In this paper we study a nonlinear evolution inclusion of subdifferential
 type in Hilbert spaces. The perturbation term is Hausdorff continuous in
the state variable and has closed but not necessarily convex values. Our result
is a stochastic generalization of an existence theorem proved by Kravvaritis and
Papageorgiou in [6]

Kravvaritis, D.

Pantelidis, G.

Kravvaritis, D

Yanev, N

DSpace@NTUA (National Technical Univ. of Athens)

A RANDOM EVOLUTION INCLUSION OF SUBDIFFERENTIAL TYPE IN HILBERT SPACES

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 22 (1996), 117-124A RANDOM EVOLUTION INCLUSIONOF SUBDIFFERENTIAL TYPE IN HILBERT SPACESD. Kravvaritis, G. PantelidisCommunicated by N. M. YanevAbstract. In this paper we study a nonlinear evolution inclusion of subdiffe-rential type in Hilbert spaces. The perturbation term is Hausdorff continuous inthe state variable and has closed but not necessarily convex values. Our resultis a stochastic generalization of an existence theorem proved by Kravvaritis andPapageorgiou in [6].1. Introduction. In this paper we study a nonlinear random multivaluedevolution inclusion of the form(∗){−x˙(ω, t) ∈ ∂ϕ(ω, x(ω, t)) + F (ω, t, x(ω, t))x(ω, 0) = x0(ω)},where F (ω, t, x) is a random multivalued perturbation term with closed but not neces-sarily convex values.Random differential inclusions have been studied by many authors (cf. Itoh [5],Papageorgiou [10], Kravvaritis and Papageorgiou [7], Nowak [9] and their references).Our result generalizes to the random case corresponding deterministic result proved byKravvaritis and Papageorgiou in [6].1991 Mathematics Subject Classification: 47H06, 47H15Key words: subdifferential, evolution inclusions, Hausdorff continuity, measurable multifunctions118 D. Kravvaritis, G. Pantelidis2. Mathematical preliminaries. Let (Ω,Σ, µ) be a complete probabilityspace and H a separable Hilbert space. By Pf (H) we will denote the nonempty closedsubsets of H. A multifunction F : Ω→ Pf (H) is said to be measurable if, for all x ∈ H,ω → d(x, F (ω)) = inf{‖z − x‖, z ∈ F (ω)}is measurable. By S2F , we will denote the set of measurable selectors of F (·), thatbelong in the Lebesgue-Bochner space L2(H), i.e.,S2F = {f ∈ L2(H) : f(ω) ∈ F (ω) µ− a.e.}.It is easy to check that this set is closed and it is nonempty if and only if inf{‖x‖ :x ∈ F (ω)} ∈ L2.A function ϕ : Ω × H → R = R ∪ {+∞} is a normal integrand if and only if(ω, x)→ ϕ(ω, x) is measurable and, for all ω ∈ Ω, x→ ϕ(ω, x) is l.s.c. and proper. If,in addition, ϕ(ω, ·) is convex, then we say that ϕ is a convex normal integrand. Recallthat for a proper, convex function ϕ : H → R the subdifferential at x is defined by∂ϕ(x) = {y ∈ H : ϕ(z) − ϕ(x) ≥ (y, z − x) for all z ∈ H}.We say that ϕ(·) is of compact type, if for every λ ∈ R+, the level set {x ∈ H :‖x‖2 + ϕ(x) ≤ λ} is compact.The generalized Hausdorff metric on Pf (H), is defined byh(A,B) = max[supa∈Ad(a,B), supb∈Bd(b,A)].We say that F : H → Pf (H) is Hausdorff continuous (h-continuous), if it is continuousfrom H into the metric space (Pf (H), h).3. The main result. Let T = [0, b] be a bounded, closed interval in R+.Consider the following initial value problem(∗∗){−x˙(t) ∈ ∂ϕ(x(t)) + f(t)x(0) = x0}where f ∈ L2(T,H) and x0 ∈ D(∂ϕ). A continuous function x : T → H is a strongsolution of (∗∗) if x(0) = x0, x(·) is absolutely continuous on every compact subset of(0, b) andx(t) ∈ D(∂ϕ) and x˙(t) ∈ ∂ϕ(x(t)) + f(t) a.e. on (0, b).A random evolution inclusion of subdifferential type . . . 119It is known [3] that (∗∗) has a unique strong solution. By a random strongsolution of (∗) we understand a stochastic process x : Ω × T → H such that for everyω ∈ Ω, x(ω, ·) is a strong solution of−x˙(ω, t) ∈ ∂ϕ(ω, x(ω, t)) + f(ω, t)x(ω, 0) = x0(ω)for some f(ω, ·) ∈ S2F (ω,·,x(ω,·)).We will make the following hypotheses:H(ϕ) : ϕ : Ω×H → R is a convex normal integrand, which is of compact type in thex-variable and the multifunctionD(ω) = D(∂ϕ(ω, ·)) = {x ∈ H : ∂ϕ(ω, x) 6= Ø}has a bounded selector.H(F ) : F : Ω× T ×H → Pf (H) is a multifunction such that(i) for all x ∈ H, (ω, t)→ F (ω, t, x) is measurable,(ii) for all (ω, t) ∈ Ω× T , x→ F (ω, t, x) is h-continuous,(iii) |F (ω, t, x)| = sup{‖z‖ : z ∈ F (ω, t, x)} ≤ a(t)+ b(t)‖x‖ a.e. for every ω ∈ Ω,with a(·), b(·) ∈ L2+.H0 : x0 : Ω→ H is measurable such thatsup{‖x0(ω)‖, ω ∈ Ω} <∞ and sup{ϕ(ω, x0(ω)), ω ∈ Ω} <∞.Theorem. If the hypotheses H(ϕ), H(F ) and H0 hold, then (∗) admits arandom strong solution.P r o o f. Let z(·) be a bounded selector of D(·) and let u(ω) ∈ ∂ϕ(ω, z(ω)). Ifwe setϕˆ(ω, x) = ϕ(ω, x)− ϕ(ω, z(ω)) − (u(ω), x − z(ω)),then (∗) is equivalent to−x˙(ω, t) ∈ ∂ϕˆ(ω, x(ω, t)) + F (ω, t, x(ω, t)) + u(ω)x(ω, 0) = x0(ω).120 D. Kravvaritis, G. PantelidisSo there is no loss of generality in assuming thatmin{ϕ(ω, x) : x ∈ H} = ϕ(ω, z(ω)) = 0.Therefore(z(ω), 0) ∈ Gr ∂ϕ(ω, ·) for all ω ∈ Ω.First let us obtain an a priori bound for the solutions of (∗). So let x(·, ·) be a randomstrong solution. By definition then there exists h(ω, ·) ∈ S2F (ω,·,x(ω,·)) such that−x˙(ω, t) ∈ ∂ϕ(ω, x(ω, t)) + h(ω, t)x(ω, 0) = x0(ω).From Lemma 3.1 (p. 64) of Brezis [3] we know that‖x(ω, t)− z(ω)‖2 ≤ ‖x0(ω)− z(ω)‖2 + 2∫ t0‖h(ω, s)‖.‖x(ω, s) − z(ω)‖ds.Invoking Lemma A.5, p. 157, of [3], we get‖x(ω, t) − z(ω)‖ ≤ ‖x0(ω)− z(ω)‖+∫ t0‖h(ω, s)‖ds for all ω ∈ Ω, t ∈ T.Now, using the growth hypothesis H(F ) (iii) we have‖x(ω, t) − z(ω)‖ ≤ ‖x0(ω)− z(ω)‖+∫ t0[a(s) + b(s)‖x(ω, s)‖] ds,and by Gronwall’s inequality we get‖x(ω, t)‖ ≤ L · exp ‖b(·)‖1 =M for all ω ∈ Ω, t ∈ T,whereL = sup{‖x0(ω)− z(ω)‖+ ‖z(ω)‖ : ω ∈ Ω}+ ‖a(·)‖1.Then consider the multifunction Fˆ : Ω× T ×H → Pf (H) defined byFˆ (ω, t, x) =F (ω, t, x) if ‖x‖ ≤MF(ω, t,Mx‖x‖)if ‖x‖ > M.We see thatFˆ (ω, t, x) = F (ω, t, pM (x)),A random evolution inclusion of subdifferential type . . . 121where pM(·) is the M -radial retraction for which we know that it is continuous. Hence(ω, t, x)→ Fˆ (ω, t, x) is measurable. Furthermore note that|Fˆ (ω, t, x)| ≤ a(t) +Mb(t) = γ(t), γ(·) ∈ L2(T ).We setB(γ) = {h ∈ L2(T,H) : ‖h(t)‖ ≤ γ(t) a.e.}.Let now q : Ω × L2(T,H) → C(T,H) be the map that to each (ω, h) ∈ Ω × L2(T,H)assigns the unique solution of(∗)′ −x˙(t) ∈ ∂ϕ(ω, x(t)) + h(t) a.e., x(0) = x0(ω).We claim that q is a Caratheodory map. Fix h ∈ L2(T,H). From Lemma 2.1 of [2] weknow thatq(ω, h) = limλ→0qλ(ω, h),where qλ(ω, h) is the unique solution ofx˙λ(t) = ∇ϕλ(ω, xλ(t)) + h(t), xλ(0) = x0(ω)withϕλ(ω, x) = inf[12λ‖x− z‖2 + ϕ(ω, z) : z ∈ H].Since ∇ϕλ(ω, x) is measurable in ω (see Theorem 2.3 of [1]), qλ(·, h) is also measurableand so q(·, h) is measurable. From Lemma 3.1 of Brezis [3], we know that, for eachω ∈ Ω, q(ω, ·) is continuous from L2(T,H) into C(T,H). So q(·, ·) is a Caratheodorymap. LetW = {q(ω, h) = x : strong solution of (∗)′, ω ∈ Ω, h ∈ B(γ)}.Then for any x(·) ∈W and t, t′ ∈ T , t < t′, we have‖x(t′)− x(t)‖=∥∥∥∥∥∫ t′tx˙(s)ds∥∥∥∥∥ ≤∫ t′t‖x˙(s)‖ds =∫ b0‖X[t,t′](s)x˙(s)‖ds≤[∫ b0‖X[t,t′](s)‖2ds] 12[∫ b0‖x˙(s)‖2ds] 12.From the estimates provided by Theorem 3.6 of Brezis [3], we know that[∫ b0‖x˙(s)‖2ds] 12≤ ‖h‖2+[ϕ(ω, x0(ω))]12 ≤ ‖γ‖2+sup{[ϕ(ω, x0(ω))]12 : ω ∈ Ω}=M1.122 D. Kravvaritis, G. PantelidisSo finally we have‖x(t′)− x(t)‖ ≤M1(t′ − t)12 ,which implies that W is equicontinuous.Also again from Theorem 3.6 of [3] we know that‖x˙(t)‖2 +ddtϕ(ω, x(t)) = (h(t), x˙(t)) a.e.,soϕ(ω, x(t))≤ϕ(ω, x0(ω)) +∫ t0(h(s), x˙(s))ds ≤ ϕ(ω, x0(ω)) + ‖h‖2‖x˙‖2≤sup{ϕ(ω, x0(ω)) : ω ∈ Ω}+ ‖γ‖2M1 =M2⇒ ϕ(ω, x(t)) ≤M2 for all t ∈ T, ω ∈ Ω and all x(·) ∈W.Recalling that ϕ(ω, ·) is of compact type, we deduce that W (t) is compact for allt ∈ T . Invoking the Arzela-Ascoli theorem, we conclude that W is relatively compactin C(T,H). As in the proof of Theorem 3.1 in [6] we can prove that W is closed, hencecompact in C(T,H). It then follows that Wˆ = convW is compact in C(T,H).Now consider the multifunction R : Ω× Wˆ → Pf (L2(T,H)) defined byR(ω, y) = S2F (ω,·,y(·)).For each y ∈ Wˆ , the multifunction ω → R(ω, y) is measurable. Indeed, since (ω, t) →F (ω, t, x) is measurable and x→ F (ω, t, x) is h-continuous F (·, ·, ·) is jointly measurable([11, Theorem 3.3]). Then for every x ∈ H,(ω, t, y)→ d(x, F (ω, t, y))is measurable. Since the distance function is continuous in x, for each h ∈ L2(T,H),(ω, t)→ d(h(t), F (ω, t, y(t)))is measurable. From Fubini’s theorem we get thatω →∫ b0d(h(t), F (ω, t, y(t)))dt = d(h, S2F (ω,·,y(·)))is measurable, which implies that ω → R(ω, y) is measurable.Next consider the multifunction G : Ω→ Pf (C(Wˆ , L2(T,H))) defined byG(ω) = {r ∈ C(Wˆ , L2(T,H)) : r(y) ∈ R(ω, y) for all y ∈ Wˆ}.A random evolution inclusion of subdifferential type . . . 123From Fryszkowski’s selection theorem [4], we know that, for all ω ∈ Ω, G(ω) 6= Ø. WehaveG(ω) = {r ∈ C(Wˆ , L2(T,H)) : d(r(y), R(ω, y)) = 0 for all y ∈ Wˆ}.Since x→ F (ω, t, x) is h-continuous, the multifunction y → R(ω, y) is also h-continuous.It then follows that y → d(r(y), R(ω, y)) is continuous. Thus if {yn} is dense in Wˆ ,thenG(ω) =∞⋂n=1{r ∈ C(Wˆ , L2(T,H)) : d(r(yn), R(ω, yn)) = 0}.Now, for fixed y ∈ Wˆ , (ω, r) → d(r(y), R(ω, y)) is a Caratheodory map. So the mul-tifunction ω → {r ∈ C(Wˆ , L2(T,H)) : d(r(y), R(ω, y)) = 0} is measurable, which inturn implies that G is measurable. By the selection theorem of Kuratowski and Ryll-Nardzewski [8], there exists r : Ω→ C(Wˆ , L2(T,H)) measurable such that r(ω) ∈ G(ω)for all ω ∈ Ω. Observe that, for every ω ∈ Ω, q(ω, r(ω)(·)) : Wˆ → Wˆ . Apply-ing Schauder’s fixed point theorem, we get x ∈ W such that q(ω, r(ω)(x)) = x. SetS(ω) = {x ∈ Wˆ : q(ω, r(ω)(x)) = x}. Then there exists s : Ω→ C(T,H) measurablesuch that s(ω) ∈ S(ω) for all ω ∈ Ω. It then follows thatx(ω, t) = q(ω, r(ω)(s(ω)))(t)is the desired random solution of (∗).REFERENCES[1] H. Attouch. Families d’operateurs maximaux monotones et mesurabilite. Ann.Mat. Pura Appl. 120 (1979), 35-111.[2] V. Barbu. Nonlinear Semigroups and Differential Equations in Banach Spaces.Noordhoff Inter. Publishing, Leyden, The Netherlands, 1976.[3] H. Brezis. Operateurs Maximaux Monotones et Semigroups de Contractionsdans les Espaces de Hilbert. North Holland, Amsterdam, 1973.[4] A. Fryszkowski. Continuous selections for a class of nonconvex multivaluedmaps. Studia Math. 76 (1983), 163-174.[5] S. Itoh. Random fixed point theorems with an application to random differentialequations in Banach spaces. J. Math. Anal. Appl. 67, (1979), 261-273.124 D. Kravvaritis, G. Pantelidis[6] D. Kravvaritis and N. S. Papageorgiou. Multivalued perturbations of sub-differential type evolution equation in Hilbert spaces. J. Differential Equations76 (1988), 238-255.[7] D. Kravvaritis and N. S. Papageorgiou. A random nonlinear multivaluedevolution equation in Hilbert space. Colloquium Math. LVIII, (1989), 129-141.[8] K. Kuratowski and C. Ryll-Nardzewski. A general theorem on selectors.Bull. Acad. Polon. Sci. Math. Astronom. Phys. 13 (1965), 397-403.[9] A. Nowak. Random differential inclusions: Measurable selection approach. Ann.Polon. Math. XLIX (1989), 291-296.[10] N. S. Papageorgiou. Random differential inclusions in Banach spaces. J. Dif-ferential Equations 65 (1986), 287-303.[11] N. S. Papageorgiou. On measurable multifunctions with applications to ran-dom multi-valued equations. Math. Japon. 32 (1987), 701-727.D. Kravvaritis, G. PantelidisDepartment of MathematicsNational Technical University of AthensZografou Campus15780 Athens, Greece Received January 15, 1996

A Random Evolution Inclusion of Subdifferential Type in Hilbert Spaces

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A Random Evolution Inclusion of Subdifferential Type in Hilbert Spaces

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