MONOTONE POSITIVE SOLUTIONS FOR p-laplacian EQUATIONS WITH SIGN CHANGING COEFFICIENTS AND MULTI-POINT BOUNDARY CONDITIONS

Transkript

1 Electronic Journal of Differential Equations, Vol. 22, No. 22, pp. 2. ISSN: URL: or ftp ejde.math.txstate.edu MONOTONE POSITIVE SOLUTIONS FOR p-laplacian EQUATIONS WITH SIGN CHANGING COEFFICIENTS AND MULTI-POINT BOUNDARY CONDITIONS JIANYE XIA, YUJI LIU Abstract. We prove the existence of three monotone positive solutions for the second-order multi-point boundary value problem, with sign changing coefficients, ptφx t] ft, xt, x t =, t,, x = x βx = lx a i x kx x mx mx i=l a i x, x mx c i x. To obtain these results, we use a fixed point theorem for cones in Banach spaces. Also we present an example that illustrates the main results.. Introduction As is well known, a differential equation defined on the interval a t b having the form ptx t] qt λrtxt =, a xa a 2 x a = a 3 xb a 4 x b =, is called a Sturm-Liouville boundary-value problem or Sturm-Liouville system. Here pt >, qt, the weighting function rt, the constants a, a 2, a 3, a 4 are given, and the eigenvalue λ is an unspecified parameter. Sturm-Liouville boundary-value problems for nonlinear second-order p-laplacian differential equations have been studied extensively; see for example, 2, 3, 4, 5, 6, 7, 8, 9,,, 5, 6]. The study of existence of positive solutions for such problems is complicated since there is no Green s function for the p-laplacian p 2. 2 Mathematics Subject Classification. 34B, 34B5, 35B. Key wor and phrases. Second order differential equation; positive solution multi-point boundary value problem. c 2 Texas State University - San Marcos. Submitted October 26, 29. Published February 4, 2. Supported by grants 6JJ58 and from the Natural Science Foundation of Hunan and Guangdong provinces, China.

2 2 J. XIA, Y. LIU EJDE-2/22 Some authors have extend Sturm-Liouville boundary conditions to nonlinear cases. For example, He, Ge and Peng 7], by means of the Leggett-Williams fixedpoint theorem, established criteria for the existence of at least three positive solutions to the one-dimensional p-laplacian boundary-value problem φy gtft, y =, t,, y B y =, y B y =,. where φv = v p2 v with v >, under the assumptions xb x, xb x and that there exist constants M i > such that B i x M i x, x R..2 In 9,,, 7], the authors extended. to a more general case. They established some existence results of at least one positive solution of Sturm-Liouville boundary value problems of higher order differential equations. Liu 2] studied the boundaryvalue problem φx n t] = ft, xt, x t,..., x n2 t, < t <, x i = for i =,,..., n 3, x n2 B x n =, B x n2 x n =..3 which is a general case of.. Liu 2] established existence of at least one positive solution of.3 without assumption.2. The Sturm-Liouville boundary conditions have been extended to multi-point cases. For example, Ma 5, 6] studied the problem ptx t] qtxt ft, xt =, t,, αx βpx = a i x, γx δpx = x,.4 where < ξ < < ξ m <, α, β, γ, δ, a i, with ρ = γβ αγ αδ >. By using Green s functions which is complicate for studying. and Guo- Krasnoselskii fixed point theorem 4, 6], the existence and multiplicity of positive solutions for.4 were given. There is no paper discussing the existence of multiple positive solutions of.4 by using Leggett-Williams fixed point theorem. Liu in, ] also studied some Sturm-Liouville type multi-point boundary value problems. In recent papers 8, 7], the authors studied the four-point boundary-value problem φx ft, xt, x t =, t,, αx βx ξ =, γx δx η =,.5 where φx = x p2 x, p >, φ x = x q2 x with /p /q =, α >, β, γ >, δ, < ξ < η <, f is continuous and nonnegative. When ξ and

3 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 3 η,.5 converges to a Sturm-Liouville boundary-value problem. So.5 can be seen as a generalized Sturm-Liouville boundary-value problem. Xu 8] proved the existence of at least one or two positive solutions of the differential equation x = φx t] atfxt =, t,, x = a i x, x x i=s c i x,.6 where < ξ < < ξ m <, a i,, c i, a and f are continuous functions, φx = x p2 x with p >. Motivated by above mentioned papers, we investigate the boundary-value problem ptφx t] ft, xt, x t =, t,, x = a i x a i x, i=l.7 x βx = x x c i x, where < ξ < < ξ m <, β, k, l m and a i,, c i for all i =,..., m; f is defined on, ] R R, continuous, nonnegative with ft,, on each subinterval of, ]; p is defined on, ], continuous and positive; φ is called p-laplacian, φx = x p2 x with p >, its inverse function is denoted by φ x with φ x = x q2 x with /p /q =. Sufficient conditions for the existence of at least three monotone positive solutions of.7 are established by using a fixed point theorem for cones in Banach spaces. We remark that the fixed point theorem used here is different form the one in 8]. Our results improve the the results in 8] since: Three positive solutions are obtained, while one or two positive solutions were obtained in 8]; the nonlinearity in.7 depen on t, x, x, while the nonlinearity in 8] depen only on t, x. The main result for this article is presented in Section 2, and an example is given in Section Main Results In this section, we first present some background definitions and state an important fixed point theorem. Then the main results are given and proved. Definition 2.. Let X be a semi-ordered real Banach space. The nonempty convex closed subset P of X is called a cone in X if x y P and ax P for all x, y P and a, and x X and x X imply x =.

4 4 J. XIA, Y. LIU EJDE-2/22 Definition 2.2. Let X be a semi-ordered real Banach space and P a cone in X. A map ψ : P, is a nonnegative continuous concave or convex functional map provided ψ is nonnegative, continuous and satisfies ψtx ty or tψx tψy for all x, y P, t, ]. Definition 2.3. Let X be a semi-ordered real Banach space. An operator T ; X X is completely continuous if it is continuous and maps bounded sets into precompact sets. Let a, a 2, a 3, a 4 > be positive constants, ψ be a nonnegative continuous functional on the cone P. Define the sets as follows: P β ; a 4 = {x P : β x < a 4 }, P β, α ; a 2, a 4 = {x P : α x a 2, β x a 4 }, P β, β 2, α ; a 2, a 3, a 4 = {x P : α x a 2, β 2 x a 3, β x a 4 } and a closed set Rβ, ψ; a, a 4 = {x P : ψx a, β x a 4 }. Theorem 2.4 3]. Let P be a cone in a real Banach space X with the norm. Suppose that T : P P is completely continuous; 2 β and β 2 be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P satisfying ψλx λψx for all x P and λ, ], and α x ψx and there exists a constant M > such that x Mβ x for all x P ; 3 there exist positive numbers a < a 2, a 3 and a 4 such that E T P β ; a 4 P β ; a 4 ; E2 α T x > a 2 for all x P β, α ; a 2, a 4 with β 2 T x > a 3 ; E3 {x P β, β 2, α ; a 2, a 3, a 4 : α x > a 2 } and α T x > b for all x P β, β 2, α ; a 2, a 3, a 4 ; E4 Rβ, ψ; a, a 4 and ψt x < a for all x Rβ, ψ; a, a 4 with ψx = a ; Then T has at least three fixed points x, x 2, x 3 P β ; a 4 such that β x i a 4, i =, 2, 3; α x > a 2, ψx 2 > a, α x 2 < a 2, ψx 3 < a. Choose X = C, ]. We call x y for x, y X if xt yt for all t, ], for x X, define its norm by x = max { max xt, max t,] t,] x t }. It is easy to see that X is a semi-ordered real Banach space. Let { P = y X : yt for all t, ], y t is decreasing on, ], yt ty for all t, ], x βpx = x x x,

5 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 5 x = a i x i=l } a i x. Then P is a nonempty cone in X. For σ, /2. Define functionals from P to R by β 2 y = β y = max t,] y t, ψx = max t,] yt, max yt, α y = min yt, y P. t σ,σ] t σ,σ] Let us list some conditions to be used in this article. A p :, ], is continuous; A2 a i,, c i for all i =,..., m satisfying i=l i=l p a i p a i φ p a i p p <, a i φ, p A2 a i, for all i =,..., m satisfying i=l i=l a i φ p p a i <, p <, a i φ, p <, A3 β ; A4 f :, ], R, is continuous with ft,, on each sub-interval of, ]; A5 θ :, ], is a continuous function and θt on each subinterval of, ]. It is easy to see that A2 hol if A2 hol. Lemma 2.5. Suppose that p :, ], with p C, ], x X, xt for all t, ] and ptφx t] on, ]. Then x is concave and xt min{t, t} max xt, t, ]. 2. t,] Proof. Suppose xt = max t,] xt. If t <, for t t,, since x t, we have pt φx t. Then ptφx t for all t t, ]. It follows that x t for all t, ]. Thus xt xt x. Let τt = t t φ ps t φ ps. Then τ Ct, ],, ] and is increasing on t, ] since dτ dt = φ pt t φ >, ps ; ;

6 6 J. XIA, Y. LIU EJDE-2/22 τt = and τ =. Thus which implies Hence φ dx dτ dx dt = dx dτ dτ dt = dx dτ φ pt t φ ps dx ptφx t = φ dτ φ t φ. ps φ pt t φ ps d 2 x dτ 2 = φ t φ ps ptφx t ] for all t t, ]. Note that φ x. It follows that d2 x dτ. Together with 2 x τ τ, ]. Then x is decreasing on, ]. We get that there exist t η t ξ such that xt x t xt x t It follows that for t t,, xt x t xt x t = t xt xt] t tx xt] t t = t t tx η t t ttx ξ t t t t tx ξ t t ttx ξ t t =. = x t t xt txt. t t If t >, for t, t, similarly to above discussion, we have xt txt, t, t. Then one gets that xt min{t, t} max t,] xt for all t, ]. The proof is complete. Consider the boundary-value problem x βx = ptφx t] θt =, t,, x = a i x a i x, x i=l x i=s c i x. 2.2 Lemma 2.6. Suppose that A A5 hold. If x is a solution of 2.2, then x is concave, decreasing and positive on,. Proof. Suppose x satisfies 2.2. It follows from the assumptions that px is decreasing on, ]. Lemma 2.5 implies that x is decreasing on, ]. Then x is concave on, ].

7 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 7 First, we prove that x. One sees from A2 and the deceasing property of ptφx t that x = i=l m = i=l m i=l m i=l It follows that a i φ φ p φx p a i φ φ pξ l φx ξ l p i=l a i φ φ pξ l φx ξ l p a i φ p a i φ p φx p a i φ φ pξ l φx ξ l p a i φ φ pξ l φx ξ l φ pξ l φx ξ l ] p a i φ p a i φ p i=l a i φ φ pξ l φx ξ l p a i φ φ pφx. p p a i p p a i x. p It follows that x. Then x t for all t, ]. Second, we prove that x. It follows from the boundary conditions in 2.2 that x βx = x x c i x x xξ k = x xξ k xξ k xξ k x. i=s xξ k xξ k ]

8 8 J. XIA, Y. LIU EJDE-2/22 Then x βx. We get x since x and β. Then xt > x for all t,. The proof is complete. Lemma 2.7. Suppose that A,A2, A3 A5 hold. Let µ = φ m i=l a iφ p. p If y is a solution of 2.2, then where and B θ = yt = B θ t A θ = φ ps pφa θ ps i=l A θ φ θudu µp θudu, t, ], ],, a i φ p pφa θ θudu p a i φ p pφa θ p θudu k m b βφ i p pφa θ p φ ps pφa θ ps φ ps pφa θ ps c i φ p pφa θ p Proof. Since y is solution of 2.2, y t = φ pt pφy pt yt = y t θudu θudu ] θudu. t φ ps pφy ps The boundary conditions in 2.2 imply y = a i φ pξ i pφy p a i φ p pφy p i=l θudu, θudu. θudu θudu θudu

9 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 9 and y βφ p pφy p = y φ y θudu ps pφy ps φ ps pφy ps c i φ p pφy p θudu. θudu It follows that y = k m b βφ i p pφy p φ ps pφy ps φ ps pφy ps c i φ p pφy p θudu θudu ] θudu. Lemma 2.6 implies that y. One fin that y = a i φ pξ i pφy p a i φ p pφy p i=l i=l a i φ p pφy p If y, one gets a i φ p p φy p Let Then Note that i=l Gc = c Gc c = i=l i=l µ = φ θudu θudu θudu θudu. θudu θudu. 2.3 a i φ p pφc θudu. p a i φ p p φc m i=l a iφ p p p. θudu.

10 J. XIA, Y. LIU EJDE-2/22 It is easy to see that Gc c is decreasing on, and on,. First, since lim t Gc/c = and Gc lim = a i φ c c p p >, i=l we get that Gc c > for all c >. Second, we have lim c Gc/c = and R G φ θudu µp R = a i φ φ θudu p p pµ ξi µp i=l θudu θudu p = i=t φ µ It follows from 2.3 that Gy y The proof is complete. a i φ ξi p p µ i=l. We get a i φ p =. p y φ θudu. µp Define the nonlinear operator T : X X by T xt = B x for t, ], where A x = B x = t φ ps pφa x ps A σ φ fu, xu, x udu φ p i=l P m i=t aiφ p p θudu θudu p p fu, xu, x udu, ],, a i φ p pφa x fu, xu, x udu p a i φ p pφa x p k m βφ p pφa x p φ ps pφa x ps fu, xu, x udu, fu, xu, x udu φ ps pφa x ps fu, xu, x udu fu, xu, x udu

11 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS c i φ p pφa x ] fu, xu, x udu. p Lemma 2.8. Suppose that A,A2, A3, A4 hold. Then i the following equalities hold: ptφt y t] ft, yt, y t =, t,, T y = a i T y a i T y, T y βt y = T y i=l T y c i T y ; ii T y P for each y P ; iii x is a positive solution of.7 if and only if x is a solution of the operator equation y = T y in P ; iv T : P P is completely continuous. Proof. The proofs of i, ii and iii are simple. To prove iv, it suffices to prove that T is continuous on P and T is relative compact. We divide the proof into two steps: Step. For each bounded subset D P, and each x D, since ft, u, v is continuous, we can prove that T is continuous at yt. Step 2. For each bounded subset D P, prove that T is relative compact on D. It is similar to that of the proof of Lemmas in 3] and are omitted. Lemma 2.9. Suppose that A, A2, A3, A4 hold. constant M > such that max T xt M max T t,] t,] x t for each x P. Then there exists a Proof. For x P, Lemma 2.6 implies that T xt and T x t for all t, ]. Lemma 2.8 implies T x βt x = T x Then there exist numbers η i, ] such that T x c i T x. T x = T x k T x m T x k m = βt x m c it x k m k T x T x] m T x T x] k m = βt x m c it x k m

12 2 J. XIA, Y. LIU EJDE-2/22 It follows that T xt k T x η i m T x η i k m b. i = T xt T x T x T x t T x ξ where ξ t, ] Then βp k m m k m max T xt t,] βp k m m k m c i c i It follows that there exists a constant M > such that for all x P, Denote max T xt Mβ T x. t,] µ = φ m i=t a iφ p p, M = βp k m m k m L = βφ µ µp c i φ µξi µp φ µs µps φ µs µps L 2 = βφ φ s m p ps c i φ p φ s. ps max T t,] x t. max T t,] x t. c i, φ µs µps, φ s ps Theorem 2.. Choose k,. Suppose that A, A2, A3, A4 hold. Let e, e 2, c be positive numbers and { k c Q = min φ m b i µpφc }, ; L µ k e2 W = φ m b i ; L 2

13 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 3 k e E = φ m b i. L If Mc > e 2 > e k > e > and A6 ft, u, v Q for all t, ], u, Mc], v c, c]; A7 ft, u, v W for all t, k], u e 2, e 2 /k], v c, c]; A8 ft, u, v E for all t, ], u, e ], v c, c]; then.7 has at least three solutions x, x 2, x 3 such that x < e, x 2 k > e 2, x 3 > e, x 3 k < e 2. Proof. To apply Theorem 2.4, we check that all its conditions are satisfied. By the definitions, it is easy to see that α is a nonnegative continuous concave functional on the cone P, β, β 2 are three nonnegative continuous convex functionals on the cone P, ψ a nonnegative continuous functional on the cone P. Lemma 2.8 implies that x = xt is a positive solution of.7 if and only if x is a solution of the operator equation y = T y and T is completely continuous. For x P, one sees that ψλx λψx for all x P and λ, ], and α x ψx for all x P. There exists a constant M > such that x Mβ x for all x P. Then and 2 of Theorem 2.4 hold. Corresponding to Theorem 2.4, we have a 4 = c, a 3 = e2 k, a 2 = e 2, a = e. Now, we check that.3 of Theorem 2.4 hol. One sees that < a < a 2, a 3 >, a 4 >. The rest is divided into four steps. Step. Prove that T P β ; a 4 P β ; a 4 ; For x P β ; a 4, we have β x a 4 = c. Then xt max xt Mc for t, ], t,] c x t c for all t, ]. So A6 implies that ft, xt, x t < Q for t, ]. Then Lemma 2.7 implies Since A x φ fu, xu, x udu φ Q. µp µp c max { e 2, La, φ 2 a, φ } a, σ p p we obtain φa min { φ c φcp,, φcp } = Q. L 2 Then max t,] T yt = T y implies T x = B x φ ps pφa x ps = k m βφ p pφa x p fu, xu, x udu fu, xu, x udu

14 4 J. XIA, Y. LIU EJDE-2/22 = = φ ps pφa x ps φ ps pφa x ps c i φ p pφa x φ ps pφa x ps p k m βφ p pφa x p φ fu, xu, x udu fu, xu, x udu ] fu, xu, x udu fu, xu, x udu ps pφa x ps fu, xu, x udu φ ps pφa x ps c i φ p pφa x p k m φ Q µps Qs ps φ ps pφa x ps βφ Q µp m c i φ Q µp Q p φ Q k m βφ µ µp φ µs µps φ φ µs µps fu, xu, x udu fu, xu, x udu fu, xu, x udu Q p µs µps ] fu, xu, x udu φ φ c i φ µξ i µp ] c. Q µps Qs ps Q µps Qs ] ps

15 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 5 On the other hand, similarly to above discussion, since T y t is decreasing and T y, from Lemma 2.7 we have T x = φ p pφa x fu, xu, x udu, p max T t,] x t = φ p pφa x p φ Q µp Q c. p fu, xu, x udu It follows that T x = max { max t,] T xt, max t,] T x t } c. Then T P β ; a 4 P β ; a 4. This completes the proof of E in Theorem 2.4. Step 2. Prove that α T x > a 2 for all x P β, α ; a 2, a 4 with β 2 T x > a 3 ; For y P β, α ; a 2, a 4 = P β, α ; e 2, c with β 2 T y > a 3 = e2 k, we have α y = min t,k] yt e 2, β y = max t,] y t c, max T yt > e 2 t,k] k. Then α T y = min T yt kβ 2T y > k e 2 t,k] k = e 2 = a 2. This completes the proof of E2 in Theorem 2.4. Step 3. Prove that {x P β, β 2, α ; a 2, a 3, a 4 : α x > a 2 } = and α T x > b for all x P β, β 2, α ; a 2, a 3, a 4 ; It is easy to check that {x P β, β 2, α ; a 2, a 3, a 4 : α x > a 2 }. For y P β, β 2, α ; a 2, a 3, a 4, one has that α y = min t,k] yt e 2, Then Thus A7 implies β 2 y = max t,k] yt e 2 k, e 2 yt e 2 k, t, k], y t c. ft, yt, y t W, t, k]. Since α T y = min T yt k max T yt, t,k] t,] we obtain α T y k max t,] T yt. Then From A x, we obtain α T y k α T y k max T yt kt y. t,] k m βφ p pφa x p φ ps pφa x ps β y = max t,] y t c. fu, xu, x udu φ ps pφa x ps fu, xu, x udu fu, xu, x udu

16 6 J. XIA, Y. LIU EJDE-2/22 = = c i φ p pφa x φ ps pφa x ps k p k m βφ p pφa x p ps φ fu, xu, x udu ] fu, xu, x udu ps pφa x ps fu, xu, x udu φ ps pφa x ps c i φ p pφa x p fu, xu, x udu k k m βφ p pφa x p ps φ φ ps pφa x ] ps pφa x ps fu, xu, x udu fu, xu, x udu fu, xu, x udu fu, xu, x udu φ ps pφa x ps c i φ p pφa x p φ ps pφa x ] fu, xu, x udu k m βφ W p fu, xu, x udu fu, xu, x udu fu, xu, x udu φ W s m ps φ W s ps

17 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 7 c i φ W p φ W s ] ps φ W = k m βφ φ s m p ps c i φ p φ φ s ] = e 2. ps s ps This completes the proof of E3 in Theorem 2.4. Step 4. Prove that Rβ, ψ; a, a 4 and that ψt x < a for all x in Rβ, ψ; a, a 4 with ψx = a ; It is easy to see that Rβ, ψ; a, a 4. For y Rβ, ψ; a, a 4 with ψx = a, one has that Hence ψy = max t,] yt = a, β y = max t,] y t a 4. yt a, t, ]; c y t c, t, ]. Then A8 implies ft, yt, y t E, t, ]. So ψt y = max T yt = T y t,] = k m βφ p pφa x p φ ps pφa x ps fu, xu, x udu φ ps pφa x ps c i φ p pφa x p k m φ ps pφa x ps βφ E φ µps Es ps E µp c i φ E µp Eξ i p fu, xu, x udu fu, xu, x udu fu, xu, x udu E p ] fu, xu, x udu φ φ E µps Es ps E µps Es ] ps

18 8 J. XIA, Y. LIU EJDE-2/22 φ E = k m φ µs µps βφ µ µp φ µs µps c i φ µ µp ] a. φ µs µps This completes the proof of E4 in Theorem 2.4. Then Theorem 2.4 implies that T has at least three fixed points x, x 2 and x 3 such that βx < e, αx 2 > e 2, βx 3 > e, αx 3 < e 2. Hence.7 has three decreasing positive solutions x, x 2 and x 3 as needed in the statement of the theorem; therefore, the proof is complete. 3. Examples Now, we present an example that illustrates our main results, and that can not be solved by results in 3, 9,, 5, 6]. Example 3.. Consider the boundary-value problem x t ft, xt, x t =, t,, x = 4 x /4 2 x /2, 3. x 2x = 4 x/4 4 x/2 2 x /4 4 x /2, where ft, u, v = f u t v 255 and u u, 4], 5 u 4, 2], f u = 42 u 4 u 2, 4], 564 u 4, 44], 564 u 44, 24], 564e u24 u 24. Corresponding to.7, one sees that φ x = x, ξ = /4, ξ 2 = /2, a = /4, a 2 = /2, b = /4, b 2 = /4, c = /2, c 2 = /4. It is easy to see that A A4 hold. Choose constants k = 2, e = 2, e 2 = and c = 2. One obtains M = βp k m m c i k m µ = φ m i=t a iφ p p =, = 5 4,

19 EJDE-2/22 MONOTONE POSITIVE SOLUTIONS 9 L = βφ µ µp φ µs µps φ µs µps φ µs, µps c i φ µξ i µp L 2 = βφ φ s p ps φ s m c i φ ps p Q = min { c φ It is easy to see that If Mc > e 2 > e k φ s, ps µpφc }, ; µ ; L 2. L k m L e 2 k m W = φ e k m E = φ > e > and ft, u, v Q for all t, ], u, Mc], v 2, 2]; ft, u, v W for all t, /2], u, 2], v 2, 2]; ft, u, v E for all t, ], u, 2], v 2, 2]; Then A6, A7 and A8 hold. Theorem 2. implies that 3. has at least three positive solutions such that x < 2, x 2 k >, x 3 > 2, x 3 k <. References ] R. I. Avery; A generalization of the Leggett-Williams fixed point theorem, MSR Hot-Line, 3999, ] R. I. Avery and A. C. Peterson; Three Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces, Computers Mathematics with Applications, 422, ] Z. Bai, Z. Gui, W. Ge; Multiple positive solutions for some p-laplacian boundary value problems. J. Math. Anal. Appl. 324, ] K. Deimling; Nonlinear Functional Analysis, Springer, Berlin, Germany, ] W. Ge; Boundary Value Problems for Ordinary Differential Equations, Science Press, Beijing, 27. 6] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cones, Academic Press, Orlando, ] X. He, W. Ge, M. Peng; Multiple positive solutions for one-dimensional p-laplacian boundary value problems, Appl. Math. Letters,

20 2 J. XIA, Y. LIU EJDE-2/22 8] H. Lian, W. Ge; Positive solutions for a four-point boundary value problem with the p- Laplacian, Nonl. Anal. 27, doi:.6/j.na ] Y. Liu, W. Ge; Multiple positive solutions to a three-point boundary value problem with p-laplacian, J. Math. Anal. Appl , ] Y. Liu; Solutions of Sturm-Liouville type multi-point boundary value problems for higherorder differential equations, J. Appl. Math. Computing, 2327, ] Y. Liu; Solutions of Sturm-liouville boundary value problems for higher-order differential equations, J. Appl. Math. Computing, 2427, ] Y. Liu; An Existence Result for Solutions of Nonlinear Sturm-Liouville Boundary Value Problems for Higher-Order p-laplacian Differential Equations, Rocky Mountain J. Math., to appear. 3] Y. Liu; Non-homogeneous boundary-value problems of higher order differential equations with p-laplacian. Electronic Journal of Diff. Eqns. 2828, no. 2, pp 43. 4] Y. Liu; Solutions to second order non-homogeneous multi-point BVPs using a fixed point theorem, Electronic Journal of Diff. Eqns. 2828, no. 96, pp 52. 5] R. Ma; Multiple positive solutions for nonlinear m-point boundary value problems. Appl. Math. Comput. 4824, ] R. Ma, B. Thompson; Positive solutions for nonlinear m-point eigenvalue problems. J. Math. Anal. Appl , ] H. Pang, et al.; Multiple positive solutions for second-order four-point boundary value problem, Comput. Math. Appl. 27, doi:.6/j.camwa ] F. Xu; Positive solutions for multipoint boundary value problems with one-dimensional p- Laplacian operator, Appl. Math. Comput., 94 27, pp Jianye Xia Department of Mathematics, Guangdong University of Finance, Guangzhou 532, China address: JianyeXia@sohu.com Yuji Liu Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 44, China address: liuyuji888@sohu.com