Applied Mathematics Letters. Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients

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1 Applied Mthemtics Letters 24 (20) Contents lists vilble t ScienceDirect Applied Mthemtics Letters journl homepge: Forced oscilltion of second-order nonliner differentil equtions with positive nd negtive coefficients A. Özbekler,, J.S.W. Wong b, A. Zfer c Deprtment of Mthemtics, Atilim University 06836, Incek, Ankr, Turkey b Deprtment of Mthemtics, City University of Hong Kong, Tt Chee Avenue, Kowloon, Hong Kong c Deprtment of Mthemtics, Middle Est Technicl University 0653 Ankr, Turkey r t i c l e i n f o b s t r c t Article history: Received 8 July 200 Received in revised form 2 Februry 20 Accepted 4 Februry 20 In this pper we give new oscilltion criteri for forced super- nd sub-liner differentil equtions by mens of nonprincipl solutions. 20 Elsevier Ltd. All rights reserved. Keywor: Nonprincipl Super-liner Sub-liner Oscilltion. Introduction The concept of the principl solution ws introduced in 936 by Leighton nd Morse [] in studying positiveness of certin qudrtic functionl ssocited with (r(t)x ) + ν(t)x = 0, t t 0. Since then the principl nd nonprincipl solutions hve been used successfully in connection with oscilltion nd symptotic theory of (.) nd relted equtions, see for instnce [ 8] nd the references cited therein. For some extensions to Hmiltonin systems, hlf-liner differentil equtions, dynmic equtions nd impulsive differentil equtions, we refer in prticulr to [4,9 2]. We recll tht nontrivil solution u of (.) is sid to be principl if for every solution v of (.) such tht u cv, c R, u(t) lim t v(t) = 0. It is well known tht principl solution u of (.) exists uniquely up to multipliction by nonzero constnt if nd only if (.) is nonoscilltory. A solution v tht is linerly independent of u is clled nonprincipl solution. Roughly speking, the wor principl nd nonprincipl my be replced by smll nd lrge or recessive nd dominnt. For other chrcteriztions of principl nd nonprincipl solutions of (.), see [5, Theorem 6.4], [3, Theorem 5.59]. In 999, Wong [8], by employing nonprincipl solution of (.), obtined the following oscilltion criterion for (.) (r(t)x ) + ν(t)x = f (t). For extensions of the theorem to impulsive differentil equtions nd dynmic equtions on time scles, see [,2]. (.2) Corresponding uthor. Fx: E-mil ddresses: ozbekler@gmil.com (A. Özbekler), jsww@chinneyhonkwok.com (J.S.W. Wong), zfer@metu.edu.tr (A. Zfer) /$ see front mtter 20 Elsevier Ltd. All rights reserved. doi:0.06/j.ml

2 226 A. Özbekler et l. / Applied Mthemtics Letters 24 (20) Theorem. (Wong s Theorem). Suppose tht (.) is nonoscilltory. Let z be positive solution of (.) stisfying < for some sufficiently lrge, i.e., nonprincipl solution. If (.3) lim H(t) = lim H(t) =, H(t) := then Eq. (.2) is oscilltory. s (.4) z(τ)f (τ)dτ, (.5) The im of our work is to extend the bove theorem to nonliner equtions of the form (r(t)x ) + p(t) x β x q(t) x γ x = f (t), t t 0, (.6) (i) 0 < γ < < β; (ii) r C([t 0, ), (0, )), p, q C([t 0, ), [0, )), f C([t 0, ), R). It is cler tht the two specil cses of (.6) re the Emden Fowler super-liner eqution (r(t)x ) + p(t) x β x = f (t), β > (.7) nd the Emden Fowler sub-liner eqution (r(t)x ) q(t) x γ x = f (t), 0 < γ <. (.8) Typiclly, nonliner results require the coefficient in n Emden Fowler eqution x + (t) x α x = f (t) to be non-negtive, see [4]. Fortuntely, we re ble to tke q to be negtive in (.8). On the other hnd, letting β + nd γ in (.6) results in (.) with ν(t) = p(t) q(t), i.e., (r(t)x ) + [p(t) q(t)]x = f (t), nd thus our result exten Theorem. by limiting process β, γ in (.6). We remrk tht the oscilltion of the solutions of (.7) nd (.8) hs been studied by mny uthors, see for instnce [8,4 24], but to the best of our knowledge there is no result in the literture similr to Theorem. for such nonliner equtions, especilly for (.6). Consider slightly more generl eqution thn (.6) (r(t)x ) + p(t)f(x) q(t)g(x) = f (t), t t 0, (.0) r, p, q, f re s in (ii), F, G C(R, R). By solution of (.0) defined on n intervl [T, ), T t 0, we men function x, x, (rx ) C[T, ), stisfying (.0). We note tht the ssumption of r, p, f being continuous is not sufficient to ensure the existence of extendble solutions of (.7) on [T, ), see [23]. However, s usul in the oscilltion theory we only consider solutions of (.0) which re extendble to [T, ) nd nontrivil in the neighborhood of infinity. Such solution is clled oscilltory if it hs rbitrrily lrge zeros, otherwise it is clled nonoscilltory. Eq. (.0) is clled oscilltory (nonoscilltory) if ll solutions re oscilltory (nonoscilltory). We shll ssume tht (C ) xf(x) > 0 nd xg(x) > 0 for x 0; (C 2 ) () lim x x F(x) >, lim x 0 x F(x) <, (b) lim x x G(x) <, lim x 0 x G(x) >. Using (C ) nd (C 2 ), it is esy to find positive constnts α 0, β 0, γ 0, δ 0 such tht mx Φ(x) = β 0, x 0 Ψ (x) = δ 0, mx x 0 min Φ(x) = α 0 ; x 0 Ψ (x) = γ 0, min x 0 Φ(x) = x F(x) nd Ψ (x) = x G(x). (.9) (.)

3 A. Özbekler et l. / Applied Mthemtics Letters 24 (20) nd In wht follows we define s N (t) := [β r(s)z 2 0 p(τ) + γ 0 q(τ)]z(τ)dτ, (.2) (s) N 2 (t) := s [α r(s)z 2 0 p(τ) + δ 0 q(τ)]z(τ)dτ. (.3) (s) 2. Min results Associted with Eq. (.0) we ssume tht the liner eqution (r(t)x ) + [p(t) q(t)]x = 0 (2.) is nonoscilltory. Denote by z(t) positive nonprincipl solution of (2.) which is defined on n intervl [, ). Noting tht we define H(t) := <, s The min result of this pper is the following theorem. (2.2) z(τ)f (τ)dτ. (2.3) Theorem 2.. Suppose tht (2.) is nonoscilltory nd let z(t) be positive solution of it stisfying (2.2), i.e. nonprincipl solution. If lim {H(t) N 2(t)} = lim {H(t) + N (t)} =, (2.4) H is given by (2.3), nd N nd N 2 re s defined by (.2) nd (.3), respectively, then Eq. (.0) is oscilltory. Proof. Suppose tht there is nonoscilltory solution x(t) of (.0). We my ssume tht x(t) 0 on [, ) for some t 0 sufficiently lrge. The chnge of vribles x = z(t)w, z(t) is positive nonprincipl solution of (2.), trnsforms (.0) into (r(t)z 2 w ) = {f (t) + p(t)φ(x) q(t)ψ (x)}z, t. (2.5) Integrtion of (2.5) le to w(t) = c + c 2 + H(t) + c = w() nd c 2 = r()z 2 ()w () re constnts. If x(t) > 0 on [, ), then using (.) we obtin w(t) c + c 2 s {p(τ)φ(x(τ)) q(τ)ψ (x(τ))}z(τ)dτ (2.6) + H(t) + N (t). (2.7) Similrly, if x(t) < 0 on [, ), then gin using (.) we obtin w(t) c + c 2 Note tht (2.2), (2.4), (2.7) nd (2.8) imply tht + H(t) N 2(t). (2.8) lim w(t) = lim w(t) = +. Becuse z(t) is positive, (2.9) implies tht x(t) cnnot hve definite sign on [, ), contrdiction. When F(x) = x β x nd G(x) = x γ x, 0 < γ < < β, then α 0 = β 0 = (β )β β/( β) > 0, δ 0 = γ 0 = ( γ )γ γ /( γ ) > 0, nd we obtin the following oscilltion criterion for Eq. (.6). (2.9)

4 228 A. Özbekler et l. / Applied Mthemtics Letters 24 (20) Theorem 2.2. Suppose tht (2.) is nonoscilltory nd let z(t) be positive solution of it stisfying (2.2), i.e. nonprincipl solution. If lim {H(t) N 0(t)} = lim {H(t) + N 0 (t)} =, (2.0) N 0 (t) = then Eq. (.6) is oscilltory. s [α r(s)z 2 0 p(τ) + δ 0 q(τ)]z(τ)dτ, (2.) (s) Corollry 2.3. Suppose tht (2.) with q(t) 0 is nonoscilltory nd let z(t) be positive solution of it stisfying (2.2), i.e. nonprincipl solution. If lim {H(t) N 0(t)} = lim {H(t) + N 0 (t)} =, (2.2) N 0 (t) := then Eq. (.7) is oscilltory. α s 0 p(τ)z(τ)dτ, (2.3) Corollry 2.4. Let z(t) be positive solution of (2.) with p(t) 0 stisfying (2.2), i.e. nonprincipl solution. If lim {H(t) N 02(t)} = lim {H(t) + N 02 (t)} =, (2.4) N 02 (t) := then Eq. (.8) is oscilltory. δ s 0 q(τ)z(τ)dτ, (2.5) Remrk. Theorem 2.2 is interesting becuse it reduces to Theorem. for the liner eqution (.9) with ν(t) = p(t) q(t) by letting β, γ in (.6). Remrk 2. Corollry 2.4 is of prticulr interest the coefficient q(t) is non-positive nd Eq. (.8) cn still be oscilltory by the forcing condition (2.4). Remrk 3. It will be interesting to improve Corollry 2.3 for Eq. (.7) by relxing the ssumption tht p(t) is non-negtive. In cse when Φ(x) is bounded, sy Φ(x) M for some M > 0 nd for ll x R, then we cn show similrly to Corollry 2.3 the following: Proposition. Under the ssumption of Corollry 2.3 nd tht the coefficient p(t) is not ssumed to be non-negtive, if lim {H(t) N 03(t)} = lim {H(t) + N 03 (t)} =, N 03 (t) := M then Eq. (.7) is oscilltory. s p(τ) z(τ)dτ, 3. Exmples Exmple 3.. Consider the forced super-liner eqution (t 2 x ) + 2t 4 x β x = (3 t 2 ) sin t + 5t cos t, β >. (3.) The corresponding liner eqution (t 2 z ) + 2t 4 z = 0

5 A. Özbekler et l. / Applied Mthemtics Letters 24 (20) is the nonoscilltory with nonprincipl solution z(t) = t 2. Then, the functions H nd N 0 become s H(t) = {(3 τ 2 ) sin τ + 5τ cos τ}τ 2 dτ, > 0 s 2 nd N 0 (t) = 2(β )β β/( β) s s 2 τ dτ, > 0. 2 After some simple clcultions, we obtin H(t) = t 2 sin t + t cos t sin t + c t + c 2, c = 3 (sin + cos ) nd c 2 = ( 2 2 ) sin ( 2 + ) cos, nd N 0 (t) = (β )β β/( β) (t 2 + c 3 t + c 4 ) c 3 = 2/ nd c 4 = / 2. Clerly, the condition (2.2) is stisfied nd hence Eq. (3.) is oscilltory for ny choice of β > by Corollry 2.3. Exmple 3.2. Consider the forced sub-liner eqution x x γ x = e µt sin(ζ t), 0 < γ < (3.2) µ > nd ζ 0 re rel constnts. The corresponding liner eqution z z = 0 is nonoscilltory with nonprincipl solution z(t) = e t. Then, the functions H(t) nd N 02 (t) red s s H(t) = e 2s e (µ+)τ sin(ζ τ)dτ, > 0 nd s N 02 (t) = ( γ )γ γ /( γ ) e 2s e τ dτ, > 0. A strightforwrd clcultion gives s e (µ+)τ sin(ζ τ)dτ = k e (µ+)s {(µ + ) sin(ζ s) ζ cos(ζ s)} + k 2 (3.3) k = {ζ 2 + (µ + ) 2 } nd k 2 = k e (µ+) {(µ + ) sin(ζ ) ζ cos(ζ )}. Using (3.3), we see tht H(t) = k 3 e (µ )t {(µ 2 ζ 2 ) sin(ζ t) 2µζ cos(ζ t)} + k 4 e 2t + k 5 k 3 = k {ζ 2 + (µ ) 2 }, k 4 = k 2 /2 nd nd tht k 5 = (k 2 /2)e 2 k 3 e (µ ) {(µ 2 ζ 2 ) sin(ζ ) 2µζ cos(ζ )}, H(t) ± N 02 (t) = k 3 e (µ )t {(µ 2 ζ 2 ) sin(ζ t) 2µζ cos(ζ t)} ± σ e t + (k 4 σ e /2)e 2t + k 5 σ e /2, σ = δ 0 = (γ )γ γ /( γ ). Therefore, the condition (2.4) hol nd hence, we conclude tht Eq. (3.2) is oscilltory for ny choice of γ (0, ), µ > nd ζ 0, by Corollry 2.4. Acknowledgement The uthors wish to express their sincere grtitude to the referee for vluble suggestions.

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