Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and solvability

Abstract

This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein–Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability.

Introduction

In the study of the existence of solutions to boundary value problems (BVPs), major advancements have been made due to the introduction, development and application of so-called “Bernstein–Nagumo” conditions [4], [16]. In particular, Bernstein–Nagumo conditions have allowed the treatment of BVPs featuring differential equations of the type

$$x^{″}=f(t,x,x^{\prime })\text{,}t\in [a,b]\text{.}$$

The significant point here is that the right-hand side of (1.1) may depend on $x^{\prime }$. The dependency of the right-hand side on $x^{\prime }$ is naturally seen in many physical phenomena and we refer the reader to [19], [7] for some nice examples.

Bernstein–Nagumo conditions for (1.1) are sufficient conditions, involving differential inequalities, that guarantee an a priori bound on the first derivative $x^{\prime }$ of potential solutions $x$, with the bound on $x^{\prime }$ being in terms of an a priori bound on $x$. Topological ideas are then usually used to gain solvability.

Bernstein–Nagumo conditions have also been of significance in the solvability of BVPs involving second-order finite-difference equations. These types of “discrete” BVPs arise as discrete approximations to “continuous” BVPs involving ordinary differential equations. For example, [10], [15], [22], [23], [24] formulated “discrete” Bernstein–Nagumo conditions so that the first finite-difference of solutions to the discrete BVP are bounded a priori, with the bound being independent of the step-size and in terms of a bound on potential solutions to the discrete BVP. Particular interest in these ideas is seen through the elimination of “spurious solutions” [18] to the discrete BVP, and the convergence of solutions of the discrete BVP to solutions of the continuous BVP as the step-size tends to zero.

In this paper we are interested in Bernstein–Nagumo conditions and the existence of solutions to the second-order dynamic equation

$$y^{\Delta \Delta }=f(t,y^{\sigma },y^{\Delta })\text{,}t\in {[a,b]}_{\mathbb{T}}\text{;}$$

subject to the boundary conditions

$$y\left(a\right)=A\text{,}y\left({\sigma }^2\left(b\right)\right)=B\text{;}$$

where: $f:{[a,b]}_{\mathbb{T}}\times {\mathbb{R}}^d\to {\mathbb{R}}^d,d\ge 1$; and $A,B\in {\mathbb{R}}^d$. Eq. (1.2) subject to (1.3) is called a boundary value problem (BVP) where $t$ comes from a so-called “time scale” $\mathbb{T}$.

The field of dynamic equations on time scales provides a natural framework for:

• establishing new insight into the theories of non-classical difference equations;

• forming novel knowledge about “differential-difference” equations;

• advancing, in their own right, each of the theories of differential equations and (classical) difference equations.

Recently, some Bernstein–Nagumo conditions for dynamic BVPs on time scales were formulated in [14], [3]. In this article, some alternate Bernstein–Nagumo conditions are established, with the new results extending and complimenting those in [14], [3] in a significant way. The extensions are demonstrated through the use of examples.

The paper is organized as follows.

In Section 2 a general existence result is presented for (1.2), (1.3). The result provides a natural motivation for the obtention of a priori bounds on solutions and greatly minimizes the proofs of the new results in the following sections. The main tool used here is Leray–Schauder topological degree.

In Section 3 some new Bernstein–Nagumo conditions are presented for (1.2), (1.3). These new conditions involve linear or quadratic growth constraints on $\Vert f(t,p,q)\Vert$ in $\Vert q\Vert$. The new results are stated for systems of BVPs on time scales, however we remark that the ideas are new even for scalar BVPs on time scales.

In Section 4 we establish some new results that guarantee a priori bounds on solutions to (1.2), (1.3). The ideas are extensions from the case $y^{\Delta \Delta }=f(t,y^{\sigma })$.

In Sections 5 Existence for scalar BVPs, 6 Existence for systems, the abstract existence theorem of Section 2 is combined with the a priori bound results of Sections 3 Nagumo conditions, 4 More on to give a number of new existence results for (1.2), (1.3).

Examples to highlight the theory and application of the new ideas are presented throughout the paper.

For more on dynamic equations on time scales we refer to [5]. To see a discussion of recent and future applications of dynamic equations on time scales see the featured front page article in New Scientist [20].

For recent related articles on solvability of dynamic BVPs on time scales, please see [11], [9], [17], [8], [21], [12], [2], [6].

To understand the concept of time scales and the above notation, some definitions are useful.

Definition 1.1

Define the forward (backward) jump operator $\sigma \left(t\right)$ at $t$ for $t<sup\mathbb{T}$ (respectively $\rho \left(t\right)$ at $t$ for $t>inf\mathbb{T}$) by

$$\sigma \left(t\right)=inf\{\tau >t:\tau \in \mathbb{T}\}\text{,}(\rho \left(t\right)=sup\{\tau <t:\tau \in \mathbb{T}\},)\text{for all }t\in \mathbb{T}\text{.}$$

Also define $\sigma (sup\mathbb{T})=sup\mathbb{T}$, if $sup\mathbb{T}<\infty$, and $\rho (inf\mathbb{T})=inf\mathbb{T}$, if $inf\mathbb{T}>-\infty$. For simplicity and clarity denote ${\sigma }^2\left(t\right)=\sigma \left(\sigma \left(t\right)\right)$ and $y^{\sigma }\left(t\right)=y\left(\sigma \left(t\right)\right)$. Define the graininess function $\mu :\mathbb{T}\to \mathbb{R}$ by $\mu \left(t\right)=\sigma \left(t\right)-t$.

Throughout this work the assumption is made that $\mathbb{T}$ has the topology that it inherits from the standard topology on the real numbers $\mathbb{R}$. Also assume throughout that $a<b$ are points in $\mathbb{T}$ and define the time scale interval

$${[a,b]}_{\mathbb{T}}≔\{t\in \mathbb{T}:a\le t\le b\}\text{.}$$

The jump operators $\sigma$ and $\rho$ allow the classification of points in a time scale in the following way: If $\sigma \left(t\right)>t$ then call the point $t$ right-scattered; while if $\rho \left(t\right)<t$ then we say $t$ is left-scattered. If $t<sup\mathbb{T}$ and $\sigma \left(t\right)=t$ then call the point $t$ right-dense; while if $t>inf\mathbb{T}$ and $\rho \left(t\right)=t$ then we say $t$ is left-dense. If $\mathbb{T}$ has a left-scattered maximum at $m$ then define ${\mathbb{T}}^{\kappa }=\mathbb{T}-\left\{m\right\}$. Otherwise ${\mathbb{T}}^{\kappa }=\mathbb{T}$.

We next define the so-called delta derivative. The novice could skip this definition and for the scalar case look at the results stated in Theorem 1.1. In particular in part (2) of Theorem 1.1 we see what the delta derivative is at right-scattered points and in part (3) of Theorem 1.1 we see that at right-dense points the derivative is similar to the definition given in calculus.

Definition 1.2

Fix $t\in {\mathbb{T}}^{\kappa }$ and let $y:\mathbb{T}\to {\mathbb{R}}^d$. Define $y^{\Delta }\left(t\right)$ to be the vector (if it exists) with the property that given $ϵ>0$ there is a neighbourhood $U$ of $t$ such that, for all $s\in U$ and each $i=1,\dots ,d$,

$$|[y_i\left(\sigma \left(t\right)\right)-y_i\left(s\right)]-y_i^{\Delta }\left(t\right)[\sigma \left(t\right)-s]|\le ϵ|\sigma \left(t\right)-s|\text{.}$$

Call $y^{\Delta }\left(t\right)$ the (delta) derivative of $y\left(t\right)$ at $t$.

Definition 1.3

If $Y^{\Delta }\left(t\right)=y\left(t\right)$ then define the integral by

$${\int }_a^{t}y\left(s\right)\Delta s=Y\left(t\right)-Y\left(a\right)\text{.}$$

Theorem 1.1 [13]

Assume that $y:\mathbb{T}\to {\mathbb{R}}^d$ and let $t\in {\mathbb{T}}^{\kappa }$ .

• If $y$ is differentiable at $t$ , then $y$ is continuous at $t$ .

• If $y$ is continuous at $t$ and $t$ is right-scattered, then $y$ is differentiable at $t$ with

  $$y^{\Delta }\left(t\right)=\frac{y\left(\sigma \left(t\right)\right)-y\left(t\right)}{\sigma \left(t\right)-t}\text{.}$$

• If $y$ is differentiable and $t$ is right-dense, then

  $$y^{\Delta }\left(t\right)=\underset{s\to t}{lim}\frac{y\left(t\right)-y\left(s\right)}{t-s}\text{.}$$

• If $y$ is differentiable at $t$ , then $y\left(\sigma \left(t\right)\right)=y\left(t\right)+\mu \left(t\right)y^{\Delta }\left(t\right)$ .

Next we define the important concept of right-dense continuity. An important fact concerning right-dense continuity is that every right-dense continuous function has a delta antiderivative [5, Theorem 1.74]. This implies that the delta definite integral of any right-dense continuous function exists.

Definition 1.4

We say that $y:\mathbb{T}\to \mathbb{R}$ is right-dense continuous (and write $y\in C_{rd}(\mathbb{T};{\mathbb{R}}^d)$) provided $y$ is continuous at every right-dense point $t\in \mathbb{T}$, and ${lim}_{s\to t^{-}}y\left(s\right)$ exists and is finite at every left-dense point $t\in \mathbb{T}$.

In this paper we will be interested in so-called “regular” time scales which we define as follows:

Definition 1.5

We say a time scale $\mathbb{T}$ is regular provided either $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}$ is an isolated time scale (i.e., all points in $\mathbb{T}$ are isolated).

In addition to $\mathbb{R}$ and $h\mathbb{Z}≔\{0,\pm h,\pm 2h,\pm 3h,\dots \},h>0$, there are many other regular time scales, e.g. $\mathbb{T}=q^{{\mathbb{N}}_0},q>1$ ([5, Example 1.41]) which is important in the theory of orthogonal polynomials and $q$-difference equations, and the harmonic numbers $\mathbb{T}=\{t_n={\sum }_{k=1}^n\frac{1}{k},n=1,2,3,\dots \}$ ([5, Example 1.45]). For numerous other examples see Bohner and Peterson [5]. If $\mathbb{T}$ is an isolated time scale and $f:\mathbb{T}\to \mathbb{R}$, then one can easily show that

$$f^{\sigma \Delta }\left(t\right)=\frac{\mu \left(\sigma \left(t\right)\right)}{\mu \left(t\right)}{f}^{\Delta \sigma }\left(t\right)\text{.}$$

As a consequence of (1.4), for all regular time scales, we have

$${\left({\Vert x\left(t\right)\Vert }^2\right)}^{\Delta \Delta }\ge 2〈x^{\sigma }\left(t\right),x^{\Delta \Delta }\left(t\right)〉+{\Vert x^{\Delta }\left(t\right)\Vert }^2\text{,}$$

which will be useful in several of the proofs in this paper.

We next define $S$ to be the set of all functions $y:\mathbb{T}\to {\mathbb{R}}^d$ given by

$$S=\{y:y\in C({[a,{\sigma }^2\left(b\right)]}_{\mathbb{T}};{\mathbb{R}}^d)\text{,}{y}^{\Delta }\in C({[a,\sigma \left(b\right)]}_{\mathbb{T}};{\mathbb{R}}^d)\text{ and }{y}^{\Delta \Delta }\in C_{rd}({[a,b]}_{\mathbb{T}};{\mathbb{R}}^d)\}\text{.}$$

A solution to (1.2) is a function $y\in S$ which satisfies (1.2) for each $t\in [a,b]$.

A widely used technique in the theory of BVPs on time scales involves reformulating the BVP (1.2), (1.3) as an equivalent delta integral equation. In particular the BVP (1.2), (1.3) is equivalent to the delta integral equation

$$y\left(t\right)={\int }_a^{\sigma \left(b\right)}G(t,s)f(s,y^{\sigma }\left(s\right),y^{\Delta }\left(s\right))\Delta s+\varphi \left(t\right)\text{,}t\in {[a,{\sigma }^2\left(b\right)]}_{\mathbb{T}}\text{;}$$

where

$$G(t,s)=\{\begin{array}{cc}-\frac{(\sigma \left(s\right)-a)({\sigma }^2\left(b\right)-t)}{{\sigma }^2\left(b\right)-a}\text{,}& \sigma \left(s\right)\le t\text{;}\\ -\frac{(t-a)({\sigma }^2\left(b\right)-\sigma \left(s\right))}{{\sigma }^2\left(b\right)-a}\text{,}& t\le s\text{;}\end{array}$$

is the Green’s function (see [5, p. 171]) for the BVP

$$y^{\Delta \Delta }=0\text{,}y\left(a\right)=0\text{,}y\left({\sigma }^2\left(b\right)\right)=0\text{;}$$

and

$$\varphi \left(t\right)=\frac{A{\sigma }^2\left(b\right)-Ba+(B-A)t}{{\sigma }^2\left(b\right)-a}\text{.}$$

One of our tools used to gain a priori bounds and existence in this work is the use of dynamic inequalities on $f$. To minimize notation in the statement of the theorems, the following notation will be used. Let $U$ be a non-negative constant and let $V$ be a positive constant. Define the sets $C_U$ and $D_V$ by:

$$C_U≔\{(t,p,q)\in {[a,b]}_{\mathbb{T}}\times {\mathbb{R}}^d\times {\mathbb{R}}^d:\Vert p\Vert \le U\}\text{;}$$$$D_V≔\{(t,p,q)\in {[a,b]}_{\mathbb{T}}\times {\mathbb{R}}^d\times {\mathbb{R}}^d:\Vert p\Vert =V,-2〈p,q〉\le \mu \left(t\right){\Vert q\Vert }^2\}\text{.}$$