MULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS

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1 Electronic Journal of Differential Equations, Vol. 27(27, No. 23, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp MULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS HANYING FENG, MEIQIANG FENG, MING JIANG, WEIGAO GE Abstract. In this paper, we study the three-point boundary-value problem for a fourth-order one-dimensional p-laplacian differential equation `φp(u (t + a(tf`u(t, t (, 1, subject to the nonlinear boundary conditions: u( ξu(1, u (1 ηu (, (φ p(u ( α 1 (φ p(u (δ, u (1 p 1p β 1 u (δ, where φ p(s s p 2 s, p > 1. Using the five functional fixed point theorem due to Avery, we obtain sufficient conditions for the existence of at least three positive solutions. 1. Introduction This paper concerns the existence of three positive solutions for the fourth-order three-point boundary-value problem (BVP for short consisting of the p-laplacian differential equation (φp (u (t + a(tf(u(t, t (, 1, (1.1 with the nonlinear boundary conditions u( ξu(1, u (1 ηu (, (φ p (u ( α 1 (φ p (u (δ, u (1 p 1 β 1 u (δ, (1.2 where f : R [, + and a : (, 1 [, + are continuous functions, φ p (s s p 2 s, p > 1, α 1, β 1, ξ 1, η 1 and < δ < 1. Two-point boundary-problems for differential equation are used to describe a number of physical, biological and chemical phenomena. For additional background and results, we refer the reader to the monograph by Agawarl, O Regan and Wong [1] as well as to the recent contributions by [2, 9, 13, 14, 2]. Boundary-value problems for n-th order differential equation [15, 16, 22] and even-order can arise, especially for fourth-order equations, in applications, see [4, 5, 6, 7, 8] and references therein. 2 Mathematics Subject Classification. 34B1, 34B15, 34B18. Key words and phrases. Fourth-order boundary-value problem; one-dimensional p-laplacian; five functional fixed point theorem; positive solution. c 27 Texas State University - San Marcos. Submitted November 3, 26. Published February 4, 27. Supported by grants from NNSF, and from SRFDP of China. 1

2 2 H. FENG, M. FENG, M. JIANG, W. GE EJDE-27/23 Recently, three-point boundary-value problems of the differential equations were presented and studied by many authors, see [1, 11, 12, 21] and the references cite there. However, three-point BVP (1.1, (1.2 have not received as much attention in the literature as Lidstone condition BVP u (t a(tf(u(t, t (, 1, u( u(1 u ( u (1, and the three-point BVP for the second-order differential equation (1.3 u (t + a(tf(u(t, t (, 1, u(, u(1 αu(η, (1.4 that were extensively considered, in [13, 14, 2] and [21], respectively. The results of existence of positive solutions of BVP (1.1, (1.2 are relatively scarce. Most recently, Liu and Ge studied two class of four-order four-point BVPs successively in [17, 18]. They proved that existence of at least two or three positive solutions. To the best of our knowledge, existence results of multiple positive solutions for fourth-order three-point BVP (1.1, (1.2 have not been found in literature. Motivated by the works in [17, 18], the purpose of this paper is to establish the existence of at least three positive solutions of (1.1, (1.2. For the remainder of the paper, we assume that: (i < a(sds < ; (ii q satisfies 1 p + 1 q 1 and φ q(z z q 2 z. 2. Background and definitions For the convenience of the reader, we provide some background material from the theory of cones in Banach spaces. We also state in this section a fixed point theorem by Avery. Definition 2.1. Let X be a real Banach space. A nonempty closed set P X is said to be a cone provided that (i x P and λ implies λx X, and (ii x P and x P implies x. Every cone P X induces an ordering in X given by x y if and only if y x P. Definition 2.2. The map ψ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that ψ : P [, is continuous and ψ(tx + (1 ty tψ(x + (1 tψ(y for all x, y P and t 1. Similarly, we say the map β is a nonnegative continuous convex functional on a cone P of a real Banach space E provided that β : P [, is continuous and for all x, y P and t 1. β(tx + (1 ty tβ(x + (1 tβ(y

3 EJDE-27/23 MULTIPLE POSITIVE SOLUTIONS 3 Let γ, β, θ be nonnegative, continuous, convex functionals on P and α, ψ be nonnegative, continuous, concave functionals on P. Then for nonnegative numbers h, a, b, d and c we define the following sets: P (γ, c {x P : γ(x < c}, P (γ, α, a, c {x P : a α(x, γ(x c}, Q(γ, β, d, c {x P : β(x d, γ(x c}, P (γ, θ, α, a, b, c {x P : a α(x, θ(x b, γ(x c}, Q(γ, β, ψ, h, d, c {x P : h ψ(x, β(x d, γ(x c}. To prove our results, we need the following Five Functionals Fixed Point Theorem due to Avery [3] which is a generalization of the Leggett-Williams fixed point theorem. Theorem 2.1. Suppose X is a real Banach space and P is a cone of X, γ, β, θ are three nonnegative, continuous, convex functionals and α, ψ are nonnegative, continuous, concave functionals such that α(x β(x, x Mγ(x for x P (γ, c and some positive numbers c, M. Again, assume that T : P (γ, c P (γ, c be a completely continuous operator and there are positive numbers h, d, a, b with < d < a such that (i {x P (γ, θ, α, a, b, c : α(x > a} and x P (γ, θ, α, a, b, c implies α(t x > a. (ii {x Q(γ, β, ψ, h, d, c : β(x < d} and x Q(γ, β, ψ, h, d, c implies β(t x < d. (iii x P (γ, α, a, c with θ(t x > b implies α(t x > a. (iv x Q(γ, β, d, c with ψ(t x < h implies β(t x < d. Then T has at least three fixed points x 1, x 2, x 3 P (γ, c such thah β(x 1 < d, a < α(x 2, d < β(x 3, with α(x 3 < a. 3. Related lemmas Lemma 3.1 ([17]. Suppose f C(R, R, then the three-point BVP y f(t, t (, 1 y ( α 1 y (δ, y(1 β 1 y(δ (3.1 has a unique solution y(t g(t, sf(sds, t (, 1,

4 4 H. FENG, M. FENG, M. JIANG, W. GE EJDE-27/23 where M (1 α 1 (1 β 1 and 1 β 1 δ t + β 1 t, if s t < δ < 1 or s δ t 1, g(t, s 1 1 β 1 δ + (1 β 1 (α 1 s s α 1 t, if t s δ < 1, M 1 α 1 β 1 s + α 1 β 1 s t +α 1 t + β 1 t + α 1 β 1 t, if δ s t 1, (1 s(t α 1, if < δ t s 1 or t < δ s 1. Lemma 3.2 ([19]. Suppose f C(R, R, then the two-point BVP has a unique solution y f(t, t (, 1 y( ξy(1, y (1 ηy ( y(t h(t, sf(sds, t [, 1], where M 1 (1 ξ(1 η and h(t, s 1 M 1 { s + η(t s + ξη(1 t, s t 1, t + ξ(s t + ξη(1 s, t s 1. (3.2 Remark 3.3. It is easy to check that if α 1 < 1, and β 1 < 1, then g(t, s for (t, s [, 1] [, 1]. If ξ, η and M 1 (1 ξ(1 η, then h(t, s for (t, s [, 1] [, 1] If u(t is a solution of BVP (1.1 and (1.2. By Lemma 3.1 and (3.1, one has φ p (u (t By Lemma 3.2 and (3.2, one obtains u(t h(t, sφ q ( g(t, sa(sf(u(sds. (3.3 g(s, τa(τf(u(τdτ ds. (3.4 Lemma 3.4 ([19]. Suppose ξ, η < 1, < < t 2 < 1 and δ (, 1. Then, for s [, 1], h(, s h(t 2, s, t 2 (3.5 h(1, s h(δ, s 1 δ. (3.6 Lemma 3.5 ([19]. Suppose ξ, η > 1, < < t 2 < 1 and δ (, 1. Then, for s [, 1], h(t 2, s h(, s 1 t 2, 1 (3.7 h(, s h(δ, s 1 1 δ. (3.8

5 EJDE-27/23 MULTIPLE POSITIVE SOLUTIONS 5 4. Triple positive solutions to (1.1, (1.2 Now let the Banach space E C[, 1] be endowed with the maximum norm u max t [,1] u(t. Let the two cones P 1, P 2 X defined by P 1 {u X : u(t is nonnegative, concave, nondecreasing on (, 1}, P 2 {u X : u(t is nonnegative, concave, nonincreasing on (, 1}. Next, choose, t 2, t 3 (, 1 and < t 2. Define nonnegative, continuous, concave functions α, ψ and nonnegative, continuous, convex functions β, θ, γ on P 1 by γ(u max t [,t 3] u(t u(t 3, u P 1, ψ(u min t [δ,1] u(t u(δ, u P 1, β(u max t [δ,1] u(t u(1, u P 1, α(u θ(u min u(t u(, u P 1, t [,t 2] max u(t u(t 2, u P 1. t [,t 2] It is easy to verify that α(u u( u(1 β(u and u u(1 1 t 3 u(t 3 1 t 3 γ(u for u P 1. Theorem 4.1. Suppose ξ, η < 1, α 1 < 1, β 1 < 1 and there exist numbers < a < b < c such that < a < b < t2 b c and f(w satisfies the following conditions: where A B C f(w < φ p ( a, w a, (4.1 C f(w > φ p ( b B, b w t 2 b, (4.2 f(w φ p ( c A, w 1 t 3 c, (4.3 h(t 3, sφ q ( h(, sφ q ( t 2 h(1, sφ q ( g(s, τa(τdτ ds, g(s, τa(τdτ ds, g(s, τa(τdτ ds. Then (1.1, (1.2 has at least three positive solutions u 1, u 2, u 3 P 1 (γ, c such that with u 3 (1 > a, u i (δ c for i 1, 2, 3. u 1 ( > b, u 2 (1 < a, u 3 ( < b (4.4 Proof. We begin by defining the completely continuous operator T : P 1 X by (3.4 as ( (T u(t h(t, sφ q g(s, τa(τf(u(τdτ ds

6 6 H. FENG, M. FENG, M. JIANG, W. GE EJDE-27/23 for u P 1. It is easy to prove that (1.1, (1.2 has a positive solution u u(t if and only if the operator T has a fixed point on P 1. Firstly, we prove T : P 1 (γ, c P 1 (γ, c. For u P 1, by Remark 3.1, it is easy to check that T u. Moreover, ( (T u (t (1 ξ η + t t φ q ( g(s, τa(τf(u(τdτ ds ( φ q g(s, τa(τf(u(τdτ ds and (T u (t φ q ( g(t, sa(sf(u(sds. So, we have T P 1 P 1. For u P 1 (γ, c, u(t u 1 t 3 γ(u 1 t 3 c. By (4.3, it follows that γ(t u max (T u(t (T u(t 3 t t 3 c A h(t 3, sφ q ( h(t 3, sφ q ( c A A c. h(t 3, sφ q ( g(s, τa(τf(u(τdτ ds g(s, τa(τdτ φ p (c/a ds g(s, τa(τdτ ds Thus, T : P 1 (γ, c P 1 (γ, c. Secondly, by taking It is immediate that u 1 (t b + ε 1 for < ε 1 < t 2 b b, u 2 (t a ε 2 for < ε 2 < a δa, u 1 (t {P (γ, θ, α, b, t 2 b, c : α(u > b}, u 2 (t {Q(γ, β, ψ, δa, a, c : β(u < a}. In the following steps, we verify the remaining conditions of Theorem 2.1. Now the proof is divided into four steps. Step 1: We prove that u P (γ, θ, α, b, t 2 b, c implies α(t u > b. (4.5

7 EJDE-27/23 MULTIPLE POSITIVE SOLUTIONS 7 In fact, u(t u( α(u b for t t 2, and u(t u(t 2 θ(u t2 b for t t 2. Thus using (4.2, one gets Step 2: We show that α(t u min (T u(t (T u( t t 2 > b B h(, sφ q ( h(, sφ q ( t 2 h(, sφ q ( t 2 b B B b. h(, sφ q ( t 2 g(s, τa(τf(u(τdτ ds g(s, τa(τf(u(τdτ ds g(s, τa(τdτ φ p (b/b ds g(s, τa(τdτ ds u Q(γ, β, ψ, δa, a, c implies β(t u < a. (4.6 In fact, u(t u(1 β(u a for t 1, Thus using (4.1, one arrives at Step 3: We verify that By Lemma 3.4, β(t u max (T u(t (T u(1 < δ t 1 a C h(1, sφ q ( h(1, sφ q ( a C C a. h(1, sφ q ( g(s, τa(τf(u(τdτ ds g(s, τa(τdτ φ p (a/c ds g(s, τa(τdτ ds u Q(γ, β, a, c with ψ(t u < δa implies β(t u < a. (4.7 β(t u max (T u(t (T u(1 Step 4: We prove that δ t 1 h(1, sφ q ( h(1, s h(δ, s h(δ, sφ q g(s, τa(τf(u(τdτ ds ( 1 δ (T u(δ 1 ψ(t u < a. δ g(s, τa(τf(u(τdτ ds u P (γ, α, b, c with θ(t u > t 2 b implies α(t u > b. (4.8

8 8 H. FENG, M. FENG, M. JIANG, W. GE EJDE-27/23 By Lemma 3.4, α(t u min (T u(t (T u( t t 2 h(, sφ q ( h(, s ( h(t 2, s h(t 2, sφ q t 2 (T u(t 2 t 2 θ(t u > b. g(s, τa(τf(u(τdτ ds g(s, τa(τf(u(τdτ ds Therefore, the hypotheses of Theorem 2.1 are satisfied and there exist three positive solutions x 1, x 2, x 3 for BVP (1.1, (1.2 satisfying (4.4. Similarly, choose, t 2, t 3 (, 1 and < t 2. Define nonnegative, continuous, concave functions α, ψ and nonnegative, continuous, convex functions β, θ, γ on P 2 by γ(u max t [t 3,1] u(t u(t 3, x P 2, ψ(x min t [,δ] u(t u(δ, u P 2, β(u max t [,δ] u(t u(, u P 2, α(u θ(u min u(t u(t 2, u P 2, t [,t 2] max u(t u(, u P 2. t [,t 2] It is easy to verify that α(u u(t 2 u( β(u and u u( 1 t 3 u(t 3 1 t 3 γ(u for u P 2. So, we obtain the following result. Theorem 4.2. Suppose ξ, η > 1, α 1 < 1, β 1 < 1 and there exist numbers < a < b < c such that < a < b < 1 t1 1 t 2 b c and f(w satisfy the following conditions: where A B C f(w < φ p ( a, w a, (4.9 C f(w > φ p ( b B, b w 1 1 t 2 b, (4.1 f(w φ p ( c A, w 1 t 3 c, (4.11 h(t 3, sφ q ( h(t 2, sφ q ( t 2 h(, sφ q ( g(s, τa(τdτ ds, g(s, τa(τdτ ds, g(s, τa(τdτ ds. Then (1.1, (1.2 has at least three positive solutions u 1, u 2, u 3 P 1 (γ, c such that u 1 (t 2 > b, u 2 ( < a, u 3 (t 2 < b,

9 EJDE-27/23 MULTIPLE POSITIVE SOLUTIONS 9 with u 3 ( > a and u i (δ c for i 1, 2, 3. Since the proof of the above theorem is similar to that of Lemma 3.5, we omit it. Acknowledgments. The authors wish to thank the referee for his (or her valuable corrections to the original manuscript. References [1] R. P. Agarwal, D. O. Regan, and P. J. Y. Wong, Positive solutions of Differential, Difference, and Integral Equations, Kluwer Academic, Dordrecht, [2] R. I. Avery, J. Henderson, Three symmetric positive solutions for a second-order boundaryvalue problem, Appl. Math. Lett. 13 ( [3] R. I. Avery, A generalization of the Leggett-Williams fixed-point theorem, Math. Sci. Res. Hot-line, 2 ( [4] L. E. Bobisud, D. O Regan, W. D. Royalty, Existence and nonexistence for a singular boundary-value problem, Appl. Anal. 28 (1998, [5] L. E. Bobisud, D. O Regan, W. D. Royalty, Solbability of some nonlinear boundary-value problems, Nonl. Anal. 12 (1998, [6] L. E. Bobisud, Existence of solutions for a singular boundary-value problem, Appl. Anal. 35 (199, [7] L. E. Bobisud, Y. S. Lee, Existence for a class of nonliear boundary-value problems, Appl. Anal. 38 (199, [8] C. J. Chyan, J. Henderson, Positive solutions of 2m th boundary-value problems, Appl. Math. Letters 15 (22, [9] P. W. Eloe, J. Henderson, Positive solutions and nonlinear (k, n k conjugate eigenvalue problem, Diff. Equ. Dynam. Syst. 6 (1998, [1] W. Feng, J. R. L. Webb, Solvability of a three-point boundary-value problem at resonance, Nonl. Appl. TMA 3(6 ( [11] C. P. Gupta, Solvability of a three-point nonlinear boundary-value problem for a second order ordinary differential equation, J. Math. Anal. Appl. 168 ( [12] C. P. Gupta, Trofimchuk, Existence of a solution to a three-point boundary-value problem and the spectral radius of o related linear operator, Nonl. Anal. TMA 34 ( [13] J. R. Graef, B. Yang, On a nonlinear boundary-value problem for fourth-order equations, Appl. Anal. 72 (1999, [14] J. R. Graef, B. Yang, Existence and non-existence of positive solutions of fourth-order nonlinear boundary-value problems, Appl. Anal. 74(2, [15] J. Henderson, H. B. Thompson, Existence of multiple positive solutions for some nth order boundary-value problems, Nonlinear Anal., 7 ( [16] B. Liu, J. Yu, Solvability of multi-point boundary-value problems at resonance(ii, Appl. Math. Comput., 136 ( [17] Y. Liu, W. Ge, Solbablity a nonlinear four-point boundary problem for a fouth order differential equation, Taiwanes J. Math. (7 ( [18] Y. Liu, W. Ge, Existence theorems of positive solutions for fourth-order a Solbablity a nonlinear four-point boundary problems, Anal. Appl. (2 ( [19] D. Ma, The existence of solution for boundary-value problem with a p-laplacian operator, Doctoral Thesis, 26. [2] R. Y. Ma, H. Y. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59 (1995, [21] R. Y. Ma, Positive solutions of nonlinear three point boundary-value problem, Electronic J. Diff. Equs (1998, No. 34, 1-8. [22] P. K. Palamides, Multi point boundary-value problems at resonance for n-order differential equations. positive and monotone solutions, Electron. J. Diff. Eqns., Vol. 24 (24, No 25 pp

10 1 H. FENG, M. FENG, M. JIANG, W. GE EJDE-27/23 Hanying Feng Department of Mathematics, Beijing Institute of Technology, Beijing 181, China. Department of Mathematics, Shijiazhuang, Mechanical Engineering College, Shijiazhuang 53, China address: Meiqiang Feng Department of Mathematics, Beijing Institute of Technology, Beijing 181, China Department of Fundamental Sciences, Beijing Information Technology Institute, Beijing, 111, China address: Ming Jiang Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 53, China address: Weigao Ge Department of Mathematics, Beijing Institute of Technology, Beijing 181, China address: