FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY AND RANDOM EFFECTS

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1 FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY AND RANDOM EFFECTS ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p AMEL BENAISSA AND MOUFFAK BENCHOHRA Absrac. In his work we sudy he exisence of mild soluions of a funcional differenial equaion wih delay and random effecs. We use a random fixed poin heorem wih sochasic domain o show he exisence of mild random soluions. Mahemaics Subjec Classificaion (21): 34G2, 34K2, 34K3 Keywords: Funcional differenial equaion, mild random soluion, finie delay, Random fixed poin, semigroup heory. Aricle hisory: Received 24 April 215 Received in revised form 28 June 215 Acceped 28 June Inroducion Funcional evoluion equaions wih sae-dependen delay appear frequenly in mahemaical modeling of several real world problems and for his reason he sudy of his ype of equaions has received grea aenion in he las few years, see for insance [8, 16, 17]. An exensive heory is developed for evoluion equaions [2, 1]. Uniqueness and exisence resuls have been esablished recenly for differen evoluion problems in he papers by Baghli and Benchohra for finie and infinie delay in [3, 4, 5]. On he oher hand, he naure of a dynamic sysem in engineering or naural sciences depends on he accuracy of he informaion we have concerning he parameers ha describe ha sysem. If he knowledge abou a dynamic sysem is precise hen a deerminisic dynamical sysem arises. Unforunaely in mos cases he available daa for he descripion and evaluaion of parameers of a dynamic sysem are inaccurae, imprecise or confusing. In oher words, evaluaion of parameers of a dynamical sysem is no wihou uncerainies. When our knowledge abou he parameers of a dynamic sysem are of saisical naure, ha is, he informaion is probabilisic, he common approach in mahemaical modeling of such sysems is he use of random differenial equaions or sochasic differenial equaions. Random differenial equaions, as naural exensions of deerminisic ones, arise in many applicaions and have been invesigaed by many auhors; see [19, 2, 21, 25, 26] and references herein. Beween hem differenial equaions wih random coefficiens (see, [7, 25]) offer a naural and raional approach (see [24], Chaper 1), since someimes we can ge he random disribuions of some main disurbances by hisorical experiences and daa raher han ake all random disurbances ino accoun and assume he noise o be whie noises. In his work we prove he exisence of mild soluions of he following funcional differenial equaion wih delay and random effecs (random parameers) of he form: (1.1) (1.2) y (, w) = Ay(, w) + f (, yρ(,y ) (, w), w), y(, w) = φ(, w), 84 a.e. J := [, T ] (, ],

2 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p where (Ω, F, P ) is a complee probabiliy space, w Ω, f : J B Ω E, φ B Ω are given random funcions which represen random nonlinear of he sysem, A : D(A) E E is he infiniesimal generaor of a srongly coninuous semigroup T (), J, of bounded linear operaors in a Banach space (E, ), B is a phase space o be specified laer, ρ : J B (, + ), and (E,. ) is a real Banach space. For any funcion y defined on (, T ] Ω and any J we denoe by y (, w) he elemen of B Ω defined by y (θ, w) = y( + θ, w), θ (, ]. Here y (, w) represens he hisory of he sae from ime, up o he presen ime. We assume ha he hisories y (, w) belong o he absrac phase B. To our knowledge, he lieraure on he local exisence of random evoluion equaions wih delay is very limied, so he presen paper can be considered as a conribuion o his quesion. 2. Preliminaries We inroduce noaions, definiions and heorems which are used hroughou his paper. Le C(J, E) be he Banach space of coninuous funcions from J ino E wih he norm kyk = sup { y() : J }. Le B(E) denoe he Banach space of bounded linear operaors from E ino E. A measurable funcion y : J E is Bochner inegrable if and only if y is Lebesgue inegrable. (For he Bochner inegral properies, see he classical monograph of Yosida [27]). Le L1 (J, E) denoe he Banach space of measurable funcions y : J E which are Bochner inegrable normed by T kykl1 = y() d. Definiion 2.1. A map f : J B Ω E is said o be Carah eodory if: (i) f (, y, w) is measurable for all y B.and for all w Ω. (ii) y f (, y, w) is coninuous for almos each J. and for all w Ω. (iii) w f (, y, w) is measurable for all y B, and almos each J. For a given se V of funcions v : (, T ] E, le us denoe by V () = {v() : v V }, (, T ] and V (J) = {v() : v V, (, T ]}. In his paper, we will employ an axiomaic definiion of he phase space B inroduced by Hale and Kao in [14] and follow he erminology used in [18]. Thus, (B, k kb ) will be a seminormed linear space of funcions mapping (, ] ino E, and saisfying he following axioms : (A1 ) If y : (, T ) E, T >, is coninuous on J and y B, hen for every J he following condiions hold : (i) y B ; (ii) There exiss a posiive consan H such ha y() Hky kb ; (iii) There exis wo funcions K( ), M ( ) : R+ R+ independen of y wih K coninuous and M locally bounded such ha : ky kb K() sup{ y(s) : s } + M ()ky kb. (A2 ) For he funcion y in (A1 ), y is a B valued coninuous funcion on J. (A3 ) The space B is complee. Denoe KT = sup{k() : J}, MT = sup{m () : J}. Remark (ii) is equivalen o φ() HkφkB for every φ B. 85

3 2. Since k kb is a seminorm, wo elemens φ, ψ B can verify kφ ψkb = wihou necessarily φ(θ) = ψ(θ) for all θ. 3. From he equivalence of in he firs remark, we can see ha for all φ, ψ B such ha kφ ψkb = : We necessarily have ha φ() = ψ(). ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p We now indicae some examples of phase spaces. For oher deails we refer, for insance o he book by Hino e al. [18]. Example 2.3. Le: BC he space of bounded coninuous funcions defined from (, ] o E; BU C hespace of bounded uniformly coninuous funcions defined from (, ] o E; C := φ BC : lim φ(θ) exis in E ; θ C := φ BC : lim φ(θ) =, endowed wih he uniform norm θ kφk = sup{ φ(θ) : θ }. We have ha he spaces BU C, C and C saisfy condiions (A1 ) (A3 ). However, BC saisfies (A1 ), (A3 ) bu (A2 ) is no saisfied. Example 2.4. The spaces Cg, U Cg, Cg and Cg. Le g be a posiive coninuous funcion on (, ]. We define: φ(θ) is bounded on (, ] ; Cg := φ C((, ], E) : g(θ) φ(θ) Cg := φ Cg : lim =, endowed wih he uniform norm θ g(θ) φ(θ) kφk = sup : θ. g(θ) Then we have ha he spaces Cg and Cg saisfy condiions (A3 ). We consider he following condiion on he funcion g. g( + θ) (g1 ) For all a >, sup sup : < θ <. g(θ) a They saisfy condiions (A1 ) and (A2 ) if (g1 ) holds. Example 2.5. The space Cγ. For any real posiive consan γ, we define he funcional space Cγ by γθ Cγ := φ C((, ], E) : lim e φ(θ) exiss in E θ endowed wih he following norm kφk = sup{eγθ φ(θ) : θ }. Then in he space Cγ he axioms (A1 ) (A3 ) are saisfied. Le Y be a separable Banach space wih he Borel σ-algebra BY. A mapping y : Ω Y is said o be a random variable wih values in Y if for each B BY, y 1 (B) F. A mapping T : Ω Y Y is called a random operaor if T (., y) is measurable for each y Y and is generally expressed as T (w, y) = T (w)y; we will use hese wo expressions alernaively. Nex, we will give a very useful random fixed poin heorem wih sochasic domain. 86

4 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p Definiion 2.6. [9] Le C be a mapping from Ω ino 2Y. A mapping T : {(w, y) : w Ω y C(w)} Y is called random operaor wih sochasic domain C if C is measurable (i.e., for all closed A Y, {w Ω : C(w) A 6= } F) and for all open D Y and all y Y, {w Ω : y C(w) T (w, y) D} F. T will be called coninuous if every T (w) is coninuous. For a random operaor T, a mapping y : Ω Y is called random (sochasic) fixed poin of T iff for p-almos all w Ω, y(w) C(w) and T (w)y(w) = y(w) and for all open D Y, {w Ω : y(w) D} F( y is measurable ). Remark 2.7. If C(w) Y, hen he definiion of random operaor wih sochasic domain coincides wih he definiion of random operaor. Lemma 2.8. [9] Le C : Ω 2Y be measurable wih C(w) closed, convex and solid (i.e., in C(w) 6= ) for all w Ω. We assume ha here exiss measurable y : Ω Y wih y in C(w) for all w Ω. Le T be a coninuous random operaor wih sochasic domain C such ha for every w Ω, {y C(w) : T (w)y = y} = 6. Then T has a sochasic fixed poin. Le y be a mapping of J Ω ino X. y is said o be a sochasic process if for each J, he funcion y(, ) is measurable. Now le us recall some fundamenal facs of he noion of Kuraowski measure of noncompacness. Definiion 2.9. [6] Le E be a Banach space and ΩE he bounded subses of E. The Kuraowski measure of noncompacness is he map α : ΩE [, ) defined by α(b) = inf{ > : B ni=1 Bi and diam(bi ) }; here B ΩE. The Kuraowski measure of noncompacness saisfies he following properies (for more deails see [6]). (a) α(b) = B is compac (B is relaively compac). (b) α(b) = α(b). (c) A B = α(a) α(b). (d) α(a + B) α(a) + α(b). (e) α(cb) = c α(b); c IR (f) α(convb) = α(b). Theorem 2.1. (M onch)[[1, 22]] Le D be a bounded, closed and convex subse of a Banach space such ha D, and le N be a coninuous mapping of D ino iself. If he implicaion V = convn (V ) or V = N (V ) = α(v ) = holds for every subse V of D, hen N has a fixed poin. Lemma 2.11 ([11]). If H C(J, E) is bounded and equiconinuous, hen α(h()) is coninuous on J and α x(s)ds : x H α(h(s))ds, J J where H(s) = {x(s) : x H}, J Lemma 2.12 ([11]). Le D be a bounded, closed and convex subse of Banach space X. If he operaor N : D D is a sric se conracion, i.e here is a consan λ < 1 such ha α(n (S)) λα(s) for any bounded se S D, hen N has a fixed poin in D. 3. Exisence of mild soluions Now we give our main exisence resul for problem (1.1)-(1.2). Before saring and proving his resul, we give he definiion of he mild random soluion. Definiion 3.1. A sochasic process y : J Ω E is said o be random mild soluion of problem (1.1)-(1.2) if y(, w) = φ(), (, ] and he resricion of y(., w) o he inerval J is coninuous and saisfies he following inegral equaion: 87

5 (3.1) T ( s)f (s, yρ(s,ys ) (, w), w)ds, J. y(, w) = T ()φ(, w) + ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p Se R(ρ ) = {ρ(s, ϕ) : (s, ϕ) J B, ρ(s, ϕ) }. We always assume ha ρ : J B (, T ] is coninuous. Addiionally, we inroduce following hypohesis: (Hφ ) The funcion φ is coninuous from R(ρ ) ino B and here exiss a coninuous and bounded funcion Lφ : R(ρ ) (, ) such ha kφ kb Lφ ()kφkb for every R(ρ ). Remark 3.2. The condiion (Hφ ), is frequenly verified by funcions coninuous and bounded. For more deails, see for insance [18]. Lemma 3.3. ([15], Lemma 2.4) If y : (, T ] E is a funcion such ha y = φ, hen kys kb (MT + Lφ )kφkb + KT sup{ y(θ) ; θ [, max{, s}]}, s R(ρ ) J, where Lφ = sup Lφ (). R(ρ ) We will need o inroduce he following hypoheses which are assumed here afer: (H1 ) The operaor soluion T () J is uniformly coninuous for >. Le M = sup{kt kb(e) : }. (H2 ) The funcion f : J B Ω E is Carah eodory. (H3 ) There exis a funcion ψ : J Ω R+ and p : J Ω R+ such ha for each w Ω, ψ(., w) is a coninuous nondecreasing funcion and p(., w) inegrable wih: f (, u, w) p(, w) ψ(kukb, w) for a.e. J and each u B. (H4 ) There exiss a funcion L : J Ω R+ wih L(., w) L1 (J, R+ ) for each w Ω such ha for any bounded B E α(f (, B, w)) l(, w)α(b). (H5 ) There exis a random funcion R : Ω R+ \{} such ha: T M kφkb + M ψ (MT + Lφ )kφkb + KT R(w), w p(s, w)ds R(w). (H6 ) For each w Ω, φ(., w) is coninuous and for each, φ(,.) is measurable. Theorem 3.4. Suppose ha hypoheses (Hφ ) and (H1 ) (H6 ) are valid, hen he problem (1.1)-(1.2) has a leas one mild random soluion on (, T ]. Proof. Le Y = {u C(J, E) : u(, w) = φ(, w) = } endowed wih he uniform convergence opology and N : Ω Y Y be he random operaor defined by (3.2) (N (w)y)() = T () φ(, w) + T ( s) f (s, y ρ(s,ys ), w) ds, J, where y : (, T ] Ω E is such ha y (, w) = φ(, w) and y (, w) = y(, w) on J. Le φ : (, T ] w) = φ(, w) = on J. Ω E be he exension of φ o (, T ] such ha φ(θ, Then we show ha he mapping defined by (3.2) is a random operaor. To do his, we need o prove ha for any y Y, N (.)(y) : Ω Y is a random variable. Then we prove ha N (.)(y) : Ω Y is measurable as a mapping f (, y, ), J, y Y is measurable by assumpions (H2 ) and (H6 ). Le D : Ω 2Y be defined by: D(w) = {y Y : kyk R(w)}. 88

6 The se D(w) bounded, closed, convex and solid for all w Ω. Then D is measurable by Lemma 17 in [12]. Le w Ω be fixed. If y D(w), from Lemma 3.3 i follows ha ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p k yρ(, y ) kb (MT + Lφ )kφkb + KT R(w) and for each y D(w), by (H3 ) and (H5 ), we have for each J (N (w)y)() M kφkb + M f (s, y ρ(s,ys ), w) ds M kφkb + M p(s, w) ψ ky ρ(s,ys ) kb, w ds p(s, w) ψ (MT + Lφ )kφkb + KT R(w), w ds M kφkb + M φ M kφkb + M ψ (MT + L )kφkb + KT R(w), w T p(s, w)ds R(w). This implies ha N is a random operaor wih sochasic domain D and N (w) : D(w) D(w) for each w Ω. Sep 1: N is coninuous. Le y n be a sequence such ha y n y in Y. Then (N (w)y n )() (N (w)y)() = + T ()φ(, w) i h T ( s) f (s, y n ρ(s,yn s ), w) f (s, y ρ(s,ys ), w) ds M f (s, y n ρ(s,yn s ), w) f (s, y ρ(s,ys ), w) ds. Since f (s,, w) is coninuous, we have by he Lebesgue dominaed convergence heorem (N (w)y n )() (N (w)y)() as n +. Thus N is coninuous. Sep 2: We prove ha for every w Ω, {y D(w) : N (w)y = y} = 6. For his we apply he M onch fixed poin heorem. 89

7 (a) N maps bounded ses ino equiconinuous ses in D(w). Le τ1, τ2 [, T ] wih τ2 > τ1, D(w) be a bounded se as in Sep 2, and y D(w). Then (N (w)y)(τ2 ) (N (w)y)(τ1 ) + T (τ2 ) T (τ1 ) kφkb τ1 [T (τ2 s) T (τ1 s)]f (s, y ρ(s,ys ), w)ds + ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p τ2 τ1 + T (τ2 s)f (s, y ρ(s,ys ), w)ds T (τ2 ) T (τ1 ) kφkb τ1 T (τ2 s) T (τ1 s) f (s, y ρ(s,ys ), w) ds τ2 + τ1 T (τ2 s)f (s, y ρ(s,ys ), w) ds T (τ2 ) T (τ1 ) kφkb + ψ (MT + Lφ )kφkb + KT R(w) τ1 T (τ2 s) T (τ1 s) p(s, w)ds τ2 φ + M ψ (MT + L )kφkb + KT R(w), w p(s, w)ds. τ1 The righ-hand of he above inequaliy ends o zero as τ2 τ1, since T () is uniformly coninuous. Nex, le w Ω be fixed (herefore we do no wrie w in he sequel) bu arbirary. (b) Now le V be a subse of D(w) such ha V conv (N (V ) {}). V is bounded and equiconinuous and herefore he funcion v v() = α(v ()) is coninuous on (, T ]. By (H4 ), Lemma 2.11 and he properies of he measure α we have for each (, T ] v() α (N (V )) () {}) α (N (V ()) α T () φ() + T ( s) f (s, y ρ(s,ys ) ) ds α T () φ() + α T ( s) f (s, y ρ(s,ys ) ) ds n o M l(s)α( y ρ(s,ys ) : y V )ds M l(s)k(s) sup α(v (τ ))ds τ s l(s)k(s)α(v (s))ds M v(s) l(s)k(s)ds = M l(s)k(s)v(s)ds. Gronwall s lemma implies ha v() = for each J, and hen V () is relaively compac in E. In view of he Ascoli-Arzel` a heorem, V is relaively compac in D(w). Applying now Theorem 2.1 we 9

8 T conclude ha N has a fixed poin y(w) D(w). Since w Ω D(w) 6=, he hypohesis ha a measurable selecor of ind exiss holds. By Lemma 2.8, he random operaor N has a sochasic fixed poin y (w), which is a mild soluion of he random problem (1.1)-(1.2). ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p Proposiion 3.5. Assume ha (Hφ ), (H1 ), (H2 ), (H5 ), (H6 ) are saisfied, hen a sligh modificaion of he proof (i.e. use he Darbo s fixed poin heorem) guaranees ha (H4 ) could be replaced by (H4 ) There exiss a nonnegaive funcion l(., w) L1 (J, IR+ ) for each w Ω, such ha α(f (, B, w)) l(, w)α(b), J. Proof. Consider he Kuraowski measure of noncompacness αc defined on he family of bounded subses of he space C(J, E) by αc (H) = sup e τ L() α(h()), J e l(s)ds, e l() = M l()k(), τ > 1. where L() = We show ha he operaor N : D(w) D(w) is a sric se conracion for each w Ω. We know ha N : D(w) D(w) is bounded and coninuous, we need o prove ha here exiss a consan λ < 1 such ha αc (N H) λαc (H) for H D(w). For each J we have α((n H)()) M α(f (s, y ρ(s,ys ), w)) : y H)ds. This implies by (H4 ) and Theorem 2.1 in [13] α((n H)()) M l(s)α(y ρ(s,ys ) : y H)ds M l(s)k(s) sup α(h(τ ))ds τ s M l(s)k(s)α(h(s))ds = e l(s)α(h(s))ds eτ L(s) e τ L(s)e l(s)α(h(s))ds = e l(s) eτ L(s) sup e τ L(s) α(h(s))ds s [,] sup e τ L() α(h()) [,T ] e l(s)eτ L(s) ds τ L(s) e ds = αc (H) τ 1 αc (H) eτ L(). τ Therefore, 1 αc (H). τ So, he operaor N is a se conracion. As a consequence of Theorem 2.12, we deduce ha N has a T fixed poin y(w) D(w). Since w Ω D(w) 6=, he hypohesis ha a measurable selecor of ind exiss holds. By Lemma 2.8, he random operaor N has a sochasic fixed poin y (w), which is a mild soluion of he random problem (1.1)-(1.2). αc (N H) 91

9 4. An example Consider he following funcional parial differenial equaion: 2 z(, x, w) + C (w)b() F (z( + σ(, z( + s, x, w)), x, w))ds, (4.1) z(, x, w) = x2 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p x [, π], [, T ], w Ω (4.2) z(,, w) = z(, π, w) =, [, T ], w Ω (4.3) z(s, x, w) = z (s, x, w), s (, ], x [, π], w Ω, where C are a real-valued random variable, b L1 (J; R+ ), F : R R is coninuous, z :), ] [, π] Ω R and σ : J R R are given funcions. Suppose ha E = L2 [, π], (Ω, F, P ) is a complee probabiliy space. Define he operaor A : E E by Av = v wih domain D(A) = {v E, v, v are absoluely coninuous, v E, v() = v(π) = }. Then Av = X n2 (v, vn )vn, v D(A) n=1 q 2 π where ωn (s) = sin ns, n = 1, 2,... is he orhogonal se of eigenvecors in A. I is well know (see [23]) ha A is he infiniesimal generaor of an analyic semigroup T (), in E and is given by T ()v = X exp( n2 )(v, vn )vn, v E. n=1 Since he analyic semigroup T () is compac, here exiss a posiive consan M such ha kt ()kb(e) M. Le B = BCU (IR ; E) be he space of uniformly bounded coninuous funcions endowed wih he following norm: kφk = sup φ(s), for φ B. s If we pu φ BCU (IR ; E), x [, π] and w Ω y(, x, w) = z(, x, w), [, T ] φ(s, x, w) = z (s, x, w), s (, ], x [, π], w Ω. Se f (, φ(x), w) = C (w)b() F (z( + σ(, z( + s, x, w)), x, w))ds, and ρ(, φ)(x) = σ(, z(, x, w)). Le φ B be such ha (Hφ ) holds, and le φ be coninuous on R(ρ ), and le f saisfies he condiions (H3 ), (H4 ), (H5 ) Then he problem (1.1)-(1.2) in an absrac formulaion of he problem (4.1)-(4.3), and condiions (H1 ) (H6 ) are saisfied. Theorem 3.4 implies ha he random problem (4.1)-(4.3) has a leas one random mild soluion. Acknowledgemen. The auhors are graeful o he referee for he careful reading of he paper. 92

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11 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 215, VOLUME 5, ISSUE 1, p [25] D. Sanescu, and B.M. Chen-Charpenier, Random coefficien differenial equaion models for bacerial growh, Mah. Compu. Model. 5 (29), [26] X. Yang, and J. Cao, Synchronizaion of delayed complex dynamical neworks wih impulsive and sochasic effecs, Nonlinear. Anal, Real World Appl. 12, (211), [27] K. Yosida, Funcional Analysis, 6h ed. Springer-Verlag, Berlin, 198. `s, PO Box 89, 22 Sidi BelLaboraory of Mahemaics, Universiy of Sidi Bel-Abbe `s, Algeria Abbe address: amel2222@yahoo.com `s, PO Box 89, 22 Sidi BelLaboraory of Mahemaics, Universiy of Sidi Bel-Abbe `s, Algeria Abbe address: benchohra@univ-sba.dz Deparmen of Mahemaics, King Abdulaziz Universiy, P.O. Box 823, Jeddah 21589, Saudi Arabia 94