Zhou and Feng Boundary Value Problems (215) 215:178 DOI 1.1186/s13661-15-436-z R E S E A R C H Open Access Triple positive solutions for a second order m-point boundary value problem with a delayed argument Jie Zhou and Meiqiang Feng * * Correspondence: meiqiangfeng@sina.com School of Applied Science, Beijing Information Science & Technology University, Beijing, 1192, People s Republic of China Abstract In this paper, we establish the new expression and properties of Green s function for an m-point boundary value problem with a delayed argument. Furthermore, using Hölder s inequality and a fixed point theorem due to Leggett and Williams, the existence of at least three positive solutions is also given. We discuss our problem with a delayed argument. In this case, our results cover m-point boundary value problems without delayed arguments and are compared with some recent results. An example is included to illustrate our main results. Keywords: triple positive solutions; delayed argument; Leggett-Williams fixed point theorem; Hölder s inequality 1 Introduction It is well known that multi-point boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics. For example, the vibrationsofaguywireofuniformcross-sectioncomposedofn parts of different densities can be set up as a multi-point boundary value problem. Many problems in the theory of elastic stability can be handled as multi-point boundary value problems too. Recently, the existence and multiplicity of positive solutions for multi-point boundary value problems of ordinary differential equations have received a great deal of attention. To identify a few, we refer the reader to [1 18] and the references therein. Recently, Sun and Liu [19] applied the Leray-Schauder nonlinear alternative to study the existence of a nontrivial solution for the problem given by { u (t)+f(t, u)=, <t <1, u() =, u(1) = αu(η), where η (, 1), α R and α 1. At the same time, a type of boundary value problems with deviating arguments has also received much attention. For example, in [2], Yang et al. studied the existence and multiplicity of positive solutions to a three-point boundary value problem with an advanced 215 Zhou and Feng. This article is distributed under the terms of the Creative Commons Attribution 4. International License (http://creativecommons.org/licenses/by/4./), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Zhouand Feng Boundary Value Problems (215) 215:178 Page 2 of 14 argument { u (t)+a(t)f(u(α(t))) =, t (, 1), u() =, bu(η)=u(1), (1.1) where < η <1,b >and1 bη >. The main tool is the fixed point index theory. It is easy to see that the solution of problem (1.1)isconcavewhena(t) on [, 1] and f (u) on[, ). However, few papers have reported the same problems where the solution is without concavity; for example, see some recent excellent results and applications of the case of ordinary differential equations with deviating arguments to a variety of problems from Jankowski [21 23], Jiang and Wei [24], Wang [25], Wang et al. [26] andhuet al. [27]. In the present paper, we shall investigate the existence of triple positive solutions for the following m-point boundary value problem with a delayed argument: { Lx = ω(t)f (t, x(α(t))), < t <1, x () =, x(1) = m 2 β i x(ξ i ), (1.2) where ξ i (, 1), β i (, + ) (i = 1,2,...,m 2) are given constants and L denotes the linear operator Lx := x ax + bx, here a C([, 1], [, + )) and b C([, 1], (, + )). Throughout this paper, we assume that α(t) t on J = [, 1]. In addition, ω, f and β i (i =1,2,...,m 2)satisfy: (H 1 ) ω L p [, 1] for some p [1, + ),andthereexistsn >such that ω(t) n a.e. on J; (H 2 ) f C([, 1] [, + ), [, + )), α C(J, J) with α(t) t on J; (H 3 ) m 2 β i φ(ξ i )<1,whereφ satisfies Lφ =, φ () =, φ(1) = 1. (1.3) Remark 1.1 By a positive solution of problem (1.2) we mean a function x C 2 (, 1) C[, 1] with x(t) > on (, 1) that satisfies (1.2). Remark 1.2 Generally, when y(t) onj, thesolutionx is not concave for the linear equation Lx y(t)=. This means that the method depending on concavity is no longer valid, and we need to introduce a new method to study this kind of problems. For the case α(t) t on J,problem(1.2) reduces to the problem studied by Feng and Ge in [11]. By using the fixed point theorem in a cone, the authors obtained some sufficient
Zhouand Feng Boundary Value Problems (215) 215:178 Page 3 of 14 conditions for the existence, nonexistence and multiplicity of positive solutions for problem (1.2) whenα(t) t on J. However, Feng and Ge did not obtain any results of triple solutions on problem (1.2). This paper will resolve this problem. In this paper, we present several new and more general results for the existence of triple positive solutions for problem (1.2) by using Leggett-Williams fixed point theorem. Another contribution of this paper is to study the expression and properties of Green s function associated with problem (1.2). The expression of the integral equation is simpler than that of [11]. The organization of this paper is as follows. In Section 2, wepresenttheexpression and properties of Green s function associated with problem (1.2). In Section 3,wepresent some definitions and lemmas which are useful to obtain our main results. In Section 4, we formulate sufficient conditions under which delayed problem (1.2) has at least three positive solutions. In Section 5, we provide an example to illustrate our main results. 2 Expression and properties of Green s function Lemma 2.1 Assume that m 2 β i φ(ξ i ) 1.Then, for any y C[, 1], the boundary value problem { Lx = y(t), < t <1, x () =, x(1) = m 2 (2.1) β i x(ξ i ) has a unique solution x(t)= H(t, s)q(s)y(s) ds, (2.2) where ( t q(t)=exp G(t, s)= 1 ) a(s) ds, H(t, s)=g(t, s)+g 1 (t, s), { φ(s)ψ(t), if s t 1, φ(t)ψ(s), if t s 1, (2.3) G 1 (t, s)= φ(t) m 2 β i G(ξ i, s) 1, m 2 β i φ(ξ i ) := φ()ψ (), here φ and ψ satisfy (1.3) and Lψ =, ψ() = 1, ψ(1) =, (2.4) respectively. Proof First suppose that x is a solution of problem (2.1). Similar to the proof of Lemma 2.3 in [1], we can get x(t)= G(t, s)q(s)y(s) ds + Aφ(t),
Zhouand Feng Boundary Value Problems (215) 215:178 Page 4 of 14 where So m 2 β i A = G(ξ i, s)q(s)y(s) ds 1. m 2 β i φ(ξ i ) x(t)= = = = = G(t, s)q(s)y(s) ds + Aφ(t) G(t, s)q(s)y(s) ds + φ(t) [ m 2 β i G(ξ i, s) G(t, s) +φ(t) 1 m 2 β i φ(ξ i ) [ G(t, s)+g1 (t, s) ] q(s)y(s) ds H(t, s)q(s)y(s) ds. m 2 β i G(ξ i, s) 1 m 2 β i φ(ξ i ) ] q(s)y(s) ds q(s)y(s) ds Then the proof is completed. Remark 2.1 The proof of Lemma 2.1 is supplementary of Theorem 3 in [28], which helps the readers to understand that (2.2)holds. Remark 2.2 The expression of the integral equation (2.2) is different from that of (2.1) in [1] and that of(2.9) in[11], which shows that we can use a completely different technique from that of [1]and[11]tostudyproblem(1.2). Remark 2.3 It is not difficult from [1, 11] to show that >andthat(i)φ is nondecreasing on J and φ >onj; (ii) ψ is strictly decreasing on J. Remark 2.4 Noticing a(t) C([,1], [,+ )), itfollowsfromthedefinitionofq(t) that 1 q(t) e M for t J, (2.5) where M = max a(t). Lemma 2.2 (See [28]) Let ξ (, 1), G(t, s), G 1 (t, s) and H(t, s) be given as in Lemma 2.1. Then we have the following results: G(t, s), G 1 (t, s), H(t, s), t, s J, (2.6) G(t, s) G(s, s), G 1 (t, s) G 1 (1, s), H(t, s) H(s) H, t, s J, (2.7) G(t, s) σ G(s, s), G 1 (t, s) φ()g 1 (1, s), H(t, s) σ H(s) σ H, t [, ξ], s J, (2.8)
Zhouand Feng Boundary Value Problems (215) 215:178 Page 5 of 14 where H(s)=G(s, s)+g 1 (1, s), H = max H(s), s J H = min H(s), σ = min { ψ(ξ), φ() }. s J (2.9) Remark 2.5 From (2.8) it follows that H(t, s)> 1 2 σ H. Remark 2.6 By (1.3), (2.4) and the definition of σ,weobtain <σ <1. 3 Preliminaries In this section, we provide some background material from the theory of cones in Banach spaces, and then we state Hölder s inequality and Legget-Williams fixed point theorem. The following definitions can be found in the book by Deimling [29]aswellasinthebook by Guo and Lakshmikantham [3]. Definition 3.1 Let E be a real Banach space over R. A nonempty closed set K E is said to be a cone provided that the following two conditions are satisfied: (i) au + bv K for all u, v K and all a, b ; (ii) u, u K implies u =. Note that every cone K E induces an ordering in E given by x y if and only if y x K. Definition 3.2 Amap is said to be a nonnegative continuous concave functional on a cone K of a real Banach space E if : K R + is continuous and ( tx +(1 t)y ) t (x)+(1 t) (y) for all x, y K and t J. Definition 3.3 An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets. Lemma 3.1 (Arzelà-Ascoli) AsetM C(J, R) is said to be a pre-compact set provided that the following two conditions are satisfied: (i) AllthefunctionsinthesetM are uniformly bounded. It means that there exists a constant r >such that u(t) r, t J, u M; (ii) All the functions in the set M are equicontinuous. It means that for every ε >, there is δ = δ(ε)>, which is independent of the function u, such that u(t1 ) u(t 2 ) < ε whenever t 1 t 2 < δ, t 1, t 2 J.
Zhouand Feng Boundary Value Problems (215) 215:178 Page 6 of 14 Lemma 3.2 (Hölder) Let u L p [a, b] and v L q [a, b], where p, q (, + ) and 1 p + 1 q =1. Then uv L 1 [a, b] and uv 1 u p v q. Let u L 1 [a, b], v L [a, b]. Then uv L 1 [a, b] and uv 1 u 1 v. ThebasicspaceusedinthispaperisE= C[, 1]. It is well known that E is a real Banach space with the norm defined by x = max x(t). Define a cone K in E by { } K = x E : x(t), min x(t) σ x. (3.1) t [,ξ] Define an operator T : K Kby (Tx)(t)=. (3.2) Lemma 3.3 Assume that (H 1 )-(H 3 ) hold. Then T(K) KandT: K K is completely continuous. Proof For x K, it follows from (2.7)and(3.2)that Tx = max. (3.3) It follows from (2.8), (3.2)and(3.3)that min (Tx)(t)= min t [,ξ] t [,ξ] σ σ Tx. Thus, T(K) K. Next we shall show that operator T is completely continuous. We break the proof into several steps. Step 1. Operator T is continuous. Since the function f (t, x) is continuous on J [, + ), this conclusion can be easily obtained.
Zhouand Feng Boundary Value Problems (215) 215:178 Page 7 of 14 Step 2. For each constant l >,letb l = {x K : x l}. ThenB l is a bounded closed convex set in K. x B l,from(3.2), we have Tx = max e M H(s)ω(s)f ( s, x ( α(s) )) ds e M H q ω p L, where L = sup, x l f (s, x(α(s))). This proves that T(B r ) is uniformly bounded. Step 3. The family {Tx : x B l } is a family of equicontinuous functions. Since H(t, s) is continuous on J J, and noticing J = [, 1], H(t, s) isuniformlycontinuousonj J. Therefore, for all ε >,thereexistsl >,when t 1 t 2 < l,suchthat H(t1, s) H(t 2, s) ε < e M ω 1 L. Then, for all x B l,when t 1 t 2 < δ,weget Tx(t1 ) Tx(t 2 ) 1 = H(t 1, s)q(s)ω(s)f ( s, x ( α(s) )) ds H(t 2, s)q(s)ω(s)f ( s, x ( α(s) )) ds [ = H(t1, s) H(t 2, s) ] q(s)ω(s)f ( s, x ( α(s) )) ds e M ω 1 L H(t1, s) H(t 2, s) ds < ε. Thus, the set {Tx : x B l } is equicontinuous. As a consequence of Step 1 to Step 3 together with Lemma 3.1, we can prove that T : K K is completely continuous. Remark 3.1 From Lemma 2.1 and (3.1), we know that x E is a solution of problem (1.2) if and only if x is a fixed point of operator T. Let < r 1 < r 2 be given and let β be a nonnegative continuous concave functional on the cone K.DefinetheconvexsetsK r1, K(β, r 1, r 2 )by K r1 = { x K : x < r 1 }, K(β, r 1, r 2 )= { x K : r 1 β(x), x r 2 }. Finally we state Leggett-Williams fixed point theorem [31].
Zhouand Feng Boundary Value Problems (215) 215:178 Page 8 of 14 Lemma 3.4 Let K be a cone in a real Banach space E, A : K c K c be completely continuous and β be a nonnegative continuous concave functional on K with β(x) x ( x K c ). Suppose that there exist <d < a < b csuchthat (i) {x K(β, a, b):β(x)>a} and β(ax)>a for x K(β, a, b); (ii) Ax < d for x d; (iii) β(ax)>a for x K(β, a, c) with Ax > b. Then A has at least three fixed points x 1, x 2, x 3 satisfying x 1 < d, a < β(x 2 ), x 3 > d and β(x 3 )<a. 4 Existence of triple positive solutions In this section, we apply Lemma 3.2 and Lemma 3.4 to establish the existence of three positive solutions for problem (1.2). We consider the following three cases for ω L p [, 1] : p >1,p =1,andp =.Casep > 1 is treated in the following theorem. For convenience, we write Ɣ = e M H q ω p, = 1 2 nσ H. Let the nonnegative continuous concave functional on the cone K be defined by (x)=min x(t). [,ξ] Note that for x K, (x) x. Theorem 4.1 Assume that (H 1 )-(H 3 ) hold. In addition, there exist constants <d < l < l σ csuchthat (H 4 ) f (t, x) c for (t, x) J [, c]; Ɣ (H 5 ) f (t, x) l for (t, x) [, ξ] [l, l σ ]; (H 6 ) f (t, x) d for (t, x) J [, d]. Ɣ Then problem (1.2) has at least three positive solutions x 1, x 2 and x 3 satisfying x 1 < d, l < (x 2 ), d < x 3 and (x 3 )<l. (4.1) For details, seefigure 1. Figure 1 The relationship of x 1, x 2 and x 3.
Zhouand Feng Boundary Value Problems (215) 215:178 Page 9 of 14 Proof By the definition of operator T and its properties, it suffices to show that the conditions of Lemma 3.4 hold with respect to T. Let x P c.then x(t) c on J.Since α(t) t 1onJ, it follows from x(t) c on J that x(α(t)) c on J. Consequently, for t J and x P c, it follows from (H 4 ), (2.8)and(3.2)that Tx = max e M H(s)ω(s)f ( s, x ( α(s) )) ds e M H q ω p f ( s, x ( α(s) )) ds e M c H q ω p Ɣ c, (4.2) which implies Tx P c.thisprovesthatt : P c P c is completely continuous. We first show that the condition (i) of Lemma 3.4 holds. Take x(t)= 1 2 (l + l ), t J.Then σ x = 1 2 ( l + l ) < l, (x)= min σ σ x(t)= 1 ( l + l ) > l. t [,ξ] 2 σ This shows that { ( x K, l, l ) } : (x)>l. σ Therefore, for all {x K(, l, l ): (x)>l} and t J,wehave σ l x(t) l σ. Since α(t) t ξ on [, ξ], it follows from l x(t) l σ on [, ξ]thatl x(α(t)) l σ for t [, ξ]. Therefore, it follows from Remark 2.1 and (H 5 )that (Tx)= min (Tx)(t) t [,ξ] σ nσ nσ ξ H(s)f ( s, x ( α(s) )) ds H(s)f ( s, x ( α(s) )) ds
Zhouand Feng Boundary Value Problems (215) 215:178 Page 1 of 14 Therefore, we have (Tx)>l, > 1 2 nσ H l = l. (4.3) ( x K, l, l ). σ This implies that condition (i) of Lemma 3.4 is satisfied. Secondly, we prove that condition (ii) of Lemma 3.4 is satisfied. If x K d,then x(t) d on J. Since α(t) t 1onJ, it follows from x(t) d on J that x(α(t)) d on J. Thus it follows from (H 6 )that Tx = max e M H(s)ω(s) ds ( ( )) f s, x α(s) e M d H q ω p Ɣ = d. (4.4) Hence, the condition (ii) of Lemma 3.4 is satisfied. Finally, we prove that the condition (iii) of Lemma 3.4 is satisfied. In fact, for all x K(, l, c)and Tx > l, it follows from (2.5), (2.9), (3.2)and(3.3)that σ (Tx)= min (Tx)(t) t [,ξ] = min t [,ξ] σ σ Tx > σ l σ = l. (4.5) This gives the proof of the condition (iii) of Lemma 3.4. To sum up, the hypotheses of Lemma 3.4 hold. Therefore, an application of Lemma 3.4 implies that problem (1.2) has at least three positive solutions x 1, x 2 and x 3 such that x 1 < d, l < (x 2 ), and x 3 > d with (x 3 )<l. The following corollary deals with the case p =.
Zhouand Feng Boundary Value Problems (215) 215:178 Page 11 of 14 Corollary 4.1 Assume that (H 1 )-(H 6 ) hold. Then problem (1.2) has at least three positive solutions x 1, x 2 and x 3 satisfying (4.1). Proof Let H 1 ω replace H q ω p and repeat the argument above, we can get the corollary. Finally we consider the case of p =1.Let (H 4 ) f (t, x) c Ɣ 1 for (t, x) J [, c]; (H 6 ) f (t, x) d Ɣ 1 for (t, x) J [, d], where Ɣ 1 = e M H ω 1. Corollary 4.2 Assume that (H 1 )-(H 3 ), (H 4 ),(H 5 ) and (H 6 ) hold. Then problem (1.2) has at least three positive solutions x 1, x 2 and x 3 satisfying (4.1). Proof Similartotheproofof(4.1), it follows from (2.3)and(H 4 ) that Tx = max e M H(s)ω(s)f ( s, x ( α(s) )) ds e M H ω(s)f ( s, x ( α(s) )) ds e M H ω 1 f ( s, x ( α(s) )) ds e M H ω 1 c Ɣ 1 c, (4.6) which shows that Tx K c, x K c. Next turning to (H 6 ),wehave Tx = max e M H(s)ω(s)f ( s, x ( α(s) )) ds e M H ω(s)f ( s, x ( α(s) )) ds e M H ω 1 f ( s, x ( α(s) )) ds
Zhouand Feng Boundary Value Problems (215) 215:178 Page 12 of 14 e M H ω 1 d Ɣ 1 c, x K d. (4.7) SimilartotheproofofTheorem4.1, we can get Corollary 4.2. Remark 4.1 Comparing with Feng and Ge [11], the main features of this paper are as follows. (i) Three positive solutions are available. (ii) α(t) t is considered throughout this paper. (iii) ω(t) is L p -integrable, not only ω(t) C(, 1) on t J. 5 Anexample In this section, we present an example. Let p =2, β 1 = e 2 1, ξ 1 = 1, a(t), b(t) 1. 2 Example 5.1 Consider the following three-point boundary value problem: { x (t)+x(t)=ω(t)f(t, x(α(t))), < t <1, x () =, x(1) = e 2 1 x( 1 2 ), (5.1) where α C(J, J), α(t) t on J and 1 ω(t)= t 1 2, 3 1 f (t, x)= d Ɣ, d Ɣ l x t J, x [, d], t J, x [d, l], l d + l x d l d, l, t J, x [l, l σ ], l c x c σ l c Ɣ, + c Ɣ x σ l, t J, x [ l c σ l σ, c], t J, x [c,+ ). (5.2) This means that problem (5.1) involves the delayed argument α. For example, we can take α(t)=t 3.Itisclearthatω is nonnegative and ω L 2 [, 1]. Conclusion 5.1 Problem (5.1) has at least three positive solutions x 1, x 2 and x 3 satisfying (4.1). Proof It follows from (1.3)and(2.4)thatφ and ψ satisfy Lφ =, φ () =, φ(1) = 1, Lψ =, ψ() = 1, φ(1) =, (5.3) where Lx = x (t)+x(t)and φ(t)= e1 t + e 1+t 1+e 2, φ() = 2e 1+e 2,
Zhouand Feng Boundary Value Problems (215) 215:178 Page 13 of 14 ψ(t)= e2 t + e t, ψ () = 1+e2 1 e 2 1 e, (5.4) 2 q(t)=1, := φ()ψ () = 2e ( ) e 2 1 >, β 1φ(ξ 1 )=e 1 1 2 φ <1. 2 On the other hand, it follows from a(t)=,m =3,α 1 = e 1 2 and ω(t)= 1 t 1 2 1 3 that e M =1, n = 3 [ ( 2, ω 2 = t 1 2 Choosing < d < l < l σ c,wehave ) 2 3 dt ] 1 2 = 6 3 2. f (t, x) c for (t, x) J [, c]; Ɣ f (t, x) l [ for (t, x), 1 ] [ l, 2l ] ; 2 σ f (t, x)= d Ɣ for (t, x) J [, d], which shows that (H 4 )-(H 6 )hold. By Theorem 4.1,problem(5.1) has least three positive solutions x 1, x 2 and x 3 satisfying (4.1). Competing interests The authors declare that they have no competing interests. Authors contributions JZ checked the proofs and verified the calculation. MF completed the main study and carried out the results of this article. All the authors read and approved the final manuscript. Acknowledgements We wish to express our thanks to Prof. Xuemei Zhang, Department of Mathematics and Physics, North China Electric Power University, Beijing, P.R. China, for her kind help, careful reading, and making useful comments on the earlier version of this paper. The authors are also grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper. This work is sponsored by the project NSFC (1131178), the Fundamental Research Funds for the Central Universities (214MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM2151123218). Received: 23 June 215 Accepted: 1 September 215 References 1. Gupta, CP: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89,133-146 (1998) 2. Feng, W, Webb, JRL: Solvability of m-point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 212, 467-48 (1997) 3. Wei, Z,Pang, C:Positive solutions of some singular m-point boundary value problems at non-resonance. Appl. Math. Comput. 171, 433-449 (25) 4. Liu, B: Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. J. Math. Anal. Appl. 35, 253-276 (25) 5. Zhao, Y, Chen, H: Existence of multiple positive solutions for m-point boundary value problems in Banach spaces. J. Comput. Appl. Math. 215, 79-9 (28) 6. Eloe, PW, Henderson, J: Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electron. J. Differ. Equ. 1997, 3 (1997) 7. Webb, JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 4319-4332 (21) 8. He, X, Ge, W: Triple solutions for second-order three-point boundary value problems. J. Math. Anal. Appl. 268, 256-265 (22) 9. Kosmatov, N: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 39, 25-36 (25) 1. Ma, R: Existence of positive solutions for a nonlinear m-point boundary value problem. Acta Math. Sin. 46, 785-794 (23)
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