Lyapunov type inequalities for even order differential equations with mixed nonlinearities

Transcription

1 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 DOI /s R E S E A R C H Open Access Lypunov type inequlities for even order differentil equtions with mixed nonlinerities Rvi P Agrwl 1,2* nd Abdullh Özbekler 1,3 * Correspondence: grwl@tmuk.edu 1 Deprtment of Mthemtics, Texs A&M University-Kingsville, 700 University Blvd., Kingsville, TX , USA 2 Deprtment of Mthemtics, Fculty of Science, King Abdulziz University, P.O. Box 80203, Jeddh, 21589, Sudi Arbi Full list of uthor informtion is vilble t the end of the rticle Abstrct In the cse of oscilltory potentils, we present Lypunov nd Hrtmn type inequlities for even order differentil equtions with mixed nonlinerities: x (2n) (t)+( 1) n 1 m q i(t) x(t) αi 1 x(t)=0,wheren, m N nd the nonlinerities stisfy 0<α 1 < < α j <1<α j+1 < < α m <2. MSC: 34C10 Keywords: Lypunov type inequlity; mixed nonliner; sub-liner; super-liner 1 Introduction Consider the Hill eqution x (t)+q(t)x(t)=0; t b, (1.1) where q(t) L 1 [, b] is rel-vlued function. If there exists nontrivil solution x(t) of (1.1) stisfying the Dirichlet boundry conditions x()=x(b)=0, (1.2) where, b R with < b nd x(t) 0fort (, b), then the inequlity q(t) dt >4/(b ) (1.3) holds. This striking inequlity ws first proved by Lypunov [1] nd it is known s the Lypunov inequlity. Lter Wintner [2] nd therefter some more uthors chieved the replcement of the function q(t) in (1.3) by the function q + (t), i.e. they obtined the following inequlity: q + (t)dt >4/(b ), (1.4) where q + (t)=mx{q(t), 0}, nd the constnt 4 in the right hnd side of inequlities (1.3) nd (1.4) is the best possible lrgest number (see [1] nd[3], Theorem 5.1) Agrwl nd Özbekler; licensee Springer. This is n Open Access rticle distributed under the terms of the Cretive Commons Attribution License ( which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly credited.

2 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 2 of 10 In [3], Hrtmn obtined n inequlity shrper thn both (1.3)nd(1.4): (b t)(t )q + (t)dt >(b ). (1.5) Clerly, (1.5)implies(1.4), since (b t)(t ) (b ) 2 /4 (1.6) for ll t (, b), nd equlity holds when t =( + b)/2. It ppers tht the first generliztion of Hrtmn s result ws obtined by Ds nd Vtsl [4], Theorem 3.1. Theorem 1.1 (Hrtmn type inequlity) If x(t) is nontrivil solution of the eqution x (2n) (t)+( 1) n 1 q(t)x(t)=0; t b, (1.7) stisfying the 2-point boundry conditions x (k) ()=x (k) (b)=0; k =0,1,...,n 1, (1.8) where, b R with < b re consecutive zeros, then the inequlity holds. (b t) 2n 1 (t ) 2n 1 q + (t)dt >(2n 1)(n 1)! 2 (b ) 2n 1 (1.9) Note tht (1.9) generlizes the Lypunov type inequlity q + (t)dt >4 2n 1 (2n 1)(n 1)! 2 (b ) 1 2n (1.10) by (1.6)(seelso[5], Corollry 3.3). The proof of Theorem 1.1 is bsed on the Green s function G n (t, s)ofthe2-pointboundry vlue problem x (2n) (t)=0; t b, (1.11) stisfying (1.8), obtined in [4]sfollows: ( G n (t, s)= ( 1)n 1 (t )(b s) (2n 1)! (b ) ( ) n 1+j (s t) n 1 j j ) n n 1 j=0 ( ) (b t)(s ) j, t s b, (1.12) (b )

3 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 3 of 10 nd ( G n (t, s)= ( 1)n 1 (s )(b t) (2n 1)! (b ) ( ) n 1+j (t s) n 1 j j ) n n 1 j=0 ( ) (t )(b s) j, (b ) s t. (1.13) Note tht ( 1) n 1 G n (t, s) 0nd mx s b ( 1)n 1 G n (t, s)=( 1) n 1 G n (t, t) (1.14) for ll t [, b](see[5]). In fct, in view of the symmetry of G n (t, s), (1.14)lsoimpliestht mx t b ( 1)n 1 G n (t, s)=( 1) n 1 G n (s, s). (1.15) In view of the lternting term ( 1) n 1 in the Green s function G n (t, s), Hrtmn nd Lypunov type inequlities for the 2-point boundry vlue problem x (2n) (t)+q(t)x(t)=0; t b, (1.16) stisfying the boundry conditions (1.8) cn be obtined by replcing the function q + (t) by q(t) in (1.9)nd(1.10), respectively. The Lypunov inequlity nd its generliztions hve been used successfully in connection with oscilltion nd Sturmin theory, symptotic theory, disconjugcy, eigenvlue problems, nd vrious properties of the solutions of (1.1) nd relted equtions; see for instnce [2, 3, 6 23] nd the references cited therein. For some of its extensions to Hmiltonin systems, higher order differentil equtions, nonliner nd hlf-liner differentil equtions, difference nd dynmic equtions, nd functionl nd impulsive differentil equtions, we refer in prticulr to [10, 11, 24 43]. The im of our pper is to extend the well-known Lypunov nd Hrtmn type inequlities for even order nonliner equtions of the form x (2n) (t)+( 1) n 1 m q i (t) x(t) α i 1 x(t)=0, (1.17) where n, m N, thepotentilsq i (t), i = 1,...,m, re rel-vlued functions nd no sign restrictions re imposed on them. Further, the exponents in (1.17)stisfy 0<α 1 < < α j <1<α j+1 < < α m < 2. (1.18) It is cler tht the two specilcses of (1.17) re the even order sub-liner eqution x (2n) (t)+( 1) n 1 q(t) x(t) γ 1 x(t)=0, 0<γ < 1, (1.19) nd the even order super-liner eqution x (2n) (t)+( 1) n 1 p(t) x(t) β 1 x(t)=0, 1<β < 2. (1.20)

4 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 4 of 10 Further, we note tht letting α i 1, i =1,...,j,ndα i 1 +, i = j+1,...,m,in(1.17)results in (1.7)withq(t)= m q i(t), i.e., ( m ) x (2n) (t)+( 1) n 1 q i (t) x(t)=0, (1.21) nd s consequence, our results extend nd improve the min results of Ds nd Vtsl [4], i.e. Theorem 1.1, nd in prticulr the clssicl Lypunov [1] nd Hrtmn s[3] results. We further remrk tht the Lypunov type inequlities hve been studied by mny uthors, see for instnce the survey pper [44] nd the references therein, but to the best of our knowledge there re no results in the literture for (1.17), nd in prticulr for (1.19) nd (1.20). 2 Min results Throughout this pper we shll ssume tht q i (t) L 1 [, b], i =1,...,m. We will need the following lemm. Lemm 2.1 If A is positive, nd B, z re nonnegtive, then Az 2 Bz μ +(2 μ)μ μ/(2 μ) 2 2/(μ 2) A μ/(2 μ) B 2/(2 μ) 0 (2.1) for ny μ (0, 2) with equlity holding if nd only if B = z =0. Proof Let H(z)=Az 2 Bz μ, z 0, (2.2) where A >0ndB 0. Clerly, when z =0orB =0,(2.1) is obvious. On the other hnd, if B >0,thenitisesytoseethtH ttins its minimum t z 0 =(μa 1 B/2) 1/(2 μ) nd H min = (2 μ)μ μ/(2 μ) 2 2/(μ 2) A μ/(2 μ) B 2/(2 μ). Thus, (2.1) holds. Note tht if B >0,then(2.1)isstrict. Now we stte nd prove our first result. Theorem 2.1 (Hrtmn type inequlity) If x(t) is nontrivil solution of (1.17) stisfying the 2-point boundry conditions (1.8), where, b R with < b re consecutive zeros, then the inequlity ( (b t) 2n 1 (t ) 2n 1 Q )( m (t)dt (b t) 2n 1 (t ) 2n 1 Q ) m (t)dt holds, where >(2n 1) 2 (n 1)! 4 (b ) 4n 2 /4 (2.3) Q m (t)= q i + (t) nd Q m (t)= θ i q i + (t)

5 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 5 of 10 with θ i =(2 α i )α i α i /(2 α i ) 2 2/(α i 2). (2.4) Proof Let x(t) be nontrivil solution of (1.17) stisfying the boundry conditions (1.8), where, b R with < b re consecutive zeros. Without loss of generlity, we my ssume tht x(t)>0fort (, b). In fct, if x(t)<0fort (, b), then we cn consider x(t), which is lso solution. Then, by using the Green s function of (1.11)-(1.8), x(t)cnbeexpressed s x(t)= ( 1) n 1 G n (t, s) q i (s)x α i (s)ds. (2.5) Let x(c)=mx t (,b) x(t). Then by (2.1) in Lemm 2.1 with A = B =1,wehve x α i (c)<x 2 (c)+θ i. Using this in (2.5), we obtin x(c)= < ( 1) n 1 G n (c, s) ( 1) n 1 G n (s, s) q i (s)x α i (s)ds which implies the qudrtic inequlity q i + (s)[ x 2 ] (c)+θ i ds, (2.6) 1 x 2 (c) x(c)+ 2 >0, (2.7) where nd 1 = 2 = ( 1) n 1 G n (s, s) Q m (t)ds ( 1) n 1 G n (s, s) Q m (t)ds. But inequlity (2.7) is possible if nd only if 1 2 > 1/4. Finlly, we note tht ( 1) n 1 G n (s, s)= (b s) 2n 1 (s ) 2n 1 (2n 1)(n 1)! 2 (b ) 2n 1. This completes the proof of Theorem 2.1. Next, we prove the following result. Theorem 2.2 (Lypunov type inequlity) If x(t) is nontrivil solution of (1.17) stisfying the 2-point boundry conditions (1.8) where, b R with < b re consecutive zeros, then

6 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 6 of 10 the inequlity ( )( ) Q m (t)dt Q m (t)dt > 44n 3 (2n 1) 2 (n 1)! 4 (2.8) (b ) 4n 2 holds, where the functions Q m nd Q m re defined in Theorem 2.1. Proof In the view of (1.6), (2.3) immeditely implies (2.8). Remrk 1 Since lim θ i = lim θ i = 1/4, α i 1 + (i>j) α i 1 (i j) where θ i is defined in (2.4), it is esy to see tht inequlities (2.3) nd(2.8) reducetoinequlities (1.9)nd(1.10), respectively, with q + (t)= m q+ i (t). Thus, Theorems 2.1 nd 2.2 reduce to Theorem 3.1 of Ds nd Vtsl [4], nd Corollry 3.3 of Yng [5], respectively. Moreover, when n = 1, they reduce to the clssicl Lypunov (1.4) ndhrtmn(1.5) inequlities with q + (t)= m q+ i (t). Remrk 2 It is of interest to find nlogs of Theorems 2.1 nd 2.2 for (1.17)-(1.8)without the term ( 1) n 1, i.e.,fortheeqution x (2n) (t)+ q i (t) x(t) α i 1 x(t)=0 (2.9) stisfying the 2-point boundry conditions (1.8). We stte these results in the following. Proposition 1 If x(t) is nontrivil solution of (2.9) stisfying the2-point boundry conditions (1.8) where, b R with < b re consecutive zeros, then the following hold: (i) Hrtmn type inequlity; ( (b t) 2n 1 (t ) 2n 1 P )( m (t)dt (b t) 2n 1 (t ) 2n 1 P ) m (t)dt >(2n 1) 2 (n 1)! 4 (b ) 4n 2 /4. (ii) Lypunov type inequlity; where ( )( ) P m (t)dt P m (t)dt > 44n 3 (2n 1) 2 (n 1)! 4, (b ) 4n 2 P m (t)= qi (t) nd P m (t)= θ qi i (t) nd θ i is defined in (2.4).

7 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 7 of 10 When q i (t)=0,forlli =2,3,...,m 1,then(1.17)nd(2.9) reduce to the equtions x (2n) (t)+( 1) n 1 p(t) x(t) β 1 x(t)+( 1) n 1 q(t) x(t) γ 1 x(t)=0 (2.10) nd x (2n) (t)+p(t) x(t) β 1 x(t)+q(t) x(t) γ 1 x(t)=0, (2.11) respectively, where p(t)=q m (t), q(t)=q 1 (t), γ = α 1 (0, 1), nd β = α m (1, 2). For these equtions we hve the following corollries. Corollry 2.3 If x(t) is nontrivil solution of (2.10) stisfying the 2-point boundry conditions (1.8) where, b R with < b re consecutive zeros, then the following hold: (i) Hrtmn type inequlity; ( (b t) 2n 1 (t ) 2n 1[ p + (t)+q + (t) ] ) dt ( (b t) 2n 1 (t ) 2n 1[ β 0 p + (t)+γ 0 q + (t) ] ) dt >(2n 1) 2 (n 1)! 4 (b ) 4n 2 /4. (ii) Lypunov type inequlity; where ( [ p + (t)+q + (t) ] b [ dt)( β0 p + (t)+γ 0 q + (t) ] ) dt > 44n 3 (2n 1) 2 (n 1)! 4, (b ) 4n 2 β 0 =(2 β)β β/(2 β) 2 2/(β 2) nd γ 0 =(2 γ )γ γ /(2 γ ) 2 2/(γ 2). Corollry 2.4 If x(t) is nontrivil solution of (2.11) stisfying the2-point boundry conditions (1.8) where, b R with < b re consecutive zeros, then the following hold: (i) Hrtmn type inequlity; ( (b t) 2n 1 (t ) 2n 1[ p(t) + q(t) ] ) dt ( (b t) 2n 1 (t ) 2n 1[ ) ] β 0 p(t) + γ0 q(t) dt >(2n 1) 2 (n 1)! 4 (b ) 4n 2 /4. (ii) Lypunov type inequlity; ( [ p(t) + q(t) ] b [ dt)( β0 p(t) + γ 0 q(t) ] ) dt > 44n 3 (2n 1) 2 (n 1)! 4, (b ) 4n 2 where the constnts β 0 nd γ 0 re defined in Corollry 2.3.

8 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 8 of 10 Remrk 3 Corollry 2.3 is of prticulr interest since it gives two new results for the even order sub-liner eqution (when p(t) = 0) nd super-liner eqution (when q(t)=0),i.e., (1.19) nd(1.20). Moreover, clssicl results cn lso be obtined by the limiting process γ 1 nd β 1 + in inequlities (i) nd (ii) given in Corollry Some specil cses In this section we consider the situtions when the potentils q i (t), i =1,...,m, re either liner, convex, or concve functions. Corollry 3.1 Let q i (t)=c i t + d i, i =1,...,m, in (1.17) be positive on [, b]. If x(t) is nontrivil solution of (1.17) stisfying the 2-point boundry conditions (1.8), where, b R with < b re consecutive zeros, then the inequlity ( )( ) (2n 1) 2 (n 1)! 4 (4n 1)! 2 ( + b)ĉm +2 D m ( + b) C m +2 D m > (3.1) (2n 1)! 4 (b ) 4n holds, where Ĉ m = c i, D m = d i, C m = θ i c i nd D m = θ i d i, nd θ i is the sme s in (2.4). Proof In this specil cse, we need to compute the integrl I := (b t) 2n 1 (t ) 2n 1 (ct + d)dt for rel constnts c nd d. Writing ct + d = c(t )+c + d nd mking the substitution t =(b )z +,weobtin I = c (b t) 2n 1 (t ) 2n dt +(c + d) (b t) 2n 1 (t ) 2n 1 dt 1 1 = c(b ) 4n (1 z) 2n 1 z 2n dz +(c + d)(b ) 4n 1 (1 z) 2n 1 z 2n 1 dz 0 = c(b ) 4n B(2n,2n +1)+(c + d)(b ) 4n 1 B(2n,2n) = B(2n,2n) [ c(b )/2 + c + d ] (b ) 4n 1, 0 where B(, ) is the Bet function. However, since we hve B(2n,2n)= Ɣ2 (2n) Ɣ(4n) I = = (2n 1)!2 (4n 1)! (2n 1)!2[ ] c( + b)/2 + d (b ) 4n 1. (4n 1)! Using this in (2.3)withq i (t)=c i t + d i the result follows.

9 Agrwl nd Özbekler Journl of Inequlities nd Applictions (2015) 2015:142 Pge 9 of 10 Corollry 3.2 Let q i (t), i =1,...,m, in (1.17) be continuous, positive, nd convex on [, b]. If x(t) is nontrivil solution of (1.17) stisfying the 2-point boundry conditions (1.8), where, b R with < b re consecutive zeros, then the inequlity holds. [ qi (b)+q i () ] > (2n 1)2 (n 1)! 4 (4n 1)! 2 (3.2) (2n 1)! 4 (b ) 4n Corollry 3.3 Let q i (t), i =1,...,m, in (1.17) be continuous, positive, nd concve on [, b]. If x(t) is nontrivil solution of (1.17) stisfying the 2-point boundry conditions (1.8), where, b R with < b re consecutive zeros, then the inequlity holds. q i [ ( + b)/2 ] > (2n 1) 2 (n 1)! 4 (4n 1)! 2 (2n 1)! 4 (b ) 4n (3.3) The proofs of Corollries 3.2 nd 3.3 re similr to those of Propositions 4.2 nd 4.3 of Ds nd Vtsl [4], nd hence they re omitted. Finlly, we conclude this pper with the following remrk. When n =1,theresultsobtined in this pper for (1.17)(or(2.11)) cn esily be extended to the second order equtions x (t) ± p(t) x(t) β 1 x(t) q(t) x(t) γ 1 x(t)=0, (3.4) i.e., for Emden-Fowler sub-liner nd Emden-Fowler super-liner equtions with positive nd negtive coefficients. The formultions of these results re left to the reder. It will be of interest to find similr results for the even order mixed nonliner equtions of the form (1.17) forsomeα k 2, or the super-liner eqution (1.20) forβ [2, ). In fct, the cse when n = 1 (Emden-Fowler super-liner) is of immense interest. Competing interests The uthors declre tht they hve no competing interests. Authors contributions All uthors contributed eqully to the writing of this pper. All uthors red nd pproved the finl mnuscript. Author detils 1 Deprtment of Mthemtics, Texs A&M University-Kingsville, 700 University Blvd., Kingsville, TX , USA. 2 Deprtment of Mthemtics, Fculty of Science, King Abdulziz University, P.O. Box 80203, Jeddh, 21589, Sudi Arbi. 3 Deprtment of Mthemtics, Atilim University, Incek, Ankr, 06836, Turkey. Acknowledgements This work ws crried out when the second uthor ws on cdemic leve, visiting TAMUK (Texs A&M University-Kingsville) nd he wishes to thnk TAMUK. This work is prtilly supported by TUBITAK (The Scientific nd Technologicl Reserch Council of Turkey). Received: 22 Jnury 2015 Accepted: 4 Mrch 2015 References 1. Lypunov, AM: Probleme générl de l stbilité du mouvement (French Trnsltion of Russin pper dted 1893). Ann. Fc. Sci. Univ. Toulouse 2, (1907); Reprinted s Ann. Mth. Studies, No. 17, Princeton (1947) 2. Wintner, A: On the nonexistence of conjugte points. Am. J. Mth. 73, (1951)

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