Even Number of Positive Solutions for 3n th Order Three-Point Boundary Value Problems on Time Scales K. R. Prasad 1 and N.

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1 Electronic Journal of Qualitative Theory of Differential Equations 011, No. 98, 1-16; Even Number of Positive Solutions for 3n th Order Three-Point Boundary Value Problems on Time Scales K. R. Prasad 1 and N. Sreedhar 1 Department of Applied Mathematics, Andhra University, Visakhapatnam, , India. rajendra9@rediffmail.com Department of Mathematics, GITAM University, Visakhapatnam, , India. sreedharnamburi@rediffmail.com Abstract We establish the existence of at least two positive solutions for the 3n th order three-point boundary value problem on time scales by using Avery-Henderson fixed point theorem. We also establish the existence of at least m positive solutions for an arbitrary positive integer m. Key words: Green s function, boundary value problem, time scale, positive solution, cone. AMS Subject Classification: 39A10, 34B05. 1 Introduction The theory of time scales was introduced and developed by Hilger [13] to unify both continuous and discrete analysis. Time scales theory presents us with the tools necessary to understand and explain the mathematical structure underpinning the theories of discrete and continuous dynamic systems and allows us to connect them. The theory is widely applied to various situations like epidemic models, the stock market and mathematical modeling of physical and biological systems. Certain economically important phenomena contain processes that feature elements of both the continuous and discrete. In recent years, the existence of positive solutions of the higher order boundary value problems (BVPs) on time scales have been studied extensively due to their striking applications to almost all area of science, engineering and technology. The existence of positive solutions are studied by many authors. A few papers along these lines are Henderson [11], Anderson [1, ], Kaufmann EJQTDE, 011 No. 98, p. 1

2 [15], Anderson and Avery [3], DaCunha, Davis and Singh [10], Peterson, Raffoul and Tisdell [18], Sun and Li [19], Luo and Ma [17], Cetin and Topal [8], Karaca [14] and Anderson and Karaca [4]. In this paper, we are concerned with the existence of positive solutions for the 3n th order BVP on time scales, ( 1) n y (3n) (t) = f(t, y(t)), t [, σ(t 3 )] (1.1) satisfying the general three-point boundary conditions, α 3i,1 y (3i 3) ( ) + α 3i, y (3i ) ( ) + α 3i,3 y (3i 1) ( ) = 0, α 3i 1,1 y (3i 3) (t ) + α 3i 1, y (3i ) (t ) + α 3i 1,3 y (3i 1) (t ) = 0, α 3i,1 y (3i 3) (σ(t 3 )) + α 3i, y (3i ) (σ(t 3 )) + α 3i,3 y (3i 1) (σ(t 3 )) = 0, (1.) for 1 i n, where n 1, α 3i,j, α 3i 1,j, α 3i,j, for j = 1,, 3, are real constants, < t < σ(t 3 ) and f : [, σ(t 3 )] R + R + is continuous. For convenience, we use the following notations. For 1 i n, let us denote β ij = α 3i 3+j,1 t j +α 3i 3+j,, γ ij = α 3i 3+j,1 t j +α 3i 3+j,(t j +σ(t j ))+α 3i 3+j,3, where j = 1, ; β i3 = α 3i,1 σ(t 3 ) + α 3i, and γ i3 = α 3i,1 (σ(t 3 )) + α 3i, (σ(t 3 ) + σ (t 3 )) + α 3i,3. Also, for 1 i n, we define m ijk = α 3i 3+j,1γ ik α 3i 3+k,1 γ ij (α 3i 3+j,1 β ik α 3i 3+k,1 β ij ), M i jk = where j, k { = 1,, 3 and let p i = max{m i1, m i13, m i3 }, } q i = min m i3 + m i 3 M i3, m i13 + m i13 M i13, β ij γ ik β ik γ ij α 3i 3+j,1 β ik α 3i 3+k,1 β ij, d i = α 3i,1 (β i γ i3 β i3 γ i ) β i1 (α 3i 1,1 γ i3 α 3i,1 γ i )+γ i1 (α 3i 1,1 β i3 α 3i,1 β i ) and l ij = α 3i 3+j,1 σ(s)σ (s) β ij (σ(s) + σ (s)) + γ ij, where j = 1,, 3. We assume the following conditions throughout this paper: (A1) α 3i,1 > 0, α 3i 1,1 > 0, α 3i,1 > 0 and α 3i, α 3i,1 1 i n, > α 3i 1, α 3i 1,1 > α 3i, α 3i,1, for all (A) p i < t < σ(t 3 ) q i and α 3i,3 α 3i,1 > α 3i,, α 3i 1,3 α 3i 1,1 < α 3i 1,, α 3i,3 α 3i,1 > α 3i,, for all 1 i n, (A3) m i 3 > M i3, m i 1 < M i1, m i 13 > M i13 and d i > 0, for all 1 i n, (A4) The point t [, σ(t 3 )] is not left dense and right scattered at the same time. EJQTDE, 011 No. 98, p.

3 This paper is organized as follows. In Section, we construct the Green s function for the homogeneous problem corresponding to (1.1)-(1.) and estimate bounds for the Green s function. In Section 3, we establish a criteria for the existence of at least two positive solutions for the BVP (1.1)-(1.) by using an Avery-Henderson fixed point theorem [5]. We also establish the existence of at least m positive solutions for an arbitrary positive integer m. Finally as an application, we give an example to illustrate our result. Green s Function and Bounds In this section, we construct the Green s function for the homogeneous problem corresponding to (1.1)-(1.) and estimate bounds for the Green s function. Let G i (t, s) be the Green s function for the homogeneous BVP, y 3 (t) = 0, t [, σ(t 3 )], (.1) satisfying the general three-point boundary conditions, α 3i,1 y( ) + α 3i, y ( ) + α 3i,3 y ( ) = 0, α 3i 1,1 y(t ) + α 3i 1, y (t ) + α 3i 1,3 y (t ) = 0, α 3i,1 y(σ(t 3 )) + α 3i, y (σ(t 3 )) + α 3i,3 y (σ(t 3 )) = 0, (.) for 1 i n. Lemma.1 For 1 i n, the Green s function G i (t, s) for the homogeneous BVP (.1)-(.) is given by G i (t, s) = where G i (t,s) t [,t ] = G i (t,s) t [t,σ(t 3 )] = G i1 (t, s), < σ(s) < t t < σ(t 3 ) G i (t, s), t < s < t < σ(t 3 ) G i3 (t, s), t < t < s < σ(t 3 ) G i4 (t, s), < t < σ(s) < t σ(t 3 ) G i5 (t, s), < t t < s < σ(t 3 ) G i6 (t, s), σ(s) < t < t < σ(t 3 ) G i1 (t, s) = 1 d i [ (β i γ i3 β i3 γ i ) + t(α 3i 1,1 γ i3 α 3i,1 γ i ) t (α 3i 1,1 β i3 α 3i,1 β i )]l i1, (.3) EJQTDE, 011 No. 98, p. 3

4 G i (t, s) = 1 d i {[ (β i1 γ i3 β i3 γ i1 ) + t(α 3i,1 γ i3 α 3i,1 γ i1 ) t (α 3i,1 β i3 α 3i,1 β i1 )]l i + [(β i1 γ i β i γ i1 ) t(α 3i,1 γ i α 3i 1,1 γ i1 )+ t (α 3i,1 β i α 3i 1,1 β i1 )]l i3 }, G i3 (t, s) = 1 d i [(β i1 γ i β i γ i1 ) t(α 3i,1 γ i α 3i 1,1 γ i1 ) + t (α 3i,1 β i α 3i 1,1 β i1 )]l i3, G i4 (t, s) = 1 d i {[ (β i γ i3 β i3 γ i ) + t(α 3i 1,1 γ i3 α 3i,1 γ i ) t (α 3i 1,1 β i3 α 3i,1 β i )]l i1 + [(β i1 γ i3 β i3 γ i1 ) t(α 3i,1 γ i3 α 3i,1 γ i1 )+ t (α 3i,1 β i3 α 3i,1 β i1 )]l i }, G i5 (t, s) = 1 d i [(β i1 γ i β i γ i1 ) t(α 3i,1 γ i α 3i 1,1 γ i1 ) + t (α 3i,1 β i α 3i 1,1 β i1 )]l i3, G i6 (t, s) = 1 d i [ (β i γ i3 β i3 γ i ) + t(α 3i 1,1 γ i3 α 3i,1 γ i ) t (α 3i 1,1 β i3 α 3i,1 β i )]l i1. Lemma. Assume that the conditions (A1)-(A4) are satisfied. Then, for 1 i n, the Green s function G i (t, s) of (.1)-(.) is positive, for all (t, s) [, σ(t 3 )] [, t 3 ]. Proof: For 1 i n, the Green s function G i (t, s) is given in (.3). We prove the result for G i1 (t, s). Then, G i1 (t, s) = g i1 (t)l i1 (s), where g i1 (t) = 1 d i [ (β i γ i3 β i3 γ i ) + t(α 3i 1,1 γ i3 α 3i,1 γ i ) t (α 3i 1,1 β i3 α 3i,1 β i )]. Using the conditions (A1) and (A4), g i1 (t) has maximum at t = m i3, and hence g i1 (t) > 0 on [, σ(t 3 )] by conditions (A) and (A3). From conditions (A) and (A4), l i1 (s) > 0 on [, t 3 ]. Therefore, G i1 (t, s) > 0, for all (t, s) [, σ(t 3 )] [, t 3 ]. Similarly, we can establish the positivity of the Green s function in the remaining cases. EJQTDE, 011 No. 98, p. 4

5 Theorem.3 Assume that the conditions (A1)-(A4) are satisfied. Then, for 1 i n, the Green s function G i (t, s) satisfies the following inequality, m i G i (σ(s), s) G i (t, s) G i (σ(s), s), for all (t, s) [, σ(t 3 )] [, t 3 ], (.4) where 0 < m i = min { } G i1 (σ(t 3 ), s) G i3 (, s), G i1 (, s) G i3 (σ(t 3 ), s), G i (, s) G i (σ(t 3 ), s), G i 4 (σ(t 3 ), s) < 1. G i4 (, s) Proof: For 1 i n, the Green s function G i (t, s) is given (.3) in six different cases. In each case, we prove the inequality as in (.4). Case 1. For < σ(s) < t t < σ(t 3 ). G i (t,s) = G i 1 (t,s) G i (σ(s),s) G i1 (σ(s),s) = [ (β i γ i3 β i3 γ i ) + t(α 3i 1,1 γ i3 α 3i,1 γ i ) t (α 3i 1,1 β i3 α 3i,1 β i )] [ (β i γ i3 β i3 γ i ) + σ(s)(α 3i 1,1 γ i3 α 3i,1 γ i ) (σ(s)) (α 3i 1,1 β i3 α 3i,1 β i )]. From (A1)-(A4), we have G i1 (t, s) G i1 (σ(s), s) and also G i (t, s) G i (σ(s), s) = G i 1 (t, s) G i1 (σ(s), s) G i 1 (t, s) G i1 (, s) G i 1 (σ(t 3 ), s). G i1 (, s) Therefore, G i (t, s) G i (σ(s), s) and G i (t, s) G i 1 (σ(t 3 ),s) G i1 (,s) (t, s) [, σ(t 3 )] [, t 3 ]. G i (σ(s), s), for all Case. For t < t < s < σ(t 3 ). G i (t,s) = G i 3 (t,s) G i (σ(s),s) G i3 (σ(s),s) = [(β i1 γ i β i γ i1 ) t(α 3i,1 γ i α 3i 1,1 γ i1 ) + t (α 3i,1 β i α 3i 1,1 β i1 )] [(β i1 γ i β i γ i1 ) σ(s)(α 3i,1 γ i α 3i 1,1 γ i1 ) + (σ(s)) (α 3i,1 β i α 3i 1,1 β i1 )]. From (A1)-(A4), we have G i3 (t, s) G i3 (σ(s), s) and also G i (t, s) G i (σ(s), s) = G i 3 (t, s) G i3 (σ(s), s) G i 3 (t, s) G i3 (σ(t 3 ), s) G i 3 (, s) G i3 (σ(t 3 ), s). Therefore, G i (t, s) G i (σ(s), s) and G i (t, s) (t, s) [, σ(t 3 )] [, t 3 ]. G i 3 (,s) G i3 (σ(t 3 ),s) G i(σ(s), s), for all EJQTDE, 011 No. 98, p. 5

6 Case 3. For t < s < t < σ(t 3 ). From (A1)-(A4) and case, we have G i (t, s) G i (σ(s), s) and also { } G i (t, s) G i (σ(s), s) min G i3 (, s) G i3 (σ(t 3 ), s), G i (, s). G i (σ(t 3 ), s) Therefore, G i (t, s) G i (σ(s), s) and { } G i3 (, s) G i (t, s) min G i3 (σ(t 3 ), s), G i (, s) G i (σ(s), s), G i (σ(t 3 ), s) for all (t, s) [, σ(t 3 )] [, t 3 ]. Case 4. For < t < σ(s) < t σ(t 3 ). From (A1)-(A4) and case 1, we have G i4 (t, s) G i4 (σ(s), s) and { } G i (t, s) G i (σ(s), s) min G i1 (σ(t 3 ), s), G i 4 (σ(t 3 ), s). G i1 (, s) G i4 (, s) Therefore, G i (t, s) G i (σ(s), s) and G i (t, s) min { for all (t, s) [, σ(t 3 )] [, t 3 ]. } G i1 (σ(t 3 ), s), G i 4 (σ(t 3 ), s) G i (σ(s), s), G i1 (, s) G i4 (, s) Case 5. For < t t < s < σ(t 3 ). From case, we have G i (t, s) G i (σ(s), s) and G i (t, s) for all (t, s) [, σ(t 3 )] [, t 3 ]. G i 3 (,s) G i3 (σ(t 3 ),s) G i(σ(s), s), Case 6. For σ(s) < t < t < σ(t 3 ). From case 1, we have G i (t, s) G i (σ(s), s) and G i (t, s) G i 1 (σ(t 3 ),s) G i1 ( G,s) i (σ(s), s), for all (t, s) [, σ(t 3 )] [, t 3 ]. From all above cases, for 1 i n, we have m i G i (σ(s), s) G i (t, s) G i (σ(s), s), for all (t, s) [, σ(t 3 )] [, t 3 ], EJQTDE, 011 No. 98, p. 6

7 where 0 < m i = min { } G i1 (σ(t 3 ), s) G i3 (, s), G i1 (, s) G i3 (σ(t 3 ), s), G i (, s) G i (σ(t 3 ), s), G i 4 (σ(t 3 ), s) < 1. G i4 (, s) Lemma.4 Assume that the conditions (A1)-(A4) are satisfied and G i (t, s) is defined as in (.3). Take H 1 (t, s) = G 1 (t, s) and recursively define H j (t, s) = H j 1 (t, r)g j (r, s) r, for j n. Then H n (t, s) is the Green s function for the homogeneous BVP corresponding to (1.1)-(1.). Lemma.5 Assume that the conditions (A1)-(A4) hold. If we define n 1 n 1 K = K j and L = m j L j, j=1 j=1 then the Green s function H n (t, s) in Lemma.4 satisfies and 0 H n (t, s) K G n (, s), for all (t, s) [, σ(t 3 )] [, t 3 ] H n (t, s) m n L G n (, s), for all (t, s) [t, σ(t 3 )] [, t 3 ], where m n is given as in Theorem.3, K j = L j = and is defined by G j (, s) s > 0, for 1 j n, t G j (, s) s > 0, for 1 j n x = max x(t). t [,σ(t 3 )] EJQTDE, 011 No. 98, p. 7

8 3 Multiple Positive Solutions In this section, we establish the existence of at least two positive solutions for the BVP (1.1)-(1.) by using an Avery-Henderson functional fixed point theorem. And then, we establish the existence of at least m positive solutions for an arbitrary positive integer m. Let B be a real Banach space. A nonempty closed convex set P B is called a cone, if it satisfies the following two conditions: (i) y P, λ 0 implies λy P, and (ii) y P and y P implies y = 0. Let ψ be a nonnegative continuous functional on a cone P of the real Banach space B. Then for a positive real number c, we define the sets P(ψ, c ) = {y P : ψ(y) < c } and P a = {y P : y < a}. In obtaining multiple positive solutions of the BVP (1.1)-(1.), the following Avery-Henderson functional fixed point theorem will be the fundamental tool. Theorem 3.1 [5] Let P be a cone in a real Banach space B. Suppose α and γ are increasing, nonnegative continuous functionals on P and θ is nonnegative continuous functional on P with θ(0) = 0 such that, for some positive numbers c and k, γ(y) θ(y) α(y) and y kγ(y), for all y P(γ, c ). Suppose that there exist positive numbers a and b with a < b < c such that θ(λy) λθ(y), for all 0 λ 1 and y P(θ, b ). Further, let T : P(γ, c ) P be a completely continuous operator such that (B1) γ(ty) > c, for all y P(γ, c ), (B) θ(ty) < b, for all y P(θ, b ), (B3) P(α, a ) and α(ty) > a, for all y P(α, a ). Then, T has at least two fixed points y 1, y P(γ, c ) such that a < α(y 1 ) with θ(y 1 ) < b and b < θ(y ) with γ(y ) < c. Let n 1 M = m n j=1 m j L j K j (3.1) EJQTDE, 011 No. 98, p. 8

9 Let B = {y : y C[, σ(t 3 )]} be the Banach space equipped with the norm Define the cone P B by y = max y(t). t [,σ(t 3 )] P = {y B : y(t) 0 on [, σ(t 3 )] and min y(t) M y }, t [t,σ(t 3 )] where M is given as in (3.1). Define the nonnegative, increasing, continuous functionals γ, θ and α on the cone P by γ(y) = min y(t), θ(y) = t [t,σ(t 3 )] We observe that for any y P, max y(t) and α(y) = t [t,σ(t 3 )] max y(t). t [,σ(t 3 )] γ(y) θ(y) α(y) (3.) and y 1 M min y(t) = 1 t [t,σ(t 3 )] M γ(y) 1 M θ(y) 1 α(y). (3.3) M Theorem 3. Suppose there exist 0 < a < b < c such that f satisfies the following conditions. c (D1) f(t, y) > Π n j=1 m jl j, for t [t, σ(t 3 )] and y [c, c ], M (D) f(t, y) < b, for t [t Π 1, σ(t n j=1 K j 3 )] and y [0, b ], M a (D3) f(t, y) > Π n j=1 m jl j, for t [t, σ(t 3 )] and y [a, a ], M where m n and M are defined in Theorem.3 and (3.1) respectively. Then the BVP (1.1)-(1.) has at least two positive solutions y 1 and y such that a < b < max y 1(t) with t [,σ(t 3 )] max y (t) with t [t,σ(t 3 )] Proof: Define the operator T : P B by Ty(t) = max y 1(t) < b, t [t,σ(t 3 )] min y (t) < c. t [t,σ(t 3 )] H n (t, s)f(s, y(s)) s. (3.4) It is obvious that a fixed point of T is the solution of the BVP (1.1)-(1.). We seek two fixed points y 1, y P of T. First, we show that T : P P. Let EJQTDE, 011 No. 98, p. 9

10 y P. From Theorem.3 and Lemma.5, we have Ty(t) 0 on [, σ(t 3 )] and also, Ty(t) = K H n (t, s)f(s, y(s)) s G n (, s) f(s, y(s)) s so that Ty K G n (, s) f(s, y(s)) s. Next, if y P, then we have Ty(t) = m n L m nl K H n (t, s)f(s, y(s)) s G n (, s) f(s, y(s)) s Ty = M Ty. Hence Ty P and so T : P P. Moreover, T is completely continuous. From (3.) and (3.3), for each y P, we have γ(y) θ(y) α(y) and y 1 M γ(y). Also, for any 0 λ 1 and y P, we have θ(λy) = max t [t,σ(t 3 )](λy)(t) = λ max t [t,σ(t 3 )] y(t) = λθ(y). It is clear that θ(0) = 0. We now show that the remaining conditions of Theorem 3.1 are satisfied. Firstly, we shall verify that condition (B1) of Theorem 3.1 is satisfied. Since y P(γ, c ), from (3.3) we have that c = min t [t,σ(t 3 )] y(t) y c M. Then γ(ty) = > min t [t,σ(t 3 )] min t [t,σ(t 3 )] t c Π n j=1 m jl j m n L H n (t, s)f(s, y(s)) s H n (t, s)f(s, y(s)) s t G n (, s) s = c, using hypothesis (D1). Now we shall show that condition (B) of Theorem 3.1 is satisfied. Since y P(θ, b ), from (3.3) we have that 0 y(t) y b M, for [, σ(t 3 )]. EJQTDE, 011 No. 98, p. 10

11 Thus θ(ty) = < max t [t,σ(t 3 )] b Π n j=1 K K j H n (t, s)f(s, y(s)) s G n (, s) s = b, by hypothesis (D). Finally, using hypothesis (D3), we shall show that condition (B3) of Theorem 3.1 is satisfied. Since 0 P and a > 0, P(α, a ). Since y P(α, a ), a = max t [t1,σ(t 3 )] y(t) y a M, for t [t, σ(t 3 )]. Therefore, α(ty) = > max t [,σ(t 3 )] H n (t, s)f(s, y(s)) s H n (t, s)f(s, y(s)) s a Π n j=1 m jl j m n L t G n (, s) s = a. Thus, all the conditions of Theorem 3.1 are satisfied and so there exist at least two positive solutions y 1, y P(γ, c ) for the BVP (1.1)-(1.). This completes the proof of the theorem. Theorem 3.3 Let m be an arbitrary positive integer. Assume that there exist numbers a r (r = 1,,, m + 1) and b s (s = 1,,, m) with 0 < a 1 < b 1 < a < b < < a m < b m < a m+1 such that f(t, y) > a r Π n j=1 m, for t [t, σ(t 3 )] and y [a r, a r ], r = 1,,, m + 1, jl j M (3.5) b s f(t, y) < Π n j=1 K j, for t [, σ(t 3 )] and y [0, b s M ], s = 1,,, m. (3.6) Then the BVP (1.1)-(1.) has at least m positive solutions in P am+1. Proof: We use induction on m. For m = 1, we know from (3.5) and (3.6) that T : P a P a, then, it follows from Avery-Henderson fixed point theorem that the BVP (1.1)-(1.) has at least two positive solutions in P a. Next, we assume that this conclusion holds for m = l. In order to prove this conclusion holds for m = l+1. We suppose that there exist numbers a r (r = 1,,, l+) EJQTDE, 011 No. 98, p. 11

12 and b s (s = 1,,, l+1) with 0 < a 1 < b 1 < a < b < < a l+1 < b l+1 < a l+ such that f(t, y) > a r Π n j=1 m, for t [t, σ(t 3 )] and y [a r, a r ], r = 1,,, l +, jl j M (3.7) b s f(t, y) < Π n j=1 K j, for t [, σ(t 3 )] and y [0, b s ], s = 1,,, l + 1. M (3.8) By assumption, the BVP (1.1)-(1.) has at least l positive solutions y i (i = 1,,, l) in P al+1. At the same time, it follows from Theorem 3., (3.7) and (3.8) that the BVP (1.1)-(1.) has at least two positive solutions y 1, y in P al+ such that a l+1 < α(y 1 ) with θ(y 1 ) < b l+1 and b l+1 < θ(y ) with γ(y ) < a l+. Obviously y 1 and y are different from y i (i = 1,,, l). Therefore, the BVP (1.1)-(1.) has at least l + positive solutions in P al+, which shows that this conclusion holds for m = l Example Let us consider an example to illustrate the usage of the Theorem 3.. Let n = and T = {0} { 1 : n N} [ 1, 3 ]. Now, consider the following BVP, n+1 y 6 (t) = 800(y + 1)4, t [0, σ(1)] T (4.1) 73(y + 999) subject to the boundary conditions, 1 y(0) y (0) + y (0) = 0, ( 1 ( 1 ) y 3y ) + y ( 1 ) = 0, y(σ(1)) + 1 y (σ(1)) + 1 (σ(1)) = 0, 3 y 3 (0) y 4 y 3 4 (0) + 3y 5 (0) = 0, y 3( 1 ) y 4( 1 ) + y 5( 1 ) = 0, y 3 (σ(1)) + 1 (σ(1)) + 1 (σ(1)) = 0. y 4 y 5 (4.) EJQTDE, 011 No. 98, p. 1

13 Then the conditions (A1)-(A4) are satisfied. The Green s function G 1 (t, s) in Lemma.1 is G 1 (t, s) = G (t, s) = G 1 (t,s) t [0, 1 ] = G 1 (t,s) t [ 1,σ(1)] = G (t,s) t [0, 1 ] = G 11 (t, s), 0 < σ(s) < t 1 < σ(1) G 1 (t, s), 0 t < s < 1 < σ(1) G 13 (t, s), 0 t < 1 < s < σ(1) G 14 (t, s), 0 < 1 < σ(s) < t σ(1) G 15 (t, s), 0 < 1 t < s < σ(1) G 16 (t, s), 0 σ(s) < 1 < t < σ(1) where G 11 (t, s) = 1 [ ][ 1 ] 6 t 5t σ(s)σ (s) + (σ(s) + σ (s)) + 4, G 1 (t, s) = 1 {[ t 7 ][ 4 t σ(s)σ (s) + (σ(s) + σ (s)) + 3 ] [ ][σ(s)σ 4 t + t (s) 3 (σ(s) + σ (s)) + 8 ]}, 3 G 13 (t, s) = 1 [ ][σ(s)σ 4 t + t (s) 3 (σ(s) + σ (s)) + 8 ], 3 G 14 (t, s) = 1 {[ ][ 1 ] 6 t 5t σ(s)σ (s) + (σ(s) + σ (s)) + 4 [ t + 7 ][σ(s)σ 4 t (s) + (σ(s) + σ (s)) + 3 ]}, G 15 (t, s) = 1 [ ][σ(s)σ 4 t + t (s) 3 (σ(s) + σ (s)) + 8 ], 3 G 16 (t, s) = 1 [ ][ 1 ] 6 t 5t σ(s)σ (s) + (σ(s) + σ (s)) + 4. The Green s function G (t, s) in Lemma.1 is G (t,s) t [ 1,σ(1)] = G 1 (t, s), 0 < σ(s) < t 1 < σ(1) G (t, s), 0 t < s < 1 < σ(1) G 3 (t, s), 0 t < 1 < s < σ(1) G 4 (t, s), 0 < 1 < σ(s) < t σ(1) G 5 (t, s), 0 < 1 t < s < σ(1) G 6 (t, s), 0 σ(s) < 1 < t < σ(1) where G 1 (t, s) = 16 [ ][ 3 ] 4 t 3t 4 σ(s)σ (s) + (σ(s) + σ (s)) + 6, EJQTDE, 011 No. 98, p. 13

14 G (t, s) = 16 {[ t 5 ][σ(s)σ 8 t (s) + 3 (σ(s) + σ (s)) + 1 ] 4 [ t + 7 ][ 8 t σ(s)σ (s) 3 ]} (σ(s) + σ (s)) + 3, G 3 (t, s) = 16 [ t + 7 ][ 8 t σ(s)σ (s) 3 ] (σ(s) + σ (s)) + 3, G 4 (t, s) = 16 {[ ][ 3 ] 4 t 3t 4 σ(s)σ (s) + (σ(s) + σ (s)) + 6 [ t + 5 ][σ(s)σ 8 t (s) + 3 (σ(s) + σ (s)) + 1 ]}, 4 G 5 (t, s) = 16 [ t + 7 ][ 8 t σ(s)σ (s) 3 ] (σ(s) + σ (s)) + 3, G 6 (t, s) = 16 [ ][ 3 ] 4 t 3t 4 σ(s)σ (s) + (σ(s) + σ (s)) + 6. From Theorem.3 and Lemma.5, we get m 1 = , K 1 = , L 1 = , m = , K = , L = Therefore, K = , L = and M = Clearly f is continuous and increasing on [0, ). If we choose a = , b = 0.04 and c = 100 then 0 < a < b < c and f satisfies c (i) f(t, y) > = Π j=1 m jl j, for t [ 1, σ(1)] and y [100, ], (ii) f(t, y) < = b, for t [0, σ(1)] and y [0, ], Π j=1 K j (iii) f(t, y) > = a Π j=1 m jl j, for t [ 1, σ(1)] and y [0.0001, ]. Then all the conditions of Theorem 3. are satisfied. Thus by Theorem 3., the BVP (4.1)-(4.) has at least two positive solutions y 1 and y satisfying < max y 1 (t) with t [0,σ(1)] 0.04 < max t [ 1,σ(1)] y (t) with max t [ 1,σ(1)] y 1 (t) < 0.04, min y (t) < 100. t [ 1,σ(1)] Acknowledgements: The authors thank the referees for their valuable suggestions and comments. EJQTDE, 011 No. 98, p. 14

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16 [14] I. Y. Karaca, Positive solutions to nonlinear three-point boundary value problems on time scales, Panamer. Math. J., 17(007), [15] E. R. Kaufmann, Positive solutions of a three-point boundary value problem on time scales, Elec. J. Diff. Eqns., 8(003), [16] M. A. Krasnosel skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, [17] H. Luo and Q. Ma, Positive solutions to a generalized second order three-point boundary value problem on time scales, Elec. J. Diff. Eqns., 17(005), [18] A. C. Peterson, Y. N. Raffoul and C. C. Tisdell, Three-point boundary value problems on time scales J. Difference Eqn. Appl., 10(004), [19] H. R. Sun and W. T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl., 99(004), (Received October 9, 011) EJQTDE, 011 No. 98, p. 16