Existence of weak solutions for a boundary value problem of a second order ordinary differential equation
 Shiyou Weng^{1} and
 Fuli Wang^{2}Email authorView ORCID ID profile
Received: 25 July 2017
Accepted: 11 January 2018
Published: 16 January 2018
Abstract
The aim of this paper is to investigate the existence of weak solutions for a boundary value problem of a second order differential equation. As the main tool, we apply a Krasnosel’skii type fixed point theorem in conjunction with the technique of measures of weak noncompactness in Banach spaces. Finally, two examples are given to illustrate our abstract results.
Keywords
MSC
1 Introduction
2 Preliminaries
Let E be a Banach space. From now on we denote by \(\mathcal{B}(E)\) the collection of all nonempty bounded subsets of E, and \(\mathcal {W}(E)\) is a subcollection of \(\mathcal{B}(E)\) consisting of all weakly compact subsets of E. Denote by \(\mathrm{B}_{r}\) the closed ball in E centered at zero with radius r. In what follows we accept the following definition (cf. [11]).
Definition 2.1
 (1)
The family \(\operatorname{ker}(\omega):=\{M\in\mathcal {B}(E):\omega(M)=0\}\) is nonempty, and \(M\in\operatorname{ker}(\omega)\) if and only if M is relatively weakly compact;
 (2)
\(N\subseteq M \Rightarrow \omega(N)\leq\omega(M)\);
 (3)
\(\omega(\overline{M}^{\omega})=\omega(M)\), where \(\overline{M}^{\omega}\) is the weak closure of M;
 (4)
\(\omega(M\cup N) = \max\{\omega(M),\omega(N)\}\);
 (5)
\(\omega(\lambda M) = \vert \lambda \vert \omega(M)\) for all \(\lambda\in\mathbb{R}\);
 (6)
\(\omega(\operatorname{co}(M)) = \omega(M)\), where \(\operatorname{co}(M)\) is the convex hull of M;
 (7)
\(\omega(M + N) \leqslant\omega(M) + \omega(N)\);
 (8)
If \((M_{n})_{n=1}^{\infty}\) is a decreasing sequence of nonempty, bounded and weakly closed subsets of E with \(\lim_{n\rightarrow\infty}\omega(M_{n})=0\), then \(M_{\infty}:=\bigcap_{n=1}^{\infty}M_{n}\) is nonempty.
The family \(\ker(\omega)\) described in (1) is called the kernel of the measure of weak noncompactness ω. Note that the intersection set \(M_{\infty}\) from (8) belongs to \(\ker(\omega)\) since \(\omega(M_{\infty})\leq\omega(M_{n})\) for all \(n\in\mathbb{N}\) and \(\lim_{n\rightarrow\infty}\omega(M_{n})=0\).
Recall that a useful characterization of relatively weakly compact sets in \(L^{1}(I)\) is provided by the following DunfordPettis theorem (cf. [14, p. 115]).
Theorem 2.2
Definition 2.3
 (1)
wscompact if \((x_{n})_{n\in\mathbb{N}} \subseteq\mathcal {D}\) is a weakly convergent sequence in \(E_{1}\), the sequence \((Tx_{n})_{n\in\mathbb{N}}\) has a strongly convergent subsequence;
 (2)
wwcompact if \((x_{n})_{n\in\mathbb{N}} \subseteq\mathcal {D}\) is a weakly convergent sequence in \(E_{1}\), the sequence \((Tx_{n})_{n\in\mathbb{N}}\) has a weakly convergent subsequence.
Remark 2.4
A continuous operator is wscompact if and only if it maps relatively weakly compact sets into relatively strongly compact ones; and it is wwcompact if and only if it maps relatively weakly compact sets into relatively weakly compact ones, since the weak compactness of a set in Banach spaces is equivalent to its weakly sequential compactness by the EberleinS̆mulian theorem (cf. [17, p. 430]).
A function \(f:I\times\mathbb{R}\rightarrow\mathbb{R}\) is said to satisfy the Carathéodory conditions if it is measurable in t for each x in \(\mathbb{R}\) and continuous in x for almost every (or a.e. for short) \(t\in I\).
Let \(\mathbf{m}(I)\) denote the collection of all measurable functions \(x:I\rightarrow\mathbb{R}\). If a function \(f:I\times\mathbb {R}\rightarrow\mathbb{R}\) satisfies the Carathéodory condition, then f defines a mapping \(\mathcal{N}_{f}: \mathbf{m}(I) \rightarrow \mathbf{m}(I)\) by \(\mathcal{N}_{f}x(t):=f(t,x(t))\). This mapping is called the superposition operator (Nemytskii operator) associated with f. Regarding its continuity, we have the following theorem (cf. [18, p. 93]).
Theorem 2.5
Definition 2.6
 (1)
contractive with ℓ if there exists \(\ell\in[0,1)\) such that \(\Vert Tx_{1}Tx_{2} \Vert \leqslant\ell \Vert x_{1}x_{2} \Vert \) for all \(x_{1},x_{2}\in\mathcal{D}\);
 (2)
ωcontractive with ℓ if it maps bounded sets into bounded sets, and there exists \(\ell\in[0,1)\) such that \(\omega (T(M))\leqslant\ell \omega(M)\) for all bounded sets M in \(\mathcal{D}\).
We end these preliminaries with the following Krasnosel’skii type fixed point result (cf. [19, Corollary 3.4]). It plays an important role in the proof of our main result.
Theorem 2.7
 (i)
A is ωcontractive with α, and A is wscompact;
 (ii)
B is contractive with β, and B is wwcompact;
 (iii)
the equality \(y=By+Ax\) with \(x\in M\) implies \(y\in M\).
3 Main results
Throughout this paper, \(L^{1}(I)\) will denote the Banach space consisting of all real functions defined and Lebesgue integrable on \(I:=[0,1]\), with the standard norm \(\Vert \cdot \Vert \); and \(L^{\infty}(I)\) will denote the Banach space consisting of all real functions defined and essentially bounded on I, with the standard norm \(\Vert \cdot \Vert _{\infty}\).
Lemma 3.1
Proof
Remark 3.2
Definition 3.3
A function \(x\in L^{1}(I)\) is said to be a weak solution of problem (1.1)(1.2) if x satisfies Eq. (3.1) on the interval I.
 (\(\mathcal{H}1\)):

The functions \(f:I\times\mathbb {R}\rightarrow\mathbb{R}\) and \(h:I\times\mathbb{R}\rightarrow \mathbb{R}\backslash\{0\}\) satisfy the Carathéodory conditions. Moreover, there exist functions \(f_{0}, h_{0} \in L^{1}_{+}(I)\) and positive numbers η and γ, respectively, such that$$ \bigl\vert f(t,x) \bigr\vert \leqslant f_{0}(t)+ \eta \vert x \vert ,\qquad \bigl\vert h(t,x) \bigr\vert \leqslant h_{0}(t)+ \gamma \vert x \vert , \quad \forall x\in\mathbb{R} \text{ and a.e. } t\in I. $$
 (\(\mathcal{H}2\)):

The function \(g:I\times\mathbb {R}\rightarrow\mathbb{R}\) satisfies the Carathéodory conditions, and there exists a positive number β such thatMoreover, \(g_{0}(t):= \vert g(t,0) \vert \) belongs to \(L^{1}_{+}(I)\).$$ \bigl\vert g(t,x)g(t,y) \bigr\vert \leqslant\beta \vert xy \vert , \quad \forall x,y\in\mathbb{R} \text{ and a.e. } t\in I. $$
 (\(\mathcal{H}3\)):

The following inequality holds:$$ 4\beta+ \gamma \Vert f_{0} \Vert + \eta \Vert h_{0} \Vert + 2\sqrt{\gamma\eta\bigl( 4 \Vert g_{0} \Vert + \Vert f_{0} \Vert \Vert h_{0} \Vert \bigr)}< 4. $$(3.6)
Remark 3.4
(1) Note that from (\(\mathcal{H}2\)) we deduce that \(\vert g(t,x) \vert \leqslant g_{0}(t)+\beta \vert x \vert \) for all \(x\in\mathbb{R}\) and a.e. \(t\in I\). Thus, by Theorem 2.5, (\(\mathcal{H}1\)) and (\(\mathcal{H}2\)) imply that the superposition operators \(\mathcal{N}_{f}\), \(\mathcal{N}_{g}\) and \(\mathcal {N}_{h}\), respectively, map \(L^{1}(I)\) into itself continuously. Further, according to [7, Lemma 3.2], \(\mathcal{N}_{f}\), \(\mathcal {N}_{g}\) and \(\mathcal{N}_{h}\) are wwcompact.
Theorem 3.5
Under assumptions (\(\mathcal{H}1\))(\(\mathcal{H}3\)), problem (1.1)(1.2) has at least one weak solution \(x\in M\), where \(M:=\{x: \Vert x \Vert \leqslant\mathbf{r}_{0}\}\) is a closed ball of \(L^{1}(I)\), and \(\mathbf{r}_{0}\) is a solution of (3.7) and satisfies (3.8) .
Proof
Let \(\mathcal{B}:=\mathcal{N}_{g}\). Define \(\mathcal{A}\) by \(\mathcal {A} x(t): =\mathcal{N}_{h} x(t)\cdot\mathbb{G}\mathcal{N}_{f} x(t)\) for \(x\in M\). For proving the operator equation \(x=\mathcal{A}x+\mathcal {B}x\) has a unique solution in \(L^{1}(I)\), our processes are divided into several steps.
(1). \(\mathcal{A}\) is wscompact.
According to Remark 3.4(1), \((\mathcal{N}_{f} x_{n})_{n\in\mathbb {N}}\) has a weakly convergent subsequence, say \((\mathcal{N}_{f} x_{n_{k}})_{k\in\mathbb{N}}\). Since the continuity of the linear operator \(\mathbb{G}\) implies its weak continuity on \(L^{1}(I)\) for a.e. \(t\in I\), then \((\mathbb{G}\mathcal{N}_{f} x_{n_{k}})_{k\in\mathbb{N}}\) converges pointwise for a.e. \(t\in I\). Now, applying Egoroff’s theorem, there exists a measurable subset \(I_{0}\subseteq I\) with \(\operatorname{meas}(I\backslash I_{0})\leqslant\delta\) such that \((\mathbb {G}\mathcal{N}_{f} x_{n_{k}})_{k\in\mathbb{N}}\) is uniformly convergent on \(I_{0}\).
Thus, we complete the proof that \(\mathcal{A}\) is wscompact.
(2). \(\mathcal{A}\) is ωcontractive.
(3). If \(y=\mathcal{B}y + \mathcal{A}x\) for \(x\in M\), then \(y\in M\).
(4). Conclusion.
The condition (i) of Theorem 2.7 is verified in (1)(2), and the condition (iii) of Theorem 2.7 is verified in (3). Moreover, \(\mathcal{B}=\mathcal{N}_{g}\) is contractive with β by (\(\mathcal{H}2\)), and \(\mathcal{B}\) is wwcompact by Remark 3.4(1). Then the condition (ii) of Theorem 2.7 is satisfied. The estimate \(\alpha+\beta<1\) is from (3.13).
Now, according to Theorem 2.7, we obtain that Eq. (3.1) has at least one solution in M, and then the existence of weak solutions in \(L^{1}(I)\) for problem (1.1)(1.2) is proved. □
4 Examples
In this section we give two examples to illustrate the existence result involved in Theorem 3.5.
Example 4.1
Example 4.2
Finally, based on Theorem 3.5, we infer that there exists \(x\in L^{1}(I)\) such that it is a weak solution of problem (4.3)(4.4). Moreover, it is easy to see that the exact solution of (4.3)(4.4) is \(x(t)=tt^{2}\).
5 Conclusions
In this work, we have established an existence result for weak solutions of the boundary value problem of nonlinear differential equations of second order. Our main assumptions about the functions being involved in the equation are the Carathéodory conditions, and the main tool is a Krasnosel’skii type fixed point theorem in conjunction with the technique of measures of weak noncompactness.
In the proof of Theorem 3.5, we avoided using the ScorzaDragoni theorem [20], the ArzelàAscoli theorem, the modulus of continuity of the function G, etc. By using Egoroff’s theorem, we have replaced this method so that the proof process is simplified. The reader can compare it with [5, 8, 10, 21, 22].
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
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Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Ethics approval and consent to participate
Both authors contributed to each part of this study equally and declare that they have no competing interests.
Competing interests
The authors declare that they have no competing interests.
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Both authors read and approved the final version of the manuscript.
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