Existence and multiplicity of solutions to boundary value problems for nonlinear high-order differential equations

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1 Jounal of pplied Matheatics & Bioinfoatics, vol.5, no., 5, 5-5 ISSN: (pint), (online) Scienpess Ltd, 5 Existence and ultiplicity of solutions to bounday value pobles fo nonlinea high-ode diffeential equations Huijun Zheng bstact In this pape, we investigate the ultiple positive solutions fo the bounday value poble of nonlinea diffeential equations. The aguents ae based upon the fixed point theoe of cone expansion and copession with no type. Matheatics Subject Classification: O75 Keywods: Highe ode bounday value poble;positive solution;cone fixed point theoe;ultiple solutions Intoduction In ecent yeas, any eseaches study the existence and ultiplicity of Huijun Zheng School of Matheatics and Statistics, Notheast Petoleu Univesity, Daqing 6338, China. E-ail:zhenghuijun@63.co ticle Info: Received : Novebe,. Revised : Decebe 9,. Published online : Januay 5, 5.

2 Existence and ultiplicity of solutions to bounday value pobles solutions to bounday value pobles fo nonlinea high-ode odinay diffeential equations, especially fo even nube ode equation. They have obtained a lot of satisfactoy esults and ost authos study the pobles unde the conjugate bounday conditions o siple bounday conditions []-[3]. Howeve,it is not coon that the study on the bounday pobles have high-ode deivative ight side bounday conditions. This pape study the bounday value pobles, fo a class of diffeential equations with ode ight-hand side bounday conditions. In ou pape, we give soe sufficient conditions fo the existence of one o two positive solutions, fo the bounday value pobles using the popeties of the coesponding Geen s function and cone expansion and copession fixed point theoe. Peliinay Notes In this pape, we concened on the following nonlinea highe-ode bounday value poble We assue that ( ) y x = ( f x y x) aex ( ) ( ), ( ),.. (,) () i y () =, i ( j) y = j (), () ( H ) f ( x, y) is continuous and nonnegative on[,] [, + ). ( H ) f ( x, y) is not equal to zeo any whee fo any copact subinteval in[,], and x f ( xy, )d x < +. Theoe. Let B be a Banach space and K B a cone in B. ssue that

3 Huijun Zheng 5 Ω, Ω ae open subset of B with Ω, Ω Ω. Let Φ: K Ω ( \ Ω) K be a copletely continuous opeato such that eithe (i) Φy y, y K Ω, Φy y, y K Ω ; o (ii) Φy y, y K Ω, Φy y, y K Ω. Then, Φ has a fixed point in K Ω ( \ Ω ). To obtain positive solutions fo the poble (). We state soe popeties of Geen s function fo (). s shown in [5], the poble () is equivalent to the integal equation whee yx ( ) = Gxs (, ) f( sys, ( ))ds s u ( u+ x s) d u, s x, [( ) ]! Gxs (, ) = x u ( u+ s x) d u, x s. [( ) ]! () Moeove, the following esults have been ecently offeed by [5]. Lea. Fo any x, s [,], we have Whee G( x,s) ( x) g( s) Gxs (,) β()() xg s, cs x, x c = ; ( x) = [( )!] ; x β ( x) = ; gs () = s [( )!]. Let y = sup y( x), = in ( x), = ax β( x). Define the cone K in Banach y 3

4 6 Existence and ultiplicity of solutions to bounday value pobles space C [,], given by We define the opeato K= { y C[,]: yx ( ), in yx ( ) y/ β }, Φ:K K by 3 ( y) ( x) Gxs (, ) f( sys, ( ))ds. Φ = Fo any y K,we have in Φyx ( ) 3 in 3 ( x)g(s) f ( s, y( s))ds ax G( x,s) f( sys, ( ))ds y β = Φ β This iplies Φ( K) K. We can easily obtain it, Φ: K Kis a copletely continuous apping. 3 Main Results Let 3, we denote { y K, y } = gs ( )d sb, = gs ( )ds Ω = < ( > ), then Ω = { y K, y = }. Theoe 3. ssue that ( H ) ) hold,if thee exist two positive nubes, and,let <, < + such that (Ⅰ) x [,], y [, ], f( xy, ) ;

5 Huijun Zheng 7 3 (Ⅱ) x [, ], y [, ], f( xy, ) B. then the poble() has at least one positive solution, satisfying in{, } y ax{, }. Poof. Let <,then Ω = { y K, y < } and Ω = { y K, y < } y Ω yx (), Φ ax y Ω, then in yx ( ) y/ β =. 3 It is easy to see β ( x)g(s) f( sys, ( ))ds g(s)ds y = Φy() x in 3 3 ( x)g(s) f( sys, ( ))ds B g(s)d s= y. 3 In view of Theoe., we know that Φ has a fixed point y Ω \ Ω.That is to say, y is a positive solution of (),and y. Infeence 3. ssue that ( H ) ) hold. Ou assuptions thoughout ae, (Ⅲ) li ax + y x [,] f( xy, ) f( xy, ) B < and li in >, y y + 3 x [, ] y then the poble() has at least one positive solution. Poof. Using (Ⅲ) we have, fo any x [,], thee exists a sufficiently sall >, such that f( xy, ), y (, ] (3) Following fo f( xy, ) B li in >, that thee exists a sufficiently lage y + 3 x [, ] y

6 8 Existence and ultiplicity of solutions to bounday value pobles >, fo any 3 x [, ] such that, B f( xy, ) y B β, y [, ] () Thus, theoe 3. iplies,then the poble() has at least one positive solution. Infeence 3. ssue that ( H ) ) hold,ou assuptions thoughout ae, f( xy, ) B (IV) li in >, and + 3 y x [, ] y f( xy, ) li ax <, y y + x [,] then the poble () has at least one positive solution. Poof. Follows fo (IV), thee exists < <,fo any f( xy, ) B, y [, ] y Consequently, we have fo any 3 x [, ], B f( xy, ) y B β, y 3 x [, ], such that [, ] (5) Following fo li ax y + x [,] f( xy, ) <, thee exists ε > y, such that ε >, and f( xy, ) ( ) ε y, y. Let c= ax{ f( xy, ) : y }, then it is obvious to see f( xy, ) c+ ( ε) yy, [, + ). Choose > ax{, c( ε) }.Let Ω = { y C[,]: y < }. then fo

7 Huijun Zheng 9 fo any y Ω, we have f( xy, ) c+ ( ε) y. (6) Thus, we obtain thee exist x [,] and, fo all x [,] and y [, ], f( xy, ). By Theoe 3. we know, then the poble() has at least one positive solution. Theoe 3. ssue that ( H ) ) hold,fo all > Let,satisfy the (Ⅱ). li ax + y x [,] f( xy, ) f( xy, ) <, li ax <, y y + x [,] y Then the poble () has two positive solutions. Poof. We can pove as the sae as Coollay and.following fo thee exist < < <,fo any x [,], f( xy, ), y (, ]. and exist x [,] and, such that f( xy, ), when x [,] and y [, ] Then by (IV),we know fo Theoe 3.,Then the poble () has two positive solutions y, y, and satisfy < < y < < y <. Theoe 3.3 ssue that ( H ) ) hold,fo all >,satisfy the (IV). Let

8 5 Existence and ultiplicity of solutions to bounday value pobles f( xy, ) li in > B, and + 3 y x [, ] y f( xy, ) B li in >. y + 3 x [, ] y Then the poble () has two positive solutions y, y, and satisfy < < y < < y <. " " Poof. Thee exist have < < <, fo any x 3 [, ], [, ] y, we Hence we obtain fo any B in f( xy, ) y B 3 x [, ] β. B in f( xy, ) y B 3 x [, ] " β, 3 x [, ], y [, ]. Then by (IV), we know fo Theoe 3.. Then the poble () has two positive solutions y, y, satisfy < < y < < y <. Conclusion We study a class of highe ode nonlinea diffeential equation bounday valu e poble and give soe sufficient conditions fo existence of positive solutions u sing the cone fixed point theoes. It eniches the esults of the elated liteatues futhe.

9 Huijun Zheng 5 Refeences [] Lingbin Kong and Junyu Wang, The Geen s Function fo (k,n-k) Conjugate Bounday Value Pobles and Its pplications, Jounal of Matheatical nalysis and pplications, 55, (), -. [] Qingliu Yao, Monotone iteative technique and positive solutions of Lidstone bounday value pobles, pplied Matheatics and Coputation, 38, (3), -9. [3] J.M. Davis, J. Hendeson and P.J. Wong, Geneal Lidstone pobles: ultiplicity and syety of solutions, Math. nal. ppl. 5, (), [] Qingliu Yao, Existence of Solution fo Geneal Lidstone Bounday Value Pobles, cta Matheatica Scientin, 5(7), (5),-. [5] Huijun Zheng, Lingbin Kong, The Geen function and positive solution fo ode nonlinea bounday value pobles, Jounal of daqing petoleu institute, 3(), (), 6-. [6] Yanping Guo, Existence of ultiple onoone positive solutions fo highe ode bounday value pobles, Jounal of Systes Science and Matheatical Sciences, 3, (3),

A question of Gol dberg concerning entire functions with prescribed zeros

A question of Gol dberg concerning entire functions with prescribed zeros A question of Gol dbeg concening entie functions with pescibed zeos Walte Begweile Abstact Let (z j ) be a sequence of coplex nubes satisfying z j as j and denote by n() the nube of z j satisfying z j.