Lyapunov-type inequalities for certain higher-order difference equations with mixed non-linearities

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1 Liu Advances in Difference Equations (08) 08:9 R E S E A R C H Open Access Lyapunov-type inequalities for certain higher-order difference equations with mixed non-linearities Haidong Liu * * Correspondence: tomlhd983@63.com School of Mathematical Sciences, Qufu Normal University, Qufu, P.R. China Abstract In this paper, we establish some new Lyapunov-type inequalities for some higher-order difference equations with boundary conditions. The obtained inequalities generalize the existing results in the literature. Keywords: Lyapunov-type inequality; Difference equation; Higher-order; Anti-periodic boundary conditions Introduction During the past decades, continuous and discrete integral inequalities have attracted the attention of many researchers (see [ 59] and the references therein). Particularly, there have been plenty of references focused on the Lyapunov-type inequality and many of its generalizations due to its broad applications in the study of various properties of solutions of differential and difference equations such as oscillation theory, disconjugacy, and eigenvalue problems (see [,, 5 7, 9, 3, 5,, 4, 7 9, 37, 39, 45, 48, 57, 59]andthe references therein). Compared with a large number of references devoted to continuous Lyapunov-type inequalities, there is not much done for discrete Lyapunov-type inequalities (see [6, 3,, 9, 39, 59] and the references therein). For example, Zhang and Tang [9] considered the following even order difference equation: k u(n)+( ) k q(n)u(n +)0, () where is the usual forward difference operator defined by u(n)u(n +) u(n), k N, n Z and q(n) is a real-valued function defined on Z. Under the following boundary conditions i u(a) i u(b)0, i 0,,...,k ; u(n) 0, n Z[a, b], () where a, b N, Z[a, b]{a, a +,...,b,b}, they obtained the following result: The Author(s) 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 LiuAdvances in Difference Equations (08) 08:9 Page of 4 Assume that k N and q(n) is a real-valued function on Z. If() hasasolutionu(n) satisfying the boundary conditions (), b [ q(n) (n a +)(b n ) ] 3(k ). (3) (b a) k 3 Recently, Liu and Tang [] studied the following m-order difference equation: m u(n) p m u(n)+r(n) u(n) p u(n)0, (4) where m N, n Z and r(n) is a real-valued function defined on Z, p > is a constant, and u(n) satisfies the following anti-periodic boundary conditions: i u(a)+ i u(b)0, i 0,,...,m ; u(n) 0, n Z[a, b], (5) and they obtained the following result: If (4) has a nonzero solution u(n) satisfying the anti-periodic boundary conditions (5), b r(n) q mp, (6) (b a) mp where q is a conjugate exponent of p. In the present paper, we shall establish a new discrete Lyapunov-type inequality for the following m-order difference equation with mixed nonlinearities: m u(n) p m m u(n)+ r i (n) i u(n) p i u(n)0, (7) with the anti-periodic boundary conditions (5), where m N, n Z, p > is a constant and r i (n) (i 0,,...,m ) are real-valued functions defined on Z. Further,wewillalso prove a new Lyapunov-type inequality for the m-order difference equation m u(n) p m u(n)+( ) m r(n) u(n +) q u(n +)0, (8) with the following boundary conditions: i u(a) i u(b)0, i 0,,...,m ; u(n) 0, n Z[a, b], (9) where m N, p q > are constants, n Z and r(n) is a real-valued function defined on Z. Our works extend the results in []and[9]. Main results Lemma. ([]) If A is positive and B, z are nonnegative, Az τ Bz σ + Ɣ στ A σ /( σ ) B τ/( σ ) 0 (0)

3 LiuAdvances in Difference Equations (08) 08:9 Page 3 of 4 for any σ (0, τ), where Ɣ στ (τ σ )σ σ /(τ σ ) τ τ/(τ σ ) τ/(τ σ ) >0 with equality holding if and only if B z 0. Lemma. ([9]) Assume that u(n) is a real-valued function on Z[a, b], u(a) u(b) 0. Then u(n) (n a)(b n) b a b u(s), n Z(a, b ), () sa b u(n) b [ (n a +)(b n ) u(n) ] We now state the main theorem of this paper. (b a) 8 b u(n). () Theorem. If u(n) is a nonzero solution of Eq.(7) satisfyingthe anti-periodic boundary conditions (5), m (b a) (m i /p)(p ) (m i)(p ) ri (n) ) /q q, (3) where q is the Hölder conjugate exponent of p, i.e., /p +/q. Proof Since the nonzero solution u(n) ofeq.(7) satisfies the anti-periodic boundary conditions (5), u(a) +u(b) 0. For n Z[a, b], wehave Then u(n)u(n) [ u(a)+u(b) ] n [ u(k +) u(k) ] b [ ] u(k +) u(k) n u(k) b u(k). (4) u(n) kn b u(k). (5) Applying discrete Hölder s inequality kn ( b b ) /α ( b /β f (k)g(k) f (k) α g(k) β) (6) to (5)withf (k),g(k) u(k), α q,andβ p,weobtainthat u(n) b u(k) (b a)/q /p u(k) p). (7)

4 LiuAdvances in Difference Equations (08) 08:9 Page 4 of 4 Similarly, we get Then i u(n) i u(n) p b i+ u(k) ( b (b a)/q i+ u(k) ) /p p, i,,...,m. (8) ( ) p b (b a) p/q i+ u(k) p, i,,...,m. (9) Summing (9)froma to b,wehave i.e., b i u(n) ( ) p p b (b a) (b a) p/q i+ u(k) p, i,,...,m, (0) i u(n) ) /p p b a From (), we obtain i u(n) ) /p p b a i+ u(k) ) /p p, i,,...,m. () i+ u(k) ) /p p ( ) ( b a b i+ u(k) ) /p p ( ) ( b a m i b m u(k) ) /p p, i,,...,m. () Then, from (7)and()fori,weobtain ( ) ( u(n) b a m b (b a)/q m u(k) ) /p p, (3) and by (8)and(), we get i u(n) ( ) ( b a m i b (b a)/q m u(k) ) /p p, i,,...,m. (4)

5 LiuAdvances in Difference Equations (08) 08:9 Page 5 of 4 Multiplying (7)by m u(n), we have m u(n) p m + r i (n) i u(n) p i u(n) m u(n) 0. (5) Then we get m u(n) p m r i (n) i u(n) p i u(n) m u(n) m r i (n) i u(n) p i u(n) m u(n) m r i (n) i u(n) p m u(n). (6) Summing (6)froma to b,wehave b m u(n) p m b ri (n) i u(n) p m u(n) b r0 (n) u(n) p m u(n) m b + ri (n) i u(n) p m u(n). (7) i For the first summation on the right-hand side of (7), from (3) and Hölder s inequality (6), we obtain that b r0 (n) u(n) p m u(n) ( ) p ( b a (b a) (p )/q ) (m )(p ) m u(n) ) (p )/p b p r0 (n) m u(n) (b a)(m /p)(p ) m(p ) (b a)(m /p)(p ) m(p ) m u(n) ) (p )/p b p r0 (n) m u(n) m u(n) ) (p )/p p r 0 (n) ) /q ( b q m u(n) ) /p p (b a)(m /p)(p ) m(p ) m u(n) )( b p r0 (n) ) /q q. (8)

6 LiuAdvances in Difference Equations (08) 08:9 Page 6 of 4 On the other hand, for the second summation on the right-hand side of (7), from (4) and Hölder s inequality (6), we have that b ri (n) i u(n) p m u(n) and ( ) p ( b a (b a) (p )/q ) (m i )(p ) m u(n) ) (p )/p b p r i (n) m u(n) (b a)(m i /p)(p ) (m i)(p ) (b a)(m i /p)(p ) (m i)(p ) m u(n) ) (p )/p b p r i (n) m u(n) m u(n) ) (p )/p p ri (n) ) /q ( b q m u(n) ) /p p (b a)(m i /p)(p ) (m i)(p ) m u(n) )( b p ri (n) ) /q q, i,,...,m, (9) m b ri (n) i u(n) p m u(n) i m i (b a) (m i /p)(p ) (m i)(p ) m u(n) )( b p ri (n) ) /q q m u(n) ) m p (b a) (m i /p)(p ) By (7), (8), and (30), we get i (m i)(p ) r i (n) ) /q q. (30) b m u(n) p (b a)(m /p)(p ) m(p ) + m u(n) )( b p r0 (n) ) /q q m u(n) ) m p (b a) (m i /p)(p ) i (m i)(p ) r i (n) ) /q q. (3) Now, we claim that b u(n) p > 0. In fact, if the above inequality is not true, we have b u(n) p 0, u(n)0forn Z[a, b ]. By the anti-periodic conditions (5), we obtain u(n)0forn Z[a, b], which contradicts u(n) 0, n Z[a, b]. From (), we

7 LiuAdvances in Difference Equations (08) 08:9 Page 7 of 4 get b m u(n) p > 0. Thus, dividing both sides of (3)by b m u(n) p,weobtain (b a)(m /p)(p ) m(p ) m + i m r0 (n) ) /q q (b a) (m i /p)(p ) (m i)(p ) (b a) (m i /p)(p ) (m i)(p ) ri (n) ) /q q r i (n) ) /q q. This completes the proof of Theorem.. Remark If r i (n) 0, i,,...,m, Theorem. coincides with Theorem in []. Let p,m k, k N in Theorem., we have the following corollary. Corollary. If u(n) is a nonzero solution of k k u(n)+ r i (n) i u(n) 0 (3) and satisfies the anti-periodic boundary conditions i u(a)+ i u(b)0, i 0,,...,k ; u(n) 0, n Z[a, b], (33) k (b a) k i / k i ri (n) ) /. Let p,m k,k N in Theorem., we have the following corollary. Corollary. If u(n) is a nonzero solution of k k u(n)+ r i (n) i u(n) 0 (34) and satisfies the anti-periodic boundary conditions i u(a)+ i u(b)0, i 0,,...,k ; u(n) 0, n Z[a, b], (35) k (b a) k i 3/ k i ri (n) ) /. Let m intheorem., we have the following corollary.

8 LiuAdvances in Difference Equations (08) 08:9 Page 8 of 4 Corollary.3 If u(n) is a nonzero solution of u(n) p u(n)+ and satisfies the anti-periodic boundary conditions r i (n) i u(n) p i u(n) 0 (36) i u(a)+ i u(b)0, i 0,; u(n) 0, n Z[a, b], (37) (b a) ( i /p)(p ) ( i)(p ) r i (n) ) /q q. Next, we establish a Lyapunov-type inequality for Eq. (8). Theorem. If u(n) is a nonzero solution of Eq.(8) satisfyingthe anti-periodic boundary conditions (9), r(n) /(p ) ) 3m >, (38) (b a) m /(p ) Ɣ q p where Ɣ q p ( p q p )( ) q (q )/(p q ) ( p)/(p q ). (39) p Proof Choose c Z[a, b] suchthat u(c) max n Z[a,b] u(n). Since(9), it follows from Lemma. that and u(c) (c a)(b c) b a b i u(n) (b a) 8 b u(n) b a 4 b u(n) (40) b i+ u(n), i,,...,m. (4) From (40)and(4), we obtain u(c) b a 4 b u(n) b a (b a) 4 8 b a 4 ( (b a) 8 b 4 u(n) ) b 6 u(n)

9 LiuAdvances in Difference Equations (08) 08:9 Page 9 of 4 b a 4 ( (b a) 8 ) m b m u(n). (4) Applying discrete Hölder s inequality (6) to the summation on the right-hand side of (4) with f (n),g(n) m u(n), α p,andβ p,weobtainthat p u(c) b a ( (b a) ) ( m b (b a) (p )/(p ) m u(n) ) /(p ) p 4 8 (b a)m /(p ) 3m m u(n) ) /(p ) p. (43) On the other hand, from (8), we have m u(n) p m u(n)( ) m r(n) u(n +) q u(n + ), (44) m u(n) p r(n) u(n +) q. (45) Summing (45)froma to b,wehave b m u(n) p b r(n) u(n +) q, (46) m u(n) ) /(p ) p r(n) u(n +) ) /(p ) q. (47) From (43) and(47), we have u(c) (b a) m /(p ) 3m (b a)m /(p ) 3m r(n) u(n +) q ) /(p ) ) /(p )u(c) r(n) (q )/(p ) K u(c) (q )/(p ), (48) where K (b a)m /(p ) 3m ) /(p ) r(n). (49)

10 LiuAdvances in Difference Equations (08) 08:9 Page 0 of 4 Using inequality (0) in Lemma. with A B,z u(c), τ,σ q p,wehave u(c) u(c) (q )/(p ) + Ɣ q > 0. (50) p From (48)and(50), we get u(c) u(c) + Ɣ q > 0. (5) K p This is possible only if i.e., K 4Ɣq p K > < 0, (5) Ɣ q p. (53) From (49) and(53), we obtain (b a) m /(p ) 3m r(n) /(p ) ) >. (54) Ɣ q p Thus, (38) holds. This completes the proof of Theorem.. For p > q, using a method similar to Theorem., we have the following theorem. Theorem.3 If u(n) is a nonzero solution of m u(n) p m u(n)+( ) m r(n)u(n + ) 0, (55) satisfying the anti-periodic boundary conditions (9), ) /(p ) r(n) 3m >, (56) (b a) m /(p ) Ɣ p where ( )( ) p 3 /(p 3) Ɣ p ( p)/(p 3). (57) p p Remark For p q, using a method similar to Theorem., wehavethattheresult coincides with Corollary.3 in [9]. Let m intheorem., we have the following corollary.

11 LiuAdvances in Difference Equations (08) 08:9 Page of 4 Corollary.4 If u(n) is a nonzero solution of u(n) p u(n)+r(n) u(n +) q u(n + ) 0 (58) and satisfies the anti-periodic boundary conditions u(a)u(b)0, u(n) 0, n Z[a, b], (59) ) /(p ) r(n) >, (b a) /(p ) Ɣ q p where Ɣ q is defined as in (39). p Let m intheorem.3, we have the following corollary. Corollary.5 If u(n) is a nonzero solution of u(n) p u(n)+r(n)u(n + ) 0 (60) and satisfies the anti-periodic boundary conditions u(a)u(b)0, u(n) 0, n Z[a, b], (6) ) /(p ) r(n) >, (b a) /(p ) Ɣ p where Ɣ p is defined as in (57). 3 Applications In this section, we investigate the nonexistence and uniqueness for solutions of certain BVPs. First, we consider the nonexistence for solutions of the BVP consisting of (7) and the boundary conditions (5). Theorem 3. Assume m (b a) (m i /p)(p ) (m i)(p ) ri (n) ) /q q <, (6) where q is the Hölder conjugate exponent of p, i.e., /p +/q.then BVP (7), (5) has no nontrivial solution.

12 LiuAdvances in Difference Equations (08) 08:9 Page of 4 Proof Assume the contrary. Then BVP (7), (5) has a nontrivial solution u(n). By Theorem.,inequality(3) holds. This contradicts assumption (6). Next, we consider the uniqueness for solutions of nonhomogeneous BVP consisting of the equation k k u(n)+ r i (n) i u(n)f (n), n Z[A, B], (63) and the boundary conditions i u(a)+ i u(b)m i, i 0,,...,k ; n Z[a, b], (64) where k N, n Z,andf, r i (n)(i 0,,...,k ) are real-valued functions defined on Z, A, B, a, b N, A < a < b < B,andM i R, i 0,,...,k. Theorem 3. Assume k (B A) (k i /) (k i) ( B ri (n) ) / <. (65) na Then BVP (63), (64) hasatmostonesolutionon(a, B) for any a, b (A, B), M i R, i 0,,...,k. Proof Let u (n) andu (n) betwosolutionsofbvp(63), (64) in(a, B). Define u(n) u (n) u (n). Then u(n) is a solution of BVP (3), (33). Then, by Theorem 3. with p and m k, wehaveu(n) 0, i.e., u (n) u (n). This shows that BVP (63), (64) hasat most one solution on (A, B). Acknowledgements The author is indebted to the anonymous referees for their valuable suggestions and helpful comments which helped improve the paper significantly. Funding This research was supported by the Natural Science Foundation of Shandong Province (China) (Grant No. ZR08MA08), A Project of Shandong Province Higher Educational Science and Technology Program (China) (Grant No. J4LI09), and the National Natural Science Foundation of China (Grant No. 677). Competing interests The author declares that there is no conflict of interests regarding the publication of this paper. Authors contributions HDL organized and wrote this paper. Further, he examined all the steps of the proofs in this research. The author read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: May 08 Accepted: 6 June 08 References. Agarwal, R.P., Özbekler, A.: Disconjugacy via Lyapunov and Vallée Poussin type inequalities for forced differential equations. Appl. Math. Comput. 65, (05). Agarwal, R.P., Özbekler, A.: Lyapunov type inequalities for even order differential equations with mixed nonlinearities. J. Inequal. Appl. 05, Article ID 4 (05)

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