Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses

Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 834158, 12 pages doi:10.1155/2009/834158 Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses Jianli Li1and Juan J. Nieto2 1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China 2Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Correspondence should be addressed to Jianli Li, [email protected] Received 7 March 2009; Revised 18 April 2009; Accepted 25 April 2009 Recommended by Donal O’Regan We consider a multi-point boundary value problem on the half-line with impulses. By using a fixed-point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained. Copyright q2009 J. Li and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Impulsive differential equations are a basic tool to study evolution processes that are subjected to abrupt changes in their state. For instance, many biological, physical, and engineering applications exhibit impulsive effects see 1–3. It should be noted that recent progress in the development of the qualitative theory of impulsive differential equations has been stimulated primarily by a number of interesting applied problems 4–24. In this paper, we consider the existence of multiple positive solutions of the following impulsive boundary value problem for short BVPon a half-line: utqtft, u0,0<t<∞,t / tk, ΔutkIkutk,k1,...,p, u0 m−2  i1 αiuξi,u ∞0, 1.1 2 Boundary Value Problems where u∞limt→∞ut,0 <ξ 1<ξ 2<··· <ξ m−2<∞,0<t 1<t 2<··· <t p<∞, Δutkut k−ut− k,andαi,f,q,andIksatisfy H10<m−2 i1αi<1; H2ft, u∈C0,∞×0,∞,0,∞,Iku∈C0,∞,0,∞, and when u/1t is bounded, ft, uand Ikuare bounded on 0,∞; H3qt∈C0,∞,0,∞ and qtis not identically zero on any compact subinterval of 0,∞. Furthermore qtsatisfies sup t∈0,∞∞ 0 Gt, sqsds < ∞,1.2 where Gt, s⎧ ⎨ ⎩ s, 0≤s≤t<∞, t, 0≤t≤s<∞. 1.3 Boundary value problems on the half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and there are many results in this area, see 8,13,14,20,25–27, for example. Lian et al. 25studied the following boundary value problem of second-order differential equation with a p-Laplacian operator on a half-line: ϕputφtft, u, u0,0<t<∞, αu0−βu00,u ∞0. 1.4 They showed the existence at least three positive solutions for 1.4by using a fixed point theorem in a cone due to Avery-Peterson 28. Yan 20, by using Leray-Schauder theorem and fixed point index theory presents some results on the existence for the boundary value problems on the half-line with impulses and infinite delay. However to the best knowledge of the authors, there is no paper concerned with the existence of three positive solutions to multipoint boundary value problems of impulsive differential equation on infinite interval so far. Motivated by 20,25, in this paper, we aim to investigate the existence of triple positive solutions for BVP 1.1. The method chosen in this paper is a fixed point technique due to Avery and Peterson 28. 2. Preliminaries In this section, we give some definitions and results that we will use in the rest of the paper. Definition 2.1. Suppose Pis a cone in a Banach. The map αis a nonnegative continuous concave functional on Pprovided α:P→0,∞is continuous and αtx 1−ty≥tαx1−tαy2.1 Boundary Value Problems 3 for all x, y ∈P,andt∈0,1. Similarly, the map βis a nonnegative continuous convex functional on Pprovided β:P→0,∞is continuous and βtx 1−ty≤tβx1−tβy2.2 for all x, y ∈P,andt∈0,1. Let γ, θ be nonnegative, continuous, convex functionals on Pand αbe a nonnegative, continuous, concave functionals on P,andψbe a nonnegative continuous functionals on P. Then, for positive real numbers a, b, c,andd, we define the convex sets Pγ,dx∈P:γx<d , Pγ,α,b,dx∈P:b≤αx,γ x≤d, Pγ,θ,α,b,c,dx∈P:b≤αx,θ x≤c, γx≤d, 2.3 and the closed set Rγ,ψ,a,dx∈P:a≤ψx,γ x≤d.2.4 To prove our main results, we need the following fixed point theorem due to Avery and Peterson in 28. Theorem 2.2. Let Pbe a cone in a real Banach space E.Letγand θbe nonnegative continuous convex functionals on a cone P,αbe a nonnegative continuous concave functional on P, and ψbe a nonnegative continuous functional on Psatisfying ψλx≤λψxfor 0≤λ≤1, such that for some positive numbers Mand d αx≤ψx,x≤Mγx2.5 for all x∈Pγ,d. Suppose Φ:Pγ,d−→ Pγ,d2.6 is completely continuous and there exist positive numbers a, d, and cwith a<bsuch that i{x∈Pγ,θ,α,b,c,d:αx>b}/ ∅and αΦx>bfor x∈Pγ,θ,α,b,c,d; iiαΦx>bfor x∈Pγ,α,b,dwith θΦx>c; iii0/ ∈Rγ,ψ,a,dand ψTx<afor x∈Rγ,ψ,a,d,withψΦxa. Then Φhas at least three fixed points x1,x 2,x 3∈Pγ,dsuch that γxi≤d, for i1,2,3,ψ x1<a, a<ψ x2with αx2<b, α x3>b. 2.7 4 Boundary Value Problems 3. Some Lemmas Define PC0,∞{u:0,∞→R|utis continuous at each t/ tk, left continuous at ttk,ut kexists, k1,...,p}. By a solution of 1.1we mean a function uin PC0,∞satisfying the relations in 1.1. Lemma 3.1. utis a solution of 1.1if and only if utis a solution of the following equation: ut∞ 0 Gt, sqsfs, usds  0<tk<t Iku m−2 i1αi 1−m−2 i1αi⎡ ⎣∞ 0 Gξi,s qsfs, usds  0<tk<ξi Iku⎤ ⎦ :Tut, 3.1 where Gt, sis defined as 1.3. The proof is similar to Lemma 3 in 9, and here we omit it. For tp<a ∗<b ∗<∞,letc∗min{a∗/1a∗,1/1b∗}. Then Gt, s 1t≥c∗Gr, s 1r,1 1t≥c∗ 1r,for t∈a∗,b∗,r∈0,∞,s∈0,∞.3.2 It is clear that 0 <c ∗<1. Consider the space Edefined by Eu∈PC0,∞:sup t∈0,∞ |ut| 1t<∞.3.3 Eis a Banach space, equipped with the norm usup0≤t<∞|ut|/1t <∞. Define the cone P⊂Eby Pu∈E:ut≥0,t∈0,∞,min t∈a∗,b∗ ut 1t≥c∗u.3.4 Lemma 3.2 see 20, Theorem 2.2.Let M⊂PC0,∞.ThenMis compact in PC0,∞,if the following conditions hold: aMis bounded in PC0,∞; bthe functions belonging to Mare piecewise equicontinuous on any interval of 0,∞; cthe functions from Mare equiconvergent, that is, given ε>0, there corresponds τε>0 such that |ft−f∞|<εfor any t≥τεand f∈M. Lemma 3.3. T:P→Pis completely continuous. Boundary Value Problems 5 Proof. Firstly, for u∈P,fromH1–H3, it is easy to check that Tu is well defined, and Tut≥0 for all t∈0,∞. For t∈a∗,b∗ 1 1tTut1 1t∞ 0 Gt, sqsfs, usds 1 1t p  k1 Iku 1 1tm−2 i1αi 1−m−2 i1αi⎡ ⎣∞ 0 Gξi,s qsfs, usds  0<tk<ξi Iku⎤ ⎦ ≥c∗∞ 0 Gr, s 1rqsfs, usds 1 1r p  k1 Iku c∗m−2 i1αi 1−m−2 i1αi⎡ ⎣∞ 0 Gξi,s  1rqsfs, usds 1 1r 0<tk<ξi Iku⎤ ⎦ ≥c∗Tur 1r,for r∈0,∞ 3.5 so min t∈a∗,b∗ Tut 1t≥c∗Tu,3.6 which shows TP ⊆P. Now we prove that Tis continuous and compact, respectively. Let un→uas n→∞ in P. Then there exists r0such that supn∈N\{0}un<r 0.ByH2we have ft, uis bounded on 0,∞×0,r 0.SetB0sup{ft, u:t, u/1t ∈0,∞×0,r 0}, and we have ∞ 0 Gt, s 1tqsfs, un−fs, uds ≤2B0∞ 0 Gt, s 1tqsds. 3.7 Therefore by the Lebesgue dominated convergence theorem and continuity of fand Ik,one arrives at Tun−Tu ≤sup t∈0,∞ 1 1t∞ 0 Gt, sqsfs, un−fs, uds  0<tk<t |Ikun−Iku|m−2 i1αi 1−m−2 i1αi ×⎡ ⎣∞ 0 Gξi,s qsfs, un−fs, uds  0<tk<ξi |Ikun−Iku|⎤ ⎦⎫ ⎬ ⎭ −→ 0asn−→ ∞. 3.8 Therefore Tis continuous. 6 Boundary Value Problems Let Ωbe any bounded subset of P. Then there exists r>0 such that u≤rfor all u∈Ω.SetB1sup{ft, u:t, u/1t ∈0,∞×0,r,B2ksup{Iku:u/1t∈0,r}, then Tusup t∈0,∞ 1 1t∞ 0 Gt, sqsfs, uds  0<tk<t |Iku| m−2 i1αi 1−m−2 i1αi⎡ ⎣∞ 0 Gξi,s qsfs, uds  0<tk<ξi |Iku|⎤ ⎦⎫ ⎬ ⎭ ≤B1∞ 0 Gt, sqsds m−2 i1αi 1−m−2 i1αi∞ 0 Gξi,s qsds 1m−2 i1αi 1−m−2 i1αip  k1 B2k. 3.9 So TΩis bounded. Moreover, for any ν∈0,∞and t,t  ∈tk,t k1⊂0,νt<t ,andu∈Ω, then  Tut 1t −Tut 1t ≤m−2 i1αi 1−m−2 i1αi⎡ ⎣B1∞ 0 Gξi,s qsds  0<tk<ξi B2k⎤ ⎦ 1 1t −1 1t B1∞ 0 Gt,s  1t −Gt,s  1tqsds  0<tk<t B2k 1 1t −1 1t −→ 0,uniformly as t−→ t. 3.10 So TΩis quasi-equicontinuous on any compact interval of 0,∞. Finally, we prove for any ε, there exists sufficiently large N1>0 such that  Tut 1t −Tut 1t<ε, ∀t,t  ≥N1,u∈Ω.3.11 Since ∞ 0Gt, sqsds < ∞, we can choose N1>0 such that m−2 i1αi N11−m−2 i1αi⎡ ⎣B1∞ 0 Gξi,s qsds  0<tk<ξi B2k⎤ ⎦<ε 6, B1∞ 0Gt, sqsds N1 <ε 6, p  k1 B2k N1 ≤ε 6. 3.12 Boundary Value Problems 7 For t,t  ≥N1, it follows that  Tut 1t−Tut 1t  ≤m−2 i1αi 1−m−2 i1αi⎡ ⎣B1∞ 0 Gξi,s qsds  0<tk<ξi B2k⎤ ⎦1 1t 1 1t B1∞ 0 Gt,s  1tqsds B1∞ 0 Gt,s  1t qsds  p  k1 B2k1 1t 1 1t <ε 3ε 6ε 6ε 3ε. 3.13 That is 3.11holds. By Lemma 3.2,TΩis relatively compact. In sum, T:P→Pis completely continuous. 4. Existence of Three Positive Solutions Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals γand θ, and the nonnegative continuous functionals ψbe defined on the cone P by γuψuθusup t∈0,∞ ut 1t,α umin t∈a∗,b∗ ut 1t.4.1 For notational convenience, we denote by Mmin t∈a∗,b∗∞ 0 Gt, s 1tqsds, M1m−2 i1αi 1−m−2 i1αi∞ 0 Gξi,s qsds. 4.2 The main result of this paper is the following. Theorem 4.1. Assume H1–H3hold. Let ak≥0,0<a<b/c ∗<cd,b/M < c∗d/2MM1 and suppose that f,Iksatisfy the following conditions: A1ft, u≤c∗d/2MM1,I ku≤dc∗/2M2for t, u/1t∈0,∞×0,d, A2ft, u>b/Mfor t, u/1t ∈a∗,b∗×b,c, A3ft, u<c ∗a/2MM1,I ku≤ac∗ak/2M2for t∈t, u/1t ∈0,∞×0,a, 8 Boundary Value Problems where M2p k1ak/1−m−2 i1αi.Then1.1has at least three positive solutions u1,u 2and u3 such that γui≤d, for i1,2,3,ψ u1<a, a<ψ u2with αu2<b,α u3>b. 4.3 Proof. Step 1. From the definition α, ψ,andγ, we easily show that αu≤ψu,u≤γufor u∈Pγ,d.4.4 Next we will show that T:Pγ,d−→ Pγ,d.4.5 In fact, for u∈Pγ,d, then sup t∈0,∞ ut 1t≤d. 4.6 From condition A1,weobtain ft, u≤dc∗ 2MM1,I ku≤dc∗ 2M2 .4.7 It follows that γTusup t∈0,∞ Tut 1t≤1 c∗min t∈a∗,b∗ Tut 1t ≤1 c∗min t∈a∗,b∗1 1t∞ 0 Gt, sqsfs, uds 1 1t p  k1 Iku 1 1t·m−2 i1αi 1−m−2 i1αi⎛ ⎝∞ 0 Gξi,s qsfs, uds  0<tk<ξi Iku⎞ ⎠⎤ ⎦ ≤1 c∗·c∗d 2MM1min t∈a∗,b∗∞ 0 Gt, s 1tqsds m−2 i1αi 1−m−2 i1αi∞ 0 Gξi,s qsds 1 c∗·c∗dp k1ak 2M21m−2 i1αi 1−m−2 i1αi ≤d 2d 2d. 4.8 Thus 4.5holds. Boundary Value Problems 9 Step 2. We show that condition iin Theorem 2.2 holds. Taking ut1tbd/2, then u∈Pγ,θ,α,b,c,dand αu>b, which shows {u∈Pγ,θ,α,b,c,d|αu>b}/ ∅.Thusfor u∈Pγ,θ,α,b,c,d, there is b≤ut 1t≤for t∈a∗,b∗.4.9 Hence by A2, we have αTumin t∈a∗,b∗ Tut 1t ≥min t∈a∗,b∗∞ 0 Gt, s 1tqsfs, uds >b M·min t∈a∗,b∗∞ 0 Gt, s 1tqsds b. 4.10 Therefore we have αTu>b, ∀u∈Pγ,θ,α,b,c,d.4.11 This shows the condition iin Theorem 2.2 is satisfied. Step 3. We now prove iiin Theorem 2.2 holds. For u∈Pγ,α,b,dwith θTu>c, we have αTumin t∈a∗,b∗ Tut 1t≥c∗Tuc∗θTu>c ∗c>b. 4.12 Hence, condition iiin Theorem 2.2 is satisfied. Step 4. Finally, we prove iiiin Theorem 2.2 is satisfied. Since ψ00<a,so0/ ∈Rγ,ψ,a,d. Suppose that u∈Rγ,θ,a,dwith ψua, then 0≤ut 1t≤a, 4.13 by the condition A3of this theorem, ψTusup t∈0,∞ Tut 1t≤1 c∗min t∈a∗,b∗ Tut 1t ≤1 c∗·c∗a 2MM1min t∈a∗,b∗∞ 0 Gt, s 1tqsds m−2 i1αi 1−m−2 i1αi∞ 0 Gξi,s qsds 1 c∗·c∗ap k1ak 2M21m−2 i1αi 1−m−2 i1αi ≤a 2a 2a. 4.14