Existence of positive solutions for second order m-point boundary value problems

Transcription

1 ANNALES POLONICI MATHEMATICI LXXIX.3 (22 Exisence of posiive soluions for second order m-poin boundary value problems by Ruyun Ma (Lanzhou Absrac. Le α, β, γ, δ and ϱ := γβ + αγ + αδ >. Le ψ( = β + α, φ( = γ + δ γ, [, ]. We sudy he exisence of posiive soluions for he m-poin boundary value problem u + h(f(u =, < <, αu( βu ( = i= a iu(ξ i, γu( + δu ( = i= b iu(ξ i, where ξ i (,, a i, b i (, (for i {,..., m 2} are given consans saisfying ϱ i= a iφ(ξ i >, ϱ i= b iψ(ξ i > and i= := a iψ(ξ i ϱ i= a iφ(ξ i ϱ i= b iψ(ξ i i= b iφ(ξ i <. We show he exisence of posiive soluions if f is eiher superlinear or sublinear by a simple applicaion of a fixed poin heorem in cones. Our resul exends a resul esablished by Erbe and Wang for wo-poin BVPs and a resul esablished by he auhor for hree-poin BVPs.. Inroducion. The sudy of muli-poin boundary value problems for linear second order ordinary differenial equaions was iniiaed by Il in and Moiseev [6]. Moivaed by [6], Gupa [4] sudied cerain hree-poin boundary value problems for nonlinear ordinary differenial equaions. Since hen, more general nonlinear muli-poin boundary value problems have been sudied by several auhors. We refer he reader o [4 6, 8 ] for some relevan references. 2 Mahemaics Subjec Classificaion: 34B, 34B8. Key words and phrases: muli-poin boundary value problems, posiive soluions, fixed poin heorem, cones. Suppored by he NSFC and GG and NWNU-KJCXGC-22. [265]

2 266 R. Y. Ma In his paper, we are ineresed in he exisence of posiive soluions of he second order m-poin boundary value problem u + h(f(u =, < <, (. αu( βu ( = γu( + δu ( = i= i= a i u(ξ i, b i u(ξ i, where ξ i (,, a i, b i (, (for i {,..., m 2} are given consans. If a i = b i = for i =,..., m 2, hen he m-poin BVP (. reduces o he wo-poin BVP u + h(f(u =, < <, (.2 αu( βu ( =, γu( + δu ( =. In 994, Erbe and Wang [3] obained he following excellen resul for (.2. Theorem A ([3, Theorem ]. Suppose ha (A f C([,, [, ; (A2 h C([, ], [, and h( on no subinerval of (, ; (A3 α, β, γ, δ, and ϱ := γβ + αγ + αδ >. Then (.2 has a leas one posiive soluion if eiher (i f = and f =, or (ii f = and f =, where f(u f := lim u + u, f f(u := lim u u. This resul has been exended and developed by many auhors (see Erbe, Hu and Wang [2] and Lian, Wong and Yeh [7] for some references. If α = γ =, β = δ =, a i = for i =,..., m 2, and b j = for j = 2,..., m 2, hen (. reduces o he hree-poin BVP { u + h(f(u =, < <, (.3 u( =, u( = bu(ξ. In 998, Ma [8] obained he following resul for (.3. Theorem B ([8, Theorem ]. Suppose ha (H < bξ < ; (H2 f C([,, [, ; (H3 h C([, ], [, and here exiss [ξ, ] such ha h( >.

3 m-poin boundary value problems 267 Then (.3 has a leas one posiive soluion if eiher (i f = and f =, or (ii f = and f =. Theorem B has been exended by Webb []. We remark ha in he proof of Theorem B, we rewrie (.3 as he equivalen inegral equaion (.4 u( = ( sh(sf(u(s ds b ξ (ξ sh(sf(u(s ds bξ + ( sh(sf(u(s ds bξ which conains one posiive erm and wo negaive erms and is no convenien for sudying he exisence of posiive soluions. In his paper, we consider he more general m-poin BVP (.. To deal wih (., we give a new inegral equaion which is equivalen o (. and only conains wo posiive erms. Our main resul (see Theorem 3. below exends and unifies he main resuls of [2, 3, 7, 8]. By a posiive soluion of (. we undersand a funcion u( which is posiive on (, and saisfies he differenial equaion and he boundary condiions in (.. The main ool of his paper is he following well-known Guo Krasnosel - skiĭ fixed poin heorem. Theorem C (see [3]. Le E be a Banach space, and le K E be a cone. Assume Ω, Ω 2 are open bounded subses of E wih Ω, Ω Ω 2, and le A : K (Ω 2 \ Ω K be a compleely coninuous operaor such ha eiher (i Au u, u K Ω, and Au u, u K Ω 2 ; or (ii Au u, u K Ω, and Au u, u K Ω 2. Then A has a fixed poin in K (Ω 2 \ Ω. 2. The preliminary lemmas. Se (2. ψ( := β + α, φ( := γ + δ γ, [, ], and := a i ψ(ξ i i= ϱ i= b i ψ(ξ i ϱ a i φ(ξ i i=. b i φ(ξ i i=

4 268 R. Y. Ma Lemma 2.. Le (A3 hold. Assume (H4. Then for y C[, ], he problem u + y( =, < <, (2.2 has a unique soluion (2.3 u( = where (2.4 (2.5 (2.6 αu( βu ( = γu( + δu ( = i= i= a i u(ξ i, b i u(ξ i G(, sy(sds + A(yψ( + B(yφ( G(, s := { φ(ψ(s, < s < <, ϱ φ(sψ(, < < s <, A(y := a i G(ξ i, sy(s ds ϱ a i φ(ξ i i= i= b i G(ξ i, sy(s ds b i φ(ξ i B(y := i= a i ψ(ξ i i= ϱ i= b i ψ(ξ i a i i= b i i= i= G(ξ i, sy(s ds. G(ξ i, sy(s ds Proof. Since ψ and φ are wo linearly independen soluions of he equaion u =, we know ha any soluion of u ( = y( can be represened as (2.7 u( = G(, sy(s ds + Aψ( + Bφ( where G is as in (2.4. I is easy o check ha he funcion defined by (2.7 is a soluion of (2.2 if A and B are defined by (2.5 and (2.6, respecively. Now we show ha he funcion defined by (2.7 is a soluion of (2.2 only if A and B are as in (2.5 and (2.6, respecively.

5 m-poin boundary value problems 269 Le u as in (2.7 be a soluion of (2.2. Then u( = u ( = φ ( u ( = φ ( so ha ϱ ψ(sφ(y(s ds + φ(sψ(y(s ds + Aψ( + Bφ(, ϱ + ψ ( ϱ ψ(sy(s ds + ψ ( ϱ ψ(sy(s ds + φ ( ϱ ψ(y( ϱ φ(sy(s ds + Aψ ( + Bφ (, ϱ φ(sy(s ds ψ ( ϱ φ(y( + Aψ ( + Bφ (, (2.8 u ( = ϱ [ψ(φ ( φ(ψ (]y( = y(. Since we have (2.9 B(γα + δα + γβ = Since we have (2. A(γα + δα + γβ = From (2.9 and (2., we ge [ ] a i ψ(ξ i A + i= i= u( = β φ(sy(s ds + Aβ + B(γ + δ, ϱ u ( = α φ(sy(s ds + Aα + B( γ, ϱ i= a i [ ] G(ξ i, sy(s ds + Aψ(ξ i + Bφ(ξ i. u( = δ ψ(sy(s ds + A(α + β + Bδ, ϱ u ( = γ ψ(sy(s ds + Aα + B( γ, ϱ i= b i [ [ ϱ i= i= ] G(ξ i, sy(s ds + Aψ(ξ i + Bφ(ξ i. ] a i φ(ξ i B = [ ] [ ] ϱ b i ψ(ξ i A b i φ(ξ i B = i= b i i= a i G(ξ i, sy(s ds, G(ξ i, sy(s ds,

6 27 R. Y. Ma which implies A and B saisfy (2.5 and (2.6, respecively. This complees he proof of he lemma. In he following, we will make he following assumpion: (H5 <, ϱ a i φ(ξ i >, i= ϱ b i ψ(ξ i >. I is easy o see ha if α = γ =, β = δ =, a i = for i =,..., m 2, b > and b j = for j = 2,..., m 2, hen (H5 reduces o < b ξ <, which is a key condiion in [8, Theorem ]. Lemma 2.2. Le (A3 and (H5 hold. Then for y C[, ] wih y, he unique soluion u of he problem (2.2 saisfies u(, [, ]. Proof. This is an immediae consequence of he facs ha G on [, ] [, ] and A(y, B(y. We noe ha if (H5 does no hold, hen y C[, ] wih y does no imply ha he unique soluion u of (2.2 is posiive. We can see his from he following resul: Lemma 2.3 ([8, Lemma 3]. Le bξ >. If y C[, ] and y, hen (.3 has no posiive soluion. Lemma 2.4. Le (A3 and (H5 hold. Le σ (, /2 be a consan. Then for y C[, ] wih y, he unique soluion u of he problem (2.2 saisfies i= min{u( [σ, σ]} Γ u where u = max{u( [, ]} and (2. Γ := min{φ( σ/φ(, ψ(σ/ψ(}. Proof. We see from (2.4 and (2.3 ha which implies (2.2 u( G(, s G(s, s, [, ], G(s, sy(s ds + A(yψ( + B(yφ(, [, ].

7 m-poin boundary value problems 27 Applying (2.4, we find ha for [σ, σ], { G(, s φ(/φ(s, s, (2.3 G(s, s = ψ(/ψ(s, s, { φ( σ/φ(, s σ, ψ(σ/ψ(, σ s, Γ, where Γ is an in (2.. Thus for [σ, σ], u( = Γ Γ G(, s G(s, sy(s ds + A(yψ( + B(yφ( G(s, s [ G(s, sy(s ds + A(yψ( + B(yφ( ] G(s, sy(s ds + A(yψ( + B(yφ( Γ u. 3. The main resul. The main resul of he paper is he following Theorem 3.. Le (H2, (A3 and (H5 hold. Assume ha (H6 h C([, ], [, and here exiss [, ] such ha h( >. Then (. has a leas one posiive soluion if eiher (i f = and f =, or (ii f = and f =. Remark 3.2. Condiion (H6 is weaker han (H3. Remark 3.3. Theorem 3. exends [3, Theorem ] and [8, Theorem ]. Proof of Theorem 3.. Since h C[, ], we may assume ha (, in (H6. Take σ (, /2 > such ha (σ, σ and le Γ be defined by (2.. Superlinear case. Suppose hen ha f = and f =. We wish o show he exisence of a posiive soluion of (.. Now (. has a soluion u = u( if and only if u solves he operaor equaion (3. u( = G(, sh(sf(u(s ds + A(h( f(u( ψ( + B(h( f(u( φ( := (T u( where φ and ψ, G, A and B are defined by (2., (2.4, (2.5 and (2.6,

8 272 R. Y. Ma respecively. Clearly (3.2 A(h( f(u( and i= i= := Ã f(u (3.3 B(h( f(u( Define := B f(u. a i b i G(ξ i, sh(s ds G(ξ i, sh(s ds a i ψ(ξ i i= ϱ i= b i ψ(ξ i a i i= b i i= ϱ a i φ(ξ i i= f(u b i φ(ξ i i= G(ξ i, sh(s ds f(u G(ξ i, sh(s ds (3.4 K = {u C[, ] u, min{u( [σ, σ]} Γ u }. I is obvious ha K is a cone in C[, ]. Moreover, by Lemmas 2.2 and 2.4, T K K. I is also easy o check ha T : K K is compleely coninuous. Now since f =, we may choose H > so ha f(u εu for < u < H, where ε > saisfies ( (3.5 ε G(s, sh(s ds + Ã ψ + B φ. Thus, if u K and u = H, hen from (3. (3.5 and he fac ha G(, s G(s, s and ψ( ψ(, we have (3.6 T u( = G(, sh(sf(u(s ds + A(h( f(u( ψ( + B(h( f(u( φ( ( G(s, sh(s ds + Ã ψ + B φ f(u ( ε G(s, sh(s ds + Ã ψ + B φ u u.

9 m-poin boundary value problems 273 Now if we le (3.7 Ω = {u C[, ] u < H }, hen (3.6 shows ha T u u for u K Ω. Furher, since f =, here exiss Ĥ 2 > such ha f(u ϱ u for u Ĥ2, where ϱ > is chosen so ha σ (3.8 ϱ γ σ G(, sh(s ds. Le H 2 = max{2h, Ĥ2/Γ } and Ω 2 = {u C[, ] u < H 2 }. Then u K and u = H 2 implies and so (3.9 T u( = min u( Γ u Ĥ2, σ σ G(, sh(sf(u(s ds + A(h( f(u( ψ( + B(h( f(u( φ( ϱ Γ G(, sh(sf(u(s ds σ σ G(, sh(s ds u. σ σ G(, sh(sϱ u(s ds Hence, T u u for u K Ω 2. Therefore, by he firs par of Theorem C, T has a fixed poin u in K (Ω 2 \ Ω such ha H u H 2. This complees he superlinear par of he heorem. Sublinear case. Suppose nex ha f = and f =. We firs choose H 3 > such ha f(y My for < y < H 3, where (3. MΓ σ σ G(, sh(s ds. By using he mehod o ge (3.9, we obain (3. T u( G(, sh(sf(u(s ds σ σ G(, sh(smu(s ds MΓ G(, sh(s ds u. σ σ

10 274 R. Y. Ma Thus, if we le Ω 3 = {u C[, ] u < H 3 } hen T u u, u K Ω 3. Now, since f =, here exiss Ĥ4 > so ha f(u λu for u Ĥ4, where λ > saisfies ( (3.2 λ G(s, sh(s ds + à ψ + B φ. We consider wo cases: Case. Suppose f is bounded, say f(y N for all y [,. In his case choose { ( } H 4 = max 2H 3, N G(s, sh(s ds + à ψ + B φ so ha for u K wih u = H 4 we have T u( = ( G(, sh(sf(u(s ds + A(h( f(u( ψ( + B(h( f(u( φ( G(s, sh(s ds + à ψ + B φ N H 4 and herefore T u u. Case 2. If f is unbounded, hen we know from (A ha here exiss H 4 > max{2h 3, Ĥ4/Γ } such ha f(y f(h 4 for < y H 4. Then for u K and u = H 4, we have T u( = ( G(, sh(sf(u(s ds + A(h( f(u( ψ( + B(h( f(u( φ( G(s, sh(s ds + ( λ G(s, sh(s ds + Therefore, in eiher case we may pu à ψ + B ψ f(u à ψ + B ψ u H 4. Ω 4 = {u C[, ] u < H 4 }, and for u K Ω 4 we have T u u. By he second par of Theorem C, i follows ha BVP (. has a posiive soluion. Thus, we have compleed he proof of Theorem 3..

11 m-poin boundary value problems 275 Remark 3.4. Erbe, Hu and Wang [2] and Lian, Wong and Yeh [7] sudied he exisence of muliple posiive soluions of he wo-poin boundary value problem u + g(, u =, < <, αu( βu ( =, γu( + δu ( =. I is easy o see from he proof of Theorem 3. ha we can apply Lemmas 2.2 and 2.4 o esablish he corresponding mulipliciy resuls under condiion (H5 for he m-poin boundary value problem u + g(, u =, < <, αu( βu ( = γu( + δu ( = i= i= a i u(ξ i, b i u(ξ i, and exend he mulipliciy resuls of [2, 7] wihou any difficulies. References [] H. Dang and K. Schmi, Exisence of posiive soluions for semilinear ellipic equaions in annular domains, Differenial Inegral Equaions 7 (994, [2] L. H. Erbe, S. C. Hu and H. Y. Wang, Muliple posiive soluions of some wo-poin boundary value problems, J. Mah. Anal. Appl. 23 (994, [3] L. H. Erbe and H. Y. Wang, On he exisence of posiive soluions of ordinary differenial equaions, Proc. Amer. Mah. Soc. 2 (994, [4] C. P. Gupa, Solvabiliy of a hree-poin nonlinear boundary value problem for a second order ordinary differenial equaion, J. Mah. Anal. Appl. 68 (992, [5], A generalized muli-poin boundary value problem for second order ordinary differenial equaions, Appl. Mah. Compu. 89 (998, [6] V. A. Il in and E. I. Moiseev, Nonlocal boundary value problem of he firs kind for a Surm Liouville operaor in is differenial and finie difference aspecs, Differ. Equ. 23 (987, [7] W. C. Lian, F. H. Wong and C. C. Yeh, On he exisence of posiive soluions of nonlinear second order differenial equaions, Proc. Amer. Mah. Soc. 24 (996, [8] R. Y. Ma, Posiive soluions of nonlinear hree-poin boundary-value problems, Elecron. J. Differenial Equaions 34 (999, 8. [9] R. Y. Ma and N. Casaneda, Exisence of soluions of nonlinear m-poin boundaryvalue problems, J. Mah. Anal. Appl. 256 (2,

12 276 R. Y. Ma [] J. R. L. Webb, Posiive soluions of some hree-poin boundary value problems via fixed poin heory, Nonlinear Anal. 47 (2, Deparmen of Mahemaics Norhwes Normal Universiy Lanzhou 737, Gansu, People s Republic of China mary@nwnu.edu.cn Reçu par la Rédacion le Révisé le (32