EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 645-654 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Stability Results in Terms of Two Measures for Set Differential Equations involving Causal Operators Ch. Appala Naidu1,∗, D. B. Dhaigude2, J. Vasundhara Devi1 1 GVP - Prof. V. Lakshmikantham Institute for Advanced Studies, Department of Mathematics, GVP College of Engineering(A), Andhra Pradesh, India 2 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Maha- rashtra, India Abstract. Differential equations involving causal operators is an area of research that unifies many types of mathematical models such as ordinary differential equations, integro differential equations, delay differential equations and so on. Also, Stability in terms of two measures is another concept that unifies various types of stability. It has been observed that set differential equations generalizes ordinary differential equations (ODEs) and the study of ODEs can be done in semilinear metric space. In this paper, combining all the fore mentioned notions an attempt is made to obtain stability results in terms of two measures for set differential equations involving causal operators. 2010 Mathematics Subject Classifications: 34D20, 34G20 Key Words and Phrases: Hukuhara difference, (h0, h)-equi stable, (h0, h)-uniformly stable, (h0, h)-equi attractive, (h0, h)-equi asymptotically stable, (h0, h)-uniformly asymptotically stable 1. Introduction During the past couple of decades the theory of set differential equations attracted the attention of many researchers and much of the basic theory is given in [ 9 ]. Some of the papers dealing with stability of set differential equations are [ 7, 5 ]. The reason for the continued interest in this area of research is that studying differential equations in metric space is gaining attention and also for the following reasons: the base space Kc(Rn), consisting of all compact convex subsets of Rn endowed with Hausdorff metric, is a semi linear metric space. If the Hukuhara derivative and the Hukuhara integral are restricted to R they become the conventional derivative and integral and one can observe that the theory of ordinary differential equations (ODE’s) can be developed in a semi linear metric space. Similarly when we restrict to Rn, then the theory of vector differential equations can ∗Corresponding author. Email addresses: appalanaidu7@gmail.com (Ch.A.Naidu), dnyanraja@gmail.com (D.B. Dhaigude), jvdevi@gmail.com (J. Vasundhara Devi) http://www.ejpam.com 645 c© 2017 EJPAM All rights reserved. Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 646 be studied in a semi linear space. Further, it is useful in studying multi valued inclusions and also is related to fuzzy differential equations. Also, one can observe that the solution U(t) of the set differential equation has the interesting nature that as time increases the diameter of U(t) is non decreasing. It has been observed that this is due to the fact that, in the generation of the set differential equation (SDE) from an ODE, certain undesirable elements may enter the solution U(t) and hence the norm used may not be suitable to develop stability without some adjustment. Therefore while studying the stability theory for SDE’s the concept of Hukuhara difference in the initial values was introduced in [ 11 ] so as to see that the properties of the solutions of the ODE’s are preserved to a certain extent in the SDE. For a detailed description through an example see [ 11 ]. The stability theory via the Lyapunov function has been extensively studied and is applied in various models due to the fact that the Lyapunov function helps to study the qualitative behavior of the solution without knowing the solution. Of late it has been observed that the Lyapunov function can be utilized to construct simpler scalar differential equations to study complex systems. Also, Lyapunov function can be used as a generalized distance and can be utilized to study the qualitative and quantitative behavior of solutions of differential equations. Owing to the developments in the study of many physical phenomena, new concepts of stability such as partial stability, practical stability, eventual stability have been intro- duced. All these developments posed the question of unifying all the definitions under one set up. This lead to the introduction of stability theory in terms of two measures [ 10 ]. The notion of causal operators has been introduced by Tonneli [ 1 ]. A causal operator or a non anticipative operator is an operator whose prior information or memory is known till the present time ’t’. Ordinary differential equations, delay differential equations, inte- gro differential equations, impulsive differential equations etc. are some of the differential equations involving causal operators. The unifying character of the causal operator makes it an interesting topic of study. In [ 2, 3, 4 ] the mathematical model of SDE involving causal operator with memory was introduced. The existence and uniqueness results, com- parison theorems and stability results has been studied. Much of the basic theory has been established in [ 8 ]. In this paper an attempt has been made to combine all the above set ups and de- velop some stability results in two measures for set differential equations involving causal operators. 2. Preliminaries In this section, we begin with the definition of Hausdorff metric, Hukuhara difference and proceed to define the semi metric space Kc(Rn). Next we proceed to define the Hukuhara derivative, Hukuhara integral and a partial order in Kc(Rn) [ 9 ]. Further, we state some important properties that are useful tools in establishing our main results. Let Kc(Rn) denote the collection of all nonempty, compact and convex subsets of Rn. Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 647 We 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 and B are bounded sets in Rn. We note that Kc(Rn) with this metric is a complete metric space. It is known that if the space Kc(Rn) is equipped with the natural algebraic operations of addition and non-negative scalar multiplication, then Kc(Rn) becomes a semi linear 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(Rn) and λ ∈ R+. Let A,B ∈ Kc(Rn). The set C ∈ Kc(Rn) 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(Rn) has a Hukuhara derivative DHF (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(Rn) and are equal to DHF (t0). Here I is any interval in R. With these preliminaries, we consider the IVP for set differential equation DHU = F (t, U), U(t0) = U0 ∈ Kc(Rn), t0 ≥ 0, (5) where F ∈ C[R+ ×Kc(Rn),Kc(Rn)]. The mapping U ∈ C1[J, Kc(Rn)], J = [t0, t0 + a] is said to be a solution of IVP (5) on J if it satisfies (5) on J . Since U(t) is continuously differentiable, we have U(t) = U0 + ∫ t t0 DHU(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,∫ F (s)ds = { ∫ f(s)ds : f is any continuous selector of F} Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 648 Observe also that U(t) is a solution of IVP (5) on J iff it satisfies (7) on J. We now proceed to define a partial order in the metric space (Kc(Rn), D). We begin with the definition of a cone in this set up. Let K(Ko) be the subfamily of Kc(Rn) consisting of set U ∈ Kc(Rn) such that any u ∈ U is a non-negative (positive) vector of n components satisfying ui ≥ 0 (ui > 0) for i=1...n. Then K is a cone in Kc(Rn) and K0 is the nonempty interior of K. Definition 1. For any U and V ∈ Kc(Rn), if there exists Z ∈ Kc(Rn) 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 introduce the causal operator [ 9, 10, 8 ] which is also known as a non anticipative operator we first introduce the following notation and notion. Let E = C[[t0, T ],Kc(Rn)], We define a norm D0 : E ×E → R+ as follows: for U, V ∈ E, D0[U, V ] = sup t0≤t≤T D[U(t), V (t)] (8) where D is 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. To develop the stability results in terms of two measures we need new concepts that are introduced in [ 5 ], which we present below. K = {a ∈ C[R+,R+] : a(u) is strictly increasing in u and a(0) = 0} L = {σ ∈ C[R+,R+], σ(u) is strictly decresing in u and limu→∞ σ(u) = 0} CK = {a ∈ C[R2 +,R+], a(t, s) ∈ K for each t and a(t, s) is continious for each s} Γ = {h ∈ C[R+ ×Kc(Rn),R+] : inf (t,U) h(t, U) = 0} Γ0 = {h ∈ Γ : inf U h(t, U) = 0, for each t ∈ R+} We next give the definitions that must be satisfied by a function V when the notion of two measures is involved, thus introducing a Lyapunov like function. Let V ∈ C[R+ × Kc(Rn), R+], then V is said to be Definition 3. h-Positive definite if there exists a ρ > 0 and a function b ∈ K such that b(h(t, U)) ≤ V (t, U), whenever h(t, U) < ρ. Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 649 Definition 4. h-decrescent if there exists a ρ > 0 and a function a ∈ K such that V (t, U) ≤ a(h(t, U)), whenever h(t, U) < ρ. Definition 5. h- weakly decrescent if there exists a ρ > 0 and a function a ∈ CK such that V (t, U) ≤ a(t, h(t, U)), whenever h(t, U) < ρ. Let h0, h ∈ Γ, Then we say that Definition 6. h0 is finer than h if there exists a ρ > 0 and a function φ ∈ CK such that h0(t, U) ≤ ρ implies h(t, U) ≤ φ(t, h0(t, U)). Definition 7. h0 is uniformly finer than h if φ is independent of t in the above definition. 3. Stability Results In this section, we develop the stability results for set differential equations involving causal operators given by DHU = (QU)(t) U(t0) = U0, (9) where Q ∈ C[E,E] is a causal operator E = C[J,Kc(Rn)], and J=[0,T]. We assume that the operator Q is smooth enough to gaurantee existence, uniqueness of solutions and continuous dependence of solutions U(t) = U(t, t0, U0) of (9) with respect to the initial values. Now we state from [ 10 ] the various stability concepts for the system (9) in terms of two measures h0, h ∈ Γ. Assume that the system (9) admits the trivial solution U(t) = θ through (t0, θ), then the differential system (9) is Definition 8. (h0, h)−equi stable if for each ε > 0, t0 ∈ R+, there exists a positive function δ = δ(t0, ε) that is continuous in t0 for each ε such that h0(t0, U0) < δ ⇒ h(t, U(t)) ≤ ε, t ≥ t0, where U(t, t0, U0) is any solution of (9). Definition 9. (h0, h)− uniformly stable if the δ in Definition 8 is independent of t0. Definition 10. (h0, h)− equi attractive, if for each ε > 0, t0 ∈ R+, there exists a positive constant δ0 = δ(t0) and T = T (t0, ε) such that h0(t0, U0) < δ0 ⇒ h(t, U(t)) < ε, t ≥ t0 + T. . Definition 11. (h0, h)− uniformly attractive, if Definition 10 holds with δ0 and T being independent of t0. Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 650 Definition 12. (h0, h)− equi asymptotically stable if the Definition 8 and Definition 10 hold simultaneously. Definition 13. (h0, h)− uniformly asymptotically stable if the Definition 9 and Definition 11 hold simultaneously. In order to use the method of Lyapunov function, it is necessary to select minimal subset of E over which the derivative of the Lyapunov function can be conveniently esti- mated. To define this set, we consider V ∈ C[R+ ×Bb,R+], where Bb = B(θ, b) = {U ∈ Kc(Rn) : D[U, θ] ≤ b} E1 = {U ∈ E;V (s, U(s)) ≤ V (t, U(t)), t0 ≤ s ≤ t}, where V ∈ C[R+ ×Kc(Rn), R+] is a Lyapunov function. In order to discuss the stability properties of (9) let us assume that the solutions of (9) exist and are unique for all t ≥ t0. In addition, in order to match the behavior of solutions of (9) with those of the corresponding ordinary differential equations with causal map, we assume that U0 = V0+W0 so that Hukuhara difference U0−V0 = W0 exists. Consequently, in what follows, we consider the solutions U(t) = U(t, t0, U0 − V0) = U(t, t0,W0) . Hence we have the initial value problem. DHU = (QU)(t) U(t0) = W0, where U0 = V0 +W0 (10) This idea is clearly explained with example in the paper [ 7 ]. Now we will state the comparison result, which is analogous to the comparison result in [ 6 ]. Theorem 1. Suppose that the following hypotheses hold (i) V ∈ C[R+ ×Kc(Rn),R+], V(t,U) is locally Lipschitzian in U, (ii) for t ≥ t0 and U ∈ E1, D+V (t, U(t)) ≤ g(t, V (t, U(t)) Where D+V (t, U(t)) = lim sup h→0+ 1 h [V (t+ h, U(t) + h(QU)(t))− V (t, U(t))] (iii) r(t) = r(t, t0, u0) is the maximal solution of the scalar differential equation u′ = g(t, u), u(t0) = u0 ≥ 0 where g ∈ C[R2 +,R+] existing on [t0,∞) then if U(t; t0, U0) is any solution of IVP (9) existing on [t0,∞) , V (t0, U0) ≤ u0 implies V (t, U(t)) ≤ r(t), t ≥ t0. Corollary 1. If, in addition to the assumptions of Theorem 1 with g(t,u)=0 and U(t) ∈ E1 then V (t, U(t)) ≤ V (t0, U0), t ≥ t0 , where U(t) is any solution of IVP (9). Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 651 Now we prove the main stability results for the system (9) Theorem 2. Suppose that the following hypothesis hold (i) V ∈ C[R+ ×Kc(Rn), R+], h ∈ Γ, V(t,U) is locally Lipschitzian in U and h-positive definite, (ii) D+V (t, U(t)) ≤ 0 for (t, U) ∈ S(h, ρ) where S(h, ρ) = {(t, U) ∈ R+ ×Kc(Rn), h(t, U) < ρ, ρ > 0} and U ∈ E1 then (A) if, in addition, h0 ∈ Γ, h0 is finer than h and V(t,U) is h0-weakly decrescent, then the system ( 9) is (h0, h)- equistable. (B) if, in addition, h0 ∈ Γ, h0 is uniformly finer than h and V(t,U) is h0- decrescent, then the system ( 9) is (h0, h)-uniformly stable. Proof. Given that V(t,U) is h0 -weakly decrescent, then by the definition for t0 ∈ R+, W0 ∈ Kc(Rn), there exist constsnt δ0 = δ0(t0) > 0 and a function a ∈ CK such that V (t0,W0) ≤ a(t0, h0(t0,W0)) (11) provided h0(t0,W0) < δ0 and W0 = U0 − V0. Also from hypothesis V(t,U) is h-positive definite, implies that there exist constant ρ0 ∈ (0, ρ) and a function b ∈ K such that b(h(t, U)) ≤ V (t, U) whenever h(t, U) ≤ ρ0 (12) and by the assumption that h0 is finer than h, there exists a constant δ1 = δ1(t0) > 0 and function φ ∈ CK such that h(t0,W0) ≤ φ(t0, h0(t0,W0)) if h0(t0,W0) < δ1, (13) where δ1 is chosen so that φ(t0, δ1) < ρ0. Let ε ∈ (0, ρ0) and t0 ∈ R+ be given, then by the assumption on ’a’, there exists a δ2 = δ2(t0, ε) > 0 that is continuous in t0 such that a(t0, δ2) < b(ε). (14) Now choose δ(t0) = min{δ0, δ1, δ2} then h0(t0,W0) < δ, where W0 = U0 − V0. b(h(t0,W0)) ≤ V (t0,W0) ≤ a(t0, h0(t0,W0)) < b(ε) Hence h(t0,W0) < ε. Now we claim that for every solution U(t) = U(t, t0,W0) = U(t, t0, U0 − V0) of (10) with h0(t0,W0) < δ implise h(t, U(t)) < ε, t ≥ t0, (15) Suppose (15) is not true, then there would exists a t1 > t0 such that h(t1, U(t1)) = ε, and h(t, U(t)) < ε for t ∈ [t0, t1) Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 652 for some U(t) = U(t, t0,W0) of (10). Set m(t) = V (t, U(t)) for t ∈ [t0, t1]. Since V(t, U) is locally Lipschitzian in U, it follows from Corollory 1 that m(t) is non increasing in [t0, t1] and V (t, U(t)) ≤ V (t0,W0), δ ≤ t ≤ t1. Thus it follows that b(ε) = b(h(t1, U(t1)) ≤ V (t1, U(t1)) ≤ V (t0,W0) < b(ε), which is a contraduction to (15) and hence the system (10) is (h0, h)-equistable. Now we prove the second part of the theorem. Since V (t, U) is h- positive definite, there exists a constant 0 < ρ0 ≤ ρ, 0 < δ0 and a function a ∈ K such that b(h(t, U)) ≤ V (t, U), (t, U) ∈ S(h, ρ0) (16) and since V(t,U) is h0-decrescent, there exist constants 0 < ρ0 ≤ ρ, δ0 > 0 and b ∈ K, such that V (t, U) ≤ a(h0(t, U)), if h0(t, U) < δ0. (17) Also we have h0 is uniformly finer than h, there exists a constant δ1 > 0, independent of ’t’ and a function φ ∈ CK such that h(t0,W0) ≤ φ(t0, h0(t0,W0)) if h0(t0,W0) < δ1, (18) where δ1 chosen so that φ(t0, δ1) < ρ0. Let ε ∈ (0, ρ0) and t0 ∈ R+ be given. By the assumption on ’a’ there exists a δ2 = δ2(ε) > 0 such that a(δ2) < b(ε). Choose δ = min{δ0, δ1, δ2} then h0(t0,W0) < δ, where W0 = U0 − V0. (19) We claim that for every solution U(t) = U(t, t0,W0) = W (t, t0, U0 − V0) of (10) with h0(t0,W0) < δ =⇒ h(t, U(t)) < ε, t ≥ t0. (20) Suppose it is not true, then there would exists a t1 > t0 such that h(t1, U(t1)) = ε and h(t, U(t)) < ε for t ∈ [t0, t1) for some U(t) = U(t, t0,W0) of (10). Set m(t) = V (t, U(t)) for t ∈ [t0, t1). Since V(t, U) is locally Lipschitzian in U, it follows from Corollary 1 that m(t) is non increasing in [t0, t1] and it follows that b(ε) = b(h(t1, U(t1)) ≤ V (t1, U(t1)) ≤ V (t0,W0) < b(ε), which is a contraduction to (15) hence the system (10) is (h0, h)-uniformly stable. Now we will prove uniform asymptotic stability result. Ch. A. Naidu,̧ D. B. Dhaigude, J. V. Devi / Eur. J. Pure Appl. Math, 10 (4) (2017), 645-654 653 Theorem 3. Suppose that (i) h0, h ∈ Γ and h0 is uniformly finer than h; (ii) V ∈ C[R+ × Kc(Rn), R+], h ∈ Γ, V(t,U) is locally Lipschitzian in U and h-positive definite, h0-decrescent and D+V (t, U(t)) ≤ −c(h0(t, U)) for (t, U) ∈ S(h, ρ) where c ∈ K S(h, ρ) = {(t, U) ∈ R+ ×Kc(Rn), h(t, U) < ρ, ρ > 0} and U ∈ E1 then the system (10) is (h0, h)-uniformly asymptotically stable. Proof. Since V (t, U) is h-positive definite and h0 decrescent, there exists a constant 0 < ρ0 ≤ ρ, and functions a, b ∈ K such that b(h(t, U)) ≤ V (t, U), (t, U) ∈ S(h, ρ0), (21) and V (t, U) ≤ a(h0(t, U)), if h0(t, U) ≤ δ0, (22) then by the Theorem 2 the system (10) is (h0.h)- uniformly stable. Now we will prove asymptotic stability. Let ε = ρ0 then there exists δ1 = δ1(ρ) > 0 such that h0(t0,W0) < δ1 ⇒ h(t, U(t)) < ρ0, t ≥ t0, (23) where U(t) = U(t, t0,W0) is any solution of (10). Let 0 < ε < ρ0 and δ = δ(ε) be the same δ as in the definition of (h0, h)- uniform stability. Assume that h0(t0, U0) < δ∗ = min{δ0, δ1} and set T = T (ε) = a(δ∗) c(δ) + 1 To prove (h0, h)- uniform asymptotic stability, it is enough to show that there exists a t∗ ∈ [t0, t0 + T ] such that h0(t ∗, U(t∗)) < δ, (24) Suppose (24) is not true, then there exists a solution U(t) = U(t, t0,W0) of (10) with h0(t0,W0) < δ∗ such that h0(t, U(t)) ≥ δ, t ∈ [t0, t0 + T ]. (25) Let m(t) = V (t, U(t)), then it follows from hypothesis (ii) that D+m(t) ≤ −c(h0(t, U(t)), t ≥ t0. (26) Then from the equation (22) we get,∫ t0+T t0 C(h0(s, U(s))ds ≤ m(t0) ≤ a(δ∗), also from (25) we have ∫ t0+T t0 C(h0(s, U(s))ds ≥ C(δ)T > a(δ∗) which is a contraduction. Hence proof of the theorem is complete. REFERENCES 654 References [1] C.Corduneanu, Functional Equations with Causal Operators, Taylor and Francis, New York, 2003. [2] J.Vasundhara Devi, Ch. Appala Naidu, Stability Results for Impulsive Set Differen- tial Equations Involving Causal Operators with Memory, GJMS, Vol 2.2, 49-53. 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