Turo, Jan

English

Turo Jan

University of Szeged

Acta Sei. Math. 44 (1982), 321—270 Existence and uniqueness of random solutions of nonlinear stochastic functional integral equations JAN TURO 1. Introduction. Stochastic or random integral equations arise quite often in the engineering, biological, chemical, and physical sciences (see, e.g., [1], [8] and [6]). The object of the present paper is to study a nonlinear stochastic functional integral equation of the type 9(0 (1.1) x{t, co) = F(t, f ft(t, s, x(s, co), co)ds, o 9«) f fi{t, s, x(s, co), co) dw(s, co), x(h(t), co), co) = (Ux)( t , co), o where (i) i €-R+=[0, +=»), and coe£2, the supporting set of a complete probability measure space (£2, J5", P); (ii) x: R+XSi—R is the unknown random function; (iii) F: R+XR3XQ-R and /}: AxRxQ-R, 7=1,2 , are given random functions, where A={(t, s): (iv) g, h: R+ —R+ are given scalar functions; (v) w: R+XQ—R is a Wiener process. The first integral of the stochastic equation (1.1) is to be understood as an ordi-nary Lebesgue integral, while the second integral is an Ito stochastic integral. We shall give sufficient conditions which will ensure the existence and uniqueness of a random solution, a second order stochastic process, of the above stochastic functio-nal integral equation. The tool which we utilize to obtain these results is the compari-son method. This method is based on the convergence of successive approximations produced by a comparison operator associated with the operator U. The abstract form of the comparison method was introduced by W A Z E W S K I [ 1 1 ] in the case of deterministic equations. Received February 2, 1981, and in revised form September 2, 1981. 322 J. Turo Almost all authors use the well-known Banach fixed point theorem or the con-cept of admissibility theory, [1], [8] and [6], in proving the existence and uniqueness of results for cases similar to equation (1.1). Unfortunately these methods involve a strong condition concerning the function F. By the comparison method this condi-tion can be slightly weakened. Consequently in this paper conditions involving some relation between the Lipschitz constants of the function F and the estimations im-posed on the functions g and h appear. Equation (1.1) is a generalization of equations considered by MANOUGIAN, R A O and TSOKOS [6] (if F(t,u1,u2,x,oS)=h(t,x)+ui+u2, f j ( t , s, x, (x>)=kj(t, s, <o) • •(pj(s,x) and g(t) = t, h{t)=t), TURO [10] (if F(t, ult u2, X, co)=F(t, ult x, coj), GIHMAN and SKOROHOD [3], and DOOB [2], among others. 2. Preliminaries. We introduce a family !F„ t£R+, of c-algebras of subsets of i2 with the following properties: (i) J ^ c J ^ , for i1<?2 , J^cJ*" tiR+; (ii) for every t, w(t, co) is !Ft-measurable; (iii) for AsO, the increments w(t+A, co) — w(t, co) are independent (in the pro-babilistic sense) of J5",. D e f i n i t i o n 2.1. We shall denote by C(R+,L2) the space of all continuous maps x: R+ ~+L2{Q,SFt, P) with the topology of uniform convergence on compacta. It may be noted that C(R+, L2) is a locally convex space whose topology is de-fined by the following family of seminorms: ||*||, = sup oOl2]}1'2 osrsn where E denotes the expected value of the random process. D e f i n i t i o n 2.2. A sequence { x j of elements of the space C(R+, Z,2) will be called a Cauchy sequence if for every e>0 and n there exists an N such that for k > N and />iV we have ||xt—x,||„<e. It is clear that the space C(R+,LJ) is complete, that is, every Cauchy sequence of its elements has a limit in C(R+,L2). D e f i n i t i o n 2.3. We shall call x a random solution of the stochastic functional integral equation (1.1) if xeC(R+, L^) and satisfies equation (1.1) P-a.e. With respect to the functions appearing in equation (1.1) we shall assume the following : (i) F(t, M1,M2,X, •) is -measurable fo r each t£R+, uly ut, xzR, and is continuous in t uniformly in mx, w2, x; 323 J. Turo: Existence and uniqueness of random solutions (ii) fj(t,s,x, •), j= 1,2, are ^-measurable for each (f, X€R, and are continuous as maps from A into L2(Q, P) \ (iii) g, h: R+—R+ are continuous and g(i) = ', h(t)^t, t£R+. 3. Somme lemmas. Let us define 9(0 (Kü)(t)= k(t) f u(s)ds, t£R+, o (L«i)(0=/(0«(fc(0), df °° Put Su= £ with the pointwise convergence of the series in R+, where n-o Ln=ZX"_ 1 , « = 1,2,. . . , L°=7 is the identity operator in C(R+,R+), the class of all continuous and nonnegative functions defined on R+. From the definition of the operator L it follows that (L"u)(t) = ln(t)u(hn(t)), where h0(t)=t, hn+1(t)=HK(t)), n = 0 ,1 , . . . , t£R+, /„(0=i, /„+i(0= nKh(t)), n = 0 ,1 , . . . , t£R+. k = 0 Lemma 3.1. ([9], [5]) Assume that (i) k,l,g,h,raC(R+,R+) and g(t), h(t)<í[0,t], tiR+; (ii) s=Sr<°°, s* = Sk*<°°, where k*(t)=k(t)g(t)\ s*(t) (iii) s, s €C(R+, R+) and sup * + * Then (a) there exists w0eC(/?+, R+) which is a unique solution of equation (3.1) u=SKu+Sr in the class Z,)oc of all non-negative and locally integrable functions on R+; (b) the function u0 is the unique solution of the equation (3.2) u = Ku + Lu+r in the class L|OC(m0)= {«: u€Lloc, ||«||o<°°}, where ||«||0=inf {c: u^cu0, c€R+}; (c) the function u = 0 is the unique solution of the inequality (3.3) u^Ku+Lu in the class Lioc(u0). 324 J. Turo P r o o f . First we prove (a). We note that if utLioc and is the solution of equa-tion (3.1), then utC(R+, R+). Thus we shall prove that equation (3.1) has a unique solution in C(R+, R+). We shall obtain a solution first on an arbitrary closed, bounded interval [0, ri\. Let C([0, n], R) be the space of all continuous functions on [0, ri\, where we introduce a norm || • ||+ in the following way: ||M||* = sup C~a ' |«(/)|, where re to,«] t Now we can prove that the operator SK is a contraction in C([0, ri\, R), i.e., ||SK||1. Indeed, from the inequality e"— 1 ^ a e ' for ae[0, 1], teR+, we have 0(*„(O) \\SKu\\* S sup e-A< 2 h m ( h n ( 0 ) / sup |h(s)| ds s [0,n] n —0 g [0,n] Hence it follows that ||5J5T||<1. Now from the Banach fixed point theorem it fol-lows that equation (3.1) has a unique solution woeC([0, n], R+). Since n is arbit-rary, w0 is a unique solution of equation (3.1) on R+. Now we prove (b). It is obvious that the function u0 satisfies equation (3.2). Next we prove that in the class Lloc (w0) the function u0 is the unique solution of equa-tion (3.2). Indeed, if u€Lloc(u0) is a solution of (3.2) then by induction we get u = "z Llh + "z L'Ku + Lnu. ¡=0 ¡=0 Because U€L1oc(M0), there exists c€R+ such that u^cu0, hence L"u^cL"u0. We easily find that L"u0— 0 since «„ is the solution of equation (3.1). As a conse-quence of this L"u~*0, and we infer that ii satisfies (3.1). In view of the uniqueness proved for this equation we conclude u=u0. Finally we prove (c). If M€L1oc(W„) is the solution of inequality (3.3) then we have L"u—0, n—and by induction we get n—1 MS Z L'Ku+L'u, n = 0 , 1 , . . . , ( = 0 325 J. Turo: Existence and uniqueness of random solutions Letting n — °o we get u^SKu, hence we conclude that 0, and the lemma is proved. R e m a r k 3.1. Now we give some effective conditions under which assumption (ii) of Lemma 3.1 is fulfilled. a) If we assume that (3.4) k(t) = const, / (0 ^ I = const, g(t) ^ gt, h(t) S Rt, g, Re[0, 1], and r(t)Srt, t€R+, for some r£R+, then assumption (ii) of Lemma 3.1 is satis-fied provided IR< 1. b) if k(t)^n, I(t)rslt, g(t)^gt, h(t)^Rt, r{t)^rt, K,l,ftR+, g€[0, 1] and /x€[0, 1), teR+, then assumption (ii) of Lemma 3.1 is satisfied. c) Finally, if we suppose (3.4) and r(t)Srtp, t£R+, for some r,peR+, then (ii) of Lemma 3.1 is satisfied provided 1RP< 1. We construct a sequence as follows: (3.5) u„+1 = Kun+Lun, n = 0,1, . . . , where u0 is defined in Lemma 3.1. Lemma 3.2. [4] If the assumptions of Lemma 3.1 are satisfied, then and u„=t 0 for n — where the sign =t denotes uniform convergence in any compact subset of R+. Proof . Relation (3.6) we get by induction. The convergence of the sequence {«„} is implied by (3.6). The limit of this sequence satisfies the inequality (3.3), and by Lemma 3.1 it must be equal to zero identically. The uniform convergence of {«„} follows from Dini's theorem. 4. Main results. In order to prove the existence of a solution of equation (1.1), we define the sequence {*„} of random functions by the relations: (3.6) 0 w„+1 = un, n = 0, 1, (4.1) *»+i = Ux„, n = 0, 1 where U is defined by (1.1) and x0 is an arbitrarily fixed element of C(R+, L^-We introduce the following 326 J. Turo A s s u m p t i o n H. We assume that (1) there exist functions Jcj, ¡£C(R+, R+), j=\, 2, such that 2 |F(f, Ui, «2, X, C0)-F(t, Mj, u2, .x, co)| s= 2 Kj(t)\Uj-üj\ + l(t)\x-x\, J-1 |fj(t, s, x, ca)-fj(t, s, x, co)| | x - x | , for t£R+, s^t, Uj,üj,x,x£R, j= 1,2; (2) F(t, 0,0,0, -)eL2{Q,^t,P) foreach and fj{t,s, 0,-)dL2(Q, FS,P) for each (/, s)e<d. R e m a r k 4.1. We note that from condition (1) of Assumption H we obtain the following estimates: |F(t, ux, u2, x, o>)|2 ^ 4 ^ ( í ) | W l | 2 + 4 ^ ( 0 | « 2 | 2 + 4 ; 2 ( í ) k | 2 + 4|F(í, 0, 0, 0, cu)|2 and |fjit, s, X, G))|2 ^ 2Fj i t ) |x |2+2 \ f j ( t , s, 0, (O)I2 for t£R+, s=.t, Uj, xiR, 7=1, 2, coiG. Put k(t) = 6№it)m)g{t)+k\(t)ki(t)i (4.2) lit) = 6P{t), r(t) = 2E[\{Ux0){t, cj)-x0(t, to)|2]. T h e o r e m 4.1. If Assumption H and assumptions (ii) and (iii) of Lemma 3.1 are satisfied with k, I and r defined by (4.2), then there exists a random solution x€C(R+, L2) of equation (1.1) such that (4.3) E[\x{t,a>)-xn{t,o>)\*} u„{t), 7i = 0 , l , . . . , tiR+. The solution x is unique in the class L*ac(u0)= {x: xíL*^, E[\x(t, a) — x0(t, a))|2]g 6LIoc(M0)}, where L*^ is the class of all locally integrable random functions defined on R+ with range in L2(Q,&„ P), and Lioc{u0) is defined in Lemma 3.1. P roof . From the assumptions of the theorem, Cauchy's inequality, and the properties of the stochastic integral ([3], [7]) it follows that the integrals in equation (4.1) exist for each n (see Remark 4.1) and x„£C{R+, L¿), n=0,1, .... To prove the existence of a solution of equation (1.1) we first prove the follow-ing estimates (4.4) £[|x„(f,a>)-x0(/,<»)|2] u0{t), n = 0 ,1 , . . . , t€R+, (4.5) E[\xn+m{t,w)-xn{t,co)\z]^un{t), n,m = 0, 1, ..., teR+. It is clear that (4.4) holds for n=Q. If we suppose that (4.4) holds for some n > 0 , 327 J. Turo: Existence and uniqueness of random solutions then from (x+y)2^2(x2+y2) and (x+y+z)2^3(x2+y2+z2), an application of Cauchy's inequality and the properties of the stochastic integral we have E[\xn+1(t, co)-x0(i, co)|2] =S S 2E[\(Ux„)(t, (o) — (UXf,)(t, <o)\2] + 2E[\(Ux0)(t, co)-x0(t, co)|2] 9(0 . si 6U2(t)E [|/ (A(/, 5, xn(s, co), <o)-fx{t, s, x0(s, co), co)) ds|2] + 0 9(0 + 6E| (t)E [| f ( f 2 (t, s, x„ (s, co), co) - f 2 (t, s, x0 (s, co), co)) dw (s, co)|2] + o + 6Ht)E[ \ x n (h ( t ) , co)-x0(h(t), co)|2] + 2£[|(t/x0)(r, a)-x0(t, co)|2] ^ 9(0 ^ (6/c?(i)^(0g(0 + 6^1(0^(0) / E[\xn(s,CO)-X0(s,CD)\Z)ds + o + 6l*{t)E[\xn(h(t), co)-x0(h(t), co)|2] + 2£[| t/(x0)(i, co)-x0(t, co)|2] ^ 9(0 S k(t) J u0(s)ds + l(t)u0(h(t)) + r(l) = u0(t). o Now (4.4) follows by induction. It follows from (4.4) that (4.5) holds for «=0 , m=0, 1, .... Now the inequality (4.5) follows from £[ |x„+ m + 1(/ , a>)-xn+1(t, co)|2] ^ (Kun)(t) + (Lun)(t) = Mn+1(i), t£R+, and by induction. Since w„=t0 for n —oo (see Lemma 3.2) and from (4.5) it follows that {x„} is a Cauchy sequence (see Definition 2.2) in C(R+, L2). Now, since C(R+, L2) is a complete space, there exists an xdC(R+, L2) such that x„ —x. If /w — then (4.5) yields estimation (4.3). By the estimation E[\x(t, co)-(Um, ">)l2] ^ ^ 2E[\x(t, co) xn(t, co)|2]+2£[|(f/xn_1)(?, co)-{Ux)(t, «)l2] ^ S4w„(0, n = 0, 1, ..., tiR+, it follows that the random function x satisfies equation (1.1). The uniqueness part of the theorem follows immediately from assertion (c) of Lemma 3.1. Indeed, if we suppose that there exists another solution x of equation (1.1) belonging to L*oc(u0) then we easily infer that i2(i)=.E[|x(/, co)—x(t, co)|2]e 6L)oc(M0), and uSKii+Lu. Hence and from (c) of Lemma 3.1 it follows that E[\x(t, co)—x(t, co)|2]=0. This completes the proof of the theorem. 328 J. Turo: Existence and uniqueness of random solutions Remark 4.2. By using the Banach fixed point theorem or the concept of ad-missibility theory it is easy to prove that there exists a unique random solution of stochastic equation (1.1) if Assumption H is fullfiled and (4.6) * ( 0 * ( 0 + / ( ' ) < 1, ' € * + , where k and I are defined by (4.2). The following theorem, which follows from part c) of Remark 3.1 and Theorem 4.1, shows that condition (4.6) is more restrictive than the assumptions of Theo-rem 4.1. Theorem 4.2. If Assumption H, assumption (iii) of Lemma 3.1 and condition (3.4) are satisfied and if r(f) = rip, t£R+, for some p, r€i?+ , then the assertion of Theorem 4.1 holds provided Itip<l. References [1] A. T. BHARUCHA-REID, Random integral equations. Academic Press (New York, 1972). [2] J. L. DOOB, Stochastic Processes, Wiley (New York, 1953). [3] 1.1. GIHMAN and A. V. SKOROHOD, Stochastic differential equations (Russian), Izdat. Naukova Dumka (Kiev, 1968). [4] M. KWAPISZ and J. TURO, Some integral-functional equations, Funkcialaj Ekvacioj, 18 (1975), 107—162. [5] M. KWAPISZ and J. TURO, Existence, uniqueness and successive approximations for a class of integral-functional equations, Aequationes Math., 14 (1976), 303—323. [6] N. M. MANOUGIAN, A. N. V. RAO and C. P. TSOKOS, On a nonlinear stochastic integral equa-tion with application to control systems, Ann. Math. Pura Appl., 110 (1976), 211—222. [7] T. T. SOONG, Random differential equations in science and engineering. Academic Press (New York, 1973). [8] C. P. TSOKOS and W. J. PADGETT, Random integral equations with applications to life sciences and engineering, Academic Press (New York, 1974). [9] J. TURO, Existence theory for a class of nonlinear random functional integral equations, Ann. Math. Pura Appl., 121 (1979), 145—155. [10] J. TURO, On a nonlinear random functional integral equation, Zeszyty Nauk. Politech. Gdansk. Mat., 11 (1978), 101—107. [11] T. WAZEWSKI, Sur une procédé de prouver la convergence des approximations successive sans utilisation des séries de comparaison, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 8 (1960), 45—52. INSTITUTE OF MATHEMATICS TECHNICAL UNIVERSITY OF GDANSK MAJAKOWSK.IEGO 11—13 80—952 GDANSK—WRZESZCZ, POLAND 

Existence and uniqueness of random solutions of nonlinear stochastic functional integral equations

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Existence and uniqueness of random solutions of nonlinear stochastic functional integral equations

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