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

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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

x Δ Δ ( t ) = f ( t , x ( t ) , x Δ ( t ) ) + e ( t ) , a . e . t [ 0 , T ] T , x Δ ( 0 ) = 0 , x ( T ) = 0 T x σ ( s ) Δ g ( s ) Open image in new window

is established, where f : [ 0 , T ] T × × Open image in new window satisfies the Carathéodory conditions and e : [ 0 , T ] T Open image in new window is a continuous function and g : [ 0 , T ] T Open image in new window is an increasing function with 0 T Δ g ( s ) = 1 Open image in new window. 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
x Δ Δ ( t ) = f ( t , x ( t ) , x Δ ( t ) ) + e ( t ) , a . e . t [ 0 , T ] T , x Δ ( 0 ) = 0 , x ( T ) = 0 T x σ ( s ) Δ g ( s ) , Open image in new window
(1.1)

where f : [ 0 , T ] T × × Open image in new window and e : [ 0 , T ] T Open image in new window are continuous functions, g : [ 0 , T ] T Open image in new window is an increasing function with 0 T Δ g ( s ) = 1 Open image in new window, 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
g ( s ) = i = 1 k a i χ ( s - t i ) , Open image in new window
where k ≥ 1 is an integer, a i ∈ [0, ∞), i = 1, ..., k, { t i } i = 1 k Open image in new window is a finite increasing sequence of distinct points in [ 0 , T ] T Open image in new window, and χ(s) is the characteristic function, that is,
χ ( s ) = 1 , s > 0 , 0 , s 0 , Open image in new window
then the nonlocal condition in BVPs (1.1) reduces to the k-point boundary condition
x ( T ) = i = 1 k a i x ( t i ) , Open image in new window

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
x ( t ) = f ( t , x ( t ) , x ( t ) ) + e ( t ) , t ( 0 , 1 ) , x ( 0 ) = 0 , x ( 1 ) = α x ( ξ ) , Open image in new window

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 U 2 ( x ) = x ( T ) - 0 T x σ ( s ) Δ g ( s ) Open image in new window. Since 0 T Δ g ( s ) = 1 Open image in new window, we have that
Q ( x ) = ( U 1 ( x 1 ) U 1 ( x 2 ) ( U 2 ( x 1 ) U 2 ( x 2 ) ) = ( 0 p 0 p T p 0 T σ ( s ) Δ g ( s ) ) . Open image in new window

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 T Open image in new window 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 σ , ρ : T T Open image in new window and the graininess μ : T + Open image in new window are defined, respectively, by
σ ( t ) : = inf { s T : s > t } , ρ ( t ) : = sup { s T : s < t } , μ ( t ) : = σ ( t ) - t . Open image in new window

The point t T Open image in new window 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 T Open image in new window has a left-scattered maximum m1, define T k = T - { m 1 } Open image in new window; otherwise, set T k = T Open image in new window. If T Open image in new window has a right-scattered minimum m2, define T k = T - { m 2 } Open image in new window; otherwise, set T k = T Open image in new window.

Definition 2.2. [3] A function f : T Open image in new window is rd-continuous (rd-continuous is short for right-dense continuous) provided it is continuous at each right-dense point in T Open image in new window and has a left-sided limit at each left-dense point in T Open image in new window. The set of rd-continuous functions f : T Open image in new window will be denoted by C r d ( T ) = C r d ( T , ) Open image in new window.

Definition 2.3. [3] If f : T Open image in new window is a function and t T k Open image in new window, 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
| f ( σ ( t ) ) - f ( s ) - f Δ ( t ) [ σ ( t ) - s ] | ε | σ ( t ) - s | , for all  s U . Open image in new window
Definition 2.4. [3] For a function f : T Open image in new window (the range ℝ of f may be actually replaced by Banach space), the (delta) derivative is defined at point t by
f Δ ( t ) = f ( σ ( t ) ) - f ( t ) σ ( t ) - t , Open image in new window
if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by
f Δ ( t ) = lim s t f ( σ ( t ) ) - f ( s ) σ ( t ) - s = lim s t f ( t ) - f ( s ) t - s Open image in new window

provided this limit exists.

Definition 2.5. [3] If FΔ(t) = f(t), then we define the delta integral by
a t f ( s ) Δ s = F ( t ) - F ( a ) . Open image in new window
Lemma 2.1. [3] Let a T k Open image in new window, b T Open image in new window and assume that f : T × T k Open image in new window is continuous at(t, t), where t T k Open image in new window 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
| f ( σ ( t ) , τ ) - f ( s , τ ) - f Δ ( t , τ ) ( σ ( t ) - s ) | ε | σ ( t ) - s | , for all  s U , Open image in new window
where f Δ denotes the derivative of f with respect to the first variable. Then
  1. (1)

    g ( t ) : = a t f ( t , τ ) Δ τ i m p l i e s g Δ ( t ) = a t f Δ ( t , τ ) Δ τ + f ( σ ( t ) , t ) Open image in new window;

     
  2. (2)

    h ( t ) : = t b f ( t , τ ) Δ τ i m p l i e s h Δ ( t ) = t b f Δ ( t , τ ) Δ τ - f ( σ ( t ) , t ) Open image in new window.

     
The construction of the Δ-measure on T Open image in new window and the following concepts can be found in [3].
  1. (i)
    For each t 0 T \ { max T } Open image in new window, the single-point set t0 is Δ-measurable, and its Δ-measure is given by
    μ Δ ( { t 0 } ) = σ ( t 0 ) - t 0 = μ ( t 0 ) . Open image in new window
     
  2. (ii)
    If a , b T Open image in new window and ab, then
    μ Δ ( [ a , b ) ) = b - a and  μ Δ ( ( a , b ) ) = b - σ ( a ) . Open image in new window
     
  3. (iii)
    If a , b T \ { max T } Open image in new window and ab, then
    μ Δ ( ( a , b ] ) = σ ( b ) - σ ( a ) and  μ Δ ( [ a , b ] ) = σ ( b ) - a . Open image in new window
     

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

2.2 The Riemann-Stieltjes integral on time scales

Let T Open image in new window be a time scale, a , b T Open image in new window, a < b, and I = [ a , b ] T Open image in new window. A partition of I is any finite-ordered

subset
P = { t 0 , t 1 , , t n } [ a , b ] T , where  a = t 0 < t 1 < < t n = b . Open image in new window

Let g be a real-valued increasing function on I. Each partition P = {t0, t1, ..., t n } of I decomposes I into subintervals I Δ j = [ t j - 1 , ρ ( t j ) ] T : = [ t j - 1 , t j ] Δ Open image in new window, j = 1, 2, ..., n, such that I Δ j I Δ k = Open image in new window 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 P ( I ) Open image in new window the set of all partitions of I.

Let P m , P n P ( I ) Open image in new window. 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 P m P n Open image in new window 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
g ( P ) = { g ( a ) = g ( t 0 ) , g ( t 1 , ) , g ( t n - 1 ) , g ( t n ) } g ( I ) , Open image in new window

where Δg j = g(t j ) - g(tj-1). We note that Δg j is positive and j = 1 n Δ g j = g ( b ) - g ( a ) Open image in new window. 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 I = [ a , b ] T Open image in new window.

Lemma 2.2. [26] Let I = [ a , b ] T Open image in new window be a closed (bounded) interval in T Open image in new window and let g be a continuous increasing function on I. For every δ > 0, there is a partition P δ = { t 0 , t 1 , , t n } P ( I ) Open image in new window such that for each j ∈ {1, 2, ..., n}, one has
Δ g j = g ( t j ) - g ( t j - 1 ) δ or Δ g j > δ ρ ( t j ) = t j - 1 . Open image in new window
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 I Δ j = [ t j - 1 , t j ] Δ Open image in new window, j = 1, 2, ..., n, and
m Δ j = inf t I Δ j f ( t ) , M Δ j = sup t I Δ j f ( t ) . Open image in new window
The upper Darboux-Stieltjes Δ-sum of f with respect the partition P, denoted by UΔ(P, f, g), is defined by
U Δ ( P , f , g ) = j = 1 n M j Δ g j , Open image in new window
while the lower Darboux-Stieltjes Δ-sum of f with respect the partition P, denoted by LΔ(P, f, g), is defined by
L Δ ( P , f , g ) = j = 1 n m Δ j Δ g j . Open image in new window
Definition 2.6. [26] Let I = [ a , b ] T Open image in new window, where a , b T Open image in new window. Let g be continuous on I. The upper Darboux-Stieltjes Δ-integral from a to b with respect to function g is defined by
a b ¯ f ( t ) Δ g ( t ) = inf P P ( I ) U Δ ( P , f , g ) ; Open image in new window
the lower Darboux-Stieltjes Δ-integral from a to b with respect to function g is defined by
a b f ( t ) Δ g ( t ) = sup P P ( I ) U Δ ( P , f , g ) . Open image in new window

If a b ¯ f ( t ) Δ g ( t ) = a b f ( t ) Δ g ( t ) Open image in new window, then we say that f is Δ-integrable with respect to g on I, and the common value of the integrals, denoted by a b f ( t ) Δ g ( t ) = a b f Δ g Open image in new window, 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 R Δ ( g , I ) Open image in new window.

Theorem 2.1. [26] Let f be a bounded function on I = [ a , b ] T Open image in new window, a , b T Open image in new window, mf (t) ≤ M for all tI, and g be a function defined and monotonically increasing on I. Then
m ( g ( b ) - g ( a ) ) a b f ( t ) Δ g ( t ) a b f ¯ ( t ) Δ g ( t ) M ( g ( b ) - g ( a ) ) . Open image in new window
If f R Δ ( g , I ) Open image in new window, then
m ( g ( b ) - g ( a ) ) a b f ( t ) Δ g ( t ) M ( g ( b ) - g ( a ) ) . Open image in new window
Theorem 2.2. [26] (Integrability criterion) Let f be a bounded function on I = [ a , b ] T Open image in new window, a , b T Open image in new window. Then, f R Δ ( g , I ) Open image in new window if and only if for every ε > 0, there exists a partition P P ( I ) Open image in new window such that
U Δ ( P , f , g ) - L Δ ( P , f , g ) < ε . Open image in new window
Theorem 2.3. [26] Let I = [ a , b ] T Open image in new window, a , b T Open image in new window. Then, the condition f R Δ ( g , I ) Open image in new window 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 T Open image in new window with at least two points. We define a a f ( t ) Δ g ( t ) = 0 Open image in new window and a b f ( t ) Δ g ( t ) = - b a f ( t ) Δ g ( t ) Open image in new window for a > b.

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

The proof of this lemma is complete.

Lemma 2.5. Let I = [ 0 , T ] T Open image in new window, 0 , T T Open image in new window. Assume that f σ is a real-bounded function on I and
g ( s ) = i = 1 k a i χ ( s - t i ) , Open image in new window
where k ≥ 1 is an integer, a i ∈ [0, ∞), i = 1, ..., k, { t i } i = 1 k Open image in new window is a finite increasing sequence of distinct points in [ 0 , T ] T Open image in new window and χ(s) is the characteristic function, that is,
χ ( s ) = 1 , s > 0 , 0 , s 0 . Open image in new window
Then
f ( T ) = 0 T f σ ( s ) Δ g ( s ) = i = 1 k a i f ( t i ) , Open image in new window

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

Proof. By Lemma 2.4, it leads to
f ( T ) = 0 T f σ ( s ) Δ g ( s ) = 0 t 1 + t 1 t 2 + + t k T f σ ( s ) Δ g ( s ) = [ f g ] 0 t 1 - 0 t 1 g ( s ) Δ f ( s ) + + [ f g ] t k T - t k T g ( s ) Δ f ( s ) = f ( T ) g ( T ) - 0 t 1 0 Δ f ( s ) + t 1 t 2 a 1 Δ f ( s ) + + t k T ( a 1 + a 2 + + a k ) Δ f ( s ) = ( a 1 + a 2 + + a k ) f ( T ) - - i = 1 k a i f ( t i ) + ( a 1 + a 2 + + a k ) f ( T ) = i = 1 k a i f ( t i ) . Open image in new window

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 Ω ̄ Open image in new window. Assume that the following conditions are satisfied:
  1. (i)

    L x λ N x f o r e v e r y ( x , λ ) ( Dom  L \ Ker  L ) Ω × [ 0 , T ] T Open image in new window;

     
  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 L Ω ̄ Open image in new window.

Definition 3.2. A mapping f : [ 0 , T ] T × × Open image in new window satisfies the Carathéodory conditions with respect to L Δ [ 0 , T ] T Open image in new window, where L Δ [ 0 , T ] T Open image in new window denotes that all Lebesgue Δ-integrable functions on [ 0 , T ] T Open image in new window, if the following conditions are satisfied:
  1. (i)

    for each (x1, x2) ∈ ℝ2, the mapping tf(t, x1, x2) is Lebesgue measurable on [ 0 , T ] T Open image in new window;

     
  2. (ii)

    for a.e. t [ 0 , T ] T Open image in new window, the mapping (x1, x2) → f (t, x1, x2) is continuous on2;

     
  3. (iii)

    for each r > 0, there exists α r L Δ ( [ 0 , T ] T , ) Open image in new window such that for a.e. t [ 0 , T ] T Open image in new window and every x1 such that |x1| ≤ r, |f (t, x1, x2)| ≤ α r .

     
Let the Banach space X = C Δ [ 0 , T ] T Open image in new window with the norm ||x|| = max{||x||, ||xΔ||}, where | | x | | = sup t [ 0 , T ] T | x ( t ) | Open image in new window. Let
L 1 oc Δ [ 0 , T ] T = { x : x | [ s , t ] T L Δ [ 0 , T ] T  for each  [ s , t ] T [ 0 , T ] T } , Open image in new window
set Y = L 1 oc Δ [ 0 , T ] T Open image in new window with the norm | | x | | L = 0 T | x ( t ) | Δ t Open image in new window. We use the space W 2 , 1 [ 0 , T ] T Open image in new window defined by
{ x : [ 0 , T ] T | x ( t ) , x Δ ( t ) is absolutely continuous on  [ 0 , T ] T with  x Δ Δ L 1 oc Δ [ 0 , T ] T } . Open image in new window
Define the linear operator L and the nonlinear operator N by
L : X Dom  L Y , L x ( t ) = x Δ Δ ( t ) , for x X Dom  L , N : X Y , N x ( t ) = f ( t , x ( t ) , x Δ ( t ) ) + e ( t ) , for  x X , Open image in new window
respectively, where
Dom  L = x W 2 , 1 [ 0 , T ] T , x Δ ( 0 ) = 0 , x ( T ) = 0 T x σ ( s ) Δ g ( s ) . Open image in new window
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
Q y = 1 Λ 0 T σ ( s ) T 0 t y ( τ ) Δ τ Δ t Δ g ( s ) , for  y Y , Open image in new window
where Λ = 0 T σ ( s ) T 0 t Δ τ Δ t Δ g ( s ) 0 Open image in new window. Linear mapping K P can be written by
K P y ( t ) = 0 t ( t - σ ( s ) ) y ( s ) Δ s , for  y Im  L . Open image in new window
Proof. It is clear that Ker L = { x ( t ) c , c } = Open image in new window, i.e., dimKer L = 1. Moreover, we have
Im  L = y Y , 0 T σ ( s ) T 0 t y ( τ ) Δ τ Δ t Δ g ( s ) = 0 . Open image in new window
(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
x Δ ( t ) = 0 t y ( τ ) Δ τ . Open image in new window
Integrating the above equation from s to T, we get
x ( s ) = x ( T ) - s T 0 t y ( τ ) Δ τ Δ t . Open image in new window
(3.2)
Substituting the boundary condition x ( T ) = 0 T x σ ( s ) Δ g ( s ) Open image in new window into (3.2), and by the condition 0 T Δ g ( s ) = 1 Open image in new window, we have
0 T σ ( s ) T 0 t y ( τ ) Δ τ Δ t Δ g ( s ) = 0 . Open image in new window
(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
x Δ ( 0 ) = 0 , x ( T ) = 0 T x σ ( s ) Δ g ( s ) . Open image in new window

Therefore, (3.1) holds.

Set Λ = 0 T σ ( s ) T 0 t Δ τ Δ t Δ g ( s ) Open image in new window. It is easy to show that Λ ≠ 0, and then we define the mapping Q : YY by
Q y = 1 Λ 0 T σ ( s ) T 0 t y ( τ ) Δ τ Δ t Δ g ( s ) , for y Y , Open image in new window

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
P x ( t ) = x ( 0 ) , for  x X . Open image in new window
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
K P y ( t ) = 0 t ( t - σ ( s ) ) y ( s ) Δ s . Open image in new window
For y ∈ Im L, we have
( L K P ) y ( t ) = 0 t ( t - σ ( s ) ) y ( s ) Δ s Δ Δ , Open image in new window
from Lemma 2.1, we obtain
0 t ( t - σ ( s ) ) y ( s ) Δ s Δ Δ = ( 0 t y ( s ) Δ s ) Δ = y ( t ) , Open image in new window
that is
( L K P ) y ( t ) = 0 t ( t - σ ( s ) ) y ( s ) Δ s Δ Δ = y ( t ) . Open image in new window
(3.4)
On the other hand, for x ∈ Dom L ⋂ Ker P,
( K P L ) x ( t ) = 0 t ( t - σ ( s ) ) x Δ Δ ( s ) Δ s , Open image in new window
using Lemma 2.4 and the boundary conditions, we get
0 t ( t - σ ( s ) ) x Δ Δ ( s ) Δ s = ( t - σ ( s ) ) x Δ Δ ( s ) 0 t + 0 t x Δ ( s ) Δ s = x ( t ) , Open image in new window
i.e.,
( K P L ) x ( t ) = 0 t ( t - σ ( s ) ) x Δ Δ ( s ) Δ s = x ( t ) , t [ 0 , T ] T . Open image in new window
(3.5)

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

Furthermore,
Q N x = 1 Λ 0 T σ ( s ) T 0 t ( N x ) ( τ ) Δ τ Δ t Δ g ( s ) , ( K P , Q N ) x ( t ) = 0 t ( t - σ ( s ) ) ( N x ) ( s ) Δ s - 0 t ( t - σ ( s ) ) 1 Λ 0 T σ ( s ) T 0 t ( N x ) ( τ ) Δ τ Δ t Δ g ( s ) Δ s . Open image in new window

Lemma 3.2. Let f : [ 0 , T ] T × × Open image in new window 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
| | N x n - N x 0 | | = sup t [ 0 , T ] T | N x n - N x 0 | = sup t [ 0 , T ] T | f ( t , x n ( t ) , x n Δ ( t ) ) - f ( t , x 0 ( t ) , x 0 Δ ( t ) ) | . Open image in new window

In view of f satisfying the Carathéodory conditions, we can obtain that for a.e. t [ 0 , T ] T Open image in new window, ||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. t [ 0 , T ] T Open image in new window, we have
| N x n | | f ( t , x n ( t ) , x n Δ ( t ) ) | + | e ( t ) | | ( α r ( t ) | + | e ( t ) | : = ψ ( t ) , | Q N x n | 1 | Λ | 0 T σ ( s ) T 0 t | ( N x n ) ( τ ) | Δ τ Δ t Δ g ( s ) 1 | Λ | 0 T σ ( s ) T 0 t | ψ ( τ ) | Δ τ Δ t Δ g ( s ) , | ( K P , Q N ) x n ( t ) | 0 t ( t σ ( s ) ) | ( N x n ) ( s ) | Δ s 0 t ( t σ ( s ) ) | Q N x n | Δ s . Open image in new window
Since functions α r ( t ) , e ( t ) L loc Δ [ 0 , T ] T Open image in new window, we get that ψ ( t ) L loc Δ [ 0 , T ] T Open image in new window. Further
| | N x n | | L 0 T | ψ ( t ) | Δ t : = χ < . Open image in new window

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

It is easy to see that { Q N x n } n = 1 Open image in new window is equicontinuous on a.e. t [ 0 , T ] T Open image in new window. So, we only show that { ( K P , Q N ) x n } n = 1 Open image in new window is equicontinuous on a.e. t [ 0 , T ] T Open image in new window. For any t 1 , t 2 [ 0 , T ] T Open image in new window with t1 < ρ(t2),
| ( K P , Q N ) x n ( t 1 ) - ( K P , Q N ) x n ( t 2 ) | t 1 t 2 | ( K P , Q N x n ) Δ ( s ) | Δ s t 1 t 2 0 s | ( N x n ) ( τ ) - ( Q N x n ( τ ) ) | Δ τ Δ s t 1 t 2 0 s | N x n ( τ ) | Δ τ Δ s + t 1 t 2 0 s | Q N x n ( τ ) | Δ τ Δ s . Open image in new window
(3.6)

Since | N x n | ψ Open image in new window with ψ L 1 oc Δ ( [ 0 , T ] T ) Open image in new window, (3.6) shows that { ( K P , Q N ) x n } n = 1 Open image in new window is equicontinuous on a.e. t [ 0 , T ] T Open image in new window. Hence, by the Arzelà-Ascoli theorem on time scales, { Q N x n } n = 1 Open image in new window and { K P , Q N x n } n = 1 Open image in new window 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 f : [ 0 , T ] T × 2 Open image in new window satisfy the Carathéodory conditions, and

(H1) There exist continuous functions r : [ 0 , T ] T + Open image in new window, g i : [ 0 , T ] T × + Open image in new window, i = 1, 2, such that
| f ( t , x 1 , x 2 ) | g 1 ( t , x 1 ) + g 2 ( t , x 2 ) + r ( t ) Open image in new window
and
lim | x | + sup t [ 0 , T ] T g i ( t , x ) | x | = r i [ 0 , + ) , i = 1 , 2 . Open image in new window

(H2) There is a constant M > 0 such that for any x ∈ Dom L\Ker L, if |x(t)| > M for all t [ 0 , T ] T Open image in new window; then, 1 Λ 0 T σ ( s ) T 0 t [ f ( τ , x ( τ ) , x Δ ( τ ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) 0 Open image in new window.

(H3) There is a constant M* > 0 such that for any c ∈ ℝ, if |c| > M*; then, we have either
c Λ 0 T σ ( s ) T 0 t [ f ( τ , c , 0 ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) > 0 Open image in new window
(3.7)
or
c Λ 0 T σ ( s ) T 0 t [ f ( τ , c , 0 ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) < 0 . Open image in new window
(3.8)
Then, problem (1.1) admits at least one solution provided that
T 2 r 1 + T r 2 < 1 . Open image in new window
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.,
Q N x = 1 Λ 0 T σ ( s ) T 0 t [ f ( τ , x ( τ ) , x Δ ( τ ) ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) = 0 . Open image in new window
Hence by (H2), we know that there exists t 0 [ 0 , T ] T Open image in new window such that |x(t0)| < M. Since
x ( t ) = x ( t 0 ) + t 0 t x Δ ( s ) Δ s . Open image in new window
So we get
| | x | | M + 0 T | x Δ ( s ) | Δ s M + T | | x Δ | | . Open image in new window
Integrating the equation
x Δ Δ ( t ) = λ [ f ( t , x ( t ) , x Δ ) ( t ) + e ( t ) ] , t [ 0 , T ] T Open image in new window
from 0 to t, we obtain
| x Δ ( t ) | = 0 t f ( s , x ( s ) , x Δ ( s ) ) + e ( s ) Δ s 0 T | f ( s , x ( s ) , x Δ ( s ) ) + e ( s ) | Δ s 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + 0 T | g 1 ( s , x ( s ) ) | Δ s + 0 T | g 2 ( s , x Δ ( s ) ) | Δ s . Open image in new window
Let ε > 0 satisfy
T [ T ( r 1 + ε ) + ( r 2 + ε ) ] < 1 . Open image in new window
For such ε, there is δ > 0 so that for i = 1, 2,
| g i ( t , x ) | < ( r i + ε ) | x | ,  uniformly for t [ 0 , T ] T  and  | x | > δ . Open image in new window
Let
Γ 1 , 0 = { t : t [ 0 , T ] T , | x ( t ) | δ } , Γ 1 , 1 = { t : t [ 0 , T ] T , | x Δ ( t ) | δ } , Γ 2 , 0 = { t : t [ 0 , T ] T , | x ( t ) | > δ } , Γ 2 , 1 = { t : t [ 0 , T ] T , | x Δ ( t ) | > δ } , i = max t [ 0 , T ] T , | x | < δ | g i ( t , x ) | , i = 1 , 2 . Open image in new window
We get
| x Δ ( t ) | 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + 0 T | g 1 ( s , x ( s ) ) | Δ s + 0 T | g 2 ( s , x Δ ( s ) ) | Δ s 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + Γ 1 , 0 | g 1 ( s , x ( s ) ) | Δ s + Γ 2 , 0 | g 1 ( s , x ( s ) ) | Δ s + Γ 1 , 1 | g 2 ( s , x Δ ( s ) ) | Δ s + Γ 2 , 1 | g 2 ( s , x Δ ( s ) ) | Δ s 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + T [ ( r 1 + ε ) | | x | | + 1 + ( r 2 + ε ) | | x Δ | | + 2 ] 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + T [ ( r 1 + ε ) M + 1 + 2 ] + T [ T ( r 1 + ε ) + ( r 2 + ε ) ] | | x Δ | | . Open image in new window
So we get
{ 1 - T [ T ( r 1 + ε ) + ( r 2 + ε ) ] } | | x Δ | | 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + T [ ( r 1 + ε ) M + 1 + 2 ] . Open image in new window
It follows from the definition of ε that there is a constant A > 0 such that
| | x Δ | | A : = 0 T | e ( s ) | Δ s + 0 T | r ( s ) | Δ s + T [ ( r 1 + ε ) M + 1 + 2 ] 1 - T [ T ( r 1 + ε ) + ( r 2 + ε ) ] . Open image in new window
Hence, we have
| | x | | = max { | | x | | , | | x Δ | | } max { M + T A , A } , Open image in new window

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
1 Λ 0 T σ ( s ) T 0 t [ f ( τ , c , 0 ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) = 0 . Open image in new window

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

If (3.7) holds, then let
Ω 3 = { x Ker  L : - λ J x + ( 1 - λ ) Q N x = 0 , λ [ 0 , 1 ] } , Open image in new window
where J : Ker L → Im Q is a linear isomorphism given by J(k) = k for any k ∈ ℝ. Since x(t) = k thus
λ k = ( 1 - λ ) Q N k = 1 - λ Λ 0 T σ ( s ) T 0 t [ f ( τ , k , 0 ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) . Open image in new window
If λ = 1, then k = 0, and in the case λ ∈ [0, 1), if |k| > M*, we have
λ k 2 = k ( 1 - λ ) Λ 0 T σ ( s ) T 0 t [ f ( τ , k , 0 ) + e ( τ ) ] Δ τ Δ t Δ g ( s ) < 0 , Open image in new window
which is a contradiction. Again, if (3.8) holds, then let
Ω 3 = { x Ker  L : - λ J x + ( 1 - λ ) Q N x = 0 , λ [ 0 , 1 ] } , Open image in new window

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 i = 1 3 Ω i Ω Open image in new window. By Lemma 3.2, we can check that K P ( I - Q ) N : Ω ̄ X Open image in new windowis compact; thus, N is L-compact on Ω ¯ Open image in new window.

Finally, we verify that the condition (iii) of Theorem 3.1 is fulfilled. Define a homotopy
H ( x , λ ) = ± λ J x + ( 1 - λ ) Q N x . Open image in new window
According to the above argument, we have
H ( x , λ ) 0 , for x Ω Ker L ; Open image in new window
thus, by the degree property of homotopy invariance, we obtain
deg ( Q N Ker L , Ω Ker  L , 0 ) = deg ( H ( , 0 ) , Ω Ker  L , 0 ) = deg ( H ( , 1 ) , Ω Ker  L , 0 ) = deg ( ± J , Ω Ker  L , 0 ) 0 . Open image in new window

Thus, the conditions of Theorem 2.4 are satisfied, that is, the operator equation Lx = Nx admits at least one solution in Dom L Ω ¯ Open image in new window. Therefore, BVPs (1.1) has at least one solution in C Δ [ 0 , T ] T Open image in new window.

4 An example

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

Example 4.1. Let T = { 0 } { 1 2 n + 1 } [ 1 2 , 1 ] Open image in new window, n = 1, 2, ..., ∞. Consider the boundary value

Problem
x Δ Δ ( t ) = 1 2 t x 2 ( t ) + 1 3 t 2 x Δ ( t ) + t , a . e . t T , x Δ ( 0 ) = 0 , x ( 1 ) = x ( 1 2 ) . Open image in new window
(4.1)
Let
g ( t ) = 0 , for 0 t 1 2 , 1 , for 1 2 t 1 , Open image in new window
then x ( 1 ) = 0 1 x σ ( s ) Δ g ( s ) Open image in new window. Let
g 1 ( t , x ) = 1 2 x 2 ( t ) , g 2 ( t , x Δ ) = 1 3 ( x Δ ( t ) ) 2 , r ( t ) = t 2 . Open image in new window

We can get that r 1 + r 2 = 5 6 < 1 Open image in new window. 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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© Li and Shu; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsYunnan University KunmingYunnanPeople's Republic of China

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