Variational method to the second-order impulsive partial differential equations with inconstant coefficients (I)

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1 Avalable onlne a Proceda Engneerng 6 ( 5 4 Inernaonal Worksho on Aomoble, Power and Energy Engneerng Varaonal mehod o he second-order mlsve aral dfferenal eqaons wh nconsan coeffcens (I Hachn L* College of Scence, Shenyang Agrclral Unversy, Shenyang, Laonng, 866, P.R.Chna. Absrac In he aer, we consder he esence of he solon of he second-order mlsve dfferenal eqaons wh nconsan coeffcens. We change he second-order mlsve aral dfferenal eqaon no he eqvalen eqaon by ransformaon. By sng he crcal on heory of varaonal mehod and La-lgram heorem, we oban new resls for he esence of he solon of he mlsve aral dfferenal eqaons. Pblshed by Elsever Ld. Selecon and/or eer-revew nder resonsbly of Socey for Aomoble, Power and Energy Engneerng Oen access nder CC BY-NC-ND lcense. Keywords:Varaonal mehod; Bondary vale roblem, Paral dfferenal eqaon, Imlse, La-lgram heorem * Corresondng ahor. el.: Pblshed by Elsever Ld. do:.6/.roeng Oen access nder CC BY-NC-ND lcense.

2 6 Hachn L / Proceda Engneerng 6 ( 5 4 E-mal address: lhc6@6.com.. an e Imlsve aral dfferenal eqaons may eress some real world rocesses whch are sbec o fed me dsrbances. In he as few years, he heory of mlsve dfferenal eqaons has been nvesgaed eensvely [-]. Varaonal mehods can be no only aled no he ordnary dfferenal eqaons, b also sded on he aral dfferenal eqaons. Usng varaonal mehod, J. J., Neo, [4], Hao Zhang [] and Y an[] sded he esence of solons of second-order mlsve ordnary dfferenal eqaons wh he Drchle roblem. However, here s a few nformaon for he mlsve aral dfferenal eqaons wh erodc bondary vale roblems and nconsan coeffcens. In he aer, we consder he solons of he followng eqaon wh nconsan coeffcens n m f (,,,,, (. (, where [, ] [, ] ;consans (,, and m, n ; ; ; ; ; he fncons f and ( are connos and bonded. For obanng or resls, he followng relmnares are sefl for analyzng he fnconals and he mlsve aral dfferenal eqaons. Defnon. (Sobolev Sace. H(,, L (, wh he nner rodc (, v dd v dd vdd and he norm Lemma. If H (, ( (, dd dd dd H s connos on bonded doman wh y, hen ( dd dd heorem. (La-lgram heorem [4, ]. Le a : H H R be a bonded blnear form. If a s coercve,.e., here ess > sch ha a(, for every H, hen, for any H (he congae sace of H, here ess a nqe H sch ha a(, (, for every v H. oreover, f s also symmerc, hen he fnconal : H R ( a( (, aans s mnmm a. defned by

3 Hachn L / Proceda Engneerng 6 ( an resls for mlsve aral dfferenal eqaon In hs secon, we consder he esence of solons of he mlsve aral dfferenal eqaon (.. Forconvenenly, we change he eqaon (. no he eqvalen eqaon. We know he frs eqaon of (. s eqvalen o he eqaon, whch s of form ( e ( Leng / n / m ( / n / m ( / n / m ( / n / m ( e e e f. C e n ( / n m / m, hen he eqaon (. s eqvalen o he eqaon ( C ( C C Cf (,,,,, (. (. Clearly, we have he followng resl. Lemma. he eqaon (. s eqvalen o he eqaon (.. And he solon of he eqaon (. s he solon of he eqaon (.. Nely, we esablsh he fnconal of (. assocaed wh he eqaon of (.. Becase s dffcl o sdy on he esence of he solon of (., we dscss he esence of he solon of (..Hence, we consder he solons of he fnconal assocaed wh he eqaon (.. ake v( H (,. llyng he frs eqaon of (. by v and negrang on, hen ( C ( vdd C vdd Cvdd Cfvdd. Snce, we have When ( C ( C vdd { [ v( ] d vdd Cvdd C v dd C v dd Cvdd., sggess we defne he blnear form a : H (, H (, R and he oeraor by a(, C v dd C v dd Cvdd (A l( Cfvdd v( d (B We defne he fnconal ( : Clearly, ( ( a( l( Cfvdd Cvdd Cv dd v( d s a Gaea dfferenable fnconal and s Gaea dervave s '( ( C v dd Defnon. A fncon Cfvdd C v dd Cvdd v( Cv dd d a(, l( H ( s sad o be a weak solon of (v, f sasfes

4 8 Hachn L / Proceda Engneerng 6 ( 5 4 ( ( = for all (, H ( v. Consderng he mlsve aral dfferenal eqaon where and ( C ( C C Cf,,,, (. are consans and ( ( y Lemma. If (, H ( Proof. Assme ha s a solon of (,. (. C, we have he followng resls. s a solon of (. s he mnmm of ( (, H (. Le ( ( ( ( C dd C dd C dd C dd hen Cdd. ( C dd C dd C dd.e., ( (. I means ha ( s mnmm. Assme ha s he mnmm of he fnconal. ake (. Snce ( mn (, we have ( ( and (, where s he frs order varaonal of (. Choose H ( I mles C dd Cfdd v wh v for every [, ] [, ] [ ] [, ]. hen C v dd C v dd Cvdd ( A ( A A Af on, (,. by v (, H ( and negrang on, hen ( ( v( d. k Cfvdd. Now, mllyng I show ha. If no, who loss of generaly, we assme here ess some sch ha (. Leng (, k k, k v hen ( v( d ( v( d. k k k k Whch conradcs above. Hence, he mlsve condons are sasfed. hs roof s comlee. Followng we wll aly La-lgram heorem o rove he esence of solon of (.,.e.,o rove ( a( l( aans s mnmm a. k d

5 Hachn L / Proceda Engneerng 6 ( heorem.. Assme ha osve sch ha mn,,,, a(, v defned by (A, hen a(,. oreover, f l( s defned by (B for every v H, and he fnconal on : H R defned by ( a( l( aans s mnmm a,.e., here s a leas one solon of (.. Proof.I s clearly ha he oeraor a (, s symmerc, blnear, and coercve. If sasfes Hence, we oban mn,,, sch ha ( (. a(, d ( ( ( d d ( d ( hen a (, s sysemc, blnear, bonded and coercve. Hence, here s a nqe solon of a(, l(.and he fnconal ( a(, l( aans s mnmm a,.e.,here s a leas one solon of (.. he roof comlees. Nely, when ( are no consans, we consder he nonlnear oeraor and he oeraor ( a(, C v dd C v dd Cvdd (A l Cfdd C d ( (, ( d. (B d We defned he fnconal : Cdd C dd ( d d C dd Clearly, ( s a Gaea dfferenable and s Gaea dervave s Cfdd

6 4 Hachn L / Proceda Engneerng 6 ( 5 4 '( ( C v dd Lemma. If (, H ( C v dd ( v( d Cvdd Cfvdd s a solon of (. s he mnmm of (. he roof s smlar wh ha of Lemma.. heorem.. Sose ha s connos osve and bonded, and ha he mlsve, f fncons are bonded. hen here s a crcal on of,.e.,he eqaon (. has a leas one solon. Proof. akng. hen here ess sch ha sasfes,, f, ( e ma Cfd ( k We can oban c d e d dd for some c, I mles ha lm (, and s coercve. Hence, ( has a leas one mnmm, whch s a crcal on of,.e., he eqaon (. has a leas one solon. Accordng o heorem. and he eqvalen of (., we have he followng resl. heorem.. Sose ha, f ( are bonded. hen here s a crcal on of s connos and bonded, and ha he mlsve fncons,.e., he eqaon (. has a leas one solon. We om he roof. Remark. Whe n, he mlsve dfferenal eqaon wh consan coeffcens has a leas one solon.

7 Hachn L / Proceda Engneerng 6 ( Acknowledgemens hs work s sored by he aor Proecs of Laonng Provnce (8-;Laonng Provncal DocoralFondaon (864; Laonng BaQanWan alens Program (997, 967; Shenyang Agrclral Unversy Docoral Fondaon. he corresondng ahor : lhc6@6.com (Hachn L References [] B. Ahmad and J.J. Neo, Esence and aromaon of solons for a class of nonlnear mlsve fnconal dfferenal eqaons wh anerodc bondary condons, Nonlnear Anal: heo eh Al., 8, 69(: [] D. D. Bano Z. Kamon and E. nche onoone erave mehods for mlsve hyerbolc dfferenal fnconal eqaons, J. Com. Al. ah., 996, 7: [] H. Zhang and Z. X. L, Varaonal aroach o mlsve dfferenal eqaons wh erodc bondary condons, Nonlnear Anal: RWA.,, : [4] J. J. Neo and D. O Regan, Varaonal aroach o mlsve dfferenal eqaons, Nonlnear Anal. RWA., 9, : [5] J. H. Chen, C. C, sdell and R. Yan, On he solvably of erodc bondary vale roblems wh mlse, J. ah. Anal. Al., 7, : 9-9. [6] L. L. H, L. S. L and Y. H. W, Posve solons of nonlnear snglar wo-on bondary vale roblems for secondorder mlsve dfferenal eqaons, Al. ah. Com., 8, 96: [7] L. Z, D. Q. Jang and D. O Regan, Esence heory for mlle solons o semosone drchle bondary vale roblems wh snglar deenden nonlneares for second-order mlsve dfferenal eqaons, Al. ah. Com., 8, 95: [8] X. L. F, X. L, and S. Svaloganahan, Oscllaon crera for mlsve arabolc sysems, Al. Anal.,, 79:9-55. [9] X. Y. Zhang, J. R. Yan and A.. Zhao, Esence of osve erodc solons for alsve dfferenal eqaon, Nonlnear. Anal., 8, 68: 9-6. [] Y. an and W. G. Ge, Posve solons for ml-on bondary vale roblem on he half-lne, J. ah. Anal. Al., 7, 5: [] Y. an and W. G. Ge, Varaonal mehods o Srm-Lovlle bondary vale roblem for mlsve dfferenal eqaons, Nonlnear Anal: A.,, 7: [] adesz Jankowsk, Posve solons o second order for-on bondary vale roblems for mlsve dfferenal eqaons, Al. ah. Com., 8,: [] X. L and S. H. Zhang, A Cell Polaon odel Descrbed by Imlsve PDEs Esence and Nmercal Aromaon, Com. ah. Al.,998, 6: -. [4]. E. aylor, Paral dfferenal eqaons I: basc heory, Srnger- Verlag, 996. [5]. Schecher, An ndcon o nonlnear analyss, Camberdge Unversy Press, 4.