, 2011:42

# Solvability of boundary value problems with Riemann-Stieltjes Δ-integral conditions for second-order dynamic equations on time scales at resonance

• Yongkun Li
• Jiangye Shu
Open Access
Research

## Abstract

In this paper, by making use of the coincidence degree theory of Mawhin, the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes Δ-integral conditions on time scales at resonance

is established, where satisfies the Carathéodory conditions and is a continuous function and is an increasing function with . An example is given to illustrate the main results.

## Keywords

boundary value problem with Riemann-Stieltjes Δ-integral conditions resonance time scales

## 1 Introduction

Hilger [1] introduced the notion of time scales in order to unify the theory of continuous and discrete calculus. The field of dynamical equations on time scales contains, links and extends the classical theory of differential and difference equations, besides many others. There are more time scales than just ℝ (corresponding to the continuous case) and ℕ (corresponding to the discrete case) and hence many more classes of dynamic equations. An excellent resource with an extensive bibliography on time scales was produced by Bohner and Peterson [2, 3].

Recently, existence theory for positive solutions of boundary value problems (BVPs) on time scales has attracted the attention of many authors; Readers are referred to, for example, [4, 5, 6, 7, 8, 9, 10, 11] and the references therein for the existence theory of some two-point BVPs and [12, 13, 14, 15, 16, 17] for three-point BVPs on time scales. For the existence of solutions of m-point BVPs on time scales, we refer the reader to [18, 19, 20].

At the same time, we notice that a class of boundary value problems with integral boundary conditions have various applications in chemical engineering, thermo-elasticity, population dynamics, heat conduction, chemical engineering underground water flow, thermo-elasticity and plasma physics. On the other hand, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two-point, three-point, multipoint and nonlocal boundary value problems as special cases [[21, 22, 23, 24], and the references therein]. However, very little work has been done to the existence of solutions for boundary value problems with integral boundary conditions on time scales.

Motivated by the statements above, in this paper, we are concerned with the following boundary value problem with integral boundary conditions
(1.1)

where and are continuous functions, is an increasing function with , and the integral in (1.1) is a Riemann-Stieltjes on time scales, which is introduced in Section 2 of this paper.

According to the calculus theory on time scales, we can illustrate that boundary value problems with integral boundary conditions on time scales also cover two-point, three-point, ..., n-point boundary problems as the nonlocal boundary value problems do in the continuous case. For instance, in BVPs (1.1), let
where k ≥ 1 is an integer, a i ∈ [0, ∞), i = 1, ..., k, is a finite increasing sequence of distinct points in , and χ(s) is the characteristic function, that is,
then the nonlocal condition in BVPs (1.1) reduces to the k-point boundary condition

where t i , i = 1, 2, ..., k can be determined (see Lemma 2.5 in Section 2).

The effect of resonance in a mechanical equation is very important to scientists. Nearly every mechanical equation will exhibit some resonance and can, with the application of even a very small external pulsed force, be stimulated to do just that. Scientists usually work hard to eliminate resonance from a mechanical equation, as they perceive it to be counter-productive. In fact, it is impossible to prevent all resonance. Mathematicians have provided more theory of resonance from equations. For the case where ordinary differential equation is at resonance, most studies have tended to the equation x″(t) = f (t, x(t), x'(t)) + e(t). For example, Feng and Webb [25] studied the following boundary value problem

when αξ = 1(ξ ∈ (0, 1)) is at resonance.

It is easy to see that x1(t) ≡ c(c ∈ ℝ) and x2(t) = pt(p ∈ ℝ) are a fundamental set of solutions of the linear mapping Lx(t) = xΔΔ(t) = 0. Let U1(x) = xΔ(0) and . Since , we have that

Thus, det Q(x) = 0, which implies that BVPs (1.1) is at resonance. By applying coincidence degree theorem of Mawhin to integral boundary value problems on time scales at resonance, this paper will establish some sufficient conditions for the existence of at least one solution to BVPs (1.1).

The rest of this paper is organized as follows. Section 2 introduces the Riemann-Stieltjes integral on time scales. Some lemmas and criterion for the existence of at least one solution to BVPs (1.1) are established in Section 3, and examples are given to illustrate our main results in Section 4.

## 2 Preliminaries

This section includes two parts. In the first part, we shall recall some basic definitions and lemmas of the calculus on time scales, which will be used in this paper. For more details, we refer to books by Bohner and Peterson [2, 3]. In the second part, we introduce the Riemann-Stieltjes Δ-integral and ∇-integral on time scales, which was first established by Mozyrska et al. in [26].

### 2.1 The basic calculus on time scales

Definition 2.1. [3] A time scale is an arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ.

The forward and backward jump operators and the graininess are defined, respectively, by

The point is called left-dense, left-scattered, right-dense or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t or σ(t) > t, respectively. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum m1, define ; otherwise, set . If has a right-scattered minimum m2, define ; otherwise, set .

Definition 2.2. [3] A function is rd-continuous (rd-continuous is short for right-dense continuous) provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by .

Definition 2.3. [3] If is a function and , then the delta derivative of f at the point t is defined to be the number fΔ(t) (provided it exists) with the property that for each ε > 0 there is a neighborhood U of t such that
Definition 2.4. [3] For a function (the range ℝ of f may be actually replaced by Banach space), the (delta) derivative is defined at point t by
if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by

provided this limit exists.

Definition 2.5. [3] If FΔ(t) = f(t), then we define the delta integral by
Lemma 2.1. [3] Let , and assume that is continuous at(t, t), where with t > a. Also assume that fΔ(t, ·) is rd-continuous on [a, σ(t)]. Suppose that for each ε > 0 there exists a neighborhood U of t, independent of τ ∈ [a, σ(t)], such that
where f Δ denotes the derivative of f with respect to the first variable. Then
1. (1)

;

2. (2)

.

The construction of the Δ-measure on and the following concepts can be found in [3].
1. (i)
For each , the single-point set t0 is Δ-measurable, and its Δ-measure is given by

2. (ii)
If and ab, then

3. (iii)
If and ab, then

The Lebesgue integral associated with the measure μΔ on is called the Lebesgue delta integral. For a (measurable) set and a function f : E → ℝ, the corresponding integral of f on E is denoted by . All theorems of the general Lebesgue integration theory hold also for the Lebesgue delta integral on .

### 2.2 The Riemann-Stieltjes integral on time scales

Let be a time scale, , a < b, and . A partition of I is any finite-ordered

subset

Let g be a real-valued increasing function on I. Each partition P = {t0, t1, ..., t n } of I decomposes I into subintervals , j = 1, 2, ..., n, such that for any kj. By Δt j = t j - t j -1, we denote the length of the j th subinterval in the partition P; by the set of all partitions of I.

Let P m , . If P m P n , we call P n a refinement of P m . If P m , P n are independently chosen, then the partition is a common refinement of P m and P n .

Let us now consider an increasing real-valued function g on the interval I. Then, for the partition P of I, we define

where Δg j = g(t j ) - g(tj-1). We note that Δg j is positive and . Moreover, g(P) is a partition of [g(a), g(b)]. In what follows, for the particular case g(t) = t we obtain the Riemann sums for delta integral. We note that for a general g the image g(I) is not necessarily an interval in the classical sense, even for rd-continuous function g, because our interval I may contain scattered points. From now on, let g be always an increasing real function on the considered interval .

Lemma 2.2. [26] Let be a closed (bounded) interval in and let g be a continuous increasing function on I. For every δ > 0, there is a partition such that for each j ∈ {1, 2, ..., n}, one has
Let f be a real-valued and bounded function on the interval I. Let us take a partition P = {t0, t1, ..., t n } of I. Denote , j = 1, 2, ..., n, and
The upper Darboux-Stieltjes Δ-sum of f with respect the partition P, denoted by UΔ(P, f, g), is defined by
while the lower Darboux-Stieltjes Δ-sum of f with respect the partition P, denoted by LΔ(P, f, g), is defined by
Definition 2.6. [26] Let , where . Let g be continuous on I. The upper Darboux-Stieltjes Δ-integral from a to b with respect to function g is defined by
the lower Darboux-Stieltjes Δ-integral from a to b with respect to function g is defined by

If , then we say that f is Δ-integrable with respect to g on I, and the common value of the integrals, denoted by , is called the Riemann-Stieltjes Δ-integral of f with respect to g on I.

The set of all functions that are Δ-integrable with respect to g in the Riemann-Stieltjes sense will be denoted by .

Theorem 2.1. [26] Let f be a bounded function on , , mf (t) ≤ M for all tI, and g be a function defined and monotonically increasing on I. Then
If , then
Theorem 2.2. [26] (Integrability criterion) Let f be a bounded function on , . Then, if and only if for every ε > 0, there exists a partition such that
Theorem 2.3. [26] Let , . Then, the condition is equivalent to each one of the following items:
1. (i)

f is a monotonic function on I;

2. (ii)

f is a continuous function on I;

3. (iii)

f is regulated on I;

4. (iv)

f is a bounded and has a finite number of discontinuity points on I.

In the following, we state some algebraic properties of the Riemann-Stieltjes integral on time scales as well. The properties are valid for an arbitrary time scale with at least two points. We define and for a > b.

Theorem 2.4. [26] Let , . Every constant function , f(t) ≡ c, is Stieltjes Δ-integrable with respect to g on I and
Theorem 2.5. [26] Let and. If f is Riemann-Stieltjes Δ-integrable with respect to g from t to σ(t), then
where g σ = g ° σ. Moreover, if g is Δ-differentiable at t, then
Theorem 2.6. [26] Let with a < b < c. If f is bounded on and g is monotonically increasing on , then
Lemma 2.3. [26] Let , . Suppose that g is an increasing function such that gΔ is continuous on and f σ is a real-bounded function on I. Then, if and only if . Moreover,
Lemma 2.4. (Delta integration by parts) Let , . Suppose that g is an increasing function such that gΔ is continuous on and f σ is a real-bounded function on I. Then
Proof. Lemma 2.3 imply that
furthermore,
Hence,

The proof of this lemma is complete.

Lemma 2.5. Let , . Assume that f σ is a real-bounded function on I and
where k ≥ 1 is an integer, a i ∈ [0, ∞), i = 1, ..., k, is a finite increasing sequence of distinct points in and χ(s) is the characteristic function, that is,
Then

where t i , i = 1, 2, ..., k can be determined.

Proof. By Lemma 2.4, it leads to

This completes the proof.

## 3 Main results

In this section, first we provide some background materials from Banach spaces and preliminary results, and then we illustrate and prove some important lemmas and theorems.

Definition 3.1. Let × and Y be Banach spaces. A linear operator L : Dom LXY is called a Fredholm operator if the following two conditions hold
1. (i)

KerL has a finite dimension;

2. (ii)

Im L is closed and has a finite codimension.

L is a Fredholm operator, and its Fredholm index is the integer Ind L = dimKer L - codimIm L. In this paper, we are interested in a Fredholm operator of index zero, i.e., dimKer L = codimIm L.

From Definition 3.1, we know that there exist continuous projector P : XX and Q : YY such that Im P = Ker L, Ker Q = Im L, X = Ker L ⊕ Ker P, Y = Im L ⊕ ImQ, and the operator L|Dom L⋂KerP: Dom L ⋂ Ker P → Im L is invertible; we denote the inverse of L|Dom L⋂KerPby K P : Im L → Dom L ⋂ Ker P. The generalized inverse of L denoted by K P , Q : Y → Dom L ⋂ Ker P is defined by K P , Q = K P (I - Q).

Now, we state the coincidence degree theorem of Mawhin [27].

Theorem 3.1. Let Ω ⊂ X be open-bounded set, L be a Fredholm operator of index zero and N be L-compact on . Assume that the following conditions are satisfied:
1. (i)

;

2. (ii)

Nx ∉ Im L for every × ∈ Ker L ⋂ ∂Ω;

3. (iii)

deg(QN|Ker L⋂∂Ω, Ω ⋂ Ker L, 0) ≠ 0 with Q : YY a continuous projector such that Ker Q = Im L.

Then, the equation Lu = Nu admits at least one nontrivial solution in Dom .

Definition 3.2. A mapping satisfies the Carathéodory conditions with respect to , where denotes that all Lebesgue Δ-integrable functions on , if the following conditions are satisfied:
1. (i)

for each (x1, x2) ∈ ℝ2, the mapping tf(t, x1, x2) is Lebesgue measurable on ;

2. (ii)

for a.e. , the mapping (x1, x2) → f (t, x1, x2) is continuous on2;

3. (iii)

for each r > 0, there exists such that for a.e. and every x1 such that |x1| ≤ r, |f (t, x1, x2)| ≤ α r .

Let the Banach space with the norm ||x|| = max{||x||, ||xΔ||}, where . Let
set with the norm . We use the space defined by
Define the linear operator L and the nonlinear operator N by
respectively, where
Lemma 3.1. L : Dom LXY is a Fredholm mapping of index zero. Furthermore, the continuous linear project operator Q : YY can be defined by
where . Linear mapping K P can be written by
Proof. It is clear that Ker , i.e., dimKer L = 1. Moreover, we have
(3.1)
If y ∈ Im L, then there exists x ∈ Dom L such that xΔΔ(t) = y(t). Integrating it from 0 to t, we have
Integrating the above equation from s to T, we get
(3.2)
Substituting the boundary condition into (3.2), and by the condition , we have
(3.3)
On the other hand, yY satisfies (3.3), we take x ∈ Dom LX as given by (3.2), then xΔΔ(t) = y(t) and

Therefore, (3.1) holds.

Set . It is easy to show that Λ ≠ 0, and then we define the mapping Q : YY by

and it is easy to see that Q : YY is a linear continuous projector.

For the mapping L and continuous linear projector Q, it is not difficult to check that Im L = Ker Q. Set y = (y - Qy) + Qy; thus, y - Qy ∈ Ker Q = Im L and Qz ∈ Im Q, so Y = ImL + Im Q. If y ∈ Im L ⋂ Q, then y(t) = 0, hence Y = Im L ⊕ Im Q. From Ker L = ℝ, we obtain that Ind L = dim Ker L - codim Im L = dim Ker L - dim Im Q = 0, that is, L is a Fredholm mapping of index zero.

Take P : XX as follows
Obviously, Im P = Ker L and X = Ker L ⊕ Ker P. Then, the inverse K P : Im L → Dom L ⋂ Ker P is defined by
For y ∈ Im L, we have
from Lemma 2.1, we obtain
that is
(3.4)
On the other hand, for x ∈ Dom L ⋂ Ker P,
using Lemma 2.4 and the boundary conditions, we get
i.e.,
(3.5)

(3.4) and (3.5) yield K P = (L|Dom L⋂Ker P)-1. The proof is completed.

Furthermore,

Lemma 3.2. Let satisfy the Carathéodory conditions, then the mapping N is L-completely continuous.

Proof. Assume that x n , x0EX satisfy ||x n - x0|| → 0, (n → ∞); thus, there exists M > 0 such that ||x n || ≤ M for any n ≥ 1. One has that

In view of f satisfying the Carathéodory conditions, we can obtain that for a.e. , ||Nx n - Nx0|| → 0, (n → ∞). This means that the operator N : EY is continuous. By the definitions of QN and K P , Q N, we can obtain that QN : EY and K P , Q : EX are continuous.

Let r = sup{||x|| : xE} < ∞ for a.e. , we have
Since functions , we get that . Further

It follows that (QN)(E) and (K P,Q N )(E) are bounded.

It is easy to see that is equicontinuous on a.e. . So, we only show that is equicontinuous on a.e. . For any with t1 < ρ(t2),
(3.6)

Since with , (3.6) shows that is equicontinuous on a.e. . Hence, by the Arzelà-Ascoli theorem on time scales, and are compact on an arbitrary bounded EX, and the mapping N : XY is L-completely continuous. The proof is completed.

Now, we are ready to apply the coincidence degree theorem of Mawhin to give the sufficient conditions for the existence of at least one solution to problem (1.1).

Theorem 3.2. Let satisfy the Carathéodory conditions, and

(H1) There exist continuous functions , , i = 1, 2, such that
and

(H2) There is a constant M > 0 such that for any x ∈ Dom L\Ker L, if |x(t)| > M for all ; then, .

(H3) There is a constant M* > 0 such that for any c ∈ ℝ, if |c| > M*; then, we have either
(3.7)
or
(3.8)
Then, problem (1.1) admits at least one solution provided that
Proof. Let Ω1 = {x ∈ Dom L\Ker L : Lx = λNx for some λ ∈ (0, 1)}. For x ∈ Ω1, we have x ∉ Ker L and Nx ∈ Im L = Ker Q; thus, QNx = 0, i.e.,
Hence by (H2), we know that there exists such that |x(t0)| < M. Since
So we get
Integrating the equation
from 0 to t, we obtain
Let ε > 0 satisfy
For such ε, there is δ > 0 so that for i = 1, 2,
Let
We get
So we get
It follows from the definition of ε that there is a constant A > 0 such that
Hence, we have

which means that Ω1 is bounded.

Let Ω2 = {x ∈ Ker L : Nx ∈ Im L}. For x ∈ Ω2, then x(t) = c for some c ∈ ℝ. Nx ∈ Im L implies QNx = 0, that is

From (H3), we know that ||x|| = |c| ≤ M *, thus Ω2 is bounded.

If (3.7) holds, then let
where J : Ker L → Im Q is a linear isomorphism given by J(k) = k for any k ∈ ℝ. Since x(t) = k thus
If λ = 1, then k = 0, and in the case λ ∈ [0, 1), if |k| > M*, we have
which is a contradiction. Again, if (3.8) holds, then let

where J as in above, similar to the above argument. Thus, in either case, ||x|| = |k| ≤ M* for any x ∈ Ω3, that is, Ω3 is bounded.

Let Ω be a bounded open subset of X such that . By Lemma 3.2, we can check that is compact; thus, N is L-compact on .

Finally, we verify that the condition (iii) of Theorem 3.1 is fulfilled. Define a homotopy
According to the above argument, we have
thus, by the degree property of homotopy invariance, we obtain

Thus, the conditions of Theorem 2.4 are satisfied, that is, the operator equation Lx = Nx admits at least one solution in Dom . Therefore, BVPs (1.1) has at least one solution in .

## 4 An example

In this section, we present an easy example to illustrate our main results.

Example 4.1. Let , n = 1, 2, ..., ∞. Consider the boundary value

Problem
(4.1)
Let
then . Let

We can get that . It is easy to check other conditions of Theorem 3.1 are satisfied. Hence, boundary value problem (4.1) has at least one solution.

## Notes

### Acknowledgements

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

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