Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations

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1 Advances in Dynamical Systems and Applications ISSN , Volume 5, Number 2, pp (21) Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations Abdullah Özbekler Atilim University Department of Mathematics 6836 Incek, Ankara, Turkey Ağacık Zafer Middle East Technical University Department of Mathematics 6531 Ankara, Turkey Abstract In this paper, we obtain new oscillation criteria for second-order half-linear impulsive differential equations having fixed moments of impulse actions. We also generalize the results to nonlinear impulsive equations. The oscillation criteria obtained in this paper can be considered as extensions of those given by Leighton, (J. London Math. Soc. 27 (1952)), Coles (Proc. Amer. Math. Soc. 19 (1968)), and Wintner (Quart. Appl. Math. 7 (1949)). AMS Subject Classifications: 34C1, 34A37. Keywords: Oscillation, half-linear, impulse. 1 Introduction Consider the half-linear impulsive differential equation (r(t)φ(x )) + q(t)φ(x) =, t θ i ; (r(t)φ(x )) + q i Φ(x) =, t = θ i (1.1) where Φ(s) := s α 1 s and z(t) := z(t + ) z(t ), z(t ± ) = lim z(τ). We assume τ t ± that (i) α is a real positive constant; (ii) {θ i } is a strictly increasing unbounded sequence of real numbers; {q i } is a real sequence; Received March 31, 21; Accepted May 6, 21 Communicated by A. Okay Çelebi

2 26 Abdullah Özbekler and Ağacık Zafer (iii) r, q PLC[t, ) := { h : [t, ) R is continuous on each interval (θ i, θ i+1 ), h(θ ± i ) exist, h(θ i) = h(θ i ) for i N} ; r(t) >. By a solution of (1.1), we mean a continuous function x(t) defined on [t, ) such that x, (rφ(x )) PLC[t, ) and (1.1) is fulfilled for all t t. Existence of such solutions can be proved in a similar manner performed for equations without impulse effect [5]. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros. The equation is called oscillatory if every solution is oscillatory. Equation (1.1) without impulse effect (r(t)φ(x )) + q(t)φ(x) =, (1.2) has been the object of intensive studies in recent years. For some studies on oscillatory behavior of solutions of equation (1.2), we refer to [8,1,11,13 19]. Related works for more general equations of the form (r(t)ψ(x)φ(x )) + q(t)f(x) = (1.3) may also be found in [7, 2]. In this paper, our aim is to obtain new oscillation criteria for the second-order halflinear impulsive differential equation (1.1) which extend well-known oscillation criteria of Leighton, Coles, and Wintner. For convenience we first state these oscillation criteria. Theorem 1.1 (See [12]). Let r and q be continuous functions defined on [, ) with r(t) >. Then (r(t)x ) + q(t)x = (1.4) is oscillatory if ds r(s) = q(s)ds =. A simple proof of Theorem 1.1 was also given by Coles [3]. In 1996, Bainov et al. [1] (see also [2, p. 3]) extended Theorem 1.1 to second-order linear impulsive differential equations of the form x + q(t)x =, t θ i ; x + q i x =, t = θ i. (1.5) Theorem 1.2 (See [2]). Suppose that q(t) for t a and q i for all i N for which and θ i a. If then equation (1.5) is oscillatory. a q(t)dt + a<θ i q i =,

3 Half-Linear Oscillations for Impulsive Differential Equations 27 In 1968, Coles [4] studied (1.4) by considering the weighted average [ ] 1 A(t) := g(τ)dτ. τ g(τ)q(s)dsdτ. (1.6) Theorem 1.3 (See [4]). Let g be a nonnegative, locally integrable function such that g, and the functions q and r are continuous functions with r >. If there is an a > such that a { ( ) k ( ) 1 } g(t) g(s)ds r(t)g 2 (s)ds dt = for some k [, 1), and then equation (1.4) is oscillatory. lim A(t) =, t If g(t) 1 and r(t) 1, then Theorem 1.3 reduces to the well-known oscillation criteria of Wintner [21]. Theorem 1.4 (See [21]). If q is continuous on [t, ) for some positive t R and 1 lim t t then equation (1.4) is oscillatory. s t t q(τ)dτds =, In the proof of Theorem 1.3, the ideas of Hartman [6] were used. We will prove a similar theorem for equation (1.1) with a simpler and direct proof. 2 Main Results 2.1 Leighton Wintner Type Oscillation Criteria In this section, we extend Theorem 1.1 and Theorem 1.2 to half-linear impulsive differential equations of the form (1.1). The results are also new when the impulses are absent. Theorem 2.1 (Leighton Type Oscillation Criteria). If q(t)dt + <θ i q i = for some positive number, then equation (1.1) is oscillatory. ds r 1/α (s) = (2.1)

4 28 Abdullah Özbekler and Ağacık Zafer Proof. Let x be a nonoscillatory solution of the equation (1.1). Without loss of generality, we assume that x(t) for t, for large enough. We define z(t) := r(t)φ(x (t)), t [, ). (2.2) Φ(x(t)) Then a solution of the Riccati type impulsive inequality z (t) + α z(t) 1+1/α + q(t) =, t θ r 1/α i ; (2.3) (t) z(t) + q i =, t = θ i ; (2.4) exists on [, ). Integrating (2.3) over [, t) and using (2.4), we see that z(t) + α [ z(s) 1+1/α ds = z() q(s)ds + q r 1/α i ]. (2.5) (s) <θ i <t Using the condition (2.1), for t sufficiently large, we have Let z(t) + α Z(t) = α Then the inequality (2.6) implies that z(s) 1+1/α ds <. (2.6) r 1/α (s) z(s) 1+1/α ds. r 1/α (s) Z 1+1/α (t) 1 α Z (t)r 1/α (t), t θ i ; (2.7) Z(t) =, t = θ i (2.8) for t and θ i for > ( sufficiently large). Since Z 1/α (θ i ) =, separation of variables and integration of (2.7) over [, t] gives which contradicts with (2.1). ds r 1/α (s) [ Z 1/α () Z 1/α (t) ] Z 1/α (), t, Remark 2.2. If r(t) 1 and α = 1, we recover Theorem 1.2. Consider the half-linear impulsive equations with damping of the form (r(t)φ(x )) + p(t)φ(x ) + q(t)φ(x) =, t θ i ; (r(t)φ(x )) + q i Φ(x) =, t = θ i (2.9)

5 Half-Linear Oscillations for Impulsive Differential Equations 29 where p PLC[t, ). Multiplying the equation (2.9) by the function ( ) e(t) = exp p(τ)/r(τ)dτ, we obtain ( r(t)φ(x )) + q(t)φ(x) =, t θ i ; ( r(t)φ(x )) + q i Φ(x) =, t = θ i, (2.1) where r(t) = r(t)e(t), q(t) = q(t)e(t) and q i = q i e(θ i ). Now we may apply Theorem 2.1 to obtain the following oscillation criteria. Corollary 2.3. If q(t)dt + <θ i q i = for some positive number, then equation (2.9) is oscillatory. ds r 1/α (s) = (2.11) When α = 1, equation (2.9) reduces to non self-adjoint impulsive equation (r(t)x ) + p(t)x + q(t)x =, t θ i ; (r(t)x ) + q i x =, t = θ i. (2.12) If p belongs to PLC[t, ), then we get the following Kreith type oscillation criterion for (2.12). See [9] for the nonimpulsive case. Theorem 2.4 (Kreith Type Oscillation Criteria). Let p PLC[t, ). If ( { q(t) p2 (t) 4r(t) p (t) )dt + <θi q i 1 } 2 2 p(θ i) = (2.13) and for some positive number, then equation (2.12) is oscillatory. dt r(t) = (2.14) Proof. Let x be a nonoscillatory solution of the equation (2.12). Without loss of generality, we assume that x(t) for t, for a sufficiently large. We define u(t) := r(t)x (t) x(t) Then we have the Riccati type impulsive equation + p(t), t [, ). 2 u (t) + u2 (t) r(t) + q(t) p2 (t) 4r(t) p (t) 2 =, t θ i ; (2.15) u(t) + q i 1 2 p(t) =, t = θ i. (2.16)

6 21 Abdullah Özbekler and Ağacık Zafer Integrating (2.15) over [, t) and using (2.16), we see that [ u 2 (s) t ( ) u(t) + ds = u() q(s) p2 (s) r(s) 4r(s) p (s) ds 2 + {q i 1 ] 2 p(θ i)}. (2.17) <θ i <t Using the condition (2.13), for a large t, we have Let u(t) + U(t) = Then the inequality (2.18) implies that u 2 (s) ds <. (2.18) r(s) u 2 (s) r(s) ds. U 2 (t) r(t)u (t), t θ i ; U(t) =, t = θ i (2.19) for t and θ i for > ( sufficiently large). Since U 1 (θ i ) =, integration of (2.19) over [, t] gives which contradicts with (2.14). ds r(s) U 1 () U 1 (t) U 1 (), t, 2.2 Coles Type Oscillation Criteria Following (1.6), we define [ ] 1 A(t) := g(s)ds { s g(s) q(τ)dτ + q i }ds. (2.2) <θ i <s Theorem 2.5 (Coles Type Oscillation Criteria). Let g be a nonnegative, locally integrable function such that g, and the functions q and r are continuous functions with r >. If there exists > such that { ( ) k/α ( g(t) g(s)ds for some k [, 1), and then equation (1.1) is oscillatory. ) 1/α } r(s)g α+1 (s)ds dt = (2.21) lim A(t) =, t

7 Half-Linear Oscillations for Impulsive Differential Equations 211 Proof. We give a proof when the function g is continuous; the proof easily modified for the case when g is locally integrable. Let x be a nonoscillatory solution of the equation (1.1). Without loss of generality, we assume that x(t) for t, for large enough. As in the proof of Theorem 2.1, we define z(t) as in (2.2) and similarly we obtain equation (2.5) by (2.3) and (2.4). Multiplying the equation (2.5) by the function g(s) and integrating over [, t), we obtain g(s)z(s)ds + α g(s) s z(τ) 1+1/α dτds = r 1/α (τ) [ ] z() A(t) g(s)ds. (2.22) By our hypothesis, the right-hand side of (2.22) tends to ; hence, for sufficiently large t, s z(τ) 1+1/α g(s)z(s)ds + α g(s) dτds. (2.23) r 1/α (τ) Using Hölder s inequality and (2.23), we obtain Let ( α s z(τ) 1+1/α α+1 ( g(s) dτds) r 1/α (τ) ( )( r(s)g α+1 (s)ds R(t) := α Since, for t >, ( R(t) α s g(s) )( g(s)ds ) α+1 g(s) z(s) ds z(s) 1+1/α α ds). (2.24) r 1/α (s) z(τ) 1+1/α dτds. r 1/α (τ) using the inequalities (2.24) and (2.25), we see that ( ) k ( ) g α z(τ) 1+1/α k ( (t) g(s)ds dτ r 1/α (τ) ) z(τ) 1+1/α dτ, (2.25) r 1/α (τ) ) 1 r(s)g α+1 (s)ds 1 α α+k Rk α 1 (t)(r (t)) α. (2.26) For >, integration of the inequality (2.26) gives [ ( s ) k/α ( s ) 1/α ] g(s) g(τ)dτ r(τ)g α+1 (τ)dτ ds K R (k 1)/α (), (2.27) where K = ( 1 α (1 k) z(τ) 1+1/α k/α dτ). r 1/α (τ) Inequality (2.27) implies that the condition (2.21) cannot hold. This contradiction completes the proof.

8 212 Abdullah Özbekler and Ağacık Zafer In case r(t) 1 and α = 1, equation (1.1) reduces to linear impulsive equation (1.5), and as a consequence of Theorem 2.5, we have the following result which is the extension of Wintner s [21] oscillation criteria to impulsive equations. Corollary 2.6 (Wintner Type Oscillation Criteria). If 1 t { s lim q(τ)dτ + q i }ds =, (2.28) t t then equation (1.5) is oscillatory. <θ i <s Proof. Take the function g(t) 1, k =, and apply Theorem 2.5. Remark 2.7. The condition (2.28) can be replaced by { 1 t lim (t τ)q(τ)dτ + } (t θ i )q i =. t t 3 Further Generalizations <θ i <t In this section we consider (1.3) under impulse effect. Specifically, (r(t)ψ(x)φ(x )) + q(t)f(x) =, t θ i ; (r(t)ψ(x)φ(x )) + q i f(x) =, t = θ i, where ψ, f C(R) with ψ(s) > and sf(s) > for s and the condition (3.1) f (s) M (3.2) (ψ(s) f(s) α 1 ) 1/α is satisfied for some real positive number M. Notice that in the special case of halflinear equation (1.1), for ψ(x) 1 and f(x) = Φ(x), the condition (3.2) is satisfied with M = α. Theorem 3.1 (Leighton Type Oscillation Criteria). If (2.1) and (3.2) are satisfied, then equation (3.1) is oscillatory. Proof. Let x be a nonoscillatory solution of the equation (3.1). Without loss of generality, we assume that x(t) for t, for large enough. We define Clearly, ν(t) satisfies ν(t) := r(t)ψ(x)φ(x ), t [, ). (3.3) f(x) ν (t) + M ν(t) 1+1/α + q(t), t θ r 1/α i ; (3.4) (t) ν(t) + q i =, t = θ i (3.5) for t [, ). The rest of the proof is the same as that of Theorem 2.1.

9 Half-Linear Oscillations for Impulsive Differential Equations 213 Theorem 3.2 (Coles Type Oscillation Criteria). In addition to the conditions of Theorem 2.5, if (3.2) holds, then equation (3.1) is oscillatory. Proof. Proceeding as in the proof of Theorem 3.1, we see that ν(t) defined by (3.3) satisfies (3.4) and (3.5). The rest of the proof is the same as that of Theorem 2.5. As a special case of equation (3.1) we may consider (r(t)φ(x )) + q(t)f(x) =, t θ i ; (r(t)φ(x )) + q i f(x) =, t = θ i. (3.6) It is easy to see that the following oscillation criterion is true. Corollary 3.3. In addition to the conditions of Theorem 2.5, if f (s) f(s) 1+1/α K for some positive number K, then equation (3.6) is oscillatory. References [1] D. D. Bainov, Y. I. Domshlak and P. S. Simeonov, Sturmian comparison theory for impulsive differential inequalities and equations, Arch. Math. (Basel) 67 (1996), [2] D. D. Bainov and P. S. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications, Orlando, Florida, [3] W. J. Coles, Shorter notes: A simple proof of a well-known oscillation theorem, Proc. Amer. Math. Soc. 19 (1967), [4] W. J. Coles, Oscillation criterion for second-order linear differential equations, Proc. Amer. Math. Soc. 19 (1968), [5] Á. Elbert, A half-linear differential equation, in: Colloq. Math. Soc. Janos Bolyai 3: Qualitative Theory of Differential Equations, Szeged (1979), [6] Philip Hartman, Ordinary Differential Equations, John Wiley and Soons, Inc., New York, London, Sydney, [7] H. L. Hong, On the oscillatory behaviour of solutions of second order nonlinear differential equations, Publ. Math. Debrecen 52 (1998), [8] H. B. Hsu and C. C. Yeh, Oscillation theorems for second-order half-linear differential equations, Appl. Math. Lett. 9 (1996),

10 214 Abdullah Özbekler and Ağacık Zafer [9] Kurt Kreith, Oscillation Theory, Lecture Notes in Mathematics, vol. 324, Springer- Verlag, Berlin, Heidelberg, New York, [1] T. Kusano and Y. Naito, Oscillation and nonoscillaton criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), [11] T. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl. 189 (1995), [12] Walter Leighton, On self-adjoint differential equations of second order, J. London Math. Soc. 27 (1952), [13] Horng Jaan Li and Cheh Chih Yeh, An integral criterion for oscillation of nonlinear differential equations, Math. Japon. 41 (1995), [14] Horng Jaan Li and Cheh Chih Yeh, Nonoscillation criteria for second order halflinear differential equations, Appl. Math. Lett. 8 (1995), [15] Horng Jaan Li and Cheh Chih Yeh, Nonoscillation theorems for second order quasilinear differential equations, Publ. Math. Debrecen 47 (1995), [16] Horng Jaan Li and Cheh Chih Yeh, Oscillation criteria for nonlinear differential equations, Houston J. Math. 21 (1995), [17] Horng Jaan Li and Cheh Chih Yeh, Oscillations of half-linear second-order differential equations, Hiroshima Math. J. 25 (1995), [18] Wei Cheng Lian, Cheh Chih Yeh, and Horng Jaan Li, The distance between zeros of an oscillatory solution to a half-linear differential equation, Comput. Math. Appl. 29 (1995), [19] J. V. Manojlovic, Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling 3 (1999), [2] J. V. Manojlovic, Oscillation theorems for nonlinear differential equations of second order, Electron. J. Qual. Theor. Differ. Equ. (2), no. 1, 1. [21] A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949),