Nontrivial Solutions for Singular Nonlinear Three-Point Boundary Value Problems 1

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1 Advances in Dynamical Systems and Applications. ISSN Volume 2 Number 2 (27), pp Research India Publications Nontrivial Solutions for Singular Nonlinear Three-Point Boundary Value Problems Jia Mu and Hong-Rui Sun School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 73, People s Republic of China Corresponding author hrsun@lzu.edu.cn. Abstract The singular nonlinear three-point boundary value problems (Lu)(t) = h(t)f (u(t)), βu() γu () =, <t<, u() = αu(η) are discussed under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where (Lu)(t) = (p(t)u (t)) +q(t)u(t),<η<, h(t) is allowed to be singular at both t = and t =, and f need not be nonnegative. The associated Green function for the problem is first given, and some existence criteria of nontrivial solutions and positive solutions are established by using topological degree theory. AMS subject classification: 34B6, 47H. Keywords: Three-point boundary value problems, Green s function, nontrivial solution, positive solution, topological degree. Supported by the NNSF of China (5778, 72649) and the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Lzu53). Received April, 27; Accepted September 24, 27

2 226 Jia Mu and Hong-Rui Sun. Introduction The existence of positive solutions for nonlinear multi-point boundary value problems has been studied by many authors, and the main tools are the method of upper and lower solutions, coincidence degree theory and fixed point theorem in cones and so on, see [,5,6,9,3 5] and the references therein. In this paper we are concerned with the singular nonlinear three-point boundary value problems (Lu)(t) = h(t)f (u(t)), <t<, βu() γu (.) () =, u() = αu(η), where Lu = (pu ) + qu and η (, ), α (, ), β, γ [, ) with β + γ =, h(t) is allowed to be singular at both t = and t =, and f is not necessarily nonnegative. The existence of nontrivial solutions and positive solutions are obtained under some conditions concerning the first eigenvalue with respect to the relevant linear operator. Our results generalize and improve those of [9, 5]. The main idea comes from [, 2, 2] and the main tool is topological degree theory. For the concepts and properties of topological degree theory, we refer the reader to [3, 4, 7]. This paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results which will be used later. It is noticed that the associated Green function for the problem is first given. In Section 3, we give two existence results of nontrivial solutions and some corollaries. Section 4 is to develop the existence criteria of multiple nontrivial solutions for the problem (.). Some examples are also given to illustrate our main results. For convenience, we list the following hypotheses: (H ) p C [, ], p(t) >, q C[, ], q(t), and the homogeneous problem (Lu)(t) =, <t<, βu() γu () =, u() = has only the trivial solution; (H 2 ) αϕ(η)<ϕ(), where ϕ is the unique solution of the problem (2.) in Section 2; (H 3 ) f : (, + ) (, + ) is continuous; (H 4 ) h : (, ) [, + ) is continuous, h(t) and where k is defined in (2.2). k(t, t)h(t)dt < +, 2. Preliminaries First, by the maximum principle, we can easily get the following result.

3 Singular Nonlinear Three-Point Boundary Value Problems 227 Lemma 2.. Suppose that (H ) holds. Let ϕ and ψ be the solutions of the linear problems (Lu)(t) =, <t<, u() = γ, u (2.) () = β, (Lu)(t) =, <t<, u() =, u () =, respectively. Then (i) ϕ C 2 [, ] is an increasing function, and ϕ(t) > for t (, ]; (ii) ψ C 2 [, ] is a decreasing function, and ψ(t) > for t [, ); (iii) p(t)(ϕ (t)ψ(t) ϕ(t)ψ (t)) = ω (a positive constant). For convenience, we set ϕ(t)ψ(s), t s, ω k(t, s) = ϕ(s)ψ(t), s t. ω Lemma 2.2. Assume that (H ) is satisfied. Then k possesses the following properties: (i) k(t, s) is continuous and k(t, s) over [, ] [, ]; (ii) k(t, s) = k(s, t), k(t, s) k(s, s) for t,s [, ]; (iii) k(t, s) ϕ(t)ψ(t) k(τ, s) for t,s,τ [, ]. ϕ()ψ() (2.2) Proof. We only prove (iii) as (i) and (ii) are easy to get by the definition of ϕ,ψ and k(t, s). By the property (ii) and (2.2), for t,s [, ], wehave ϕ(t)ψ(t)k(τ,s) ϕ(t)ψ(t)ϕ(s)ψ(s) ϕ()ψ()k(t, s) for t,s,τ [, ]. ω The proof is completed. Lemma 2.3. Assume that (H ) and (H 2 ) hold. Then Green s function of the problem (Lu)(t) =, <t<, βu() γu () =, u() = αu(η) is given by G(t, s) = k(t, s) + αk(η, s) ϕ(t), (2.3) ϕ() αϕ(η)

4 228 Jia Mu and Hong-Rui Sun where k(t, s) is given in (2.2). Proof. It is easy to check that the function u defined by u(t) = G(t, s)y(s)ds, (2.4) is a solution of (Lu)(t) = y(t), <t<, βu() γu (2.5) () =, u() = αu(η), where y C[, ] and G(t, s) is defined by (2.3). Now we show that the function defined by (2.4) is a solution of (2.5) only if G(t, s) is the same as in (2.3). By the method of variation of constants, we can obtain that any solution of (2.5) can be represented by In view of (2.2), we have u(t) = k(t,s)y(s)ds + Aϕ(t). (2.6) and t u(t) = ψ(t) t (pu ) (t) = (pψ ) (t) + (pϕ ) (t) ϕ(s)y(s)ds + ϕ(t) ω t t ψ(s)y(s)ds + Aϕ(t) (2.7) ω ω ϕ(s)y(s)ds + p(t)ψ (t) ω ϕ(t)y(t) + A(pϕ ) (t) ω ψ(s)y(s)ds p(t)ϕ (t) ψ(t)y(t), (2.8) ω so that by (2.7), (2.8) and the definitions of ϕ, ψ and ω, we obtain (pu ) (t) + q(t)u(t) = p(t)ψ (t) ω ϕ(t)y(t) p(t)ϕ (t) ω ψ(t)y(t) = p(t) ω [ϕ (t)ψ(t) ϕ(t)ψ (t)]y(t) = y(t). Observe that since u() = ϕ() ψ(s)y(s)ds + Aϕ(), ω u () = ϕ () ω ψ(s)y(s)ds + Aϕ (),

5 Singular Nonlinear Three-Point Boundary Value Problems 229 and ϕ() = γ,ϕ () = β,weget βu() γu () =. Since u() = Aϕ() and u(η) = condition of (2.5), we get that k(η,s)y(s)ds + Aϕ(η), by the second boundary A = α k(η,s)y(s)ds, ϕ() αϕ(η) so by (2.6) the proof is finished. By Lemma 2., Lemma 2.2 and the monotone properties ϕ,ψ, we have the following lemma. Lemma 2.4. Assume (H ) and (H 2 ) hold. Then G(t, s) defined by (2.3) has the following properties: (i) G(t, s) is nonnegative and continuous on [, ] [, ]; (ii) ϕ(t)ψ(t) k(τ, s) G(t, s) Dk(s, s) for t,s,τ [, ], where ϕ()ψ() D = + αϕ() ϕ() αϕ(η). In the Banach space C[, ] with norm defined by u = max u(t), weset t P =u C[, ] : u(t), t [, ]}, P = u P : u(t) ϕ(t)ψ(t) } u, t [, ]. ϕ()ψ() Then P is a positive cone in C[, ] and P P. For r>, we denote B r =u C[, ] : u <r}. By the above discussion, we know that the problem (.) is equivalent to the integral equation u(t) = G(t, s)h(s)f (u(s))ds =: Au(t), t [, ]. It is easy to see that u is a solution of (.) if and only if u is a fixed point of A. For the sake of convenience, we let (T u)(t) = G(t, s)h(s)u(s)ds, t [, ]. (2.9)

6 23 Jia Mu and Hong-Rui Sun Lemma 2.5. [7] Suppose that E is a Banach space, A n : E E(n=, 2, 3,...)are completely continuous operators, A : E E, and Then A is completely continuous. lim max A nu Au = for all r>. n u <r Lemma 2.6. If (H ) (H 4 ) hold, then A : C[, ] C[, ] is completely continuous. Proof. By (H ) (H 4 ), it is easy to see that A : C[, ] C[, ]. For any natural number n(n 2),weset inf h(s), t /n, t<s /n h n (t) = h(t), /n t /n, (2.) inf h(s), /n t. /n s<t Then h n :[, ] [, + ) is continuous and h n (t) h(t), t (, ). Let (A n u)(t) := G(t, s)h n (s)f (u(s))ds. (2.) It is obvious that A n : C[, ] C[, ] is completely continuous. For every r> and u B r, by (2.), (2.), Lemma 2.4 (ii) and the absolute continuity of the integral, we have lim A nu Au lim max n n t G(t, s)(h n (s) h(s))f (u(s))ds ( ) max lim Dk(s, s)(h(s) h n (s))ds r u r n ( ) D max r u r lim n e(n) k(s, s)h(s)ds =, where e(n) =[, /n] [ /n,]. Then by Lemma 2.5, A : C[, ] C[, ] is completely continuous. By virtue of Krein Rutmann s theorem [8] and similar as in the proof of [2, Lemma 5], we get the following lemma. Lemma 2.7. If (H ), (H 2 ), and (H 4 ) hold, then for the operator T defined by (2.9), (i) T : C[, ] C[, ] is a completely continuous linear operator and T(P) P ; (ii) the spectral radius r(t) is not zero and T has a positive eigenfunction corresponding to its first eigenvalue λ = (r(t )).

7 Singular Nonlinear Three-Point Boundary Value Problems 23 Lemma 2.8. [3] Let E be a Banach space and be a bounded open set in E. Suppose that A : E is a completely continuous operator. If there exists u = θ such that u Au = µu for u, µ, then the topological degree deg(i A,, θ) is equal to zero. Lemma 2.9. [3] Let E be a Banach space and be a bounded open set in E with θ. Suppose that A : E is a completely continuous operator. If then deg(i A,, θ) =. Au = µu for u, µ, Lemma 2.. [7] Let A be a completely continuous operator which is defined on a Banach space E. Let x E be a fixed point of A and assume that A is defined in a neighborhood of x and Fréchet differentiable at x. If is not an eigenvalue of the linear operator A (x ), then x is an isolated singular point of the completely continuous vector field I A, and for small enough r>, deg(i A, B(x,r),θ)= ( ) k, where k is the sum of the algebraic multiplicities of the real eigenvalues of A (x ) in (, + ). 3. Existence of Nontrivial Solutions Theorem 3.. Suppose (H ) (H 4 ) hold. Assume that there exists a constant b such that b, u (, + ), (3.) lim inf u + u >λ, (3.2) lim sup u u <λ, (3.3) where λ is the first eigenvalue of T defined by (2.9). Then the problem (.) has at least one nontrivial solution. Proof. It follows from (3.2) that there exists ε> such that (λ + ε)u when u is sufficiently large. Then by (3.) there exists b such that (λ + ε)u b for u (, + ). (3.4) Let u be a positive eigenfunction of T corresponding to its first eigenvalue λ. Then u = λ Tu, and u P by Lemma 2.7. Take R > max 2, [(εη2 + D(λ + ε)) ũ +Db ] } k(s, s)h(s)ds εη, 2 σ σ k(s, s)h(s)ds

8 232 Jia Mu and Hong-Rui Sun where σ (, /2) such that h(t) for t (σ, σ), η = min σ t σ ϕ(t)ψ(t) ϕ()ψ() = ϕ(σ)ψ( σ) ϕ()ψ() >, ũ(t) = b G(t, s)h(s)ds, and D is given in Lemma 2.4. We may assume that A has no fixed point on B R (otherwise the proof is completed). In the following we prove u Au = τu for u B R, τ. (3.5) If otherwise, there exist u B R and τ > such that u Au = τ u. So we have that u (t) +ũ(t) = Au (t) +ũ(t) + τ u (t) = G(t, s)h(s)[f(u (s)) + b]ds + τ u (t). (3.6) Then by T(P) P and u P,weget u +ũ P. (3.7) By (3.4), (3.7), we have that Au (t) +ũ(t) (λ + ε) = λ T(u +ũ)(t) + G(t, s)h(s)u (s)ds + (b b ) G(t, s)h(s)[ε(u (s) +ũ(s)) + b]ds G(t, s)h(s)ds G(t, s)h(s)[(λ + ε)ũ(s) + b ]ds σ ( ) ϕ(t)ψ(t) 2 λ T(u +ũ)(t) + k(s, s)h(s)ε u +ũ ds ϕ()ψ() σ Dk(s, s)h(s)[(λ + ε) ũ +b ]ds λ T(u +ũ)(t) + εη 2 ( u ũ ) σ Dk(s, s)h(s)[(λ + ε) ũ +b ]ds λ T(u +ũ)(t) + εη 2 u σ [(εη 2 + D(λ + ε)) ũ +Db ] λ T(u +ũ)(t). σ σ k(s, s)h(s)ds k(s, s)h(s)ds k(s, s)h(s)ds

9 Singular Nonlinear Three-Point Boundary Value Problems 233 Then in view of (3.6) and (3.7), one arrives at u +ũ λ T(u +ũ) + τ u τ u. (3.8) Put τ = supτ : u +ũ τu }. It is easy to see that τ τ > and u +ũ τ u. We get from the property of the operator T that λ T(u +ũ) τ λ Tu = τ u. Therefore by (3.8), we have u +ũ λ T(u +ũ) + τ u (τ + τ )u, which contradicts the definition of τ. Hence (3.5) is true and we have from Lemma 2.8 that deg(i A, B R,θ)=. (3.9) It follows from (3.3) that there exists ε>and <R 2 < 2 such that ( ε)λ u for u R 2. (3.) Now we show that Au = τu for u B R2, τ. (3.) If not, there exist u 2 B R2 and τ 2 such that Au 2 = τ 2 u 2. We may suppose τ 2 > (otherwise the proof is finished). Let u = u 2. Then u = P B R2. From (3.), it follows that u (t) τ 2 u (t) = G(t, s)h(s)f (u 2 (s))ds ( ε)λ G(t, s)h(s) u 2 (s) ds = ( ε)λ T(u )(t). (3.2) Put τ = infτ : u τu }. It is easy to show that τ > and u τ u. We have from the property of the operator T that λ Tu λ T(τ u ) = τ u. (3.3) Therefore by (3.2) and (3.3), we get u ( ε)τ u, which contradicts the definition of τ. Hence (3.) is true and we have from Lemma 2.9 that deg(i A, B R2,θ)=. (3.4)

10 234 Jia Mu and Hong-Rui Sun By (3.9) and (3.4), we have deg(i A, B R \ B R2,θ)= deg(i A, B R,θ) deg(i A, B R2,θ)=. Thus A has at least one fixed point in B R \ B R2. This means that the problem (.) has at least one nontrivial solution. By Theorem 3., we can get the following results. Corollary 3.2. Assume the conditions (H ) (H 4 ) hold. If there exists a constant b such that b M for u b, (3.5) where M = max G(t, s)h(s)ds, and in addition, (3.2) and (3.3) hold, then the t [,] problem (.) has at least one nontrivial solution. Corollary 3.3. Suppose (H ) (H 4 ) are satisfied. If there is a constant b such that b, u (, + ), (3.6) lim inf u u >λ, (3.7) and in addition, (3.3) holds, then the problem (.) has at least one nontrivial solution. Theorem 3.4. If (H ) (H 4 ), (3.) and (3.2) hold, f() =, and lim = λ, (3.8) u u where λ is not an eigenvalue of T, then (.) has at least one nontrivial solution. Proof. By (3.8), we get (A θ (u))(t) = λ G(t, s)h(s)u(s)ds = λ(t u)(t), (3.9) which implies that is not an eigenvalue of A θ. In addition, from f() =, we get Aθ = θ. According to Lemma 2., we can find a positive number R such that deg(i A, B R,θ)=±. (3.2) Similar as in the proof of Theorem 3., we have from (3.) and (3.2) that there is R 2 >R such that deg(i A, B R2,θ)=. (3.2) By (3.2), (3.2) and the additivity of the Leray Schauder degree, we have deg(i A, B R2 \ B R,θ)= deg(i A, B R2,θ) deg(i A, B R,θ)=.

11 Singular Nonlinear Three-Point Boundary Value Problems 235 Thus A has at least one fixed point in B R2 \ B R. This means that the problem (.) has at least one nontrivial solution. The following result is an immediate consequence of Theorem 3.4. Corollary 3.5. Suppose (H ) (H 4 ) are satisfied. If (3.2), (3.5), (3.8), and f() = hold, then the problem (.) has at least one nontrivial solution. 4. Existence of Positive Solutions Theorem 4.. Assume that the conditions (H ) (H 4 ) are satisfied. If u, u (, + ), (4.) lim inf u + u >λ, (4.2) lim sup u u <λ, (4.3) where λ is the first eigenvalue of the operator T defined by (2.9), then the problem (.) has at least one positive solution and one negative solution. Proof. By (4.) and (H 3 ), we have f() =. If we denote, u, f 3 (u) =, u <, (4.4) then f 3 is continuous. Define (A 3 u)(t) := G(t, s)h(s)f 3 (u(s))ds, t [, ]. From (4.) we know that A 3 (C[, ]) P. Similar to the proof of Theorem 3. in which b =, we have that A 3 has a fixed point ũ P \θ}. So it follows from (4.4) that ũ is the fixed point of A and (.) has at least one positive solution. Denote f 4 (u) = f( u) for u (, + ). Then (4.), (4.2), (4.3) are satisfied in which f is replaced by f 4. Define (A 4 u)(t) := G(t, s)h(s)f 4 (u(s))ds, t [, ]. By the same method as above, we have that A 4 has a fixed point ṽ P \θ}, i.e., A 4 ṽ =ṽ. Since f 4 (ṽ(t)) = f( ṽ(t)), t [, ], and ṽ(t) = G(t, s)h(s)f ( ṽ(s))ds = (A( ṽ))(t), t [, ],

12 236 Jia Mu and Hong-Rui Sun we obtain that the problem (.) has at least one negative solution. Theorem 4.2. Let (H ) (H 4 ) hold. If α<, β>, f() =, q(t) < for t [, ], (3.2), and lim sup u + u <λ (4.5) hold, where λ is the first eigenvalue of T defined by (2.9), then the problem (.) has at least one positive solution. Proof. It follows from (3.2) and (H 3 ) that there is a constant b such that b for u [, + ). (4.6) Denote, u, f 5 (u) =, u <. Then by (4.6), f 5 is bounded below and continuous. Define (4.7) (A 5 u)(t) := G(t, s)h(s)f 5 (u(s))ds, t [, ]. It is easy to see that all conditions of Theorem 3. are satisfied in which f is replaced by f 5. By Theorem 3. we know that A 5 has at least one nontrivial fixed point ũ, i.e., A 5 ũ =ũ. In the following we prove ũ(t), t [, ]. (4.8) If otherwise, there exists t [, ] such that ũ(t ) = min ũ(t) <. By the assumption t [,] that α<,β > and the boundary condition of (.), we have that t =,. Thus Hence However, by (4.7) we have ũ(t )<, ũ (t ) =, ũ (t ). (Lũ)(t ) = p(t )ũ (t ) + p (t )ũ (t ) + q(t )ũ(t )>. (4.9) (Lũ)(t ) = h(t )f 4 (ũ(t )) =, which contradicts (4.9). Therefore, (4.8) holds. Then Aũ = ũ, and the problem (.) has at least one positive solution. By Theorem 4.2, we easily get the following result.

13 Singular Nonlinear Three-Point Boundary Value Problems 237 Corollary 4.3. Suppose that the conditions (H ) (H 4 ) are satisfied. If α<, β>, f() =, q(t) <, for t [, ], and lim sup u u <λ, lim inf u u >λ hold, where λ is the first eigenvalue of T defined by (2.9), then the problem (.) has at least one negative solution. Corollary 4.4. Assume that the conditions (H ) (H 4 ) hold. If α<,β >, f() =, q(t) < for t [, ], (3.3), and lim inf u + u >λ hold, where λ is the first eigenvalue of T defined by (2.9), then the problem (.) has at least one positive solution and one negative solution. Remark 4.5. In most papers, the nonlinear term f is a nonnegative function defined on [, + ) to guarantee the operator A generated by f is a cone mapping, so that one can apply the fixed point theory in cone. In our main results, f may be a sign-changing function, and consequently, A is not necessarily a cone mapping. Thus the theory of fixed point index on a cone becomes invalid, and in order to obtain the existence of nontrivial solutions, we make use of topological degree theory which is not confined in a cone. Remark 4.6. In Theorems 4. and 4.2, f is not required to be bounded below, and in particular, the existence of positive solutions is obtained in Theorem 4.2 and Corollary 4.4 although A may not be a cone mapping. Now we give several examples to illustrate our main results. For convenience, in the following examples, we always choose the functions p and q to satisfy the assumption (H ), α, β, γ, and η to satisfy (H 2 ), and let h(t) = t p ( t) q, where p,q (, ). It is obvious that h(t) is singular at both t = and t =, and h satisfies (H 4 )by convergence of Euler s integral. Example 4.7. If we let = n a i u i, where n is a positive even number, a i R i= for i =, 2,...,n, and λ <a <λ, a n >, then is bounded below and usually sign changing for u, and all conditions of Theorem 3. are satisfied. So the singular BVP (.) has at least one nontrivial solution. Example 4.8. If we let = u 2 sin u, then is bounded below and sign changing for u. Theorem 3.4 can be applied since lim = (which is not an eigenvalue u u

14 238 Jia Mu and Hong-Rui Sun of T ), and lim u + u =+. Example 4.9. If we choose = n a i u 2i, where n is a positive integer, a i > i= (i =, 2,...,n)and a <λ, then u, lim = a, lim =+. u u u + u So by Theorem 4., the problem (.) has at least one positive solution and one negative solution. n Example 4.. If we let α <,β >, and = a i u i, where n 3isan odd number, a i R for i =, 2,...,n, a < λ and a n >, then is bounded below and usually sign changing for u. In addition, lim = a, u + u lim =+. So by Theorem 4.2, the problem (.) has at least one positive u + u solution. i= References [] Bruce Calvert and Chaitan P. Gupta. Multiple solutions for a super-linear three-point boundary value problem, Nonlinear Anal., 5(, Ser. A: Theory Methods):5 28, 22. [2] Yujun Cui and Yumei Zou. Nontrivial solutions of singular superlinear m-point boundary value problems, Appl. Math. Comput., 87(2): , 27. [3] Klaus Deimling. Nonlinear functional analysis, Springer-Verlag, Berlin, 985. [4] Da Jun Guo and V. Lakshmikantham. Nonlinear problems in abstract cones, volume5ofnotes and Reports in Mathematics in Science and Engineering, Academic Press Inc., Boston, MA, 988. [5] Chaitan P. Gupta. Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 68(2):54 55, 992. [6] V. A. Il in and E. I. Moiseev. Nonlocal boundary value problem of the first kind for a Sturm Liouville operator in its differential and finite different aspects, Differential Equations, 23(7):83 8, 987. [7] M.A. Krasnosel skiĭ and P. P. Zabreĭko. Geometrical methods of nonlinear analysis, volume 263 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 984. Translated from the Russian by Christian C. Fenske.

15 Singular Nonlinear Three-Point Boundary Value Problems 239 [8] M. G. Kreĭn and M. A. Rutman. Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 95(26):28, 95. [9] Bing Liu. Positive solutions of a nonlinear three-point boundary value problem, Appl. Math. Comput., 32(): 28, 22. [] Ruyun Ma. Positive solutions of a nonlinear three-point boundary-value problem, Electron. J. Differential Equations, pages No. 34, 8 pp. (electronic), 999. [] Donal O Regan. Theory of singular boundary value problems, World Scientific Publishing Co. Inc., River Edge, NJ, 994. [2] Jingxian Sun and Guowei Zhang. Nontrivial solutions of singular superlinear Sturm-Liouville problems, J. Math. Anal. Appl., 33(2):58 536, 26. [3] Yongping Sun and Lishan Liu. Solvability for a nonlinear second-order three-point boundary value problem, J. Math. Anal. Appl., 296(): , 24. [4] Guowei Zhang and Jingxian Sun. Positive solutions of m-point boundary value problems, J. Math. Anal. Appl., 29(2):46 48, 24. [5] Zhongxin Zhang and Junyu Wang. Positive solutions to a second order three-point boundary value problem, J. Math. Anal. Appl., 285(): , 23.