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Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales

Abstract

In this paper, four functionals fixed point theorem is used to verify the existence of at least one positive solution for third-order boundary value problems with integral boundary conditions for an increasing homeomorphism and homomorphism on time scales. We also provide an example to demonstrate our results.

MSC:34B18, 34N05.

1 Introduction

The theory of time scales was introduced by Hilger [1] in his PhD thesis in 1988. Theoretically, this new theory has not only unified continuous and discrete equations, but has also exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic equations on time scales has led to several important applications, for example, insect population models, biology, neural networks, heat transfer, and epidemic models, see [212].

Recently, scientists have noticed that the boundary conditions in many areas of applied mathematics and physics come down to integral boundary conditions. For instance, the models on chemical engineering, heat conduction, thermo-elasticity, plasma physics, and underground water flow can be reduced to the nonlocal problems with integral boundary conditions. For more information about this subject, we refer the readers to the excellent survey by Corduneanu [13], and Agarwal and O’Regan [14]. In addition, such kind of boundary value problem in a Banach space has been studied by some researchers, we refer the readers to [1518] and the references therein. However, to the best of our knowledge, little work has been done on the existence of positive solutions for third-order boundary value problem with integral boundary conditions on time scales. This paper attempts to fill this gap in literature.

In [19], Ma considered the existence and multiplicity of positive solutions for the m-boundary value problems

{ ( p ( t ) u ) q ( t ) u + f ( t , u ) = 0 , 0 < t < 1 , a u ( 0 ) b p ( 0 ) u ( 0 ) = i = 1 m 2 α i u ( ξ i ) , c u ( 1 ) + d p ( 1 ) u ( 1 ) = i = 1 m 2 β i u ( ξ i ) .

The main tool is Guo-Krasnoselskii fixed point theorem.

In [15], Boucherif considered the second-order boundary value problem with integral boundary conditions

{ y = f ( t , y ( t ) ) , 0 < t < 1 , y ( 0 ) a y ( 0 ) = 0 1 g 0 ( s ) y ( s ) d s , y ( 1 ) b y ( 1 ) = 0 1 g 1 ( s ) y ( s ) d s .

By using Krasnoselskii’s fixed point theorem, he obtained the existence criteria of at least one positive solution.

In [17], Li and Zhang were concerned with the second-order p-Laplacian dynamic equations with integral boundary conditions on time scales

{ ( ϕ p ( x ( t ) ) ) + λ f ( t , x ( t ) , x ( t ) ) = 0 , t ( 0 , T ) T , x ( 0 ) = 0 , α x ( T ) β x ( 0 ) = 0 T g ( s ) x ( s ) s .

By using Legget-Williams fixed point theorem, they obtained the existence criteria of at least three positive solutions.

In [18], Zhao et al. considered the third-order differential equation

x (t)+f ( t , x ( t ) ) =0,tJ,

subject to one of the following integral boundary conditions

x ( 0 ) = 0 , x ( 0 ) = 0 , x ( 1 ) = 0 1 g ( t ) x ( t ) d t , x ( 0 ) = 0 1 g ( t ) x ( t ) d t , x ( 0 ) = 0 , x ( 1 ) = 0 .

They investigated the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear boundary value problems of third-order differential equations with integral boundary conditions in ordered Banach spaces by means of fixed-point principle in cone and the fixed-point index theory for strict set contraction operator.

In [16], Fu and Ding considered the third-order boundary value problems with integral boundary conditions

{ ( φ ( x ( t ) ) ) = f ( t , x ( t ) ) , t J , x ( 0 ) = 0 , x ( 0 ) = 0 , x ( 1 ) = 0 1 g ( t ) x ( t ) d t .

The arguments were based upon the fixed-point principle in cone for strict set contraction operators.

In [7], Han and Kang were concerned with the existence of multiple positive solutions of the third-order p-Laplacian dynamic equation on time scales

{ ( ϕ p ( u ( t ) ) ) + f ( t , u ( t ) ) = 0 , t [ a , b ] , α u ( ρ ( a ) ) β u ( ρ ( a ) ) = 0 , γ u ( b ) + δ u ( b ) = 0 , u ( ρ ( a ) ) = 0 ,

where ϕ p (s)= is p-Laplacian operator, i.e., ϕ p (s)=|s | p 2 s, p>1, ϕ p 1 = ϕ q , 1 p + 1 q =1. By using fixed point theorems in cones, they obtained the existence of multiple positive solutions for singular nonlinear boundary value problem.

Motivated by the results above, in this study, we consider the following third-order boundary value problem (BVP) on time scales:

{ ( ϕ ( u ( t ) ) ) + q ( t ) f ( t , u ( t ) , u ( t ) ) = 0 , t [ 0 , 1 ] T , a u ( 0 ) b u ( 0 ) = 0 1 g 1 ( s ) u ( s ) s , c u ( 1 ) + d u ( 1 ) = 0 1 g 2 ( s ) u ( s ) s , u ( 1 ) = 0 ,
(1.1)

where is a time scale, 0,1T, [ 0 , 1 ] T =[0,1]T, ϕ:RR is an increasing homeomorphism and positive homomorphism with ϕ(0)=0. A projection ϕ:RR is called an increasing homeomorphism and positive homomorphism if the following conditions are satisfied:

  1. (i)

    If xy, then ϕ(x)ϕ(y) for all x,yR;

  2. (ii)

    ϕ is a continuous bijection, and its inverse mapping is also continuous;

  3. (iii)

    ϕ(xy)=ϕ(x)ϕ(y) for all x,yR.

Throughout this paper, we assume that the following conditions hold:

(C1) a,b,c,d[0,) with ac+ad+bc>0,

(C2) fC( [ 0 , 1 ] T × R + × R + , R + ),

(C3) q, g 1 and g 2 C( [ 0 , 1 ] T , R + ).

By using the four functionals fixed point theorem [20], we get the existence of at least one positive solution for BVP (1.1). In fact, our result is also new when T=R (the differential case) and T=Z (the discrete case). Therefore, the result can be considered as a contribution to this field.

This paper is organized as follows. In Section 2, we provide some definitions and preliminary lemmas, which are the key tools for our main result. We give and prove our main result in Section 3. Finally, in Section 4, we give an example to demonstrate our result.

2 Preliminaries

In this section, to state the main results of this paper, we need the following lemmas.

We define B= C [0,1], which is a Banach space with the norm

u=max { max t [ 0 , 1 ] T | u ( t ) | , max t [ 0 , 1 ] T | u ( t ) | } .

Define the cone PB by

P = { u B : u ( t )  is nonnegative, nondecreasing on  [ 0 , 1 ] T , u ( t )  is nonincreasing on  [ 0 , 1 ] T ,  and  a u ( 0 ) b u ( 0 ) = 0 1 g 1 ( s ) u ( s ) s } .

Denote by θ and φ, the solutions of the corresponding homogeneous equation

( ϕ ( u ( t ) ) ) =0,t [ 0 , 1 ] T ,
(2.1)

under the initial conditions,

{ θ ( 0 ) = b , θ ( 0 ) = a , φ ( 1 ) = d , φ ( 1 ) = c .
(2.2)

Using the initial conditions (2.2), we can deduce from equation (2.1) for θ and φ the following equations:

θ(t)=b+at,φ(t)=d+c(1t).
(2.3)

Set

Δ:= | 0 1 g 1 ( s ) ( b + a s ) s ρ 0 1 g 1 ( s ) ( d + c ( 1 s ) ) s ρ 0 1 g 2 ( s ) ( b + a s ) s 0 1 g 2 ( s ) ( d + c ( 1 s ) ) s |
(2.4)

and

ρ:=ad+ac+bc.
(2.5)

Lemma 2.1 Let (C1)-(C3) hold. Assume that

(C4) Δ0.

If u C [0,1] is a solution of the equation

u ( t ) = 0 1 G ( t , s ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s + A ( f ) ( b + a t ) + B ( f ) ( d + c ( 1 t ) ) ,
(2.6)

where

G(t,s)= 1 ρ { ( b + a σ ( s ) ) ( d + c ( 1 t ) ) , σ ( s ) t , ( b + a t ) ( d + c ( 1 σ ( s ) ) ) , t s ,
(2.7)
A(f):= 1 Δ | 0 1 g 1 ( s ) H f ( s ) s ρ 0 1 g 1 ( s ) ( d + c ( 1 s ) ) s 0 1 g 2 ( s ) H f ( s ) s 0 1 g 2 ( s ) ( d + c ( 1 s ) ) s | ,
(2.8)
B(f):= 1 Δ | 0 1 g 1 ( s ) ( b + a s ) s 0 1 g 1 ( s ) H f ( s ) s ρ 0 1 g 2 ( s ) ( b + a s ) s 0 1 g 2 ( s ) H f ( s ) s |
(2.9)

and

H f (s):= 0 1 G(s,r) ϕ 1 ( r 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) r,
(2.10)

then u is a solution of the boundary value problem (1.1).

Proof Let u satisfy the integral equation (2.6), then u is a solution of problem (1.1). Then we have

u ( t ) = 0 1 G ( t , s ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s + A ( f ) ( b + a t ) + B ( f ) ( d + c ( 1 t ) ) ,

i.e.,

u ( t ) = 0 t 1 ρ ( b + a ( σ ( s ) ) ) ( d + c ( 1 t ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( t ) = + t 1 1 ρ ( b + a t ) ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( t ) = + A ( f ) ( b + a t ) + B ( f ) ( d + c ( 1 t ) ) , u ( t ) = 0 t c ρ ( b + a ( σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( t ) = + t 1 a ρ ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( t ) = + A ( f ) a B ( f ) c .

Therefore,

u ( t ) = 1 ρ ( c ( b + a ( σ ( t ) ) ) a ( d + c ( 1 σ ( t ) ) ) ) ϕ 1 ( t 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) u ( t ) = 1 ρ ( ( a d + a c + b c ) ) ϕ 1 ( t 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) u ( t ) = ϕ 1 ( t 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) , ϕ ( u ( t ) ) = t 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ , ( ϕ ( u ( t ) ) ) = q ( t ) f ( t , u ( t ) , u ( t ) ) .

So, we get

( ϕ ( u ( t ) ) ) +q(t)f ( t , u ( t ) , u ( t ) ) =0.

Since

u ( 0 ) = 0 1 b ρ ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( 0 ) = + A ( f ) b + B ( f ) ( d + c ) , u ( 0 ) = 0 1 a ρ ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( 0 ) = + A ( f ) a B ( f ) c ,

we have

a u ( 0 ) b u ( 0 ) = B ( f ) ρ = 0 1 g 1 ( s ) ( 0 1 G ( s , r ) ϕ 1 ( r 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) r + A ( f ) ( b + a s ) + B ( f ) ( d + c ( 1 s ) ) ) s .
(2.11)

Since

u ( 1 ) = 0 1 d ρ ( b + a ( σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( 1 ) = + A ( f ) ( b + a ) + B ( f ) d , u ( 1 ) = 0 1 c ρ ( b + a ( σ ( s ) ) ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s u ( 1 ) = + A ( f ) a B ( f ) c ,

we have

c u ( 1 ) + d u ( 1 ) = A ( f ) ρ = 0 1 g 2 ( s ) ( 0 1 G ( s , r ) ϕ 1 ( r 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) r + A ( f ) ( b + a s ) + B ( f ) ( d + c ( 1 s ) ) ) s .
(2.12)

From (2.11) and (2.12), we get

{ [ 0 1 g 1 ( s ) ( b + a s ) s ] A ( f ) + [ ρ 0 1 g 1 ( s ) ( d + c ( 1 s ) ) s ] B ( f ) = 0 1 g 1 ( s ) H f ( s ) s , [ ρ 0 1 g 2 ( s ) ( b + a s ) s ] A ( f ) + [ 0 1 g 2 ( s ) ( d + c ( 1 s ) ) s ] B ( f ) = 0 1 g 2 ( s ) H f ( s ) s ,

which implies that A(f) and B(f) satisfy (2.8) and (2.9), respectively. □

Lemma 2.2 Let (C1)-(C3) hold. Assume that

(C5) Δ<0, ρ 0 1 g 2 (s)(b+as)s>0, a 0 1 g 1 (s)s>0.

Then for u C [0,1], the solution u of problem (1.1) satisfies

u(t)0for t [ 0 , 1 ] T .

Proof It is an immediate subsequence of the facts that G0 on [ 0 , 1 ] T × [ 0 , 1 ] T and A(f)0, B(f)0. □

Lemma 2.3 Let (C1)-(C3) and (C5) hold. Assume that

(C6) c 0 1 g 2 (s)s<0.

Then the solution u C [0,1] of problem (1.1) satisfies u (t)0 for t [ 0 , 1 ] T .

Proof Assume that the inequality u (t)<0 holds. Since u (t) is nonincreasing on [ 0 , 1 ] T , one can verify that

u (1) u (t),t [ 0 , 1 ] T .

From the boundary conditions of problem (1.1), we have

c d u(1)+ 1 d 0 1 g 2 (s)u(s)s u (t)<0.

The last inequality yields

cu(1)+ 0 1 g 2 (s)u(s)s<0.

Therefore, we obtain that

0 1 g 2 (s)u(1)s< 0 1 g 2 (s)u(s)s<cu(1),

i.e.,

( c 0 1 g 2 ( s ) s ) u(1)>0.

According to Lemma 2.2, we have that u(1)0. So, c 0 1 g 2 (s)s>0. However, this contradicts to condition (C6). Consequently, u (t)0 for t [ 0 , 1 ] T . □

Lemma 2.4 If (C1)-(C6) hold, then max t [ 0 , 1 ] T u(t)M max t [ 0 , 1 ] T u (t) for uP, where

M=1+ b + 0 1 s g 1 ( s ) s a 0 1 g 1 ( s ) s .
(2.13)

Proof For uP, since u (t) is nonincreasing on [ 0 , 1 ] T , one arrives at

u ( t ) u ( 0 ) t u (0),

i.e., u(t)u(0)t u (0). Hence,

0 1 g 1 (s)u(s)s 0 1 g 1 (s)su(0) 0 1 s g 1 (s)s u (0).

By au(0)b u (0)= 0 1 g 1 (s)u(s)s, we get

u(0) b + 0 1 s g 1 ( s ) s a 0 1 g 1 ( s ) s u (0).

Hence,

u ( t ) = 0 t u ( s ) s + u ( 0 ) t u ( 0 ) + u ( 0 ) t u ( 0 ) + b + 0 1 s g 1 ( s ) s a 0 1 g 1 ( s ) s u ( 0 ) ( 1 + b + 0 1 s g 1 ( s ) s a 0 1 g 1 ( s ) s ) u ( 0 ) = M u ( 0 ) ,

i.e.,

max t [ 0 , 1 ] T u(t)M max t [ 0 , 1 ] T u (t).

The proof is finalized. □

From Lemma 2.4, we obtain

u = max { max t [ 0 , 1 ] T | u ( t ) | , max t [ 0 , 1 ] T | u ( t ) | } max { M max t [ 0 , 1 ] T u ( t ) , max t [ 0 , 1 ] T u ( t ) } = M max t [ 0 , 1 ] T u ( t ) .

Now, define an operator T:PB by

T u ( t ) = 0 1 G ( t , s ) ϕ 1 ( 0 s q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s + A ( f ) ( b + a t ) + B ( f ) ( d + c ( 1 t ) ) ,
(2.14)

where G, A(f) and B(f) are defined as in (2.7), (2.8) and (2.9), respectively.

Lemma 2.5 Let (C1)-(C6) hold. Then T:PP is completely continuous.

Proof By Arzela-Ascoli theorem, we can easily prove that operator T is completely continuous. □

3 Main results

Let α and Ψ be nonnegative continuous concave functionals on , and let β and Φ be nonnegative continuous convex functionals on , then for positive numbers r, j, l and R, we define the sets

Q ( α , β , r , R ) = { u P : r α ( u ) , β ( u ) R } , U ( Ψ , j ) = { u Q ( α , β , r , R ) : j Ψ ( u ) } , V ( Φ , l ) = { u Q ( α , β , r , R ) : Φ ( u ) l } .
(3.1)

Lemma 3.1 [20]

If is a cone in a real Banach space , α and Ψ are nonnegative continuous concave functionals on , β and Φ are nonnegative continuous convex functionals on , and there exist positive numbers r, j, l and R such that

T:Q(α,β,r,R)P

is a completely continuous operator, and Q(α,β,r,R) is a bounded set. If

  1. (i)

    {uU(Ψ,j):β(u)<R}{uV(Φ,l):r<α(u)};

  2. (ii)

    α(Tu)r for all uQ(α,β,r,R) with α(u)=r and l<Φ(Tu);

  3. (iii)

    α(Tu)r for all uV(Φ,l) with α(u)=r;

  4. (iv)

    β(Tu)R for all uQ(α,β,r,R) with β(u)=R and Ψ(Tu)<j;

  5. (v)

    β(Tu)R for all uU(Ψ,j) with β(u)=R.

Then T has a fixed point u in Q(α,β,r,R).

Suppose that ω,zT with 0<ω<z<1. For the convenience, we take the notations

A : = 1 Δ | 0 1 g 1 ( s ) H ( s ) s ρ 0 1 g 1 ( s ) ( d + c ( 1 s ) ) s 0 1 g 2 ( s ) H ( s ) s 0 1 g 2 ( s ) ( d + c ( 1 s ) ) s | , H ( s ) : = 0 1 G ( s , r ) ϕ 1 ( r 1 q ( τ ) τ ) r , Ω = ω z G ( ω , s ) ϕ 1 ( s z q ( τ ) τ ) s , Λ = 0 1 1 ρ a ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( 0 1 q ( τ ) τ ) s + A a , N = a 0 1 g 1 ( s ) s b + 0 1 s g 1 ( s ) s ,

and define the maps

α(u)= min t [ ω , z ] T u(t),Φ(u)= max t [ 0 , 1 ] T u(t),β(u)=Ψ(u)= max t [ 0 , 1 ] T u (t).
(3.2)

Let Q(α,β,r,R), U(Ψ,j) and V(Φ,l) be defined by (3.1).

Theorem 3.1 Assume that (C1)-(C6) hold. If there exist constants r, j, l, R with max{ r ω ,R}l, max{ N + 1 N j, N + 1 N ω + 1 r}<R, and suppose that f satisfies the following conditions:

(C7) f(t,u, u )ϕ( r Ω ) for (t,u, u ) [ ω , z ] T ×[r,l]×[0,R];

(C8) f(t,u, u )ϕ( R Λ ) for (t,u, u ) [ 0 , 1 ] T ×[0,MR]×[0,R].

Then BVP (1.1) has at least one positive solution uP such that

min t [ ω , z ] T u(t)r, max t [ 0 , 1 ] T u(t)R.

Proof The boundary value problem (1.1) has a solution u=u(t) if and only if u solves the operator equation u=Tu. Thus, we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T.

We first show that Q(α,β,r,R) is bounded, and T:Q(α,β,r,R)P is completely continuous. For all uQ(α,β,r,R) with Lemma 2.4, we have

uM max t [ 0 , 1 ] T u (t)=Mβ(u)MR,

which means that Q(α,β,r,R) is a bounded set. According to Lemma 2.5, it is clear that T:Q(α,β,r,R)P is completely continuous.

Let

u 0 = R N + 1 (Nt+1).

Clearly, u 0 P. By direct calculation,

α ( u 0 ) = u 0 ( ω ) = R N + 1 ( N ω + 1 ) > r , β ( u 0 ) = R N + 1 N < R , Ψ ( u 0 ) = β ( u 0 ) = R N + 1 N j , Φ ( u 0 ) = u 0 ( 1 ) = R N + 1 ( N + 1 ) = R l .

So, u 0 {uU(Ψ,j):β(u)<R}{uV(Φ,l):r<α(u)}, which means that (i) in Lemma 3.1 is satisfied.

For all uQ(α,β,r,R) with α(u)=r and l<Φ(Tu), since u is nonincreasing on [ 0 , 1 ] T , we have

α(Tu)=Tu(ω)ωTu(1)=ωΦ(Tu)>ωlr.

So, α(Tu)>r. Hence, (ii) in Lemma 3.1 is fulfilled.

For all uV(Φ,l) with α(u)=r,

α ( T u ) = min t [ ω , z ] T T u ( t ) = ( T u ) ( ω ) = 0 1 f G ( ω , s ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s + A ( f ) ( b + a ω ) + B ( f ) ( d + c ( 1 ω ) ) 0 1 G ( ω , s ) ϕ 1 ( s 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s ω z G ( ω , s ) ϕ 1 ( s z q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s r Ω ω z G ( ω , s ) ϕ 1 ( s z q ( τ ) τ ) s = r ,

and for all uU(Ψ,j) with β(u)=R,

β ( T u ) = max t [ 0 , 1 ] T ( T u ) ( t ) = ( T u ) ( 0 ) 0 1 1 ρ a ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( 0 1 q ( τ ) f ( τ , u ( τ ) , u ( τ ) ) τ ) s + A ( f ) a R Λ ( 0 1 1 ρ a ( d + c ( 1 σ ( s ) ) ) ϕ 1 ( 0 1 q ( τ ) τ ) s + A a ) = R .

Thus, (iii) and (v) in Lemma 3.1 hold. We finally prove that (iv) in Lemma 3.1 holds.

For all uQ(α,β,r,R) with β(u)=R and Ψ(Tu)<j, we have

β(Tu)=Ψ(Tu)<j< N N + 1 R<R.

Thus, all conditions of Lemma 3.1 are satisfied. T has a fixed point u in Q(α,β,r,R). Therefore, BVP (1.1) has at least one positive solution uP such that

min t [ ω , z ] T u(t)r, max t [ 0 , 1 ] T u(t)R.

The proof is completed. □

4 An example

Example 4.1 In BVP (1.1), suppose that T=[0,1], q(t)= g 1 (t)= g 2 (t)=1, a=4, b=2, c= 1 2 , and d=8, i.e.,

{ ( ϕ ( u ( t ) ) ) + f ( t , u ( t ) , u ( t ) ) = 0 , t [ 0 , 1 ] , 4 u ( 0 ) 2 u ( 0 ) = 0 1 u ( s ) d s , 1 2 u ( 1 ) + 8 u ( 1 ) = 0 1 u ( s ) d s , u ( 1 ) = 0 ,
(4.1)

where

f ( t , u , u ) = { 82 , u [ 0 , 5 ] , 17 179 u + 14 , 593 179 , u 5

and

ϕ(u)= { u 2 , u 0 , u , u 0 .

By simple calculation, we get ρ=35, θ(t)=2+4t, φ(t)= 17 2 1 2 t, Δ= 3 , 185 4 , M= 11 6 , N= 6 5 , A=B= 32 1 , 911 , and

G(t,s)= 1 35 { ( 2 + 4 s ) ( 17 2 1 2 t ) , s t , ( 2 + 4 t ) ( 17 2 1 2 s ) , t s .

Set ω= 1 3 , z= 2 3 , then we get

Ω= 5 81 ,Λ= 9 , 649 9 , 555 .

Choose r=5, l=150, j=10 and R=100, it is easy to check that max{15,100}150, max{ 55 3 , 55 12 }<100, and conditions (C1)-(C6) are satisfied. Now, we show that (C7) and (C8) are satisfied:

f ( t , u ( t ) , u ( t ) ) 82 > ϕ ( r Ω ) = 81 for  ( t , u ( t ) , u ( t ) ) [ 1 3 , 2 3 ] × [ 5 , 150 ] × [ 0 , 100 ] ; f ( t , u ( t ) , u ( t ) ) < 99 ϕ ( R Λ ) = 99.0258 for  ( t , u ( t ) , u ( t ) ) [ 0 , 1 ] × [ 0 , 550 3 ] × [ 0 , 100 ] .

So, all conditions of Theorem 3.1 hold. Thus, by Theorem 3.1, BVP (4.1) has at least one positive solution u such that

min t [ 1 3 , 2 3 ] u(t)5, max t [ 0 , 1 ] u(t)100.

References

  1. Hilger, S: Ein masskettenkalkül mit anwendug auf zentrumsmanningfaltigkeiten. PhD thesis, Universitat Würzburg (1988)

    MATH  Google Scholar 

  2. Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results Math. 1999, 35: 3–22. 10.1007/BF03322019

    MathSciNet  Article  MATH  Google Scholar 

  3. Agarwal RP, O’Regan D: Nonlinear boundary value problems on time scales. Nonlinear Anal. 2001, 44: 527–535. 10.1016/S0362-546X(99)00290-4

    MathSciNet  Article  MATH  Google Scholar 

  4. Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Comput. Math. Appl. 2008, 56: 2429–2443. 10.1016/j.camwa.2008.05.018

    MathSciNet  Article  MATH  Google Scholar 

  5. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.

    Book  MATH  Google Scholar 

  6. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2002.

    MATH  Google Scholar 

  7. Han W, Kang S: Multiple positive solutions of nonlinear third-order BVP for a class of p -Laplacian dynamic equations on time scales. Math. Comput. Model. 2009, 49: 527–535. 10.1016/j.mcm.2008.08.002

    MathSciNet  Article  MATH  Google Scholar 

  8. Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains. Kluwer Academic, Dordrecht; 1996.

    Book  MATH  Google Scholar 

  9. Liu J, Xu F: Triple positive solutions for third-order m -point boundary value problems on time scales. Discrete Dyn. Nat. Soc. 2009., 2009: Article ID 123283

    Google Scholar 

  10. Sun TT, Wang LL, Fan YH: Existence of positive solutions to a nonlocal boundary value problem with p -Laplacian on time scales. Adv. Differ. Equ. 2010., 2010: Article ID 809497

    Google Scholar 

  11. Tokmak F, Karaca IY: Existence of positive solutions for third-order m -point boundary value problems with sign-changing nonlinearity on time scales. Abstr. Appl. Anal. 2012., 2012: Article ID 5067163

    Google Scholar 

  12. Tokmak F, Karaca IY: Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales. Bound. Value Probl. 2013., 52: Article ID 12

    Google Scholar 

  13. Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge; 1991.

    Book  MATH  Google Scholar 

  14. Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.

    Book  MATH  Google Scholar 

  15. Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364–371. 10.1016/j.na.2007.12.007

    MathSciNet  Article  MATH  Google Scholar 

  16. Fu D, Ding W: Existence of positive solutions of third-order boundary value problems with integral boundary conditions in Banach spaces. Adv. Differ. Equ. 2013., 2013: Article ID 65

    Google Scholar 

  17. Li Y, Zhang T: Multiple positive solutions for second-order p -Laplacian dynamic equations with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 867615

    Google Scholar 

  18. Zhao J, Wang P, Ge W: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 402–413. 10.1016/j.cnsns.2009.10.011

    MathSciNet  Article  MATH  Google Scholar 

  19. Ma R: Multiple positive solutions for nonlinear m -point boundary value problems. Appl. Math. Comput. 2004, 148: 249–262. 10.1016/S0096-3003(02)00843-3

    MathSciNet  Article  MATH  Google Scholar 

  20. Avery R, Henderson J, O’Regan D: Four functionals fixed point theorem. Math. Comput. Model. 2008, 48: 1081–1089. 10.1016/j.mcm.2007.12.013

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

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Correspondence to Fatma Tokmak.

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Karaca, I.Y., Tokmak, F. Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales. J Inequal Appl 2013, 498 (2013). https://doi.org/10.1186/1029-242X-2013-498

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Keywords

  • Green’s function
  • time scales
  • fixed point theorem
  • increasing homeomorphism and positive homomorphism
  • integral boundary conditions
  • positive solution