nd Internatonal Conference on Electroncs, Network and Computer Engneerng (ICENCE 6) Postve solutons of the fourth-order boundary value problem wth dependence on the frst order dervatve YuanJan Ln, a, Fe Yang,b Nanchang Insttute of Scence and Technology, Nanchang 338, Jangx Nanchang Insttute of Scence and Technology, Nanchang 338, Jangx a lnyuanzhou@6.com, bfexu6@6.com Keywords: The frst order dervatve; Fourth-order boundary value problem; Postve soluton bstract: In ths paper, By the use of a new fxed pont theorem and the Green functon. The exstence of at least one postve solutons for the fourth-order boundary value problem wth the frst order dervatve u () (t ) + u (t ) = λ f (t, u (t ), u (t )) () u= () u= () u= () u= < t < s consdered, where f s a nonnegatve contnuous functon and λ >, < < π. Introducton Recently, there has been much attenton focused on the queston of postve soluton of fourth-order dfferental equaton wth one or two parameters. For example, astronomy, bology, physcs, chemcal engneerng and nformaton scence and other felds. So, the fourth-order boundary value problems has very mportant n real lfe applcatons, see for example [-, 6-9 ]. L [6] nvestgated the exstence of postve solutons for the fourth-order boundaryvalue problem. ll the above works were done under the assumpton that the frst order dervatve u s not nvolved explctly n the nonlnear term f. In ths paper, we are concerned wth the exstence of postve solutons for the fourth-order boundary value problem u () (t ) + u (t ) = λ f (t, u (t ), u (t )) () u= () u= () u= () u= < t < () The followng condtons are satsfed throughout ths paper: (H ) λ >, < < π ; (H ) f :[,] [, ) R [, ) s contnuous. The prelmnary lemmas Suppose Y = C[,] be the Banach space equpped wth the norm u = max u (t ). t [,] 6. The authors - Publshed by tlants Press 75
Let λ, λ be the roots of the polynomal P( λ) = λ + λ, namely, λ =, λ =. By (H ) t s easy to see that π < λ <. Let G ( ts, )( =, ) be the Green s functon of the lnear boundary value problem: u () t + λ ut () =, u() = u() =. Then, carefully calculaton yeld: s( t), s t G (, ts) = t( s), t s sn ssn ( t), s t sn G (, ts) = sn t sn ( s), t s sn Lemma.: Suppose (H ) (H ) hold. Then for any gt C[,], BVP () u t + u t = g t < t < () () (), u() = u() = u () = u () = () the unque soluton where ut = G( tsg, ) ( s, t) g( t)dtds. (3) s( t), s t G (, ts) =, t( s), t s sn τ sn ( s), τ s sn G (, s τ ) = sn ssn ( τ ), s τ sn Lemma. [5] : ssume (H ) (H ) hold. Then one has: () G( ts, ), ts, [,] ; () G( ts, ) CG( ss, ), ts, [,]; () G(, ts) δ G(,) ttg( ss, ), ts, [,]. Where: C =, δ = ; C =, δ = sn. sn Lemma.3: ssume (H ) (H ) hold and are gven as above, Then one has: mn ut du 3 t 76
sn CG where: d =, M C = G(,) ssg(,)d s, Proof: By(3)and () of Lemma.,we get: Therefore, M = G( ss, )ds, G = mn G 3 ( tt, ). u( t) CC G( s, s) G ( tt, ) g( t)dtd s CC M G ( tt, ) g( t)dt u CC M G ( ττ, ) g( τ)dτ By ()of Lemma., we have: ut dd G( ttg, ) ( ssg, ) ( ssg, ) ( t, t) g( t)dtds = dd CG( tt, ) G( t, t) g( t)dt δδ C G (,) tt u CC M Let G = mn G 3 ( tt, ), we have: t [, ] t [, ] δδ CG mn ut t [ 3, ] CC M u = sn CG u M = du Theorem. [] : Let r > r >, L > be constants and { α β } Ω = u X : ( u) < r, ( u) < L, =, two bounded open sets n X. Set { α } D = u X : ( u) = r,, =, ; ssume T : K K s a completely contnuous operator satsfyng: ( ) α α ( Tu) < r, u D K; ( Tu) > r, u D K; ( ) β ( Tu) < L, u K; ( 3 )there s p ( K)\ { } Ω, such that α( p) and α( u+ λp) α( u),for all u K, λ. Then T has at least one fxed pont n ( Ω \ Ω) K. 77
3. The man results Let X = C [,] be the Banach space equpped wth the norm { :, mn t [ } 3, ] K= u X u ut du s a cone n X. max max t [,] t [,] u = ut + u t, and Defne functonals a( u) = max ut, β ( u) = max u ( t), u X. t [,] t [,] then, u max { a( u), β( u) }, α( λu) = λα( u), βλ ( u) = λβ( u), u X, λ R, α( u) α( v), uv, Ku, v. ssume (H ) hold, the greens functon of the problem () G (, ts). let gt () =, we have t + t t t sn sn sn (, ) (, t)dtd = + G tsg s s we denote: t [,] M max G ( tsg, ) ( s, t )d t d s =, sn Q = [6 ( cos ) 3sn ] = t [, 3 ] 3 m max G ( tsg, ) ( s, t )d t d s We wll suppose that there are L > b > db > c >, such that f(, tuv, ) f(, tuv, ) satsfes the followng growth condtons: c (H 3 ) f( tuv,, ) <, ( tuv,, ) [,] [, c] [ LL, ]; λm b 3 (H ) f(, tuv, ), (, tuv, ) [, ] [ dbb, ] [ LL, ]; λm L (H 5 ) f( tuv,, ) <, ( tuv,, ) [,] [, b] [ LL, ]. λq Let f( tuv,, ),( tuv,, ) [,] [, b] (, ) f (, tuv, ) = f(, tbv, ),(, tuv, ) [,] ( b, ) (, ) f ( tuv,, ),( tuv,, ) [,] [, ) [ LL, ] f(, tuv, ) = f (, tu, L),(, tuv, ) [,] [, ) (, L] f ( tul,, ),( tuv,, ) [,] [, ) [ L, ) Defne: λ (, ) (, t) ( t, ( t), ( t))dtd Tu t = G t s G s f u u s () = t t (5) ( Tu) ( t) λ[ G ( s, t) f ( t, u( t), u ( t))dtd s sg ( s, t) f ( t, u( t), u ( t))dtd s] Lemma 3.: Suppose (H ) (H ) hold, then T : K K s completely contnuous. Proof:For u K, by (5) and Lemma.,there s Tu. 78
so, = max λ (, ) (, t) ( t, ( t), ( t))dtd t [,] Tu G t s G s f u u s we have : λ CCG s s G ττ f τuτ u τ τ s (, ) (, ) (,, )d d (, ) (,, )d λcc M G ττ f τuτ u τ τ mn = mn (, ) (, ) (,, )d 3 3 t [, ] t [, ] Tu t λ G t s G s t f t u t u t tds (, ) (, ) (, ) (, ) (,, )d d λd d G ttg ssg ssg t t f t ut u t t s CG ( tt, ) G (, ) f(, u, u)d λd d t t t t t t CG G (, ) f(, u, u)d λd d τ τ τ τ τ τ λδ δ CG CC M = d Tu Tu Therefore, we get T( K) K. So we can get T( K) K.Let B K s bounded, t s clear that T( B ) s bounded. Usng f, G(, ts), G(, tss ) contnuous, we show that T( Bs ) equcontnuous. By the rzela-scol theorem, a standard proof yelds T : K K s completely contnuous. Theorem 3.: Suppose condton (H ) (H 5 ) hold, Then BVP () has at least one postve soluton ut () satsfyng: c< α( u) < b, u () t < L Proof : Take Ω = { u X: ut () < c, u() t < L}, Ω = { u X: ut () < b, u() t < L} two bounded open sets n X and D = { u X α u = c}, D = { u X α u = b} : : such that α( u+ λp) α( u), u K, λ, u D K, α( u) = c, From (H 3 ) we have: = max (, ) (, ) (,, )d d t [,] a Tu λ G t s G s t f t u t u t t s max (, ) (, ) c < λ G d d tsg st t s t [,] λm c = max G (, ) (, )d d tsg st t s M t [,] = c 3 u D K, α( u) = b. From Lemma.3, we have u() t dα( u) = db, t [, ], so, from (H ) we get: = max (, ) (, ) (,, )d d t [,] a Tu λ G t s G s t f t u t u t t s max 3 (, ) (, ) b λ G d d tsg st t s λm > t [, 3 ] 79
3 b = max G ( 3 tsg, ) ( s, t )d t d s m t [, ] = b u K, from (H 5 ) we get: = max (, ) (,, )d d t [,] t β Tu λ G s t f t u t u t t s λ (, ) (, τ) ( τ, ( τ), ( τ))dτd d Q ssq s f u u s < 6 ( cos ) 3sn L = = L Q sn λ sg s τ f τ u τ u τ τ s (, ) (,, )d d Theorem. mples there s u Ω ( \ Ω) K, such that u = Tu. for BVP(), satsfyng : c< α( u) < b, u () t < L so, ut () s a postve soluton Thus, Theorem 3. s completed. References [] R. M, H.WNG. On the Exstence of Postve Solutons of Fourth-Order Ordnary Dfferental Equatons. ppl. nal. 995, 59: 5 3 [] B. LIU. Postve Solutons of Fourth-Order Two-Pont Boundary Value Problems. ppl. Math. Comput,, 8: 7 [3] R.Y. M. Exstence of Postve Solutons of a Fourth-Order Boundary Value Problem. ppl. Math. Comput, 5,68: 9 3 [] Z.B. BI, H.Y. Wang. On the Postve Solutons of Some Nonlnear Fourth-Order Beam Equatons. J. Math. nal. ppl., 7: 357 368 [5] Y.X. LI. Postve Solutons of Fourth-Order Boundary Value Problems wth Two Parameters. J. Math. nal. ppl, 3, 8:77 8 [6] Q.L. YO. Local Exstence of Multple Postve Solutons to a Sngular Cantlever Beam Equaton. J. Math. nal. ppl., 363: 38 5 [7] H.Y. FENG, D.H. JI, W.G. GE. Exstence and Unqueness of Solutons for a Fourth-Order Boundary Value Problem. Nonlnear nal, 9, 7: 356 3566 [8] G. Q. CHI. Exstence of Postve Solutons for Fourth-Order Boundary Value Problem wth Varable Parameters. Nonlnear nal, 7, 66:87 88 [9] J. f. Xu, Z. l. Yang. Postve Solutons for a Fourth order p-laplacan Boundary Value Problem. Nonlnear nal,, 7: 6-63 [] Y. P. GUO, W. G. GE. Postve Solutons for Three-Pont Boundary Value Problems wth Dependence on the Frst Order Dervatves. Journal of Mathematcal nalyss and pplcatons,, 9: 9 3 75