POSITIVE SOLUTIONS FOR A FUNCTIONAL DELAY SECOND-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM

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1 Electronic Journal of Differential Equations, Vol. 26(26, No. 4, pp.. ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp POSITIVE SOLUTIONS FOR A FUNCTIONAL DELAY SECOND-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM CHUANZHI BAI, XINYA XU Abstract. We establish criteria for the existence of positive solutions to the three-point boundary-value problems expressed by second-order functional delay differential equations of the form x (t = f(t, x(t, x(t τ, x t, < t <, x = φ, x( = x(η, where φ C[ τ, ], < τ <, and τ < η <.. Introduction In recent years, many authors have paid attention to the research of boundaryvalue problems for functional differential equations because of its potential applications (see, for example, [, 3, 5, 6, 7, 8, 9,, ]. In a recent paper [7], by applying a fixed-point index theorem in cones, Jiang studied the existence of multiple positive solutions for the boundary-value problems of second-order delay differential equation y (x + f(x, y(x τ =, < x <, (. y(x =, τ x, y( =, where < τ < and f C([, ] [, +, [,. For τ >, let C(J be the Banach space of all continuous functions ψ : [ τ, ] =: J R endowed with the sup-norm ψ J := sup{ ψ(s : s J}. For any continuous function x defined on the interval [ τ, ] and any t I =: [, ], the symbol x t is used to denote the element of C(J defined by Set x t (s = x(t + s, s J. C + (J =: {ψ C(J : ψ(s, s J}. 2 Mathematics Subject Classification. 34K, 34B. Key words and phrases. Functional delay differential equation; boundary value problem; positive solutions; fixed point index. c 26 Texas State University - San Marcos. Submitted August, 25. Published March 26, 26. Supported by grant 3KJD56 from the Natural Science Foundation of Jiangsu Education Office and Jiangsu Planned Projects for Postdoctoral Research Funds.

2 2 C. BAI, X. XU EJDE-26/4 In this paper, motivated and inspired by [, 4, 7, 8], we apply a fixed point theorem in cones to investigate the existence of positive solutions for three point boundary-value problems of second-order functional delay differential equation x (t = f(t, x(t, x(t τ, x t, < t <, x = φ, x( = x(η, (.2 where < τ <, τ < η <, f : I R + R + C + (J R + is a continuous function, and φ is an element of the space C + (J =: {ψ C+ (J : ψ( = }. We need the following well-known lemma. See [2] for a proof and further discussion of the fixed-point index i(a, K r, K. Lemma.. Assume that E is a Banach space, and K E is a cone in E. Let K r = {x K : u < r}. Furthermore, assume that A : K K is a compact map, and Ax x for x K r = {x K; x = r}. Then, one has the following conclusions. ( If x Ax for x K r, then i(a, K r, K =. (2 If x Ax for x K r, then i(a, K r, K =. 2. Preliminaries and some lemmas In the sequel we shall denote by C (I the space all continuous functions x : I R with x( =. This is a Banach space when it is furnished with usual sup-norm We set x I := sup{ x(s : s I}. C + (I := {x C (I : x(t, t I}. For each φ C + (J and x C+ (I we define { φ(t + s, t + s, t I, s J, x t (s; φ := x(t + s, t + s, t I, s J, and observe that x t ( ; φ C + (J. It is easy to check that ϕ (t = sin η+t is the eigenfunction related to the smallest eigenvalue λ = 2 (η+ 2 of the eigenproblem x = λx, x( =, x( = x(η. By [4], the Green s function for the three-point boundary-value problem is given by x =, x( =, x( = x(η, t, t s η, s, s t and s η, G(t, s = s η t, t s and s η, s + η s η t, η s t. Lemma 2.. Suppose that G(t, s is defined as above. Then we have the following results: ( G(t, s G(s, s, t, s,

3 EJDE-26/4 POSITIVE SOLUTIONS FOR DELAY THREE-POINT BVP 3 (2 G(t, s ηtg(s, s, t, s. Proof. It is easy to see that ( holds. To show that (2 holds, we distinguish four cases: If t s η, then If s t and s η, then If t s and s η, then G(t, s = t ηts = ηtg(s, s. G(t, s = s ηts = ηtg(s, s. G(t, s = s η t ηst s = ηtg(s, s. η Finally, if η s t, then G(t, s = s s η η t s s η η ts η( s η = ηt s( s η η( s = η = ηtg(s, s. Remark 2.2. If s η and s η, then G(s, s = s and G(s, s = s( s η, respectively. For convenience, let x(t; φ := { φ(t, τ t, x(t, t. Suppose that x(t is a solution of BVP (.2, then it can be written as x(t = Let K C (I be a cone defined by For each x K and t I, we have x t ( ; φ J = G(t, sf(s, x(s, x(s τ; φ, x s ( ; φds, t I. K = {x C + (I : x(t ηt x I, t I}. sup s [ τ,] = max x t (s; φ { sups [ τ,] x(t + s, if t + s I, sup s [ τ,] φ(t + s, if t + s max{ x I, φ J }, } (2. and x t ( ; φ J sup s [ τ,] {x(t + s : t + s I} x(t ηt x I. (2.2 Define an operator A φ : K C (I as follows: (A φ x(t := Firstly, we have the following result. Lemma 2.3. A φ (K K. G(t, sf(s, x(s, x(s τ; φ, x s ( ; φds, t I.

4 4 C. BAI, X. XU EJDE-26/4 Proof. For any x K, we observe that (A φ x( =. By Lemma 2. (, we have (A φ x(t, t I. It follows from Lemma 2. ( and (2 that Thus, A φ (K K. (A φ x(t ηt G(s, sf(s, x(s, x(s τ; φ, x s ( ; φds ηt A φ x I, t I. Secondly, similar to the proof of Theorem 2. in [6], we get that Lemma 2.4. A φ : K K is completely continuous. We formulate some conditions for f(t, u, v, ψ as follows which will play roles in this paper. (H lim inf min f(t, u, v, ψ =. u+v+ ψ J + t [,] u + v + ψ J f(t, u, v, ψ (H2 lim sup max =. u+v+ ψ J + t [,] u + v + ψ J (H3 lim inf min f(t, u, v, ψ > 2 ( + M, u+v+ ψ J + t [,] u + v + ψ J 3 (η + 2 where M = 2 τ(η τ + 3( η 2 + τ(η + + 3(η + ( 2 τ η(η + t sin η+ (t + τdt + 2. η t sin η+ tdt (H4 There is a h > such that u h, v max{h, φ J }, ψ J max{h, φ J }, and t implies ( 6 f(t, u, v, ψ < µh, where µ = G(s, sds = + η + η 2. (H5 There is a h 2 > such that 4 ηh 2 u h 2, ( 4 τηh 2 v h 2, 4 ηh 2 ψ J h 2, and t implies ( ( f(t, u, v, ψ > bh 2, where b = G dt 2, t. In the following, we give some lemmas which will be used in this paper. Lemma 2.5. If (H is satisfied, then there exist < r < such that Proof. Choose L > such that i(a φ, K r, K =, r r. η ( 3 4 τ L 2, s ds >. For u, v and ψ C + (J, (H implies that there is r > such that f(t, u, v, ψ L(u + v + ψ J, u + v + ψ J r, t. (2.3 Choose r > (2.2 that 4r 3( 4τη. For x K r, r r, we have by the definition of K and x(t τ η(t τ x I η ( 4 τ r > 3 r, 4 t 3 4,

5 EJDE-26/4 POSITIVE SOLUTIONS FOR DELAY THREE-POINT BVP 5 x t ( ; φ J x(t ηt x I 4 ηr > 3 r, So by (2.3, we have for such x, This shows that (A φ x ( 2 = L η ( 3 4 τ Lr 4 t , t f(t, x(t, x(t τ; φ, x t ( ; φdt 2, t f(t, x(t, x(t τ, x t ( ; φdt 2, t [x(t + x(t τ + x t ( ; φ J ]dt 2, t dt > r = x I. A φ x I > x I, x K r. It is obvious that A φ x x for x K r. Therefore, by Lemma., we conclude that i(a φ, K r, K =. Lemma 2.6. If (H2 is satisfied, then there exists < R < such that i(a φ, K R, K = for R R. Proof. By (H2, for any < ε < 3( G(s, sds, u, v and ψ C + (J, there exists R > such that Putting then Choose f(t, u, v, ψ ε(u + v + ψ J, u + v + ψ J R, t. C := max max f(t, u, v, ψ ε(u + v + ψ J +, t u,v, u+v+ ψ J R f(t, u, v, ψ ε(u + v + ψ J + C, foru, v, ψ C + (J, t I. (2.4 R > (C + 2ε φ J /( G(s, sds 3ε G(s, sds. Let R R and consider a point x K R. By the definition of x(t; φ, we get x(s τ; φ max{ x I, φ J }, s I. (2.5 By (2., (2.4 and (2.5, for x K R, R R, and t I, (A φ x(t = G(t, sf(s, x(s, x(s τ; φ, x s ( ; φds G(s, sf(s, x(s, x(s τ; φ, x s ( ; φds G(s, s[ε(x(s + x(s τ; φ + x s ( ; φ J + C]ds G(s, s[ε( x I + 2 max{ x I, φ J } + C]ds

6 6 C. BAI, X. XU EJDE-26/4 = 3εR G(s, s[ε(3 x I + 2 φ J + C]ds < R = x I. G(s, sds + (C + 2ε φ J G(s, sds Thus, A φ x I < x I for x K R. Hence, by Lemma., i(a φ, K R, K =. Lemma 2.7. If (H4 is satisfied, then i(a φ, K h, K =. Proof. Let x K h, then we have by (2. and (2.5 that x(t τ; φ max{h, φ J }, t, and x t ( ; φ J max{h, φ J }, t. Thus, from (H4 we obtain (A φ x(t G(s, sf(s, x(s, x(s τ; φ, x s ( ; φds < µh G(s, sds = h = x I, t. This shows that A φ x I < x I, x K h. Hence, Lemma. implies i(a φ, K h, K =. Lemma 2.8. If (H5 is satisfied, then i(a φ, K h2, K =. Proof. For x K h2, we have and h 2 = x I x(t τ η(t τ x I η ( 4 τ h 2, 4 t 3 4, h 2 = x I sup x(t + s = x t ( ; φ J x(t ηt x I s [ τ,] 4 ηh 2, for 4 t 3 4. It follows from (H5 that This shows that (A φ x ( 2 > bh 2 2, t f(t, x(t, x(t τ, x t ( ; φdt 2, t dt = h 2 = x I. A φ x I > x I, x K h2. Therefore, by Lemma., we conclude that i(a φ, K h2, K =.

7 EJDE-26/4 POSITIVE SOLUTIONS FOR DELAY THREE-POINT BVP 7 3. Main results Theorem 3.. Assume that (H3 and (H4 are satisfied, then BVP (.2 has at least one positive solution. Proof. According to Lemma 2.7, we have that i(a φ, K h, K =. (3. Fix m >, and let g(t, u, v, ψ = (u + v + ψ J m for u, v and ψ C + (J. Then g(t, u, v, ψ satisfy (H. Define B φ : K K by (B φ x(t := G(t, sg(s, x(s, x(s τ; φ, x s ( ; φds, t I. Then B φ is a completely continuous operator. One has from Lemma 2.5 that there exists < h < r <, such that r r implies i(b φ, K r, K =. (3.2 Define H φ : [, ] K K by H φ (s, x = ( sa φ x + sb φ x, then H φ is a completely continuous operator. By the condition (H3 and the definition of g, for u, v, ψ C + (J, and t I, there are ε > and r > r such that where λ = f(t, u, v, ψ 3 (λ ( + M + ε(u + v + ψ J, u + v + ψ J > r, g(t, u, v, ψ 3 (λ ( + M + ε(u + v + ψ J, u + v + ψ J > r, 2 (η+. We define 2 C := max max f(t, u, v, ψ t u,v,u+v+ ψ J r 3 [λ ( + M + ε](u + v + ψ J + max max g(t, u, v, ψ t u,v,u+v+ ψ J r 3 [λ ( + M + ε](u + v + ψ J +. It follows that f(t, u, v, ψ 3 [λ (+M+ε](u+v+ ψ J C, u, v, ψ C + (J, t I, (3.3 g(t, u, v, ψ 3 [λ (+M+ε](u+v+ ψ J C, u, v, ψ C + (J, t I. (3.4 We claim that there exists r r such that H φ (s, x x, s [, ], x K, x r. (3.5 In fact, if H φ (s, z = z for some z K and s, then z(t satisfies the equation and the boundary condition z (t = ( s f(t, z(t, z(t τ; φ, z t ( ; φ + s g(t, z(t, z(t τ; φ, z t ( ; φ, < t <, (3.6 z( =, z( = z(η. (3.7

8 8 C. BAI, X. XU EJDE-26/4 From the above condition, there exists ξ (η, such that z (ξ =. Multiplying left side of (3.6 by ϕ (t = sin η+t and then integrating from to ξ, after integrating two times by parts, we get from z (ξ = that ξ ξ z (tϕ (tdt = ϕ (ξz(ξ + λ z(tϕ (tdt. (3.8 By (3.6 and (3.7, we have that z (t for each t I. Thus we obtain from (3.6, (3.8 and (H3 that ξ λ z(tϕ (tdt λ z(tϕ (tdt = ξ = ( s z (tϕ (tdt ϕ (ξz(ξ z (tϕ (tdt ϕ I z I f(t, z(t, z(t τ; φ, z t ( ; φϕ (tdt + s g(t, z(t, z(t τ; φ, z t ( ; φϕ (tdt. Combining (2.2, (3.3, (3.4 and (3.9, we get λ z(tϕ (tdt 3 ( s (λ ( + M + ε ( s C ϕ (tdt + 3 s (λ ( + M + ε s C = 3 (λ ( + M + ε C ϕ (tdt ϕ (tdt 3 (λ ( + M + ε [z(t + z(t τ; φ + z t ( ; φ J ]ϕ (tdt [z(t + z(t τ; φ + z t ( ; φ J ]ϕ (tdt [z(t + z(t τ; φ + z t ( ; φ J ]ϕ (tdt [2z(t + z(t τ; φ]ϕ (tdt C ϕ (tdt ( 3 (λ ( + M + ε 2 z(tϕ (tdt + C ϕ (tdt τ z(t τ; φϕ (tdt (3.9

9 EJDE-26/4 POSITIVE SOLUTIONS FOR DELAY THREE-POINT BVP 9 = ( 3 (λ ( + M + ε 2 z(tϕ (tdt + C ϕ (tdt = ( 3 (λ ( + M + ε 2 z(tϕ (tdt + C ϕ (tdt, then we have ( τ (λ M + ε z(tϕ (t + τdt + 2 We also have τ τ τ λ z(t[ϕ (t ϕ (t + τ]dt + λ + 2λ z(tϕ (tdt + 3C η ϕ (tdt + z(t τϕ (tdt z(tϕ (t + τdt z(tϕ (tdt η τ z(tϕ (tdt 3 λ τ(η τ ϕ I z I + λ ( η + τ ϕ I z I + 2λ ( η ϕ I z I + 3Cη ϕ I + 3 [ 3(η + = λ τ(η τ + 3( η + τ + η + τ z(tϕ (t + τdt + 2 z(tϕ (tdt ] z I + 3Cη. τ η z I tϕ (t + τdt + 2η z I tϕ (tdt, which together with (3. leads to ( τ (λ M + εη tϕ (t + τdt + 2 i.e., tϕ (tdt z I [ 3(η + ] λ τ(η τ + 3( η + τ + z I + 3Cη η + [ = λ 2 τ(η τ + 3( η 2 + τ(η + + 3(η + 2] z I + 3Cη; (η + 3C z I ε ( τ tϕ (t + τdt + 2 tϕ (tdt 3C = ε ( τ t sin η+ (t + τdt + 2 t sin := r. η+tdt (3. (3. Let r = + max{r, r}. We obtain (3.5 and consequently, by (3.2 and homotopy invariance of the fixed-point index, we have i(a φ, K r, K = i(h φ (,, K r, K = i(h φ (,, K r, K = i(b φ, K r, K =. (3.2

10 C. BAI, X. XU EJDE-26/4 Use (3. and (3.2 to conclude that i(a φ, K r \ K h, K =. Hence, A φ has fixed points x in K r \ K h, which means that x (t is a positive solution of BVP (.2 and x I > h. Thus, the proof is complete. By Lemmas 2.6 and 2.8, we have the following result. Theorem 3.2. Assume that (H2 and (H5 are satisfied, then BVP (.2 has at least one positive solution. Finally, we obtain from Lemmas 2.7 and 2.8 the following result. Theorem 3.3. If (H4 and (H5 are satisfied, then BVP (.2 has at least one positive solution. References [] C. Z. Bai, J.Fang; Existence of multiple positive solutions for functional differential equations, Comput. Math. Appl. 45 ( [2] K. Deimling; Nonlinear Functional Analysis, Springer-Verlag, New York, 985. [3] L.H. Erbe, Q.K. Kong; Boundary value problems for singular second order functional differential equations, J. Comput. Appl. Math. 53 ( [4] J. Henderson; Double solutions of three-point boundary-value problems for second-order differential equations, Electron. J. of Differential Equations 24 (24, No. 5, -7. [5] J. Henderson; Boundary Value Problems for Functional Differential Equations, World Scientific, 995. [6] C.H. Hong, C.C. Yeh, C.F. Lee, F.H. Wong; Existence of positive solutions for functional differential equations, Comput. Math. Appl. 4 ( [7] D.Q. Jiang; Multiple positive solutions for boundary-value problems of second-order delay differential equations, Appl.Math. Lett. 5 ( [8] G. L. Karakostas, K. G. Mavridis, P. Ch. Tsamatos; Multiple positive solutions for a functional second-order boundary-value problem, J. Math. Anal. Appl. 282 ( [9] J.W. Lee, D. O Regan; Existence results for differential delay equations-i, J. Differential Equations 2 ( [] J.W. Lee, D. O Regan; Existence results for differential delay equations-ii, Nonlinear Analysis 7 ( [] S.K. Ntouyas, Y, Sficas, P.Ch. Tsamatos; An existence principle for boundary-value problems for second order functional differential equations, Nonlinear Analysis 2 ( Chuanzhi Bai Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsi 223, China. and Department of Mathematics, Nanjing University, Nanjing 293, China address: czbai8@sohu.com Xinya Xu Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsi 223, China address: xinyaxu@hytc.edu.cn