Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p, p 2,..., p n )-LAPLACIAN DEVRIM ÇAKMAK, MUSTAFA FAHRI AKTAŞ, AYDIN TIRYAKI Abstrct. We prove some generlized Lypunov-type inequlities for n-dimensionl Dirichlet nonliner systems. We extend the results obtined by Çm nd Tiryi [6] for prmeter < p < 2. As n ppliction, we obtin lower bounds for the eigenvlues of the corresponding system.. Introduction In 907, Lypunov [9] obtined the remrble inequlity if Hill s eqution f (s) ds 4 b, (.) x + f (t)x = 0 (.2) hs rel nontrivil solution x (t) such tht x () = 0 = x (b), where, b R with < b re consecutive zeros nd x is not identiclly zero on [, b], where f is rel-vlued continuous function defined on R. We now tht the constnt 4 in the right hnd side of inequlity (.) cnnot be replced by lrger number (see [7, p. 345]). Since this result hs proved to be useful tool in oscilltion theory, disconjugcy, eigenvlue problems nd numerous other pplictions in the study of vrious properties of solutions for differentil equtions, mny proofs nd generliztions or improvements of it hve ppered in the literture. For uthors, who contributed to the Lypunov-type inequlities, we refer to [-9]. Here, we give some inequlities which re useful in the comprison of our min results. We now tht since the function h(x) = x p is concve for x > 0 nd < p < 2, Jensen s inequlity h( ω+v 2 ) 2 v = b c for =, 2,..., n implies [h(ω) + h(v)] with ω = c 2 2 p [ c + ] p b c (c ) p + (b c ) p = m (c ) (.3) nd 2000 Mthemtics Subject Clssifiction. 26D0, 34A40, 34C0. Key words nd phrses. Lypunov-type inequlity; lower bound; (p, p 2,..., p n)-lplcin. c 203 Texs Stte University - Sn Mrcos. Submitted August 20, 202. Published My 27, 203.
2 D. ÇAKMAK, M. F. AKTAŞ, A. TIRYAKI EJDE-203/28 for < p < 2, =, 2,..., n. If p > 2 for =, 2,..., n, then the function h(x) = x p is convex for x > 0. Thus, the inequlity (.3) is reversed, i.e. (c ) p + (b c ) p [ 22 p c + ] p = m2 (c ) (.4) b c for p > 2, =, 2,..., n. Moreover, if we obtin the minimum of the right hnd side of inequlities (.3) nd (.4) for c (, b), =, 2,..., n, then it is esy to see tht min m i(c ) = m i ( + b p <c <b 2 ) = 2 (b ) p (.5) for i =, 2 nd =, 2,..., n. In 2006, Npoli nd Pinsco [0] obtined the following inequlity ( ) α/p ( ) α2/p 2 2 α+α2 f (s)ds f 2 (s)ds, (.6) (b ) α+α2 if the qusiliner system (φ p (x )) = f (t) x α 2 x x 2 α2 (φ p2 (x 2)) = f 2 (t) x α x 2 α2 2 x 2 (.7) hs rel nontrivil solution (x (t), x 2 (t)) such tht x () = x (b) = 0 = x 2 () = x 2 (b) where, b R with < b consecutive zeros, nd x for =, 2 re not identiclly zero on [, b], where φ α (u) = u α 2 u, f nd f 2 re rel-vlued positive continuous functions defined on R, < p, p 2 < + nd the nonnegtive prmeters α, α 2 stisfy α p + α2 p 2 =. In 200, Çm nd Tiryi [6] obtined the following inequlity n ( ) f + (s)ds α /p n [ (c ) p + ] α /p (b c ) p, (.8) = = where x (c ) = mx <t<b x (t) nd f + (t) = mx {0, f (t)} for =, 2,..., n, if the n-dimensionl problem (φ p (x )) = f (t) x α 2 x x 2 α2... x n αn (φ p2 (x 2)) = f 2 (t) x α x 2 α2 2 x 2... x n αn... (.9) (φ pn (x n)) = f n (t) x α x 2 α2... x n αn 2 x n hs rel nontrivil solution (x (t), x 2 (t),..., x n (t)) stisfying the Dirichlet boundry conditions x () = 0 = x (b) (.0) where, b R with < b consecutive zeros, x 0 for =, 2,..., n on [, b]. Here, n N, φ α (u) = u α 2 u, f re rel-vlued continuous functions defined on R, < p < + nd the nonnegtive prmeters α stisfy n α = p = for =, 2,..., n. Using (.5) in the inequlity (.8), Çm nd Tiryi [6] lso obtined the inequlity n ( f + (s)ds) α /p 2 P n = α (b ). (.) (P n = α ) =
EJDE-203/28 LYAPUNOV-TYPE INEQUALITIES 3 Recently, Yng et l [9] obtined the inequlity where f (s)ds H = 2 p (b ) p H, (.2) M p g (M, M 2,..., M n ) (.3) with M = x (c ) = mx <t<b x (t) for =, 2,..., n, t lest one inequlity in (.2) is lso strict, if the following nonliner system involving (p, p 2,..., p n )- Lplcin opertors (φ p (x )) + F (t, x, x 2,..., x n ) = 0 (φ p2 (x 2)) + F 2 (t, x, x 2,..., x n ) = 0... (φ pn (x n)) + F n (t, x, x 2,..., x n ) = 0 (.4) hs rel nontrivil solution (x (t), x 2 (t),..., x n (t)) stisfying the boundry condition (.0), where n N, φ α (u) = u α 2 u, < p < + nd F C([, b] R n, R) for =, 2,..., n, under the following hypothesis: (C) There exist the functions f C([, b], [0, )) nd g C(R n, [0, )) for =, 2,..., n such tht nd F (t, x, x 2,..., x n ) f (t)g (x, x 2,..., x n ) (.5) g (x, x 2,..., x n ) is monotonic nondecresing in ech vrible (.6) for =, 2,..., n. Yng et l [9] clim tht the inequlity (.) with f (t) > 0 for =, 2,..., n of Çm nd Tiryi [6] cn be obtined by using the inequlity (.2) under the following conditions F (t, x, x 2,..., x n ) = f (t)g (x, x 2,..., x n ), =, 2,..., n, (.7) where g (x, x 2,..., x n ) = z (x, x 2,..., x n ) with z (x, x 2,..., x n ) = x α 2 x x 2 α2... x n αn z 2 (x, x 2,..., x n ) = x α x 2 α2 2 x 2... x n αn... z n (x, x 2,..., x n ) = x α x 2 α2... x n αn 2 x n, (.8) where α 0 for =, 2,..., n such tht n α = p =. It is esy to see from (.6) tht the nondecresing condition on ech vrible of g with (.8) for =, 2,..., n is not stisfied. Therefore, [9, Remrs 3, Corollry 3] fil. So, [9, Corollry 3] does not pply to this exmple. Now, we present the following hypothesis insted of (C): (C*) There exist the functions f C([, b], [0, )) nd g C(R n, [0, )) for =, 2,..., n such tht F (t, x, x 2,..., x n ) f (t)g ( x, x 2,..., x n ) (.9)
4 D. ÇAKMAK, M. F. AKTAŞ, A. TIRYAKI EJDE-203/28 nd g (u, u 2,..., u n ) is monotonic nondecresing in ech vrible u i, such tht either g (0, 0,..., 0) = 0 or g (u, u 2,..., u n ) > 0 for t lest one u i 0 for i =, 2,..., n, for =, 2,..., n. It is cler tht if the hypothesis (C) is replced by (C*) for system (.4), then (.) with f (t) > 0 for =, 2,..., n of Çm nd Tiryi [6] obtin by using inequlity (.2) under the condition α for =, 2,..., n. In this rticle, our purpose is to obtin Lypunov-type inequlities for system (.4) similr to the ones given in Yng et l [9] by imposing somewht different conditions on the function F for =, 2,..., n, nd improve nd generlize the results of Çm nd Tiryi [6] when < p < 2 for =, 2,..., n. In ddition, the positivity conditions on the function f for =, 2,..., n in hypothesis (C) re dropped. We lso obtin better lower bound for the eigenvlues of corresponding system s n ppliction. We derive some Lypunov-type inequlities for system (.4), where ll components of the solution (x (t), x 2 (t),..., x n (t)) hve consecutive zeros t the points, b R with < b in I = [t 0, ) R. For system (.4), we lso derive some Lypunov-type inequlities which relte not only points nd b in I t which ll components of the solution (x (t), x 2 (t),..., x n (t)) hve consecutive zeros but lso point in (, b) where ll components of the solution (x (t), x 2 (t),..., x n (t)) re mximized. Since our ttention is restricted to the Lypunov-type inequlities for system of differentil equtions, we shll ssume the existence of the nontrivil solution (x (t), x 2 (t),..., x n (t)) of system (.4). 2. Min results We give the following hypothesis for system (.4). (C2) There exist the functions f C([, b], R) nd g C(R n, [0, )) such tht F (t, x, x 2,..., x n )x f (t)g ( x, x 2,..., x n ) (2.) nd g (u, u 2,..., u n ) is monotonic nondecresing in ech vrible u i such tht either g (0, 0,..., 0) = 0 or g (u, u 2,..., u n ) > 0 for t lest one u i 0, i =, 2,..., n, for =, 2,..., n. (2.2) One of the min results of this rticle is the following theorem, whose proof is different from the tht of [9, Theorem ] nd modified tht of [3, Theorem 2.]. Theorem 2.. Assume tht hypothesis (C2) is stisfied. If (.4) hs rel nontrivil solution (x (t), x 2 (t),..., x n (t)) stisfying the boundry condition (.0), then the inequlities f + (s)ds [ 22 p c + ] p M H (2.3) b c hold, where f + (t) = mx{0, f (t)}, nd H, M for =, 2,..., n re s in (.3). Moreover, t lest one inequlity in (2.3) is strict. Proof. Let the boundry condition (.0) hold; i.e., x () = 0 = x (b) for =, 2,..., n where n N,, b R with < b consecutive zeros nd x for =
EJDE-203/28 LYAPUNOV-TYPE INEQUALITIES 5, 2,..., n re not identiclly zero on [, b]. Thus, by Rolle s theorem, we cn choose c (, b) such tht M = mx <t<b x (t) = x (c ) nd x (c ) = 0 for =, 2,..., n. By using x () = 0 nd Hölder s inequlity, we obtin nd hence x (c ) c ( c x (s) ds (c ) (p )/p x (s) p /p ds) (2.4) c x (c ) p (c ) p x (s) p ds (2.5) for =, 2,..., n nd c (, b). Similrly, by using x (b) = 0 nd Hölder s inequlity, we obtin x (c ) p (b c ) p x (s) p ds c (2.6) for =, 2,..., n nd c (, b). Multiplying the inequlities (2.5) nd (2.6) by (b c ) p nd (c ) p for =, 2,..., n, respectively, we obtin nd (b c ) p x (c ) p [(c )(b c )] p c (c ) p x (c ) p [(c )(b c )] p x (s) p ds (2.7) c x (s) p ds (2.8) for =, 2,..., n nd c (, b). Thus, dding the inequlities (2.7) nd (2.8), we hve x (c ) p [(b c ) p + (c ) p ] [(c )(b c )] p x (s) p ds (2.9) for =, 2,..., n nd c (, b). It is esy to see tht the functions z (x) = (b x) p + (x ) p te the minimum vlues t +b 2 ; i.e., z (x) min <x<b z (x) = z ( + b 2 ) = 2(b 2 )p for =, 2,..., n. Thus, we obtin nd hence x (c ) p [2( b 2 )p ] [(c )(b c )] p 2M p = 2 x (c ) p 2 [ b (c )(b c )] p x (s) p ds (2.0) x (s) p ds (2.) for =, 2,..., n nd c (, b). Multiplying the -th eqution of system (.4) by x (t), integrting from to b by using integrtion by prts nd ting into
6 D. ÇAKMAK, M. F. AKTAŞ, A. TIRYAKI EJDE-203/28 ccount tht x () = 0 = x (b) nd the inequlities (2.) for =, 2,..., n, then the monotonicity of g yields x (s) p ds = F (s, x (s), x 2 (s),..., x n (s))x (s)ds f (s)g ( x (s), x 2 (s),..., x n (s) )ds f + (s)g ( x (s), x 2 (s),..., x n (s) )ds = g (M, M 2,..., M n ) Then, using (2.2) in (2.), we hve f + (s)ds f + (s)ds. 2M p [ b ] p g (M, M 2,..., M n ) 2(c )(b c ) (2.2) (2.3) for =, 2,..., n. Since (x (t), x 2 (t),..., x n (t)) is nontrivil solution of system (.4), it is esy to see tht t lest one inequlity in (2.3) is strict, which completes the proof. Another min result of this pper is the following theorem whose proof is lmost the sme to tht of [9, Theorem ]; hence it is omitted. Theorem 2.2. Let ll the ssumptions of Theorem 2. hold. Then the inequlity f + (s)ds [ (c ) p + ] M (b c ) p H (2.4) holds, where f + (t), H nd M for =, 2,..., n re s in Theorem 2.. Moreover, t lest one inequlity in (2.4) is strict. Remr 2.3. The right-hnd side of inequlities (2.3) in Theorem 2. or (2.4) in Theorem 2.2 shows tht c, for =, 2,..., n, cnnot be too close to or b, since the exponents stisfy < p < + for =, 2,..., n. We hve f + (s)ds < for =, 2,..., n, but [ lim c +, c b c + ] p =, or b c [ lim c +, c b (c ) p + ] = (b c ) p for =, 2,..., n. Now, ccording to the vlue of p, we compre Theorem 2. with Theorem 2.2 s follows. Remr 2.4. It is esy to see from inequlity (.3) tht if we te < p < 2, for =, 2,..., n, then inequlity (2.3) is better thn (2.4) in the sense tht (2.4) follows from (2.3), but not conversely. Similrly, from inequlity (.4), if p > 2, for =, 2,..., n, then inequlity (2.4) is better thn (2.3) in the sense tht (2.3) follows from (2.4), but not conversely. Moreover, if p = 2 or c = +b 2 for =, 2,..., n, then Theorem 2. is exctly the sme s Theorem 2.2. By using (.5) in Theorem 2. or 2.2, we obtin the following result.
EJDE-203/28 LYAPUNOV-TYPE INEQUALITIES 7 Theorem 2.5. Let ll the ssumptions of Theorem 2. hold. Then the inequlity f + (s)ds 2 p (b ) p M H (2.5) holds, where f + (t), H nd M for =, 2,..., n re s in Theorem 2.. Moreover, t lest one inequlity in (2.5) is strict. Now, we present the following hypothesis which gives the importnce of our theorems for system (.9). (C3) There exist the functions f C([, b], R) nd g C(R n, [0, )) such tht F (t, x, x 2,..., x n )x = f (t)g ( x, x 2,..., x n ) (2.6) nd g (u, u 2,..., u n ) is monotonic nondecresing in ech vrible u i such tht either g (0, 0,..., 0) = 0 or g (u, u 2,..., u n ) > 0 for t lest one u i 0 for i =, 2,..., n, (2.7) where g ( x, x 2,..., x n ) = x z (x, x 2,..., x n ) with (.8) for =, 2,..., n such tht α 0 nd n α = p =. It is esy to see tht system (.4) with hypothesis (C3) reduces to system (.9). Since n (M H ) α /p =, (2.8) = we hve the following results from Theorems 2. nd 2.2, respectively. Theorem 2.6. Assume tht hypothesis (C3) is stisfied. If (.4) hs rel nontrivil solution (x (t), x 2 (t),..., x n (t)) stisfying the boundry condition (.0), then n ( ) f + (s)ds α /p n [ 2 2 p ( = = c + ) p ] α /p, (2.9) b c where x (c ) = mx <t<b x (t) nd f + (t) = mx{0, f (t)} for =, 2,..., n. Moreover, t lest one inequlity in (2.9) is strict. Theorem 2.7. Let ll the ssumptions of Theorem 2.6 hold. Then the inequlity n ( ) f + (s)ds α /p n [ (c ) p + ] α /p (2.20) (b c ) p = = holds, where c nd f + (t) for =, 2,..., n re s in Theorem 2.6. Moreover, t lest one inequlity in (2.20) is strict. By using (.5) in Theorem 2.6 or 2.7 nd (2.8) in Theorem 2.5, we hve the following result. Corollry 2.8. Let ll the ssumptions of Theorem 2.6 hold. Then the inequlity n ( ) f + (s)ds α /p 2 P n = α (b ) (P n (2.2) = α ) = holds, where f + (t) for =, 2,..., n is s in Theorem 2.6. Moreover, t lest one inequlity in (2.2) is strict.
8 D. ÇAKMAK, M. F. AKTAŞ, A. TIRYAKI EJDE-203/28 Remr 2.9. It is esy to see from (.3) tht if we te < p < 2 for =, 2,..., n, then (2.9) is better thn (.8) in the sense tht (.8) follows from (2.9), but not conversely. Similrly, from (.4), if p > 2 for =, 2,..., n, then (.8) is better thn (2.9) in the sense tht (2.9) follows from (.8), but not conversely. Remr 2.0. It is esy to see tht inequlity (2.20) is exctly the sme s (.8), nd (2.2) is exctly the sme s (.). Remr 2.. When α = p for =, 2,..., n, nd for i, α i = 0 for i =, 2,..., n in system (.9), we obtin the result for the cse of single eqution from Theorems 2.6, 2.7 or Corollry 2.8. Remr 2.2. Since f(x) f + (x), the integrls of f + (s)ds for =, 2,..., n in the bove results cn lso be replced by f (s) ds for =, 2,..., n, respectively. 3. Applictions In this section, we present some pplictions of the Lypunov-type inequlities obtined in Section 2. Firstly, we give the sme exmple of Yng et l [9] which gives the importnce of our results. Note tht our Corollry 2.8 is pplicble to the following exmple, but [9, Corollry 3] is not pplicble to it, since the nondecresing condition on ech vrible of g for =, 2,..., n is not stisfied. Exmple 3.. Consider the qusiliner system (φ p (x )) + f (t)(3 + sin 2x ) x α 2 x x 2 α2 x 2 = 0 (φ p2 (x 2)) + f 2 (t)( + sin 2 2x 2 ) x α x x 2 α2 2 x 2 = 0, (3.) where φ α (u) = u α 2 u, p, p 2 >, α, α 2 0 with α p re nonnegtive continuous functions on [, b]. + α2 p 2 =, f nd f 2 Assume tht system (3.) hs rel nontrivil solution (x (t), x 2 (t)) stisfying the Dirichlet boundry condition x () = x (b) = 0 = x 2 () = x 2 (b). Since F (t, x, x 2 )x 4f (t) x α x 2 α2 F 2 (t, x, x 2 )x 2 2f 2 (t) x α x 2 α2, nd (3.2) where g (u, u 2 ) = u α uα2 2 for =, 2 which re stisfied the nondecresing condition on ech vrible u i for i =, 2, we hve the following inequlities 4 f (s)ds > 2 p (b ) p M H, 2 f 2 (s)ds > 2 p 2 (b ) p2 M 2H 2 (3.3) with M H = M p α M α2 2 nd M 2 H 2 = M α M p2 α2 2 from Theorem 2.6. Hence, we hve ( from Corollry 2.8. ) α f (s)ds p ( ) α 2 p 2 2 α+α2 α p f 2 (s)ds > (b ) α+α2 (3.4)
EJDE-203/28 LYAPUNOV-TYPE INEQUALITIES 9 Secondly, we give nother ppliction of the Lypunov-type inequlities obtined for system (.9). Note tht the lower bounds re found by using inequlity (2.20) in Theorem 2.7 coincide with tht of [6, Theorem 9]. Now, we present new lower bounds by using inequlity (2.9) in Theorem 2.6 which give better lower bound for the eigenvlues of following system thn tht of [6, Theorem 9] when < p < 2 for =, 2,..., n. Let λ for =, 2,..., n be generlized eigenvlues of system (.9), nd r(t) be positive function for ll t R. Therefore, system (.9) with f (t) = λ α r(t) > 0 for =, 2,..., n nd ll t R reduces to the system ( x p 2 x ) = λ α r(t) x α 2 x x 2 α2... x n αn ( x 2 p2 2 x 2) = λ 2 α 2 r(t) x α x 2 α2 2 x 2... x n αn... ( x n pn 2 x n) = λ n α n r(t) x α x 2 α2... x n αn 2 x n. (3.5) By using similr techniques to the technique in [6], we obtin the following result which gives lower bounds for the n-th eigenvlue λ n. The proof of following theorem is bsed on bove generliztion of the Lypunov-type inequlity, s in tht of [6, Theorem 9] nd hence is omitted. Theorem 3.2. There exist function (λ, λ 2,..., λ n ) such tht λ n (λ, λ 2,..., λ n ) (3.6) for every generlized eigenvlue (λ, λ 2,..., λ n ) of system (3.5), where x (c ) = mx <t<b x (t) for =, 2,..., n nd (λ, λ 2,..., λ n ) = α n { n = [ 2 2 p ( c + b c ) p ] α /p [ n (λ α ) α /p r(s)ds ] } p n/α n. = (3.7) Remr 3.3. Let < p < 2 for =, 2,..., n. If we compre Theorem 3.2 with [6, Theorem 9], we obtin (λ, λ 2,..., λ n ) h (λ, λ 2,..., λ n ) since the inequlity (.3) holds. Thus, Theorem 3.2 gives better lower bound thn [6, Theorem 9]. Remr 3.4. Since is continuous function, (λ, λ 2,..., λ n ) + s ny eigenvlue of λ 0 + for =, 2,..., n. Therefore, there exists bll centered in the origin such tht the generlized spectrum is contined in its exterior. Also, by rerrnging terms in (3.6) we obtin n = λ α /p n [2 2 p ( c + [ n ) p ] α /p b c = = α α /p r(s)ds]. (3.8) It is cler tht when the intervl collpses, right-hnd side of (3.8) pproches infinity. Acnowledgments. The uthors wnt to thn the nonymous referee for his/her vluble suggestions nd comments tht helped us improve this rticle.
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