A NECESSARY AND SUFFICIENT CONDITION FOR THE GLOBAL ASYMPTOTIC STABILITY OF DAMPED HALF-LINEAR OSCILLATORS
|
|
- Laurence Wright
- 5 years ago
- Views:
Transcription
1 Acta Math. Hungar., 138 (1-2 (213, DOI: 1.17/s First published online September 5, 212 A NECESSARY AND SUFFICIEN CONDIION FOR HE GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS S. HAA 1 and J. SUGIE 2 1 Department of Mathematics and Computer Science, Shimane University, Matsue , Japan hata@math.shimane-u.ac.jp 2 Department of Mathematics and Computer Science, Shimane University, Matsue , Japan jsugie@riko.shimane-u.ac.jp Abstract. his paper is concerned with the global asymptotic stability of the equilibrium of the half-linear differential equation with a damped term, ( ϕp (x + h(tϕp (x + (ϕ p (x =. A necessary and sufficient growth condition on h(t is obtained by using the generalized Prüfer transformation and the generalized Riccati transformation. Equivalent planar systems to the above-mentioned equation are also considered in the proof of our main result. 1. Introduction We consider the second-order differential equation (HL (ϕ p (x + h(tϕ p (x + (ϕ p (x =, where the prime denotes d/dt, the function ϕ p (z is defined by ϕ p (z = z p 2 z, z R Corresponding author. Supported in part by Grant-in-Aid for Scientific Research, No from the Japan Society for the Promotion of Science. Keywords and phrases: Global asymptotic stability, half-linear differential equations, damped oscillator, growth condition, Prüfer transformation, Riccati transformation. 21 Mathematics Subject Classification: Primary 34D23, 34D45; Secondary 34C15, 37C7.
2 2 S. HAA and J. SUGIE with p > 1, and the damping coefficient h(t is continuous and nonnegative for t. he only equilibrium of (HL is the origin (x, x = (,. Equation (HL includes the damped linear oscillator (L x + h(tx + x = as a special case in which p = 2. By the demand from a theoretical aspect and a practical aspect, there are a lot of reports concerning equation (L or its general type. For example, see [1, 3, 6, 11, 12, 15, 16, 18, 2, 22]. As is well known, the solution space of (L is additive and homogeneous; namely, (i the sum of two solutions is a solution and (ii the product of a solution and any constant is also a solution. Since the solution space of (HL has only the characteristic (ii mentioned above, equation (HL is often called half-linear. In particular, we will call equation (HL the damped half-linear oscillator because it is a generalization of the damped linear oscillator (L. Let x(t = (x(t, x (t and x R 2, and let be any suitable norm. We denote the solution of (HL through (t, x by x(t; t, x. he global existence and uniqueness of solutions of (HL are guaranteed for the initial value problem. For details, see Došlý [4, p. 17] or Došlý and Řehák [5, pp. 8 1]. he equilibrium of (HL is said to be stable if, for any ε > and any t, there exists a δ(ε, t > such that x < δ implies x(t; t, x < ε for all t t. he equilibrium is uniformly stable if it is stable and δ can be chosen independent of t. he equilibrium is said to be attractive if, for any t, there exists a δ (t > such that x < δ implies x(t; t, x as t. he equilibrium is uniformly attractive if δ in the definition of attractivity can be chosen independent of t, and for every η > there is a (η > such that t and x < δ imply x(t; t, x < η for t t + (η. he equilibrium is said to be globally attractive if, for any t, any η > and any x R 2, there is a (t, η, x > such that x(t; t, x < η for all t t + (t, η, x. he equilibrium is said to be uniformly globally attractive if, for any ρ > and any η >, there is a (ρ, η > such that t and x < ρ imply x(t; t, x < η for all t t + (ρ, η. he equilibrium is globally asymptotically stable if it is stable and globally attractive. he equilibrium is uniformly globally asymptotically stable if it is uniformly stable and is uniformly globally attractive. With respect to the various definitions of stability and boundedness, the reader may refer to the books [2, 8, 9, 17, 21, 27] for example. he purpose of this paper is to establish a criterion for judging whether the equilibrium of (HL is globally asymptotically stable or not. Very recently, Onitsuka and Sugie [19, heorem 3.1] have considered a system of differential equations of the form x = a(tx + b(tϕ p (y, y = (c(tϕ p (x (d(ty, where p is the conjugate number of p; namely, 1 p + 1 p = 1,
3 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 3 and presented some sufficient conditions for the uniform global asymptotic stability. As a new variable, let y be ϕ p (x. Note that ϕ p is the inverse function of ϕ p. hen, equation (HL becomes the above system with a(t =, b(t = c(t = 1 and d(t = h(t/(. By means of their result, we see that if h(t is bounded for t and (1 lim inf t +γ t h(sds > holds for every γ >, then the equilibrium of (HL is uniformly globally asymptotically stable. Condition (1 is a restriction that is considerably stronger than H(t def = h(sds as t. If h(t satisfies condition (1, then it is said to be integrally positive. For example, the function sin 2 t is integrally positive. About the integral positivity, see [1, 23 26]. Needless to say, the concept of the global asymptotic stability is stronger than that of the uniform global asymptotic stability. herefore, to seek a good criterion for judging whether the equilibrium of (HL is globally asymptotically stable, we have to discuss under the assumption that is weaker than condition (1. So far as the special case in which p = 2 is concerned, Hatvani and otik [14, heorem 3.1] have already proved that if there exists a γ with < γ < π such that (2 lim inf t +γ t h(sds >, then the equilibrium of (L is asymptotically stable if and only if (3 eh(s ds dt =. e H(t Besides, they pointed out that < γ < π was the best requirement. Condition (2 is weaker than condition (1. For example, let h(t = max {, a + sin t }. hen, condition (1 never holds unless a 1. On the other hand, condition (2 is satisfied as long as a >. A strong point of condition (2 is that the set {t : h(t = } is permitted to be the union of infinitely many disjoint intervals whose length are less than π. he criterion (3 is the so-called growth condition on h(t. It has first been given by Smith [22, heorems 1 and 2]. He showed that if there exists an h > such that h(t h for t, then condition (3 is necessary and sufficient for the equilibrium of (L to be asymptotically stable. he above-mentioned result of Hatvani and otik is its natural generalization. It is known that (i if h(t is bounded for t or h(t = t, then condition (3 is satisfied;
4 4 S. HAA and J. SUGIE (ii if h(t = t 2, then condition (3 is not satisfied. (for details, see [12]. In this paper, we intend to extend their result to be able to apply to the damped half-linear oscillator (HL. o this end, we define ω p by sup{(tan p θ p 2 (π p 2θ: < θ < π p /2} if p 2, ω p = sup{(tan p (π p /2 θ p 2 (π p 2θ: < θ < π p /2} if 1 < p < 2 (the constant π p and the function tan p θ are explained in Section 2. hen, we can state our main theorem as follows: heorem 1. Suppose that there exists a γ with < γ < ω p such that (4 lim inf t +γ t h(sds >. hen the equilibrium of (HL is globally asymptotically stable if and only if ( (5 ϕ eh(s ds p dt =. e H(t Remark 1. In the special case in which p = 2, it is clear that π p = ω p = π and ϕ p (z = ϕ 2 (z = z. Hence, assumption (4 becomes assumption (2 and our criterion (5 coincides with the well-known criterion (3. he function tan p θ is continuous and increasing for θ ( π/2, π/2. Hence, there exists the function arctan p that is defined as the inverse function of tan p θ. It turns out that ω p is more than at least π p 2 arctan p 1 for any p > 1. In the proof of Hatvani and otik s result, it is important that every solution x(t of (L is either oscillatory or monotone for t sufficiently large. his important property is easily led from only the form of (L. Although equation (HL has the same property as equation (L, we have to use the so-called Riccati technique to confirm this property (see Lemma 2. Riccati s technique is required also in order to show that condition (5 is necessary for the equilibrium of (HL to be globally asymptotically stable. It is comparatively easy to transform the damped linear oscillator (L into its polar coordinates system and to examine the asymptotic behavior of solutions of (L. In the damped half-linear oscillator (HL, we adopt the generalized Prüfer transformation mentioned in Section 2. At that time, we need to note whether an angle of each solution of (HL becomes nπ p with n Z. heorem 1 tells us that Hatvani and otik s ideas is also available for nonlinear differential equations such as (HL by combining with other techniques and that the Smith-type growth condition (5 becomes a criterion for the equilibrium of (HL to be globally asymptotically stable.
5 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 5 From heorem 1, we can conclude that the growth condition (3 in Smith s result and Hatvani and otik s result is not caused by the special characteristics of the linear solution space, namely, addition and homogeneity. For this reason, there may be a possibility that a criterion such as the growth condition (5 can be found for more general nonlinear differential equations than (HL. 2. Preliminaries We introduce the generalized sine function sin p θ, the generalized cosine function cos p θ and the generalized circular constant π p according to the book of Došlý and Řehák[5, pp. 4 5]. Let S(t be the solution of an elementary half-linear differential equation without the damping term, (ϕ p (x + (ϕ p (x = satisfying the initial condition (S(, S ( = (, 1. hen, it is easy to verify that the generalized Pythagorean identity S(t p + S (t p 1 holds and the function S(t is positive and increasing on [, π p /2] with S(π p /2 = 1 and S (π p /2 =, where 1 2 π p = (1 s p ds = 2π 1/p p sin(π/p. Combining the arc of S(t from to π p /2 as a basic pattern, we define the generalized sine function sin p θ as follows: { S(θ if θ πp /2 sin p θ = S(π p θ if π p /2 < θ π p and { sinp (θ π p if π p θ < 2π p sin p θ = sin p (θ 2nπ p if 2nπ p θ < 2(n + 1π p for n Z. he generalized cosine function cos p θ is defined as cos p θ = d dθ sin p θ. hen, from the above Pythagorean identity it follows that sin p θ p + cos p θ p = 1 for θ R. We define the generalized tangent function tan p θ by tan p θ = sin p θ cos p θ.
6 6 S. HAA and J. SUGIE About generalized trigonometric functions, see also [4, 7, 13]. Before we advance to the proof of heorem 1, it is very useful to change equation (HL into planar equivalent systems. Because ϕ p is the inverse function of ϕ p as mentioned in Section 1, by putting y = ϕ p (x as a new variable, equation (HL becomes the half-linear system (HS Let x = ϕ p (y, y = (ϕ p (x h(ty. x = r cos p θ and y = ϕ p (r sin p θ (this is slightly different from the generalized Prüfer transformation given in [4, pp ] and [5, p. 7]. hen, by a straightforward calculation, we can transform system (HS into the system (6 r = h(t r sin p θ p, sin p θ p 2 θ = cos p θ p 2 h(t ϕ p(sin p θϕ p (cos p θ. In particular, when θ nπ p with n Z, system (6 becomes the generalized polar coordinates system (HP r = h(t r sin p θ p, θ = cos p θ p 2 h(t sin p θ p 2 sin p θϕ p (cos p θ. 3. Fundamental characteristic of solutions As is customary, we say that a nontrivial solution x(t of (HL is oscillatory, if there exists a sequence {t n } tending to such that x(t n =. Otherwise, we say that it is nonoscillatory. Sturm s separation theorem holds for equation (HL as well as for equation (L. Equation (HL can be classified as oscillatory or nonoscillatory according to whether all nontrivial solutions of (HL have or do not have a sequence of zeros tending to. Lemma 2. Every nonoscillatory solution x(t of (HL is monotone for t sufficiently large. Proof. Since x(t is nonoscillatory, there exists a such that x(t for t. We may assume without loss of generality that x(t > for t. Let w(t = ϕ p (x (t/x(t for t. Since d dz ϕ p(z = ( z p 2, z R,
7 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 7 we have w (t = ( = ( x (t x(t x (t x(t p 2 x (tx(t (x (t 2 x 2 (t p 2 x (t ( x(t x (t x(t p. aking into account that p = p ( and ( x (t p 2 x (t + h(t x (t p 2 x (t + ( x(t p 2 x(t = (this is an alternative expression of (HL, we obtain the generalized Riccati equation (7 w (t + + h(tw(t + ( w(t p = for t. Suppose that there exists a 1 such that w(t > for t 1. hen, w (t < ( for t 1. Integrating this inequality from 1 to t, we obtain w(t < w( 1 ((t 1, which tends to as t. his is a contradiction. hus, we see that w(t = for some t. From (7 it turns out that w (t = ( <, and hence, we can choose an ε > so that w(t < for t < t t + ε. Suppose that there exists a t > t + ε such that w(t >. Let ˆt be the supremum of all t [t + ε, t ] for which w(t. hen, w(ˆt = and (8 w(t > for ˆt < t t. Using (7 again, we see that w (ˆt = ( <. Hence, w(t < on the rightneighborhood of ˆt. his contradicts (8. hus, we get ( x (t ϕ p = w(t for t t. x(t From the positivity of x(t it follows that x (t for t t ; namely, x(t is monotone for a sufficiently large t. As defined in Section 1, the concept of the global asymptotic stability is the union of the concept of the stability and the concept of the global attractivity. We can show the stability of the equilibrium of (HL only by assuming that h(t for t. Lemma 3. he equilibrium of (HL is uniformly stable. Proof. Let x(t be any solution of (HL with the initial time t. Define v(t = x(t p + ϕ p (x (t p.
8 8 S. HAA and J. SUGIE Recall that ϕ p is the inverse function of ϕ p. aking into account that d dz z q = qϕ q (z, z R with q = p or p, we obtain ( v (t = pϕ p (x(tx (t + p ϕ p ϕp (x (t ( ϕ p (x (t for t t. Hence, it follows that = pϕ p (x(tx (t p x (t ( h(tϕ p (x (t + (ϕ p (x(t = p h(tϕ p (x (tx (t = p h(t x (t p (9 v(t v(t for t t. Since ϕ p (x (t p = x (t p (p 1 = x (t p, we can rewrite v(t as v(t = x(t p + x (t p. Let be the p-norm p. hen, from (9 it is easy to confirm that for any ε >, there exists a δ(ε > such that t and x < δ imply x(t; t, x < ε for all t t ; that is, the equilibrium of (HL is uniformly stable. Remark 2. By virtue of Lemma 3, to prove our main theorem, we have only to show that the growth condition (5 is necessary and sufficient for the equilibrium of (HL to be globally attractive under the assumption (4. 4. Proof of heorem 1 We first prove that if the equilibrium of (HL is globally attractive, then the growth condition (5 holds. We then prove the converse. Necessity. Suppose that (5 does not hold. hen, we can choose so large that ( (1 ϕ eh(s ds 1 p dt < e H(t 2(. p 1 Consider the solution x(t of (HL satisfying the initial condition (x(, x ( = (1,. We will show that x(t > 1/2 for all t. By way of contradiction, we suppose that there exists a 1 > such that x( 1 = 1/2 and x(t > 1/2 for t < 1. Let w(t = ϕ p (x (t/x(t for t 1. hen, w( =. As shown in the proof of Lemma 2, we see that w(t satisfies the generalized Riccati equation (7. aking into account that w( = and following the same process as in the proof of Lemma 2, we conclude that x(t is nonincreasing, and hence, ( = x( 1 x(t x( = 1 for t 1.
9 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 9 From (11 it turns out that ( ϕp (x (t + h(tϕp (x (t = (ϕ p (x(t ( for t 1. Hence, ( ϕp (x (te H(t (e H(t for t 1. Integrating both sides of this inequality from to t, we get ϕ p (x (te H(t ϕ p (x ( e H( ( for t 1. Since ϕ p x (t ϕ p e H(s ds = ( is the inverse function of ϕ p, we see that ( ( eh(s ds e H(t for t 1. Integrate both sides of this inequality from to 1 to obtain 1 ( x( 1 x( ϕ p ( eh(s ds dt 1 ( p 1 1 ( p 1 ϕ p ϕ p ϕ p e H(t ( eh(s ds dt ( e H(t eh(s ds e H(t dt. e H(s ds Hence, it follows from (1 that x( 1 > 1/2, which contradicts the assumption that x( 1 = 1/2. hus, the solution x(t does not tend to zero as t, and hence, the equilibrium of (HL is not globally attractive. Sufficiency. Let x(t be any solution of (HL with the initial time t and let (r(t, θ(t be the solution of (6 corresponding to x(t. We prove that if (4 and (5 hold, then (x(t, x (t tends to the origin (, as t. For this purpose, it is enough to show that r(t = p x(t p + x (t p as t. Since (12 r (t = h(t r(t sin p θ(t p for t t, r(t is nonincreasing for t t and therefore, it has the limiting value r. Suppose that r is positive. hen, we get a contradiction as shown below.
10 1 S. HAA and J. SUGIE If x(t is nonoscillatory, then by means of Lemma 2 we see that it is monotone for t sufficiently large. Hence, we can find a t so large that or Since x(t > and x (t for t, x(t < and x (t for t. tan p θ(t = sin p θ(t cos p θ(t = x (t x(t for t, we see that π p /2 < θ(t (mod 2π p or π p /2 < θ(t π p (mod 2π p for t. If x (t = for t, then x(t is a nonzero constant. his is impossible, because the only equilibrium of (HL is the origin (,. Hence, in either case, there exists a such that θ(t nπ p for t with n Z. Hence, (r(t, θ(t is a solution of (HP for t. Since x(t is nonincreasing for t, there exists a c R with c r such that x(t c as t. Hence, it follows that x (t p r p c p as t. If c < r, then we can choose a 2 so that x (t p > rp c p 2 for t 2. Hence, by (12 we have (r p (t = pr p 1 (tr (t = p h(trp (t sin p θ(t p = p h(t x (t p < p (r p c p h(t 2 for t 2. Integrating this inequality from 2 to t, we obtain r p r p ( 2 r p (t r p ( 2 < p (r p c p t h(sds, 2 2 which tends to as t. his is a contradiction. hus, we see that c = r. We therefore conclude that x(t r and x (t as t. here are two cases that should be considered: (i < r x(t x( and x (t for t and (ii x( x(t r < and x (t for t. Note that (13 ( ϕp (x (te H(t = (ϕp (x(te H(t for t.
11 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 11 In the former case, integrate both sides of (13 from to t to obtain ϕ p (x (te H(t ϕ p (x (te H(t ϕ p (x ( e H( (ϕ p (r for t. Since eh(s ds as t, there exists a 3 > such that e H(s ds = Hence, we can estimate that namely, e H(s ds ϕ p (x (te H(t < 1 p ϕ p(r e H(s ds > 1 p ( p ( 1 1 t x (t < r p ϕ eh(s ds p e H(t e H(s ds for t 3. e H(s ds for t 3 ; for t 3. Integrating this inequality from 3 to t, we get ( p 1 ( u r x( 3 x(t x( 3 < 1r p ϕ eh(s ds p du 3 e H(u e H(s ds for t 3. his estimation and condition (5 lead to a contradiction. In the latter case, we have ( p 1 ( u r x( 3 x(t x( 3 > 1r eh(s ds p du, e H(u 3 ϕ p which tends to as t. his is a contradiction. hus, x(t have to oscillate. Since x(t = r(t cos p θ(t, there exist two sequences {τ n } and {σ n } with t τ n < σ n < τ n+1 and τ n as n such that θ(τ n = π p /2 (mod 2π p and θ(σ n = π p /2 (mod 2π p. Recall that y(t = ϕ p (r(t sin p θ(t and (x(t, y(t is the solution of (HS corresponding to (r(t, θ(t. From the second equation of (HP it is clear that θ = 1 h(t tan p θ p 2 sin p θϕ p (cos p θ < for < θ < π p /2 and π p < θ < π p /2 (mod 2π p ; namely, θ is negative in the first quadrant and the third quadrant. his means that the solution (x(t, y(t moves clockwise in the first quadrant and the third quadrant. We divide our argument into two cases: (i p 2 and (ii 1 < p < 2. Case (i: p 2. From the definition of ω p and the assumption that < γ < ω p, there exists a θ (, π p /2 such that (14 γ < (tan p θ p 2 (π p 2θ < ω p.
12 12 S. HAA and J. SUGIE We can choose two divergent sequences {t n } and {s n } with τ n < t n < σ n < s n < τ n+1 such that θ(t n = π p θ, θ(s n = θ and θ < θ(t < π p θ for t n < t < s n. Since θ is not necessarily negative in the second quadrant and the fourth quadrant, the behavior of (x(t, y(t may be not so simple. For this reason, the point in the set {t (τ n, σ n : θ(t = π p θ } might not be only one. In such a case, we should select the supremum of all t (τ n, σ n for which θ(t π p θ as the point t n. Suppose that there exists an N N such that s n t n γ for n N. By (12 we have r (t r (sin p θ p h(t for t n t s n. Needless to say, r (t otherwise. Hence, we can estimate that r(s n r(t n r (sin p θ p sn t n h(tdt r (sin p θ p n +γ t n h(tdt for n N and r(t n+1 r(s n for n N. Adding these two evaluations, we obtain his inequality yields that r(t n+1 r(t n r (sin p θ p n +γ r r(t N r(t n+1 r(t N r (sin p θ p t n h(tdt for n N. n i=n i +γ t i h(tdt. Since t n tends to as n, this is a contradiction. hus, there exists a sequence {n k } with n k N and n k as k such that (15 s nk t nk < γ. Since sin p θ 1 and cos p θ 1 for θ R, we see that θ (t = 1 h(t tan p θ(t p 2 sin 1 h(t p θ(tϕ p (cos p θ(t (tan p θ p 2 for t nk t s nk. It follows from (15 that θ (π p θ = θ(s nk θ(t nk s n k t nk (tan p θ 1 p 2 > γ (tan p θ 1 p 2 snk t nk snk t nk h(tdt h(tdt
13 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 13 for each k N. Hence, from (14 it turns out that 1 snk t nk h(tdt > π p 2θ Using (12 and this estimation, we obtain namely, snk γ > for k N. (tan p θ p 2 log r(s n k snk r(t nk = r (t t nk r(t dt = 1 h(t sin p θ(t p dt t nk (sin p θ p snk ( h(tdt (sin p θ p t nk π p 2θ ( } r(s nk { (sin r(t nk exp p θ p π p 2θ γ < 1 (tan p θ p 2 γ ; (tan p θ p 2 for each k N. his shows that r(t tends to zero as t. Because the limiting value r of r(t is positive, this is a contradiction. Case (ii: 1 < p < 2. From the definition of ω p and the assumption that < γ < ω p, there exists a θ (, π p /2 such that γ < Let ε > be so small that (16 γ < ( ( πp p 2(πp tan p 2 θ 2θ < ω p. ( ( πp p 2(πp tan p 2 θ 2θ 2ε < ω p. Since the solution (x(t, y(t of (HS rotates around the origin (, infinite many times, we can choose four divergent sequences { t n }, {t n }, {s n } and { s n } with t n < t n < s n < s n such that θ( t n = π p ε, θ(t n = π p /2 + θ, θ(s n = π p /2 θ, θ( s n = ε, and π p 2 + θ < θ(t < π p ε for t n < t < t n, ε < θ(t < π p 2 θ for s n < t < s n. Here, we select the supremum of all t (τ n, σ n for which θ(t π p ε as the point t n, and the infimum of all t ( t n, σ n for which θ(t π p /2 + θ as the point t n. Hence, it naturally follows that ε < θ(t < π p ε for t n < t < s n. As in the proof of case (i, we can find a sequence {n k } with n k N and n k as k such that (17 s nk t nk < γ.
14 14 S. HAA and J. SUGIE Since sin p θ 1 and cos p θ 1 for θ R, we see that θ 1 h(t (t = tan p θ(t p 2 sin p θ(tϕ p (cos p θ(t 1 h(t (tan p (π p /2 θ p 2 for t nk t t nk and s nk t s nk. Integrating this inequality, we obtain and π p 2 + θ (π p ε = θ(t nk θ( t nk t nk t nk (tan p (π p /2 θ 1 p 2 nk t nk h(tdt ε (π p /2 θ = θ( s nk θ(s nk s nk s nk (tan p (π p /2 θ 1 p 2 for each k N. Hence, it follows from (16 and (17 that 1 snk t nk h(tdt > π p 2θ 2ε Using (12 and this estimation, we obtain snk s nk h(tdt γ > for k N. (tan p (π p /2 θ p 2 namely, log r( s n k snk r( t nk = r (t t nk r(t dt = 1 snk t nk (sin p ε p (π p 2θ 2ε snk h(t sin p θ(t p dt (sin p ε p h(tdt t nk γ ; (tan p (π p /2 θ p 2 ( } r( s nk { (sin r( t nk exp p ε p π p 2θ γ 2ε < 1 (tan p (π p /2 θ p 2 for each k N. his shows that r(t tends to zero as t. Because the limiting value r of r(t is positive, this is a contradiction. he proof of heorem 1 is thus complete.
15 GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS 15 References [1] Z. Artstein and E. F. Infante, On the asymptotic stability of oscillators with unbounded damping, Quart. Appl. Math., 34 (1976/77, [2] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control heory, 2nd ed. Springer-Verlag (Berlin, Heidelberg and New York, 25. [3] R. J. Ballieu and K. Peiffer, Attractivity of the origin for the equationẍ+f(t, x, ẋ ẋ α ẋ +g(x =, J. Math. Anal. Appl., 65 (1978, [4] O. Došlý, Half-linear differential equations, in Handbook of Differential Equations: Ordinary Differential Equations I (eds. A. Cañada, P. Drábek and A. Fonda, Elsevier (Amsterdam, 24, pp [5] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies 22, Elsevier (Amsterdam, 25. [6] L. H. Duc, A. Ilchmann, S. Siegmund and P. araba, On stability of linear timevarying second-order differential equations, Quart. Appl. Math., 64 (26, [7] Á. Elbert, A half-linear second order differential equation, in Qualitative heory of Differential Equations I (ed. M. Farkas, Colloq. Math. Soc. János Bolyai, 3, North- Holland (Amsterdam and New York, 1981, pp [8] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience (New York, London and Sydney, 1969; (revised Krieger (Malabar, 198. [9] A. Halanay, Differential Equations: Stability, Oscillations, ime Lags, Academic Press (New York and London, [1] L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal., 25 (1995, [11] L. Hatvani, Integral conditions on the asymptotic stability for the damped linear oscillator with small damping, Proc. Amer. Math. Soc., 124 (1996, [12] L. Hatvani,. Krisztin and V. otik, A necessary and sufficient condition for the asymptotic stability of the damped oscillator, J. Differential Equations, 119 (1995, [13] L. Hatvani and L. Székely, Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients, E. J. Qualitative heory of Diff. Equ., 38 (211, [14] L. Hatvani and V. otik, Asymptotic stability of the equilibrium of the damped oscillator, Diff. Integral Eqns., 6 (1993,
16 16 S. HAA and J. SUGIE [15] A. O. Ignatyev, Stability of a linear oscillator with variable parameters, Electron. J. Differential Equations, 1997 (1997, No. 17, pp [16] J. J. Levin and J. A. Nohel, Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics, Arch. Rational Mech. Anal., 5 (196, [17] A. N. Michel, L. Hou and D. Liu, Stability Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Birkhäuser (Boston, Basel and Berlin, 28. [18] M. Onitsuka, Non-uniform asymptotic stability for the damped linear oscillator, Nonlinear Anal., 72 (21, [19] M. Onitsuka and J. Sugie, Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients, to appear in Proc. Roy. Soc. Edinburgh Sect. A 141 (211, [2] P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta Math., 17 (1993, [21] N. Rouche, P. Habets and M. Laloy, Stability heory by Liapunov s Direct Method, Applied Math. Sciences 22, Springer-Verlag (New York, Heidelberg and Berlin, [22] R. A. Smith, Asymptotic stability of x + a(tx + x =, Quart. J. Math. Oxford (2, 12 (1961, [23] J. Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull Austral. Math. Soc., 78 (28, [24] J. Sugie, Global asymptotic stability for damped half-linear oscillators, Nonlinear Anal., 74 (211, [25] J. Sugie and S. Hata, Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients, to appear in Monatsh. Math. [26] J. Sugie and M. Onitsuka, Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients, Proc.Amer.Math.Soc., 138 (21, [27]. Yoshizawa, Stability heory by Liapunov s Second Method, Math. Soc. Japan (okyo, 1966.
Nonresonance for one-dimensional p-laplacian with regular restoring
J. Math. Anal. Appl. 285 23) 141 154 www.elsevier.com/locate/jmaa Nonresonance for one-dimensional p-laplacian with regular restoring Ping Yan Department of Mathematical Sciences, Tsinghua University,
More informationSturm-Liouville Problem on Unbounded Interval (joint work with Alois Kufner)
(joint work with Alois Kufner) Pavel Drábek Department of Mathematics, Faculty of Applied Sciences University of West Bohemia, Pilsen Workshop on Differential Equations Hejnice, September 16-20, 2007 Pavel
More informationLeighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 2, pp. 25 214 (21) http://campus.mst.edu/adsa Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential
More informationApplied Mathematics Letters
Applied Mathematics Letters 24 (211) 219 223 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Laplace transform and fractional differential
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationOscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments
Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations
More informationON STABILITY OF LINEAR TIME-VARYING SECOND-ORDER DIFFERENTIAL EQUATIONS
QUARTERLY OF APPLIED MATHEMATICS VOLUME, NUMBER XXXX XXXX, PAGES S -(XX)- ON STABILITY OF LINEAR TIME-VARYING SECOND-ORDER DIFFERENTIAL EQUATIONS By LUU HOANG DUC (Fachbereich Mathematik, J.W. Goethe Universität,
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More informationOscillation Theorems for Second-Order Nonlinear Dynamic Equation on Time Scales
Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) 289 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/0.2785/amis/070608 Oscillation heorems for Second-Order Nonlinear
More informationBoundedness of solutions to a retarded Liénard equation
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 24, 1-9; http://www.math.u-szeged.hu/ejqtde/ Boundedness of solutions to a retarded Liénard equation Wei Long, Hong-Xia Zhang
More informationALMOST PERIODIC SOLUTIONS OF NONLINEAR DISCRETE VOLTERRA EQUATIONS WITH UNBOUNDED DELAY. 1. Almost periodic sequences and difference equations
Trends in Mathematics - New Series Information Center for Mathematical Sciences Volume 10, Number 2, 2008, pages 27 32 2008 International Workshop on Dynamical Systems and Related Topics c 2008 ICMS in
More informationOn the generalized Emden-Fowler differential equations
On the generalized Emden-Fowler differential equations Joint research with Mauro Marini Topological and Variational Methods for ODEs Dedicated to Massimo Furi Professor Emeritus at the University of Florence,
More informationRESOLVENT OF LINEAR VOLTERRA EQUATIONS
Tohoku Math. J. 47 (1995), 263-269 STABILITY PROPERTIES AND INTEGRABILITY OF THE RESOLVENT OF LINEAR VOLTERRA EQUATIONS PAUL ELOE AND MUHAMMAD ISLAM* (Received January 5, 1994, revised April 22, 1994)
More informationBASIS PROPERTIES OF EIGENFUNCTIONS OF THE p-laplacian
BASIS PROPERTIES OF EIGENFUNCTIONS OF THE p-laplacian Paul Binding 1, Lyonell Boulton 2 Department of Mathematics and Statistics, University of Calgary Calgary, Alberta, Canada T2N 1N4 Jan Čepička3, Pavel
More informationOSCILLATION OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS
Journal of Applied Analysis Vol. 5, No. 2 29), pp. 28 298 OSCILLATION OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS Y. SHOUKAKU Received September,
More informationSZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART
Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics
More informationHOMOLOGICAL LOCAL LINKING
HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION NAEEM M.H. ALKOUMI
More informationNonlinear Control Lecture 5: Stability Analysis II
Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationNontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
Advances in Dynamical Systems and Applications ISSN 973-532, Volume 6, Number 2, pp. 24 254 (2 http://campus.mst.edu/adsa Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
More informationEXISTENCE AND APPROXIMATION OF SOLUTIONS OF SECOND ORDER NONLINEAR NEUMANN PROBLEMS
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 03, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationD. D. BAINOV AND M. B. DIMITROVA
GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 2, 1999, 99-106 SUFFICIENT CONDITIONS FOR THE OSCILLATION OF BOUNDED SOLUTIONS OF A CLASS OF IMPULSIVE DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH A CONSTANT
More informationA LaSalle version of Matrosov theorem
5th IEEE Conference on Decision Control European Control Conference (CDC-ECC) Orlo, FL, USA, December -5, A LaSalle version of Matrosov theorem Alessro Astolfi Laurent Praly Abstract A weak version of
More information2 Statement of the problem and assumptions
Mathematical Notes, 25, vol. 78, no. 4, pp. 466 48. Existence Theorem for Optimal Control Problems on an Infinite Time Interval A.V. Dmitruk and N.V. Kuz kina We consider an optimal control problem on
More informationImpulsive stabilization of two kinds of second-order linear delay differential equations
J. Math. Anal. Appl. 91 (004) 70 81 www.elsevier.com/locate/jmaa Impulsive stabilization of two kinds of second-order linear delay differential equations Xiang Li a, and Peixuan Weng b,1 a Department of
More informationA note on linear differential equations with periodic coefficients.
A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69. 251 Lleida, Spain.
More informationExistence Results for Multivalued Semilinear Functional Differential Equations
E extracta mathematicae Vol. 18, Núm. 1, 1 12 (23) Existence Results for Multivalued Semilinear Functional Differential Equations M. Benchohra, S.K. Ntouyas Department of Mathematics, University of Sidi
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationNonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 20, Number 1, June 2016 Available online at http://acutm.math.ut.ee Nonhomogeneous linear differential polynomials generated by solutions
More informationEXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian
More informationPeriodic solutions of weakly coupled superlinear systems
Periodic solutions of weakly coupled superlinear systems Alessandro Fonda and Andrea Sfecci Abstract By the use of a higher dimensional version of the Poincaré Birkhoff theorem, we are able to generalize
More informationAsymptotic theory of second order nonlinear differential equations: quadro completo
Asymptotic theory of second order nonlinear differential equations: quadro completo Zuzana Došlá Joint research with Mauro Marini Convegno dedicato a Mauro Marini, Firenze, Decembre 2-3, 2016 Table of
More informationTHE ANALYSIS OF SOLUTION OF NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
THE ANALYSIS OF SOLUTION OF NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS FANIRAN TAYE SAMUEL Assistant Lecturer, Department of Computer Science, Lead City University, Ibadan, Nigeria. Email :
More informationRELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 147 161. RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL
More informationResearch Article Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument
Journal of Applied Mathematics Volume 2012, Article ID 498073, 18 pages doi:10.1155/2012/498073 Research Article Oscillation Criteria of Certain hird-order Differential Equation with Piecewise Constant
More informationDYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 7 pp. 36 37 DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department
More informationNONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality
M athematical Inequalities & Applications [2407] First Galley Proofs NONLINEAR DIFFERENTIAL INEQUALITY N. S. HOANG AND A. G. RAMM Abstract. A nonlinear differential inequality is formulated in the paper.
More informationJUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k
Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS
More informationOscillation of second-order differential equations with a sublinear neutral term
CARPATHIAN J. ATH. 30 2014), No. 1, 1-6 Online version available at http://carpathian.ubm.ro Print Edition: ISSN 1584-2851 Online Edition: ISSN 1843-4401 Oscillation of second-order differential equations
More informationDynamic Systems and Applications xx (200x) xx-xx ON TWO POINT BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL INCLUSIONS
Dynamic Systems and Applications xx (2x) xx-xx ON WO POIN BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENIAL INCLUSIONS LYNN ERBE 1, RUYUN MA 2, AND CHRISOPHER C. ISDELL 3 1 Department of Mathematics,
More information154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.
54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and
More informationProof. We indicate by α, β (finite or not) the end-points of I and call
C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending
More informationEcon 204 Differential Equations. 1 Existence and Uniqueness of Solutions
Econ 4 Differential Equations In this supplement, we use the methods we have developed so far to study differential equations. 1 Existence and Uniqueness of Solutions Definition 1 A differential equation
More informationA NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.
Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone
More informationẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.
4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R
More informationAsymptotic behaviour of solutions of third order nonlinear differential equations
Acta Univ. Sapientiae, Mathematica, 3, 2 (211) 197 211 Asymptotic behaviour of solutions of third order nonlinear differential equations A. T. Ademola Department of Mathematics University of Ibadan Ibadan,
More informationNODAL PROPERTIES FOR p-laplacian SYSTEMS
Electronic Journal of Differential Equations, Vol. 217 (217), No. 87, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NODAL PROPERTIES FOR p-laplacian SYSTEMS YAN-HSIOU
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationExistence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order Differential Equations with Two Deviating Arguments
Advances in Dynamical Systems and Applications ISSN 973-531, Volume 7, Number 1, pp. 19 143 (1) http://campus.mst.edu/adsa Existence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationOSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY DIFFERENTIAL EQUATION WITH RICHARD S NONLINEARITY
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 21 27. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
More informationNONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS
NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS PAUL W. ELOE Abstract. In this paper, we consider the Lidstone boundary value problem y (2m) (t) = λa(t)f(y(t),..., y (2j)
More informationOn the discrete boundary value problem for anisotropic equation
On the discrete boundary value problem for anisotropic equation Marek Galewski, Szymon G l ab August 4, 0 Abstract In this paper we consider the discrete anisotropic boundary value problem using critical
More informationSingular boundary value problems
Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: irena.rachunkova@upol.cz Malá Morávka, May 2012 Overview of our common research
More informationAbsolutely convergent Fourier series and classical function classes FERENC MÓRICZ
Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary, e-mail: moricz@math.u-szeged.hu Abstract.
More informationBIBO stabilization of feedback control systems with time dependent delays
to appear in Applied Mathematics and Computation BIBO stabilization of feedback control systems with time dependent delays Essam Awwad a,b, István Győri a and Ferenc Hartung a a Department of Mathematics,
More informationUpper and lower solution method for fourth-order four-point boundary value problems
Journal of Computational and Applied Mathematics 196 (26) 387 393 www.elsevier.com/locate/cam Upper and lower solution method for fourth-order four-point boundary value problems Qin Zhang a, Shihua Chen
More informationOn Some Mean Value Results for the Zeta-Function and a Divisor Problem
Filomat 3:8 (26), 235 2327 DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some Mean Value Results for the
More informationThe complex periodic problem for a Riccati equation
The complex periodic problem for a Riccati equation Rafael Ortega Departamento de Matemática Aplicada Facultad de Ciencias Universidad de Granada, 1871 Granada, Spain rortega@ugr.es Dedicated to Jean Mawhin,
More informationHOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS
Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationA discrete analogue of Lyapunov-type inequalities for nonlinear systems
Computers Mathematics with Applications 55 2008 2631 2642 www.elsevier.com/locate/camwa A discrete analogue of Lyapunov-type inequalities for nonlinear systems Mehmet Ünal a,, Devrim Çakmak b, Aydın Tiryaki
More informationA FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,
More informationA TWO PARAMETERS AMBROSETTI PRODI PROBLEM*
PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationThe zeros of the derivative of the Riemann zeta function near the critical line
arxiv:math/07076v [mathnt] 5 Jan 007 The zeros of the derivative of the Riemann zeta function near the critical line Haseo Ki Department of Mathematics, Yonsei University, Seoul 0 749, Korea haseoyonseiackr
More informationFRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
More informationMEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW
MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationPassivity-based Stabilization of Non-Compact Sets
Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained
More informationSHADOWING PROPERTY FOR INDUCED SET-VALUED DYNAMICAL SYSTEMS OF SOME EXPANSIVE MAPS
Dynamic Systems and Applications 19 (2010) 405-414 SHADOWING PROPERTY FOR INDUCED SET-VALUED DYNAMICAL SYSTEMS OF SOME EXPANSIVE MAPS YUHU WU 1,2 AND XIAOPING XUE 1 1 Department of Mathematics, Harbin
More informationOscillation Criteria for Delay and Advanced Difference Equations with General Arguments
Advances in Dynamical Systems Applications ISSN 0973-531, Volume 8, Number, pp. 349 364 (013) http://campus.mst.edu/adsa Oscillation Criteria for Delay Advanced Difference Equations with General Arguments
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics OPTIMAL OSCILLATION CRITERIA FOR FIRST ORDER DIFFERENCE EQUATIONS WITH DELAY ARGUMENT GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE AND IOANNIS P. STAVROULAKIS Volume 235 No. 1
More informationMULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Fixed Point Theory, 4(23), No. 2, 345-358 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
More informationINTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A JOHN DOMAIN. Hiroaki Aikawa
INTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A OHN OMAIN Hiroaki Aikawa Abstract. The integrability of positive erharmonic functions on a bounded fat ohn domain is established. No exterior conditions are
More informationCzechoslovak Mathematical Journal
Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899
More informationSTRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH SPACES
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 237 249. STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi
More informationON WEAK CONVERGENCE THEOREM FOR NONSELF I-QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES
BULLETIN OF INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN 1840-4367 Vol. 2(2012), 69-75 Former BULLETIN OF SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) ON WEAK CONVERGENCE
More informationPOSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SINGULAR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
International Journal of Pure and Applied Mathematics Volume 92 No. 2 24, 69-79 ISSN: 3-88 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/.2732/ijpam.v92i2.3
More informationAsymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments
Bull. Math. Soc. Sci. Math. Roumanie Tome 57(15) No. 1, 14, 11 13 Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments by Cemil Tunç Abstract
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationA note on stability in three-phase-lag heat conduction
Universität Konstanz A note on stability in three-phase-lag heat conduction Ramón Quintanilla Reinhard Racke Konstanzer Schriften in Mathematik und Informatik Nr. 8, März 007 ISSN 1430-3558 Fachbereich
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationOn the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations
DOI: 1.2478/awutm-213-12 Analele Universităţii de Vest, Timişoara Seria Matematică Informatică LI, 2, (213), 7 28 On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationLYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.457 DYNAMICAL SYSTEMS Volume 25, Number 2, October 2009 pp. 457 466 LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
More informationGLOBAL ATTRACTIVITY IN A NONLINEAR DIFFERENCE EQUATION
Sixth Mississippi State Conference on ifferential Equations and Computational Simulations, Electronic Journal of ifferential Equations, Conference 15 (2007), pp. 229 238. ISSN: 1072-6691. URL: http://ejde.mathmississippi
More informationWeakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 37, 1851-1856 Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Hong Bin Bai School of Science Sichuan University of Science
More informationAn asymptotic ratio characterization of input-to-state stability
1 An asymptotic ratio characterization of input-to-state stability Daniel Liberzon and Hyungbo Shim Abstract For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic
More informationA Note on Some Properties of Local Random Attractors
Northeast. Math. J. 24(2)(2008), 163 172 A Note on Some Properties of Local Random Attractors LIU Zhen-xin ( ) (School of Mathematics, Jilin University, Changchun, 130012) SU Meng-long ( ) (Mathematics
More informationOscillation theorems for the Dirac operator with spectral parameter in the boundary condition
Electronic Journal of Qualitative Theory of Differential Equations 216, No. 115, 1 1; doi: 1.14232/ejqtde.216.1.115 http://www.math.u-szeged.hu/ejqtde/ Oscillation theorems for the Dirac operator with
More informationDISSIPATIVE PERIODIC PROCESSES
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 77, Number 6, November 1971 DISSIPATIVE PERIODIC PROCESSES BY J. E. BILLOTTI 1 AND J. P. LASALLE 2 Communicated by Felix Browder, June 17, 1971 1. Introduction.
More informationOn boundary value problems for fractional integro-differential equations in Banach spaces
Malaya J. Mat. 3425 54 553 On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb
More informationCONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
More informationON THE OSCILLATION OF THE SOLUTIONS TO LINEAR DIFFERENCE EQUATIONS WITH VARIABLE DELAY
Electronic Journal of Differential Equations, Vol. 008(008, No. 50, pp. 1 15. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp ON THE
More informationExistence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1
Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 2 (26), pp. 29 217 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Positive Periodic Solutions
More information