Positive solutions for a multi-point eigenvalue. problem involving the one dimensional

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1 Elecronic Journal of Qualiaive Theory of Differenial Equaions 29, No. 4, -3; h:// Posiive soluions for a muli-oin eigenvalue roblem involving he one dimensional -Lalacian Youyu Wang Weigao Ge 2 Sui Sun Cheng 3. Dearmen of Mahemaics, Tianjin Universiy of Finance and Economics, Tianjin 3222, P. R. China 2. Dearmen of Mahemaics, Beijing Insiue of Technology, Beijing 8, P. R. China 3. Dearmen of Mahemaics, Tsing Hua Universiy, Hsinchu, Taiwan 343. Absrac A muli-oin boundary value roblem involving he one dimensional -Lalacian and deending on a arameer is sudied in his aer and exisence of osiive soluions is esablished by means of a fixed oin heorem for oeraors defined on Banach saces wih cones. Keywords Posiive soluions; Boundary value roblems; One-dimensional -Lalacian. Inroducion In his aer we sudy he exisence of osiive soluions o he boundary value roblem BVP for he one-dimensional -Lalacian φ u + λqf, u =,,,. u = α i u, u = u,.2 where φ s = s 2 s, >,, wih < ξ < ξ 2 < < ξ < and α i,, f saisfy H α i, [, saisfy < H 2 f C[, ] [,, [, ; α i <, and < ; This work is sonsored by he Naional Naural Science Foundaion of China No Mahemaics Subjec Classificaion 34B, 34B5 wang youyu@63.com EJQTDE, 29 No. 4,.

2 H 3 q C, L [, ] wih q > on,. The number λ is regarded as a arameer and is o be deermined among osiive numbers. The sudy of mulioin boundary value roblems for linear second-order ordinary differenial equaions was iniiaed by Il in and Moiseev [,2]. Since hen here has been much curren aenion focused on he sudy of nonlinear mulioin boundary value roblems, see [3,4,5,7]. The mehods include he Leray-Schauder coninuaion heorem, nonlinear alernaives of Leray-Schauder, coincidence degree heory, and fixed oin heorem in cones. For examle, In [6], R. Ma and N. Casaneda sudied he following BVP x + afx =,, x = m 2 α i x, x = m 2 x, where < ξ < < ξ m 2 <, α i, wih < m α i < and m <. They showed he exisence of a leas one osiive soluion if f is eiher suerlinear or sublinear by alying he fixed oin heorem in cones. The auhors in [7] considered he muli-oin BVP for one dimensional -Lalacian φ u + f, u =,,, φ u = α i φ u, u = u. Using a fixed oin heorem in a cone, we rovided sufficien condiions for he exisence of mulile osiive soluions o he above BVP. In aer [8], we invesigaed he following more general muli-oin BVPs φ u + qf, u, u =,,, u = α i u, u = u, u = α i u, u = u. The main ool is a fixed oin heorem due o Avery and Peerson[9], we rovided sufficien condiions for he exisence of mulile osiive soluions. In view of he common concern abou muli-oin boundary value roblems as exhibied in [6,7,8] and heir references, i is of ineress o coninue he invesigaion and sudy he roblem. and.2. EJQTDE, 29 No. 4,. 2

3 Moivaed by he works of [7] and [8], he aim of his aer is o show he exisence of osiive soluions u o BVP. and.2. For his urose, we consider he Banach sace E = C[, ] wih he maximum norm x = max x. By a osiive soluion of. and.2, we means a funcion u C [, ], φ u C, L[, ] which is osiive on [, ] and saisfies he differenial equaion. and he boundary condiions.2. Our main resuls will deend on he following Guo-Krasnoselskii fixed-oin heorem. Theorem A[][]. Le E be a Banach sace and le K E be a cone in E. Assume Ω, Ω 2 are oen subses of E wih Ω, Ω Ω 2, and le T : K Ω 2 \ Ω K be a comleely coninuous oeraor such ha eiher i Tx x, x K Ω and Tx x, x K Ω 2, or ii Tx x, x K Ω and Tx x, x K Ω 2. Then T has a fixed oin in K Ω 2 \ Ω. In secion 3, we shall resen some sufficien condiions wih λ belonging o an oen inerval o ensure he exisence of osiive soluions o roblems. and.2. To he auhor s knowledge, no one has sudied he exisence of osiive soluions for roblems. and.2 using he Guo- Krasnoselskii fixed-oin heorem. 2. The reliminary lemmas Le C + [, ] = {ω C[, ] : ω, [, ]}. Lemma 2. Le H H 3 hold. Then for x C + [, ], he roblem φ u + λqf, x =,,, 2. has a unique soluion u = α i u, u = u, 2.2 u = B x A x λ qτfτ, xτdτ ds, 2.3 EJQTDE, 29 No. 4,. 3

4 where A x, B x saisfy R = A x = α i B x = A x λ ξi A x λ qsfs, xsds qτfτ, xτdτ ds., 2.4 Lemma 2.2 Suose, r i i =, 2,, and <, denoe r = min i r i, max r i, hen here exiss a unique i such ha Proof. Define x φ Then, Hx C, +, R. r, φ x = φ x + r i. Hx = φ x φ x + r i. R Le hen we have x = r, x = R, Hx = φ x φ x + r i φ x φ x + r = φ x φ [ = φ x φ x =, ] x + r and Hx = φ x φ x + r i φ x φ x + R = φ x φ [ = φ x φ x =. ] x + R EJQTDE, 29 No. 4,. 4

5 The zero oin heorem guaranees ha here exiss an x [x, x] such ha Hx =. If here exis wo consans x, x 2 [x, x] saisfying Hx = Hx 2 =, hen Case. x =. i.e., H =,, i =, 2,, n 2. Therefore, φ r i =. So, φ r i =, i =, 2,,n 2, hen, r i = Hx = φ x φ x + r i = φ x φ φ x + β ir i = φ x φ x = φ x. Obviously, here exiss a unique x = saisfying Hx =. So, x = x 2 =. Case 2. x. i If x,, hen Hx = φ x φ x + r i = φ x φ x φ x <. So, when, x,, Hx. ii If x, +, hen Hx = φ x φ x + r i [ = φ x φ + r ] i x = φ x Hx, where Hx = φ + r i x As Hx = and x, so here mus exis i {, 2,, n 2}, such ha φ r i >. Thus, we ge Hx is sricly increasing on, +. If Hx = Hx 2 =, hen Hx = Hx 2 =. So, x = x 2 since Hx is sricly increasing on, +.. EJQTDE, 29 No. 4,. 5

6 Therefore, Hx = has a unique soluion on, +. Combining case, case 2, we obain Hx = has a unique soluion on, + and x [x, x]. φ α i Lemma 2.3 Le k =, hen here exiss a unique real number A x ha φ α i saisfies 2.4. Furhermore, A x is conained in he inerval Proof The equaion is equivalen o A x = α i φ By Lemma 2.2, we can easily obain A x So he conclusion is obvious. [ λk ξ [, λk ξi [ λk ξ qsfs, xsds, λk ] qsfs, xsds. qsfs, xsds ] qsfs, xsds,. qsfs, xsds Lemma 2.4 Le H H 3 hold. If x C + [, ] and λ >, hen he unique soluion of roblem saisfies u for [, ]. Proof: According o Lemmas 2. and 2.3, we firs have. ] u = B x = =, β i β i A x λ λ qτfτ, xτdτ ds qτfτ, xτdτ ds qτfτ, xτdτ ds and u = B x = u + u +. A x λ λ qτfτ, xτdτ ds qτfτ, xτdτ ds qτfτ, xτdτ ds EJQTDE, 29 No. 4,. 6

7 If,, we have u = B x = u + u +. A x λ λ qτfτ, xτdτ ds qτfτ, xτdτ ds qτfτ, xτdτ ds So u, [, ]. Lemma 2.5 Le H H 3 hold. If x C + [, ] and λ >, hen he unique soluion of roblem saisfies where min u γ u, [,] γ = β. i Proof: Clearly u = A x λ qsfs, xsds. This imlies ha = qsfs, xsds u = u and min u = u. [,] Furhermore, i is easy o see ha u 2 u for any, 2 [, ] wih 2. Hence u is a decreasing funcion on [,]. This means ha he grah of u is concave down on,. For each i {, 2,, n 2} we have i.e., u u u u, u u u so ha u u and, by means of he boundary condiion u = u u, we have u u. This comlees he roof. EJQTDE, 29 No. 4,. 7

8 Now we define K = {ω ω C + [, ], min ω γ ω }, where γ is defined in Lemma 2.5. For any λ >, define oeraor T λ : C + [, ] K by T λ x = A x λ A x λ qτfτ, xτdτ ds qτfτ, xτdτ ds. 2.5 By Lemmas 2. and 2.3, we know T λ x is well defined. Furhermore, we have he following resul. Lemma 2.6[Lemma 2.4, 7] T : K K is comleely coninuous. Le f, x min f := lim inf min x [,] φ x, min f := lim inf min x + [,] f, x φ x, max f f, x := lim su max x + [,] φ x, maxf := lim su max x [,] f, x φ x. 3. Main resuls where We now give our resuls on he exisence of osiive soluions of BVP. and.2. Theorem 3. Le H H 3 hold and minf >, maxf <. If γ min f N < λ <. 3. k + maxf M M = β i β qsds, i N = β i φ qτdτ ds + hen he roblem..2 has a leas one osiive soluion. qτdτ ds, Proof. Define he oeraor T λ as 2.5. Under he condiion 3., here exiss an ε > such ha γ min f ε N λ. 3.2 k + maxf + ε M i. Since maxf <, here exiss an H > such ha for x : x H, f, x max f + εφ x. 3.3 EJQTDE, 29 No. 4,. 8

9 So, Le Ω = {x E : x < H }, hen for x K Ω, we have Therefore, qsfs, xsds λk T λ x = T λ x = λk + qsfs, xsds + λ qsfs, xsds λk + maxf + ε λk + maxf + εφ x qsfs, xsds qsφ xsds qsds. qsfs, xsds [λk + maxf + ε] q qsds x. + qτfτ, xτdτ ds β [λk + maxf + ε] q i = M[λk + max f + ε] q x qτfτ, xτdτ ds qsds x x. Thus T λ x x. ii. Nex, since min f >, here exiss an H 2 > such ha for x > H 2, f, x min f εφ x. 3.4 Take H 2 = max{h 2, 2H } and Ω 2 = {x E : x < H 2 }. Then for x K Ω 2, we have qτfs, xτdτ λ qτfτ, xτdτ λmin f ε qτφ xτdτ λφ γmin f εφ x qτdτ. So, qsfs, xsds [λmin f ε] q qτdτ γ x. EJQTDE, 29 No. 4,. 9

10 Therefore, T λ x = T λ x = + β i [λmin f ε] q γ x + [λmin f ε] q γ x = N [λmin f ε] q γ x qτfτ, xτdτ ds qτfτ, xτdτ ds qτdτ ds qτdτ ds x. So T λ x x. Therefore, by he firs ar of Theorem A, T λ has a fixed oin x K Ω 2 \ Ω such ha H x H 2. I is easily checked ha x is a osiive soluion of roblems. and.2. The roof is comlee. Theorem 3.2. Le H H 3 hold and min f >, maxf <. If γ min f N < λ <. 3.5 k + maxf M hen he roblem..2 has a leas one osiive soluion. Proof. Under he condiion 3.5, here exiss an ε > such ha γ min f ε N λ. 3.6 k + maxf + ε M i. Since min f >, here exiss an H > such ha for x : x H, f, x min f εφ x. 3.7 Le Ω = {x E : x < H }, hen for x K Ω, we have qτfs, xτdτ λ qτfτ, xτdτ λmin f ε qτφ xτdτ λφ γmin f εφ x qτdτ. EJQTDE, 29 No. 4,.

11 So, qsfs, xsds [λmin f ε] q qτdτ γ x. Therefore, T λ x = T λ x = + [λmin f ε] q γ x + [λmin f ε] q γ x = N [λmin f ε] q γ x qτfτ, xτdτ ds β i qτfτ, xτdτ ds qτdτ ds qτdτ ds x. So T λ x x. ii. Since maxf <, here exiss an H 2 > such ha for x > H 2, f, x maxf + εφ x. 3.8 There are wo cases : Case, f is bounded, and Case 2, f is unbounded. Case. Suose ha f is bounded, i.e., here exiss N > such ha f, x φ N for [, ] and x <. Define H 2 = max { 2H, β [λk + ] q i and Ω 2 = {x E : x < H 2 }. Then for x K Ω 2, we have qsfs, xsds λk λk + qsfs, xsds + λ λk + φ N qsfs, xsds qsds. } qsds N, qsfs, xsds So, qsfs, xsds [λk + ] q qsds N. EJQTDE, 29 No. 4,.

12 Therefore, T λ x = T λ x = β i + β [λk + ] q i H 2 = x. qτfτ, xτdτ ds qτfτ, xτdτ ds qsds N Case 2. We choose H 2 > max{2h, H 2 } such ha f, x f, H 2 for [, ] and < x < H 2. Le Ω 2 = {x E : x < H 2 }. Then for x K Ω 2, we have So, Therefore, qsfs, xsds λk λk + λk + qsfs, xsds + λ qsfs, xsds qsfs, H 2 ds λk + maxf + εφ H 2 qsfs, xsds qsds. qsfs, xsds [λk + maxf + ε] q qsds H 2. T λ x = T λ x = β i + qτfτ, xτdτ ds β [λk + max f + ε] q i qτfτ, xτdτ ds qsds H 2 = M [λk + maxf + ε] q H 2 H 2 = x. Therefore, by he second ar of Theorem A, T λ has a fixed oin x K Ω 2 \Ω such ha H x H 2. I is easily checked ha x is a osiive soluion of roblems. and.2. EJQTDE, 29 No. 4,. 2

13 REFERENCES. V. A. Il in, E. I. Moiseev, Nonlocal boundary value roblem of he firs kind for a Surm-Liouville oeraor in is differenial and finie difference asecs, Differenial Equaions, V. A. Il in, E. I. Moiseev, Nonlocal boundary value roblem of he second kind for a Surm-Liouville oeraor, Differenial Equaions, W. Feng, On an m-oin boundary value roblem, Nonlinear Anal W. Feng, J. R. L. Webb, Solvabiliy of m-oin boundary value roblem wih nonlinear growh, J. Mah. Anal. Al C. P. Gua, A generalized muli-oin boundary value roblem for second order ordinary differenial equaions, Al. Mah. Comu R. Ma and N. Casaneda, Exisence of soluions of nonlinear m-oin boundary-value roblems, J. Mah. Anal. Al Y. Wang and C. Hou, Exisence of mulile osiive soluions for one dimensional -Lalacian, J. Mah. Anal. Al Y. Wang and W. Ge, Mulile osiive soluions for muli-oin boundary value roblem wih one dimensional -Lalacian, J. Mah. Anal. Al R. I. Avery, A. C. Peerson, Three osiive fixed oins of nonlinear oeraors on ordered Banach saces, Comu. Mah. Al D. Guo and V. Lakshmikanham, Nonlinear Problems in Absrac Cones, Academic Press, New York, M.A. Krasnosel skii, Posiive Soluions of Oeraor Equaions, Noordhoff, Gronignen, 964. Received February 4, 29 EJQTDE, 29 No. 4,. 3