13_xxx_jonalgedda.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 4, 2010, 737-747 ISSN 1307-5543 – www.ejpam.com Existence, Uniqueness of solutions for Set Differential Equations involving causal Operators with Memory J. Vasundhara Devi GVP-Prof.V.Lakshmikantham Institute for Advanced Studies, Department of Mathematics, GVP Col- lege of Engineering, Visakhapatnam, AP, India. Abstract. In this paper, we obtain existence and uniqueness results of IVP for set differential equations involving causal operators with memory. This paper is 2nd in sequel. In the first one we obtained inequality results and existence results for set causal operators involving memory. 2000 Mathematics Subject Classifications: 34A12 Key Words and Phrases: Set differential equations, Existence, Uniqueness and Comparison theorems and Extremal solutions. 1. Introduction Owing to the generalization encompassed in both set differential equations and the causal operators, the study of set differential equations involving causal operators with memory has been initiated in [6]. The basic differential inequality and existence results have been de- veloped for both set causal operators with memory and set differential equations involving causal operators with memory. The Set differential equations [4] have certain advantages that dictate the continued in- terest in them. They are useful to study multivalued differential inclusions or multivalued differential equations. Moreover, they include the theory of ordinary differential equations and ordinary differential systems as special cases. This yields the theory of ordinary differen- tial equations and that of systems in a semilinear metric space instead of a linear metric space which is an additional benefit. A causal operator [1,3] or a non anticipative operator is a term adopted from engineering literature. The study of causal or Volterra operators envelopes the study of several dynamic systems such as ordinary differential equations [2], delay differential equations[2], integro Email address: jvdevi�gmail. om http://www.ejpam.com 737 c© 2010 EJPAM All rights reserved. J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 738 differential equations[5] and integral equations to name a few. In this paper we continue to combine these two areas and study set differential equa- tions involving causal operators with memory. This will provide a unified treatment of the basic theory of set differential equations (SDE’s), SDE’s with delay and set integro differential equations which in turn include ordinary dynamic systems of the corresponding type. 2. Preliminaries We begin with the definitions of Kc(R n), the semilinear space in which we work. We next define the Hausdorff Metric, the Hukuhara difference, the Hukuhara derivative and the Hukuhara Integral. We also state all the important properties that are useful in this paper. We further define a partial order in Kc(R n) . We also state all the required results developed in [6] that will be used in this paper. Let Kc(R n) denote the collection of all nonempty, compact and convex subsets of Rn. Define the Hausdorff metric by D[A, B] =max[sup x∈B d(x ,A), sup y∈A d(y, B)], (1) where d(x ,A) = inf[d(x , y) : y ∈ A], A, B are bounded sets in Rn. We note that Kc(R n) with this metric is a complete metric space. It is known that if the space Kc(R n) is equipped with the natural algebraic operations of addition and non-negative scalar multiplication, then Kc(R n) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space. The Hausdorff metric (1) satisfies the following properties: D[A+ C , B+ C] = D[A, B] and D[A, B] = D[B,A], (2) D[λA,λB] = λD[A, B], (3) D[A, B] ≤ D[A, C] + D[C , B], (4) for all A, B, C ∈ Kc(R n) and λ ∈ R+. Let A, B ∈ Kc(R n). The set C ∈ Kc(R n) satisfying A = B + C is known as the Hukuhara difference of the sets A and B and is denoted by the symbol A− B. We say that the mapping F : I → Kc(R n) has a Hukuhara derivative DH F(t0) at a point t0 ∈ I , if lim h→0+ F(t0 + h)− F(t0) h and lim h→0+ F(t0)− F(t0 − h) h exist in the topology of Kc(R n) and are equal to DH F(t0). Here I is any interval in R. With these preliminaries, we consider the set differential equation DH U = F(t, U), U(t0) = U0 ∈ Kc(R n), t0 ≥ 0, (5) J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 739 where F ∈ C[R+ × Kc(R n), Kc(R n)]. The mapping U ∈ C1[J , Kc(R n)], J = [t0, t0 + a] is said to be a solution of (5) on J if it satisfies (5) on J . Since U(t) is continuously differentiable, we have U(t) = U0 + ∫ t t0 DH U(s)ds, t ∈ J . (6) Hence, we can associate with the IVP (5) the Hukuhara integral U(t) = U0 + ∫ t t0 F(s, U(s))ds, t ∈ J . (7) where the integral is the Hukuhara integral which is defined as, intF(s)ds = { ∫ f (s)ds : f is any continuous selector of F} Observe that U(t) is a solution of (5) on J iff it satisfies (7) on J . We now define a partial order in the metric space Kc(R n). To do so, we need the definition of a cone in Kc(R n), which is given below. Let K(K0) be the subfamily of Kc(R n) consisting of sets U ∈ Kc(R n) such that any u ∈ U is a nonnegative (positive) vector of n-components satisfying ui ≥ 0(ui > 0) for i = 1,2,3, . . . , n. Then K is a cone in Kc(R n) and K0 is the nonempty interior of K. Definition 1. For any U and V ∈ Kc(R n), if there exists a Z ∈ Kc(R n) such that Z ∈ K(K0) and U = V + Z then we say that U ≥ V (U > V ). Similarly we can define U ≤ V (U < V ). To define the causal operator we introduce the following notation. Let E = C[[t0, T], Kc(R n)] and E0 = C[[t0 − h1, T], Kc(R n)], where U ∈ E0 implies U(t) = Φ0(t), t0 − h1 ≤ t ≤ t0 and U(t) is any arbitrarily continuous function on [t0, T]. We define a norm on E as follows: for U , V ∈ E D0[U , V ] = Supt0≤t≤T D[U(t), V (t)] where D denotes the Hausdorff Metric. Definition 2. By a causal operator or a Volterra operator or a nonanticipative operator we mean a mapping Q : E → E satisfying the property that if U(s) = V (s),t0 ≤ s ≤ t < T then (QU)(s) = (QV )(s),t0 ≤ s ≤ t < T. By a causal operator with memory we mean a mapping Q : E0 → E such that for U(s) = V (s),t0 ≤ s ≤ t < T, Q(U ,Φ0)(s) =Q(V,Φ0)(s), t0 ≤ s ≤ t < T and Φ0 ∈ C1 = C[[t0 − h1, t0], Kc(R n)]. J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 740 We now state the following theorems from [6] which are needed to prove results in the next sections. Before proceeding further, we set D0[U , V ] = sup t0≤s≤T D[U(s), V (s)] Theorem 1. Assume that (i) Q is nondecreasing in U for each t ∈ I = [t0, T]. (ii) DH V (t) ≤ (QV )(t) DHW (t) ≥ (QW )(t) where V,W ∈ C1[I , Kc(R n)] and (iii) V (t0) 0, t ∈ I where Ω = {U , V ∈ E0 : max s∈[t0−h1,t] D[U(s), V (s)] = D[U(t), V (t)]t ∈ I} Then there exists a unique solution U(t) of the IVP(8) and (9) provided T − t0 < 1 L . Proof. Define D0[U , V ](t) = max s∈[t0−h1,t] D[U(s), V (s)] For any U ∈ E0, define the Hukuhara integral operator T on I by (T U)(t) = Φ0(t0) + ∫ t t0 Q(U ,Φ0)(s)ds (12) Ut0 = Φ0 ∈ C1 on [t0 − h1, t0] (13) Now for U , V ∈ Ω, using the properties of Hausdorff metric and hypothesis of the theorem, D[T U(t), T V (t)] = D[Φ0(t0) + ∫ t t0 Q(U ,Φ0)(s)ds,Φ0(t0) + ∫ t t0 Q(V,Φ0)(s)ds] = D[ ∫ t t0 Q(U ,Φ0)(s)ds, ∫ t t0 Q(V,Φ0)(s)ds] ≤ ∫ t t0 D[Q(U ,Φ0)(s),Q(V,Φ0)(s)]ds ≤ ∫ t t0 Lmax t0≤s≤t D[U(s), V (s)]ds = L ∫ t t0 D[U(t), V (t)]ds ≤ LD0[U , V ](t − t0) ≤ L(T − t0)D0[U , V ] which is a contraction, when L(T − t0)< 1 or (T − t0)< 1 L . Thus, since (T− t0)< 1 L , T is contraction from E to E and hence by contraction mapping theorem there exists a U ∈ E such that T U = U, Ut0 = Φ0 we get that U is unique solution for the IVP (8) and (9) whenever (T − t0)< 1 L . Hence the proof is complete. J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 742 Remark 1. The restriction (T − t0)< 1 L can be avoided by using a weighted norm. We define D0[U , V ] = maxs∈[t0,T]D[U(s), V (s)]e−λt . where λ is to be chosen suitably. Now using the relation (12), and using the properties of Hausdorff metric we arrive at, D[(T U)(t), (T V )(t)] = D[Φ0(t0) + ∫ t t0 Q(U ,Φ0)(s)ds,Φ0(t0) + ∫ t t0 Q(V,Φ0)(s)ds] = D[ ∫ t t0 Q(U ,Φ0)(s)ds, ∫ t t0 Q(V,Φ0)(s)ds] ≤ ∫ t t0 D[Q(U ,Φ0)(s),Q(V,Φ0)(s)]ds ≤ ∫ t t0 L maxs∈[t0,T]D[U(s), V (s)]e−λseλsds = D0[U , V ]L ∫ t t0 eλsds = LD0[U , V ] ∫ t t0 eλsds = L λ D0[U , V ][eλt − eλt0] ≤ L λ D0[U , V ]eλt This gives e−λt D[T U(t), T V (t)] ≤ L λ D0[U , V ] Thus, D0[T U , T V ]≤ L λ D0[U , V ] Now to choose λ, we observe that L λ < 1 2 , yields that T is a contraction, so we can choose λ < 2 L . Hence we get that T is a contraction and that there exists a unique U ∈ E0 such that U is a solution of the IVP (8) and (9). In order to establish existence and uniqueness result using generalized Lipschitz condition. We need the following comparison theorem on R+ from [4]. Theorem 4. Assume that m ∈ C[I ,R+], g ∈ C[I ×R+,R+] and for t ∈ I , D−m(t) ≤ g[t| m |0(t)], (14) where | m |0(t) = supt0≤s≤t | m(s) | . J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 743 Suppose that r(t) = r(t, t0, w0) is the maximal solution of the scalar differential equation w′ = g(t, w), w(t0) = w0 ≥ 0 (15) existing on I. Then m(t0)≤ w0 implies m(t) ≤ r(t), t ∈ I . We now state a Comparison Theorem that connects an estimate on the solution of the IVP (8) and (9) with the maximal solution of the initial value problem (15). Theorem 5. Let Q ∈ C[E0, E] be a causal map such that for t ∈ I , D[(QU)(t), (QV )(t)]≤ g(t, D0[U , V ](t)], where g ∈ C[I ×R+,R+]. Suppose further that the maximal solution r(t, t0, w0) of the scalar differential equation (15) exists on I. Then, if U(t), V (t) are any two solutions of (8) and (9) with initial function Ut0 = Vt0 = Φ0 ∈ C1, then we have D[U(t), V (t)] ≤ r(t, t0, w0), t ∈ I Proof. We first observe that D0[Ut0 , Vt0 ] ≤ w0 is satisfied automatically. The proof of the theorem is exactly same as that of Theorem 5.7.2 in [4]. Hence we avoid the proof. We are now in a position to state the existence and uniqueness result using successive ap- proximations and generalized Lipschitz condition. Once again the proof is very much similar to the corresponding theorem, Theorem 5.7.3 in [4]. Hence we omit it. Observe that the only difference between the two results is that the following theorem has memory included in its set up. Theorem 6. Suppose that (i) Q ∈ C[B, E] be a causal map, where B ⊆ E0 with B = {U ∈ E0 : D0[U ,Φ0(t0)]≤ b, D0[Ut0 ,Φ0] = 0, t ∈ I} and D0[Q(U ,Φ0),θ] ≤ M1 on B. (ii) g ∈ C[I ×R+,R+]g(t,u) ≤ M2 on I × [0,2b], g(t, 0) = 0, g(t,u) be nondecreasing in u for each t ∈ I and w(t) ≡ 0 is the only solution of w′ = g(t, w), w(t0) = 0 on I (16) (iii) D[Q(U ,Φ0)(t),Q(V,Φ0)(t)] ≤ g(t, D0[U , V ](t)], on B Then the successive approximations defined by Un+1(t) = Φ0(t0) + ∫ t t0 Q(Un,Φ0)(s)ds. Un+1t0 = Φ0 ∈ C1, n= 0,1,2,3 . . . exist on I0 = [t0, t0 + η] where η = min[T − t0, b 2M ] M = max[M1, M2] and converge uniformly to a unique solution U(t) of (8) and (9). J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 744 4. Global Existence Let bE = C[[t0,∞], Kc(R n)] and cE0 = C[[t0 − h1,∞], Kc(R n]. We now state and prove a global existence result. Theorem 7. Assume that Q ∈ C[cE0, bE] and is smooth enough to guarantee local existence of solutions of IVP (8) and (9) for any (t0,Φ0(t0)) ∈ R+× Kc(R n). Further Q(U ,Φ0) be such that D[Q(U ,Φ0)(t),θ] ≤ g(t, D0[U ,θ](t)] (17) where g ∈ C[R+ 2,R], g(t, w) is nondecreasing in w for each t ∈ R+, and the maximal solution r(t) = r(t, t0, w0) of scalar IVP (15) exists on [t0,∞). Then the largest interval of existence for any solution U(t) of (8) and (9) is [t0,∞), whenever D0[Φ0,θ] ≤ w0 Proof. Suppose that U(t) = U(t, t0,Φ0(t0)) with Ut0 = Φ0 be any solution of (8) and (9) existing on [t0,β), t0 < β <∞, with D0[Φ0,θ] ≤ w0 and the value of β cannot be increased. Set m(t) = D[U(t),θ] then m(t0) = D[U(t0),θ] = D[Φ0(t0),θ]≤ D0[Φ0,θ] ≤ w0 Consider D+m(t) ≤ D[DH U(t),θ] ≤ D[Q(U ,Φ0)(t),θ] ≤ g(t, D0[U ,θ](t)). Now using the comparison theorem, Theorem 4, we obtain that m(t) ≤ r(t), t0 ≤ t ≤ β . For any t1, t2 such that t0 < t1 < t2 < β , we obtain, using the properties of the Hausdorff metric, and relation (17), D[U(t1), U(t2)] = D[ ∫ t t0 Q(U ,Φ0)(s)ds, ∫ t2 t0 Q(U ,Φ0)(s)ds] ≤ ∫ t2 t1 D[Q(U ,Φ0)(s),θ]ds ≤ ∫ t2 t1 g(s, D0[U ,θ])(s)ds = ∫ t2 t1 g(s, | m |0 (s)])ds. Using the fact that m(t) = D[U(t),θ] ≤ r(t) and g(t,u) is nondecreasing in u for each t, we get D[U(t1), U(t2)]≤ r(t2)− r(t1) Since lim t→β r(t, t0, w0) J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 745 exists, taking the limit as t1, t2 → β −, we conclude that {U(tk)} is a Cauchy sequence and therefore the l imt→β−U(t, t0,Φ0) = Uβ exists. Now define Φβ (t) =    Φ0(t), t0 − h1 ≤ t ≤ t0 U(t, t0,Φ0), t0 ≤ t < β Uβ , t = β and consider the IVP DH U(t) = Q(U ,Φ0)(t), t ≥ β , Uβ = Φβ on [t0 − h1,β], t0 ≥ 0 (18) setcE0 = C[[t0 − h1,β + a], Kc(R n)] and bE = C[[t0,β + a], Kc(R n)], a > 0, bB ⊂cE0 where bB = {U ∈cE0 : D0[U ,Φ0(t0)]≤ b, D0[U(t0),Φ0(t0)] = 0, t ∈ J} Then Q : bB→ bE is a causal map such that it guarantees the local existence of a solution, hence there exists U(t,β , Uβ ) satisfying (17) on some interval [β ,β +α], 0< α < a. Thus U(t, t0,Φ0) can be extended beyond β , contradicting our assumption that β cannot be increased. Thus every solution U(t, t0,Φ0) of (8), (9) such that D0[Φ0,θ] ≤ w0 exists globally on [t0 − h1,∞). Hence the proof is complete. Theorem 8. Let Q ∈ C1[cE0, bE] and satisfy the estimate. D[Q(U ,Φ0)(t),θ] ≤ g(t, D[U(t),θ]], U ∈ Ω (19) where Ω = {U ∈ E0 : max t0−h1≤s≤t D[U(s),θ] = D[U(t),θ], t ∈ I} and g ∈ C[[t0,∞)×R+,R+], g(t,u) is monotone nondecreasing in u for each t ∈ [t0,∞). Assume that for every t0 > 0, the scalar differential equation u′ = g(t,u),u(t0) = u0 ≥ o, (20) has a solution u(t) existing on [t0,∞). Then for Φ0 ∈ C1 such that D0[Φ0,θ] ≤ u0 there exists a solution U(t) of (8), (9) on [t0,∞) satisfying D[U(t),θ] ≤ u(t), t ∈ [t0,∞) (21) Proof. Consider the space bE, of all continuous functions from [t0,∞) to Kc(R n), and a family of pseudonorms {pn(U)} ∞ n=1 be defined for U ∈ bE, pn(U) = sup t0≤t≤n D[U(t),θ]. J. Devi / Eur. J. Pure Appl. Math, 3 (2010), 737-747 746 Let the topology on bE be generated by this family. A fundamental system of neighborhoods is then given by {Vn(U} ∞ n=1, where Vn(U) = {U ∈ bE : pn(U)≤ 1} Under this topology, bE becomes a complete, locally convex linear space. Now define a subset E ⊂ bE as follows. E = {U ∈ Ω : D[U(t),θ] ≤ u(t), t ≥ t0} where u(t) is a solution of (20) existing on [t0,∞). Then under the topology of bE, E is closed convex and bounded. Consider the integral operator defined by (T U)(t) = Φ0(t0) + ∫ t t0 Q(U ,Φ0)(s)ds and Ut0 = Φ0 ∈ C1 It is obvious that a fixed point of T will be a solution of the IVP (8),(9). The operator T is compact in the topology of bE and therefore closure of T E is compact since bE is bounded. The proof of the theorem is complete, if we show that T E ⊆ E. Hence consider U ∈ E. Then D[(T U)(t),θ] = D[Φ0(t0) + ∫ t t0 Q(U ,Φ0)(s)ds,θ] ≤ D0[Φ0,θ] + ∫ t t0 D[Q(U ,Φ)(t),θ] ≤ D0[Φ0,θ] + ∫ t t0 g(t, D[U(t),θ])ds ≤ D0[Φ0,θ] + ∫ t t0 g(t,u(s))ds because of the relation (18),(20) and (21) and the monotonic nature of g, the definition of the set E and the fact that u(t) is a solution of (20), with D0[Φ0,θ] ≤ u0. This yields D[(T U)(t),θ] ≤ u0 + ∫ t t0 g(s,u(s))ds = u(t) Hence T U ∈ E or T E ⊆ E. Thus the proof is complete. ACKNOWLEDGEMENTS This work has been done under the project no. SR/S4/MS: 491/07 sanctioned by Department of Science and Technology, Government of India. The author ac- knowledges their support REFERENCES 747 References [1] C.Corduneanu, Functional Equations with Causal Operators, Taylor and Fran- cis,Newyork (2003). [2] V.Lakshmikantham and S.Leela, Differential and Integral Inequalities, Vol.I and II , Aca- demic press, Newyork, (1969). [3] V.Lakshmikantham and S.Leela, Z.Drici and McRae FA, Theory of Causal Differential Equations, Atlantis Press and World Scientific, (2009). [4] V.Lakshmikantham, T.GnanaBhaskar and J.Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, 2(006). [5] V.Lakshmikantham and M.Rama Mohan Rao, Theory of Integro Differential equations, Gordan and Breach Science Publishers,Amsterdam, (1995). [6] J.Vasundhra Devi, Comparison Theorems and Existence Results for Set Causal Operators with Memory submitted to NonLinear Analysis, TMA.