Lyapunov-type inequalities for Laplacian systems and applications to boundary value problems

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1 Avilble online t J. Nonliner Sci. Appl Reserch Article Journl Hoepge: Lypunov-type inequlities for Lplcin systes nd pplictions to boundry vlue probles Qio-Lun Li Wing-Su Cheungb College of Mthetics nd Infortion Science Hebei Norl University Shijizhung Chin. b Deprtent of Mthetics The University of Hong Kong Hong Kong Chin. Counicted by S. H. Wu Abstrct In this pper we estblish soe new Lypunov-type inequlities for clss of Lplcin systes. With these sufficient conditions for the non-existence of nontrivil solutions to certin boundry vlue probles re obtined. A lower bound for the eigenvlues is lso deduced. Keywords: Lypunov type inequlity boundry vlue proble Lplcin systes MSC: 26D15 26D10 34B15. c 2018 All rights reserved. 1. Introduction For the second-order liner differentil eqution x 00 t + qtxt = where q C b] R the following result is known: Theore 1.1. Assue xt is solution of Eq. 1.1 such tht x = xb = 0 nd xt 6= 0 for t b. Then Zb b qs ds > 4. Eqution 1.1 is known s the Lypunov inequlity. Since this inequlity hs found extensive pplictions in the study of vrious properties of solutions of ny differentil nd difference equtions including oscilltion theory disconjugcy nd eigenvlue probles uch efforts hve been devoted to the iproveent nd extensions of the inequlity. These include for exple works on dely differentil equtions higher order differentil equtions discrete differentil equtions nd Hiltonin systes. See for exple ] nd the references therein. Corresponding uthor Eil ddresses: qll71125@163.co Qio-Lun Li wscheung@hku.hk Wing-Su Cheung doi: /jns Received: Revised: Accepted:

2 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl In 2003 Yng 11] considered the following second order hlf liner differentil eqution rt x t γ 2 x t + qt xt γ 2 xt = nd obtined the following inequlity q + b γ 1 tdt rt] dt 1/γ 1 > 2 γ provided tht Eq. 1.2 hs nontrivil solution xt which stisfies x = xb = 0 nd xt 0 t b. Here q + t := xqt 0} r > 0 γ > 1. In 2006 Npoli nd Pinsco in 5] estblished Lypunov-type inequlities for the qusiliner syste of resonnt type u t p 2 u t = ft ut µ 2 vt ν ut v t q 2 v t = gt ut µ vt ν 2 vt on the intervl b with Dirichlet boundry conditions u = ub = v = vb = 0. In 2017 Jleli nd Set 7] obtined Lypunov type inequlities for the following syste involving one-diensionl -Lplcin opertors i = 1 2: u t p 1 2 u t u t q 1 2 u t = ft ut α 2 vt β ut v t p 2 2 v t v t q 2 2 v t = gt ut α vt β 2 vt on the intervl b under the Dirichlet boundry conditions u = ub = v = vb = 0. In this pper we shll estblish soe new Lypunov-type inequlities which re interesting in their own right nd s pplictions we obtin sufficient conditions for the non-existence of nontrivil solutions to certin boundry vlue probles nd lower bound for the eigenvlues. 2. Lypunov type inequlities Theore 2.1. If nontrivil continuous solution to the boundry vlue proble u t 2 u t = rt ut α 2 ut u + ub = 0 u + u b = exists where r is positive continuous function on b] > 1 nd α = / then 2 } b 1 rtdt. Proof. Let u be nontrivil solution to 2.1. Multiplying eqution 2.1 by u nd integrting over b we obtin t dt = rt ut α dt.

3 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl For ny t 0 b] we hve So 2ut 0 = t0 u tdt u tdt. t 0 2ut0 t dt. By using Hölder inequlity with preters nd p i = hence 2 ut0 b 1 p i 2 t0 b p we hve 1 t dt 1 pi t dt i = Now choosing t 0 s the point where ut ttins its xiu we hve t0 α rtdt > rt ut α 2 dt t0 b p 2 b 1 } t0. Using the inequlity we get The proof is coplete. i } b 1 rtdt. Theore 2.2. If nontrivil continuous solution u 1 u 2 to the Lplcin syste u 1 t 2 u 1 t = r1 t t α t α2 u 1 t u 2 t 2 u 2 t = r2 t t β 1 u2 t β 2 2 u2 t u j + u j b = 0 j = 1 2 u j + u j b = 0 j = exists where r 1 r 2 re nonnegtive rel-vlued functions > 1 i = α j β j > 1 j = 1 2 nd α 1 + α 2 = 1 β 1 + β 2 = then 2 b 1 }] β 1 2 b 1 }] α 2 1 r 1 tdt β 1 1 r 2 tdt α 2. Proof. Let u 1 u 2 be nontrivil solution to 2.2. Multiplying the first eqution of 2.2 by u 1 nd

4 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl integrting over b we obtin 1 t dt = r 1 t u1 t α 1 u2 t α2 dt. 2.4 Multiplying the second eqution of 2.2 by u 2 nd integrting over b we obtin 2 t dt = Fro the proof of Theore 2.1 we get for ny t 0 x 0 b] 2 b p t b q x 0 Fro 2.4 nd 2.6 we obtin 2 b p t 0 Fro 2.5 nd 2.7 we hve 2 2 b q x 0 r 2 t u1 t β 1 u2 t β2 dt t dt i = t dt i = r 1 t u1 t α 1 u2 t α2 dt. 2.8 r 2 t u1 t β 1 u2 t β2 dt. 2.9 Let t 0 x 0 b] be such tht t 0 = x u1 t : t b} u2 x 0 = x u2 t : t b}. Fro 2.8 we hve nd so By using the inequlity we get 2 b p t 0 t 0 α 1 u2 x 0 α 2 Siilrly fro 2.9 we obtin 2 } b p t 0 t 0 α 1 u2 x 0 α 2 i } b 1 u 1 t 0 α 1 u 2 x 0 α 2 b r 1 tdt r 1 tdt. r 1 tdt } b 1 u 1 t 0 β 1 u 2 x 0 β 2 r 2 tdt. 2.11

5 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl Rising inequlity 2.10 to power e 1 > 0 inequlity 2.11 to power e 2 > 0 nd ultiplying the resulting inequlities we obtin 2 b 1 α 1 u 1 t 0 }] e1 e 1 +β 1 e 2 2 b 1 u 2 x 0 α 2e 1 + β 2 }] e2 e 1 +e 2 e 2 e1 e2. r 1 tdt r 2 tdt Choose e 1 nd e 2 such tht u1 t 0 u2 x 0 cncel out i.e. e1 e 2 solve the hoogeneous liner syste α 1 α 2 e 1 + β 2 e 1 + β 1 e 2 = 0 e 2 = 0. Using 2.3 we y tke Then we hve e 1 = β 1 e 2 = α 2. nd so 2 b 1 2 b 1 The proof is coplete. }] β1 }] β 1 2 }] b 1 2 b 1 α 2 }] α β1 1 r 1 tdt r 1 tdt β 1 1 r 2 tdt r 2 tdt α 2 α 2. Theore 2.3. If nontrivil continuous solution u 1 u 2 u 3 to the Lplcin syste u 1 t pi 2 u 1 t = r 1 t u 1 t α1 2 u 2 t α 2 u 3 t α 3u 1 t u 2 t qi 2 u 2 t = r 2 t u 1 t α 1 u 2 t α2 2 u 3 t α 3u 2 t u 3 t γi 2 u 3 t = r 3 t u 1 t α 1 u 2 t α 2 u 3 t α3 2 u 3 t u j + u j b = 0 j = u j + u j b = 0 j = exists where r 1 r 2 r 3 re nonnegtive rel-vlued functions > 1 i = α j > 1 j = nd α 1 + α 2 + α 3 =

6 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl then 2 b 1 r 1 tdt α 1 }] α 1 r 2 tdt α 2 2 b 1 r 3 tdt }] α 2 α 3. 2 b γ }] α Proof. Let u 1 u 2 u 3 be nontrivil solution to Multiplying the first eqution of 2.12 by u 1 nd integrting over b we obtin 1 t dt = r 1 t u1 t α 1 u2 t α 2 u3 t α3 dt Let t 0 x 0 y 0 b] be such tht t 0 = x u1 t : t b } 2 x 0 = x u2 t : t b } 3 y 0 = x u3 t : t b }. Fro the proof of Theore 2.1 we get Fro 2.15 nd 2.16 we obtin which yields By using the inequlity we get Siilrly we obtin 2 b p t 0 2 b p t 0 t 0 α 1 u2 x 0 α 2 u3 y 0 α 3 1 t dt i = } b p t 0 t 0 α 1 u2 x 0 α 2 u3 y 0 α 3 2 } b 1 2 } b 1 2 } b γ i 1 2 u1 t 0 α 1 u1 t 0 α 1 u2 x 0 α 2 u2 x 0 α 2 u3 y 0 α 3 u3 y 0 α 3 u1 t 0 α 1 u2 x 0 α 2 u3 y 0 α 3 r 1 tdt b r 1 tdt. r 1 tdt r 2 tdt 2.18 r 3 tdt. 2.19

7 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl Rising inequlity 2.17 to power e 1 > 0 inequlity 2.18 to power e 2 > 0 inequlity 2.19 to power e 3 > 0 nd ultiplying the resulting inequlities we obtin 2 b 1 α 1 u 1 t 0 u 3 y 0 α 3e 1 +α 3 e 2 + }] e1 e 1 +α 1 e 2 +α 1 e 3 α 3 2 b 1 u 2 x 0 α 2e 1 + }] e2 α 2 e 2 +α 2 e 3 2 b γ e 3 e1 e2 e3. r 1 tdt r 2 tdt r 3 tdt }] e3 e 1 +e 2 +e 3 Choose e 1 e 2 nd e 3 such tht u 1 t 0 u 2 x 0 u 3 y 0 cncel out i.e. e 1 e 2 e 3 solve the hoogeneous liner syste α 1 e 1 + α 1 e 2 + α 1 e 3 = 0 α 2 e 1 + α 2 e 2 + α 2 e 3 = 0 Fro 2.13 we y tke nd get 2 b 1 }] α1 α 3 e 1 + α 3 e 2 + α1 r 1 tdt r 2 tdt Fro 2.13 we get r 1 tdt 2 b 1 α 1 The proof is then coplete. }] α 1 r 2 tdt α 3 e 1 = α 1 e 2 = e 3 = α 2 α 3 2 }] b 1 α 2 α 2 e 3 = 0. α 2 r 3 tdt 2 b 1 r 3 tdt α 3. }] α 2 α 3. 2 }] b γ 2 b γ α 3 }] α 3 As iedite consequences of Theores the following corollry gives sufficient conditions for the non-existence of nontrivil solutions to the bove boundry vlue probles.

8 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl Corollry 2.4. If then BVP 2.1 hs no nontrivil solution. b If 1 < r 1 tdt β 1 then BVP 2.2 hs no nontrivil solution. c If r 1 tdt < α 1 r 2 tdt 2 b 1 rtdt < 1 2 b 1 α 2 }] α 1 then BVP 2.12 hs no nontrivil solution. r 2 tdt }] β 1 r 3 tdt 2 b 1 α 2 α 3 2 b 1 } 2 }] α 2 b q }] α 2 2 }] α 3 b γ Theore 2.5. Let λ µ γ be generlized eigenvlue of the following syste u 1 t pi 2 u 1 t = λα 1 vt u 1 t α1 2 u 2 t α 2 u 3 t α 3u 1 t u 2 t qi 2 u 2 t = µα 2 vt u 1 t α 1 u 2 t α2 2 u 3 t α 3u 2 t u 3 t γi 2 u 3 t = γα 3 vt u 1 t α 1 u 2 t α 2 u 3 t α3 2 u 3 t u j + u j b = 0 j = u j + u j b = 0 j = where α j stisfy condition 2.13 v 0 v L 1 b. Then γ hλ µ where h : is the function defined by with nd Cα σ 1 αθ 2 = hx y = 1 α 3 C η 1 x σ y θ b vtdt σ = α 1 θ = α 2 η = α 3 2 b 1 }] σ 2 b 1 }] θ 2 }] η. b γ

9 Q.-L. Li W.-S. Cheung J. Nonliner Sci. Appl Proof. Let λ µ γ be generlized eigenvlue with corresponding nontrivil solutions u 1 u 2 u 3. By substituting into 2.14 the functions nd using condition 2.13 we hve Hence The proof is coplete. r 1 t = λα 1 vt r 2 t = µα 2 vt r 3 t = γα 3 vt 2 b 1 λ σ α σ 1 µθ α θ 2 γ η α η 3 }] σ vtdt. γ η 2 b 1 C α η 3 λσ µ θ vtdt. }] θ 2 }] η b γ Acknowledgent The first uthor s reserch ws supported by NNSF of Chin The uthors would like to thnk the referees for their extreely creful reding nd thoughtful suggestions. References 1] T. Abdeljwd A Lypunov type inequlity for frctionl opertors with nonsingulr Mittg-Leffler kernel J. Inequl. Appl pges. 1 2] R. P. Agrwl A. Özbekler Lypunov type inequlities for second order forced ixed nonliner ipulsive differentil equtions Appl. Mth. Coput ] M. F. Aktş D. Çkk A Tiryki On the Lypunov-type inequlities of three-point boundry vlue proble for third order liner differentil equtions Appl. Mth. Lett ] L.-Y. Chen C.-J. Zho W.-S. Cheung On Lypunov-type inequlities for two-diensionl nonliner prtil systes J. Inequl. Appl pges. 1 5] P. L. De Nápoli J. P. Pinsco Estites for eigenvlues of qusiliner elliptic systes J. Differentil Equtions ] S. Dhr Q.-K. Kong Lypunov-type inequlities for higher order hlf-liner differentil equtions Appl. Mth. Coput ] M. Jleli B. Set On Lypunov-type inequlities for p q-lplcin systes J. Inequl. Appl pges. 1 8] A. Tiryki D. Çkk M. F. Aktş Lypunov-type inequlities for certin clss of nonliner systes Coput. Mth. Appl ] X.-P. Wng Lypunov type inequlities for second-order hlf-liner differentil equtions J. Mth. Anl. Appl ] Y.-Y. Wng Y.-N. Li Y.-Z. Bi Lypunov-type inequlities for qusiliner systes with ntiperiodic boundry conditions Russin version ppers in Ukrïn. Mt. Zh Ukrinin Mth. J ] X.-J. Yng On inequlities of Lypunov type Appl. Mth. Coput