ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay"

Transcription

1 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 functional, Lyapunov's direct method, process, nonautonomous dierential systems, partial stability. 1. INTRODUCTION Denote by C the space of the continuous functions : [?r; 0]! R n, where r > 0 is a constant. In the function space C we use the norms kk = maxfj(s)j : s 2 [?r; 0]g; jjjjjj = s Z 0 j(s)j 2 ds ;?r where j j denotes an arbitrary norm in R n. If x : [t 0? r; T )! R n (0 t 0 < T 1) is continuous and t 2 [t 0 ; T ), we dene x t 2 C by x t (s) := x(t + s) for s 2 [?r; 0]. Consider the functional dierential equation _x = f(t; x t ); (1:1) where f : R + C! R n is continuous and f(t; 0) = 0 for all t 0. We suppose, that for each t 0 2 R + and each 2 C there is a unique solution x(; t 0 ; ) dened on an interval [t 0 ; t 0 + ); > 0. We assume, that the function x continuously depends on the initial data t 0 ;. Denote by K the set of strictly increasing continuous functions w : R +! R + with w(0) = 0. If V is a continuous functional from R + C into R, then _ V denotes the derivative of functional V with respect to equation (1:1) dened by _V (t; ) = _ V (1:1) (t; ) = lim sup!0+ V (t + ; x t+ (; t; ))? V (t; ) : In this paper we study the asymptotic stability of the zero solution of equation (1:1) using Lyapunov functionals. In this topic the following two theorems are basic : 1

2 Theorem A (see [5, Theorem 4.1]). Suppose, that for every M > 0 there is an L(M) > 0 such that [t 2 R + ; 2 C M := f 2 C : kk Mg] =) jf(t; )j L(M): (1:2) If there are M > 0, a continuous functional V : R + C M! R and functions w 1 ; w 2 ; w 3 2 K such that w 1 (j(0)j) V (t; ) w 2 (kk) ; (1:3) _V (t; )?w 3 (jjjjjj) (1:4) for all t 2 R +, 2 C M, then the zero solution of equation (1.1) is uniformly asymptotically stable. Theorem B ([3, Theorem 5.2.1]). Suppose that conditions (1.2) and (1.3) of Theorem A hold, and, instead of condition (1.4), the inequality _V (t; )?w 3 (j(0)j) (1:5) holds. Then the zero solution of equation (1.1) is uniformly asymptotically stable. It is an old problem (see [1, p. 252]), whether the boundedness condition can be dropped from Theorems A and B. Note, that - as is usual in the stability theory - the proofs of these theorems need conditions (1.4) or (1.5) only along the solutions of (1.1). We will show by examples that if conditions (1.4) and (1.5) are required only along the solutions of (1.1), then the boundedness condition cannot be dropped. At rst we replace the norms jj; kk and jjjjjj in Theorems A and B by abstract "measures" (see [6],[7] and [8]). It will be pointed out that, under these general circumstances, to the boundedness condition there corresponds an estimate between the measures. Our examples will show that these estimates are essential. 2. THEOREMS AND COROLLARIES Let X be a Banach-space. A continuous function h : R X! R + is called a measure in X, if h(t; 0) = 0. The continuous function u : R X R +! X is said to be a process (see e.g. [3, Chapter 4.1]), if u(t 0 ; x; 0) = x and u(t 0 + t 1 ; u(t 0 ; x; t 1 ); t) = u(t 0 ; x; t 1 + t) for all t 0 2 R; t 1 ; t 2 R +. For example, let U(t 0 ; ; t? t 0 ) := x t (; t 0 ; ), where x(t; t 0 ; ) (t t 0 ) is the solution of equation (1.1) with x(t 0 ; t 0 ; ) =. It is easy to see, that U is a process. 2

3 Let h 0 and h be measures. If u is a process and t 0 2 R, x 0 2 X, then the function u(t 0 ; x 0 ; ) is said to be a motion. Let x denote the state of this motion at t t 0, i.e. x := u(t 0 ; x; t? t 0 ). The motion is (h 0 ; h)-stable or stable in measures (h 0 ; h), if for each t t 0 and > 0 there is a (x 0 ; t 0 ; t ; ) > 0 such that h 0 (t ; x? x ) < implies h(t + t; u(t ; x; t)? u(t ; x ; t)) < for all t 0. The stability is uniform, if can be chosen independently of t. The motion u(t 0 ; x 0 ; ) is said to be (h 0 ; h)-attractive, if there is a (x 0 ; t 0 ; t ) > 0 such that, if h 0 (t ; x? x ) <, then h(t + t; u(t ; x; t)? u(t ; x ; t))! 0 (t! 1). The attractivity is uniform, if is independent of t and the convergence is uniform in t. The motion is said to be (uniformly) asymptotically stable, if it is (uniformly) stable and (uniformly) attractive. For a V : R X! R we dene the derivative _ V with respect to the process u by _V (t; x) := lim sup!0+ V (t + ; u(t; x; ))? V (t; x) : Examples: The zero solution of equation (1.1) is stable in the usual sense (see e.g. [3, Chapter 5.1]), if and only if the zero motion (U(t 0 ; 0; )) is stable in measures h 0 (t; ) = kk and h(t; ) = j(0)j. The solution x(; t 0 ; 0 ) of equation (1.1) is stable if and only if the motion U(t 0 ; 0 ; ) is stable in the previous measures. The zero solution of the ordinary dierential equation _x = F (t; x) (x 2 R n ) is partially stable (see [4]), if the zero motion of this equation is stable in the measures h 0 (t; x) = p x x2 2 + ::: + x2 n and h(t; x) = p x ::: + x2 s, where 0 < s n and x = (x 1 ; x 2 ; :::; x n ). The stability of an invariant set A with respect to this equation is equivalent to the stability in measures h 0 (t; x) = h(t; x) = d(x; A), where d(x; A) means the distance between x and A in R n. (For further examples for processes, among them partial dierential equations, see e.g. [9].) Proposition 1. Let the measures h 0 ; h be given. Suppose that there are a continuous functional V : R X! R, functions w 1 ; w 2 ; w 3 ; w 4 2 K and a measure h 1 satisfying the following conditions: w 1 (h(t; x)) V (t; x) w 2 (h 0 (t; x)) (i) _V (t; x)?w 3 (h 1 (t; x)) h 0 (t; x) w 4 (h 1 (t; x)) (ii) (iii) for all t 2 R and x 2 X. Then the zero motion of the process u is uniformly asymptotically (h 0 ; h)-stable. Proof: At rst we prove the uniform stability. Let an > 0 be given and dene () := w?1 2 (w 1()). Now if h 0 (t ; x) <, then V (t ; x) w 1 () by (i). Condition (ii) and the continuity of function V (t + 3

4 t; u(t ; x; t)) implies, that this function is nonincreasing in t, so w 1 (h(t + t; u(t ; x; t))) V (t + t; u(t ; x; t)) V (t ; u(t ; x; t? t )) = V (t ; x) w 1 (): Consequently, h(t + t; u(t ; x; t)) which proves the uniform stability. The conditions imply _V (t + t; u(t ; x; t))?w 3 (h 1 (t + t; u(t ; x; t)))?w 3 (w 4 (h 0 (t + t; u(t ; x; t))))?w 3 (w 4 (w?1 2 (V (t + t; u(t ; x; t))))); which is a dierential inequality for V (t + t; u(t ; x; t)). By [6, Theorem 3.1.1] we get lim V (t + t; u(t ; x; t)) = 0 t!1 uniformly in t. Obviously, inequality (i) proves the uniform asymptotic stability. Note, that inequalities (ii) and (iii) could be replaced by _V (t; x)?w 5 (h 0 (t; x)); with an appropriate w 5 2 K. In spite of this fact we separated them because it is inequality (iii) that corresponds to the boundedness condition in Theorem A. In order to formulate the problem corresponding to that of omitting the boundedness condition from Theorem A, we weaken some conditions of Proposition 1. It can be seen that conditions (ii) and (iii) may be asked only along the motions, even we can assume on the motions in condition (iii) that h(t + t; u(t ; x; t)) B for all t 0, where B is a constant independent of t ; x. Conditions (ii) and (iii) can be further weakened if we need only (nonuniform) asymptotic stability. The process u is said to be h 0 -continuous (with respect to x) if for every > 0, t 0 2 R, t 1 0 and x 0 2 X there is = (t 0 ; t 1 ; x 0 ; ) > 0 such that h 0 (t 0 ; x?x 0 ) < implies h 0 (t 0 +t 1 ; u(t 0 ; x; t 1 )?u(t 0 ; x 0 ; t 1 )) <. If the process u is h 0 -continuous, then conditions (ii) and (iii) can be asked only for suciently large values of t. This modications result in the following. 4

5 Theorem 1. Let measures h 0, h be given. Assume that the process u is h 0 -continuous with respect to x. Suppose that there are continuous functional V : R X! R, functions w 1 ; w 2 ; w 3 ; w 4 2 K, measure h 1, and constants T; B > 0 satisfying the following conditions: w 1 (h(t; x)) V (t; x) w 2 (h 0 (t; x)) (i) for all t t 0 and x 2 X, _V (t + t; u(t ; x; t))?w 3 (h 1 (t + t; u(t ; x; t))) (ii) for all t t 0, t T and x 2 X, and h 0 (t + t; u(t ; x; t)) w 4 (h 1 (t + t; u(t ; x; t))) (iii) for each t t 0, t T and x 2 X such that h(t + s; u(t ; x; s)) < B for all s T. Then the zero motion of the process u is asymptotically (h 0 ; h)-stable. Corollary 1 (Theorem A revisited). Suppose that all but condition (1.4) of Theorem A are satised. Suppose, in addition, that there is a T = T (t 0 ) > 0 such that the inequality _V (t; x t )?w 3 (jjjx t jjj) (1:4 0 ) holds for all t t 0 + T and for every solution x : [t 0? r; 1)! R n of (1.1). Then the zero solution of equation (1.1) is asymptotically stable. Proof: Let h 0 (t; ) := kk, h(t; ) := j(0)j and h 1 (t; ) := jjjjjj in Theorem 1. It is enough to prove, that condition (iii) in Theorem 1 follows from the boundedness condition of Theorem A. Inequalities (i) and (ii) imply the stability, so we can assume, that the solutions are bounded above in the measure h with an arbitrary B. Suppose that T r. We have from the boundedness (see condition (1.2)), that the absolute value of the derivative of function x(t; t 0 ; ) less or equal than L(B). For every t t 0 there exists t 1 2 [t? r; t] with jx(t 1 )j = kx t k. So if t t 0 + r, then s 2 [t 1? kx t k=2l(b); t 1 + kx t k=2l(b)] implies jx(s)j kx t k=2 and we have the inequality jjjx t jjj s kx t k 3 8L(B) : This means that condition (iii) is satised and the proof is complete. Theorem B cannot be deduced from Theorem 1, because one cannot estimate below the measure j(0)j by the measure kk. So we must replace inequality (iii) with a more general condition. 5

6 Theorem 2. Assume that all but condition (iii) of Theorem 1 are satised. Suppose, in addition that if w 2 (h 0 (t + t; u(t ; x; t))) > > 0 for all t T, then Z 1 w 3 (h 1 (t + t; u(t ; x; t)))dt = 1: T Then the zero motion of the process u is asymptotically (h 0 ; h)-stable. Proof: By h 0 -continuity of the process u, for every > 0 there exists a (t 0 ; x 0 ; t ; T; ) > 0 such that h 0 (t ; x? x ) < implies h 0 (t + T; u(t ; x; T )? u(t ; x ; T )) <. From the proof of Proposition 1, for each > 0 we get an (t 0 ; x 0 ; t ; T; ) > 0 such that if h 0 (t + T; u(t ; x; T )? u(t ; x ; T )) <, then h(t + t; u(t ; x; t)? u(t ; x ; t)) < for all t T. This proves the stability of the zero motion. To complete the proof it is enough to show that lim t!1 V (t + t; u(t ; x; t)) =: v 0 = 0: Suppose v 0 > 0. Then v 0 w 2 (h 0 (t + t; u(t ; x; t))) for all t T and the last condition of the theorem gives Z 1 w 3 (h 1 (t + t; u(t ; x; t)))dt = 1: By condition (ii) we have 0 V (t + t; u(t ; x; t)) (t! 1), which is a contradiction. V (t + T; u(t ; x; T ))? T Z t T w 3 (h 1 (t + t; u(t ; x; t)))dt!?1 Corollary 2 (Theorem B revisited). Suppose that all but condition (1.5) of Theorem B are satised. Suppose, in addition, that there is a T = T (t 0 ) > 0 such that the inequality _V (t; x t )?w 3 (jx(t)j) (1:5 0 ) holds for all t t 0 + T and for every solution x : [t 0? r; 1)! R n of (1.1). Then the zero solution of equation (1.1) is asymptotically stable. Proof: Let h 0 (t; ) := kk and h(t; ) = h 1 (t; ) := j(0)j in Theorem 2. We use the boundedness condition to prove the new condition in Theorem 2. As we saw in the proof of Corollary 1, for every t t 0 + r we have t 1 2 [t? r; t] such that jx(s)j kx t k=2 for all s 2 [t 1? kx t k=2l(b); t 1 + kx t k=2l(b)]. If kx t k for all t t 0, then it follows from the last property that Z 1 t 0 w 3 (jx(t)j)dt = 1; which proves the new condition. 6

7 3. EXAMPLES The following two examples show that the boundedness condition cannot be dropped from the Corollary 1 and 2. Consider the ordinary scalar dierential equation _x = _ (t) x; (x 2 R) (3:1) (t) where : [?1; 1)! (0; 1) is continuously dierentiable (r = 1; t 0 = 0). Obviously, the functions x(t) = c (t) (c 2 R) are the solutions of equation (3.1). In both examples we choose the functional V (t; ) = kk + jjjjjj. We have to construct a function which is bounded on [?1; 1) and satises an inequality _V (t; c t )?w 3 (jjjc t jjj) (Cor: 1) (3:2) respectively _V (t; c t )?w 3 (jc (t)j) (Cor: 2) (3:3) for all t 1; c 2 R and, at the same time, (t) 6! 0 as t! 1. Let a function : [?1; 1)! (0; 1) be such that (a) (n) = 1 + 1=2 n, (b) (t? 1)? (t) 1=2 n+1 for all t 2 [n; n + 1), (c) jjj jjj 1=2 n?1 for all t 2 [n; n + 1) for n =?1; 0; 1; :::. (Continuously dierentiable functions with these properties can be constructed from pieces of lines and parabolas.) Condition (b) implies that k t k is monotone nonincreasing. So _V (t; c t ) jjjc t jjj?jcjw( 1 2 n)?w 3(jjjc t jjj) for all t 0; c 2 R, i.e. inequality (3.2) holds. Condition (a) implies, that (t) 6! 0 as t! 1, so the zero solution of equation (3.1) is not asymptotically stable. We now construct a function satisfying (3.3). Let a sequence of intervals f[a n ; b n ]g 1 n=0 be given such that [a n+1?1; b n+1?1] [a n ; b n ], a 0 = 0, b 0 = 1=4. Consider a function having the following properties: (a) (t) = 1=2 n+1 for t 2 [b n ; a n+1 ], (b) (t? 1)? (t) 1=2 n+1 for all t 2 [a n ; a n+1 ), (c) _ (t)?1 and (t) 1 + 1=2 n+2 for t 2 [a n+1? 1; b n+1? 1], 7

8 (d) (t) monotone decreasing on interval [b n+1? 1; b n ], (e) maxf (t) : t 2 [a n ; a n+1? 1]g = 1 + 1=2 n+1 and (f) (t) 4 for all t?1. Since k t k is nonincreasing, properties (a) and (b) implies _V (t; c t ) jcj jjj t jjj?jcjw( 1 2 n )?w 3(jc (t)j) for all t 2 [b n ; a n+1 ] (n = 0; 1; :::) and c 2 R with appropriate functions w; w 3 2 K. If t 2 [a n ; b n ], then _V (t; c t ) jcj k t k?jcj?jcj (t) 4 =? 1 jc (t)j: 4 Consequently, (3.3) is satised for all t 1. On the other hand, condition (e) guarantees (t) 6! 0 (t! 1), i.e. the zero solution of equation (3.1) is not asymptotically stable. REFERENCES 1. Burton T. A., Volterra Integral and Dierential Equations, Academic Press (1983). 2. Burton T. & Hatvani L., Stability theorems for nonautonomous functional dierential equations by Lyapunov functionals, Tohoku Math. J. 41, (1989). 3. Hale J., Theory of functional dierential equations, Springer-Verlag New York-Heidelberg-Berlin (1977). 4. Hatvani L., On partial asymptotic stability and instability, Acta Sci. Math. 45, (1983). 5. Hatvani L., On the asymptotic stability of the solutions of functional dierential equations, Coll. Math. Soc. J. Bolyai 53, (1988). 6. Lakshmikantham V., Leela S. & Martynyuk A. A., Stability analysis of nonlinear systems, Marcel Dekker, Inc. New York and Basel (1989). 7. Lakshmikantham V. & Xin Zhi Liu, Perturbing families of Lyapunov functions and stability in terms of two measures, J. Math. Anal. Appl. 140, (1989). 8. Movchan A. A., Stability of processes with respect to two metrics, J. Appl. Math. Mech. 24, (1961). 9. Stephen H. Saperstone, Semidynamical Systems in Innite Dimensional Spaces, Springer-Verlag New York- Heidelberg-Berlin (1981). 8

Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle

Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle Malaya J. Mat. 4(1)(216) 8-18 Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle B. C. Dhage a,, S. B. Dhage a and S. K. Ntouyas b,c, a Kasubai,

More information

Congurations of periodic orbits for equations with delayed positive feedback

Congurations of periodic orbits for equations with delayed positive feedback Congurations of periodic orbits for equations with delayed positive feedback Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday Gabriella Vas 1 MTA-SZTE Analysis and Stochastics

More information

EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS

EVANESCENT 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 information

Converse Lyapunov Functions for Inclusions 2 Basic denitions Given a set A, A stands for the closure of A, A stands for the interior set of A, coa sta

Converse Lyapunov Functions for Inclusions 2 Basic denitions Given a set A, A stands for the closure of A, A stands for the interior set of A, coa sta A smooth Lyapunov function from a class-kl estimate involving two positive semidenite functions Andrew R. Teel y ECE Dept. University of California Santa Barbara, CA 93106 teel@ece.ucsb.edu Laurent Praly

More information

GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS

GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS Communications in Applied Analysis 19 (215), 679 688 GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS TINGXIU WANG Department of Mathematics, Texas A&M

More information

ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (1996), 13 { 27. ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION

ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (1996), 13 { 27. ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (996), 3 { 27 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION P. J. Y. Wong and R. P. Agarwal Abstract. We oer sucient conditions for the

More information

DISSIPATIVE PERIODIC PROCESSES

DISSIPATIVE 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 information

ALMOST PERIODIC SOLUTIONS OF NONLINEAR DISCRETE VOLTERRA EQUATIONS WITH UNBOUNDED DELAY. 1. Almost periodic sequences and difference equations

ALMOST 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 information

Disconjugate operators and related differential equations

Disconjugate 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 information

610 S.G. HRISTOVA AND D.D. BAINOV 2. Statement of the problem. Consider the initial value problem for impulsive systems of differential-difference equ

610 S.G. HRISTOVA AND D.D. BAINOV 2. Statement of the problem. Consider the initial value problem for impulsive systems of differential-difference equ ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 23, Number 2, Spring 1993 APPLICATION OF THE MONOTONE-ITERATIVE TECHNIQUES OF V. LAKSHMIKANTHAM FOR SOLVING THE INITIAL VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL-DIFFERENCE

More information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1 Yong Zhou Abstract In this paper, the initial value problem is discussed for a system of fractional differential

More information

Patryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński

Patryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF

More information

HYBRID DHAGE S FIXED POINT THEOREM FOR ABSTRACT MEASURE INTEGRO-DIFFERENTIAL EQUATIONS

HYBRID DHAGE S FIXED POINT THEOREM FOR ABSTRACT MEASURE INTEGRO-DIFFERENTIAL EQUATIONS HYBRID DHAGE S FIXED POINT THEOREM FOR ABSTRACT MEASURE INTEGRO-DIFFERENTIAL EQUATIONS Dr. Sidheshw. S. Bellale Dept. of Mathematics, Dayanand Science College, Latur, Maharashtra (IND) Email: sidhesh.bellale@gmail.com

More information

EXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS GENERATED BY IMPULSES FOR SECOND-ORDER HAMILTONIAN SYSTEM

EXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS GENERATED BY IMPULSES FOR SECOND-ORDER HAMILTONIAN SYSTEM Electronic Journal of Differential Equations, Vol. 14 (14), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND MULTIPLICITY

More information

An important method in stability and ISS analysis of continuous-time systems is based on the use class-kl and class-k functions (for classical results

An important method in stability and ISS analysis of continuous-time systems is based on the use class-kl and class-k functions (for classical results Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems D. Nesic Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, 3052,

More information

Spurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics

Spurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics

More information

is a new metric on X, for reference, see [1, 3, 6]. Since x 1+x

is a new metric on X, for reference, see [1, 3, 6]. Since x 1+x THE TEACHING OF MATHEMATICS 016, Vol. XIX, No., pp. 68 75 STRICT MONOTONICITY OF NONNEGATIVE STRICTLY CONCAVE FUNCTION VANISHING AT THE ORIGIN Yuanhong Zhi Abstract. In this paper we prove that every nonnegative

More information

TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J.

TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J. RGMIA Research Report Collection, Vol. 2, No. 1, 1999 http://sci.vu.edu.au/ rgmia TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES S.S. Dragomir and

More information

QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those

QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.

More information

Midterm 1. Every element of the set of functions is continuous

Midterm 1. Every element of the set of functions is continuous Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions

More information

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary

More information

THE NEARLY ADDITIVE MAPS

THE 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 information

SYNCHRONIZATION OF NONAUTONOMOUS DYNAMICAL SYSTEMS

SYNCHRONIZATION OF NONAUTONOMOUS DYNAMICAL SYSTEMS Electronic Journal of Differential Equations, Vol. 003003, No. 39, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp SYNCHRONIZATION OF

More information

FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS

FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS Fixed Point Theory, Volume 5, No. 2, 24, 181-195 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS CEZAR AVRAMESCU 1 AND CRISTIAN

More information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOURTH-ORDER BOUNDARY-VALUE PROBLEMS IN BANACH SPACES

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOURTH-ORDER BOUNDARY-VALUE PROBLEMS IN BANACH SPACES Electronic Journal of Differential Equations, Vol. 9(9), No. 33, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS

More information

RESOLVENT OF LINEAR VOLTERRA EQUATIONS

RESOLVENT 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 information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS

A 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 information

ON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

ON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY Georgian Mathematical Journal Volume 11 (24), Number 2, 337 348 ON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY I.-G. E. KORDONIS, CH. G. PHILOS, I. K.

More information

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem

EXISTENCE 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 information

ON THE CONVERGENCE OF THE ISHIKAWA ITERATION IN THE CLASS OF QUASI CONTRACTIVE OPERATORS. 1. Introduction

ON THE CONVERGENCE OF THE ISHIKAWA ITERATION IN THE CLASS OF QUASI CONTRACTIVE OPERATORS. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXIII, 1(2004), pp. 119 126 119 ON THE CONVERGENCE OF THE ISHIKAWA ITERATION IN THE CLASS OF QUASI CONTRACTIVE OPERATORS V. BERINDE Abstract. A convergence theorem of

More information

Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory

Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory Hacettepe Journal of Mathematics and Statistics Volume 46 (4) (2017), 613 620 Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory Chris Lennard and Veysel Nezir

More information

A NECESSARY AND SUFFICIENT CONDITION FOR THE GLOBAL ASYMPTOTIC STABILITY OF DAMPED HALF-LINEAR OSCILLATORS

A NECESSARY AND SUFFICIENT CONDITION FOR THE GLOBAL ASYMPTOTIC STABILITY OF DAMPED HALF-LINEAR OSCILLATORS Acta Math. Hungar., 138 (1-2 (213, 156 172. DOI: 1.17/s1474-12-259-7 First published online September 5, 212 A NECESSARY AND SUFFICIEN CONDIION FOR HE GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS

More information

Tomasz Człapiński. Communicated by Bolesław Kacewicz

Tomasz Człapiński. Communicated by Bolesław Kacewicz Opuscula Math. 34, no. 2 (214), 327 338 http://dx.doi.org/1.7494/opmath.214.34.2.327 Opuscula Mathematica Dedicated to the Memory of Professor Zdzisław Kamont GLOBAL CONVERGENCE OF SUCCESSIVE APPROXIMATIONS

More information

Chapter 3 Pullback and Forward Attractors of Nonautonomous Difference Equations

Chapter 3 Pullback and Forward Attractors of Nonautonomous Difference Equations Chapter 3 Pullback and Forward Attractors of Nonautonomous Difference Equations Peter Kloeden and Thomas Lorenz Abstract In 1998 at the ICDEA Poznan the first author talked about pullback attractors of

More information

NONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality

NONLINEAR 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 information

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy

More information

Boundedness of solutions to a retarded Liénard equation

Boundedness 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 information

Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions

Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions Mouffak Benchohra a,b 1 and Jamal E. Lazreg a, a Laboratory of Mathematics, University

More information

Fractional differential equations with integral boundary conditions

Fractional differential equations with integral boundary conditions Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 (215), 39 314 Research Article Fractional differential equations with integral boundary conditions Xuhuan Wang a,, Liping Wang a, Qinghong Zeng

More information

On (h, k) trichotomy for skew-evolution semiflows in Banach spaces

On (h, k) trichotomy for skew-evolution semiflows in Banach spaces Stud. Univ. Babeş-Bolyai Math. 56(2011), No. 4, 147 156 On (h, k) trichotomy for skew-evolution semiflows in Banach spaces Codruţa Stoica and Mihail Megan Abstract. In this paper we define the notion of

More information

LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan

LYAPUNOV-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 information

A Fixed Point Theorem and its Application in Dynamic Programming

A Fixed Point Theorem and its Application in Dynamic Programming International Journal of Applied Mathematical Sciences. ISSN 0973-076 Vol.3 No. (2006), pp. -9 c GBS Publishers & Distributors (India) http://www.gbspublisher.com/ijams.htm A Fixed Point Theorem and its

More information

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University

More information

THE PERRON PROBLEM FOR C-SEMIGROUPS

THE PERRON PROBLEM FOR C-SEMIGROUPS Math. J. Okayama Univ. 46 (24), 141 151 THE PERRON PROBLEM FOR C-SEMIGROUPS Petre PREDA, Alin POGAN and Ciprian PREDA Abstract. Characterizations of Perron-type for the exponential stability of exponentially

More information

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical

More information

NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL. 1. Introduction

NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXXVI, 2 (2017), pp. 287 297 287 NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL PINGPING ZHANG Abstract. Using the piecewise monotone property, we give a full description

More information

Abstract. Previous characterizations of iss-stability are shown to generalize without change to the

Abstract. Previous characterizations of iss-stability are shown to generalize without change to the On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag and Yuan Wang Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Mathematics,

More information

PATA TYPE FIXED POINT THEOREMS OF MULTIVALUED OPERATORS IN ORDERED METRIC SPACES WITH APPLICATIONS TO HYPERBOLIC DIFFERENTIAL INCLUSIONS

PATA TYPE FIXED POINT THEOREMS OF MULTIVALUED OPERATORS IN ORDERED METRIC SPACES WITH APPLICATIONS TO HYPERBOLIC DIFFERENTIAL INCLUSIONS U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 4, 216 ISSN 1223-727 PATA TYPE FIXED POINT THEOREMS OF MULTIVALUED OPERATORS IN ORDERED METRIC SPACES WITH APPLICATIONS TO HYPERBOLIC DIFFERENTIAL INCLUSIONS

More information

Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph

Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid

More information

Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings

Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of

More information

WITH REPULSIVE SINGULAR FORCES MEIRONG ZHANG. (Communicated by Hal L. Smith) of repulsive type in the sense that G(u)! +1 as u! 0.

WITH REPULSIVE SINGULAR FORCES MEIRONG ZHANG. (Communicated by Hal L. Smith) of repulsive type in the sense that G(u)! +1 as u! 0. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 2, February 1999, Pages 41{47 S 2-9939(99)512-5 PERIODIC SOLUTIONS OF DAMPED DIFFERENTIAL SYSTEMS WITH REPULSIVE SINGULAR FORCES MEIRONG

More information

Czechoslovak Mathematical Journal

Czechoslovak 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 information

Alfred O. Bosede NOOR ITERATIONS ASSOCIATED WITH ZAMFIRESCU MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES

Alfred O. Bosede NOOR ITERATIONS ASSOCIATED WITH ZAMFIRESCU MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES F A S C I C U L I M A T H E M A T I C I Nr 42 2009 Alfred O. Bosede NOOR ITERATIONS ASSOCIATED WITH ZAMFIRESCU MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES Abstract. In this paper, we establish some fixed

More information

Discrete Population Models with Asymptotically Constant or Periodic Solutions

Discrete Population Models with Asymptotically Constant or Periodic Solutions International Journal of Difference Equations ISSN 0973-6069, Volume 6, Number 2, pp. 143 152 (2011) http://campus.mst.edu/ijde Discrete Population Models with Asymptotically Constant or Periodic Solutions

More information

AW -Convergence and Well-Posedness of Non Convex Functions

AW -Convergence and Well-Posedness of Non Convex Functions Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it

More information

SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES

SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert

More information

On the simplest expression of the perturbed Moore Penrose metric generalized inverse

On the simplest expression of the perturbed Moore Penrose metric generalized inverse Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated

More information

Periodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca

Periodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca Periodic solutions of a class of nonautonomous second order differential systems with (q, p) Laplacian Daniel Paşca Abstract Some existence theorems are obtained by the least action principle for periodic

More information

Some notes on a second-order random boundary value problem

Some notes on a second-order random boundary value problem ISSN 1392-5113 Nonlinear Analysis: Modelling and Control, 217, Vol. 22, No. 6, 88 82 https://doi.org/1.15388/na.217.6.6 Some notes on a second-order random boundary value problem Fairouz Tchier a, Calogero

More information

ON THE CONTINUITY OF GLOBAL ATTRACTORS

ON THE CONTINUITY OF GLOBAL ATTRACTORS ON THE CONTINUITY OF GLOBAL ATTRACTORS LUAN T. HOANG, ERIC J. OLSON, AND JAMES C. ROBINSON Abstract. Let Λ be a complete metric space, and let {S λ ( ) : λ Λ} be a parametrised family of semigroups with

More information

A converse Lyapunov theorem for discrete-time systems with disturbances

A converse Lyapunov theorem for discrete-time systems with disturbances Systems & Control Letters 45 (2002) 49 58 www.elsevier.com/locate/sysconle A converse Lyapunov theorem for discrete-time systems with disturbances Zhong-Ping Jiang a; ; 1, Yuan Wang b; 2 a Department of

More information

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space

More information

Influence of the Stepsize on Hyers Ulam Stability of First-Order Homogeneous Linear Difference Equations

Influence of the Stepsize on Hyers Ulam Stability of First-Order Homogeneous Linear Difference Equations International Journal of Difference Equations ISSN 0973-6069, Volume 12, Number 2, pp. 281 302 (2017) ttp://campus.mst.edu/ijde Influence of te Stepsize on Hyers Ulam Stability of First-Order Homogeneous

More information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS

More information

On the effect of α-admissibility and θ-contractivity to the existence of fixed points of multivalued mappings

On the effect of α-admissibility and θ-contractivity to the existence of fixed points of multivalued mappings Nonlinear Analysis: Modelling and Control, Vol. 21, No. 5, 673 686 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.5.7 On the effect of α-admissibility and θ-contractivity to the existence of fixed points

More information

Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form

Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form arxiv:1206.3504v1 [math.ds] 15 Jun 2012 P. Pepe I. Karafyllis Abstract In this paper

More information

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

HAIYUN 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 information

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL

More information

PROJECTIONS ONTO CONES IN BANACH SPACES

PROJECTIONS ONTO CONES IN BANACH SPACES Fixed Point Theory, 19(2018), No. 1,...-... DOI: http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html PROJECTIONS ONTO CONES IN BANACH SPACES A. DOMOKOS AND M.M. MARSH Department of Mathematics and Statistics

More information

BIBO stabilization of feedback control systems with time dependent delays

BIBO 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 information

Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang

Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations

More information

INEQUALITIES IN METRIC SPACES WITH APPLICATIONS. Ismat Beg. 1. Introduction and preliminaries

INEQUALITIES IN METRIC SPACES WITH APPLICATIONS. Ismat Beg. 1. Introduction and preliminaries Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 17, 001, 183 190 INEQUALITIES IN METRIC SPACES WITH APPLICATIONS Ismat Beg Abstract. We prove the parallelogram inequalities

More information

On a functional equation connected with bi-linear mappings and its Hyers-Ulam stability

On a functional equation connected with bi-linear mappings and its Hyers-Ulam stability Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 017, 5914 591 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On a functional equation connected

More information

DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS. 1. Introduction

DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXIII, 2(2004), pp. 97 205 97 DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS A. L. SASU Abstract. The aim of this paper is to characterize the uniform exponential

More information

Existence Results for Multivalued Semilinear Functional Differential Equations

Existence 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 information

Spectrum of one dimensional p-laplacian Operator with indefinite weight

Spectrum of one dimensional p-laplacian Operator with indefinite weight Spectrum of one dimensional p-laplacian Operator with indefinite weight A. Anane, O. Chakrone and M. Moussa 2 Département de mathématiques, Faculté des Sciences, Université Mohamed I er, Oujda. Maroc.

More information

Stability Properties With Cone Perturbing Liapunov Function method

Stability Properties With Cone Perturbing Liapunov Function method Theoretical Mathematics & Applications, vol. 6, no., 016, 139-151 ISSN: 179-9687(print), 179-9709(online) Scienpress Ltd, 016 Stability Properties With Cone Perturbing Liapunov Function method A.A. Soliman

More information

Mathematical Journal of Okayama University

Mathematical Journal of Okayama University Mathematical Journal of Okayama University Volume 46, Issue 1 24 Article 29 JANUARY 24 The Perron Problem for C-Semigroups Petre Prada Alin Pogan Ciprian Preda West University of Timisoara University of

More information

On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist

On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist Nonlinear Analysis 49 (2002) 603 611 www.elsevier.com/locate/na On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist

More information

ON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad

ON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad ON TRIVIAL GRAIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -gradient Young measure supported on K must be trivial the

More information

A RETRACT PRINCIPLE ON DISCRETE TIME SCALES

A RETRACT PRINCIPLE ON DISCRETE TIME SCALES Opuscula Mathematica Vol. 26 No. 3 2006 Josef Diblík, Miroslava Růžičková, Barbora Václavíková A RETRACT PRINCIPLE ON DISCRETE TIME SCALES Abstract. In this paper we discuss asymptotic behavior of solutions

More information

J. Korean Math. Soc. 37 (2000), No. 4, pp. 593{611 STABILITY AND CONSTRAINED CONTROLLABILITY OF LINEAR CONTROL SYSTEMS IN BANACH SPACES Vu Ngoc Phat,

J. Korean Math. Soc. 37 (2000), No. 4, pp. 593{611 STABILITY AND CONSTRAINED CONTROLLABILITY OF LINEAR CONTROL SYSTEMS IN BANACH SPACES Vu Ngoc Phat, J. Korean Math. Soc. 37 (2), No. 4, pp. 593{611 STABILITY AND CONSTRAINED CONTROLLABILITY OF LINEAR CONTROL SYSTEMS IN BANACH SPACES Vu Ngoc Phat, Jong Yeoul Park, and Il Hyo Jung Abstract. For linear

More information

THE SEMI ORLICZ SPACE cs d 1

THE SEMI ORLICZ SPACE cs d 1 Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 269 276. THE SEMI ORLICZ SPACE cs d 1 N. SUBRAMANIAN 1, B. C. TRIPATHY 2, AND C. MURUGESAN 3 Abstract. Let Γ denote the space of all entire

More information

Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ

Absolutely 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 information

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets George Isac Department of Mathematics Royal Military College of Canada, STN Forces Kingston, Ontario, Canada

More information

Application of Measure of Noncompactness for the System of Functional Integral Equations

Application of Measure of Noncompactness for the System of Functional Integral Equations Filomat 3:11 (216), 363 373 DOI 1.2298/FIL161163A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Application of Measure of Noncompactness

More information

On the fixed point theorem of Krasnoselskii and Sobolev

On the fixed point theorem of Krasnoselskii and Sobolev Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/ On the fixed point theorem of Krasnoselskii and Sobolev Cristina G. Fuentes and

More information

3 Stability and Lyapunov Functions

3 Stability and Lyapunov Functions CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

More information

STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES

STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES WATARU TAKAHASHI, NGAI-CHING WONG, AND JEN-CHIH YAO Abstract. In this paper, we study nonlinear analytic

More information

Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces

Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Nonlinear Analysis 42 (2000) 561 572 www.elsevier.nl/locate/na Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Pavel Drabek a;, Nikos M. Stavrakakis b a Department

More information

Initial value problems for singular and nonsmooth second order differential inclusions

Initial value problems for singular and nonsmooth second order differential inclusions Initial value problems for singular and nonsmooth second order differential inclusions Daniel C. Biles, J. Ángel Cid, and Rodrigo López Pouso Department of Mathematics, Western Kentucky University, Bowling

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics ON A HYBRID FAMILY OF SUMMATION INTEGRAL TYPE OPERATORS VIJAY GUPTA AND ESRA ERKUŞ School of Applied Sciences Netaji Subhas Institute of Technology

More information

Renormings of c 0 and the minimal displacement problem

Renormings of c 0 and the minimal displacement problem doi: 0.55/umcsmath-205-0008 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVIII, NO. 2, 204 SECTIO A 85 9 ŁUKASZ PIASECKI Renormings of c 0 and the minimal displacement problem Abstract.

More information

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous

More information

ACCURATE SOLUTION ESTIMATE AND ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE SYSTEM

ACCURATE SOLUTION ESTIMATE AND ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE SYSTEM Sutra: International Journal of Mathematical Science Education c Technomathematics Research Foundation Vol. 1, No. 1,9-15, 2008 ACCURATE SOLUTION ESTIMATE AND ASYMPTOTIC BEHAVIOR OF NONLINEAR DISCRETE

More information

The Existence of Maximal and Minimal Solution of Quadratic Integral Equation

The Existence of Maximal and Minimal Solution of Quadratic Integral Equation The Existence of Maximal and Minimal Solution of Quadratic Integral Equation Amany M. Moter Lecture, Department of Computer Science, Education College, Kufa University, Najaf, Iraq ---------------------------------------------------------------------------***--------------------------------------------------------------------------

More information

Oscillatory Mixed Di erential Systems

Oscillatory Mixed Di erential Systems Oscillatory Mixed Di erential Systems by José M. Ferreira Instituto Superior Técnico Department of Mathematics Av. Rovisco Pais 49- Lisboa, Portugal e-mail: jferr@math.ist.utl.pt Sra Pinelas Universidade

More information

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1 Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium

More information