Oscillation Criteria for Delay and Advanced Difference Equations with General Arguments

Transcription

1 Advances in Dynamical Systems Applications ISSN , Volume 8, Number, pp (013) Oscillation Criteria for Delay Advanced Difference Equations with General Arguments I. P. Stavroulakis University of Ioannina Department of Mathematics Ioannina, Greece Dedicated to the memory of Professor Panayiotis D. Siafarikas, a very good friend colleague. Abstract Consider the first-order delay difference equation x(n) + p(n)x(τ(n)) = 0, n 0, the first-order advanced difference equation x(n) p(n)x(µ(n)) = 0, n 1, [ x(n) p(n)x(ν(n)) = 0, n 0], where denotes the forward difference operator x(n) = x(n + 1) x(n), denotes the backward difference operator x(n) = x(n) x(n 1), {p(n)} is a sequence of nonnegative real numbers, {τ(n)} is a sequence of positive integers such that τ(n) n 1, for all n 0, {µ(n)} [{ν(n)}] is a sequence of positive integers such that µ(n) n + 1 for all n 1, [ν(n) n + for all n 0]. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given. AMS Subject Classifications: 39A1, 39A1. Keywords: Oscillation, delay, advanced difference equations. Received October 31, 01; Accepted November 0, 01 Communicated by Ondřej Došlý

2 350 I. P. Stavroulakis 1 Introduction The problem of establishing sufficient conditions for the oscillation of all solutions to the first-order delay difference the advanced difference equation with variable arguments of the form x(n) + p(n)x(τ(n)) = 0, n 0 (1.1) x(n) p(n)x(µ(n)) = 0, n 1, [ x(n) p(n)x(ν(n)) = 0, n 0], (1.) where denotes the forward difference operator x(n) = x(n +1) x(n), denotes the backward difference operator x(n) = x(n) x(n 1), {p(n)} is a sequence of nonnegative real numbers, {τ(n)} is a sequence of positive integers such that τ(n) n 1, for all n 0, {µ(n)} [{ν(n)}] is a sequence of positive integers such that µ(n) n + 1 for all n 1, [ν(n) n + for all n 0] has been recently investigated by several authors. See, for example, [1 9, 11, 18,, 31, 3, 36 39, 41] the references cited therein. Strong interest in equations (1.1) (1.) is motivated by the fact that they represent the discrete analogues of the delay the advanced differential equation x (t) + p(t)x(τ(t)) = 0, t t 0, (1.3) x (t) p(t)x(µ(t)) = 0, t t 0, (1.4) where p, τ, µ C([t 0, ), R + ), R + = [0, ), τ(t), µ(t) are nondecreasing τ(t) < t µ(t) > t for t t 0 (see, for example, [10, 1 1, 3 30, 33 35, 40, 4, 43] the references cited therein). By a solution of Eq. (1.1), we mean a sequence x(n) which is defined for n min {τ(n) : n 0} which satisfies Eq. (1.1) for all n 0. By a solution of Eq. (1.), we mean a sequence of real numbers {x(n)} which is defined for n 0 satisfies (1.) for all n 1 [n 0]. As usual, a solution {x(n)} is said to be oscillatory if for every positive integer n 0 there exist n 1, n n 0 such that x(n 1 )x(n ) 0. In other words, a solution {x(n)} is oscillatory if it is neither eventually positive nor eventually negative. Otherwise, the solution is called nonoscillatory. In this paper our purpose is to present the most interesting sufficient conditions for the oscillation of all solutions to the above equations (1.1) (1.), especially in the case where 0 < n 1 i=τ(n) p(i) 1 e n p(i) < 1 i=τ(n)

3 Oscillation Criteria for Delay Advanced Difference Equations 351 for Eq. (1.1), for Eq. (1.). µ(n) p(i) ν(n) 1 p(i) < 1 Oscillation Criteria for Eq. (1.1) In this section we study the delay difference equation with variable argument x(n) + p(n)x(τ(n)) = 0, n = 0, 1,,..., (.1) where x(n) = x(n+1) x(n), {p(n)} is a sequence of nonnegative real numbers {τ(n)} is a nondecreasing sequence of integers such that τ(n) n 1 for all n 0 lim τ(n) =. In 008, Chatzarakis, Koplatadze Stavroulakis [, 3] investigated for the first time the oscillatory behavior of Eq. (.1) in the case of a general delay argument τ(n) derived the following theorems. Theorem.1 (See [3]). If then all solutions of Eq. (.1) oscillate. Theorem. (See []). Assume that n i=τ(n) n 1 i=τ(n) := Then all solutions of Eq. (.1) oscillate. p(i) > 1, (.) p(i) < + (.3) n 1 i=τ(n) p(i) > 1 e. (.4) Remark.3. It should be mentioned that in the case of the delay differential equation x (t) + p(t)x(τ(t)) = 0, t t 0 (.5) it has been proved (see [3, 7]) that either one of the conditions t τ(t) p(s)ds > 1 (.6)

4 35 I. P. Stavroulakis or t τ(t) p(s)ds > 1 e, (.7) implies that all solutions of Eq. (.5) oscillate. Therefore, conditions (.) (.4) are the discrete analogues of conditions (.6) (.7). Remark.4 (See []). Note that condition (.3) is not a limitation since, if (.) holds then all solutions of Eq. (.1) oscillate. Remark.5 (See []). Condition (.4) is optimal for Eq. (.1) under the assumption that (n τ(n)) =, since in this case the set of natural numbers increases infinitely lim n + in the interval [τ(n), n 1] for n. Now, we are going to present an example to show that condition (.4) is optimal, in the sense that it cannot be replaced by the nonstrong inequality. Example.6 (See []). Consider Eq. (.1), where τ(n) = [βn], p(n) = ( n λ (n + 1) λ) ([βn]) λ, β (0, 1), λ = ln 1 β (.8) [βn] denotes the integer part of βn. It is obvious that n 1+λ ( n λ (n + 1) λ) λ for n. Therefore n ( n λ (n + 1) λ) ([βn]) λ λ e for n. (.9) Hence, in view of (.8) (.9), we have n 1 i=τ(n) p(i) = λ e = λ e n 1 i=[βn] n 1 i=[βn] e λ i ( i λ (i + 1) λ) ([βi]) λ. 1 i 1 i = λ e ln 1 β = 1 e or n 1 i=τ(n) p(i) = 1 e. (.10) Observe that all conditions of Theorem. are satisfied except condition (.4). In this case it is not guaranteed that all solutions of Eq. (.1) oscillate. Indeed, it is easy to see that the function u = n λ is a positive solution of Eq. (.1).

5 Oscillation Criteria for Delay Advanced Difference Equations 353 As it has been mentioned above, it is an interesting problem to find new sufficient conditions for the oscillation of all solutions of the delay difference equation (.1), in the case where neither (.) nor (.4) is satisfied. In Chatzarakis, Koplatadze Stavroulakis [3] Chatzarakis, Philos Stavroulakis [4, 5] derived the following theorem. Theorem.7 (See [3 5]). Assume that 0 < 1. Then either one of the conditions e or n j=τ(n) n j=τ(n) n j=τ(n) p (j) > 1 ( 1 1 ), (.11) p (j) > 1 1 p (j) > 1 1 implies that all solutions of Eq. (.1) oscillate. If 0 < 1 in addition, e ( 1 1 ), (.1) (1 1 ) (.13) p(n) 1 1 for all large n, (.14) n j=τ(n) p (j) > (.15) or if 0 < 6 4 in addition, p(n) for all large n, (.16) n j=τ(n) p (j) > then all solutions of Eq. (.1) oscillate. ( ), (.17) Remark.8. In the case where the sequence {τ(n)} is not assumed to be nondecreasing, define [ 5] σ(n) = max {τ(s) : 0 s n, s N}. Clearly, the sequence of integers {σ(n)} is nondecreasing. In this case, Theorems.1,..7 can be formulated in a more general form. More precisely in conditions (.), (.4), (.11), (.1), (.13), (.15) (.17), the term τ(n) is replaced by σ(n).

6 354 I. P. Stavroulakis Remark.9. Observe the following: (i) When 0 < 1, it is easy to verify that e 1 1 > > > 1 1 > (1 1 ) therefore condition (.13) is weaker than conditions (.15), (.1) (.11). (ii) When 0 < 6 4, it is easy to show that 1 ( 3 ) > 1 (1 ) 1, therefore in this case when (.16) holds, inequality (.17) improves inequality (.13) especially, when = , the lower bound in (.13) is while in (.17) is Example.10 (See [4]). Consider the equation where x(n) + p(n)x(n ) = 0, p(3n) = , p(3n + 1) =, p(3n + ) =, n = 0, 1,, Here τ(n) = n it is easy to see that = n 1 j=n p(j) = = 0.96 < n j=n ( ) , 3 p(j) = = Observe that > 1 1 ( ) , that is, condition (.1) of Theorem.7 is satisfied therefore all solutions oscillate. Also, condition (.13) is satisfied. Observe, however, that ( ) < 1, = 0.96 < , < 1 ( 1 1 )

7 Oscillation Criteria for Delay Advanced Difference Equations 355 therefore none of the conditions (.), (.4) (.11) are satisfied. If, on the other h, in the above equation p(3n) = p(3n + 1) = , p(3n + ) =, n = 0, 1,,..., it is easy to see that = n 1 j=n p(j) = = 0.96 < ( ) n j=n p(j) = = Furthermore, it is clear that p(n) for all large n. In this case 0.91 > 1 1 ( 3 ) , that is, condition (.17) of Theorem.7 is satisfied therefore all solutions oscillate. Observe, however, that ( ) < 1, = 0.96 < , < 1 ( 1 1 ) , 0.91 < 1 1 ( 1 ) , 0.91 < 1 1 (1 ) therefore, none of the conditions (.), (.4), (.11), (.1) (.13) are satisfied. 3 Oscillation Criteria for Eq. (1.) In this section, we study the advanced difference equation with variable argument x(n) p(n)x(µ(n)) = 0, n 1, [ x(n) p(n)x(ν(n)) = 0, n 0], (3.1) where denotes the backward difference operator x(n) = x(n) x(n 1), denotes the forward difference operator x(n) = x(n + 1) x(n), {p(n)} is a sequence of nonnegative real numbers {µ(n)} [{ν(n)}] is a sequence of positive integers such that µ(n) n + 1 for all n 1, [ν(n) n + for all n 0].

8 356 I. P. Stavroulakis In the special case where µ(n) = n + k, [ν(n) = n + σ] the advanced difference equations (3.1) take the form x(n) p(n)x(n + k) = 0, n 1, [ x(n) p(n)x(n + σ) = 0, n 0], where k is a positive integer greater or equal to one σ is a positive integer greater or equal to two. In [19], the advanced difference equation with constant argument is studied proved that if or x(n) p(n)x(n + σ) = 0, n 0, n+σ 1 p(i) > 1, (3.) n+σ 1 +1 ( ) σ σ 1 p(i) >, (3.3) σ then all solutions oscillate. Very recently, Chatzarakis Stavroulakis [6, 8] investigated for the first time the oscillatory behavior of Eq. (3.1) with variable argument established the following theorems: Theorem 3.1 (See [8]). Assume that the sequence {µ(n)} [{ν(n)}] is nondecreasing. If µ(n) ν(n) 1 p(i) p(i) > 1, (3.4) then all solutions of Eq. (3.1) oscillate. Theorem 3. (See [6, 8]). Assume that the sequence {µ(n)} [{ν(n)}] is nondecreasing µ(n) +1 p (i) ν(n) 1 +1 p (i) =. (3.5) If 0 < 1 µ(n) p(i) ν(n) 1 p(i) > 1 ( 1 1 ), (3.6)

9 Oscillation Criteria for Delay Advanced Difference Equations 357 or if 0 < 1/ µ(n) p(i) ν(n) 1 p(i) > 1 1 ( ) 1 1, (3.7) then all solutions of Eq. (3.1) oscillate. If 0 < < (3 5 5)/, in addition p(n) 1 1 for all large n µ(n) ν(n) 1 ( ) p(i) p(i) 1 > , (3.8) or if 0 < 6 4, in addition p(n) µ(n) ν(n) 1 p(i) p(i) > then all solutions of Eq. (3.1) oscillate. for all large n ( ), (3.9) Remark 3.3. In the case where the sequence {µ(n)} [ν(n)] is not assumed to be nondecreasing, define (cf. [6, 8]) ξ(n) = max {µ(s) : 1 s n, s N}, [ρ(n) = max {ν(s) : 1 s n, s N}]. Clearly, the sequence of integers {ξ(n)} [{ρ(n)}] is nondecreasing. In this case, Theorems can be formulated in a more general form. More precisely, in conditions (3.4), (3.6), (3.7), (3.8) (3.9)), the term µ(n) [ν(n)] is replaced by ξ(n) [ρ(n)]. Remark 3.4. Observe the following: When 0, then conditions (3.7) (3.9) reduce to µ(n) ν(n) 1 p(i) p(i) > 1, that is, to condition (3.4). However, when 0 < 1/, then we have 1 ( ) ( ) 1 1 > 1 1, which means that condition (3.7) improves condition (3.6). In the case where 0 < 6 4 ( since 1 1 > /), we can show that 1 ( 3 ) ( ) > , which means that condition (3.9) improves condition (3.8).

10 358 I. P. Stavroulakis Example 3.5. Consider the difference equation with x(n) p(n)x(n [n ])) = 0, (3.10) p(n) = (n + 1) ln(n + 1), n N 0\B d, n B where is a positive real number with 0 < 1/, [n ] denotes the integer part of n, d is a positive real number such that 1 1 ( 1 1 ) < + d < 1 ( 1 1 ) B = {terms of the sequence {b(n)}}, [ ] b(n) = (b(n 1) + 1) 1/ + 1, n 1 b(0) = 0 [ ] where (b(n 1) + 1) 1/ + 1 denotes the integer part of (b(n 1) + 1) 1/ + 1. Equation (3.10) is of type (3.1) with µ(n) = n [n ]. Here, {p(n)} is a sequence of positive real numbers, {µ(n)} is a sequence of positive integers such that µ(n) n + 1 for all n 1. Moreover, we note that the sequence {µ(n)} is nondecreasing. We will first show that Since lim (i + 1) ln(i + 1) +1, =. (3.11) is nonincreasing, taking into account the fact that (i + 1) ln(i + 1) c c 1 f(x)dx f(c) c+1 where f(x) is a nonincreasing positive function, we have +1 (i + 1) ln(i + 1) = c +1 i n++[n ] n+1 i+1 f(x)dx, = ln ln(n [n ]) ln(n + ) ds (s + 1) ln(s + 1) ds (s + 1) ln(s + 1)

11 Oscillation Criteria for Delay Advanced Difference Equations (i + 1) ln(i + 1) = i +1 i 1 n ds (s + 1) ln(s + 1) ds (s + 1) ln(s + 1) = ln ln(n + + [n ]). ln(n + 1) It is easy to see that ( lim ln ln(n ) ( [n ]) = lim ln ln(n + + ) [n ]) ln(n + ) ln(n + 1) = 1 =. From the above it is obvious that (3.11) holds true. In particular, since b(n) + 1 b(n) [(b(n)) ] it follows from (3.11) that b(n)+1+[(b(n)) ] lim (i + 1) ln(i + 1) i=b(n)+1 Observe (it is a matter of elementary calculations) that =. (3.1) b(n) < b(n) + 1 b(n) [(b(n)) ] < b(n + 1) for large n. (3.13) Now, in view of (3.13), we get b(n)+1+[(b(n)) ] i=b(n)+1 p(i) = b(n)+1+[(b(n)) ] i=b(n)+1 (i + 1) ln(i + 1) for all large n consequently, because of (3.1) Furthermore, since d b(n)+1+[(b(n)) ] lim p(i) =. (3.14) i=b(n)+1 for all large i, we obtain (i + 1) ln(i + 1) +1 p(i) +1 (i + 1) ln(i + 1) for all large n,

12 360 I. P. Stavroulakis which, by virtue of (3.11), gives From (3.14) (3.15) it follows that Next, we shall prove that Observe that b(n)+1+[(b(n)) ] i=b(n) so, because of (3.14), p(i). (3.15) p(i) =. (3.16) p(i) = + d. (3.17) p(i) = d + b(n)+1+[(b(n)) ] i=b(n)+1 p(i) for all large n, b(n)+1+[(b(n)) ] lim p(i) = d +. (3.18) i=b(n) But it is easy to prove that, for each large n, there exists at most one integer n so that By taking into account this fact, we obtain p(i) = n + 1 b(n ) n [n ]. (i + 1) ln(i + 1) + d for all large n. Thus, by using (3.11), we derive n+1+[n ] (n + 1) ln(n + 1) + (i + 1) ln(i + 1) + d +1 p(i) + d. (3.19) From (3.18) (3.19) we conclude that (3.17) is always valid. Thus, 1 1 µ(n) ( ) 1 1 < p(i) = + d < 1 ( 1 1 ) < 1 that is, condition (3.7) of Theorem 3. is satisfied therefore all solutions of (3.10) oscillate. Observe, however, that none of the conditions (3.6) (3.4) is satisfied.

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