Existence of solutions for a coupled system of difference equations with causal operators

Abstract

In this paper, we discuss a coupled system of difference equations with causal operators. Sufficient conditions under which such problems have solutions are given, by using a monotone iterative method. Difference inequalities with causal operators are also investigated. An example illustrates the results.

Introduction

By ${\mathbb{N}}_0\text{,}$ we denote the set of nonnegative integer numbers. Let $Z[a\text{,}b]=\{a\text{,}a+1\text{,}\dots \text{,}b\}\text{,}E=C(Z[a\text{,}b]\times Z[a\text{,}b]\text{,}\mathbb{R})\text{,}$ and $Q\in C(E\text{,}E)$, where $a\text{,}b\in {\mathbb{N}}_0$ and $a<b$. We shall say that $Q$ is a causal operator, or nonanticipative, if the following property holds: for each couple of elements of $E$ such that $u_i(s)=v_i(s)\text{,}i=1\text{,}2$ for $a⩽s⩽k\text{,}s\text{,}k\in {\mathbb{N}}_0\text{,}$ there results $Q(u_1\text{,}{u}_2)(s)=Q(v_1\text{,}{v}_2)(s)$ for $0⩽s⩽k$ with $k<b$ arbitrary; for details about causal operators, see [6].

Put $E_0=C(Z[0\text{,}T-1]\times Z[0\text{,}T-1]\text{,}\mathbb{R})\text{,}{E}_1=C(Z[1\text{,}T]\times Z[1\text{,}T]\text{,}\mathbb{R})\text{,}{\overline{E}}_0=C(Z[0\text{,}T-1]\text{,}\mathbb{R})\text{,}{\overline{E}}_1=C(Z[1\text{,}T]\text{,}\mathbb{R})$. In this paper, we investigate two-point boundary value problems for first order difference equations with causal operators $Q_1\text{,}{Q}_2$ of the form:

$$\left\{\begin{array}{c}\mathrm{\Delta }y(k-1)=Q_1(y\text{,}z)(k)\text{,}k\in Z[1\text{,}T]\text{,}\hfill \\ \mathrm{\Delta }z(k-1)=Q_2(z\text{,}y)(k)\text{,}k\in Z[1\text{,}T]\text{,}\hfill \\ h_1(y\text{,}z)=0\text{,}{h}_2(z\text{,}y)=0\text{,}\hfill \end{array}\right.$$

where $\mathrm{\Delta }y(k-1)=y(k)-y(k-1)\text{,}{h}_i(y\text{,}z)=g_i(y(0)\text{,}y(T)\text{,}z(0)\text{,}z(T))\text{,}i=1\text{,}2$ with assumption

$H_1:$ $Q_1\text{,}{Q}_2\in C(E_1\text{,}{E}_1)\text{,}{g}_1\text{,}{g}_2\in C({\mathbb{R}}^4\text{,}\mathbb{R})\text{,}{Q}_1\text{,}{Q}_2$ are bounded, and the next type of equations:

$$\left\{\begin{array}{c}\mathrm{\Delta }y(k)=Q_1(y\text{,}z)(k)\text{,}k\in Z[0\text{,}T-1]\text{,}\hfill \\ \mathrm{\Delta }z(k)=Q_2(z\text{,}y)(k)\text{,}k\in Z[0\text{,}T-1]\text{,}\hfill \\ h_1(y\text{,}z)=0\text{,}{h}_2(z\text{,}y)=0\text{,}\hfill \end{array}\right.$$

where $\mathrm{\Delta }y(k)=y(k+1)-y(k)$ with assumption

$H_2:$ $Q_1\text{,}{Q}_2\in C(E_0\text{,}{E}_0)\text{,}{g}_1\text{,}{g}_2\in C({\mathbb{R}}^4\text{,}\mathbb{R})\text{,}{Q}_1\text{,}{Q}_2$ are bounded.

For example, causal operators $Q_1\text{,}{Q}_2$ may take forms:

$$Q_1(y\text{,}z)(k)=f_1(k\text{,}y(k)\text{,}y({\alpha }_1(k))\text{,}\dots \text{,}y({\alpha }_r(k))\text{,}z(k)\text{,}z({\beta }_1(k))\text{,}\dots \text{,}z({\beta }_s(k)))\text{,}$$

$$Q_2(z\text{,}y)(k)=f_2(k\text{,}z(k)\text{,}z({\alpha }_1(k))\text{,}\dots \text{,}z({\alpha }_r(k))\text{,}y(k)\text{,}y({\beta }_1(k))\text{,}\dots \text{,}y({\beta }_s(k)))\text{,}$$

where $f_1\text{,}{f}_2\in C(Z[1\text{,}T]\times {\mathbb{R}}^{r+s+2}\text{,}\mathbb{R})\text{,}{\alpha }_i\text{,}{\beta }_j\in C(Z[1\text{,}T]\text{,}$ $Z[0\text{,}T])\text{,}{\alpha }_i(k)⩽k$ and ${\beta }_j(k)⩽k\text{,}i=1\text{,}2\text{,}\dots \text{,}r\text{,}j=1\text{,}2\text{,}\dots \text{,}s$.

Note that problems (1), (2) represent two discrete analogues of a coupled system of differential equations with causal operators $Q_1\text{,}{Q}_2$ of the form

$$\left\{\begin{array}{c}{x}^{\prime }(t)=Q_1(x\text{,}y)(t)\text{,}t\in [0\text{,}T]\text{,}\hfill \\ y^{\prime }(t)=Q_2(y\text{,}x)(t)\text{,}t\in [0\text{,}T]\text{,}\hfill \\ g_1(x(0)\text{,}x(T))=0\text{,}{g}_2(y(0)\text{,}y(T))=0\text{.}\hfill \end{array}\right.$$

Let $\mathcal{C}=C(Z[0\text{,}T]\text{,}\mathbb{R})$ denote the class of maps $w$ continuous on $Z[0\text{,}T]$ (discrete topology) with the norm

$$\Vert w\Vert ={\displaystyle \underset{k\in Z[0\text{,}T]}{\max }}|w(k)|\text{.}$$

Note that $\mathcal{C}$ is a Banach space. By a solution of (1), we mean a pair $(w\text{,}v)\in \mathcal{C}\times \mathcal{C}$ such that it satisfies problem (1). Similarly we define the solution of problem (2).

To find a solution of problem (1) or (2) we are going to apply a monotone iterative method combined with lower and upper solutions of discrete problems. For some results on lower and upper approximations of solutions of difference equations and their applications, usually in showing the existence of a specific solution, see, e.g. [1], [3], [4], [5], [12], [14], [15], [16], [18], [19] and the related references therein. The monotone iterative method is a well known one in case of continuous problems, see for example [11]. It was also applied to discrete problems, see for example papers [8], [9], [10], [17], [18]. The study of a coupled system of difference equations with causal operators is also significant because this kind of system may occur in applications. Note that difference equations with delayed arguments belong to difference equations with causal operators. Difference equations are also discussed, for example, in papers [2], [13].

This paper is organized as follows. In Section 2, we present some basic materials needed to prove the main result for problem (1). In Section 3, we prove the existence of solutions for system (1), by applying a theorem from [7]. An example is also included in this section to illustrate theoretical results. In the last section, we discuss problem of type (2) giving sufficient conditions which guarantee the existence of solutions.