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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 260409, 9 pages
A Survey of Recent Results for the Generalizations of Ordinary Differential Equations
1Department of Mathematics and Computer Science, Belmont University, 1900 Belmont Boulevard, Nashville, TN 37212, USA
2Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-9701900 São Carlos, SP, Brazil
3Department of Mathematical Analysis, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain
Received 23 October 2013; Accepted 5 December 2013; Published 10 February 2014
Academic Editor: Ferhan M. Atici
Copyright © 2014 Daniel C. Biles et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation.
1. Existence Theory for Differential Equations and Inclusions
There was a series of results which progressively weakened the continuity in the state variable of the classical Carathéodory existence theorem for first-order differential equations; these include [1–7]. Biles and Schechter posed the open problem of proving an existence result for discontinuous systems of differential equations lacking a quasimonotonicity property; see [7, page 3352]. Motivated by that question, Cid and Pouso  explored an alternative approach to discontinuous equations which consisted, roughly speaking, of inserting the differential equation into a semicontinuous differential inclusion for which existence results were available, and then positing assumptions on the discontinuities of the former differential equation so that every solution of the inclusion is a solution of the equation. Besides getting an existence result for nonquasimonotone discontinuous systems, the approach in  came to unify and extend previous similar results for autonomous equations proven in  and for nonautonomous equations proven in .
Here and henceforth we work in a real interval with .
Theorem 1 (see [8, Theorem 2.4]). Assume that and the null set satisfy the following conditions.(i)There exists such that for all and all one has , where is a norm in .(ii)For all , is measurable.(iii)For all , is continuous in , where , and for each and one has
where denotes the closed convex hull, is the unit ball centered at the origin, and is the contingent derivative of the multivalued map (see  for details).
Then the set of all Carathéodory solutions of the initial value problem is a nonempty, compact, and connected subset of .
Moreover, in the scalar case , one has the following.(1) has pointwise maximum, , and minimum, , which are the extremal solutions of the initial value problem. Furthermore for each one has (2) is a funnel; that is, for all and there exists such that .
The simplest case of Theorem 1 occurs in the one-dimensional case, that is, , and when the discontinuity sets are single-valued, that is, for, let us say, absolutely continuous functions . In this situation we have and then condition (1) reads simply as follows: and it is helpful to note that, in the one-dimensional case, we have The first alternative in (5) means that coincides neither with nor with any limit value of when the variables tend to . This is a sort of transversality condition between and the discontinuity curve , and it is immediately satisfied in case has a sufficiently big (or sufficiently small) slope.
The second alternative is much clearer: it simply means that solves the differential equation at the point . The moral is that we do not have to worry about discontinuities of when they are located over graphs of solutions of the differential equation (even though these solutions do not satisfy the initial condition or they are not defined on the whole interval ).
For simplicity, we have often called admissible discontinuity curve any function satisfying (5), and they have proven to be useful in other contexts; see  for singular and discontinuous problems and  for a revision of Perron’s method using similar curves.
Let us now turn our attention to differential inclusions. The rest of this section is devoted to a somewhat inverse approach to that in the first part: one can get new results for inclusions by means of known results for discontinuous equations.
To start with, we quote  where we can find necessary and sufficient conditions for the existence of Carathéodory solutions to where is an arbitrary multifunction. Biles proves that the necessary and sufficient conditions for (7) to have at least one solution are that have a selection such that either or exists (in Lebesgue’s sense) for some . This uses and generalizes a theorem for differential equations by binding in .
We also used known results for equations to study nonautonomous first-order inclusions in . Let us proceed to review the main ideas in that paper.
For a given multifunction we consider the initial value problem and we look for solutions in the Carathéodory sense, that is, absolutely continuous solutions.
A very usual assumption on the multifunction is that it assumes compact values for and all , hence the set has minimum and maximum. We simply impose the following condition. (H1) For and all the set has a minimum; and now we introduce the following definition.
Definition 2. A superfunction (or upper solution) of (8) is any such that and for one has .
We also impose the following. (H2) There exists such that for and all we have and we restrict, for technical convenience, the set of superfunctions to the following one.
Definition 3. The set of admissible superfunctions of (8) is
Notice that , , is an admissible superfunction of (8). Thus we can define Standard arguments reveal that and that .
For simplicity of notation, we also define and we consider the ordinary problem Plainly, solutions of (13) are solutions of (8) by virtue of (H1), but the converse is false in general. Moreover, superfunctions of (8) in the sense of Definition 2 are nothing but the usual superfunctions of (13), and so is a reasonable candidate for being a solution to (8). Note also that solutions of (8) need not be admissible superfunctions in the sense of Definition 3, so might not be the least solution of (8).
Definition 4. A lower admissible nonquasisemicontinuity curve for (8) (LAD curve, for short) is an absolutely continuous function for which there exist disjoint sets such that and for one has and for one has
Remark 5. The sets or in Definition 4 might be empty.
A particularly clear case of a LAD curve corresponds to , which means that , so in that case the LAD curve is nothing but a solution of the differential inclusion on .
In turn, let us point out the following sufficient condition for an absolutely continuous function to be a LAD curve with : there exist and such that for we have or for we have Notice that (16) (or (17)) implies that crosses each solution of at most once, so (16) (or (17)) is a transversality condition for with respect to the differential equation .
We are now in a position to present the main result in .
Theorem 6 (see [16, Theorem 2.5]). Assume that conditions (H1) and (H2) hold. Suppose moreover that the following condition is fulfilled.(H3)Either for and all one has
or there exist countably many LAD curves for (8), , , such that for and all one has (18).
Then one has the following results.(a)There exists a null measure set such that where , and for each the set contains no positive measure subset.(b)The function is a solution of (8) provided that for all the set is measurable.(c)If is measurable for every , then is the least solution of (8) provided that one of the following conditions hold: either for , all , and all one has or the first alternative in (H3) holds, which, furthermore, guarantees that is the least solution to (13).
The following result is Lemma 2 in , and it is very useful to prove that the ’s are measurable in practical situations.
Lemma 7. Let be a null measure set and let be such that is measurable for each .
If, moreover, for all and all one has then the mapping is measurable for each pair such that for all .
Notice that our multifunctions need not satisfy the usual hypotheses such as monotonicity or upper/lower semicontinuity. Moreover, need not assume closed or convex values.
Another example where we used known results for equations to deduce new result for inclusions is , which concerns second-order inclusions and relies on the results proven for equations in . In order to present the main result in  we need some notations and preliminaries.
Let , and For each we define its “pseudoinverse” as We notice that is nondecreasing but not necessarily continuous. Moreover, if is increasing in , then for all .
Theorem 8 (see [18, Theorem 4.1]). Suppose that for some the following hypotheses hold.(F1)For each the multifunction defined as has an admissible selection on the right of , that is, a selection such that(i);(ii) for ;(iii);(iv).(F2)There exists such that for all and all one has
(F3)For every , the relation on implies on .
Then the initial value problem has an increasing solution in .
2. Dynamic Equations on Time Scales
The study of time scales was formalized in the Ph.D. thesis of Hilger in 1988 . The notions of derivative from differential calculus and the forward jump operator from difference calculus are unified and extended to the delta derivative on a time scale (an arbitrary set on the real line). These lead to the study of dynamic equations on time scales, unifying differential and difference equations. In addition, these ideas can be applied in situations more general than those for differential and difference equations, such as population problems in which the species alternates between time frames in which they are active and periods of dormancy. The study of time scales yields interesting insight into the special cases. For example, one realizes that the only reason we have the simple derivative from elementary calculus of is is because the graininess of real line is zero.
Much of the earlier history of time scales can be found in the books by Bohner and Peterson [21, 22] which are on the bookshelf of every time scales analyst. We refer the reader to these sources for the basic concepts and definitions for time scales. Reference  collects much of the information for the linear case. As an example, we will overview the first-order linear case. We define the cylinder transformation on by for , where is the principal logarithm function and . We call a function regressive if for all , where is the graininess of the time scale. We can now define the time scales (or generalized) exponential function by , where . The exponential function thus defined enjoys many properties analogous to that of the standard exponential function on the real line. The following can now be proven.
Theorem 9. Suppose is rd-continuous and regressive, and let . Then, is the unique solution to
Note that this yields the corollaries that is the unique solution to , on the real line and is the unique solution to , on the integers, where represents the forward difference operator from difference calculus.
We note that the nabla derivative on time scales was defined by Atici and Guseinov  in 2002, which generalizes the backward difference operator. One might think that the results for nabla derivatives mirror those for the delta case, but this is not true; see, for example, . Recently, work has progressed for dynamic equations with the diamond-alpha derivative initiated in  and furthered in [26–28]. In the remainder of this section, without making a claim to being complete, we overview some of the recent work in dynamic equations on time scales to illustrate how many of the ideas from differential and difference equations have been generalized and extended.
Existence of solutions has been proven in a number of cases, such as  using fixed point theory,  proving a Nagumo-type existence result, and  using a fixed point theorem due to Avery and Peterson. (A number of other existence theorems are mentioned in specific contexts below.) As an example, here is the theorem from .
Theorem 10. Assume there exist a lower solution and an upper solution with on and(a) satisfies for all , and ,(b)there exists a such that for all right scattered , and ,(c) is nonincreasing for all right scattered and ,(d) is nondecreasing in its third variable, nonincreasing in its fourth variable, and nondecreasing in its fifth variable,(e) is nonincreasing in its first variable, and(f) satisfies a Nagumo condition with respect to the pair and .
Then, there exists a solution to the problem
Singular problems have been studied in [32, 33]. Green’s functions have been considered in [23, 34]. A Sturm-Liouville eigenvalue problem was studied by . Periodic solutions were investigated in . OscillationS of solutions have been considered in [37–39] using the time scales Taylor formula, [40–45]. Asymptotic behavior of solutions has been studied in [46, 47] using Taylor monomials and in . Laplace transforms on time scales were studied by .
Delay equations were studied in [40, 42, 44]. Impulsive problems have been studied in [37, 38, 49–51]. Functional dynamic equations have been studied in [50, 52] using Lyapunov functions [41, 46]. Fractional derivatives have been considered in [53, 54]. Problems in abstract spaces were studied in . Dynamic inclusions have been studied in [24, 37, 50, 56]. Partial differentiation on time scales was introduced in  and was continued in .
3. Generalized Ordinary Differential Equations
In order to generalize some results on continuous dependence of solutions of ordinary differential equations with respect to the initial data, Jaroslav Kurzweil introduced, in 1957, the notion of generalized ordinary differential equations for functions taking values in Euclidean and Banach spaces. This generalization of the notion of ordinary differential equations uses the concept of the Perron generalized integral, also known as the Kurzweil integral. We refer to these equations as generalized ODEs. See [62–66].
The correspondence between generalized ODEs and classic ODEs is very simple. It is known that the ordinary system where , is an open set and , has the integral representation whenever the integral exists in some sense. It is also known that if the integral in (29) is in the sense of Riemann, Lebesgue (with the equivalent definition given by E. J. McShane), or Henstock-Kurzweil, for instance, then such an integral can be approximated by a Riemann-type sum of the form where is a fine partition of the interval and, for each , is sufficiently “close” to the interval .
Alternatively, if we define then the integral in (29) can be approximated by In such a case, the right-hand side of (32) approximates the nonabsolute Kurzweil integral which, when considered in (29), gives rise to a “differential equation” of type (28), however in a wider sense. Such type of equation is known as generalized ordinary differential equation or Kurzweil equation. See [67, 68].
Let be a compact interval and consider a function (called a gauge on ). A tagged partition of the interval with division points and tags , , is called -fine if
Definition 11. Let be a Banach space. A function is called Kurzweil integrable over , if there is an element such that, given , there is a gauge on such that for every -fine tagged partition of . In this case, is called the Kurzweil integral of over and it will be denoted by .
The Kurzweil integral has the usual properties of linearity, additivity with respect to adjacent intervals, and integrability on subintervals. See, for instance, , for these and other interesting properties.
Now, consider a subset and a function .
Any function is called a solution of the generalized ordinary differential equation (we write simply generalized ODE) on the interval , provided where the integral is obtained by setting in the definition of the Kurzweil integral.
As it was done in [69, 70], but using different assumptions, namely, Carathéodory and Lipschitz-type conditions on the indefinite integral, we proved in  that retarded functional differential equations (we write RFDEs, for short) can be regarded as abstract generalized ODEs and some applications were investigated.
In , together with professor Štefan Schwabik, we proved that RFDEs subject to impulse effects can also be regarded as generalized ODEs taking values in a Banach space.
Recently, in , together with Federson et al., we proved that a solution of a measure RFDEs of the form where and are distributional derivatives in the sense of L. Schwartz with respect to and , respectively, can be related to a solution of an abstract generalized ODE. More precisely, we considered the integral form of (37) as follows: where are given real numbers, with , and , for . We also considered as being an open set and where by we mean the Banach space of all regulated functions endowed with the usual supremum norm and we assumed that : is a function such that, for each , the mapping is Henstock-Kurzweil integrable (or Perron integrable) over with respect to a nondecreasing function . Then, we defined a function by and proved the correspondence between a solution of (38) and a solution of the generalized ODE with initial condition just by requiring the following conditions.(A)The integral exists, for every .(B)There exists a function which is Lebesgue integrable with respect to such that, for all , , we have (C)There exists a function which is Lebesgue integrable with respect to , such that for all , , we have
In [77, 78], together with Federson et al., we proved that measure RFDEs are useful tools in the study of impulsive RFDEs and functional dynamic equations on time scales with or without impulse action. In other words, it was proved that the unique solution of the Cauchy problem for a measure RFDE of type can be regarded, in a one-to-one relation, with the unique solution of the Cauchy problem for the measure RFDE with impulses given by
Still in [77, 78], we related the solution of problem (47) to the solution of the following impulsive functional dynamic equation on time scales where is defined as being the extension of defined by , for .
In order to obtain the correspondences presented in [77, 78], the requirement was mainly that is Henstock-Kurzweil integrable with respect to a nondecreasing function . Therefore many discontinuities are allowed. Moreover, does not need to be rd-continuous nor regulated, and yet good results for impulsive functional dynamic equations on time scales can be obtained through these correspondences.
Even more recently, together with Federson et al., we studied, in , measure neutral functional differential equations (we write measure NFDEs) of type where is a nonautonomous linear operator (i.e., ). Besides, we assume that admits a representation where is a normalized measurable function satisfying which is continuous to the left on of bounded variation in , and the variation of in , tends to zero as .
In order to obtain a correspondence between solutions of measure NFDEs and solutions of a certain class of generalized ODEs of the form whose right-hand side is given by with as (47) and given by
Besides conditions (A), (B), and (C) presented above, we required that the normalized function satisfies the following:(D)there exists a Lebesgue integrable function : such that
Thus, in , results on the local existence and uniqueness of solutions, as well as continuous dependence of solutions on the initial data, were established.
Clearly there is still much to do to develop the theory of abstract generalized ODEs and to apply the results to other types of differential equations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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