Nonlinear Analysis: Theory, Methods & Applications
Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and 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 The significant point here is that the right-hand side of (1.1) may depend on . The dependency of the right-hand side on 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 of potential solutions , with the bound on being in terms of an a priori bound on . 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 subject to the boundary conditions where: ; and . Eq. (1.2) subject to (1.3) is called a boundary value problem (BVP) where comes from a so-called “time scale” .
The field of dynamic equations on time scales provides a natural framework for:
- (1)
establishing new insight into the theories of non-classical difference equations;
- (2)
forming novel knowledge about “differential-difference” equations;
- (3)
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 in . 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 .
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 at for (respectively at for ) by Also define , if , and , if . For simplicity and clarity denote and . Define the graininess function by .
Throughout this work the assumption is made that has the topology that it inherits from the standard topology on the real numbers . Also assume throughout that are points in and define the time scale interval The jump operators and allow the classification of points in a time scale in the following way: If then call the point right-scattered; while if then we say is left-scattered. If and then call the point right-dense; while if and then we say is left-dense. If has a left-scattered maximum at then define . Otherwise .
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 and let . Define to be the vector (if it exists) with the property that given there is a neighbourhood of such that, for all and each , Call the (delta) derivative of at . Definition 1.3 If then define the integral by Theorem 1.1 [13] Assume that and let . If is differentiable at , then is continuous at . If is continuous at and is right-scattered, then is differentiable at with If is differentiable and is right-dense, then If is differentiable at , then .
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 is right-dense continuous (and write ) provided is continuous at every right-dense point , and exists and is finite at every left-dense point .
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 is regular provided either or is an isolated time scale (i.e., all points in are isolated).
In addition to and , there are many other regular time scales, e.g. ([5, Example 1.41]) which is important in the theory of orthogonal polynomials and -difference equations, and the harmonic numbers ([5, Example 1.45]). For numerous other examples see Bohner and Peterson [5]. If is an isolated time scale and , then one can easily show that As a consequence of (1.4), for all regular time scales, we have which will be useful in several of the proofs in this paper.
We next define to be the set of all functions given by A solution to (1.2) is a function which satisfies (1.2) for each .
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 where is the Green’s function (see [5, p. 171]) for the BVP and
One of our tools used to gain a priori bounds and existence in this work is the use of dynamic inequalities on . To minimize notation in the statement of the theorems, the following notation will be used. Let be a non-negative constant and let be a positive constant. Define the sets and by:
Section snippets
General existence
In this section an abstract existence result is presented for (1.2), (1.3). This new result emphasizes the natural search for a priori bounds on solutions to BVPs, which will be conducted in the following sections. Theorem 2.1 Let and be positive constants in and let , be continuous. Consider the family of BVPs:If all potential solutions to(2.1), (2.2)that satisfy:also
Nagumo conditions
In this section some new Bernstein–Nagumo conditions are presented for the dynamic BVP (1.2), (1.3).
The first result allows linear growth on in . Theorem 3.1 Let and be non-negative constants in and let satisfy:If is a solution to(1.2), (1.3)that satisfies for then for , where is a constant involving and . Proof Let be a solution to (1.2), (1.3) that
More on a priori bounds
In this section we obtain more a priori bounds on solutions to (1.2), (1.3). The ideas may be viewed as extensions from the case where .
The following result allows linear growth of in and . Theorem 4.1 Let and be non-negative constants in . If satisfies:then all solutions to(1.2), (1.3)satisfy for , where is a constant involving
Existence for scalar BVPs
In this section an existence result is obtained for scalar BVPs on time scales. Theorem 5.1 Let and be non-negative constants in and let be a positive constant. If is continuous, scalar-valued and satisfies:then, for , the scalar BVP(1.2), (1.3)has at least one solution. Proof We want to show that the conditions of Theorem 2.1 hold for some positive constants and
Existence for systems
In this section a number of existence theorems are presented. The ideas combine the results of Sections 2 General existence, 3 Nagumo conditions, 4 More on in a natural way. Theorem 6.1 Let and be non-negative constants in . If is continuous and satisfiesthen the BVP(1.2), (1.3)has at least one solution. Proof Consider the family (2.1), (2.2). We want to show that the conditions of Theorem 2.1 hold for
Acknowledgement
Research supported by The Australian Research Council’s Discovery Projects (DP0450752).
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