ON FRACTIONAL RANDOM DIFFERENTIAL EQUATIONS WITH DELAY. Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong

Opuscula Mah. 36, no. 4 (26), 54 556 hp://dx.doi.org/.7494/opmah.26.36.4.54 Opuscula Mahemaica ON FRACTIONAL RANDOM DIFFERENTIAL EQUATIONS WITH DELAY Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Communicaed by Palle E.T. Jorgensen Absrac. In his paper, we consider he exisence and uniqueness of soluions of he fracional random differenial equaions wih delay. Moreover, some kind of boundedness of he soluion is proven. Finally, he applicabiliy of he heoreical resuls is illusraed wih some real world examples. Keywor: sample pah fracional inegral, sample pah fracional derivaive, fracional differenial equaions, sample fracional random differenial equaions, Capuo fracional derivaive, delay. Mahemaics Subjec Classificaion: 26A33, 47H4, 6H25.. INTRODUCTION The heory of fracional differenial equaions is an imporan branch of differenial equaion heory, which has been applied in physics, chemisry, biology and engineering and has emerged an imporan area of invesigaion in he las few decades. For some fundamenal resuls in he heory of fracional calculus and fracional differenial equaions (see [4, 6, 8 2]). Random differenial equaions and random inegral equaions have been sudied sysemaically in Ladde and Lakshmikanham [7] and Bharucha-Reid [3], respecively. They are good models in various branches of science and engineering since random facors and uncerainies have been aken ino consideraion. Hence, he sudy of he fracional differenial equaions wih random parameers seem o be a naural one. We refer he reader o he monographs [3, 7, 26, 27], he papers [6 9, 2, 4] and he references herein. Iniial value problems for fracional differenial equaions wih random parameers have been sudied by V. Lupulescu and S.K. Nouyas [24]. The basic ool in he sudy of he problems for random fracional differenial equaions is o rea i as a fracional differenial equaion in some appropriae Banach space. In [25], auhors proved he exisence resuls for a random fracional equaion under a Carahéodory c AGH Universiy of Science and Technology Press, Krakow 26 54

542 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong condiion. Exisence resuls for exremal random soluion are also proved. For several oher research resuls see [22, 23]. Coninuing he work of he auhors, in his paper, we consider he fracional random differenial equaions wih delay as follows: D α [,a], P a.e. x(, ω) = f ω (, x ), x(, ω) [ σ,] (.) = ϕ(, ω). The random fracional differenial equaions wih delay (.) is no new o he heory of random differenial equaions. When he random parameer ω is absen, he random equaion (.) reduces o he fracional differenial equaions wih delay, { D α x() = f(, x ) for [, a], (.2) x() = ϕ() for [ σ, ]. The classical fracional differenial equaions wih delay (.2) has been sudied in he lieraure by several auhors. See for example, M. Benchohra, e al. [2], J. Deng, e al. [5] and he references herein. In his paper, inspired and moivaed by Lupulescu e al. [22 25] and Dhage [6 9], under he condiion of he Lipschizean righ-hand side we obain he exisence and uniqueness of he soluions of he fracional random differenial equaions wih delay. To prove his asserion we use an idea of successive approximaions which has been applied in [2] for he fracional differenial equaions for his problem. The paper is organized as follows. In Secion 2, we se up he appropriae framework on random processes and on fracional calculus. In Secion 3, we prove he exisence and uniqueness of soluions of he fracional random differenial equaions wih delay. Moreover, some kind of boundedness of he soluion is proven. Finally, in Secion 4, an example is given o illusrae our resuls 2. PRELIMINARIES In his secion, we give some noaions and properies relaed o he sample pah fracional inegral, he sample pah fracional derivaive and he meric space of random elemens. The reader can see he deailed resuls in he monographs [7, 26, 27], he papers [24] and he references herein. Le (Ω, A, P) be a complee probabiliy space. Le ([, a], L, λ) be a Lebesguemeasure space where a > and le x(, ) : [, a] Ω R d be a produc measurable funcion. A mapping x(, ) : [, a] Ω R d is called sample pah Lebesgue inegrable on [, a] if x(, ω) : [, a] R d is Legesgue inegrable on [, a] for a.e. ω Ω. Le α >. If x(, ) : [, a] Ω R d is sample pah Lebesgue inegrable on [, a], hen he fracional inegral I α x(, ω) = x(s, ω)

On fracional random differenial equaions wih delay 543 which will be called he sample pah fracional inegral of x, where Γ is Euler s Gamma funcion. If x(, ω) : [, a] R d is Lebesgue inegrable on [, a] for a.e. ω Ω, hen I α x(, ω) is also Lebesgue inegrable on [, a] for a.e. ω Ω. A mapping x(, ) : [, a] Ω R d is said o be a Carahéodory funcion if x(, ω) is coninuous for a.e. ω Ω and ω x(, ω) is measurable for each [, a]. We recall ha a Carahéodory funcion is a produc measurable funcion (see [3]), hen funcion (, ω) I α x(, ω) is also a Carahéodory funcion (see [24]). A mapping x(, ) : [, a] Ω R d is said o have a sample pah derivaive a (, a) if he funcion x(, ω) has a derivaive a for a.e. ω Ω. We will denoe by d d x(, ω) or by x (, ω) he sample pah derivaive of x(, ω) a. We say ha x(, ) : [, a] Ω R d is sample pah differeniable on [, a] if x(, ) has a sample pah derivaive for each (, a) and possesses a one-sided sample pah derivaive a he end poins and a. If x(, ) : [, a] Ω R d is sample pah absoluely coninuous on [, a], ha is, x(, ω) is absoluely coninuous on [, a] for a.e. ω Ω, hen he sample pah derivaive x (, ω) exiss for λ-a.e. [, a]. Le x(, ) : [, a] Ω R d be sample pah absoluely coninuous on [, a] and le α (, ]. Then, for λ-a.e. and for a.e. ω Ω, we define he Capuo sample pah fracional derivaive of x by D α x(, ω) = I x (, ω) = Γ( α) x (s, ω) α. Obviously, if x(, ) : [, a] Ω R is sample pah differeniable on [, a] and x (, ω) is coninuous on [, a], hen D α x(, ω) exiss for every [, a] and D α x(, ω) is coninuous on [, a]. If x(, ) : [, a] Ω R d is a Carahéodory funcion, hen D α I α x(, ω) = x(, ω) for all [, a] and for a.e. ω Ω. If x(, ) : [, a] Ω R is sample pah absoluely coninuous on [, a], hen I α D α x(, ω) = x(, ω) x(, ω) (2.) for λ-a.e. and for a.e. ω Ω. If x(, ) : [, a] Ω R d is sample pah differeniable on [, a] and x (, ω) is coninuous on [, a], hen (2.) hol for all [, a] and for a.e. ω Ω. If x(, ) : [, a] Ω R d is sample pah absoluely coninuous on [, a], hen y(, ω) = Γ( α) x(s, ω) α

544 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong is also sample pah absoluely coninuous on [, a]. Moreover, for λ-a.e. and for a.e. ω Ω and a.e. ω Ω, we have ha d y(, ω) = Γ( α) d x(s, ω) α = x(, ω) Dα x(, ω) + α Γ( α). Le he space C = C([, a], R d ) of all coninuous funcions from [, a] from R d endowed wih he uniform meric d(x, y) = sup x() y(), [,a] where is he Euclidean norm on R d. If x : Ω C is a random elemen, hen he funcion x(, ) : [, a] Ω R d is a Carahéodory funcion. If x : Ω C is a random elemen, hen he funcion x(, ω) : [, a] R d is said o be a realizaion or a rajecory of he random elemen x, corresponding o he oucome ω Ω. Le M(C) be he space of all probabiliy measures on B(C). If x : Ω C is a random elemen in C, hen he probabiliy measure µ x, defined by µ x = P(x (B)) = P({ω Ω, x(ω) B}), B B(C), is called he disribuion of x. Since C has been a complee and separable meric space, hen i is well known ha M(C) is a complee and a separable meric space wih respec o he Prohorov meric D : M(C) [, ) given by D(µ, η) = inf{ɛ >, µ(b) η(b ɛ ) + ɛ}, B B(C), where B α = {x C; inf d(x, y) < ɛ}. y B Le R(Ω, C) be he meric space of all random elemens in C. A sequence of random variables {x n } R(Ω, C) is said o converge almos everywhere (a.e.) o x R(Ω, C) if here exiss N R such ha P(N) =, and lim n d(x n (ω), x(ω)) = for every ω Ω \ N. We use he noaion x n x a.e. for almos everywhere convergence. If {x n } R(Ω, C) is a ρ convergen sequence, hen i is known ha {x n } is a ρ-cauchy sequence, where ρ(x, y) = D(µ, η) is a meric on he se of random elemens in C (see []). Lemma 2. (see [24]). Le e n : [, a] Ω R d be a sequence of Carahéodory posiive funcions and le u, v be posiive consans such ha e (, ω) u for every n, e n (, ω) u + v for all [, a] and ω Ω. Then for every n, where E α = k= z k Γ(kα + ) e n (s, ω) e n (, ω) ue α (va α ), ω Ω, is he Miag-Leffler funcion.

On fracional random differenial equaions wih delay 545 For a posiive number σ, we denoe by C σ he space C([ σ, ], R d ). Also, we denoe by x y Cσ = sup s [ σ,] x y he meric on he space C σ. Le x( ) C([ σ, ), R d ). Then, for each [, )we denoe by x he elemen of C σ defined by x (s) = x( + s) for s [ σ, ]. 3. MAIN RESULTS In his secion, we consider he fracional random differenial equaions wih delay as follows : D α [,a], P a.e. x(, ω) = f ω (, x ), x(, ω) [ σ,] (3.) = ϕ(, ω). where x : Ω R d is a random vecor, D α x is he Capuo fracional derivaive of x wih respec o he variable, and f : Ω [, a] C σ R d is a given funcion. Lemma 3.. Le a (λ P) - measurable funcion x : [ σ, a] Ω R d be a sample pah Lebesgue inegrable on [ σ, a]. Then x(, ω) is a sample soluion of (3.) if and only if x(, ω) is a coninuous on [ σ, a] for P a.e. ω Ω and i saisfies he following random inegral equaion: x(, ω) [ σ,] = ϕ(, ω), [,a], P a.e. x(, ω) = ϕ(, ω) + f ω (s, x s ). (3.2) We shall consider he fracional random differenial equaions wih delay (3.) assuming ha he following assumpions are saisfied. (f) The mapping f ω (, ) : [, a] C σ R d are measurable and coninuous for each ω Ω. (f2) There exiss a Carahéodory funcion L : [, a] Ω R d such ha f ω (, Φ) f ω (, Ψ) L(, ω) Φ Ψ Cσ for every [, a], Φ, Ψ C σ and P a.e. ω Ω. (f3) There exiss a non-negaive consan M such ha f ω (, Φ) M for every [, a], Φ C σ and P a.e. ω Ω. Theorem 3.2. Le f : Ω [, a] C σ R d saisfy he assumpions (f) (f2). Then he fracional random differenial equaions wih delay (3.) has a unique soluion on [, a].

546 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Proof. To prove he heorem we apply he mehod of successive approximaions. So, we define he funcions x n : [ σ, a] R d, n =,, 2, 3,... as follows: for every [, a], every ω Ω le us pu { x ϕ(, ω) for [ σ, ], (, ω) = (3.3) ϕ(, ω) for [, a]. and ϕ(, ω) for [ σ, ], x n (, ω) = ϕ(, ω) + f ω (s, x n s ) for [, a]. (3.4) For n = and every [, a], every ω Ω, we have x (, ω) x (, ω) f ω (s, x (ω)) In paricular, M Mα Γ( + α) Maα <. (3.5) Γ( + α) x 2 (, ω) x (, ω) fω (s, x s) f ω (s, x (ω)) = ˆL(ω) = ˆL(ω) where ˆL(ω) = sup L(, ω). [,a] Furher, if we assume ha ˆL(ω)M Γ( + α) x s(, ω)) x (, ω) Cσ sup x (r, ω)) x (r, ω) s [ σ,] ˆL(ω)M Γ( + 2α) a2α, [, a], s α ˆL(ω)M Γ( + 2α) 2α x n (, ω) x n (, ω) nα M[ˆL(ω)] Γ( + nα) nα ˆM[L(ω)a], [, a], Γ( + nα)

On fracional random differenial equaions wih delay 547 hen we have x n+ (, ω) x n (, ω) ˆL(ω) M[ˆL(ω)s] nα Γ( + nα) (n+)α (n+)α M[ˆL(ω)] M[ˆL(ω)a] Γ( + (n + )α) Γ( + (n + )α) (3.6) for every [, a] and P a.e. ω Ω. For every n =,, 2,..., he funcions x n (, ω) : [ σ, a] R d are coninuous on [ σ, a] for every ω Ω. Indeed, since ϕ C σ, x (, ω) is coninuous on [ σ, a] for every ω Ω. Nex, we assume ha x k (, ω), k =,, 2,..., n, are coninuous on [ σ, a] for every ω Ω. Then for [ σ, a] and h > small enough such ha + h (, a] we have x n ( + h, ω) x n (, ω) = + +h +h f ω (s, x n s ) ( + h s) f ω (s, x n s ) ( + h s) f ω (s, x n s ) ( + h s) fω (s, x n s ). (3.7) On he oher hand, by assumpions (f2)-(f3) we have f ω (, x n ) f ω (, x n ) f ω (, x ) + f ω (, x ) M + ˆL(ω) x n (, ω) x (, ω) Cσ = M + ˆL(ω) sup x n (r, ω) x (r, ω) r [ σ,] = M + ˆL(ω) sup r [ σ,] = M + ˆL(ω) sup s [ σ,] M + ˆL(ω) x n ( + r, ω) x ( + r, ω) x n (s, ω) x (s, ω) For every n, every [, a] and every ω Ω, we ge sup [,a] fω (, x n ) ˆL(ω) M + sup fω (γ, x n 2 (γ, ω)). γ [,s] sup fω (γ, x n 2 (γ, ω)). (3.8) γ [,s]

548 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Using Lemma 4. of [24] and (3.8), we obain f ω (, x n ) ME α (ˆL(ω)a α ). (3.9) From (3.7) and (3.9) we ge x n ( + h, ω) x n (, ω) ME α (ˆL(ω)a α ) + ME α (ˆL(ω)a α ) ( + h s) +h ME α(ˆl(ω)a α ) ( 2h α + ( + h) α α ) Γ( + α) P.a.e as h +. Similarly, for [, a] and h > small enough such ha h [, a) we obain ha x n ( + h, ω) x n (, ω) P.a.e as h +. Therefore, for n =,, 2,... he funcion x n (, ω) : [ σ, a] Ω R d is coninuous on [, a] for every ω Ω. Since ϕ C σ is a random variable and for [, a], he mapping ω f ω(s, x n s ) are measurable for n =,, 2,.... From he assumpions (f) and (f2), i follows ha x n (, ) are Carahéodory funcions. Now for any n =,, 2,... and [ σ, ] we shall show ha he sequence {x (, ω)} is a Cauchy sequence uniformly on he variable wih P a.e. and hen {x n (, ω)} is uniformly convergen for all ω Ω. For n > m, from (3.6) we have sup x n (, ω) x m (, ω) [,a] On he oher hand, by he inequaliy, i follows ha n sup k=m [,a] M n k=m Γ((k + )α) k!α 2k Γ k+ (α), k, x k+ (, ω) x k (, ω) [ˆL(ω)a] (k+)α Γ( + (k + )α). (3.) [ˆL(ω)a] (k+)α Γ( + (k + )α) [ˆL(ω)a] (k+)α (k + )!α 2k+ Γ k+ (α) = α [ [ˆL(ω)a] α ] k+. (3.) (k + )! αγ( + α) From (3.) and (3.) we ge sup x n (, ω) x m (, ω) αm [,a] n k=m [ [ˆL(ω)a] α ] k+. (k + )! αγ( + α)

On fracional random differenial equaions wih delay 549 The convergence of he series [ k= [ α k ˆL(ω)a] k! αγ(+α)] implies ha for any ɛ > we find n N large enough such ha for n, m n, sup x n (, ω) x m (, ω) ɛ. (3.2) [,a] Therefore, he sequence x n (, ω) is uniformly convergen on [, a] for all ω Ω. For ω Ω denoe is limi by ˆx(, ω). Define x : [ σ, a] Ω R d by x(, ω) = ϕ(, ω) for [ σ, ] and x(, ω) = ˆx(, ω) for [, a]. Since ϕ C σ is a random variable and x n (, ω) x(, ω) P.a.e as n for [, a], we see [ σ, a] and x(, ω) is a measurable funcion. Hence, he funcion x : [ σ, a] Ω R d is a Carahéodory funcion. We shall show ha x(, ) is a soluion of he fracional random inegral equaion (3.2). Le n N. For any ɛ >, here exis n large enough such ha for every n n we derive f ω (, x n ) f ω (, x ) L(, ω) x n (, ω) x (, ω) Cσ = ˆL(ω) sup x n (r, ω) x (r, ω) r [ σ,] = ˆL(ω) sup x n ( + r, ω) x( + r, ω) r [ σ,] = ˆL(ω) sup x n (γ, ω) x(γ, ω) γ [ σ,] = ˆL(ω) sup x n (γ, ω) x(γ, ω) ɛ γ [,] for any [, a], because he sequence {x n (, ω)} is uniformly convergen on [, a] for all ω Ω. Thus, for any [, a] we have f ω (s, x n s ) f ω (s, x s ) ˆL(ω) f ω (s, x n s ) f ω (s, x s ) sup x n (γ, ω) x(γ, ω) γ [,s] ˆL(ω)a α Γ( + α) sup x n (γ, ω) x(γ, ω). γ [,] By he Lebesgue dominaed convergence heorem, we infer ha f ω (s, x n s ) f ω (s, x s ) P.a.e, as n, for all [, a] and ω Ω.

55 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Consequenly, we have x(, ω) ϕ(, ω) f ω (s, x s ) x n (, ω) x(, ω) + xn (, ω) ϕ(, ω) f ω (s, x n + f ω (s, x n s ) f ω (s, x s ). s ) Thus, in view of he wo previous convergences and he fac ha he second erm of he righ-hand side is equal o zero, one obains x(, ω) = ϕ(, ω) + f ω (s, x s ) for all [, a] and ω Ω. Therefore, x(, ω) is a soluion of problem (3.). For he uniqueness of he soluion, le us assume ha x, y : [ σ, a] Ω R d are wo Carahéodory funcions which are soluions of he problem (3.). Then we have x(, ω) y(, ω) = If we ake ˆL(ω) = ˆL(ω) = ˆL(ω) = ˆL(ω) ξ(s, ω) = f ω (s, x s ) f ω (s, x s ) f ω (s, y s ) x s (, ω) y s (, ω) Cσ f ω (s, y s ) sup x s (r, ω) y s (r, ω) r [ σ,] sup x(s + r, ω) y(s + r, ω) r [ σ,] sup x(γ, ω) y(γ, ω). (3.3) γ [s σ,s] sup x(γ, ω) y(γ, ω) γ [s σ,s]

On fracional random differenial equaions wih delay 55 for all s [, ] and ω Ω, hen from (3.3) we see ha ξ(, ω) ˆL(ω) Applying Lemma 2. we obain ha ξ(s, ω). x(, ω) y(, ω) =, for all [, a] and ω Ω, which complees he proof. The nex wo heorems presen boundedness ype resuls for he soluions of (3.). Theorem 3.3. Suppose ha he funcion f : Ω [, a] C σ R d and ϕ, ˆϕ C σ saisfy he assumpions as in Theorem 3.2. Then he soluion x o he problem (3.) saisfies ( Ma α ) x(, ω) ϕ(, ω) E α for all s [, ] and ω Ω. Proof. Since for all [, a] and ω Ω, x(, ω) = ϕ(, ω) + ϕ(, ω) + ϕ(, ω) + ϕ(, ω) + M M M f ω (s, x s ) x s (, ω) Cσ sup x(s + r, ω) r [ σ,] sup x(γ, ω). γ [s σ,s] Applying Lemma 2., for all s [, ] and ω Ω, we obain ha ( Ma α ) x(, ω) ϕ(, ω) E α, where E α (z) is he Miag- Leffler funcion. This proof is compleed. Theorem 3.4. Suppose ha he funcion f : Ω [, a] C σ R d and ϕ C σ saisfy he assumpions as in Theorem 3.2. Then he soluion x and y are soluions of he problem (3.) wih x(, ω) = ϕ(, ω) and y(, ω) = ˆϕ(, ω) for [ σ, ], saisfies ( α ˆL(ω)a ) x(, ω) y(, ω) ϕ ˆϕ E α for all s [, ] and ω Ω.

552 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Proof. Noe ha for all [, a] and ω Ω, x(, ω) y(, ω) = ϕ(, ω) + ϕ(, ω) ˆϕ(, ω) Cσ + ˆL(ω) ϕ(, ω) ˆϕ(, ω) Cσ + ˆL(ω) ϕ ˆϕ + ˆL(ω) f ω (s, x s ) ˆϕ(, ω) f ω (s, y s ) x s (, ω) y s (, ω) Cσ sup x(s + r, ω) y(s + r, ω) r [ σ,] sup x(γ, ω) y(γ, ω). γ [s σ,s] Applying Lemma 2., for all s [, ] and ω Ω, we obain ha x(, ω) y(, ω) ϕ ˆϕ E α ( ˆL(ω)a α where E α (z) is he Miag- Leffler funcion. This proof is complee. ), 4. ILLUSTRATIVE EXAMPLES In his secion we give some examples o illusrae he usefulness of our main resuls. Example 4.. Le us consider he class of delay random fracional differenial equaions wih disribued delay. For m N and < σ < σ 2 <... < σ m < σ. Consider he random fracional differenial equaions wih delay as follows: x(, ω) = ϕ(, ω) for [ σ, ], D α x(, ω) = σ g,ω (s, x( + s, ω)) + m g i,ω (, x( σ i, ω)) for [, a], (4.) i= where g i : Ω [, a] C σ R d, i =,, 2,..., m are some random mapping, Ω is a complee probabiliy space and C σ is he space C([ σ, ] Ω, R d ). Assume ha g i,ω : [, a] C σ R d saisfy he following assumpions: (g) The mapping g i,ω (, ) : [, a] C σ R d are measurable and coninuous, for every (, ϕ ) [, a] C σ, each ω Ω;

On fracional random differenial equaions wih delay 553 (g2) There exiss a Carahéodory funcion L i : [, a] Ω R d such ha g i,ω (, Φ) g i,ω (, Ψ) L i (, ω) Φ Ψ Cσ for i =,, 2,..., m, every [, a], Φ, Ψ C σ and P a.e. ω Ω; (g3) There exiss some non-negaive consans M i such ha g i,ω (, Φ) M i for i =,, 2,..., m, every [, a], Φ C σ and P a.e. ω Ω. Observe ha, if g i,ω : [, a] C σ R d saisfy he following assumpions (g) (g3), hen he problem (4.) has a unique soluion. Indeed, le he mapping g : Ω [, b] C σ R d given by g ω (, ϕ) = σ g,ω (τ, ϕ(τ))dτ + m g i,ω (, ϕ( σ i )) saisfies assumpions (f), (f2) and (f3). From assumpion (g) one can infer ha assumpion (f) is saisfied. And by assumpion (g2), we have where g ω (, ϕ) g ω (, ψ) L(, ω) = σ + σ i= g,ω (τ, ϕ(τ)) g,ω (τ, ψ(τ)) dτ m g i,ω (, ϕ( σ i )) g i,ω (, ψ( σ i )) i= L(, ω) ϕ ψ Cα, sup τ [ σ,] L (τ, ω) + m L i (, ω). Hence, g saisfies (f2). Moreover, by assumpion (g3), we ge g ω (, ϕ) σ g,ω (τ, ϕ(τ)) dτ + where M = σm + m M i. So, g saisfies (f3). i= i= m g i,ω (, ϕ( σ i ) M, Example 4.2. In he classical populaion models, i is considered ha he birh rae changes immediaely as soon as a change in he number of individuals is produced. However, he members of he populaion mus reach a cerain degree of developmen o give birh o new individuals and his suggess inroducing a delay erm ino he i=

554 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong sysem. Now, le (Ω, A, P) be complee probabiliy space measure space, we consider a random ime-delay Malhusian model as follows: x(, ω) = ϕ(, ω) = ( + ω 2 ),, D α x(, ω) = ω2 x(, ω),, (4.2) + ω2 where ω symbolizes a random facor and x : [, ) Ω R is he populaion a ime. Le us he mapping f ω (, ) : [, ) C σ R given by f ω (, x ) = ω2 x(, ω) for all ω Ω. + ω2 I is easy o check ha f ω (, x ) saisfies he assumpion (f2) (f3). Indeed, we have (i) for all (, ω) [, a] Ω, fω (, x ) f ω (, y ) ω 2 = + ω 2 x(, ω) ω 2 x(, ω) y(, ω) + ω 2 = sup x(r, ω) y(r, ω) = x y Cσ r [,] i.e. f ω (, ) saisfies he assumpion (f2), where L(, ω) =. (ii) for all (, ω) [, a] Ω, fω (, x ) ω 2 = + ω 2 x(, ω) ω 2 x(, ω) + ω 2 sup x(r, ω) a 2 ( + ω 2 ). r [,] i.e. f ω (, ) saisfies he assumpion (f3), where M = a 2 ( + ω 2 ). Hence, by Theorem 3.2, he problem (4.2) has a random soluion defined on [, ). REFERENCES [] W. Arend, K. Bay, M. Hiber, F. Neubrander, Vecor-valued Laplace Transforms and Cauchy Problems, Monographs in Mahemaics, Vol. 96, Birkhäuser, Basel, 2. [2] M. Benchohra, J. Henderson, S.K. Nouyas, A. Ouahab, Exisence resuls for fracional order funcional differenial equaions wih infinie delay, J. Mah. Anal. Appl. 338 (28), 34 35. [3] A.T. Bharucha-Reid, Random Inegral Equaions, Academic Press, New York, 972. [4] J. Deng, L. Ma, Exisence and uniqueness of soluions of iniial value problems for nonlinear fracional differenial equaions, App. Mah Le. 6 (2), 676 68.

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556 Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong [24] V. Lupulescu, S.K. Nouyas, Random fracional differenial equaions, In. Elecron. J. Pure Appl. Mah. 4 (22), 9 36. [25] V. Lupulescu, D. O Regan, Ghaus ur Rahman, Exisence resuls for random fracional differenial equaions, Opuscula Mah. 34 (24) 4, 83 825. [26] T.T. Soong, Random Dfferenial Equaions in Science and Engineering, Academic Press, New York, 973. [27] C.P. Tsokos, W.J. Padge, Random Inegral Equaions wih Applicaions o Life Sciences and Engineering, Academic Press, New York, 974. [28] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequaliy and is applicaion o a fracional differenial equaion, J. Mah. Anal. Appl. 328 (27), 75 8. Ho Vu (corresponding auhor) hovumah@gmail.com Banking Universiy Faculy of Economic Mahemaics Ho Chi Minh Ciy, Vienam Nguyen Ngoc Phung Nguyen Phuong Received: November 5, 24. Revised: December 24, 25. Acceped: December 24, 25.