Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 609143, 11 pages doi:10.1155/2009/609143 Research Article Existence of Solutions for m-point Boundary Value Problems on a Half-Line Changlong Yu, Yanping Guo, and Yude Ji College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, China Correspondence should be addressed to Changlong Yu, changlongyu@126.com Received April 2009; Accepted July 2009 Recommended by Paul Eloe By using the Leray-Schauder continuation theorem, we establish the existence of solutions for f t, x t , x t 0, < t < ∞, x m-point boundary value problems on a half-line x t m−2 0, where αi ∈ R, m−2 αi / and < η1 < η2 < · · · < ηm−2 < ∞ are i αi x ηi , limt → ∞ x t i given Copyright q 2009 Changlong Yu et al 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 Introduction Multipoint boundary value problems BVPs for second-order differential equations in a finite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions are obtained by use of the Leray-Schauder continuation theorem, Guo-Krasnosel’skii fixed point theorem, and so on; for details see 1–4 and the references therein In the last several years, boundary value problems in an infinite interval have been arisen in many applications and received much attention; see 5, Due to the fact that an infinite interval is noncompact, the discussion about BVPs on the half-line is more complicated, see 5–14 and the references therein Recently, in 15 , Lian and Ge studied the following three-point boundary value problem: x t f t, x t , x t 0, < t < ∞, 1.1 x αx η , lim x t t→ ∞ 0, Advances in Difference Equations where α ∈ R, α / 1, and η ∈ 0, ∞ are given In this paper, we will study the following m-point boundary value problems: x t f t, x t , x t 0, < t < ∞, 1.2 m−2 αi x ηi , x lim x t 0, t→ ∞ i where αi ∈ R, m−2 αi / 1, αi have the same signal, and < η1 < η2 < · · · < ηm−2 < ∞ are i given We first present the Green function for second-order multipoint BVPs on the half-line and then give the existence results for 1.2 using the properties of this Green function and the Leray-Schauder continuation theorem {x ∈ C1 0, ∞ , limt → ∞ x t exists, limt → ∞ x t exists} We use the space C∞ 0, ∞ with the norm x max{ x ∞ , x ∞ }, where · ∞ is supremum norm on the half-line, and {x : 0, ∞ → R is absolutely integrable on 0, ∞ } with the norm x L1 L1 0, ∞ ∞ |x t |dt We set ∞ P ∞ p s ds, P1 and we suppose αi , i m−2 α i αi ∞ sp s ds, Q q s dt, 1.3 1, 2, , m − are the same signal in this paper and we always assume Preliminary Results In this section, we present some definitions and lemmas, which will be needed in the proof of the main results e Definition 2.1 see 15 It holds that f : 0, ∞ × R2 −→ R is called an S-Carath´ odory function if and only if i for each u, v ∈ R2 , t → f t, u, v is measurable on 0, ∞ , ii for almost every t ∈ 0, ∞ , u, v → f t, u, v is continuous on R2 , iii for each r > 0, there exists ϕr t ∈ L1 0, ∞ with tϕr t ∈ L1 0, ∞ , ϕr t > on 0, ∞ such that max{|u|, |v|} ≤ r implies |f t, u, v | ≤ ϕr t , for a.e t ∈ 0, ∞ Lemma 2.2 Suppose BVP, m−2 i αi / 1, if for any v t ∈ L1 0, ∞ with tv t ∈ L1 0, ∞ , then the x t v t 0, < t < ∞, 2.1 m−2 αi x ηi , x i lim x t t→ ∞ 0, Advances in Difference Equations has a unique solution Moreover, this unique solution can be expressed in the form ∞ x t G t, s v s ds, 2.2 where G t, s is defined by ⎧m−2 ⎪ ⎪ α s Λs, ⎪ ⎪ i ⎪ ⎪i ⎪ ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ α s Λt, ⎪ i ⎪ ⎪ ⎪i ⎪ ⎪ ⎪ ⎪ ⎪ i m−2 ⎪ ⎪ ⎪ ⎪ αk ηk αk s ⎪ 1⎨ k G t, s s ≤ η1 , s ≤ t, s ≤ η1 , t ≤ s, Λs, < ηi ≤ s ≤ ηi , s ≤ t, i 1, 2, , m − 3, k i m−2 Λ⎪ i ⎪ ⎪ ⎪ αk ηk αk s ⎪ ⎪ ⎪k ⎪ k i ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ α η Λs, ⎪ ⎪ i i ⎪ ⎪i ⎪ ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ α η Λt, ⎪ i i ⎩ 2.3 Λt, < ηi ≤ s ≤ ηi , t ≤ s, i 1, 2, , m − 3, s ≥ ηm−2 , s ≤ t, s ≥ ηm−2 , t ≤ s, i here note Λ m−2 i 1− αi Proof Integrate the differential equation from t to ∞, noticing that v t , tv t ∈ L1 0, ∞ , then from to t and one has ∞ t x t x m−2 i Since x x t s αi x ηi , from 2.4 , it holds that ∞ m−2 m−2 i 1− 2.4 v τ dτ ds αi αi ηi αi ηi i ∞ ηi m−2 v s ds sv s ds t t i t v s ds sv s ds 2.5 For ≤ t ≤ η1 , the unique solution of 2.1 can be stated by t x t m−2 i αi s − m−2 αi i m−3 ηi i ηi i k η1 s v s ds t αk ηk 1− m−2 k i αk s m−2 i αi m−2 i αi s − m−2 αi i Λt t v s ds ∞ v s ds ηm−2 m−2 i αi ηi − m−2 αi i t v s ds 2.6 Advances in Difference Equations If ηi ≤ t ≤ ηi , ≤ i ≤ m − 3, the unique solution of 2.1 can be stated by η1 x t m−2 i αi s − m−2 αi i i−1 ηj j ⎛ s v s ds j k ⎝ i k t 1− 1− i k 1 m−3 ηj ⎛ 1− ⎝ j k ηm−2 Λs m−2 k j αk ηk m−2 i 1− j i ηj ∞ αi m−2 k i αk s m−2 i αi αk ηk t m−2 i αk s m−2 k i αk s m−2 i αi αk ηk ηi ηi m−2 k j αk ηk ηj m−2 i αi ηi − m−2 αi i Λt αk s αi Λs ⎞ ⎠v s ds v s ds 2.7 v s ds Λt ⎞ ⎠v s ds t v s ds If ηm−2 ≤ t < ∞, the unique solution of 2.1 can be stated by η1 x t m−2 i αi s − m−2 αi i t ηm−2 We note Λ G t, s 1− m−3 ηi m−2 i αi ηi − m−2 αi i m−2 i i αk ηk 1− ηi ∞ s v s ds t m−2 i αi ηi − m−2 αi i m−2 k i αk s m−2 i αi Λs v s ds 2.8 t v s ds αi , then ⎧m−2 ⎪ ⎪ αi s Λs, ⎪ ⎪ ⎪ ⎪i ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ αi s Λt, ⎪ ⎪ ⎪ ⎪i ⎪ ⎪ ⎪ i ⎪ m−2 ⎪ ⎪ ⎪ αk ηk ⎪ αk s ⎪ ⎪ k i 1 ⎨k m−2 Λ⎪ i ⎪ ⎪ α η ⎪ αk s k k ⎪ ⎪ ⎪k ⎪ k i ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ αi ηi Λs, ⎪ ⎪ ⎪i ⎪ ⎪ ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ αi ηi Λt, ⎩ i i k 1 s v s ds s ≤ η1 , s ≤ t, s ≤ η1 , t ≤ s, Λs, < ηi ≤ s ≤ ηi , s ≤ t, i 1, 2, , m − 3, Λt, < ηi ≤ s ≤ ηi , t ≤ s, i 1, 2, , m − 3, 2.9 s ≥ ηm−2 , s ≤ t, s ≥ ηm−2 , t ≤ s Advances in Difference Equations ∞ Therefore, the unique solution of 2.1 is x t proof G t, s v s ds, which completes the Remark of Lemma 2.2.Obviously G t, s satisfies the properties of a Green function, so we call G t, s the Green function of the corresponding homogeneous multipoint BVP of 2.1 on the half-line Lemma 2.3 For all t, s ∈ 0, ∞ , it holds that ⎧ ⎪ ⎪s, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪s ⎨ |G t, s | ≤ Λ , ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪max ⎪ ⎪ ⎩ ⎩ m−2 αi < 0, i m−2 i αi s −Λ 0≤ ⎫ m−2 αi ηm−2 ⎬ i , αi < 1, i 2.10 m−2 , ⎭ −Λ m−2 αi > i Proof For each s ∈ 0, ∞ , G t, s is nondecreasing in t Immediately, we have m−2 i αi s Λ , i k m−2 k i αk ηk Λ ⎧ ⎪s, ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎨ αk s , m−2 i αi ηi ≤ G t, s ≤ G s, s Λ s ≤ η1 , m−2 αk ηk k Λ ⎪k ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ α η Λs, ⎪ i i ⎩ Λ s, ηi ≤ s ≤ ηi αk < ∞, i 1, 2, , m − 3, 2.11 i s ≥ ηm−2 i Further, we have m−2 i Λ < min m−2 i αi s Λ m−2 i Therefore, we get the result Λ αi s , , αi s ≤ G t, s ≤ s, m−2 αi < 0, i m−2 i αi η1 Λ m−2 i αi ηm−2 Λ ≤ G t, s ≤ s , Λ ≤ G t, s ≤ s, m−2 0≤ αi < 1, i m−2 αi > i 2.12 Advances in Difference Equations Lemma 2.4 For the Green function G t, s , it holds that lim G t, s t→ ∞ G s ⎧ ⎪s, ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ m−2 ⎪ ⎪ αk ⎨ αk ηk k i ⎪k Λ⎪ ⎪ ⎪m−2 ⎪ ⎪ ⎪ ⎪ αi ηi Λs, ⎪ ⎩ s ≤ η1 , Λ s, ηi ≤ s ≤ ηi < ∞, i 1, 2, , m − 2.13 s ≥ ηm−2 i Lemma 2.5 For the function x ∈ C1 0, ∞ , it is satisfied that m−2 x αi x ηi 2.14 i and αi i 1, 2, , m − have the same signal, < η1 < η2 < · · · < ηm−2 < ∞, then there exists η ∈ η1 , ηm−2 satisfying x where α m−2 i αx η , 2.15 αi Proof Let αi i 1, 2, , m − are positive, and note M∗ max{x t | t ∈ η1 , ηm−2 }, m∗ min{x t | t ∈ η1 , ηm−2 }, then for every i i 1, 2, , m − , we have m∗ ≤ x ηi ≤ M∗ , so m∗ m−2 αi ≤ m−2 αi x ηi ≤ M∗ m−2 αi , that is, m∗ ≤ m−2 αi x ηi / m−2 αi x ≤ M∗ i i i i i Because x t is continuous on the interval η1 , ηm−2 , there exists η ∈ η1 , ηm−2 satisfying m−2 x αx η , where α i αi Theorem 2.6 see Let M ⊂ C∞ 0, ∞ {x ∈ C 0, ∞ , limt → relatively compact in X if the following conditions hold: ∞x t exists} Then M is a M is uniformly bounded in C∞ 0, ∞ ; b the functions from M are equicontinuous on any compact interval of 0, ∞ ; c the functions from M are equiconvergent, that is, for any given T > such that |f t − f ∞ | < , for any t > T, f ∈ M > 0, there exists a T Main Results Consider the space X {x ∈ C∞ 0, ∞ , x the operator T : X × 0, → X by m−2 i αi x ηi , limt → ∞ T x, λ t λ G t, s f s, x s , x s ds, The main result of this paper is following ∞ x t ≤ t < ∞ 0} and define 3.1 Advances in Difference Equations e Theorem 3.1 Let f : 0, ∞ × R2 → R be an S-Carath´ odory function Suppose further that there exists functions p t , q t r t ∈ L1 0, ∞ with t , tq t tr t ∈ L1 0, ∞ such that ≤ p t |u| f t, u, v q t |v| r t 3.2 for almost every t ∈ 0, ∞ and all u, v ∈ R2 Then 1.2 has at least one solution provided: ηm−2 P αηm−2 P 1−α max αηm−2 P α−1 P1 Q < 1, P1 P1 Q < 1, Q, αP1 α−1 α < 0, ≤ α < 1, αηm−2 P α−1 < 1, 3.3 α > Lemma 3.2 Let f : 0, ∞ × R2 → R be an S-Carath´ odory function Then, for each λ ∈ e 0, , T x, λ is completely continuous in X Proof First we show T is well defined Let x ∈ X; then there exists r > such that x ≤ r For each λ ∈ 0, , it holds that ∞ T x, λ t G t, s f s, x s , x s ds λ ∞ ≤ 3.4 |G t, s |ϕr s ds < ∞, ∀t ∈ 0, ∞ Further, G t, s is continuous in t so the Lebesgue dominated convergence theorem implies that ∞ |T x, λ t1 − T x, λ t2 | ≤ λ |G t1 , s − G t2 , s | f s, x s , x s ds ∞ ≤λ |G t1 , s − G t2 , s |ϕr s ds 3.5 −→ 0, T x, λ t1 − T x, λ t2 as t1 −→ t2 , t2 ≤λ f s, x s , x s ds t1 ≤ t2 ϕr s ds −→ t1 where ≤ t1 , t2 < ∞ Thus, T x ∈ C1 0, ∞ 3.6 as t1 −→ t2 , Advances in Difference Equations m−2 i Obviously, T x, λ αi T x, λ ηi Notice that ∞ lim T x, λ t→ ∞ t f s, x s , x s ds lim t→ ∞ t 3.7 0, so we can get T x, λ t ∈ X We claim that T x, λ is completely continuous in X, that is, for each λ ∈ 0, , T x, λ is continuous in X and maps a bounded subset of X into a relatively compact set Let xn → x as n → ∞ in X Next we prove that for each λ ∈ 0, , T xn , λ → T x, λ as n → ∞ in X Because f is a S-Carath´ odory function and e ∞ G s f s, xn s , xn s − f s, x s , x s ∞ ds ≤ G s ϕr0 s ds < ∞, 3.8 where r0 > is a real number such that max{maxn∈N\{0} xn , x } ≤ r0 , we have |T xn , λ ∞ − T x, λ ∞ ∞ |≤λ G s −→ 0, f s, xn s , xn s − f s, x s , x s ds 3.9 as n −→ ∞ Also, we can get |T xn , λ t − T xn , λ ∞ ∞ |≤λ G t, s − G s f s, xn s , xn s ∞ ≤ G t, s − G s ϕr0 s ds ds 3.10 −→ 0, T xn , λ t − T xn , λ ∞ as t −→ ∞, ∞ ≤ f s, xn s , xn s t ∞ ≤ ds 3.11 ϕr0 s ds −→ 0, as t −→ ∞ t Similarly, we have |T x, λ t − T x, λ T x, λ t − T x, λ ∞ | −→ 0, ∞ −→ 0, as t −→ ∞, as t −→ ∞ 3.12 Advances in Difference Equations For any positive number T0 < ∞, when t ∈ 0, T0 , we have ∞ |T xn , λ t − T x, λ t | ≤ −→ 0, T xn , λ t − T x, λ as n −→ ∞, t − f s, x s , x s f s, xn s , xn s −→ 0, ds 3.13 ∞ ≤ t − f s, x s , x s |G t, s | f s, xn s , xn s ds as n −→ ∞ Combining 3.9 – 3.13 , we can see that T ·, λ is continuous Let B ⊂ X be a bounded subset; it is easy to prove that T B is uniformly bounded In the same way, we can prove 3.5 , 3.6 , and 3.12 , we can also show that T B is equicontinuous and equiconvergent Thus, by Theorem 2.6, T ·, λ : X× 0, → X is completely continuous The proof is completed Proof of Theorem 3.1 In view of Lemma 2.2, it is clear that x ∈ X is a solution of the BVP 1.2 if and only if x is a fixed point of T ·, Clearly, T x, 0 for each x ∈ X If for each λ ∈ 0, the fixed points T ·, λ in X belong to a closed ball of X independent of λ, then the LeraySchauder continuation theorem completes the proof We have known T ·, λ is completely continuous by Lemma 3.2 Next we show that the fixed point of T ·, λ has a priori bound M independently of λ Assume x T x, λ and set ∞ ∞ sq s ds, Q1 R ∞ r s ds, R1 sr s dt 3.14 According to Lemma 2.5, we know that for any x ∈ X, there exists η ∈ η1 , ηm−2 satisfying x αx η Hence, there are three cases as follow Case α < For any x ∈ X, x x η ≤ holds and, therefore, there exists a t0 ∈ 0, η such Then, we have that x t0 |x t | t x s ds ≤ t η x ∞ t0 ≤ t x ηm−2 ∞ , t ∈ 0, ∞ , 3.15 and so it holds that ∞ ≤ λf t, x, x L1 ≤ p t |x t | x q t |x t | ≤ ηm−2 P P1 ≤ f t, x, x Q x ∞ r t L1 L1 3.16 R, therefore, x ∞ ≤ R − ηm−2 P − P1 − Q M1 3.17 10 Advances in Difference Equations At the same time, we have ∞ |x t | ≤ λ G t, s f s, x s , x s ds ≤ ∞ 3.18 ds sf s, x s , x s ≤ P1 x Q1 M1 ∞ R1 , t ∈ 0, ∞ , and so x Set M ∞ ≤ Q1 M1 R1 − P1 3.19 M1 max{M1 , M1 }, which is independent of λ Case ≤ α < For any x ∈ X, we have t |x t | αx η x s ds ≤ α x η t x which implies that |x t | ≤ αη/ − α t x In the same way as for Case 1, we can get x ∞ ≤ ∞ t ∈ 0, ∞ , , ≤ αηm−2 / − α 1−α R − α − P1 − Q − αηm−2 P x Set M ∞ ∞ Q1 M2 R1 ≤ − α − P1 t x ∞ 3.20 for all t ∈ 0, ∞ M2 , 3.21 M2 max{M2 , M2 }, which is independent of λ and is what we need Case α > For x ∈ X, we have |x t | t x η x s ds ≤ η and so |x t | ≤ αη/ α − t x Similarly, we obtain x x Set M ∞ ∞ ≤ ≤ ∞ |x | α ≤ αηm−2 / α − t−η x t x ∞ ∞ α−1 R α − 1 − P1 − Q − αηm−2 P α Q1 M3 R1 αηm−2 QM3 α − − αηm−2 P , t ∈ 0, ∞ , 3.22 for all t ∈ 0, ∞ M3 , R 3.23 M3 max{M3 , M3 } and which is we need So 1.2 has at least one solution Advances in Difference Equations 11 Acknowledgment The Natural Science Foundation of Hebei Province A2009000664 and the Foundation of Hebei University of Science 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