Existence theorems for a nonlinear second-order distributional differential equation

1 Introduction

The first-order distributional differential equation (DDE) in the form

The DDE (1.1) provides a good model for many physical processes, biological neural nets, pulse frequency modulation systems and automatic control problems (Das and Sharma, 1971, 1972; Leela, 1974). Particularly, when u is an absolute continuous function, then (1.1) reduces to an ODE. However, in physical systems, one cannot always expect the perturbations to be well-behaved. For example, if u is a function of boundary variation, Du can be identified with a Stieltjes measure and will have the effect of suddenly changing the state of the system at the points of discontinuity of u, that is, the system could be controlled by some impulsive force. In this case, (1.1) is also called a measure differential equation (MDE), see Das and Sharma (1971, 1972), Dhage and Bellale (2009), Federson and Mesquita (2013), Federson et al. (2012), Leela (1974), Antunes Monteiro and Slavík (2016), Satco (2014), Slavík (2013), Slavík (2015). Results concerning existence, uniqueness, and stability of solutions, were obtained in those papers. However, this situation is not the worst, because it is well-known that the solutions of a MDE, if exist, are still functions of bounded variation. The case when u is a continuous function has also been considered in Liu et al. (2012) and Zhou et al. (2015). The integral there is understood as a Kurzweil–Henstock integral (Krejčí, 2006; Kurzweil, 1957; Lee, 1989; Pelant and Tvrdý, 1993; Schwabik and Ye, 2005; Talvila, 2008; Tvrdý, 1994; Tvrdý, 2002; Ye and Liu, 2016) (or Kurzweil–Henstock–Stieltjes integral, or distributional Kurzweil–Henstock integral), which is a generalization of the Lebesgue integral. Especially, if u denotes a Wiener process (or Brownian motion), then (1.1) becomes a stochastic differential equation (SDE), see, for example, Boon and Lam (2011) and Mao (2008). In this case, u is continuous but pointwise differentiable nowhere, and the Itô integral plays an important role there. As for the relationship between the Kurzweil–Henstock integral and the Itô integral, we refer the interested readers to Boon and Lam (2011), Chew et al. (2001) and Toh and Chew (2012) and references therein.

It is well-known that regulated functions (that is, a function whose one-side limits exist at every point of its domain) contain continuous functions and functions of bounded variation as special cases (Fraňková, 1991). Therefore, it is natural to consider the situation when u is a regulated function, see Pelant and Tvrdý (1993) and Tvrdý (1994). Denote by $G[0\text{,}1]$ the space of all real regulated functions on $[0\text{,}1]$ , endowed with the supremum norm $\Vert ·\Vert$ . Since the DDE allows both the inputs and outputs of the systems to be discontinuous, most conventional methods for ODEs are inapplicable, and thus the study of DDEs becomes very interesting and important.

The purpose of our paper is to apply the Leray–Schauder nonlinear alternative and Kurzweil–Henstock–Stieltjes integrals to establish existence of a solution to the second order DDE of type

• $(H_1)$

  $f(t\text{,}x)$ is Kurzweil–Henstock integrable with respect to t for all $x\in G[0\text{,}1]$ ;

• $(H_2)$

  $f(t\text{,}x)$ is continuous with respect to x for all $t\in [0\text{,}1]$ ;

• $(H_3)$

  there exist nonnegative Kurzweil–Henstock integrable functions k and h such that $$-k\Vert x\Vert -h⩽f(·\text{,}x)⩽k\Vert x\Vert +h\forall x\in B_r\text{,}$$ where $B_r=\{x\in G[0\text{,}1]:\Vert x\Vert ⩽r\}\text{,}r>0$ ;

• $(H_4)$

  $g(t\text{,}x)$ is a function with bounded variation on $[0\text{,}1]$ and $g(0\text{,}x)=0$ for all $x\in G[0\text{,}1]$ ;

• $(H_5)$

  $g(t\text{,}x)$ is continuous with respect to x for all $t\in [0\text{,}1]$ ;

• $(H_6)$

  there exists $M>0$ such that $${\displaystyle \underset{x\in B_r}{\sup }}{\displaystyle \underset{[0\text{,}1]}{\mathrm{var}}}g⩽M\text{,}$$ where $${\displaystyle \underset{[0\text{,}1]}{\mathrm{var}}}g=\sup {\displaystyle \sum _n}|g(s_n\text{,}x(s_n))-g(t_n\text{,}x(t_n))|\text{,}$$ the supremum taken over every sequence $\{(t_n\text{,}{s}_n)\}$ of disjoint intervals in $[0\text{,}1]$ , is called the total variation of g on $[0\text{,}1]$ .

Now, we state our main result.

If $k(t)\equiv 0$ on $[0\text{,}1]$ , then $(H_3)$ can be reduced to.

• $(H_3^{\prime })$ there exists a nonnegative Kurzweil–Henstock function h such that $$-h⩽f(·\text{,}x)⩽h\forall x\in B_r\text{.}$$

Thus, the following result holds as a direct consequence.

It is worth to mention that condition $(H_3^{\prime })$ , together with $(H_1)$ and $(H_2)$ , was firstly proposed by Chew and Flordeliza (1991), to deal with first-order Cauchy problems.

The paper is organized as follows. In Section 2, we give two useful lemmas: we prove that under our hypotheses problem (1.2) and (1.3) can be rewritten in an equivalent integral form (Lemma 2.1) and we recall the Leray–Schauder theorem (Lemma 2.2). Then, in Section 3, we prove our existence result (Theorem 1.1). We end with Section 4, providing two illustrative examples. Along all the manuscript, and unless stated otherwise, we always assume that $x\text{,}u\in G[0\text{,}1]$ . Moreover, we use the symbol ${\int }_a^b$ to mean ${\int }_{[a\text{,}b]}$ .

2 Auxiliary Lemmas

By $(H_1)$ and $(H_4)$ , we define

Now, we present the well-known Leray–Schauder nonlinear alternative theorem.

We prove existence of a solution to problem (1.2) and (1.3) with the help of the preceding two lemmas.

3 Proof of Theorem 1.1

Let

On the other hand, by (Tvrdý, 2002, Proposition 2.3.16) and $(H_4)$ , $G_u$ is regulated on $[0\text{,}1]$ . Further, from $(H_6)$ and the Hölder inequality (Tvrdý, 2002, Theorem 2.3.8 ), it follows that $$\begin{array}{cc}\hfill \Vert G_u\Vert ⩽& \left(|g(0\text{,}x(0))|+|g(1\text{,}x(1))|+\underset{[0\text{,}1]}{\mathrm{var}}g\right)\Vert u\Vert \hfill \\ \hfill ⩽& 2M\Vert u\Vert \text{.}\hfill \end{array}$$

Let

For each $x\in B_r$ and $t\in [0\text{,}1]$ , define the operator $\mathcal{T}:G[0\text{,}1]\to G[0\text{,}1]$ by

We prove that $\mathcal{T}$ is completely continuous in three steps. Step 1: we show that $\mathcal{T}:B_r\to B_r$ . Indeed, for all $x\in B_r$ , one has

Moreover, $(H_6)$ , together with the convergence Theorem 1.7 of Krejčí (2006), yields that $${\displaystyle \underset{n\to \infty }{\lim }}{\int }_0^{t}g(s\text{,}{x}_n(s))\mathit{du}(s)={\int }_0^{t}g(s\text{,}x(s))\mathit{du}(s)\text{,}$$ $t\in [0\text{,}1]$ . Hence, $$\mathcal{T}{x}_n(t)-\mathcal{T}x(t)=\frac{\beta +t}{2}\left[\left(F(1\text{,}{x}_n)+F(\eta \text{,}{x}_n)+G_u(1\text{,}{x}_n)+G_u(\eta \text{,}{x}_n)\right)-(F(1\text{,}x)+F(\eta \text{,}x)+G_u(1\text{,}x)+G_u(\eta \text{,}x))\right]-{\int }_0^{t}F(s\text{,}{x}_n(s))-F(s\text{,}x(s))\mathit{ds}-{\int }_0^t{G}_u(s\text{,}{x}_n)-G_u(s\text{,}x)\mathit{ds}\text{,}t\in [0\text{,}1]\text{.}$$

Therefore, ${\lim }_{n\to \infty }\mathcal{T}{x}_n(·)=\mathcal{T}x(·)$ , and thus $\mathcal{T}$ is a completely continuous operator. Finally, let $$\mathrm{\Omega }=\left\{x\in G[0\text{,}1]:\Vert x\Vert <r\right\}$$ and assume that $x\in \partial \mathrm{\Omega }$ such that $\mathcal{T}x=\lambda x$ for $\lambda >1$ . Then, by (3.4), one has $$\lambda r=\lambda \Vert x\Vert =\Vert \mathcal{T}x\Vert ⩽r\text{,}$$ which implies that $\lambda ⩽1$ . This is a contradiction. Therefore, by Lemma 2.2, there exists a fixed point of $\mathcal{T}$ , which is a solution of problem (1.2) and (1.3). The proof of Theorem 1.1 is complete.

4 Illustrative examples

We now give two examples to illustrate Theorem 1.1 and Corollary 1.2, respectively. Let $g^{\ast }(t\text{,}x(t))=0$ if $t=0$ and $g^{\ast }(t\text{,}x(t))=1$ if $t\in (0\text{,}1]$ for all $x\in B_r$ . Then, it is easy to see that $g^{\ast }$ satisfies hypotheses $(H_4)$ – $(H_6)$ with $M=1$ .