Homoclinic solutions of difference equations with variable exponents

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1 Homoclinic solutions of difference equations with variable exponents Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Stepan Tersian d a Department of Mathematics, University of Craiova, Craiova, Romania b Department of Mathematics, Central European University, 1051 Budapest, Hungary c Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, Bucharest, Romania d Department of Mathematical Analysis, University of Rousse, 7017 Rousse, Bulgaria addresses: mmihailes@yahoo.com vicentiu.radulescu@imar.ro sterzian@ru.acad.bg Abstract. We study the existence of homoclinic solutions for a class of non-homogeneous difference equation with periodic coefficients. Our proofs rely on the critical point theory combined with adequate variational techniques, which are mainly based on the mountain-pass lemma Mathematics Subject Classification. Primary: 39A14. Secondary: 37C29, 58E05. Key words: homoclinic solution, non-homogeneous difference equation, variable exponent, mountain-pass lemma. Dedicated to Professor Gheorghe Moroşanu on the occasion of his 60-th birthday 1 Introduction Partial difference equations usually describe the evolution of certain phenomena over the course of time. Elementary but relevant examples of partial difference equations are concerned with heat diffusion, heat control, temperature distribution, population growth, cellular neural networks, etc. We are concerned in this paper with a class of partial difference equations involving a nonhomogeneous operator. Our Correspondence address: Vicenţiu Rădulescu, Department of Mathematics, University of Craiova, Craiova, Romania. vicentiu.radulescu@imar.ro 1

2 interest for problems of this type is motivated by major applications of differential difference operators to various applied fields, such as electrorheological (smart) fluids, space technology, robotics, image processing, etc. Let T > 0 be a given natural number let p( ), q( ) : Z [2, ), V ( ) : Z R be three T -periodic functions f(k, t) : Z R R be a continuous function in t R T -periodic in k. This paper is devoted to the study of the difference non-homogeneous equations of type 2 p(k 1) u(k 1) V (k) u(k) q(k) 2 u(k) + f(k, u(k)) = 0 for k Z, u(k) 0 as k, (1) where 2 p( ) sts for the p( )-Laplace difference operator, that is, 2 p(k 1) u(k 1) = u(k) p(k) 2 u(k) u(k 1) p(k 1) 2 u(k 1), (2) for each k Z. We have denoted by the difference operator, which is defined by u(k) = u(k + 1) u(k), for each k Z. The goal of the present paper is to establish the existence of nontrivial homoclinic solutions for problem (1). In order to explain the notion of homoclinic solution we go back to the definition of homoclinic orbit, which was introduced by H. Poincaré [14] in the context of Hamiltonian systems. More exactly, Poincaré called a trajectory x(t) a homoclinic orbit (or doubly asymptotic trajectory) if it is asymptotic to a constant as t. Since we are seeking solutions u(k) for problem (1) satisfying lim k u(k) = 0, in accord with the above discussion, we are interested in finding nontrivial homoclinic solutions for problem (1). Throughout the years the study of existence of homoclinic orbits by means of variational methods has captured a special attention (see, e.g., [5], [13], [8], [15], [16], [17] the references therein). Particularly, we point out the very recent advances in this field obtained by C. Li [10] A. Cabada, C. Li & S. Tersian [4]. More exactly, in [10] a unified approach to the existence of homoclinic orbits for some classes of ODE s with periodic potentials is established while in [4] the authors extend the ideas from [10] to the case of homoclinic orbits for discrete p-laplacian type equations. Motivated by the studies in [10] [4], we focus in the present paper on the case of non-homogeneous difference equations. An important feature of this paper is the presence of the nonconstant potential p( ). The study of difference equations involving non-homogeneous difference operators of type (2) was initiated by M. Mihăilescu, V. Rădulescu & S. Tersian in [12], where some eigenvalue problems were investigated. After our best knowledge the present paper represents a first attempt in finding homoclinic solutions for non-homogeneous difference equations. Finally, we remember that boundary value problems involving difference operators have been intensively studied in the last decade. In this context we point out the results obtained in the papers of 2

3 R. P. Agarwal, K. Perera & D. O Regan [1], A. Cabada, A. Iannizzotto & S. Tersian [3], H. Fang & D. Zhao [7], M. Ma & Z. Guo [11], A. Kristály, M. Mihăilescu, V. Rădulescu & S. Tersian [9]. 2 Main result Throughout this paper we denote p + := sup p(k) p := inf p(k) we assume that q + := sup q(k) q := inf q(k), 1 < q q + < p p +. (3) We also assume that the T -periodic function V satisfies the supplementary conditions (V1) 0 < V 0 := min{v (0),..., V (T 1)}; (V2) V 0 < q +. while the continuous function f = f(k, t) : Z R R which is assumed to be T -periodic in k verifies (F1) there exist α > p + r > 0 such that αf (k, t) := α t 0 f(k, s) ds tf(k, t), k Z, t 0, F (k, t) > 0, k Z, t r ; (F2) f(k, t) = o( t q+ 1 ) as t 0. In order to present our main result, we introduce for each p( ) : Z (1, ) the space { l p( ) := u : Z R; ρ p( ) (u) := } u(k) p(k) <. On l p( ) we introduce the Luxemburg norm { u p( ) := inf µ > 0; u(k) µ p(k) 1 }. It is easy to check that the following relations hold true u p( ) < 1 u p+ p( ) ρ p( )(u) u p p( ), (4) u p( ) > 1 u p p( ) ρ p( )(u) u p+ p( ), (5) u p( ) 0 ρ p( ) (u) 0. (6) 3

4 We also consider the space { } l = u : Z R; u := sup u (k) <. We start with the following embedding property. Proposition 1. Assume condition (3) is fulfilled. Then l q( ) l p( ). Proof. Indeed, if u(k) p(k) < then there exists S > 0 such that u(k) q(k) 1, k > S. It follows that u(k) 1, k > S. By relation (3) we infer that q(k) < p(k) for all k Z. That fact the above inequality assure that u(k) p(k) u(k) q(k), k > S, the conclusion of Proposition 1 is obvious now. By Proposition 1, relation (3) the hypotheses on functions V f we infer that the natural space where we should seek homoclinic solutions for (1) is l q( ). Thus, we say that u l q( ) is a homoclinic solution for (1) if u(k 1) p(k 1) 2 u(k 1) v(k 1) + V (k) u(k) q(k) 2 u(k)v(k) f(k, u(k))v(k) = 0, for all v l q( ) lim u(k) = 0. k The main result of this paper is given by the following theorem. Theorem 1. Assume hypotheses (3), (V1)-(V2) (F1)-(F2) are fulfilled. Then problem (1) possesses at least a nontrivial homoclinic solution. Moreover, given a nontrivial homoclinic solution u of problem (1), there exist two integers S 1 S 2 with S 1 S 2 such that for all k > S 2 all k < S 1 the sequence u(k) is strictly monotone. 3 Auxiliary results The basic idea in proving Theorem 1 is to consider the associate energetic functional of problem (1) to show that it possesses a nontrivial critical point by using the mountain-pass lemma, see [2]. In order to do that we first introduce the following notations: ϕ p(t) (t) := t p(t) 2 t Φ p(t) (t) := t p(t) p(t). 4

5 Note that 2 p(k 1) u(k 1) = (ϕ p(k 1)( u(k 1))). Next, we introduce the functional A : l q( ) R defined by A(u) := Φ p(k 1) ( u(k 1)) + V (k)φ q(k) (u(k)). Now we define the energetic functional associate to problem (1) as J : l q( ) R defined by J(u) := A(u) F (k, u(k)). Stard arguments show that J C 1 (l q( ), R) with the derivative given by J (u), v = [ u(k 1) p(k 1) 2 u(k 1) v(k 1) + V (k) u(k) q(k) 2 u(k)v(k) f(k, u(k))v(k)], for all u, v l q( ). Thus, we found that the critical points of J correspond to the weak solutions of equation (1). In order to facilitate further computations we point out that on l q( ) we can introduce an equivalent norm with q( ), namely u q( ) := inf { µ > 0; V (k) q(k) It is easy to check the following properties of the above norm u(k) µ u q( ) < 1 u q+ q( ) V (k) q(k) u(k) q(k) u q q( ), (7) u q( ) > 1 u q q( ) V (k) q(k) u(k) q(k) u q+ q( ), (8) u q( ) 0 The next lemma assures that J has a mountain-pass geometry. q(k) 1 }. V (k) q(k) u(k) q(k) 0. (9) Lemma 1. Assume the hypotheses of Theorem 1 are fulfilled. Then there exists ϱ > 0 ν > 0 e l q( ) with e q( ) > ϱ such that i) J(u) ν for all u l q( ) with u q( ) > ϱ; ii) J(e) < 0. 5

6 Proof. i) Condition (F2) implies that there exists δ (0, 1) such that for all t δ all k Z. Define F (k, t) V 0 2q + t q+ V 0 2q + t q(k), (10) ϱ := ( ) 1/q V0 q + δ q+ /q. By condition (V2) we deduce that ϱ (0, 1). Then for any u q( ) = ϱ relation (7) yields ϱ q = V 0 q + δq+ = u q q( ) V (i) q(i) u(i) q(i) V 0 q + u(k) q(k), i Z for all k Z. It follows that 1 > δ q+ u(k) q(k), for all k Z. Consequently, u(k) < 1 for every k Z thus u(k) q(k) u(k) q+, for all k Z. The above inequalities show that actually for all k Z. Next, by (10) we deduce δ u(k), F (k, u(k)) V 0 2q + u(k) q(k) 1 2 provided u q( ) = ϱ. Defining V (k) q(k) u(k) q(k), ν := ϱq+ 2, we get by (7) that for each u with u q( ) = ϱ it holds true the following estimates J(u) = A(u) F (k, u(k)) 1 2 V (k) q(k) u(k) q(k) 1 V (k) 2 q(k) u(k) q(k) V (k) q(k) u(k) q(k) 1 2 u q+ q( ) = ν. Thus, the first part of the lemma is verified. ii) We infer by condition (F1) that there exist two constants c 1 > 0 c 2 > 0 such that F (k, t) c 1 t α c 2, 6

7 for all k Z all t R. Define v l q( ) by v(0) = a > 0 v(k) = 0 if k 0. We find J(ηv) = A(ηv) F (k, ηv(k)) = 2 p(0) ηp(0) a p(0) + V (0) ηq(0) a q(0) c 1 η α a α + c 2, q(0) for each η > 0. Since by relation (3) we have α > p + p(0) p > q + q(0) the above relation shows that for any η > 0 sufficiently large we have J(ηv) < 0. The proof of Lemma 1 is complete. In order to prove the next result, we recall that given c R, we say that a sequence (u(k)) l q( ) is said to satisfy the (P S) c condition if J (u(k)) c J (u(k)) 0. Lemma 2. Assume the hypotheses of Theorem 1 are fulfilled. Then, there exists c > 0 a bounded (P S) c sequence for J in l q( ). Proof. Lemma 1 the mountain-pass lemma imply that there exists a sequence {u n } l q( ) such that J(u n ) c, J (u n ) 0, (11) where c := inf γ Γ max t [0,1] J(γ(t)) Γ := {γ C([0, 1], l q( ) ); γ(0) = 0, γ(1) = e}, with e given in Lemma 1 ii). We verify that {u n } is bounded in l q( ). Indeed, assuming that u n q( ) > 1 for each n we deduce by condition (F1) relation (8) that αj(u n ) J (u n ), u n = ( ) α p(k) 1 u n (k 1) p(k 1) + ( ) α q(k) 1 V (k) u n (k) q(k) [αf (k, u n (k)) f(k, u n (k))u n (k)] (α q + ) u n q q( ), for all n. The above estimates condition (11) show that {u n } is bounded in l q( ). The proof of Lemma 2 is complete. 7

8 4 Proof of Theorem 1 Assume {u n } is the sequence given by Lemma 2. Then for each n N the sequence { u n (k) ; k Z} l q( ) is bounded u n (k) 0 as k. Assume that { u n (k) } achieves its maximum in k n Z. Undoubtedly, there exists j n Z such that j n T k n < (j n + 1)T, define w n (k) := u n (k j n T ). Then { w n (k) } attains its maximum in The T -periodicity of p( ), q( ) V ( ) implies i n := k n j n T [0, T ]. V (k) q(k) u n(k) q(k) = V (k) q(k) w n(k) q(k), J(u n ) = J(w n ). Since {u n } n N is bounded in l q( ) the above estimates relations (7) (8) yield that {w n } n N is bounded in l q( ), too. Then there exists w l q( ) such that w n converges weakly to w in l q( ) as n. We claim that w n (k) w(k) as n for each k Z. Indeed, defining the test function v m l q( ) by 1 if j = m v m (j) := 0 if j m, taking into account the weak convergence of w n to w in l q( ) we find for all k Z. The claim is clear now. lim w n(k) = lim w n, v k = w, v k = w(k), n n Next, we point out that for each v l q( ) we have J (w n ), v = J (u n ), v( + j n T ) J (u n ) v q( ), which in view of relation (11) from Lemma 2 implies J (w n ) 0 as n. It follows that for each v l q( ) we have [ϕ p(k 1) ( w n (k 1)) v(k 1) + V (k)ϕ q(k) (w n (k))v(k) f(k, w n (k))v(k) 0] 0, (12) 8

9 as n. Consider v l q( ) has compact support, hence there exist a, b Z, a < b such that v (k) = 0 if k Z \ [a, b] v (k) 0 if k {a + 1, b 1}. The set of compact support functions, denoted by l q( ) 0 is dense in l q( ). That fact can be easily explained since for each v l q( ) we can define v n l q( ) 0 by V (j) q(j) v(j) v n(j) 0 as n, v n (j) = 0 if j n + 1 v n (j) = v(j) if j = n we have j Z or, by relation (9), v v n q( ) 0 as n. Now, for each v l q( ) 0 in (12) taking into account the finite sums the continuity of f(k, ) we obtain by passing to the limit as n that [ϕ p(k 1) ( w(k 1)) v(k 1) + V (k)ϕ q(k) (w(k))v(k) f(k, w(k))v(k) 0] 0. We found that w is a critical point of J consequently a solution of (1). We show that w 0. Assume by contradiction the contrary, that means w = 0. Then we have u n = w n = max{ w n (k) ; k Z} 0, as n. On the other h, condition (F2) implies that for a given ɛ > 0 there exists δ (0, 1) such that F (k, t) ɛ t q+ (13) f(x, t)t ɛ t q+, for all k {0, 1,..., T 1} all t < δ. The above inequalities show that for every k {0, 1,..., T 1} there exists M k such that for n > M k we have w n (k) < δ. Since i n {0, 1,..., T 1} it follows that for n > M := max{m n ; k {0, 1,..., T 1}} every k Z we have That fact relation (13) imply w n (k) w n (i n ) < δ < 1. F (k, w n (k)) ɛ w n (k) q+ ɛ w n (k) q(k), f(k, w n (k))w n (k) ɛ w n (k) q+ ɛ w n (k) q(k). 9

10 We infer that for each n > M every k Z the following estimates hold true 0 < q J (w n ) = q 1 p(k) w n (k 1) p(k 1) + q V (k) q(k) w n (k) q(k) q F (k, w n (k)) w n (k 1) p(k 1) + V (k) w n (k) q(k) f (k, w n (k)) w n (k) ( q F (k, w n (k)) f (k, w n (k)) w n (k) ) J (w n ), w n + q F (k, w n (k)) + f (k, w n (k)) w n (k) J (w n ), w n + q ɛ w n (k) q(k) + ɛ w n (k) q(k) ( ) J (w n ), w n + q ɛ q+ + ɛ q+ V (k) V 0 V 0 q(k) w n(k) q(k) J (w n ) w n q( ) + ɛ q+ (q + 1) [ ] w n q+ V q( ) + w n q q( ). 0 Taking into account that u n q( ) is bounded, J (u n ) 0 as n ɛ > 0 is arbitrary we find by the above estimates a contradiction with J(w n ) c > 0 as n. Thus, we have that w is a nontrivial solution of problem (1). Next, let u be a nonzero homoclinic solution of problem (1). Assume that it attains positive local maximums negative local minimums at infinitely many points k n. In particular we can assume that { k n }. Consequently, 2 p(k n 1) u (k n 1) u(k n ) 0, u(k n ) 0, as n. Using that facts multiplying in (1) by u(k n )/ u(k n ) q(kn), we have f (k n, u (k n )) u(k n ) u(k n ) q(kn) 2 p(kn 1) u (k n 1) u(k n ) u(k n ) q(kn) + f (k n, u (k n )) u(k n ) u(k n ) q(kn) = V (k n ) V 0 > 0. (14) Using (14) condition (F2) we deduce f (k n, u (k n )) u(k n ) 0 = lim n u(k n ) q(k V 0 > 0, n) which represents a contradiction. Consequently u does not attain positive local maximums negative local minimums at infinitely many points. Assume now that function u vanishes at infinitely many points l n. By condition (F2) we find that 2 p(l n 1) u (l n 1) = 0, consequently, u(l n 1) u(l n + 1) < Therefore it has an unbounded

11 sequence of positive local maximums negative local minimums, in contradiction with the previous assertion. We proved that, for k large enough, function u has constant sign it is strictly monotone. The proof of Theorem 1 is complete. Remark 1. Note, that the homogeneous problem 2 p(k 1) u(k 1) V (k) u(k) q(k) 2 u(k) = 0 for k Z, u(k) 0 as k, (15) has only the trivial solution. Indeed, if u is positive or negative, let k 0 be the point of his positive maximum or negative minimum. Then 2 p(k 0 1) u(k 0 1)u(k 0 ) 0 0 = 2 p(k 0 1) u(k 0 1)u(k 0 ) V (k 0 ) u(k 0 ) q(k 0) < 0, which is a contradiction. The same conclusion can be made if u is sign-changing. Note that under assumptions of Theorem 1, for every λ > 0 one can prove, following the same approach, the existence of a nontrivial solution for the eigenvalue problem 2 p(k 1) u(k 1) V (k) u(k) q(k) 2 u(k) + λf(k, u(k)) = 0 for k Z, u(k) 0 as k, Above remark shows that λ = 0 is a bifurcation point of problem (16). holds: Remark 2. We can prove that if in addition to conditions (F1) (F2) the following condition (F3) f(k, t) 0 for any t < 0 all k Z, the homoclinic solution of the problem (1) is positive. Indeed, let u be a homoclinic solution of the problem assume that (F3) holds. Suppose that there exists k 0 such that u(k 0 ) < 0 let k 1 be such that u(k 1 ) = min {u(k), k Z} < 0. In consequence 2 p(k 1 1) u (k 1 1) 0, which implies that f (k 1, u (k 1 )) = 2 p(k 1 1) u (k 1 1) + V (k 1 ) u (k 1 ) q(k 1) 2 u (k 1 ) < 0, in contradiction with (F3). Then u(k) 0 for every k Z. Suppose that u(k 2 ) = 0 for some k 2 Z. Note, that if u(k 2 + 1) = 0 or u(k 2 1) = 0, the solution is identically zero by a recursion f(k, 0) = 0. Hence 2 p(k 2 1) u(k 2 1) > 0 we arrive to a contradiction. Hence, the solution u is positive. We summarize in what follows the above remark. Theorem 2. Suppose that the functions V : Z R f : Z R satisfy assumptions of Theorem 1 (F3). Then, problem (1) has a positive homoclinic solution. 11 (16)

12 Remark 3. In the case q(k) = 2 we can estimate the maximum of the positive solution u, provided the additional assumption (F4) Let f(k, t) has the form f(k, t) = tg(k, t), where g(k, t) is T -periodic in k, g(k, 0) = 0 for each k, g(k, t) is increasing in t for t > 0. Let g 1 (k, t) be the inverse function of g(k, t) for t > 0. We have that g 1 (k, t) is increasing in t for t > 0. Let u be a positive homoclinic solution of the problem (1) u(k 0 ) > 0 is its maximum. Note that, in view of the periodicity of coefficients, if u(.) is a solution of problem (1), then u(. + jt ), j Z is also a solution of (1). Hence, we may assume that k 0 [0, T 1]. We have 2 p(k 0 1) u (k 0 1) 0 u(k 0 )g (k 0, u (k 0 )) V (k 0 ) u (k 0 ) 0, by properties of g V u(k 0 ) g 1 (k 0, V 0 ). Thus max{u(k) : k [0, T 1]} min{g 1 (k, V 0 ) : k [0, T 1]}. (17) Thus, we have obtained the following result. Corollary 1. Suppose that the functions V : Z R f : Z R satisfy assumptions of Theorem 2 (F4). Then the positive homoclinic solution of the equation 2 pu (k 1) V (k)u(k) + u(k)g(k, u(k)) = 0 satisfies relation (17). Example. Consider the equations 2 p(k 1) u(k 1) V (k)u(k) + au2 (k) + bu 3 (k) = 0, k Z, (18) 2 p(k 1) u(k 1) V (k)u(k) + au2 (k) + bu 3 +(k) = 0, k Z, (19) where u + (k) = max{u(k), 0}, 2 < p(k) < 3, p(k) V (k) are T -periodic, V 0 < 2, a > 0 b > 0. All assumptions of Theorem 2 are satisfied for f(k, t) = at 2 + bt 5 + there exists a positive homoclinic solution u of equation (19), which is a homoclinic solution of problem (18). Moreover, we can estimate max{u(k) : k Z}. Let u(k 0 ) > 0 be the maximum of u. Then 2 p(k 0 1) u(k 0 1) 0 by equation (18) we have V (k 0 )u(k 0 ) + au 2 (k 0 ) + bu 3 (k 0 ) 0, which implies au(k 0 ) + bu 2 (k 0 ) V 0 0. Hence max{u(k) : k Z} 12 a 2 + 4bV 0 a. 2b

13 Acknowledgments. The research of M. Mihăilescu was partially supported by Grant CNCSIS PN II-RU PD-117/2010 Probleme neliniare modelate de operatori diferenţiali neomogeni. V. Rădulescu acknowledges the support through Grant CNCSIS PCCE 55/2008 Sisteme diferenţiale în analiza neliniară şi aplicaţii. References [1] R. P. Agarwal, K. Perera D. O Regan, Multiple positive solutions of singular nonsingular discrete problems via variational methods, Nonlinear Analysis 58 (2004), [2] A. Ambrosetti P. H. Rabinowitz, Dual variational methods in critical point theory applications, J. Functional Analysis 14 (1973), [3] A. Cabada, A. Iannizzotto S. Tersian, Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009), [4] A. Cabada, C. Li S. Tersian, On homoclinic solutions of a semilinear p-laplacian difference equation with periodic cofficients, in preparation. [5] V. Coti Zelati P. H. Rabinowitz, Homoclinic orbits for Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), [6] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, [7] H. Fang D. Zhao, Existence of nontrivial homoclinic orbits for fourth-order difference equations,applied Mathematics Computation, 214 (2009), [8] M. R. Grossinho, F. Minhos S. Tersian, Positive homoclinic solutions for a class of second order differential equations, J. Math. Anal. Appl. 240(1) (1999), [9] A. Kristály, M. Mihăilescu, V. Rădulescu S. Tersian, Spectral estimates for a nonhomogeneous difference problem, Communications in Contemporary Mathematics, in press. [10] C. Li, A unified proof on existence of homoclinic orbits for some semilinear ordinary differential equations with periodic potentials, J. Math. Anal. Appl. 365 (2010), [11] M. Ma Z. Guo, Homoclinic orbits for second order self-adjoint differeence equations, J. Math. Anal. Appl. 323 (2006), [12] M. Mihăilescu, V. Rădulescu S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, Journal of Difference Equations Applications 15 (2009), [13] W. Omana M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5 (1992), [14] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, [15] P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), [16] K. Tanaka, Homoclinic orbits in a first-order superquadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), [17] S. Tersian J. Chaparova, Periodic homoclinic solutions of extended Fisher Kolmogorov equations, J. Math. Anal. Appl., 260 (2001),