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On second order nonlinear boundary value problems and the distributional HenstockKurzweil integral
Boundary Value Problems volumeÂ 2015, ArticleÂ number:Â 73 (2015)
Abstract
In the present paper, we investigate the existence of solutions to second order nonlinear boundary value problems (BVPs) involving the distributional HenstockKurzweil integral. The present results in this article are generalizations of previous results in the literature.
1 Introduction
New existence results are derived for solutions of the second order differential equation
subject to the boundary conditions
where \(x''\), \(x'\) stand for the distributional derivative of the function \(x \in C[0,1]\), \(C[0, 1]\) denotes the space where the functions \(x: [0, 1] \rightarrow\mathbb{R}\) are continuous, f is a distribution (generalized function), Î² is a positive constant and \(\eta\in[0,1]\). The space \(C[0, 1]\) is considered with the uniform norm \(\ \cdot \_{\infty}\).
In recent years, the existence of solutions of boundary value problems have been studied by many authors [1â€“4]. Chew and Flordeliza, in [5], generalized the classical CarathÃ©odoryâ€™s existence theorem on the Cauchy problem \(x^{\prime}=f(t, x)\) with \(x(0)=0\). Particularly, in [6, 7], the differential equations involving the approximate derivatives are considered. BVP (1.1)(1.2), which has been studied in [4] by using ordinary derivatives, came from the steadystate of a heat bar model. The boundary conditions model the behavior of a thermostat where the sensor measures the temperature. The heat bar is insulated at \(t=0\), and the controller releases heat at \(t=1\) depending on the temperature at \(t=\eta\). However, it is well known that the notion of a distributional derivative is very general, including ordinary derivatives and approximate derivatives. Without loss of generality, we use the distributional derivatives to discuss (1.1)(1.2) in a general form. The existence result obtained under weaker conditions extends the previous results in the literatures.
This paper is organized as follows. In SectionÂ 2, we introduce fundamental concepts and basic results of the distributional HenstockKurzweil integral. In SectionÂ 3, we apply Schauderâ€™s fixed point theorem to verify the existence of BVP (1.1) and (1.2). In SectionÂ 4, we give an example to illustrate TheoremÂ 3.1 in this paper.
2 Distributional HenstockKurzweil integral
In this section, the definition of distributional HenstockKurzweil integral and its main properties needed in this paper are presented.
Define the space
where the support of a function Ï• is the closure of the set on which Ï• does not equal zero. A sequence \(\{\phi_{n}\}\subset C_{c}^{\infty}\) converges to \(\phi\in C_{c}^{\infty}\) if there is a compact set K such that all \(\phi_{n}\) have support in K and the sequence of derivatives \(\phi_{n}^{(m)}\) converges to \(\phi^{(m)}\) uniformly for every \(m\in\mathbb{N}\cup\{0\}\). Denote \(C_{c}^{\infty}\) endowed with this convergence property by \(\mathcal{D}\). Also, if \(\phi\in\mathcal{D}\), we call Ï• a test function. The dual space to \(\mathcal{D}\) is denoted by \(\mathcal{D}'\) and if \(f\in\mathcal{D}'\) then \(f:\mathcal{D}\rightarrow \mathbb{R}\), \(\langle f,\phi\rangle\in \mathbb{R}\) for \(\phi\in\mathcal{D}\).
For all \(f\in\mathcal{D}'\), we define the distributional derivative \(f'\) of f to be a distribution satisfying \(\langle f',\phi\rangle=\langle f,\phi'\rangle\), where Ï• is a test function and \(\phi'\) is the ordinary derivative of Ï•. With this definition, it is easy to get that all distributions have derivatives of all orders and each derivative is a distribution.
Let \((a,b)\) be an open interval in \(\mathbb{R}\). We define
\({\mathcal{D}}'((a, b))\) denotes the dual space of \({\mathcal{D}}((a, b))\).
Let \(C[a,b]\) be the space of continuous functions on \([a,b]\) and \(B_{C} = \{F \in C[a, b] \mid F(a) = 0\}\). Note that \(B_{C}\) is a Banach space with the uniform norm
We give an introduction about the definition of the \(D_{HK}\)integral.
Definition 2.1
A distribution f is distributionally HenstockKurzweil integrable or briefly \(D_{HK}\)integrable on \([a,b]\) if f is the distributional derivative of a continuous function \(F \in B_{C}\).
The \(D_{HK}\)integral of f on \([a,b]\) is defined by \((D_{HK})\int _{a}^{b} f = F(b)\), where \(F\in B_{C}\) is the primitive of f and â€˜\((D_{HK})\int \)â€™ denotes the \(D_{HK}\)integral. For succinctness, we refer to â€˜\((D_{HK})\int \)â€™ as simply â€˜âˆ«â€™. Moreover, the space of \(D_{HK}\)integrable distributions is defined by
With this definition, if \(f\in D_{HK}\) then, for all \(\phi\in \mathcal{D}((a, b))\),
Now we give an example showing that the \(D_{HK}\)integral includes the HKintegral, and hence includes Lebesgue and Riemann integrals (see [8â€“10] for details).
Example 2.1
In [8], Lee points out that if F is a continuous function and pointwise differentiable nearly everywhere on \([a, b]\), then F is \(ACG^{\ast}\) (generalized absolutely continuous), and if F is a continuous function which is differentiable nowhere on \([a, b]\), then F is not \(ACG^{\ast}\). A primitive F of the HKintegrable function f is \(ACG^{\ast}\) (see [8, 10] for details). Therefore, if \(F\in C[a,b]\) but is differentiable nowhere on \([a, b]\), the distributional derivative of F exists and is \(D_{HK}\)integrable but not HKintegrable. On another aspect, if F is \(ACG^{\ast}\) then it belongs to \(C[a,b]\). Hence \(F'\) is not only HKintegrable but also \(D_{HK}\)integrable.
Let us introduce some basic results of the distributional HenstockKurzweil integral needed later.
Lemma 2.1
([11], TheoremÂ 4, fundamental theorem of calculus)

(a)
Let \(f\in D_{HK}\), and define \(F(t)=\int_{a}^{t} f\). Then \(F\in B_{C}\) and \(F'=f\).

(b)
Let \(F\in C[a, b]\). Then \(\int_{a}^{t} F'=F(t)F(a)\) for all \(t\in[a, b]\).
For \(f \in D_{HK}\) and \(F\in B_{C}\) with \(F'=f\), we define the Alexiewicz norm by
We say a sequence \(\{f_{n}\}\subset D_{HK}\) converges strongly to \(f\in D_{HK}\) if \(\f_{n}f\\to0\) as \(n\to\infty\). Then the following result holds.
Lemma 2.2
([11], TheoremÂ 2)
With the Alexiewicz norm, \(D_{HK}\) is a Banach space.
Now we impose a partial ordering on \(D_{HK}\): for \(f, g \in D_{HK}\), we say that \(f\succeq g \) (or \(g\preceq f\)) if and only if \(fg\) is a positive measure on \([a,b]\). By the definition, if \(f, g \in D_{HK}\), then \(\int_{I} f\geq\int_{I} g \) for every \(I=[c,d] \subset [a,b]\), whenever \(f\succeq g\). See [12] for details.
It is shown that the following results hold.
Lemma 2.3
([11], DefinitionÂ 6, integration by parts)
Let \(f\in D_{HK}\), and g is a function of bounded variation. Define \(fg=DH\), where \(H(t)=F(t)g(t)\int_{a}^{t} F\,dg\). Then \(fg\in D_{HK}\) and
Lemma 2.4
([12], CorollaryÂ 5, dominated convergence theorem for the \(D_{HK}\)integral)
Let \(\{{f_{n}}\}_{n=0}^{\infty}\) be a sequence in \(D_{HK}\) such that \(f_{n}\rightarrow f\) as \(n\to\infty\) in \(\mathcal{D}'\). Suppose that there exist \(f_{},f_{+}\in D_{HK}\) satisfying \(f_{}\preceq f_{n}\preceq f_{+}\) for \(\forall n\in\mathbb{N}\). Then \(f\in D_{HK}\) and \(\lim_{n\rightarrow\infty}\int_{a}^{b} f_{n} = \int_{a}^{b} f\).
The next statement is modified from [13] and [11].
Lemma 2.5
Let \(f\in D_{HK}\) and \(\{{f_{n}}\}_{n=0}^{\infty}\) be a sequence in \(D_{HK}\) such that \(f_{n}\rightarrow f\) as \(n\to\infty\) inÂ \(\mathcal{D}'\). Define \(F_{n}(x)=\int_{a}^{x} f_{n}\) and \(F(x)=\int_{a}^{x} f\). If g is a function of bounded variation and \(F_{n} \rightarrow F\) as \(n\to\infty\) uniformly on \([a,b]\), then \(\int_{a}^{b} f_{n}g\rightarrow\int_{a}^{b} fg\) as \(n\to \infty\).
3 Main results
In this section, we firstly assume that f satisfies the following assumptions:
 (C_{1}):

\(f(t,x)\) is \(D_{HK}\)integrable with respect to t for all \(x\in C[0,1]\);
 (C_{2}):

\(f(t,x)\) is continuous with respect to x for all \(t\in[0,1]\), i.e., for each \(t\in[0,1]\), \(\f(t,x_{n})f(t,x)\ \to0\) as \(\x_{n}x\_{\infty}\to0\) for \(\{x_{n}\}\subset C[0,1]\);
 (C_{3}):

There exist \(f_{}, f_{+}\in D_{HK}\) such that \(f_{}(\cdot)\preceq f(\cdot,x)\preceq f_{+}(\cdot)\) for all \(x\in C[0,1]\).
Lemma 3.1
BVP (1.1)(1.2) is equivalent to the integral equation
where Î· is a constant with \(0\leq\eta\leq1\).
Proof
In view of Eq. (1.1), condition (C_{1}) and LemmaÂ 2.1, we have \(x'\in C[0,1]\), and for all \(t\in [0,1]\), \(s\in[0,1]\),
Then
According to the boundary conditions, one has
Then
It is easy to calculate that BVP (1.1)(1.2) holds by taking the derivative of both sides of (3.2). This completes the proof.â€ƒâ–¡
Lemma 3.2
([14], TheoremÂ 6.15)
Let M be a convex, closed subset of a normed space X. Let T be a continuous map of M into a compact subset K of M. Then T has a fixed point.
With the help of the preceding two lemmas, we can now prove the existence of solutions of BVP (1.1)(1.2).
Theorem 3.1
Under assumptions (C_{1})(C_{3}), there exists at least one solution of BVP (1.1)(1.2).
Proof
Suppose that
Then, for each \(t\in[0, 1]\), we have
Let \(B = \{{x \in C[0,1]:\x\_{\infty}\leq l}, l=(\beta+6)M >0\}\). For each \(x\in B\), \(t\in[0,1]\), define
Now we prove this theorem in three steps.
Step 1: \(\mathcal{A}:B\rightarrow B\).
For all \(x\in B\), by (3.4), one has
Furthermore, let \(F(t)=\int_{0}^{t}f(s,x(s))\,ds\) for \(t\in[0,1]\), and
Then, for all \(t\in[0,1]\), one has
In particular, for \(t=\eta\), we have
In view of (3.5)(3.7), one has
By (3.3), we also obtain \(\F\_{\infty}\leq M\), then \(\\mathcal{A}x\_{\infty}\leq(\beta+6)M=l\). Hence, \(\mathcal{A}(B)\subseteq B\).
Step 2: \(\mathcal{A}(B)\) is equicontinuous.
Let \(t_{1}, t_{2}\in[0,1]\), \(x\in B\)
For every \(t\in[0,1]\), we let \(F_{+}(t)=\int_{0}^{t}f_{+}(s)\,ds\), \(F_{}(t)=\int_{0}^{t}f_{}(s)\,ds\). By (C_{3}), we obtain
i.e.,
Moreover,
So,
Since \(f_{}(s), f_{+}(s), F_{}(s), F_{+}(s)\in D_{HK}\), then their primitives are continuous on \([0,1]\) and hence uniformly continuous on \([0,1]\). Then by (3.8), \(\mathcal{A}(B)\) is equiuniformly continuous on \([0,1]\) for all \(x\in B\).
In view of Step 1, Step 2 and the AscoliArzelÃ theorem, \(\mathcal{A}(B)\) is relatively compact.
Step 3: \(\mathcal{A}\) is a continuous mapping.
Let \(x\in B\), \(\{x_{n}\}_{n\in\mathbb{N}}\) be a sequence in B and \(x_{n} \rightarrow x\) as \(n\rightarrow\infty\).
By (C_{2}), one has
According to assumption (C_{3}) and LemmaÂ 2.4, we have
It is easy to verify, by LemmaÂ 2.5, that
Hence, \(\mathcal{A}\) is continuous.
Thus, \(\mathcal{A}\) satisfies the hypotheses of LemmaÂ 3.2, then there exists a fixed point of \(\mathcal{A}\) which is a solution of (3.1). By LemmaÂ 3.1, BVP (1.1)(1.2) has at least one solution.â€ƒâ–¡
4 Example
In this section, we give an example for the application of TheoremÂ 3.1.
Example 4.1
Consider the initial value problem
where r is the distributional derivative of the Riemann function \(R(t)=\sum_{n=1}^{\infty}\frac{\sin n^{2}\pi t}{n^{2}}\) inÂ [15].
It is easy to see that \(R(t)\in C[0,1]\) and \(R(0)=0\), hence r is \(D_{HK}\)integrable. Let \(f(t,x)=t^{2}\sin x+r\), then \(f(t,x)\) satisfies (C_{1}), (C_{2}). Moreover, let \(f_{}(t)=t^{2}+r\) and \(f_{+}(t)=t^{2}+r\), then
i.e., (C_{3}) holds. Therefore, the initial value problem (4.1) has a solution.
Remark 4.1
It is well known that the function \(R(t)\) given by Riemann is continuous but pointwise differentiable nowhere on \([0,1]\), then the distributional derivative r in (4.1) is neither HK nor Lebesgue integrable. Hence, this example is not covered by any result using HK or Lebesgue integral. Thus, TheoremÂ 3.1 is more extensive.
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Liang, B., Ye, G., Liu, W. et al. On second order nonlinear boundary value problems and the distributional HenstockKurzweil integral. Bound Value Probl 2015, 73 (2015). https://doi.org/10.1186/s1366101503380
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DOI: https://doi.org/10.1186/s1366101503380