{ u (t)+a(t)f(u) =0, 0 <t<1; { u (t)+a(t)u (t)+b(t)u(t)+h(t)f(u) =0, 0 <t<1;

Transcript

1 4~% K x z ff Vol.4, No. 2#2μ ADVANCES IN MATHEMATICS Feb., 2 TG p-laplaian AS<7n fi%ν,, zφ 2, e` (. iψw Jwe ρ Zψ 423; 2. ψ 'wjww Tv 37 haπ tχ#$ffi, "±w ffi ρ "flψ p-laplaian ß-»ffΞ u ψffλ+(fi &! ~ 'fluψff*μ &»±(fi xπflω v y "±οfflλ CF;Π p-laplaian ß-±Ξ uψff±w ffi±(fi MR(2 qwbkπ 34B5 / oybkdπo75.8 Z]9UOΠ A Zi8DΠ -97(2-7-8 ` [] ΩΩ Shauder μ524φ =oa'"52!ar { u (t+a(tf(u =, <t<; u( =, u( u(η = flr-%νp &&, >,η (,,η <. 5 ` [2] ΩΩ.Π=.zOμ524Φ ffi=oa'"52!ar { u (t+a(tu (t+b(tu(t+h(tf(u =, <t<; u( =, u( u(η = flr-%νp && <η<, <η<, [,. fl`xrm Il-N]C@fl ΦΩY.7- Iμ524Φ [O 7qο Q ο f 5o p-laplaian A'"52!aR { (φp (u (t + a(tf(t, u(t,u (t =, <t<; u( =, u( u(η =, u (Q ( = :'flr-%νp && φ p (s D p-laplian N φ p (s = s p 2 s, p >,φ q =(φ p, p + q =, <η<, <η<, [,. S p-laplaian N]C@fl-!aR νa$9fiv *dφ ymarvpφο /+: 6fiO?-ΦΩ [3 7]. (D 8S p-laplaian NA' "!ar- 8H9i fl`ff fi μ4 FG%Π regf+%π ujχπ Zψ?n W jχ (No. 8C826; Zψ?(3l<w jχ±zψ?f χmz7d[ /-Λ Pf2. w u(3jχ (No. 535; ο 973 jχ (No. 26CB8593; `Pf2. w uχmz7jχ/-λ tys73@63.om; liug@nanai.edu.n

2 72 J w y ρ 4~ νfl`& ρ3g4 fvjfiff (C f C([, ] [, + R, [, + ; (C 2 a L [, ] DAD- a(t ν [, ] -Ξ*h7μX^ f6mv ^ χflfl`-,~qq ρs~3f- ffψa?. ; E D I@ Banah Λh 3Q P D E &ffiaλyffl^»ß4ffiffl IVjΞ (a 4 x P, λ, ß λx P ; (b 4 x P, x P, ß x =. ßff P D E &- I.?.2 Ψ: ψ fiff^ P 7-ADfltΦ?T #~ ψ : P [, flt ψ(tx +( ty tψ(x+( tψ(y 8Pfi- x, y P _ t fiff M ffψ: ϕ D P 7-ADfltY?T #~ ϕ : P [, flt ϕ(tx +( ty tϕ(x+( tϕ(y 8Pfi- x, y P _ t fiff?.3 J4ßI r>a>,l>. ; ψ D P 7-ADfltΦ?T ϕ, ω D P 7-A DfltY?T 4ΠY^ P (ϕ, r; ω, L ={y P ϕ(y <r,ω(y <L, P (ϕ, r; ω, L ={y P ϕ(y r, ω(y L, P (ϕ, r; ω, L; ψ, a ={y P ϕ(y <r,ω(y <L,ψ(y >a, P (ϕ, r; ω, L; ψ, a ={y P ϕ(y r, ω(y L, ψ(y a. fg; P 7-ADfltY?T ϕ, ω ß4 (C 3 %ν M>, B,8Ξ x P fi x M {ϕ(x,ω(x; (C 4 P (ϕ, r; ω, L, 8Ξ- r>,l>. ffijffifl`,~qο-χflp~ω*-,~ Φ dl. [8] E D Banah Λh P E D I. ßI r 2 d>b>r >, L 2 L > J4 g4 ϕ, ω D P 7-ADfltY?Tß4 (C 3, (C 4, ψ D P 7-ADfltΦ?T ψ(y ϕ(y, 8Pfi y P (ϕ, r 2 ; ω, L 2 fiff T : P (ϕ, r 2 ; ω, L 2 P (ϕ, r 2 ; ω, L 2 D,fltN g; (A {y P (ϕ, d; ω, L 2 ; ψ, b ψ(y >b,ψ(ty >b, 8 y P (ϕ, d; ω, L 2 ; ψ, b; (A 2 ϕ(ty <r,ω(ty <L, 8 y P (ϕ, r ; ω, L ; (A 3 ψ(ty >b, 8 y P (ϕ, r 2 ; ω, L 2 ; ψ, b ϕ(ty >d. ß T ν P (ϕ, r 2 ; ω, L 2 &$9fi5Iμ52 y,y 2,y 3, fi y P (ϕ, r ; ω, L, y 2 {P (ϕ, r 2 ; ω, L 2 ; ψ, b ψ(y >b, y 3 P (ϕ, r 2 ; ω, L 2 \ (P (ϕ, r 2 ; ω, L 2 ; ψ, b P (ϕ, r ; ω, L.

3 % U»> μψlπ6p p-laplaian B(# "bs;(ffis.&οq 73 2 rbhnerlq E = C [, ] D Banah Λh &>I4Π^Ξ { u = u(t, t t u (t. P = {u E u(t,uν [,] 7Φ, ß P D E 7- I. ffvj (C 2 ψ %ν-i 3, B, < a(tdt < +. 8 u P, 4Π?T ϕ(u = u(t, ω(u = t t u (t, ψ(u = t u(t. ß ϕ, ω, ψ : P [, D5IADflt?T ß4 u ={ϕ(u,ω(u (C 3, (C 4 fi ff ϕ, ω DY?T ψ DΦ?T ψ(u ϕ(u, 8Pfi u P. dl 2. [5] ; u P, -I 3, ß t u(t t u(t. G(t, s = { t( s, t s, s( t, s t. Λψ G(t, s G(s, s, t, s. dl 2.2 ;Vj (C, (C 2 fiff u(t E D]C@fl u(t = G(t, sφ q + t { a(τf(τ,u(τ,u (τdτ ds G(η, sφ q a(τf(τ,u(τ,u (τdτ ds + (2. - Ir ß u(t 4D!aR (Q - Ir P 3Q u(t D]C@fl (2. - Ir ß8Ξ t [, ], ρfi u (t = G (t, sφ q a(τf(τ,u(τ,u (τdτ ds = + t + { G(η, sφ q ( sφ q t ( sφ q a(τf(τ,u(τ,u (τdτ ds + a(τf(τ,u(τ,u (τdτ ds a(τf(τ,u(τ,u (τdτ ds + { G(η, sφ q ( t u (t = φ q a(τf(τ,u(τ,u (τdτ a(τf(τ,u(τ,u (τdτ ds +,.

4 74 J w y ρ 4~, (φ p (u (t + a(tf(t, u(t,u (t =, < u ( =.» ff G(,s=G(,s=_ (2. C, u( =. u( = { G(η, sφ q a(τf(τ,u(τ,u (τdτ ds + = G(η, sφ q a(τf(τ,u(τ,u (τdτ u(η = G(η, sφ q a(τf(τ,u(τ,u (τdτ ds + η { G(η, sφ q = G(η, sφ q a(τf(τ,u(τ,u (τdτ ds +, a(τf(τ,u(τ,u (τdτ ds + ds + η. P u( u(η =. $< u(t D!aR (Q - Ir χffi 8Ξ u P, 4ΠN T : P E (Tu(t = G(t, sφ q + t { a(τf(τ,u(τ,u (τdτ ds G(η, sφ q a(τf(τ,u(τ,u (τdτ ds +. (2.2 dl 2.3 T : P P D,flt- P 8 u P, ff (2.2 C_Vj (C, (C 2, h- (Tu(t, t [, ],Tu E. (Tu (t = φ q ( t a(τf(τ,u(τ,u (τdτ. P Tuν*h [, ] 7DΦTI N T (P ; D D P - fit^ ß%ν M>, B, D {u P u M. N = (t,u,v [,] [,M] [ M,M] f(t, u, v. $< u D, ρfi (Tu(t = + t { G(t, sφ q a(τf(τ,u(τ,u (τdτ ds a(τf(τ,u(τ,u (τdτ ds + a(τdτ ds G(η, sφ q { N q G(s, sφ q + 6 N q ( + G(η, sφ q φ q ( a(τdτ ds + a(τdτ +,

5 % U»> μψlπ6p p-laplaian B(# "bs;(ffis.&οq 75 (Tu (t = G (t, sφ q a(τf(τ,u(τ,u (τdτ ds + { G(η, sφ q a(τf(τ,u(τ,u (τdτ ds + t = ( sφ q a(τf(τ,u(τ,u (τdτ ds + ( sφ q a(τf(τ,u(τ,u (τdτ ds t + { G(η, sφ q a(τf(τ,u(τ,u (τdτ ds + ( { t N q φ q a(τdτ sds + ( sds + G(s, sds + t ( N (+ q φ q a(τdτ + 6(, ( t ( (Tu (t = φ q a(τf(τ,u(τ,u (τdτ N q φ q a(τdτ. ffid K Arzela-Asoli 4Φψ T (D ^ w^ &"ff±ωhξ%effi4φμ!χfl T ν P 7Dflt- ±! T : P P D,flt- χffi ρ ΩffieU N = ( ( s G 2,s φ q ( φ q M = ( + 3 ( ( H = + φ q 3( = a(τdτ ds, a(τdτ, a(τdτ { 2 (r, 2 (L ρ-,~qο3fξ {?L 2. g;%νßi r 2 b>b>r >,L 2 L >, B, b N r2 M, L2 H, [,, 3fg;fiff { (B f(t, u, v < φ p ( r M,φ p ( L H, 8 (t, u, v [.] [,r ] [ L,L ]; (B 2 f(t, u, v >φ p ( b N, 8 (t, u, v [, ] [b, b] [ L 2,L 2 ]; { (B 3 f(t, u, v φ p ( r2 M,φ p ( L2 H, 8 (t, u, v [, ] [,r 2 ] [ L 2,L 2 ]. ß!aR (Q $9fi5Iflr u,u 2,u 3 B, b< t t u (t <r, t t u (t <L, u 2 (t t u 2(t r 2, u 3 (t <b, u 3(t <b, t. t u 2 (t L 2, t u 3 (t L 2.,

6 76 J w y ρ 4~ P ff Φ 2.2, Φ 2.3 ψ ρ#sχ Φ. -PfiVj6ß4 ffiρcl νxoχ ( Ggρχ T : P (ϕ, r 2 ; ω, L 2 P (ϕ, r 2 ; ω, L 2. 3Q u P (ϕ, r 2 ; ω, L 2, ß ϕ(u r 2,ω(u L 2. ffid ffg4 (B 3, fi ±! f(t, u(t,u (t ϕ(tu= (Tu(t = t t + t { G(η, sφ q { G(s, sφ q r 2 M + r 2 M 6 r r 2 2 = r 2. ( + { φ p ( r2 M,φ p ( L2 H. G(t, sφ q a(τdτ G(η, sφ q φ q ( a(τf(τ,u(τ,u (τdτ ds a(τdτ ds + a(τdτ + a(τf(τ,u(τ,u (τdτ ds ds + 8 u P, fi Tu P. ff Tuν [, ] 7Φ ψ t (Tu (t ={ (Tu (, (Tu (. ±! ω(tu= t (Tu (t ={ (Tu (, (Tu ( { ( sφ q a(τf(τ,u(τ,u (τdτ + ( G(η, sφ q sφ q a(τf(τ,u(τ,u (τdτ ds + ( G(η, sφ q L ( ( 2 H 2 + φ q a(τdτ + 6( L L 2 2 = L 2. ±! T : P (ϕ, r 2 ; ω, L 2 P (ϕ, r 2 ; ω, L 2. (2 ρχ Φ. -Vj (A fiff u+ u(t = b 2 ds a(τf(τ,u(τ,u (τdτ ds +, a(τf(τ,u(τ,u (τdτ ds +, <t<. 2Λχfl u(t = b 2 P (ϕ, b; ω, L 2; ψ, b ψ(u =ψ( b ±! {u P (ϕ, b; ω, L 2 ; ψ, b. 4 u P (ϕ, b; ω, L 2 ; ψ, b, ß8 t b, u (t L 2. ffvj (B 2, ρfi f(t, u(t,u (t >ϕ p ( b N, t. 2 >b,, b u(t

7 % U»> μψlπ6p p-laplaian B(# "bs;(ffis.&οq 77 ff?t ψ -4Π_ Φ 2., fi ψ(tu= t t > b N = b. (Tu(t (Tu(t t G G(t, sφ q a(τf(τ,u(τ,u (τdτ ds ( s φ q a(τf(τ,u(τ,u (τdτ ds ( 2,s ( G 2,s ( s φ q a(τdτ ds ±! ψ(tu >b, 8Pfi- u P (ϕ, b; ω, L 2 ; ψ, b, fi fl Φ. -Vj (A fiß4 (3 ρχ Φ. -Vj (A 2 fiff 3Q u P (ϕ, r ; ω, L, ßVj (B fl { ( f(τ,u(τ,u r ( L (τ < φ p,φ p, τ. M H M ( χfl T : P (ϕ, r ; ω, L P (ϕ, r ; ω, L. ffid Φ. -Vj (A 2 fiß4 (4 5Yρχ Φ. -Vj (A 3 fiff g; u P (ϕ, r 2 ; ω, L 2 ; ψ, b ϕ(tu >b, ßff?T ψ -4Π_ Φ 2., fi ψ(tu= t (Tu(t (Tu(t = t ϕ(tu > b = b. ffid Φ. -Vj (A 3 fiß4 ±!ff Φ. ψ!ar (Q $9fi5Iflr ß4 u P (ϕ, r ; ω, L, u 2 {P (ϕ, r 2 ; ω, L 2 ; ψ, b ψ(u >b, u 3 P (ϕ, r 2 ; ω, L 2 \ (P (ϕ, r 2 ; ω, L 2 ; ψ, b P (ϕ, r ; ω, L. ffffi u 3 ß4 ϕ(u 3 ψ(u 3, P t u 3 (t <b.χffi 3 Ms p = 3 2,a(t,η = = 2 _ = 5. ρ νffi!arξ {( u (t 2 u (t + f(t, u(t,u (t =, <t<; u( =, u( 2 u( 2 = 5, u ( =, (R fiψ { f(t, u, v = 23 t +24u8 +( v 77 2, u, 23 t +24+( v 77 2, u >. u+ r = 2 3,b=,=4,L =2,r 2 = 35, L 2 = 77. fi N = , M = , H =

8 78 J w y ρ 4~ fl b N < { r2 M, L2 H 63.8, = 5 < ={ 2 (r, 2 (L = 4, < f(t, u, v ß4 f(t, u, v < < {φ 3 ( r,φ3 2 M ( L 2 H.95, 8 t, u 2 3, u 4, v 77; 2 f(t, u, v > 24 >φ3 2 ( b N 23.94, 8 4 t 3 4, v 2; 3 f(t, u, v < 25. < {φ 3 2 ( r2 M,φ3 2 ( L2 H 25.6, 8 t, u 35, v 77. a4φ 2. -PfiVj6ß4 ±!!ar (R $9fi5Iflr u,u 2 ffl u 3, ß4 :J[^ u (t < 2 t 3, < u 2 (t u 2(t 35, 4 t 3 t 4 u 3 (t <, 4 t 3 4 u 3(t < 4, t t u (t < 2; t u 2 (t 77; t u 3 (t 77. [] Ma R.Y., Positive solutions for seond-order three-point boundary value problem, Appl. Math. lett., 2, 4: -5. [2] Chen H.B., Positive solutions for the nonhomogeneous three-point boundary value problem of seond-order differential equations, Math. Comp. Mode, 27, 45: [3] He X.M., Ge W.G., Twin positive solutions for the one-dimensional p-laplaian boundary value problems, Nonlinear Anal., 24, 56: [4] Ma D.X., Han J.X., Chen X.G., Positive solution of boundary value problem for one-dimensional p-laplaian with singularities, J. Math. Anal. Appl., 26, 324: [5] Su H., Wei Z.L., Wang B.H., The existene of positive solutions for a nonlinear four-point singular boundary value problem with a p-laplaian operator, Nonlinear Anal., 27, 66: [6] Sun B., Ge W.G., Zhao D.X., Three positive solutions for multipoint one-dimensional p-laplaian boundary value problem with dependene on the first order derivative, Math. Comp. Mode, 27, 45: [7] Pang H.H., Feng M.Q., Ge W.G., Existene and monotone iteration of positive solution for a three-point boundary value problem, Appl. Math. Lett., 28, 2: [8] Bai Z.B., Ge W.G., Existene of three positive solutions for some seond-order boundary value problem, Comput. Math. Appl., 24, 48: Multiple Positive Solutions of the Nonhomogeneous Boundary Value Problem for a Class of Third-order p-laplaian Equations TIAN Yuansheng, LIU Chungen (. Department of Mathematis, Xiangnan University, Chenzhou, Hunan, 432, P. R. China; 2. Shool of Mathematis, Nanai University, Tianjin, 37, P. R. China Abstrat: In this paper, by using a fixed point theorem on onvex one, we onsider the nonhomogeneous boundary value problem for a lass of third order p-laplae equations, the multipliity result of three positive solutions are obtained, an example is that inluded to illustrate the importane of results obtained. Key words: p-laplaian operator; nonhomogeneous boundary value problem; fixed point theorem; positive solution