Nonoscillation of the solutions of impulsive differential equations of third order

$(2001), 309-316. Technical University Department of Mathematics Sliven 8800, Bulgaria (Received June, 1998; Revised November, 1999) Necessary and sufficient conditions are found for existence of at least one bounded nonoscillatory solution of a class of impulsive differential equations of third order and fixed moments of impulse effect. Some asymptotic properties nonoscillating are investigated. Key words: Oscillation, Impulsive Differential Equations. AMS subject classifications: 34A37. 1. Introduction The impulsive differential equations with deviating argument are adequate mathematical models of numerous processes and phenomena in physics, biology and electrical engineering. In spite of wide possibilities for their application, the theory se equations is developing rather slowly because of considerable difficulties in technical and theoretical character related to their study. In the recent twenty years, the number of investigations devoted to the oscillatory and nonoscillatory behavior of functional differential equations has considerably increased. A large part works of this subject published in 1977 is presented in [5]. In monographs [2] and [3], published in 1987 and 1991, respectively, the oscillatory and asymptotic properties of various classes of functional differential equations were systematically studied. A pioneering work devoted to the investigation oscillatory properties of impulsive differential equations with deviating argument was rendered by Gopalsamy and Zhang [1]. In the present paper, necessary and sufficient conditions are found for existence of at least one bounded nonoscillatory solution of a class of impulsive differential equations of third order and fixed moments of impulse effect. Some asymptotic properties nonoscillating are investigated. Printed in the U.S.A. @2001 by North Atlantic Science Publishing Company 2. Preinary Notes Consider the impulsive differential equation yâ"(t) + f(t,y(t)) O, 5 rk, k E N, 0, x"(,) + f(()) e N, eN (1) zxâ()- a()- 0, with initial conditions ()(0) where Yi R, O, 1,2. Here Ay"(âk) y"(r k + O)- y"(r k -0). We suppose that y(v k -O)= y(rk); yâ(v k --0) yâ(7"k); R + (0, + oc); 7"1, 72,... are the moments of impulse effect. Introduce the following conditions: 111. 0 1t2. f for I"11 < I"1,"1," e t 113. f k C(R,R), uf k(u > 0 for u (2) Definition 1: A function y C(R+,R) is called a solution equation (1) with initial conditions (2) if it satisfies the following conditions" (a) If 0 r 0 rl, then the function y coincides with the solution equation __ e R+. :/= 0 and f k(ul) <_ f k(u2) for yâ"(t)+f(t,y(t))=O with initial conditions (2). If v k < t <_ 7 k+l, then the function y coincides with the solution equation yâ"(t)+f(t,y(t))-O with initial conditions Y"(âk + O) Y"(âk O) f k(y(-k)), â( 0) (), â(+0) ( + 0) (,- 0) (,). Definition 2: The nonzero solution y(t) problem (1), (2) is said to be nonoscillatory if there exists a point o >_ 0 such that y(t) has a constant sign for >_ t Otherwise, the solution y(t) is said to oscillate. Definition 3: ([4]) A set of real-valued functions defined on the interval [to, + oe) is said to be equiconvergent at oc if all functions in are convergent in R as t---oc and for any c > 0 there exists _> o such that for each function f G f, the inequality If(t)- s_ocf(s) < c is valid for >_ Lemma 1" ([4]) Let f be uniformly bounded and an equicontinuous subset Banach space B([t0, oc)) and let f be equiconvergent at oc. then the set f is relatively compact. t. 3. Main Results Theorem 1" Let the following conditions hold: (a) Conditions H1-H3 are met. There exists a point T >_ 0 such that (b) T 0 for rk _> t 1. Then Ay"(-k) < O, 7k >_ (3) From y(t) > O, >_ t I and condition H2 implies f(t,y(t)) > 0, t > t 1. Therefore, From (3) and (4) it follows that y"(t)is a decreasing function for t _> t 1. The following two cases are possible: Case 1: y"(t) > 0 for t >_ t 1. Then yâ(t) is an increasing function for t >_ t 1. 1.1" If yâ(t) > 0 for t _> t 2 _> tl, then yâ(t) >_ yâ(t2) > 0. We integrate the last inequality from t 2 to t(t >_ t2) and conclude y(t) >_ yâ(t2)(t- t) + y(t). It follows from the above inequality as t + oc, that ty(t + oc which contradicts the assumption that y is a bounded solution equation (1). 1.2: If yâ(t)< 0 for >_ t >_ 1. Then y(t) is decreasing and bounded, so there exists a it, t__ + cy(t) c 1 >_ From y"(t) > 0, Ayâ(rk) 0 for t, r k >_ t 1 to see that yâ(t) is an increasing negative function. Therefore, t yâ(t)- c 2 <_ + Let us suppose c 2<0. Then there exists a constant c a<0 and a point 3>_t 2 such that yâ(t) _< c a for >_ 3. Now, we integrate the above inequality from t 3 to t, (t >_ t3) and arrive at the inequality y(t)<_ cat + y(t3). It follows from the above inequality after taking the it as t + oc, that t__ + y(t) cx, which contradicts the assumption that y is a positive bounded solution equation (1). Therefore, yâ(t) From yâ"(t) < 0, Ay"(-k) < 0 for t, k tl we see that y"(t) is a decreasing positive function. Therefore, t__+y"(t -c 4 >_ We want to prove that c 4-0. Assume that c 4>0. Then there exists aconstant c 5>0 and apoint 4_>t 1 such that y"(t) > c 5 for >_ 4. Now, we integrate the above inequality from 4 to t, >- and arrive at the inequality (t >_ t4) yâ(t) c5(t- t4)/ yâ(t4). It follows from (5) and after taking the it as t + + cx, which contradicts that t_ + yâ(t) Therefore, y"(t) , that t +yâ(t)- Let us suppose t_ + cy(t) c > But y is a bounded, continuous, decreasing and positive function. Thus, there exists constants c > 0, c 6 > 0 and point 5 >_ t 1 such that c <_ y(t) <_ c 6 for _> t 5. We integrate (1) from t to + cx, (t >_ t5) and arrive at the equality t- + oc y"(t) y"(t) From y(t) >_ c for >_ t 5 and conditions 1t2 and 113, y"(t) E AY"(rk) Vk>-t / f(u y(u))du / f(u,c)du j j f k(c). We integrate the last inequality from s to oc and conclude t-, T cx Vk>-t i.e. (6) Integrating (6) from T to t, (t >_ T), (X) we obtain or rk>_s T-rk-t (â- T)2fk(c). T<_r k The last inequality contradicts condition 2 orem 1. Therefore, y( t This contradicts the assumption Case 2" y"(t) _t 1. It is easy to see that t__++My(t) y(t) > 0. The proof is complete. Theorem 2: Let the following conditions hold: (1) Conditions 111-1t3 are met. (2) t2 f(t, c) dt + k-1 â2 f(c)] < + c k for some constant c 7 Then the equation (1) has a bounded nonoscillatory solution. Proof: From condition (2), one can find a sufficiently large T Let X be the space of bounded continuous functions fined by {y E X; Then Y is a bounded convex closed subset of X. Define the operator S: Y---,X as follows" (sv)(t) (a) maps Y into itself. In fact --c c + 1 f (s t) 2f (s, y(s))ds + 1 E r k > t(rk t)2f k(y(-k)), + f (s T)2f(s, y(s ds + T >_ 0 such that (7) rkT on [T, oe). Let Y C X be de- <_ y signc _< _> T âk > T(k T)2f k(Y(k)) <_ <_ T. due to conditions 1t2, tl3 and (7). (b) S is continuous. To prove this, let {Yn} be a Cauchy sequence in Y, and let t--+ + c [[ Y,- Y [[ 0. Because Y is closed, y E Y. To prove the continuity of S, we see that (8) Set r(s) Ln(,r-k) Then s2 f(s, y(s))- f(s, y(s)) 7"2k f k(yn(Zâk))- (8) reduces to noting the fact that (s- t) 2 _< s 2 for s > >_ It is obvious that r() 821f(8,c) 1, L(k) From the definition of 2T2 1t3, f k(c) (10) Gn, L n and conditions 1t2, lira F n s we obtain O, li__,rnL, âk (11) From (9), (10), (11)and the Lebesgue convergence theorem, which means that S is continuous. (c) To show SY is precompact, we see that (Sy)(t), y e Y, is uniformly bounded. Now we will prove that coy is an equicontinuous family of functions on R+. y E Y and t For > 0, (Sy)(t2)- (Sy)(tl) (s- t2)2f(s V(s))dsI (s- + 1/21 E >_ (âk t2)2fk(Y(k))2 E (k >_ tl)fk(Y(k))[ Vk>-tl <_ For any given / s21f(s,c) lds/rk-tl lf(c) l. exists T > 0, there > T such that rk >- T1 Hence, for pY. For T any >_ T1, from (8), tl < t7 T, (y)(t)- ()(t) s() <2 (s- t2)2f(s y(s))ds- (s- f(s, y(s))ds +1/21 cx _> (- t2)2I(u())2 (- tl)2fk(Y(Tk))l _> t2 âk -> t2 cx tl -< âk < t2 (7- k tl)2f k(Y(ârk))t2 t2 <_v k t1 (ârk--tl )2fk(Y(r k _< 1/21 [(.- (s y(s))ds + 1/2 If(s, y(s)) ds -t-1/2 E >_ vk [(7"k t2)2 (7"k tl)2]f k(Y(Yk ))1 mr1 <_ r k <_ (- tl)21f (V(r)) <_ It2-ta i[j s21f(s,c) Ids+ âk >- t2 rlf(c) l] i s21f(sâc) lds+<- rlf(c) <MIt-tll+l s21f(sâc) lds+ tl "rk < t2 Hence, for any given c > 0, there exists a 5 tl _< v k < 2 > 0 such that (Sy)(t2)-(SY)(tl) < C, for all yEY. That is, the interval It2- tl < 5, IT, + oo) can be divided into a finite number of subintervals on which every (Sy)(t), y E Y, has oscillation less than Therefore, SY is an equicontinuous family on [T, We prove that the set SY is equiconvergent to The definition operator S implies Hence, for any given > 0, there exists a point T, > T such that s: Te f(s,c) ds + lfa(c) rk >_T e (12) < 2. (13) From (12) and (13), for t >_ T e we get I(Sy)(t)-l < e for all y E Y. Therefore SY is equiconvergent at oe. Lemma 1 implies that the set SY is relatively compact. According to the Schauder fixed point theorem, there exists a y Y such that y- SY. This y is a bounded nonoscillatory solution of (1). The proof is complete. Theorem 3" Assume that conditions H1-H3 hold. Then condition talf(t,c) ldt+ T f k(c) < + k-1 for some constant c 0 is necessary and oscillatory solution y such that toy(t sufficient for the d, d :/: existence of a bounded$

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Nonoscillation of the solutions of impulsive differential equations of third order

Hindawi Publishing Corporation – Aug 21, 2008

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