Existence of positive solution for a third-order three-point BVP with sign-changing Green s function

Transcription

1 Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp:// Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion Xing-Long Li, Jian-Ping Sun, Fang-Di Kong Deparmen of Applied Mahemaics, Lanzhou Universiy of Technology, Lanzhou, Gansu 735, People s Republic of China Absrac By using he Guo-Krasnoselskii fixed poin heorem, we invesigae he following hirdorder hree-poin boundary value problem u () = f(,u()), [,1], u () = u(1) =, u () + αu() =, where α [,) and [ 11+4α 5 3(4+α), 1). The emphasis is mainly ha alhough he corresponding Green s funcion is sign-changing, he soluion obained is sill posiive. Keywords: Third-order hree-poin boundary value problem; Sign-changing Green s funcion; Posiive soluion; Exisence; Fixed poin 1 AMS Subjec Classificaion: 34B1, 34B18 Suppored by he Naural Science Foundaion of Gansu Province of China (18RJZA4). Corresponding auhor. jpsun@lu.cn EJQTDE, 13 No. 3, p. 1

2 1 Inroducion Third-order differenial equaions arise from a variey of differen areas of applied mahemaics and physics, e.g., in he deflecion of a curved beam having a consan or varying cross secion, a hree-layer beam, elecromagneic waves or graviy driven flows and so on [3]. Recenly, he exisence of single or muliple posiive soluions o some hird-order hree-poin boundary value problems (BVPs for shor) has received much aenion from many auhors, see [1,, 5, 1, 15, 16] and he references herein. However, all he above-menioned papers are achieved when he corresponding Green s funcions are posiive, which is a very imporan condiion. A naural quesion is ha wheher we can obain he exisence of posiive soluions o some hird-order hree-poin BVPs when he corresponding Green s funcions are sign-changing. In 8, Palamides and Smyrlis [11] sudied he exisence of a leas one posiive soluion o he singular hird-order hree-poin BVP wih an indefiniely signed Green s funcion u () = a()f(, u()), (, 1), u() = u(1) = u () =, where ( 17 4, 1). Their echnique was a combinaion of he Guo-Krasnoselskii fixed poin heorem and properies of he corresponding vecor field. In 1, by using he Guo-Krasnoselskii and Legge-Williams fixed poin heorems, Sun and Zhao [13,14] discussed he hird-order hree-poin BVP wih sign-changing Green s funcion u () = f(, u()), [, 1], (1.1) u () = u(1) = u () =, where ( 1, 1). They obained he exisence of single or muliple posiive soluions o he BVP (1.1) and proved ha he obained soluions were concave on [, ] and convex on [, 1]. I is worh menioning ha here are oher ype of works on sign-changing Green s funcions which prove he exisence of sign-changing soluions, posiive in some cases, see Infane and Webb s papers [6 8]. EJQTDE, 13 No. 3, p.

3 In his paper we sudy he following hird-order hree-poin BVP u () = f(, u()), [, 1], u () = u(1) =, u () + αu() =. (1.) Throughou his paper, we always assume ha α [, ) and [ 11+4α 5, 1). Obviously, 3(4+α) he BVP (1.1) is a special case of he BVP (1.). However, i is necessary o poin ou ha his paper is no a simple exension of [13]. In fac, if we le α =, hen [ 1, 1), which is differen from he resricion in [13]. On he oher hand, compared wih [13], we can only prove ha he obained soluion is concave on [, ]. Our main ool is he following well-known Guo-Krasnoselskii fixed poin heorem [4,9]: Theorem 1.1 Le E be a Banach space and K be a cone in E. Assume ha Ω 1 and Ω are bounded open subses of E such ha Ω 1, Ω 1 Ω, and le T : K (Ω \Ω 1 ) K be a compleely coninuous operaor such ha eiher (1) Tu u for u K Ω 1 and Tu u for u K Ω, or () Tu u for u K Ω 1 and Tu u for u K Ω. Then T has a fixed poin in K (Ω \ Ω 1 ). Preliminaries For he BVP u () =, [, 1], u () = u(1) =, u () + αu() =, we have he following lemma. (.1) Lemma.1 The BVP (.1) has only rivial soluion. Proof. I is simple o check. In he remainder of his paper, we always assume ha Banach space C [, 1] is equipped wih he norm u = max [,1] u(). EJQTDE, 13 No. 3, p. 3

4 Now, for any y C [, 1], we consider he BVP u () = y(), [, 1], u () = u(1) =, u () + αu() =. (.) Afer a direc compuaion, one may obain he expression of Green s funcion G(, s) of he BVP (.) as follows: where and G(, s) = g 1 (, s) + g (, s) + g 3 (,, s), g 1 (, s) = ( α )(1 s), (, s) [, 1] [, 1], ( α), s 1, g (, s) = ( s), s 1, s, g 3 (,, s) = 1, s <. α I is no difficul o verify ha he G(, s) has he following properies: G(, s) for s and G(, s) for s 1. Moreover, for s, and for s <, max{g(, s) : [, 1]} = G(1, s) =, (1 s) min{g(, s) : [, 1]} = G(, s) = α max{g(, s) : [, 1]} = G(, s) = s s α, min{g(, s) : [, 1]} = G(1, s) =. Le K = {y C [, 1] : y() is nonnegaive and decreasing on [, 1]}. Then K is a cone in C [, 1]. EJQTDE, 13 No. 3, p. 4

5 Lemma. Le y K and u() = G(, s)y(s)ds, [, 1]. Then u is he unique soluion of he BVP (.) and u K. Moreover, u() is concave on [, ]. Proof. For, we have u() = [g 1 (, s) + ( s) + 1 ] α [g 1 (, s) + 1 ] α g 1 (, s)y(s)ds. Since 11+4α 5 implies ha α, we ge 3(4+α) 3α+6 u () = α (s s )y(s)ds α [ y() α (s s )ds α [ α(1 3) = y() 3( α) + ] [ α(1 3) y() 3( α) ]. sy(s)ds sds α α α (1 s) y(s)ds α ] ds + (1 s) ds A he same ime, 11+4α 5 3(4+α) > 1 3 shows ha u () = α α αy() α. αy()(1 3) 3( α) (s s )y(s)ds (s s )ds y() α α ds + αy() α (1 s) y(s)ds (1 s) ds For < 1, we have u() = [g 1 (, s) + ( s) + 1 ] α [g 1 (, s) + ] ( s) g 1 (, s)y(s)ds. EJQTDE, 13 No. 3, p. 5

6 Since 11+4α 5 3(4+α) u () = α α αy() α = y(). implies ha 6 α, we ge 1 (s s ) ( s)y(s)ds (s s y()( ) )ds + y() ] [ α(1 3) 3( α) + s sds + α α αy()(1 )3 3( α) (1 s) y(s)ds Obviously, u () = y() for [, 1], u () = u(1) = and u () + αu() =. This shows ha u is a soluion of he BVP (.). The uniqueness follows immediaely from Lemma.1. Since u () for [, 1] and u(1) =, we have u() for [, 1]. So, u K. In view of u () for [, ], we know ha u() is concave on [, ]. Lemma.3 Le y K. Then he unique soluion u of he BVP (.) saisfies where θ (, 1 3 ] and θ = θ. min u() θ u, [,θ] Proof. By Lemma., we know ha u() is concave on [, ], hus for [, ], u() (1 )u() + u(). (.3) In view of u K, we know ha u = u(), which ogeher wih (.3) implies ha u() u,. Consequenly, min u() = u(θ) θ u = θ u. [,θ] EJQTDE, 13 No. 3, p. 6

7 3 Main resuls For convenience, we denoe A = Then i is obvious ha < B < A. G(, s)ds and B = θ G(, s)ds. Theorem 3.1 Assume ha f : [, 1] [, + ) [, + ) is coninuous and saisfies he following condiions: (H1) For each u [, + ), he mapping f(, u) is decreasing; (H) For each [, 1], he mapping u f(, u) is increasing; (H3) There exis wo posiive consans r and R wih r R such ha f(, r) r A and f(θ, θ R) R B. Then he BVP (1.) has a posiive and decreasing soluion u saisfying min{r, R} u max{r, R}. Moreover, he obained soluion u() is concave on [, ]. Proof. Le K = { } u K : min u() [,θ] θ u. Then i is easy o see ha K is a cone in C [, 1]. Now, we define an operaor T on K by (Tu)() = G(, s)f(s, u(s))ds, [, 1]. Obviously, if u is a fixed poin of T in K, hen u is a nonnegaive and decreasing soluion of he BVP (1.). In wha follows, we will seek a fixed poin of T in K by using Theorem 1.1. Firs, by Lemma. and Lemma.3, we know ha T : K K. Furhermore, alhough G(, s) is no coninuous, i follows from known exbook resuls, for example see [1], ha T : K K is compleely coninuous. Nex, for any u K, we claim ha θ G(, s)f(s, u(s))ds + G(, s)f(s, u(s))ds. (3.1) EJQTDE, 13 No. 3, p. 7

8 In fac, if u K, recall ha G(, s) for s and G(, s) for s 1, hen i follows from 11+4α 5 3(4+α) θ ha G(, s)f(s, u(s))ds + [ f(, u()) G(, s)ds + θ G(, s)f(s, u(s))ds ] G(, s)ds [ ( s) = f(, u()) (g 1 (, s) ) ] ds + g 1 (, s)ds θ α (1 )f(, u()) [ = (4 + α) + (4 + αθ 3 3αθ ) 6θ + αθ 3 ] 6( α) [ (1 )f(, u()) (4 + α) + 1 6( α) 3 8 ] 3. Now, wihou loss of generaliy, we assume ha r < R. Le Ω 1 = {u C [, 1] : u < r} and Ω = {u C [, 1] : u < R}. ha For any u K Ω 1, we ge u(s) r for s [, 1], which ogeher wih (H3) implies (Tu)() = max G(, s)f(s, u(s))ds + [,1] G(, s)f(s, u(s))ds G(, s)f(, r)ds r = u, [, 1]. max [,1] G(, s)f(s, u(s))ds This shows ha Tu u for u K Ω 1. (3.) For any u K Ω, we ge θ R u(s) R for s [, θ], which ogeher wih (3.1) and EJQTDE, 13 No. 3, p. 8

9 (H3) implies ha Tu() = = θ θ θ R = u, This indicaes ha G(, s)f(s, u(s))ds G(, s)f(s, u(s))ds + G(, s)f(s, u(s))ds G(, s)f(θ, θ R)ds θ G(, s)f(s, u(s))ds + G(, s)f(s, u(s))ds Tu u for u K Ω. (3.3) Therefore, i follows from Theorem 1.1, (3.) and (3.3) ha he operaor T has a fixed poin u K (Ω \ Ω 1 ), which is a desired posiive and decreasing soluion of he BVP (1.) wih r u R. Moreover, similar o he proof of Lemma., we can prove ha he obained soluion u() is concave on [, ]. Example 3. We consider he BVP u () = u () + 9(1 ), [, 1], 4 (3.4) u () = u(1) =, u ( 1) + u() =. Since α = 1 and = 1, if we choose θ = 1, hen a simple calculaion shows ha 3 θ = 1 3, A = 5 4 and B = Le f(, u) = u 4 + 9(1 ), (, u) [, 1] [, + ). Then (H1) and (H) are saisfied. Moreover, i is easy o verify ha and f(θ, θ 4 ) 1 4B, f(, 1) 1 A f(, 18) 18 A, f(θ, 556θ ) 556 B. EJQTDE, 13 No. 3, p. 9

10 Therefore, i follows from Theorem 3.1 ha he BVP (3.4) has posiive and decreasing soluions u 1 and u saisfying 1 4 u 1 1 < 18 u 556. References [1] D. Anderson, Green s funcion for a hird-order generalized righ focal problem, J. Mah. Anal. Appl. 88 (3), no. 1, [] Z. Bai, X. Fei, Exisence of riple posiive soluions for a hird order generalized righ focal problem, Mah. Inequal. Appl. 9 (6), no. 3, [3] M. Gregus, Third Order Linear Differenial Equaions, Reidel, Dordrech, [4] D. Guo, V. Lakshmikanham, Nonlinear Problems in Absrac Cones, Academic Press, New York, [5] L.-J. Guo, J.-P. Sun, Y.-H. Zhao, Exisence of posiive soluion for nonlinear hird-order hree-poin boundary value problem, Nonlinear Anal. 68 (8), no. 1, [6] G. Infane, J. R. L. Webb, Nonzero soluions of Hammersein inegral equaions wih disconinuous kernels, J. Mah. Anal. Appl. 7 (), no. 1, 3-4. [7] G. Infane, J. R. L. Webb, Three-poin boundary value problems wih soluions ha change sign, J. Inegral Equaions Appl. 15 (3), no. 1, [8] G. Infane, J. R. L. Webb, Loss of posiiviy in a nonlinear scalar hea equaion. NoDEA Nonlinear Differenial Equaions Appl. 13 (6), no., [9] M. Krasnoselskii, Posiive Soluions of Operaor Equaions, Noordhoff, Groningen, [1] R. H. Marin, Nonlinear Operaors & Differenial Equaions in Banach Spaces, Wiley, New York, EJQTDE, 13 No. 3, p. 1

11 [11] Alex P. Palamides, George Smyrlis, Posiive soluions o a singular hird-order hree-poin boundary value problem wih indefiniely signed Green s funcion, Nonlinear Anal. 68 (8), no. 7, [1] Alex P. Palamides, Nikolaos M. Savrakakis, Exisence and uniqueness of a posiive soluion for a hird-order hree-poin boundary-value problem, Elecron. J. Differenial Equaions 1 (1), no. 155, 1-1. [13] J.-P. Sun, J. Zhao, Posiive soluion for a hird-order hree-poin boundary value problem wih sign-changing Green s funcion, Commun. Appl. Anal. 16 (1), no., [14] J.-P. Sun, J. Zhao, Muliple posiive soluions for a hird-order hree-poin BVP wih sign-changing Green s funcion, Elecron. J. Differenial Equaions 1 (1), no. 118, 1-7. [15] Y. Sun, Posiive soluions of singular hird-order hree-poin boundary value problem, J. Mah. Anal. Appl. 36 (5), no., [16] Q. Yao, The exisence and mulipliciy of posiive soluions for a hird-order hree-poin boundary value problem, Aca Mah. Appl. Sin. Engl. Ser. 19 (3), no. 1, (Received February 3, 13) EJQTDE, 13 No. 3, p. 11