Research Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point Boundary Value Problems

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1 Hindawi Publishing Corporation Boundary Value Problems Volume 9, Article ID 575, 8 pages doi:.55/9/575 Research Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point Boundary Value Problems Yude Ji and Yanping Guo College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 58, China Correspondence should be addressed to Yude Ji, jiyude-98@63.com Received 5 July 9; Revised 3 August 9; Accepted 3 October 9 Recommended by Kanishka Perera We consider the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem u n t a t f u t, t,, u αu η, u u n, u βu η, wheren,α,β, <η<,α β α η n <, a t L p, for some p and has countably many singularities in, /. The associated Green s function for the nth-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity f which yield the existence of countably many positive solutions by using the Krasnosel skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone. Copyright q 9 Y. Ji and Y. Guo. 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 The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see and the references therein. However, there are a few papers dealing with the existence of positive solutions for the nth-order multipoint boundary value problems with infinitely many singularities. Hao et al. 3 discussed the existence and multiplicity of positive solutions for the following nth-order nonlinear singular boundary value problems: u n t a t f t, u, t,, u, u u n, u αu ( η ),. where < η <, < αη n <, a t may be singular at t and/ort. Hao et al. established the existence of at least two positive solution for the boundary value problems

2 Boundary Value Problems if f is either superlinear or sublinear by applying the Krasnosel skii-guo theorem on cone expansion and compression. In 4, Kaufmann and Kosmatov showed that there exist countably many positive solutions for the two-point boundary value problems with infinitely many singularities of following form: u t a t f u t, u, u, <t<,. where a t L p, for some p and has countably many singularities in, /. In 5, Ji and Guo proved the existence of countably many positive solutions for the nth-order ordinary differential equation u n t a t f u t, t,,.3 with one of the following m-point boundary conditions: m u k i u ξ i, u u n, u, i m u, u u n, u k i u ξ i, i.4 where n, k i > i,,...,m, <ξ <ξ < <ξ m <, f C,,,, a t L p, for some p and has countably many singularities in, /. Motivated by the result of 3 5, in this paper we are interested in the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem u n t a t f u t, t,, u αu ( η ), u u n, u βu ( η ),.5 where n, α, β, <η<, α β α η n <, f C,,,, a t L p, for some p and has countably many singularities in, /. We show that the problem.5 has countably many solutions if a and f satisfy some suitable conditions. Our approach is based on the Krasnosel skii fixed point theorem and Leggett-Williams fixed point theorem in cones. Suppose that the following conditions are satisfied. H There exists a sequence {t k } k such that t k <t k k N, t < /, lim k t k t, and lim t tk a t for all k,,... H There exists m>such that a t m for all t t, t. Assuming that a t satisfies the conditions H - H we cite 5, Example 6. to verify existence of a t and imposing growth conditions on the nonlinearity f, it will be shown that problem.5 has infinitely many solutions.

3 Boundary Value Problems 3 The paper is organized as follows. In Section, we provide some necessary background material such as the Krasnosel skii fixed-point theorem and Leggett-Williams fixed point theorem in cones. In Section 3, the associated Green s function for the nth-order threepoint boundary value problem is first given and we also look at some properties of the Green s function associated with problem.5. InSection 4, we prove the existence of countably many positive solutions for problem.5 under suitable conditions on a and f. In Section 5, we give two simple examples to illustrate the applications of obtained results.. Preliminary Results Definition.. Let E be a Banach space over R. A nonempty convex closed set P E is said to be a cone provided that i au P for all u P and for all a ; ii u, u P implies u. Definition.. The map α : P, is said to be a nonnegative continuous concave functional on P provided that α is continuous and α ( tx t y ) tα x t α ( y ),. for all x, y P and t. Similarly, we say that the map γ : P, is a nonnegative continuous convex functional on P provided that γ is continuous and γ ( tx t y ) tγ x t γ ( y ),. for all x, y P and t. Definition.3. Let < a < b be given and let α be a nonnegative continuous concave functional on P. Define the convex sets P r and P α, a, b by P r {x P x <r}, P α, a, b {x P a α x, x b}..3 The following Krasnosel skii fixed point theorem and Leggett-Williams fixed point theorem play an important role in this paper. Theorem.4 6, Krasnosel skii fixed point theorem. Let E be a Banach space and let P E be a cone. Assume that Ω, Ω are bounded open subsets of E such that Ω Ω Ω. Suppose that T : P ( Ω \ Ω ) P.4

4 4 Boundary Value Problems is a completely continuous operator such that, either i Tu u,u P Ω, and Tu u,u P Ω,or ii Tu u,u P Ω, and Tu u,u P Ω. Then T has a fixed point in P Ω \ Ω. Theorem.5 7, Leggett-Williams fixed point theorem. Let A : P c P c be a completely continuous operator and let α be a nonnegative continuous concave functional on P such that α x x for all x P c. Suppose there exist <a<b<d csuch that C {x P α, b, d α x >b} /, and α Ax >b for x P α, b, d, C Ax <a for x a, C 3 α Ax >b for x P α, b, c,with Ax >d. Then A has at least three fixed points x,x, and x 3 such that x <a, b<α x, x 3 >a with α x 3 <b..5 In order to establish some of the norm inequalities in Theorems.4 and.5 we will need Holder s inequality. We use standard notation of L p a, b for the space of measurable functions such that f s p ds<,.6 where the integral is understood in the Lebesgue sense. The norm on L p a, b,, is defined by ( /p f p f s ds) p..7 Theorem.6 8, Holder s inequality. Let f L p a, b and g L q a, b, wherep> and /p /q. Thenfg L a, b and, moreover f s g s ds f p g q..8 Let f L a, b and g L a, b.thenfg L a, b and f s g s ds f g..9

5 Boundary Value Problems 5 3. Preliminary Lemmas To prove the main results, we need the following lemmas. Lemma 3. see 5. For y t C,, the boundary value problem u n t y t, t,, u, u u n, u 3. has a unique solution t t s n u t y s ds tn n! s n n! y s ds. 3. Lemma 3. see 5. The Green s function for the boundary value problem u n t, t,, u, u u n, u 3.3 is given by t n s n t s n, s t, g t, s n! t n s n, t s. 3.4 Lemma 3.3 see 5. The Green s function g t, s defined by 3.4 satisfies that i g t, s is continuous on,, ; ii g t, s g θ s,s for all t, s, and there exists a constant γ τ, / such that > for any τ min g t, s γ τg θ s,s γ τ g ( t,s ), t,s,, 3.5 t τ, τ where { ( ) τ n } τ γ τ min,, θ s θ s θ s s s n / n s<θ s <. 3.6

6 6 Boundary Value Problems Lemma 3.4. Suppose α β α η n /, then for y t C,, the boundary value problem u n t y t, t,, u αu ( η ), u u n, u βu ( η ) 3.7 has a unique solution t t s n u t n! y s ds α α ( β α ) η n α t n α ( β α ) η n η ( ) n η s αη n y s ds n! α ( β α ) η n s n n! y s ds ( β α ) t n α ( β α ) η n η s n n! y s ds ( ) n η s y s ds. n! 3.8 Proof. The general solution of u n t y t can be written as t t s n u t n! y s ds n Atn A i t i B. i 3.9 Since u i fori,,...,n, we get A i fori,,...,n. Now we solve for A, B by u αu η and u βu η, it follows that B α η ( ) n η s y s ds αaη n αb n! s n η ( ) n η s y s ds A B β y s ds βaη n βb. n! n! 3. By solving the above equations, we get ( A α ( β α ) s n α η n n! y s ds ( β α ) η ( ) n ) η s y s ds, n! ( η ( ) n η s ) B α ( β α ) α y s ds αη n s n η n n! n! y s ds. 3.

7 Boundary Value Problems 7 Therefore, 3.7 has a unique solution t t s n u t n! y s ds α α ( β α ) η n α t n α ( β α ) η n η ( ) n η s αη n y s ds n! α ( β α ) η n s n n! y s ds ( β α ) t n α ( β α ) η n η s n n! y s ds ( ) n η s y s ds. n! 3. Lemma 3.5. Suppose <α β α η n <, the Green s function for the boundary value problem u n t y t, t,, u αu ( η ), u u n, u βu ( η ) 3.3 is given by G t, s g t, s ( β α ) t n α α ( β α ) η n g( η, s ), 3.4 where g t, s is defined by 3.4. We omit the proof as it is immediate from Lemma 3.4 and 3.4. Lemma 3.6. Suppose <α β α η n <, the Green s function G t, s defined by 3.4 satisfies that i G t, s is continuous on,, ; ii G t, s J s for all t, s, and there exists a constant γ τ > for any τ, / such that min G t, s γ τj s γ τ G ( t,s ), t,s,, 3.5 t τ, τ where J s g θ s,s γ τ min τ n, max { α, β } α ( β α ) η n g( η, s ), min{ (β α ) τ n, ( β α ) τ n } α max { α, β }

8 8 Boundary Value Problems { ( ) (β ) τ n τ min α τ n, ( β α ) τ n } α min, θ s θ s, max { α, β } min γ τ, (β ) min{ α τ n, ( β α ) τ n } α max { α, β }. 3.6 Proof. i From Lemma 3.3 and 3.4,weget G t, s is continuous on,,. 3.7 ii From Lemma 3.3 and 3.4, we have G t, s g t, s g θ s,s ( β α ) t n α α ( β α ) η n g( η, s ) max { α, β } α ( β α ) η n g( η, s ) J s. 3.8 Next, we prove that 3.5 holds. From Lemma 3.3 and 3.4,fort τ, τ, we have G t, s g t, s γ τ g θ s,s ( β α ) t n α α ( β α ) η n g( η, s ) min{ (β α ) τ n, ( β α ) τ n } α α ( β α ) η n g ( η, s ) (β ) min{ α τ n, ( β α ) τ n } α { } γ τ g θ s,s max { α, β } max α, β α ( β α ) η g( η, s ) n max γ τ (g θ { α, β } s,s α ( β α ) η g( η, s ) ) n γ τ J s γ τ G ( t,s ), 3.9 for all t,, where γ τ min{τ n, min{ β α τ n, β α τ n } α / max{α, β}}, τ, /.

9 Boundary Value Problems 9 We use inequality 3.5 to define our cones. Let E C,, then E is a Banach space with the norm u max t, u t. For a fixed τ, /, define the cone P E by P { } u E u t on,, and min u t γ τ u. 3. t τ, τ Define the operator T by Tu t G t, s a s f u s ds, t. 3. Obviously, u t is a solution of.5 if and only if u t is a fixed point of operator T. Theorems.4 and.5 require the operator T to be completely continuous and cone preserving. If T is continuous and compact, then it is completely continuous. The next lemma shows that T : P P for τ, / and that T is continuous and compact. Lemma 3.7. The operator T is completely continuous and T : P P for each τ, /. Proof. Fix τ, /. Since a s f u s for all s,, u P and since G t, s for all t, s,, then Tu t for all t,,u P. Let u P, by 3.5 and 3. we have min u t min t τ, τ t τ, τ min t τ, τ G t, s a s f u s ds G t, s a s f u s ds 3. γ τ G ( t,s ) a s f u s ds γ τ Tu ( t ), for all t,. Thus min t τ, τ u t γ τ Tu. 3.3 Clearly operator 3. is continuous. By the Arzela-Ascoli theorem T is compact. Hence, the operator T is completely continuous and the proof is complete. 4. Main Results In this section we present that problem.5 has countably many solutions if a and f satisfy some suitable conditions.

10 Boundary Value Problems For convenience, we denote Λ τ max t, τ G t, s ds m, Λ. J q a p 4. Theorem 4.. Suppose conditions H and H hold, let {τ k } k be such that t k <τ k <t k,k,,...let {R k } k and {r k} k be such that R k <γ τk r k <r k <R k, Mr k <LR k, k,,..., 4. where M Λ,, L, Λ. Furthermore, for each natural number k, assume that f satisfies the following two growth conditions: H 3 f u LR k for all u,r k, H 4 f u Mr k for all u γ τk r k,r k. Then problem.5 has countably many positive solutions {u k } k such that r k u k R k for each k,,... Proof. Consider the sequences {Ω,k } k and {Ω,k} k of open subsets of E defined by Ω,k {u E u <R k }, Ω,k {u E u <r k }. 4.3 Let {τ k } k be as in the hypothesis and note that t <t k <τ k <t k < /, for all k N. For each k N, define the cone P k by P k { } u E u t on,, and min u t γ τ k u. 4.4 t τ k, τ k Fixed k and let u P k Ω,k. For s τ k, τ k, we have γ τk r k γ τk u min u s u s u r k. s τ k, τ k 4.5

11 Boundary Value Problems By condition H 4,weget Tu max t, max t, τk τ k G t, s a s f u s ds G t, s a s f u s ds max t, τk τ k G t, s a s ds Mr k 4.6 mmr k max t, r k u. τ τ G t, s ds get Now let u P k Ω,k, then u s u R k for all s,. By condition H 3,we Tu max t, G t, s a s f u s ds J s a s ds LR k 4.7 J q a p LR k R k u. It is obvious that Ω,k Ω,k Ω,k. Therefore, by Theorem.4, the operator T has at least one fixed point u k P k Ω,k \ Ω,k such that r k u k R k. Since k N was arbitrary, Theorem 4. is completed. Let τ k is defined by Theorem 4.. We define the nonnegative continuous concave functionals α k u on P by α k u min u t. t τ k, τ k 4.8 We observe here that, for each u P, α u u. For convenience, we denote τk Λ J q a p, Γ k min t τ k, τ k τ k G t, s ds m. 4.9

12 Boundary Value Problems Theorem 4.. Suppose conditions H and H hold, let {τ k } k be such that t k <τ k <t k,k,,...let {a k } k, {b k} k, and {c k} k be such that { c k <a k <b k min γ τk, M } c k <c k, k,,..., 4. L where L Λ,, M, Γ k. Furthermore, for each natural number k, assume that f satisfies the following growth conditions: H 5 f u c k /L for all u,c k, H 6 f u <a k /L for all u,a k, H 7 f u b k /M for all u b k,b k /γ τk. Then problem.5 has three infinite families of solutions {u k } k, {u k} k, and {u 3k} k such that u k <a k, min u k t >b k, u 3k >a k, with min u 3k t <b k, t τ k, τ k t τ k, τ k 4. for each k,,... Proof. We note first that T : P ck P ck is completely continuous operator. If u P, then from properties of G t, s, Tu t, and by Lemma 3.7, min t τk, τ k Tu t γ τk Tu. Consequently, T : P P. If u P ck, then u c k, and by condition H 5, we have Tu max Tu t t, max G t, s a s f u s ds t, c k J s a s ds L 4. c k L J q a p c k. Therefore, T : P ck P ck. Standard applications of Arzela-Ascoli theorem imply that T is completely continuous operator. In a completely analogous argument, condition H 6 implies that condition C of Theorem.5 is satisfied. We now show that condition C of Theorem.5 is satisfied. Clearly, { ( u P α k,b k, b ) } k α k u >b k /. 4.3 γ τk

13 Boundary Value Problems 3 If u P α k,b k,b k /γ τk, then b k u s b k /γ τk,fors τ k, τ k. By condition H 7,weget α k Tu min t τ k, τ k G t, s a s f u s ds min t τ k, τ k τk τ k G t, s a s f u s ds 4.4 m bk M min t τ k, τ k τk τ k G t, s ds b k. Therefore, condition C of Theorem.5 is satisfied. Finally, we show that condition C 3 of Theorem.5 is also satisfied. If u P α k,b k,c k and Tu >b k /γ τk, then α k Tu min Tu t γ τ k Tu >b k. t τ k, τ k 4.5 Therefore, condition C 3 is also satisfied. By Theorem.5, There exist three infinite families of solutions {u k } k, {u k} k,and{u 3k} k for problem.5 such that u k <a k, min u k t >b k, u 3k >a k, with min u 3k t <b k, t τ k, τ k t τ k, τ k 4.6 for each k,,...thus, Theorem 4. is completed. 5. Example In this section, we cite an example see 5 to verify existence of a t, and two simple examples are presented to illustrate the applications for obtained conclusion of Theorems 4. and 4.. Example 5.. As an example of problem.5, we mention the boundary value problem u 3 t a t f u t, u u ( t,, ), u, u u ( ), 5.

14 4 Boundary Value Problems where a t is defined by 5, Example 6. and ε /4, f u 3 4k / 4 k ( u 8 k ) /65 8k 4 8 k 4 k, 8 4k u /5 4k sin π 4k /5 4k 3 4k, 3 4k / 4k 8k 4 8k ( u 8k) 4k, u u [ ] 4 k, 5 4k, [ ] 5 4k, 4k, u [ 4k, 4k] k,,..., 4, u [ 4, ). 5. We notice that n 3, α /, β, η /. If we take t 5/6, t k t k i / i 4, τ k / t k t k, k,,..., then t k <τ k <t k,and/5 <t <τ k <τ /4 / 3 4 < /4,γ τk min{τ k, min{ β α τ k, β α τ k } α / max{α, β}} > /5, k,,... It follows from a direct calculation that τ τ G t, s ds > /4 /4 3/4 G t, s ds s ds /4g t, 4 ( t ) 3/4 ( ) g 3 /4,s ds { t 4 3 /4 [ t s t s ] ds ( t )[ / /4 ( 96t 3 t 8t 5 ), 576 3/4 t t s ds ( ( ) ) 4 s s ds 3/4 / ]} 4 s ds 5.3

15 Boundary Value Problems 5 so τ max t, τ G t, s ds max t /4, /4 ( ) / J J s ds 5 6, /4 /4 3 7 G t, s ds > 3, a ( π 3 9 ) In addition, if we take r k 4k, R k 4k, M 3, L /, m 4/3 /4, then a t ( ) 4 /4 m, t t, t, 3 R k 4 k < 5 4k <γ τk r k <r k 4k <R k 4k, Mr k 3 4k <LR k 4k, k,,..., Λ τ max t, τ G t, s ds m < 3 M, /4 /3 4/3 Λ J a π >L, 5/6 /3 9/4 5.5 and f u satisfies the following growth conditions: f u LR k 4k, f u Mr k 3 4k, u [, 4k], u [ ] 5 4k, 4k. 5.6 Then all the conditions of Theorem 4. are satisfied. Therefore, by Theorem 4. we know that problem 5. has countably many positive solutions {u k } k such that 4k u k 4k for each k,,... Example 5.. As another example of problem.5, we mention the boundary value problem u 3 t a t f u t, t,, u ( ) ( ) u, u, u u, 5.7

16 6 Boundary Value Problems where a t is defined by 5, Example 6. and ε /4, f u u u [ 4 k, 4k 3 ], / 4k k ( u 8k 4 ) 8k 6 8k k, u [ 4k 3, 4k ], 5 4k u 4k sin π 5 4k 4k 45 4k, u [ 4k, 5 4k ], 45 4k / 4k 65 8k 4 8k ( u 8k) 4k, u [ 5 4k, 4k], k,,..., 4, u [ 4, ). 5.8 We notice that n 3, α /, β, η /. If we take t 5/6, t k t k i / i 4, τ k / t k t k, k,,..., then t k <τ k <t k,and/5 <t <τ k <τ /4 / 3 4 < /4, γ τk min{τ k, min{ β α τ k, β α τ k } α / max{α, β}} > /5, k,,... It follows from a direct calculation that Λ J a 5 ( π ), 4 τk min G t, s ds min t τ k, τ k τ k t /5, /5 /4 /4 G t, s ds > In addition, if we take a k 4k 3, b k 4k, c k 4k, M /45, L, m 4/3 /4, then ( ) 4 /4 a t m, t t, t, 3 c k 4 k <a k 4k 3 <b k 4k < { 9 4k min γ τk, M } c k <c k 4k, L

17 Boundary Value Problems 7 M 45 < 45 ( ) 4 /4 τk < min 3 t τ k, τ k Λ J a 5 ( π ) < L, 4 τ k G t, s ds m Γ k, k,,..., 5. and f u satisfies the following growth conditions: f u c k L f u < a k L [ 4k, u, 4k], [ 4k 3, u, 4k 3 ], f u b k 4k 45 4k, M /45 u [ 4k, 5 4k ]. 5. Then all the conditions of Theorem 4. are satisfied. Therefore, by Theorem 4. we know that problem 5.7 has countably many positive solutions {u k } k such that u k < 4k 3, min u k t > 4k t τ k, τ k u 3k > 4k 3, with min t τ k, τ k u 3k t < 4k, 5. for each k,,... Remark 5.3. In 8, the existence of solutions for local or nonlocal boundary value problems of higher-order nonlinear ordinary fractional differential equations that has been treated did not discuss problems with singularities. In 3, the singularity only allowed to appear at t and/ort, the existence and multiplicity of positive solutions were asserted under suitable conditions on f. Although, 4, 5 seem to have considered the existence of countably many positive solutions for the second-order and higher-order boundary value problems with infinitely many singularities in, /. However, in 5, only the boundary conditions u oru have been considered. It is clear that the boundary conditions of Examples 5. and 5. are u / andu /. Hence, we generalize second-order and higher-order multipoint boundary value problem. Acknowledgments The project is supported by the Natural Science Foundation of Hebei Province A9664, the Foundation of Hebei Education Department 853, the Foundation of Hebei University of Science and Technology XL64, and the National Natural Science Foundation of PR China 9745.

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