Oscillations of all solutions of iterative equations

Наведено достатнi умови коливання всiх розв’язкiв лiнiйних функцiональних iтерацiйних рiвнянь.

Збережено в:

Бібліографічні деталі
Дата:	2007
Автори:	Nowakowska, W., Werbowski, J.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2007
Назва видання:	Нелінійні коливання
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/177203
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Oscillations of all solutions of iterative equations / W. Nowakowska, J. Werbowski // Нелінійні коливання. — 2007. — Т. 10, № 3. — С. 348-364. — Бібліогр.: 13 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-177203
record_format	dspace
spelling	irk-123456789-1772032021-02-12T01:25:39Z Oscillations of all solutions of iterative equations Nowakowska, W. Werbowski, J. Наведено достатнi умови коливання всiх розв’язкiв лiнiйних функцiональних iтерацiйних рiвнянь. The paper contains sufficient conditions for the oscillation of all solutions of linear functional iterative equations. 2007 Article Oscillations of all solutions of iterative equations / W. Nowakowska, J. Werbowski // Нелінійні коливання. — 2007. — Т. 10, № 3. — С. 348-364. — Бібліогр.: 13 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177203 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Наведено достатнi умови коливання всiх розв’язкiв лiнiйних функцiональних iтерацiйних рiвнянь.
format	Article
author	Nowakowska, W. Werbowski, J.
spellingShingle	Nowakowska, W. Werbowski, J. Oscillations of all solutions of iterative equations Нелінійні коливання
author_facet	Nowakowska, W. Werbowski, J.
author_sort	Nowakowska, W.
title	Oscillations of all solutions of iterative equations
title_short	Oscillations of all solutions of iterative equations
title_full	Oscillations of all solutions of iterative equations
title_fullStr	Oscillations of all solutions of iterative equations
title_full_unstemmed	Oscillations of all solutions of iterative equations
title_sort	oscillations of all solutions of iterative equations
publisher	Інститут математики НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/177203
citation_txt	Oscillations of all solutions of iterative equations / W. Nowakowska, J. Werbowski // Нелінійні коливання. — 2007. — Т. 10, № 3. — С. 348-364. — Бібліогр.: 13 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT nowakowskaw oscillationsofallsolutionsofiterativeequations AT werbowskij oscillationsofallsolutionsofiterativeequations
first_indexed	2025-07-15T15:14:19Z
last_indexed	2025-07-15T15:14:19Z
_version_	1837726391627415552
fulltext	UDC 517 . 9 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS КОЛИВАННЯ ВСIХ РОЗВ’ЯЗКIВ IТЕРАЦIЙНИХ РIВНЯНЬ W. Nowakowska, J. Werbowski Poznań Univ. Technology, Inst. Math. ul. Piotrowo 3A, 60-965 Poznań, Poland e-mails: wnowakow@math.put.poznan.pl jwerbow@math.put.poznan.pl The paper contains sufficient conditions for the oscillation of all solutions of linear functional iterative equations. Наведено достатнi умови коливання всiх розв’язкiв лiнiйних функцiональних iтерацiйних рiв- нянь. 1. Introduction. Many authors investigate oscillatory properties of solutions of difference equati- ons (see [1] and the references cited therein) with “advanced” arguments, ∆y(n) = m∑ i=0 pi(n)y(n + i + 1), (1) or with "delayed"arguments, ∆y(n) = l∑ j=1 qj(n)y(n− j), (2) where l, m, n ∈ ℵ = {1, 2, . . .}, pi, qj : ℵ → <, i = 1, 2, . . . ,m; j = 1, 2, . . . , l, are given functions and the difference operator ∆y is defined by ∆y(n) = y(n + 1)− y(n). If we “join” equations (1) and (2) we obtain a difference equation with “advanced” and “delayed” arguments, ∆y(n) = m∑ i=0 pi(n)y(n + i + 1) + l∑ j=1 qj(n)y(n− j), (3) where m,n and pi, qj are as above. Some kind of generalization of difference and recurrence equations are iterative functional equations. In this paper we consider iterative functional equation of the form ∆gx(t) = m∑ i=0 ai(t)x(gi+1(t)) + l∑ j=1 bj(t)x(g−j(t)), (4) c© W. Nowakowska, J. Werbowski, 2007 348 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 349 where t ∈ =, = is an unbounded subset of <+ = [0,∞), m ≥ 0, l ≥ 1. The difference operator ∆g is defined by ∆gx(t) = x(g(t)) − x(t). The functions ai, bj : = → <+, i = 0, 1, . . . ,m; j = 1, 2, . . . , l, and g : = → = are given and x is an unknown real-valued function. By gm we mean the m-th iterate of the function g, i.e., g0(t) = t, gm+1(t) = g(gm(t)), t ∈ =, m = 0, 1, . . . . By g−1 we mean the inverse function to g and g−m−1(t) = g−1(g−m(t)). In the whole paper the upper indices at the sign of a function will denote iterations. In each instance we have the relation g1(t) = g(t). We also assume that g(t) 6= t and lim t→∞ g(t) = ∞, t ∈ =. (5) Moreover we assume that g has an inverse function. By a solution of equation (4) we mean a function x : = → < such that sup{\|x(s)\| : s ∈ ∈ =t0 = [t0,∞) ∩ =} > 0 for any t0 ∈ <+ and x satisfies (4) on =. A solution x of equation (4) is called oscillatory if there exists a sequence of points {tn}∞n=1, tn ∈ =, such that limn→∞tn = ∞ and x(tn)x(tn+1) ≤ 0 for n = 1, 2, . . . . Otherwise it is called nonoscillatory. As usual we take k−1∑ j=k aj = 0 and k−1∏ j=k aj = 1. In this paper we investigate oscillatory properties of solutions of equation (4). The same problem for functional equations has been considered in [2 – 7] and for equations (1) and (2) for example in [1, 8 – 13]. The aim of this paper is to present new oscillation criteria for equation (4). Let us observe that in the particular case, i.e., = = ℵ and g(t) = t + 1 from equation (4) we get equation (3). In the end of this paper we give an application of the obtained results to recurrence equations. In our considerations the following lemma will be useful. Lemma 1. Consider the functional inequality x(g(t)) ≥ p(t)x(t) + q(t)x(gk+1(t)) (6) where k ≥ 1, p, q : = → <+, and g satisfies condition (5). If lim inf =3t→∞ k−1∑ i=0 q(gi(t)) k∏ j=1 p(gi+j(t)) > ( k k + 1 )k+1 (7) or lim sup =3t→∞ k∑ i=0 q(gi(t)) k∏ j=1 p(gi+j(t))× × { 1 + i∑ l=1 q(gk+l(t)) k∏ m=1 p(gk+l+m(t)) } > 1, (8) then the functional inequality (6) has not positive solutions for large t ∈ =. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 350 W. NOWAKOWSKA, J. WERBOWSKI Condition (7) comes from [3] and (8) follows from Theorem 2 of [4]. Similarly we have the following lemma (see [5]). Lemma 2. Consider the functional inequality x(gk(t)) ≥ p(t)x(gk+1(t)) + q(t)x(t) (9) where k ≥ 1, and p, q and g are as previously. If lim inf =3t→∞ k−1∑ i=0 q(g−i(t)) k∏ j=1 p(g−i−j(t)) > ( k k + 1 )k+1 (10) or lim sup =3t→∞ k∑ i=0 q(g−i(t)) k∏ j=1 p(g−i−j(t))× × { 1 + i∑ l=1 q(g−k−l(t)) k∏ m=1 p(g−k−l−m(t)) } > 1, (11) then the functional inequality (9) has not positive solutions for large t ∈ =. To prove our main results we will also need the following. Lemma 3. Let, for sufficiently large t ∈ =t1 , k−1∑ i=0 q(gi(t)) k∏ j=1 p(gi+j(t)) ≥ δ > 0, δ < ( k k + 1 )k+1 . (12) Then every nonoscillatory solution x(t) > 0, t ∈ =t2 , t2 ≥ t1 of inequality (6) satisfies the following inequality: p(t)x(t) ≥ δx(g(t)) for t ∈ =t3 , t3 ≥ t2. Proof. Suppose that x(t) > 0, t ∈ =t2 , is a nonoscillatory solution of inequality (6). Then also in view of assumption (5) imposed on the function g there exists a point t3 ≥ t2 such that x(gi(t)) > 0, i ∈ {1, 2, . . . , k + 1}, and t ∈ =t3 . Thus from inequality (6) we get x(g(t)) ≥ p(t)x(t) which gives, for i ∈ {1, 2, . . . , k + 1}, x(gi(t)) ≥ x(t) i−1∏ j=0 p(gj(t)) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 351 and x(gi+k+1(t)) ≥ x(gk+1(t)) k+i∏ j=k+1 p(gj(t)), i = 1, 2, . . . , k + 1. (13) From (6) we obtain, for i = 1, 2, . . . , k + 1 and t ∈ =t4 , t4 ≥ t3, x(gi+1(t)) ≥ p(gi(t))x(gi(t)) + q(gi(t))x(gk+i+1(t)). (14) Multiplying both sides of this inequality by k−1∏ j=i+1 p(gj(t)) and summing up from i = 0 to k − 1 we obtain x(gk(t)) ≥ x(t) k−1∏ j=0 p(gj(t)) + k−1∑ i=0 q(gi(t)) k−1∏ j=i+1 p(gj(t))x(gk+i+1(t)). Multiplying both sides of above inequality by p(gk(t)) we get p(gk(t))x(gk(t)) ≥ k−1∑ i=0 q(gi(t)) k∏ j=i+1 p(gj(t))x(gk+i+1(t)) and from (13) p(gk(t))x(gk(t)) ≥ k−1∑ i=0 q(gi(t)) k+i∏ j=i+1 p(gj(t))x(gk+1(t)). In view of assumption (12) we have p(gk(t))x(gk(t)) ≥ δx(gk+1(t)). Hence p(t)x(t) ≥ δx(g(t)). Above inequality concludes the proof. Similarly we can prove the following lemma. Lemma 4. Suppose that for sufficiently large t ∈ =t1 inequality k−1∑ i=0 q(g−i(t)) k∏ j=1 p(g−i−j(t)) ≥ δ > 0, δ < ( k k + 1 )k+1 , is true. Then every nonoscillatory solution x(t) > 0, t ∈ =t2 , of inequality (9) satisfies for sufficiently large t ∈ =t3 , t3 ≥ t2, the following inequality: p(t)x(gk+1(t)) ≥ δx(gk(t)). ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 352 W. NOWAKOWSKA, J. WERBOWSKI 2. Main results. Notice that if in equation (4) one of the coefficients satisfies ak(t) > 1, k = 0, 1, . . . ,m, then equation (4) has only oscillatory solutions. So, further we will consider equation (4) with the assumption ak(t) < 1, k = 0, 1, . . . ,m. We may observe that equation (4) has the form [1− a0(t)]x(g(t)) = a1(t)x(g2(t)) + a2(t)x(g3(t)) + . . . + am(t)x(gm+1(t))+ + x(t) + b1(t)x(g−1(t)) + b2(t)x(g−2(t)) + . . . + bl(t)x(g−l(t)), where the coefficients ak(t), bj(t) and l, m are as before. Thus, x(g(t)) = m∑ i=1 Ai(t)x(gi+1(t)) + l∑ j=0 Bj(t)x(g−j(t)), l ≥ 0, m ≥ 1, (15) where Ai(t) = ai(t) 1− a0(t) ≥ 0, i = 1, 2, . . . ,m, and Bj(t) = bj(t) 1− a0(t) ≥ 0, j = 1, 2, . . . , l, B0(t) = 1 1− a0(t) > 0. Further we will assume that inequalities are satisfied for sufficiently large t ∈ =. Now we present sufficient conditions for all solutions of equation (15) to by oscillatory. Let us start with the following. Theorem 1. If l ≤ m and lim inf =3t→∞ m−1∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t)) > ( m m + 1 )m+1 (16) or lim sup =3t→∞ m∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t))× × { 1 + i∑ k=1 Q(gk+m(t)) m∏ s=1 P (gm+k+s(t)) } > 1, (17) where P (t) = B0(t) + l∑ k=1 Bk(t)Ak(g−k−1(t)) (18) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 353 and Q(t) = m−1∑ k=1 Ak(t)Am−k(gk(t)) + Am(t), (19) then equation (15) possesses only oscillatory solutions. Proof. Suppose that x is a nonoscillatory solution of (15) and let x(t) > 0. Then, in view of assumption (5) about the function g and positivity of the functions Ak(t) and Bk(t), from equation (15) we have x(g(t)) ≥ Ai(t)x(gi+1(t)), i = 1, 2, . . . ,m. (20) Hence, x(gk+1(t)) ≥ Ai(gk(t))x(gk+1+i(t)) for 1 ≤ k ≤ m. Thus x(gk+1(t)) ≥ Am−k(gk(t))x(gm+1(t)) for 0 ≤ k ≤ m− 1. (21) Similarly from inequality (20) we have for l ≤ m that x(g−k(t)) ≥ Ak(g−k−1(t))x(t) for 1 ≤ k ≤ m. (22) Using now inequalities (21) and (22) in (15) we obtain x(g(t)) ≥ { m−1∑ k=1 Ak(t)Am−k(gk(t)) + Am(t) } x(gm+1(t))+ + { B0(t) + l∑ k=1 Bk(t)Ak(g−k−1(t)) } x(t) and x(g(t)) ≥ P (t)x(t) + Q(t)x(gm+1(t)). (23) Applying now Lemma 1 to the above inequality, in view of assumptions (16) and (17) we obtain a contradiction to the fact that x(t) is a positive solution of equation (15). Thus the theorem is proved. Remark 1. From conditions (16) and (17) of Theorem 1 it follows that the coefficients ai, i = 0, 1, . . . ,m, make an essential influence on oscillation of solutions of equation (15) the coefficients bj , j = 1, 2, . . . , l. Let us observe that if in equation (4) all coefficients bj = 0 for j = 1, 2, . . . , l (then in equation (15) Bj(t) ≡ 0 for k = 1, 2, . . . , l and B0 6≡ 0 ) conditions (16) and (17) take the following respective forms: lim inf =3t→∞ m−1∑ i=0 Q(gi(t)) m∏ j=1 B0(gi+j(t)) > ( m m + 1 )m+1 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 354 W. NOWAKOWSKA, J. WERBOWSKI or lim sup =3t→∞ m∑ i=0 Q(gi(t)) m∏ j=1 B0(gi+j(t))× × { 1 + i∑ k=1 Q(gk+m(t)) m∏ s=1 B0(gm+k+s(t)) } > 1, where Q(t) is as before. The above conditions could be satisfied and depend only on the coeffi- cients ai. On the other hand, in case where in equation (4) all the coefficients ai ≡ 0 for i = 0, 1, . . . ,m, the left-hand sides of conditions (16) and (17) equal to zero independly of the coefficients bj . Now we give sufficient conditions for all solutions of equation (15) to be oscillatory which can be applied when inequality l ≤ m is not satisfied. Theorem 2. Suppose that l ≥ m− 2, m ≥ 3, and lim inf =3t→∞ l∑ i=0 S(g−i(t)) l+1∏ j=1 R(g−i−j(t)) > ( l + 1 l + 2 )l+2 (24) or lim sup =3t→∞ l+1∑ i=0 S(g−i(t)) l+1∏ j=1 R(g−i−j(t))× × { 1 + i∑ k=1 S(g−k−l−1(t)) l+1∏ s=1 R(g−k−l−s−1(t)) } > 1, (25) where R(t) = A1(gl(t)) + m∑ k=2 Ak(gl(t))Bk−2(gl+k(t)) (26) and S(t) = l−1∑ k=0 Bk(gl(t))Bl−k(gl−k−1(t)) + Bl(gl(t)). (27) Then every solution of equation (15) oscillates. Proof. Suppose that x(t) > 0 is a nonoscillatory solution of equation (15). Then, similarly as in the proof of Theorem 1, in view of assumption (5) on the function g and positivity of the functions Ak(t) and Bk(t) from equation (15) we obtain x(g(t)) ≥ Bi(t)x(g−i(t)), i = 0, 1, . . . , l. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 355 Hence we get x(g−k(t)) ≥ Bl−k−1(g−k−1(t))x(g−l(t)), 0 ≤ k ≤ l − 1, (28) and, for m ≤ l + 2, x(gk+1(t)) ≥ Bk−2(gk(t))x(g2(t)), 2 ≤ k ≤ m. (29) Applying now inequalities (28) and (29) in (15) we have x(g(t)) ≥ { A1(t) + m∑ k=2 Ak(t)Bk−2(gk(t)) } x(g2(t))+ + { l−1∑ k=0 Bk(t)Bl−k−1(g−k−1(t)) + Bl(t) } x(g−l(t)) and x(gl+1(t)) ≥ R(t)x(gl+2(t)) + S(t)x(t). (30) Thus, in view of (24), (25) and Lemma 2, the above inequality cannot possess a positive solution. We get a contradiction which completes the proof. Remark 2. Let us observe that Theorems 1 and 2 have "common area", i.e., both could be applied for l = m− 2, l = m− 1 and l = m. But Theorems 1 and 2 are independent. To prove this, we consider a functional equation of the form x(t + 1) = 1 10 x(t + 2) + 1 t x(t + 3) + [t]2x(t + 4) + tx(t) + 1 2[t]2 x(t− 1), t ≥ 2. The above equation has only oscillatory solutions because condition (16) of Theorem 1 is sati- sfied. However assumption (24) of Theorem 2 is not satisfied. On the other hand, for the functi- onal equation x(t + 1) = 1 [t]2 x(t + 2) + t 2 x(t + 3) + 1 3 x(t + 4) + 1 5 x(t) + t 2 x(t− 1), t ≥ 2, condition (16) of Theorem 1 is not fulfilled but the above equation has only oscillatory solutions because condition (24) of Theorem 2 is true. We present now conditions for all solutions of equation (15) to be oscillatory. These condi- tions can be applied in the case where the assumptions of Theorems 1 and 2, respectively, are not satisfied. Theorem 3. Let l ≤ m and m−1∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t)) ≥ δ > 0, δ < ( m m + 1 )m+1 , (31) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 356 W. NOWAKOWSKA, J. WERBOWSKI where P (t) and Q(t) are given by (18) and (19). If lim sup =3t→∞ m∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t))× × { 1 + i∑ k=1 Q(gk+m(t)) m∏ s=1 P (gm+k+s(t)) } > 1− δm+1, (32) then every solution of equation (15) is oscillatory. Proof. Suppose that x is a nonoscillatory solution of (15) and let x(t) > 0. Then, similarly as in the proof of Theorem 1, inequality (23) is true. Hence we have x(g(t)) ≥ P (t)x(t) and similarly as in the proof of Lemma 3, x(gi+m+1(t)) ≥ x(gm+1(t)) m+i∏ j=m+1 P (gj(t)), i = 1, 2, . . . ,m + 1. (33) From (23) for i ∈ {1, 2, . . . ,m} we get x(gi+1(t)) ≥ P (gi(t))x(gi(t)) + Q(gi(t))x(gm+i+1(t)). (34) Multiplying both sides of the above inequality by m∏ j=i+1 P (gj(t)) and a subsequent summation from i = 1 to m we obtain x(gm+1(t)) ≥ x(g(t)) m∏ j=1 P (gj(t)) + m∑ i=1 Q(gi(t)) m∏ j=i+1 P (gj(t))x(gm+i+1(t)). Applying now inequality (23) we get x(gm+1(t)) ≥ x(t) m∏ j=0 P (gj(t)) + Q(t) m∏ j=1 P (gj(t))x(gm+1(t))+ + m∑ i=1 Q(gi(t)) m∏ j=i+1 P (gj(t))x(gm+i+1(t)). (35) From (34) we obtain for i ∈ {1, 2, . . . ,m} and j ∈ {0, 1, . . . ,m} that x(gm+i+1−j(t)) ≥ P (gm+i−j(t))x(gm+i−j(t)) + Q(gm+i−j(t))x(g2m+i+1−j(t)) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 357 and, in view of (33), x(gm+i+1−j(t)) ≥ ≥ P (gm+i−j(t))x(gm+i−j(t)) + Q(gm+i−j(t)) 2m+i−j∏ l=m+1 P (gl(t))x(gm+1(t)). (36) Applying now (36) for j = 0 in (35) we have x(gm+1(t)) ≥ x(t) m∏ j=0 P (gj(t)) + Q(t) m∏ j=1 P (gj(t))x(gm+1(t))+ + m∑ i=1 Q(gi(t)) m∏ j=i+1 P (gj(t))× × { P (gm+i(t))x(gm+i(t)) + Q(gm+i(t)) 2m+i∏ l=m+1 P (gl(t))x(gm+1(t)) } . Hence, x(gm+1(t)) ≥ x(t) m∏ j=0 P (gj(t))+ + x(gm+1(t))  1∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t))+ + m∑ i=1 Q(gi(t))Q(gm+i(t)) 2m∏ j=1 P (gi+j(t))  + + m∑ i=2 Q(gi(t)) m∏ j=i+1 P (gj(t))P (gm+i(t))x(gm+i(t)). ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 358 W. NOWAKOWSKA, J. WERBOWSKI Using now (36) for j = 1 in the above inequality we obtain x(gm+1(t)) ≥ x(t) m∏ j=0 P (gj(t)) + x(gm+1(t))× ×  2∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t)) + m∑ i=1 Q(gi(t))Q(gm+i(t)) 2m∏ j=1 P (gi+j(t))+ + m∑ i=2 Q(gi(t))Q(gm+i−1(t))P (gm+i(t)) 2m−1∏ j=1 P (gi+j(t))  + + m∑ i=3 Q(gi(t)) 1∏ l=0 P (gm+i−l(t)) m∏ j=i+1 P (gj(t))x(gm+i−1(t)). Finally we have x(gm+1(t)) ≥ x(t) m∏ j=0 P (gj(t)) + x(gm+1(t))× ×  m∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t)) + m∑ i=1 Q(gi(t))Q(gm+i(t)) 2m∏ j=1 P (gi+j(t)) + . . . . . . + m∑ i=m−1 Q(gi(t))Q(gi+2(t)) m+2∏ j=1 P (gi+j(t)) m∏ s=3 P (gi+s(t))+ + Q(gm(t))Q(gm+1(t)) m+1∏ j=1 P (gm+j(t)) m∏ s=2 P (gm+s(t))  . (37) From assumption (31), in view of Lemma 3, we have that a nonoscillatory solution of (23) satisfies the following inequality: P (t)x(t) ≥ δx(g(t)). Hence m∏ j=0 P (gj(t))x(t) ≥ δm+1x(gm+1(t)). ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 359 Using the above inequality in (37) we obtain x(gm+1(t)) ≥ δm+1x(gm+1(t)) + x(gm+1(t))× ×  m∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t)) + m∑ i=1 Q(gi(t))Q(gm+i(t)) 2m∏ j=1 P (gi+j(t))+ . . . . . . + m∑ i=m−1 Q(gi(t))Q(gi+2(t)) m+2∏ j=1 P (gi+j(t)) m∏ s=3 P (gi+s(t))+ + Q(gm(t))Q(gm+1(t)) m+1∏ j=1 P (gm+j(t)) m∏ s=2 P (gm+s(t))  . Dividing now the above inequality by x(gm+1(t)) we obtain m∑ i=0 Q(gi(t)) m∏ j=1 P (gi+j(t))× × { 1 + i∑ k=1 Q(gk+m(t)) m∏ s=1 P (gm+k+s(t)) } ≤ 1− δm+1. The last inequality contradicts assumption (32). Thus the proof is complete. Theorem 4. Let l ≥ m− 2, m ≥ 3, and l∑ i=0 S(g−i(t)) l+1∏ j=1 R(g−i−j(t)) ≥ δ > 0, δ < ( l + 1 l + 2 )l+2 , (38) where R(t) and S(t) are given by (26) and (27). If lim sup =3t→∞ l+1∑ i=0 S(g−i(t)) l+1∏ j=1 R(g−i−j(t))× × { 1 + i∑ k=1 S(g−k−l−1(t)) l+1∏ s=1 R(g−k−l−s−1(t)) } > 1− δl+2, (39) then equation (15) has only oscillatory solutions. Proof. Let x(t) > 0 be a nonoscillatory solution of equation (15). Then, as in the proof of Theorem 2, inequality (30) is satisfied. Thus for i ∈ {1, 2, . . . , l + 1} we get x(gl+1−i(t)) ≥ R(g−i(t))x(gl+2−i(t)) + S(g−i(t))x(g−i(t)). ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 360 W. NOWAKOWSKA, J. WERBOWSKI Multiplying both sides of the above inequality by l+1∏ j=i+1 R(g−j(t)) and then summing from i = 1 to l + 1 we obtain x(t) ≥ x(gl+1(t)) l+1∏ j=1 R(g−j(t)) + l+1∑ i=1 S(g−i(t)) l+1∏ j=i+1 R(g−j(t))x(g−i(t)). Applying now inequality (30) we get x(t) ≥ x(gl+2(t)) l+1∏ j=0 R(g−j(t)) + S(t) l+1∏ j=1 R(g−j(t))x(t)+ + l+1∑ i=1 S(g−i(t)) l+1∏ j=i+1 R(g−j(t))x(g−i(t)). Further in the same way as in the proof of Theorem 3 from the above inequality we have x(t) ≥ x(gl+2(t)) l+1∏ j=0 R(g−j(t))+ + x(t)  l+1∑ i=0 S(g−i(t)) l+1∏ j=1 R(g−i−j(t))+ + l+1∑ i=1 S(g−i(t))S(g−l−1−i(t)) 2l+2∏ j=1 R(g−i−j(t)) + . . . . . . + l+1∑ i=l S(g−i(t))S(g−i−2(t)) l+3∏ j=1 R(g−i−j(t)) l−2∏ s=0 R(gs−l−1−i(t))+ + S(g−l−1(t))S(g−l−2(t)) l+2∏ j=1 R(g−l−1−j(t)) l−1∏ s=0 R(gs−2l−2(t))  . (40) From assumption (38), in view of Lemma 4, we get R(t)x(gl+2(t)) ≥ δx(gl+1(t)). Thus l+1∏ j=0 R(g−j(t))x(gl+2(t)) ≥ δl+2x(t). ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 361 Applying now the above inequality in (40) and dividing both sides of the obtained inequality by x(t) we obtain S(g−i(t)) l+1∏ j=1 R(g−i−j(t))× × { 1 + i∑ k=1 S(g−k−l−1(t)) l+1∏ s=1 R(g−k−l−s−1(t)) } ≤ 1− δl+2. This contradicts assumption (39). Thus the proof is complete. 3. Final remarks. As it was mentioned, functional equations are a generalization of recurren- ce equations. So, from oscillation criteria given for the functional equations we also obtain suffi- cient conditions for oscillations of solutions of the recurrence equations. Consider a recurrence equation of the form x(n− 1) = m∑ i=1 Ai(n)x(n− i− 1) + l∑ j=0 Bj(n)x(n + j), l ≥ 0, m ≥ 1. (41) Applying now the results obtained, for example, in Theorem 3 we obtain the following conditi- on for equation (41). Let l ≤ m and m−1∑ i=0 Q(n− i) m∏ j=1 P (n− i− j) ≥ δ > 0, δ < ( m m + 1 )m+1 , where P (n) = B0(n) + l∑ k=1 Bk(n)Ak(n + k + 1) (42) and Q(n) = m−1∑ k=1 Ak(n)Am−k(n− k) + Am(n). (43) If for l ≤ m lim sup n→∞ m∑ i=0 Q(n− i) m∏ j=1 P (n− i− j)× × { 1 + i∑ k=1 Q(n− k −m) m∏ s=1 P (n−m− k − s) } > 1− δm+1, (44) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 362 W. NOWAKOWSKA, J. WERBOWSKI then equation (41) possesses only oscillatory solutions. Conditions similar to the above were presented by Chatzarakis and Stavroulakis [8] and Stavroulakis [13] for a difference equation of the form x(n + 1)− x(n) + p(n)x(n−m) = 0, m > 0, n = 0, 1, 2, . . . , (45) where p : ℵ → <+ \ {0}. From [8] and [13] it follows that if lim inf n→∞ m∑ i=1 p(n− i) = α ≤ ( m m + 1 )m+1 and one of the conditions lim sup n→∞ m∑ i=1 p(n− i) > 1− α2 4 , (46) lim sup n→∞ m∑ i=1 p(n− i) > 1− αm, (47) or lim sup n→∞ m∑ i=1 p(n− i) > 1− α2 2(2− α) , (48) hold, then all solutions of equation (45) oscillate. It was shown in [8] that for any m condition (48) is better than (46) and for m = 1, 2 condition (48) implies (47), for m ≥ 4 condition (47) implies (48) but for m = 3 conditions (47) and (48) are independent. Now we show that our condition (44), in many cases, is better than conditions (47) and (48). For m = 1 condition (47) is better than (48), so it suffices to prove that condition (44) is better than (47). Let us consider an equation of the form x(n + 1)− x(n) + p(n)x(n− 1) = 0, n = 0, 1, 2, . . . , where p(n) = 4(2 + (−1)n) 19 + 1 n . For this equation condition (44) is fulfilled but condition (47) is not satisfied because α = 4 19 and lim sup n→∞ p(n− 1) = 12 19 < 1− α. Now let m = 3. In [8] it was shown that conditions (47) and (48) are independent because, for the difference equation x(n + 1)− x(n) + p(n)x(n− 3) = 0, n = 0, 1, 2, . . . , (49) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 OSCILLATION OF ALL SOLUTIONS OF ITERATIVE EQUATIONS 363 where p(2n) = 1 10 and p(2n + 1) = 1 10 + 6731 10000 sin2 nπ 2 , condition (47) is satisfied and (48) is not but if, in equation (49), p(2n) = 8 100 and p(2n + 1) = 8 100 + 744 1000 sin2 nπ 2 , then condition (48) is fulfilled and (47) is not. Observe that our condition (44) for equation (49) of the form 2∑ i=0 p(n− i) ≥ δ, δ < ( 3 4 )4 , and lim sup n→∞ 3∑ i=0 p(n− i) { 1 + i∑ k=1 p(n− k − 3) } > 1− δ4 (50) is satisfied for both sequences p(n) defined above. But if we take p(2n) = 9 100 and p(2n + 1) = 9 100 + 632 1000 sin2 nπ 2 , then conditions (47) and (48) are not satisfied but condition (50) is true. On the other hand, for m ≥ 4 it suffices to show that condition (44) is better than (48). For example, the difference equation x(n + 1)− x(n) + p(n)x(n− 4) = 0, n = 0, 1, 2, . . . , where p(5n) = p(5n + 1) = p(5n + 2) = p(5n + 3) = 3 40 and p(5n + 4) = 27 40 , has only oscillatory solutions since condition (44) is satisfied. However condition (48) is not fulfilled. 1. Agarwal R. P., Bohner M., Grace S. R., O’Regan D. Discrete oscillation theory. — Hindawi Publ. Corporation, 2005. 2. Golda W., Werbowski J. Oscillation of linear functional equations of the second order // Funkc. ekvacioj. — 1994. — 37. — P. 221 – 227. 3. Nowakowska W., Werbowski J. Oscillation of linear functional equations of higher order // Arch. Math. — 1995. — 31.— P. 251 – 258. 4. Nowakowska W., Werbowski J. Oscillatory behavior of solutions of functional equations // Nonlinear Anal. — 2001. — 44. — P. 767 – 775. 5. Nowakowska W., Werbowski J. Oscillatory solutions of linear iterative functional equations // Indian J. Pure and Appl. Math. — 2004. — 35, № 4. — P. 429 – 439. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3 364 W. NOWAKOWSKA, J. WERBOWSKI 6. Shen J., Stavroulakis I. P. Oscillation criteria for second order functional equations // Acta Math. Sci. Ser. B. — 2002. — 22, № 1. — P. 56 – 62. 7. Thandapani E., Ravi K.. Oscillation of nonlinear functional equations // Indian J. Pure and Appl. Math. — 1999. — 30, № 12. — P. 1235 – 1241. 8. Chatzarakis G. E., Stavroulakis I. P. Oscillation criteria for first order linear delay difference equations // Techn. Rept. Univ. Ioannina. — 2005. — 15. — P. 1 – 16. 9. Erbe L. H., Zhang B. G. Oscillation of discrete analogues of delay equations // Different. Integr. Equat. — 1989. — 2. — P. 300 – 309. 10. Györi I., Ladas G. Oscillation theory of delay differential equations with applications. — Oxford: Clarendon Press, 1991. 11. Ladas G., Philos Ch. G., Sficas Y. G. Sharp conditions for the oscillation for delay difference equations // J. Appl. Math. Simulat. — 1989. — 2. — P. 101 – 111. 12. Nowakowska W., Werbowski J. Oscillatory behavior of solutions of linear recurrence equations // J. Different. Equat. and Appl. — 1995. — 1. — P. 239 – 247. 13. Stavroulakis I. P. Oscillation criteria for delay and difference equations // Stud. Univ. Žilina. Math. Ser. — 2003. — 17, № 1. — P. 161 – 176. Received 05.01.2007 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 3

Oscillations of all solutions of iterative equations

Репозитарії

Схожі ресурси