ( τ ) + 1) dτ (cid:1) √ (cid:15) + 2 t (cid:90) T ( K ( s ) + 1) exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) λ ( s ) ds. (cid:112) ρ ( t ) ≤ (cid:112) ρ ( t ) + λ ( t ) = z ε ( t ) we obtain (cid:112) ρ ( t ) ≤ (cid:112) ρ ( T ) + (cid:15) exp (cid:0) t (cid:90) T ( K ( s ) + 1) ds (cid:1) + √ (cid:15) t (cid:90) T exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) ds + 2 λ ( t ) − (cid:0) t (cid:90) T ( K ( τ ) + 1) dτ (cid:1) √ (cid:15) + 2 t (cid:90) T ( K ( s ) + 1) exp (cid:0) t (cid:90) s ( K ( τ ) + 1) dτ (cid:1) λ ( s ) ds, which completes the proof of the lemma . In this section, we give and prove our main results in the study of the integro-differentialsweeping process ( P f ,f ). They concern the existence, uniqueness, and continuous depen-dence of the solution with respect to the initial data. We state first in the next propositiona result which will be utilized in our development. Clearly, when the sets C ( t ) are bounded,the hypothesis ( H ) is ensured by the usual Hausdorff variation hypothesishaus (cid:0) C ( s ) , C ( t ) (cid:1) ≤ | υ ( s ) − υ ( t ) | , ∀ t, s ∈ [ T , T ] , which according to the equality haus( S, S (cid:48) ) = sup y ∈ H | d S ( y ) − d S (cid:48) ( y ) | for bounded sets S, S (cid:48) ,amounts to requiring for the bounded sets C ( s ) , C ( t ) that | d C ( s ) ( y ) − d C ( t ) ( y ) | ≤ | υ ( s ) − υ ( t ) | , ∀ t, s ∈ [ T , T ] , ∀ y ∈ H. (11)The result is proved in [7, 13] under the hypothesis (11) but the proof in [7, 13] is validwith the hypothesis ( H ). One of the advantages of ( H ) is that the sets C ( t ) there neednot be bounded. Proposition 4.1
Let H be a real Hilbert space, suppose that C ( · ) satisfies ( H ) . Let h :[ T , T ] −→ H be a single-valued mapping in L ([ T , T ] , H ) . Then for any x ∈ C ( T ) thereexists a unique absolutely continuous solution x ( · ) for the following differential inclusion (cid:26) − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + h ( t ) a.e t ∈ [ T , T ] ,x ( T ) = x . Moreover x ( · ) satisfies the following inequality (cid:107) ˙ x ( t ) + h ( t ) (cid:107) ≤ (cid:107) h ( t ) (cid:107) + | ˙ υ ( t ) | a.e. t ∈ [ T , T ] . (12)3 Theorem 4.2
Let H be a real Hilbert space and assume that ( H ) , ( H ) and ( H ) aresatisfied. Then for any initial point x ∈ H, with x ∈ C ( T ) there exists a uniqueabsolutely continuous solution x : [ T , T ] −→ H of the differential inclusion ( P f ,f ) . Thissolution satisfies:1. For a.e. t ∈ [ T , T ] (cid:107) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:107) ≤ | ˙ υ ( t ) | + (cid:107) f ( t, x ( t )) (cid:107) + t (cid:90) T (cid:107) f ( t, s, x ( s )) (cid:107) ds. (13)
2. If T (cid:90) T (cid:20) β ( τ ) + τ (cid:90) T β ( τ, s ) ds (cid:21) dτ < , one has (cid:107) f ( t, x ( t )) (cid:107) ≤ (1 + M ) β ( t ) , for all t ∈ [ T , T ] , (14) (cid:107) f ( t, s, x ( s )) (cid:107) ≤ (1 + M ) β ( t, s ) , for all ( t, s ) ∈ Q ∆ , (15) and for almost all t ∈ [ T , T ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (1 + M ) (cid:18) β ( t ) + t (cid:90) T β ( t, s ) ds (cid:19) + | ˙ υ ( t ) | , (16) where M := 2 (cid:18) (cid:107) x (cid:107) + T (cid:90) T | ˙ υ ( τ ) | dτ + 12 (cid:19) .3. Assume the following strengthened form of assumption ( H , ) on the function f holds: ( H (cid:48) , ) : there exist non-negative functions α ( · ) ∈ L ([ T , T ] , R ) and g ( · ) ∈ L ( P ∆ , R ) such that (cid:107) f ( t, s, x ) (cid:107) ≤ g ( t, s ) + α ( t ) (cid:107) x (cid:107) , for any ( t, s ) ∈ Q ∆ and any x ∈ (cid:91) t ∈ [ T ,T ] C ( t ) . Then we have (cid:107) f ( t, x ( t )) (cid:107) ≤ (1 + (cid:102) M ) β ( t ) , for all t ∈ [ T , T ] , (17) (cid:107) f ( t, s, x ( s )) (cid:107) ≤ g ( t, s ) + α ( t ) (cid:102) M , a.e. ( t, s ) ∈ Q ∆ , (18) and for almost all t ∈ [ T , T ] (cid:107) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:107) ≤| ˙ υ ( t ) | + (1 + (cid:102) M ) β ( t ) + t (cid:90) T g ( t, s ) ds + T α ( t ) (cid:102) M , (19)4 where (cid:102) M := (cid:107) x (cid:107) exp (cid:18) T (cid:90) T ( b ( τ ) + 1) dτ (cid:19) + exp (cid:18) T (cid:90) T ( b ( τ ) + 1) dτ (cid:19) T (cid:90) T (cid:18) | ˙ υ ( s ) | + 2 β ( s ) + 2 T (cid:90) T g ( s, τ ) dτ (cid:19) ds, and b ( t ) := 2 max { β ( t ) , α ( t ) } for all t ∈ [ T , T ] . Proof . The proof of existence of solution is divided in several steps.
Step 1. Discretization of the interval I = [ T , T ] . For each n ∈ N , divide the interval I into n intervals of length h = T − T n and define, forall i ∈ { , · · · , n − } (cid:26) t ni +1 := t ni + h = T + ih,I ni := (cid:2) t ni , t ni +1 (cid:3) , (20)so that T = t n < t n < · · · < t ni < t ni +1 < · · · < t nn = T. (21) Step 2. Construction of the sequence x n ( · ) . We construct a sequence of mappings ( x n ( · )) n ∈ N in C ( I, H ) which converges uniformly toa solution x ( · ) of ( P ).Our method consists in establishing a sequence of discrete solutions ( x nk ( · )) n ∈ N in eachinterval I nk := (cid:2) t nk , t nk +1 (cid:3) (0 ≤ k ≤ n −
1) by using Proposition 4.1. Indeed, we proceed asfollows.Given the following problem( P ) : − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ) + t (cid:90) T f ( t, s, x ) ds a.e. t ∈ [ T , t n ] ,x ( T ) = x . (22)Then ( P ) is a perturbed sweeping process with the perturbation depending only on time.Let h : [ T , t n ] → H be defined by h ( t ) := f ( t, x ) + t (cid:90) T f ( t, s, x ) ds for all t ∈ [ T , t n ].We see by integrable linear growth condition that T (cid:90) T (cid:107) h ( t ) (cid:107) dt ≤ (1 + (cid:107) x (cid:107) ) T (cid:90) T β ( t ) dt + (1 + (cid:107) x (cid:107) ) T (cid:90) T t (cid:90) T β ( t, s ) ds dt, and since β ( · ) ∈ L ([ T , T ] , R + ) and β ( · ) ∈ L ( Q ∆ , R + ), then h ( · ) is an integrable func-tion. Therefore, by Proposition 4.1 the differential inclusion ( P ) has a unique absolutelycontinuous solution denoted by x n ( · ) : [ T , t n ] −→ H, (23)5satisfying the following inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x n ( t ) + f ( t, x ) + t (cid:90) T f ( t, s, x ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( t, x ) + t (cid:90) T f ( t, s, x ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | ˙ υ ( t ) | (24)a.e. t ∈ [ T , t n ] . Next, let us consider the following problem( P ) : − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x n ( t n )) + t n (cid:90) T f ( t, s, x ) ds t (cid:90) t n f ( t, s, x n ( t n )) ds a.e. t ∈ [ t n , t n ] ,x ( t n ) = x n ( t n ) . (25)Let h : [ t n , t n ] → H be defined by h ( t ) := f ( t, x n ( t n )) + t n (cid:90) T f ( t, s, x ) ds + t (cid:90) t n f ( t, s, x n ( t n )) ds for all t ∈ [ t n , t n ] . We can see by integrable linear growth conditions that T (cid:90) T (cid:107) h ( t ) (cid:107) dt ≤ (1 + (cid:107) x n ( t n ) (cid:107) ) T (cid:90) T β ( t ) dt + (1 + (cid:107) x (cid:107) ) T (cid:90) T t n (cid:90) T β ( t, s ) ds dt + (1 + (cid:107) x n ( t n ) (cid:107) ) T (cid:90) T t (cid:90) t n β ( t, s ) ds dt ≤ (1 + max {(cid:107) x n ( t n ) (cid:107) , (cid:107) x (cid:107)} ) (cid:18) T (cid:90) T β ( t ) dt + T (cid:90) T t (cid:90) T β ( t, s ) ds dt (cid:19) . We know from the above problem ( P ) that the mapping x n ( · ) is absolutely continuous,then in particular bounded on [ T , T ]. Further, since β ( · ) ∈ L ([ T , T ] , R + ) and β ( · ) ∈ L ( Q ∆ , R + ), then h ( · ) is an integrable mapping. The same arguments as above showthat ( P ) has a unique absolutely continuous solution denoted by x n ( · ) : [ t n , t n ] −→ H, (26)and this solution satisfies the following inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x n ( t ) + f ( t, x n ( t n )) + t n (cid:90) T f ( t, s, x ) ds + t (cid:90) t n f ( t, s, x n ( t n )) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( t, x n ( t n )) + t n (cid:90) T f ( t, s, x ) ds + t (cid:90) t n f ( t, s, x n ( t n )) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | ˙ υ ( t ) | a.e. t ∈ [ t n , t n ] . (27)6Successively, for each n , we have a finite sequence of absolutely continuous mappings( x nk ( · )) ≤ k ≤ n − with for each k ∈ { , · · · , n − } x nk ( · ) : (cid:2) t nk , t nk +1 (cid:3) −→ H (28)such that( P k − ) : − ˙ x nk ( t ) ∈ N C ( t ) ( x nk ( t )) + f ( t, x nk − ( t nk )) + k − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x nj − ( t nj )) ds + t (cid:90) t nk f ( t, s, x nk − ( t nk )) ds a.e. t ∈ (cid:2) t nk , t nk +1 (cid:3) .x nk ( t nk ) = x nk − ( t nk ) , (29)where for k = 0 we put x n − ( T ) := x . Moreover (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x nk ( t ) + f ( t, x nk − ( t nk )) + k − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x nj − ( t nj )) ds + t (cid:90) t nk f ( t, s, x nk − ( t nk )) ds, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( t, x nk − ( t nk )) + k − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x nj − ( t nj )) ds + t (cid:90) t nk f ( t, s, x nk − ( t nk )) ds, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | ˙ υ ( t ) | , (30)a.e. t ∈ (cid:2) t nk , t nk +1 (cid:3) . Defining for each k ∈ { , , · · · , n − } the mapping h k : [ t nk , t nk +1 ] → H by h k ( t ) := f ( t, x nk − ( t nk )) + k − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x nj − ( t nj )) ds + t (cid:90) t nk f ( t, s, x nk − ( t nk )) ds, for all t ∈ (cid:2) t nk , t nk +1 (cid:3) . Clearly, by integrable linear growth conditions we have T (cid:90) T (cid:107) h k ( t ) (cid:107) dt ≤ (1 + (cid:107) x nk − ( t nk ) (cid:107) ) T (cid:90) T β ( t ) dt + k − (cid:88) j =0 (1 + (cid:107) x nj − ( t nj ) (cid:107) ) t nj +1 (cid:90) t nj β ( t, s ) ds dt + (1 + (cid:107) x nk − ( t nk ) (cid:107) ) T (cid:90) T t (cid:90) t nk β ( t, s ) ds dt ≤ (1 + max ≤ j ≤ k − (cid:13)(cid:13) x nj − ( t nj ) (cid:13)(cid:13) ) (cid:18) T (cid:90) T β ( t ) dt + T (cid:90) T t (cid:90) T β ( t, s ) ds dt (cid:19) . P j ) ≤ j ≤ k − that the mapping x nk − ( · ) is absolutelycontinuous, then in particular bounded on [ T , T ]. Further, since β ( · ) ∈ L ([ T , T ] , R + )and β ( · ) ∈ L ( P ∆ , R + ); the mapping h k ( · ) is integrable on [ t nk , t nk +1 ].Now, we define the sequence ( x n ( · )) n from the discrete sequences ( x nk ( . )) as follows.For each n ∈ N , let x n ( · ) : [ T , T ] −→ H such that x n ( t ) := x nk ( t ) , if t ∈ (cid:2) t nk , t nk +1 (cid:3) . (31)It is obvious from this definition that x n ( · ) is absolutely continuous.Let θ n ( · ) : [ T , T ] −→ [ T , T ] be defined by (cid:26) θ n ( T ) := T ,θ n ( t ) := t nk , if t ∈ (cid:3) t nk , t nk +1 (cid:3) . (32)We obtain from (29), (30), (31), (32), that − ˙ x n ( t ) ∈ N C ( t ) ( x n ( t )) + f ( t, x n ( θ n ( t ))) + t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds a.e. t ∈ [ T , T ] ,x n ( T ) = x , (33)and a.e. t ∈ [ T , T ] we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x n ( t ) + f ( t, x n ( θ n ( t ))) + t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( t, x n ( θ n ( t ))) + t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | ˙ υ ( t ) | . (34) Step 3. We show that the sequence ( ˙ x n ( · )) is uniformly dominated by an inte-grable function. Since β ( · ) ∈ L ([ T , T ] , R + ) and β ( · , · ) ∈ L ( P ∆ , R + ) we suppose without loss of gener-ality that T (cid:90) T (cid:20) β ( τ ) + τ (cid:90) T β ( τ, s ) ds (cid:21) dτ < . (35)By construction we have for each i ∈ { , · · · , n − } and for a.e. t ∈ (cid:2) t ni , t ni +1 (cid:3)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x n ( t ) + f ( t, x n ( t ni )) + i − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x n ( t nj )) ds + t (cid:90) t ni f ( t, s, x n ( t ni )) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( t, x n ( t ni )) + i − (cid:88) j =0 t nj +1 (cid:90) t nj f ( t, s, x n ( t nj )) ds + t (cid:90) t ni f ( t, s, x n ( t ni )) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | ˙ υ ( t ) | . H , ) and ( H , ) we have (cid:107) ˙ x n ( t ) (cid:107) ≤ (cid:107) f ( t, x n ( t ni )) (cid:107) + 2 i − (cid:88) j =0 t nj +1 (cid:90) t nj (cid:13)(cid:13) f ( t, s, x n ( t nj )) (cid:13)(cid:13) ds + 2 t (cid:90) t ni (cid:107) f ( t, s, x n ( t ni )) (cid:107) ds + | ˙ υ ( t ) |≤ ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) β ( t ) + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) i − (cid:88) j =0 t nj +1 (cid:90) t nj β ( t, s ) ds + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t (cid:90) t ni β ( t, s ) ds + | ˙ υ ( t ) | = | ˙ υ ( t ) | + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) β ( t ) + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t (cid:90) T β ( t, s ) ds, and then (cid:13)(cid:13) x n ( t ni +1 ) (cid:13)(cid:13) ≤ (cid:107) x n ( t ni ) (cid:107) + t ni +1 (cid:90) t ni | ˙ υ ( τ ) | dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t ni +1 (cid:90) t ni β ( τ ) dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t ni +1 (cid:90) t ni τ (cid:90) T β ( τ, s ) ds dτ. Iterating, it follows that (cid:13)(cid:13) x n ( t ni +1 ) (cid:13)(cid:13) ≤ (cid:107) x (cid:107) + i (cid:88) k =0 t nk +1 (cid:90) t nk | ˙ υ ( τ ) | dτ + 2(1 + max ≤ j ≤ n (cid:13)(cid:13) x n ( t nj ) (cid:13)(cid:13) ) i (cid:88) k =0 t nk +1 (cid:90) t nk β ( τ ) dτ + 2(1 + max ≤ j ≤ n (cid:13)(cid:13) x n ( t nj ) (cid:13)(cid:13) ) i (cid:88) k =0 t nk +1 (cid:90) t nk τ (cid:90) T β ( τ, s ) ds dτ This yields the following inequality (cid:13)(cid:13) x n ( t ni +1 ) (cid:13)(cid:13) ≤ (cid:107) x (cid:107) + t ni +1 (cid:90) T | ˙ υ ( τ ) | dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t ni +1 (cid:90) T β ( τ ) dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) t ni +1 (cid:90) T τ (cid:90) T β ( τ, s ) ds dτ. (36)9The inequality (36) being true for all i ∈ { , · · · , n − } , we havemax ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ≤ (cid:107) x (cid:107) + T (cid:90) T | ˙ υ ( τ ) | dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) T (cid:90) T β ( τ ) dτ + 2(1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) T (cid:90) T τ (cid:90) T β ( τ, s ) ds dτ, which gives by (35)max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ≤ (cid:107) x (cid:107) + T (cid:90) T | ˙ υ ( τ ) | dτ + 12 (1 + max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ) . This can be rewritten as max ≤ k ≤ n (cid:107) x n ( t nk ) (cid:107) ≤ M, (37)where M := 2 (cid:18) (cid:107) x (cid:107) + T (cid:82) T | ˙ υ ( τ ) | dτ + 12 (cid:19) .On one hand, from the growth condition of f , f and (37) we have, for almost all t andfor all n , (cid:107) f ( t, x n ( θ n ( t ))) (cid:107) ≤ β ( t )(1 + (cid:107) x n ( θ n ( t )) (cid:107) ) ≤ (1 + M ) β ( t ) . (38) (cid:107) f ( t, s, x n ( θ n ( s ))) (cid:107) ≤ β ( t, s )(1 + (cid:107) x n ( θ n ( s )) (cid:107) ) ≤ (1 + M ) β ( t, s ) . (39)Hence, (34) implies for almost all t and for all n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ x n ( t ) + f ( t, x n ( θ n ( t ))) + t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (1 + M ) (cid:18) β ( t ) + t (cid:90) T β ( t, s ) ds (cid:19) + | ˙ υ ( t ) | , (40)and thus (cid:107) ˙ x n ( t ) (cid:107) ≤ M ) (cid:18) β ( t ) + t (cid:90) T β ( t, s ) ds (cid:19) + | ˙ υ ( t ) | . (41) Step 4. We show that x n ( · ) converges. It suffices to show that x n ( · ) is a Cauchy sequence in the Banach space ( C ( I, H ) , (cid:107)·(cid:107) ∞ ).Let m, n ∈ N . For almost all t ∈ [ T , T ], we have − ˙ x n ( t ) − f ( t, x n ( θ n ( t ))) − t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds ∈ N C ( t ) ( x n ( t )) , − ˙ x m ( t ) − f ( t, x m ( θ m ( t ))) − t (cid:90) T f ( t, s, x m ( θ m ( s ))) ds ∈ N C ( t ) ( x m ( t )) . (42)0Let us set α ( t ) := (1 + M ) (cid:18) β ( t ) + t (cid:90) T β ( t, s ) ds (cid:19) + | ˙ υ ( t ) | ,γ ( t ) := 2(1 + M ) (cid:18) β ( t ) + t (cid:90) T β ( t, s ) ds (cid:19) + | ˙ υ ( t ) | . The absolute continuity of x n ( · ) gives by (41) (cid:107) x n ( t ) (cid:107) ≤ η for all t ∈ [ T , T ] , (43)with η := (cid:107) x (cid:107) + T (cid:90) T γ ( s ) ds. Using (40) and the hypomonotonicity of the normal cone N ( C ( t ); · ), we get that (cid:104) ˙ x n ( t ) + f ( t, x n ( θ n ( t ))) + t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds − ˙ x m ( t ) − f ( t, x m ( θ m ( t ))) − t (cid:90) T f ( t, s, x m ( θ m ( s ))) ds, x n ( t ) − x m ( t ) (cid:105) ≤ α ( t ) r (cid:107) x n ( t ) − x m ( t ) (cid:107) Therefore (cid:104) ˙ x n ( t ) − ˙ x m ( t ) , x n ( t ) − x m ( t ) (cid:105) ≤ α ( t ) r (cid:107) x n ( t ) − x m ( t ) (cid:107) + (cid:104) f ( t, x n ( θ n ( t ))) − f ( t, x m ( θ m ( t ))) , x m ( t ) − x n ( t ) (cid:105) + (cid:42) t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds − t (cid:90) T f ( t, s, x m ( θ m ( s ))) ds, x m ( t ) − x n ( t ) (cid:43) . Applying the Lipschitz continuity of f ( t, · ) and f ( t, s, · ) with Lipschitz radius L η ( · ) , L η ( · ) ∈ L ( I, R + ) on the bounded subset B [0 , η ], it follows that12 ddt (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ α ( t ) r (cid:107) x n ( t ) − x m ( t ) (cid:107) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) (cid:16) (cid:107) x n ( θ n ( t )) − x n ( t ) (cid:107) + (cid:107) x n ( t ) − x m ( t ) (cid:107) + (cid:107) x m ( t ) − x m ( θ m ( t )) (cid:107) (cid:17) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) (cid:16) t (cid:90) T (cid:107) x n ( θ n ( s )) − x n ( s ) (cid:107) ds + t (cid:90) T (cid:107) x n ( s ) − x m ( s ) (cid:107) ds + t (cid:90) T (cid:107) x m ( t ) − x m ( θ m ( t )) (cid:107) ds (cid:17) n ∈ N and for all t , (cid:107) x n ( t ) − x n ( θ n ( t )) (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t (cid:90) θ n ( t ) ˙ x n ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ t (cid:90) θ n ( t ) (cid:107) ˙ x n ( τ ) (cid:107) dτ ≤ t (cid:90) θ n ( t ) γ ( τ ) dτ. Therefore12 ddt (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ α ( t ) r (cid:107) x n ( t ) − x m ( t ) (cid:107) + L r ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) (cid:16) t (cid:90) θ n ( t ) γ ( τ ) dτ + t (cid:90) θ m ( t ) γ ( τ ) dτ (cid:17) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) (cid:16) t (cid:90) T s (cid:90) θ n ( s ) γ ( τ ) dτ ds + t (cid:90) T s (cid:90) θ m ( s ) γ ( τ ) dτ ds (cid:17) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) t (cid:90) T (cid:107) x n ( s ) − x m ( s ) (cid:107) ds. Moreover, noting by (43) that (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ (cid:107) x n ( t ) (cid:107) + (cid:107) x m ( t ) (cid:107) ≤ η, we deduce that12 ddt (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ α ( t ) r (cid:107) x n ( t ) − x m ( t ) (cid:107) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) + 2 η (cid:32)L η ( t ) (cid:16) t (cid:90) θ n ( t ) γ ( τ ) dτ + t (cid:90) θ m ( t ) γ ( τ ) dτ (cid:17) + 2 ηL η ( t ) (cid:16) t (cid:90) T (cid:20) s (cid:90) θ n ( s ) γ ( τ ) dτ + s (cid:90) θ m ( s ) γ ( τ ) dτ (cid:21) ds (cid:17) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) t (cid:90) T (cid:107) x n ( s ) − x m ( s ) (cid:107) ds. (44)Let us put G n,m ( t ) := 2 ηL r ( t ) (cid:16) t (cid:90) θ n ( t ) γ ( τ ) dτ + t (cid:90) θ m ( t ) γ ( τ ) dτ (cid:17) , (cid:101) G n,m ( s ) := s (cid:90) θ n ( s ) γ ( τ ) dτ + s (cid:90) θ m ( s ) γ ( τ ) dτ Since γ ( · ) ∈ L ( I, R + ) and for each t ∈ I , we have θ n ( t ) , θ m ( t ) −→ t , thenlim n,m → + ∞ G n,m ( t ) = 0 and lim n,m → + ∞ (cid:101) G n,m ( t ) = 0 . (45)2On the other hand, for each n ∈ N writing (cid:90) tθ n ( t ) γ ( s ) ds ≤ (cid:90) TT γ ( s ) ds, we see that | G n,m ( t ) | ≤ ηL η ( t ) (cid:18)(cid:90) TT γ ( s ) ds (cid:19) and (cid:12)(cid:12)(cid:12) ˜ G n,m ( s ) (cid:12)(cid:12)(cid:12) ≤ (cid:18)(cid:90) TT γ ( s ) ds (cid:19) . Therefore, for all t ∈ [ T , T ] by (45) and the dominated convergence theorem, we obtainlim n,m → + ∞ (cid:90) TT G n,m ( t ) dt = 0 . (46)lim n,m → + ∞ (cid:90) TT ˜ G n,m ( s ) ds = 0 . (47)Note also by (44) that12 ddt (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ (cid:18) α ( t ) r + L η ( t ) (cid:19) (cid:107) x n ( t ) − x m ( t ) (cid:107) + G n,m ( t ) + 2 ηL η ( t ) (cid:16) T (cid:90) T (cid:101) G n,m ( s ) ds (cid:17) + L η ( t ) (cid:107) x n ( t ) − x m ( t ) (cid:107) t (cid:90) T (cid:107) x n ( s ) − x m ( s ) (cid:107) ds. Applying Lemma 3.3 with ρ ( t ) = (cid:107) x n ( t ) − x m ( t ) (cid:107) , K ( t ) = 2 (cid:18) α ( t ) r + L η ( t ) (cid:19) , K ( t ) = 2 L η ( t ) ε ( t ) := ε n,m ( t ) = 2 G n,m ( t ) + 4 ηL η ( t ) (cid:16) T (cid:90) T (cid:101) G n,m ( s ) ds (cid:17) , (cid:15) > , we then see that (cid:107) x n ( t ) − x m ( t ) (cid:107) ≤ (cid:112) (cid:107) x n ( T ) − x m ( T ) (cid:107) + (cid:15) exp (cid:18) t (cid:90) ( K ( s ) + 1) ds (cid:19) + √ (cid:15) t (cid:90) T exp (cid:18) t (cid:90) s ( K ( τ ) + 1) dτ (cid:19) ds + 2 (cid:18)(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) t (cid:90) T ε n,m ( s ) ds + (cid:15) − exp (cid:18) t (cid:90) T ( K ( τ ) + 1) dτ (cid:19) √ (cid:15) (cid:19) + 2 t (cid:90) T ( K ( s ) + 1) exp (cid:18) t (cid:90) s ( K ( τ ) + 1) dτ (cid:19)(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) s (cid:90) T ε n,m ( τ ) dτ + (cid:15) ds. K ( t ) := max (cid:26) α ( t ) η + L r ( t ) , L η ( t ) (cid:27) , for almost all t ∈ [ T , T ].This, along with the fact that (cid:107) x n ( T ) − x m ( T ) (cid:107) = 0 and taking (cid:15) →
0, we getlim n,m → + ∞ (cid:107) x n ( · ) − x m ( · ) (cid:107) ∞ = 0 . Therefore, the sequence ( x n ( · )) is a Cauchy sequence in ( C ([ T , T ] , H ) , (cid:107)·(cid:107) ∞ ) and thereforeconverges uniformly to a function x ( · ) ∈ C ([ T , T ] , H ) . Step 5. We show that x ( · ) is absolutely continuous. We have for almost all t ∈ I and for any n , (cid:107) ˙ x n ( t ) (cid:107) ≤ γ ( t ) . So we can extract a subsequence of ( ˙ x n ( · )) (that, without loss of generality, we do notrelabel) which converges weakly in L ( I, H ) to a function g ( · ) ∈ L ( I, H ) . This meansthat (cid:90) TT (cid:104) ˙ x n ( s ) , h ( s ) (cid:105) ds −→ (cid:90) TT (cid:104) g ( s ) , h ( s ) (cid:105) ds, ∀ h ∈ L ∞ ( I, H ) . Now observe that for all z ∈ H (cid:90) TT (cid:10) ˙ x n ( s ) , z · [ T ,t ] ( s ) (cid:11) ds = (cid:90) tT (cid:104) ˙ x n ( s ) , z (cid:105) ds = (cid:104) z, (cid:90) tT ˙ x n ( s ) ds (cid:105) . and (cid:90) TT (cid:10) g ( s ) , z · [ T ,t ] ( s ) (cid:11) ds = (cid:90) tT (cid:104) g ( s ) , z (cid:105) ds = (cid:104) z, (cid:90) tT g ( s ) ds (cid:105) . So from the weak convergence we deduce that (cid:90) tT ˙ x n ( s ) ds −→ (cid:90) tT g ( s ) ds weakly in H. This implies that x n ( T ) + (cid:90) tT ˙ x n ( s ) ds −→ x ( T ) + (cid:90) tT g ( s ) ds weakly in H. But x n ( · ) is absolutely continuous, so x n ( t ) = x n ( T ) + (cid:90) tT ˙ x n ( s ) ds −→ x ( T ) + (cid:90) tT g ( s ) ds weakly in H. On the other hand, we have for all t ∈ [ T , T ] x n ( t ) −→ x ( t ) strongly in H, hence we get x ( t ) = x ( T ) + (cid:90) tT g ( s ) ds. x ( · ) is absolutely continuous and ˙ x ( t ) = g ( t ) a.e. t ∈ [ T , T ], so in particular (cid:107) x ( t ) (cid:107) ≤ ˜ η for all t ∈ [ T , T ] , (48)with ˜ η := (cid:107) x (cid:107) + T (cid:90) T g ( s ) ds. Step 6. We show that x ( · ) is a solution of ( P f ,f ) . For each t ∈ I , since θ n ( t ) −→ t for all t ∈ I and x n ( · ) converges uniformly to x ( · ), wehave x n ( θ n ( t )) −→ x ( t ).Let us set for each t ∈ Iy n ( t ) := t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds, and y ( t ) := t (cid:90) T f ( t, s, x ( s )) ds. We have shown in the above step that ˙ x n ( · ) converges weakly to ˙ x ( · ) in L ( I, H ).Moreover, by (43) and (48) we can choose some real c > n , (cid:107) x n ( θ n ( t )) (cid:107) , (cid:107) x ( t ) (cid:107) ≤ c for all t ∈ [ T , T ]. Therefore, by assumption, there exists L c ( · ) , L c ( · ) ∈ L ([ T , T ] , R + ) such that f ( t, · ) and f ( t, s, · ) are L c ( t )-Lipschitz and L c ( t )-Lipschitz respectively on B [0 , c ]. It follows that T (cid:90) T (cid:107) f ( t, x n ( θ n ( t ))) − f ( t, x ( t )) (cid:107) dt ≤ T (cid:90) T L c ( t ) (cid:107) x n ( θ n ( t )) − x ( t ) (cid:107) dt (49) T (cid:90) T (cid:107) y n ( t ) − y ( t ) (cid:107) dt ≤ T (cid:90) T L c ( t ) t (cid:90) T (cid:107) x n ( θ n ( s )) − x ( s ) (cid:107) ds dt. (50)Note that for every ( t, s ) ∈ Q ∆ L c ( t ) (cid:107) x n ( θ n ( t )) − x ( t ) (cid:107) ≤ cL c ( t ) ,L c ( t ) t (cid:90) T (cid:107) x n ( θ n ( s )) − x ( s ) (cid:107) ds ≤ c ( T − T ) L c ( t ) . Then by (49), (50) and by the Lebesgue dominated convergence theorem f ( · , x n ( θ n ( · ))) −→ f ( · , x ( · )) strongly in L ( I, H ) .y n ( · ) −→ y ( · ) strongly in L ( I, H ) . ζ n ( · ) := ˙ x n ( · ) + f ( · , x n ( θ n ( · ))) + y n ( · ) −→ ζ ( · ) := ˙ x ( · ) + f ( · , x ( · )) + y ( · )weakly in L ( I, H ).By Mazur’s lemma we can find a convex combination r ( n ) (cid:80) k = n S k,n ζ k ( · ), with r ( n ) (cid:80) k = n S k,n = 1and S k,n ∈ [0 ,
1] for all k, n , which converges strongly in L ( I, H ) to ζ ( · ). Extracting asubsequence, we may suppose that r ( n ) (cid:80) k = n S k,n ζ k ( · ) converges almost everywhere on I to somemapping ζ ( · ).Further, we know that there is a negligible set N ⊂ I such that for each t ∈ I \ N one hasfor all n ∈ N − ζ n ( t ) := − ˙ x n ( t ) − f ( t, x n ( θ n ( t ))) − t (cid:90) T f ( t, s, x n ( θ n ( s ))) ds ∈ N C ( t ) ( x n ( t )) . Fix any t ∈ I \ N and any n ∈ N . From Definition 2.1 of the normal cone, one has forevery z ∈ C ( t ) (cid:104)− ζ n ( t ) , z − x n ( t ) (cid:105) ≤ γ ( t )2 r (cid:107) z − x n ( t ) (cid:107) for all z ∈ C ( t ) , hence (cid:104)− ζ n ( t ) , z − x n ( t ) (cid:105) ≤ γ ( t )2 r ( (cid:107) z − x ( t ) (cid:107) + (cid:107) x ( t ) − x n ( t ) (cid:107) ) := λ n ( t ) , (51)with lim n −→∞ λ n ( t ) = γ ( t )2 r (cid:107) z − x ( t ) (cid:107) . Therefore, (cid:104)− ζ ( t ) , z − x ( t ) (cid:105) = (cid:104)− ζ ( t ) + r ( n ) (cid:88) k = n S k,n ζ k ( t ) , z − x ( t ) (cid:105) + r ( n ) (cid:88) k = n S k,n (cid:104)− ζ k ( t ) , z − x k ( t ) (cid:105) + r ( n ) (cid:88) k = n S k,n (cid:104)− ζ k ( t ) , − x ( t ) + x k ( t ) (cid:105) . The first expression of the second member of the latter equality tends to zero by whatprecedes, and keeping in mind that | ζ k ( t ) | ≤ γ ( t ), we also see that the third expressiontends to zero. Concerning the second expression, thanks to (51), it satisfies the estimate r ( n ) (cid:88) k = n S k,n (cid:104)− ζ k ( t ) , z − x k ( t ) (cid:105) ≤ r ( n ) (cid:88) k = n S k,n λ k ( t ) . Thus, passing to the limit we obtain (cid:104)− ζ ( t ) , z − x ( t ) (cid:105) ≤ γ ( t )2 r (cid:107) z − x ( t ) (cid:107) , ∀ z ∈ C ( t ) . − ˙ x ( t ) − f ( t, x ( t )) − t (cid:90) T f ( t, s, x ( s )) ds ∈ N C ( t ) ( x ( t )) , a.e. t ∈ [ T , T ] , and thus − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds, a.e. t ∈ [ T , T ] . Now consider the situation when T (cid:90) T (cid:20) β ( τ ) + τ (cid:90) T β ( τ, s ) ds (cid:21) dτ ≥ . We fix a subdivision of [ T , T ] given by T , T , ..., T k = T such that, for any0 ≤ i ≤ k − T i +1 (cid:90) T i (cid:20) β ( τ ) + τ (cid:90) T β ( τ, s ) ds (cid:21) dτ < . Then, by what precedes, there exists an absolutely continuous map x : [ T , T ] −→ H such that x ( T ) = x , x ( t ) ∈ C ( t ) for all t ∈ [ T , T ], and − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds, a.e. t ∈ [ T , T ] . Similarly, there is an absolutely continuous map x : [ T , T ] −→ H such that x ( T ) = x ( T ), x ( t ) ∈ C ( t ) for all t ∈ [ T , T ], and − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds, a.e. t ∈ [ T , T ] . By induction, we obtain for each 0 ≤ i ≤ k − x i : [ T i , T i +1 ] −→ H such that for each 0 ≤ i ≤ k − x i ( T i ) = x i − ( T i ) and x i ( t ) ∈ C ( t ) for all t ∈ [ T i , T i +1 ], and − ˙ x i ( t ) ∈ N C ( t ) ( x i ( t )) + f ( t, x i ( t )) + t (cid:90) T f ( t, s, x i ( s )) ds, a.e. t ∈ [ T i , T i +1 ] . We set x − (0) = x and define the mapping x : [ T , T ] −→ H given by x ( t ) = x i ( t ) , textif t ∈ [ T i , T i +1 ] , ≤ i ≤ k − . x ( · ) is an absolutely continuous mapping satisfying x ( T ) = x , x ( t ) ∈ C ( t ) forall t ∈ [ T , T ] and − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds, a.e. t ∈ [ T , T ] . (52) Step 7. We prove the estimations.
Let x ( · ) be a solution of ( P f ,f ).Take N ⊂ [ T , T ] such that λ ( N ) = 0 and the inclusion (52) holds for every t ∈ [ T , T ] \ N .Fix any t ∈ [ T , T ] \ N .By definition of proximal normal cone, there is some real a > a ∈ (0 , a ] x ( t ) ∈ Proj C ( t ) ( x ( t ) − a ˙ x ( t ) − af ( t, x ( t )) − a t (cid:90) T f ( t, s, x ( s )) ds ) . We derive from the latter inclusion that a (cid:107) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:107) = d C ( t ) (cid:16) x ( t ) − a ˙ x ( t ) − aεf ( t, x ( t )) − a t (cid:90) T f ( t, s, x ( s )) ds (cid:17) ≤ | υ ( t ) − υ ( τ ) | + (cid:13)(cid:13)(cid:13) x ( t ) − x ( τ ) − a ˙ x ( t ) − af ( t, x ( t )) − a t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13) , since x ( τ ) ∈ C ( τ ) for all τ ∈ [ T , T ]. For any τ ∈ T , t [ with t − a < τ < t , taking a = t − τ one obtains (cid:13)(cid:13)(cid:13) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13) ≤ | υ ( t ) − υ ( τ ) | t − τ + (cid:13)(cid:13)(cid:13) x ( t ) − x ( τ ) t − τ − ˙ x ( t ) − f ( t, x ( t )) − t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13) . Making τ ↑ t yields (cid:13)(cid:13)(cid:13) ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˙ υ ( t ) | + (cid:107)− f ( t, x ( t )) − t (cid:90) T f ( t, s, x ( s )) ds (cid:13)(cid:13)(cid:13) ≤ | ˙ υ ( t ) | + (cid:107) f ( t, x ( t )) (cid:107) + t (cid:90) T (cid:107) f ( t, s, x ( s )) (cid:107) ds. (53)8This justifies (13).Now assume t (cid:90) T (cid:20) β ( τ ) + τ (cid:90) T β ( τ, s ) ds (cid:21) dτ < . We have from (38), (39) and (40) that the estimates (14), (15) and (16) are obviouslyfulfilled.If in addition (cid:107) f ( t, s, x ) (cid:107) ≤ g ( t, s ) + α ( t ) (cid:107) x (cid:107) we have from (53) that (cid:107) ˙ x ( t ) (cid:107) ≤| ˙ υ ( t ) | + 2 (cid:107) f ( t, x ( t )) (cid:107) + 2 t (cid:90) T (cid:107) f ( t, s, x ( s )) (cid:107) ds ≤| ˙ υ ( t ) | + 2 β ( t )(1 + (cid:107) x ( t ) (cid:107) ) + 2 t (cid:90) T g ( t, s ) ds + 2 α ( t ) t (cid:90) T (cid:107) x ( s ) (cid:107) ds = | ˙ υ ( t ) | + 2 β ( t ) + 2 t (cid:90) T g ( t, s ) ds + 2 β ( t ) (cid:107) x ( t ) (cid:107) + 2 α ( t ) t (cid:90) T (cid:107) x ( s ) (cid:107) ds. (54)Putting ρ ( t ) := (cid:107) x (cid:107) + t (cid:90) T (cid:107) ˙ x ( s ) (cid:107) ds and noting that (cid:107) x ( t ) (cid:107) ≤ ρ ( t ), the inequality (54)ensures that˙ ρ ( t ) ≤| ˙ υ ( t ) | + 2 β ( t ) + 2 t (cid:90) T g ( t, s ) ds + 2 β ( t ) ρ ( t ) + 2 α ( t ) t (cid:90) T ρ ( s ) ds. Applying Gronwall Lemma 3.2 with ρ ( · ), one obtains (cid:107) x ( t ) (cid:107) ≤ ρ ( t ) ≤ (cid:107) x (cid:107) exp (cid:18) t (cid:90) T ( b ( τ ) + 1) dτ (cid:19) + t (cid:90) T (cid:18) | ˙ υ ( s ) | + 2 β ( s ) + 2 s (cid:90) T g ( s, τ ) dτ (cid:19) exp (cid:18) t (cid:90) s ( b ( τ ) + 1) dτ (cid:19) ds, where b ( τ ) := 2 max { β ( τ ) , α ( τ ) } for almost all τ ∈ [ T , T ]. This yields the validity of(17), (18) and (19). Step 8. Uniqueness. x ( · ) , x ( · ) are two solutions, the hypo-monotonicityproperty of the normal cone yields for almost all t ∈ [ T , T ] (cid:104)− ˙ x ( t ) − f ( t, x ( t )) − t (cid:90) T f ( t, s, x ( s )) ds + ˙ x ( t ) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds,x ( t ) − x ( t ) (cid:105) ≤ r (cid:107) x ( t ) − x ( t ) (cid:107) (cid:88) i =1 (cid:18) (cid:107) ˙ x i ( t ) (cid:107) + (cid:107) f ( t, x i ( t )) (cid:107) + t (cid:90) T (cid:107) f ( s, x i ( s )) (cid:107) ds (cid:19) , from which we obtain (cid:104) ˙ x ( t ) − ˙ x ( t ) , x ( t ) − x ( t ) (cid:105)≤ r (cid:107) x ( t ) − x ( t ) (cid:107) (cid:88) i =1 (cid:18) (cid:107) ˙ x i ( t ) (cid:107) + (cid:107) f ( t, x i ( t )) (cid:107) + t (cid:90) T (cid:107) f ( t, s, x i ( s )) (cid:107) ds (cid:19) + (cid:104) f ( t, x ( t )) − f ( t, x ( t )) , x ( t ) − x ( t ) (cid:105) + (cid:104) t (cid:90) T f ( t, s, x ( s )) ds − t (cid:90) T f ( t, s, x ( s )) ds, x ( t ) − x ( t ) (cid:105) . Since the absolutely continuous mappings x ( · ) and x ( · ) are in particular bounded on[ T , T ], we can choose some real η > i = 1 , (cid:107) x i ( t ) (cid:107) ≤ η for all t ∈ [ T , T ]. The latter inequality assures us that ddt (cid:107) x ( t ) − x ( t ) (cid:107) ≤ L η ( t ) (cid:107) x ( t ) − x ( t ) (cid:107) t (cid:90) T (cid:107) x ( s ) − x ( s ) (cid:107) ds + (cid:18) L η ( t ) + 12 r (cid:88) i =1 (cid:18) (cid:107) ˙ x i ( t ) (cid:107) + (cid:107) f ( t, x i ( t )) (cid:107) + t (cid:90) T (cid:107) f ( t, s, x i ( s )) (cid:107) ds (cid:19)(cid:19) (cid:107) x ( t ) − x ( t ) (cid:107) . Finally, setting ρ ( t ) := (cid:107) x ( t ) − x ( t ) (cid:107) we get˙ ρ ( t ) ≤ (cid:18) L η ( t ) + 1 r (cid:88) i =1 (cid:18) (cid:107) ˙ x i ( t ) (cid:107) + (cid:107) f ( t, x i ( t )) (cid:107) + t (cid:90) T (cid:107) f ( t, s, x i ( s )) (cid:107) ds (cid:19)(cid:19) ρ ( t )+ 2 L η ( t ) (cid:112) ρ ( t ) t (cid:90) T (cid:112) ρ ( s ) ds, hence it suffices to invole Lemma 3.3 with ε ( · ) , (cid:15) > Proposition 4.3
Assume that the assumptions of Theorem 4.2 ( in case ) holds. Foreach a ∈ C ( T ) , denote by x a ( · ) the unique solution of the integro-differential sweepingprocess − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds a.e in [ T , T ] x ( T ) = a ∈ C ( T ) Then, the map ψ : a −→ x a ( · ) from C ( T ) to the space C ([ T , T ] , H ) endowed with theuniform convergence norm is Lipschitz on any bounded subset of C ( T ) . Proof . Let M be any fixed positive real number. We are going to prove that ψ isLipschitz on C ( T ) ∩ M B .According to Theorem 4.2 ( case 3 ), there exists a real number M depending only on M such that, for all z ∈ C ( T ) ∩ M B and for almost all ( t, s ) ∈ Q ∆ (cid:107) ˙ x z ( t )+ f ( t, x z ( t ))+ t (cid:90) T f ( t, s, x z ( s )) ds (cid:107) ≤ α ( t ) := | ˙ υ ( t ) | +(1+ M ) β ( t )+ t (cid:90) T g ( t, s ) ds + T α ( t ) M , Thanks to this last inequality, for some η > M , for all z ∈ C ( T ) ∩ M B and for all t ∈ [ T , T ], we have x z ( t ) ∈ B [0 , η ] . (55)Fix any a, b ∈ C ( T ) ∩ M B . By the hypomonotonicity property of the normal cone, wehave for almost all ( t, s ) ∈ Q ∆ (cid:104)− ˙ x a ( t ) − f ( t, x a ( t )) − t (cid:90) T f ( t, s, x a ( s )) ds + ˙ x b ( t ) + f ( t, x b ( t )) + t (cid:90) T f ( t, s, x b ( s )) ds, x ( t ) − x ( t ) ≤ α ( t ) r (cid:107) x b ( t ) − x a ( t ) (cid:107) , from which we obtain (cid:104) ˙ x b ( t ) − ˙ x a ( t ) , x b ( t ) − x a ( t ) (cid:105) ≤ α ( t ) r (cid:107) x b ( t ) − x a ( t ) (cid:107) + (cid:104) f ( t, x a ( t )) − f ( t, x b ( t )) , x b ( t ) − x a ( t ) (cid:105) + (cid:104) t (cid:90) T f ( t, s, x a ( s )) ds − t (cid:90) T f ( t, s, x b ( s )) ds, x b ( t ) − x a ( t ) (cid:105) . Since, by assumption ( H , ) and ( H , ), there are a non-negative functions L η ( · ), L η ( · ) ∈ L ([ T , T ] , R ) such that f ( t, · ) and f ( t, s, · ) are L η ( · )-Lipschitz, L η ( · )-Lipschitz (re-spectively) on B [0 , η ], the above inequality along with (55), entails that for almost all t ∈ [ T , T ], ddt (cid:107) x b ( t ) − x a ( t ) (cid:107) ≤ (cid:18) L η ( t ) + α ( t ) r (cid:19) (cid:107) x b ( t ) − x a ( t ) (cid:107) + 2 L η ( y ) (cid:107) x b ( t ) − x a ( t ) (cid:107) t (cid:90) T (cid:107) x b ( s ) − x a ( s ) (cid:107) ds. t ∈ [0 ,T ] (cid:107) x b ( t ) − x a ( t ) (cid:107) ≤ (cid:107) b − a (cid:107) exp (cid:18) t (cid:90) T ( K ( s ) + 1) ds (cid:19) , where K ( t ) := max (cid:26) L η ( t ) + α ( t ) r , L η ( t ) (cid:27) , for almost all t ∈ [ T , T ]. The proof is thencomplete. In this section, as a consequence of Theorem 4.2, we obtain the existence and uniquenessof solutions for nonlinear integro-differential complementarity systems. Our results gener-alize those from [3].Let
T > T be real numbers, I = [ T , T ], n, m ∈ N , f : I × R n −→ R n , f : Q ∆ × R n −→ R n and g : I × R n −→ R m be given mappings. Assuming that g ( t, · ) is differentiable foreach t ∈ I , the NIDCS (associated with f , f and g ) can be described as( NIDCS ): − ˙ x ( t ) = f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds + ∇ g ( t, · )( x ( t )) T z ( t )0 ≤ z ( t ) ⊥ g ( t, x ) ≤ , where z : I −→ R m is unknown mapping. The term ∇ g ( t, · )( x ( t )) T z ( t ) can be seen as thegeneralized reactions due to the constraints in mechanics.Of course, the behaviour of a solution with respect to t is connected to the variation withrespect to t of the set constraint C ( t ) = { x ∈ R n : g ( t, x ) ≤ , g ( t, x ) ≤ , ..., g m ( t, x ) ≤ } , (56)where we set g ( t, · ) = ( g ( t, · ) , g ( t, · ) , ..., g m ( t, · )) for each t ∈ I. Theorem 5.1 [3] Let C ( t ) be defined as in (56) and assume that, there exists an extendedreal ρ ∈ ]0 , ∞ ] such that1. for all t ∈ I , for all k ∈ { , ..., m } , g k ( t, · ) is continuously differentiable on U ρ ( C ( t )) := { y ∈ R n : d C ( t ) ( y ) < ρ } ;2. there exists a real γ > such that, for all t ∈ I , for all k ∈ { , ..., m } , g k ( t, · ) , for all x, y ∈ U ρ ( C ( t )) (cid:104)∇ g k ( t, · )( x ) − ∇ g k ( t, · )( y ) , x − y (cid:105) ≥ − γ (cid:107) x − y (cid:107) , that is, ∇ g k ( t, · ) is γ -hypomonotone on U ρ ( C ( t )) ;
3. there is a real δ > such that for all ( t, x ) ∈ I × R n with x ∈ bdry ( C ( t )) , thereexists ¯ υ ∈ B satisfying, for all k ∈ { , ..., m }(cid:104)∇ g ( t, · )( x ) , ¯ υ (cid:105) ≤ − δ. (57) Then for all t ∈ I , the set C ( t ) is r-prox-regular with r = min { ρ, δγ } . The nonlinear differential complementarity systems (NDCS) (i.e., (NIDCS) with f ≡
0) was studied in [3], where the authors transform the (NDCS) involving inequality con-straints C ( t ) to a perturbed sweeping process . We extend this approach by transforming(NIDCS) into an integro-differential sweeping process of the form (4). Also, in contrast to[3], we do not assume that the moving set C ( t ) described by a finite number of inequali-ties is absolutely continuous with respect to the Hausdorff distance. Rather, we providesufficient verifiable conditions ensuring this regularity needed on C ( · ). Proposition 5.2
Let C ( t ) be defined as in (56) . Assume that there exist an absolutelycontinuous function w , a real δ > and a vector y ∈ R n with (cid:107) y (cid:107) = 1 such that for each i = 1 , ...m g i ( t, x ) ≤ g i ( s, x ) + | w ( t ) − w ( s ) | , f or all x ∈ U r ( C ( s )) , (58) (cid:104)∇ g i ( t, · )( x ) , y (cid:105) ≤ − δ, f or, all t ∈ I, x ∈ U r ( C ( t )) , (59) where r denotes the prox-regularity constant of all sets C ( t ) . Then C ( · ) is υ ( · ) − absolutelycontinuous on I with υ ( · ) := δ − w ( · ) . Proof . Let s, t ∈ I , let x ∈ C ( s ) and choose a subdivision T < T < ... < T p = T suchthat T k (cid:82) T k − | ˙ υ ( τ ) | dτ < r for every k = 1 , · · · , p . Fix any k = 1 , ..., p and s, t ∈ [ T k − , T k ].Take any i = 1 , ..., m and note that g i ( t, x + | υ ( t ) − υ ( s ) | y ) = ( g i ( t, x + | υ ( t ) − υ ( s ) | y ) − g i ( s, x + | υ ( t ) − υ ( s ) | y ))+ g i ( s, x + | υ ( t ) − υ ( s ) | y ) ≤ | w ( t ) − w ( s ) | + g i ( s, x + | υ ( t ) − υ ( s ) | y )= | w ( t ) − w ( s ) | + g i ( s, x )+ (cid:90) (cid:104)∇ g i ( s, x + θy | υ ( t ) − υ ( s ) | ) , y | υ ( t ) − υ ( s ) |(cid:105) d θ. (60)According to (59) and to the inclusion x ∈ C ( s ) it ensues that g i ( t, x + | υ ( t ) − υ ( s ) | y ) ≤ | w ( t ) − w ( s ) | − δ | υ ( t ) − υ ( s ) | ≤ . This being true for every i = 1 , ..., m , it follows that x + | υ ( t ) − υ ( s ) | y belongs to C ( t ),otherwise stated, x ∈ C ( t ) + | υ ( t ) − υ ( s ) | ( − y ). It results that C ( s ) ⊂ C ( t ) + | υ ( t ) − υ ( s ) | B . Since the variables s and t play symmetric roles, the set-valued mapping C ( · ) has anabsolutely continuous variation on [ T k − , T k ]. From this we clearly derive that C ( · ) hasan absolutely continuous variation on I .3 Example 5.3
Let m = 1 , n = 2 , T = 1 , g ( t, x ) = t − x − x , and define C ( t ) = { x ∈ R : g ( t, x ) ≤ } . Clearly that C ( t ) is r-prox-regular, since g ( t, · ) satisfies all assumptions of theorem 5.1 forall t ∈ I . Now we check (58) and (59) . Let x ∈ R , t, s ∈ I . Fix any δ ∈ (0 , and put y = (1 , . Then g ( t, x ) − g ( s, x ) = t − s ≤ | t − s | = | w ( t ) − w ( s ) | , (cid:104)∇ g ( t, · )( x ) , y (cid:105) = − ≤ − δ. We see that w ( t ) = t / is not Lipschitz on I but it is absolutely continuous there. Then C ( · ) has an absolutely continuous variation υ on I , with υ ( t ) = t / /δ . Theorem 5.4
Assume that the assumptions in Theorem 5.1, Proposition 5.2 and condi-tions ( H ) , ( H ) are satisfied. Then, for every initial data x with g (0 , x ) ≤ , problem(NIDCS) has one and only one solution x ( · ) . Proof . In the same arguments like in [3] one has the equivalent between the problem(NIDCS) and the integro-differential sweeping process − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) T f ( t, s, x ( s )) ds Therefore, all assumptions of Theorem 4.2 are satisfied and the conclusion follows.
The aim of this section is to illustrate the integro-differential sweeping process in thetheory of non-regular electrical circuits. Electrical devices like diodes are described interms of Ampere-Volt characteristic which is (possibly) a multifunction expressing thedifference of potential v D across the device as a function of current i D going through thedevice[5].Let us consider the electrical system shown in Fig. 1 that is composed of three resistors R ≥ R ≥ V R k = R k x k ( k = 1 , L ≥ L ≥ V L k = R k ˙ x k ( k = 1 , C ( t ) (cid:54) = 0, C ( t ) (cid:54) = 0 and C ( t ) (cid:54) = 0 with voltage/current laws V C k = C k ( t ) (cid:82) x k ( t ) dt, k = 1 , ,
3, two ideal diodes with characteristics 0 ≤ − V D k ⊥ i k ≥ i : [0 , T ] → R .Using Kirchhoff’s laws, we have (cid:26) V R + V R + V L + V C + V C = − V D ∈ − N ( R + ; x − i ) V R − V R + V L + V C − V C = − V D ∈ − N ( R + ; x ) . − ˙ x ( t ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) − ˙ x ( t ) − ˙ x ( t ) (cid:19) ∈ N [ i ( t ) , + ∞ [ × [0 , + ∞ [ ( x ( t )) + A (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) R + R L − R L − R L R + R L (cid:33) x ( t ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) x ( t ) x ( t ) (cid:19) + t (cid:90) (cid:20) A (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) L C ( t ) + L C ( t ) − L C ( t ) − L C ( t ) 1 L C ( t ) + L C ( t ) (cid:33) x ( s )) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) x ( s ) x ( s ) (cid:19) + (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) L C ( t ) i ( s )0 (cid:19) (cid:21) ds. (61) Proposition 6.1
Assume that i : [0 , T ] −→ R is an absolutely continuous function and C k : [0 , T ] −→ R ∗ , k = 1 , , are continuous functions. Then for any initial condition x (0) = x ∈ C (0) , problem (61) has one and only one absolutely continuous solution x ( · ) . Proof . Put w ( t ) = ( i ( t ) , t , C ( t ) := w ( t ) + [0 , + ∞ [ × [0 , + ∞ [, f ( t, x ) = A x , f ( t, s, x ) = A ( t ) x + 1 L C ( t ) w ( s ). So (61) can be rewritten in the frame of our problem( P f ,f ) as − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( t, x ( t )) + t (cid:90) f ( t, s, x ( s )) ds a.e. in [0 , T ] x (0) = x ∈ C (0)Then the above data satisfying all the assumptions of Theorem 4.2 ( precisely case 3 ),5with υ ( t ) = t (cid:90) (cid:107) ˙ w ( s ) (cid:107) ds, β ( t ) = (cid:107) A (cid:107) , g ( t, s ) = 1 L C ( t ) (cid:107) w ( s ) (cid:107) , α ( t ) = (cid:107) A ( t ) (cid:107) . This finishes the proof.
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