Research Article The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non-C 1 Perturbation Term
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1 Hndaw Publshng Corporaton Boundary Value Problems Volume 9, Artcle ID 87834, 1 pages do:1.1155/9/87834 Research Artcle The Soluton of Two-Pont Boundary Value Problem of a Class of Duffng-Type Systems wth Non-C 1 Perturbaton Term Jang Zhengxan and Huang Wenhua School of Scences, Jangnan Unversty, 18 Lhu Dadao, Wux Jangsu 141, Chna Correspondence should be addressed to Huang Wenhua, hpjangyue@163.com Receved 14 June 9; Accepted 1 August 9 Recommended by Vel Shakhmurov Ths paper deals wth a two-pont boundary value problem of a class of Duffng-type systems wth non-c 1 perturbaton term. Several exstence and unqueness theorems were presented. Copyrght q 9 J. Zhengxan and H. Wenhua. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1. Introducton Mnmax theorems are one of powerful tools for nvestgaton on the soluton of dfferental equatons and dfferental systems. The nvestgaton on the soluton of dfferental equatons and dfferental systems wth non-c 1 perturbaton term usng mnmax theorems came nto beng n the paper of Stepan A.Tersan n Tersan proved that the equaton Lut ft, ut L d /dt exsts exactly one generalzed soluton under the operators B j j 1, related to the perturbaton term ft, ut beng selfadjont and commutng wth the operator L d /dt and some other condtons n 1. Huang Wenhua extended Tersan s theorems n 1 n 5 and 6, respectvely, and studed the exstence and unqueness of solutons of some dfferental equatons and dfferental systems wth non-c 1 perturbaton term 4, the condtons attached to the non-c 1 perturbaton term are that the operator Bu related to the term s self-adjont and commutes wth the operator A where A s a selfadjont operator n the equaton Au ft, u. Recently, by further research, we observe that the condtons mposed upon Bu can be weakened, the self-adjontness of Bu can be removed and Bu s not necessarly commutng wth the operator A. In ths note, we consder a two-pont boundary value problem of a class of Duffngtype systems wth non-c 1 perturbaton term and present a result as the operator Bu related to the perturbaton term s not necessarly a selfadjont and commutng wth the operator L. We obtan several valuable results n the present paper under the weaker condtons than those n 4.
2 Boundary Value Problems. Prelmnares Let H be a real Hlbert space wth nner product, and norm, respectvely, let X and Y be two orthogonal closed subspaces of H such that H X Y.LetP : H X, Q : H Y denote the projectons from H to X and from H to Y, respectvely. The followng theorem wll be employed to prove our man theorem. Theorem.1. Let H be a real Hlbert space, f : H R an everywhere defned functonal wth Gâteaux dervatve f : H H everywhere defned and hemcontnuous. Suppose that there exst two closed subspaces X and Y such that H X Yand two nonncreasng functons α :,,,β :,, satsfyng s αs, s βs, as s.1 and f h1 y ) f h y ),h 1 h α h1 h h 1 h,. for all h 1,h X, y Y, and fx k1 fx k,k 1 k β k1 k k 1 k,.3 for all x X, k 1,k Y. Then a f has a unque crtcal pont v H such that fv ; b fv max x X mn y Y fx y mn y Y max x X fx y. We also need the followng lemma n the present work. To the best of our knowledge, the lemma seems to be new. Lemma.. Let A and B be two dagonalzaton n n matrces, let μ 1 μ μ n and λ 1 λ λ n be the egenvalues of A and B, respectvely, where each egenvalue s repeated accordng to ts multplcty. If A commutes wth B, that s, AB BA, thena B s a dagonalzaton matrx and μ 1 λ 1 μ λ μ n λ n are the egenvalues of A B. Proof. Snce A s a dagonalzaton n n matrx, there exsts an nverse matrx P such that P 1 AP dag μ 1 E 1, μ E,...,μ s E s, where μ 1 < μ < < μ s 1 s n are the dstnct egenvalues of A, E 1,,...,s are the r r r 1 r r s n dentty matrces. And snce AB BA, thats, P dag μ 1 E 1, μ E,...,μ s E s ) P 1 B BP dag μ 1 E 1, μ E,...,μ s E s ) P 1,.4 we have dag μ 1 E 1, μ E,...,μ s E s ) P 1 BP P 1 BP dag μ 1 E 1, μ E,...,μ s E s )..5
3 Boundary Value Problems 3 Denote P 1 BP C j, where C j are the submatrces such that E C j and C j E 1,,...,s are defned, then, by.5, μ C j μ j C j, j 1,,...,s )..6 Notced that μ / μ j / j, we have C j O / j, and hence P 1 BP dag C 11, C,...,C ss,.7 where C and E 1,,...,s are the same order square matrces. Snce B s a dagonalzaton n n matrx, there exsts an nvertble matrx Q dag Q 1, Q,...,Q s such that Q 1 ) ) P 1 BP Q dag Q 1 1, Q 1,...,Q 1 s dag C 11, C,...,C ss dag Q 1, Q,...,Q s ) dag Q 1 1 C 11Q 1, Q 1 C Q,...,Q 1 s C ss Q s dag λ 1,λ,...,λ n,.8 where λ 1 λ λ n are the egenvalues of B. Let R PQ, then R s an nvertble matrx such that R 1 BR dag λ 1,λ,...,λ n and R 1 A BR R 1 AR R 1 BR Q 1 ) P 1 AP Q R 1 BR dag ) Q 1 1, Q 1,...,Q 1 s dag ) μ 1 E 1, μ E,...,μ s E s dag Q1, Q,...,Q s dag λ 1,λ,...,λ n dag μ 1 E 1, μ E,...,μ s E s ) dag λ1,λ,...,λ n dag μ 1,μ,...,μ n ) dag λ1,λ,...,λ n dag μ 1 λ 1,μ λ,...,μ n λ n )..9 A B s a dagonalzaton matrx and μ 1 λ 1 μ λ μ n λ n are the egenvalues of A B. The proof of Lemma. s fulflled. Let, denote the usual nner product on R n and denote the correspondng norm by u n 1 u 1/, where u u 1,u,...,u n T.Let, denote the nner product on L,, R n. It s known very well that L,, R n s a Hlbert space wth nner product u, v ut, vtdt, ) u, v L,, R n.1 and norm u u, u ut, utdt1/, respectvely.
4 4 Boundary Value Problems Now, we consder the boundary value problem u Au gt, u ht, u a, u b, t,,.11 where u :, R n, A s a real constant dagonalzaton n n matrx wth real egenvalues μ 1 μ μ n each egenvalue s repeated accordng to ts multplcty, g :, R n R n s a potental Carathéodory vector-valued functon, h :, R n s contnuous, a a 1,a,...,a n T, b b 1,b,...,b n T, a,b R, 1,,...,n. Let ut vt ωt, ωt 1 t/a t/b, t,, then.11 may be wrtten n the form v Av g t, v h t, v v,.1 where g t, v gt, v ω, h t ht Aωt. Clearly, g t, v s a potental Carathéodory vector-valued functon, h :, R n. Clearly, f v s a soluton of.1, u v ω wll be a soluton of.11. Assume that there exsts a real bounded dagonalzaton n n matrx Bt, u t,, u R n such that for a.e. t, and ξ, η L,, R n gt, η gt, ξ Bt, ξ τη ξη ξ,.13 where τ dagτ 1,τ,...,τ n,τ, 1 1,,...,n, Bt, u commutes wth A and s possessed of real egenvalues λ 1 t, u λ t, u λ n t, u. In the lght of Lemma., A Bt, u s a dagonalzaton n n matrx wth real egenvalues μ 1 λ 1 t, u μ λ t, u μ n λ n t, u each egenvalue s repeated accordng to ts multplcty. Assume that there exst postve ntegers N 1,,...,n such that for u L,, R n N μ <λ t, u < N 1 μ 1,,...,n..14 Let ξ 1,,...,n be n lnearly ndependent egenvectors assocated wth the egenvalues μ λ t, u 1,,...,n and let γ 1,,...,n be the orthonormal vectors obtaned by orthonormalzng to the egenvectors ξ 1,,...,n of μ λ t, u 1,,...,n. Then for every u R n A Bt, uγ μ λ t, u ) γ 1,,...,n..15 And let the set γ 1, γ,...,γ n be a bass for the space R n, then for every u R n, u u 1 γ 1 u γ u n γ n..16
5 Boundary Value Problems 5 It s well known that each v L,, R n can be represented by the absolutely convergent Fourer seres v C k sn ktγ, C k v t sn ktdt 1,,...,n; k 1,, Defne the lnear operator L d /dt : DL L,, R n L,, R n, DL v L,, R n v v, vt Lv C k k C k sn ktγ, v t sn ktdt, 1,,...,n, σl n n N. C k sn ktγ, C k k4 <,.18 Clearly, L d /dt s a selfadjont operator and DL s a Hlbert space for the nner product u, v and the norm nduced by the nner product s v [ u t, v t ) ut, vt ] dt, u, v DL,.19 [ v t, v t ) vt, vt ] dt, v DL.. Defne X x L,, R n xt N C k sn ktγ,t,, C k x t sn ktdt, Y y L,, R n yt C k y t sn ktdt, C k sn ktγ,t,, C k k4 <..1. Clearly, X and Y are orthogonal closed subspaces of DL and DL X Y.
6 6 Boundary Value Problems Defne two projectve mappngs P : DL X and Q : DL Y by Pv x X and Qv y Y, v x y DL, then S P Q s a selfadjont operator. Usng the Resz representaton theorem, we can defne a mappng T : L,, R n L,, R n by Tu, v [ u, v ) Au, v gt, u, v ht, v ] dt, v L,, R n..3 We observe that T n.3 s defned mplcty. Let Tu Fu n.3, we have Fu, v [ u, v ) Au, v gt, u, v ht, v ] dt, v DL L,, R n..4 Clearly, F and hence F s defned mplcty by.4. It can be proved that u s a soluton of.11 f and only f u satsfes the operator equaton Fu The Man Theorems Now, we state and prove the followng theorem concernng the soluton of problem.11. Theorem 3.1. Assume that there exsts a real dagonalzaton n n matrx Bt, u u L,, R n wth real egenvalues λ 1 t, u λ t, u λ n t, u satsfyng.14 and commutng wth A. Denote α u mn mn mn λ t, ũ μ N >, 3.1 ũ u 1 n t β u mn ũ u mn 1 n mn t N 1 μ λ t, ũ >. 3. If α :,,, β :,,, s αs, s βs, as s, 3.3 problem.11 has a unque soluton u, and u satsfes Fu, and Fu max mn x X y Y Fx y ω mn max y Y x X Fx y ω, 3.4 where F s a functonal defned n.4 and ω 1 t/a t/b, t,.
7 Boundary Value Problems 7 Proof. Frst, by vrtue of.1 and., we have x, x ) dt x, x ) dt N N C k sn ktγ, ) max N x, xdt, 1 n y, y ) dt y, y ) dt N C k sn ktγ dt 3.5 k C k sn ktγ, C k sn ktγ dt, y, y ) dt max 1 n N 1 k max 1 n N 1 C k sn ktγ, y dt 3.7 y, ydt. Denote Fu Fv ω F v. By.4,.13, 3.5, 3.6, 3.7, 3.1, and3., for all x 1, x X, y Y, letv 1 x 1 y DL, v x y DL, v v 1 v x 1 x x X, x 1 Pv 1 X, x Pv X, y Qv 1 Qv Y, we have F v 1 F v, x 1 x Fu 1 Fu, x 1 x Fu 1, x Fu, x [ ] u 1, x ) Au 1, x gt, u 1, x ht, x dt [ ] u, x ) Au, x gt, u, x ht, x dt [ u1 u, x ) Au 1 u, x gt, u 1 gt, u, x ] dt [ v, x ) Av, x Bt, v ω τvv, x ] dt [ x, x ) Ax, x Bt, v ω τvx, x ] dt
8 8 Boundary Value Problems n N N C k sn ktγ, x 1 k1 n N C k sn kta Bt, ṽγ, x dt n ) N N μ λ t, ṽ C k sn ktγ, x dt α v 1 x, xdt k1 1 α v 1 v max 1 n N 1 [ max 1 n N ) ] x, x x, x dt α v 1 v x 1 x, α v 1 v ) α v 1 v, max 1 n N for all x X, y 1, y Y,letv 1 x y 1 DL, v x y DL, v v 1 v y 1 y y Y, y 1 Qv 1 Y, y Qv Y, x Pv 1 Pv X, we have F v 1 F v, y 1 y Fu 1 Fu, y 1 y [ u1 u, y ) Au 1 u, y gt, u 1 gt, u, y ] dt [ v, y ) Av, y Bt, ṽv, y ] dt [ y, y ) A Bt, ṽy, y ] dt y, y ) 1 y, y ) 1 kn 1 max N 1 1 n k max 1 n N 1 C k sn kt μ λ t, ṽ ) γ, y dt k C k sn kt μ λ t, ṽ ) γ, y dt
9 Boundary Value Problems 9 k C k sn ktγ, y 1 max 1 n N max 1 n N 1 kn 1 k C k sn kt μ λ t, ṽ ) γ, y dt k C k sn kt N 1 μ λ t, ṽ )) γ, y dt mn ṽ v mn 1 n mn t, N 1 μ λ t, ṽ > max 1 n N 1 1 ) 1 1 y, y ) dt max 1 n N 1 β v [ y, y ) y, y ] dt max 1 n N 1 1 β v y β v 1 v y 1 y, ) β β v 1 v v 1 v. max 1 n N By 3.3, s α s,s β s, as s. Clearly, α and β are nonncreasng. Now, all the condtons n the Theorem.1 are satsfed. By vrtue of Theorem.1, there exsts a unque v DL such that F v Fv ω Fu and F v Fv ω Fu max x X mn y Y Fx y ω mn y Y max x X Fx y ω, where F s a functonal defned mplcty n.4 and ωt 1 t/a t/b,t,. v t s just a unque soluton of.1 and u t v tωt s exactly a unque soluton of.11. The proof of Theorem 3.1 s completed. Now, we assume that there exsts a postve nteger N such that N μ <λ t, u < N 1 μ 1,,...,n 3.1 for u L,, R n,t,. Defne X x L,, R n xt C k x t sn ktdt, N C k sn ktγ,t,, 3.11
10 1 Boundary Value Problems Y y L,, R n yt C k sn ktγ,t,, α u mn β u mn C k mn mn ũ u 1 n t mn mn ũ u 1 n t y t sn ktdt, 1 kn1 1 kn1 C k k4 <, 3.1 λ t, ũ μ N >, 3.13 N 1 μ λ t, ũ > Replace the condton.14 by 3.1 and replace.1,., 3.1, and 3. by 3.11, 3.1, 3.11, and3.14, respectvely. Usng the smlar provng technques n the Theorem 3.1, we can prove the followng theorem. Theorem 3.. Assume that there exsts a real dagonalzaton n n matrx Bt, u t,, u R n wth real egenvalues λ 1 t, u λ t, u λ n t, u satsfyng.13 and 3.1 and commutng wth A. If the functons α and β defned n 3.11) and 3.14 satsfy 3.3, problem.11 has a unque soluton u, and u satsfes Fu and 3.4. It s also of nterest to the case of A O. Corollary 3.3. Let ht, gt, u, a and b be as n.11. Assume that there exsts a real dagonalzaton n n matrx Bt, u t,, u R n wth real egenvalues λ 1 t, u λ t, u λ n t, u satsfyng.13 and N <λ t, u < N 1 N Z, 1,,...,n. Denote α u mn β u mn mn mn ũ u 1 n t mn mn ũ u 1 n t λ t, ũ N >, N 1 λ t, ũ > If α and β satsfy 3.3, the problem u gt, u ht, t,, u a, u b 3.16 has a unque soluton u, and u satsfes Fu and 3.4,whereF s a functonal defned n Fu, v [ u, v ) gt, u, v ht, v ] dt, v DL. 3.17
11 Boundary Value Problems 11 Corollary 3.4. Let ht, gt, u, a, b, and Bt, u be as n Corollary 3.3. The egenvalues of Bt, uλ 1 t, u λ t, u λ n t, u satsfy N <λ t, u < N 1 N Z. Denote α u mn β u mn mn mn ũ u 1 n t mn mn ũ u 1 n t λ t, ũ N >, N 1 λ t, ũ > If α and β satsfy 3.3, problem 3.16 has a unque soluton u, and u satsfes Fu and 3.4,whereF s a functonal defned n If there exsts a C functonal G :, R n.13 should be R such that gt, u Gt, u, then gt, η gt, ξ Gt, η Gt, ξ 1 D Gt, ξ τη ξη ξdτ, 3.19 where D G s just a Hessan of G. In ths case, the followng corollary follows from Theorem 3.1. Corollary 3.5. Let the egenvalues of 1 D Gt, ξ τη ξdτλ 1 t, u λ t, u λ n t, u satsfy.14. Ifα and β defned n 3.1 and 3. satsfy 3.3, problem.11where gt, u Gt, u) has a unque soluton u, and u satsfes Fu and 3.4. Usng the smlar technques of the present paper, we can also nvestgate the twopont boundary value problem u Au gt, u ht, u a, u b, t,, 3. where u, A, ht, gt, u, a and b are as n problem.11. The correspondng results are smlar to the results n the present paper. The specal case of A Oandn 1 n problem 3. has been studed by Zhou Tng and Huang Wenhua 5. Zhou and Huang adopted the technques dfferent from ths paper to acheve ther research. References 1 S. A. Tersan, A mnmax theorem and applcatons to nonresonance problems for semlnear equatons, Nonlnear Analyss: Theory, Methods & Applcatons, vol. 1, no. 7, pp , H. Wenhua and S. Zuhe, Two mnmax theorems and the solutons of semlnear equatons under the asymptotc non-unformty condtons, Nonlnear Analyss: Theory, Methods & Applcatons, vol. 63, no. 8, pp , 5. 3 H. Wenhua, Mnmax theorems and applcatons to the exstence and unqueness of solutons of some dfferental equatons, Journal of Mathematcal Analyss and Applcatons, vol. 3, no., pp , 6.
12 1 Boundary Value Problems 4 H. Wenhua, A mnmax theorem for the quas-convex functonal and the soluton of the nonlnear beam equaton, Nonlnear Analyss: Theory, Methods & Applcatons, vol. 64, no. 8, pp , 6. 5 Z. Tng and H. Wenhua, The exstence and unqueness of soluton of Duffng equatons wth non- C perturbaton functonal at nonresonance, Boundary Value Problems, vol. 8, Artcle ID , 9 pages, 8.
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