Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments
© Figueroa and Pouso; licensee Springer. 2012
Received: 13 May 2011
Accepted: 20 January 2012
Published: 20 January 2012
We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this article, and we show with an example that similar results with the classical definitions are false. We also introduce an example showing that the problems considered need not have the least (or the greatest) solution between given lower and upper solutions, but we can prove that they do have minimal and maximal solutions in the usual set-theoretic sense. Sufficient conditions for the existence of lower and upper solutions, with some examples of application, are provided too.
where f : I × ℝ2 → ℝ and are Carathéodory functions, is a continuous nonlinear operator and . Here denotes the set of real functions which are continuous on the interval J.
We define a solution of problem (1) to be a function such that (i.e., is absolutely continuous on I0) and x fulfills (1).
In the space we consider the usual pointwise partial ordering, i.e., for we define γ1 ≤ γ2 if and only if γ1(t) ≤ γ2(t) for all t ∈ I. A solution of (1), x*, is a minimal (respectively, maximal) solution of (1) in a certain subset if x* ∈ Y and the inequality x ≤ x*, (respectively, x ≥ x*) implies x = x*, whenever x is a solution to (1) and x ∈ Y. We say that x* is the least (respectively the greatest) solution of (1) in Y if x* ≤ x (respectively x* ≥ x) for any other solution x ∈ Y. Notice that the least solution in a subset Y is a minimal solution in Y, but the converse is false in general, and an analgous remark is true for maximal and greatest solutions.
Interestingly, we will show that problem (1) may have minimal (maximal) solutions between given lower and upper solutions and not have the least (greatest) solution. This seems to be a peculiar feature of equations with deviated arguments, see  for an example with a second-order equation. Therefore, we are obliged to distinguish between the concepts of minimal solution and least solution (or maximal and greatest solutions), unfortunately often identified in the literature on lower and upper solutions.
First-order differential equations with state-dependent deviated arguments have received a lot of attention in the last years. We can cite the recent articles [2–7] which deal with existence results for this kind of problems. For the qualitative study of this type of problems we can cite the survey of Hartung et al.  and references therein.
The deviating argument τ depends at each moment t on the global behavior of the solution, and not only on the values that it takes at the instant t.
Delay problems, which correspond to differential equations of the form x'(t) = f(t, x(t), x(t - r)) along with a functional start condition, are included in the framework of problem (1). This is not allowed in articles [3–6].
No monotonicity conditions are required for the functions f and τ, and they need not be continuous with respect to their first variable.
This article is organized as follows. In Section 2, we state and prove the main results in this article, which are two existence results for problem (1) between given lower and upper solutions. The first result ensures the existence of maximal and minimal solutions, and the second one establishes the existence of the greatest and the least solutions in a particular case. The concepts of lower and upper solutions introduced in Section 2 are new, and we show with an example that our existence results are false if we consider lower and upper solutions in the usual sense. We also show with an example that our problems need not have the least or the greatest solution between given lower and upper solutions. In Section 3, we prove some results on the existence of lower and upper solutions with some examples of application.
2 Main results
We begin this section by introducing adequate new definitions of lower and upper solutions for problem (1).
Remark 1 Definition 1 requires implicitly that Λ be bounded in [α, β].
are really attained for almost every fixed t ∈ I0 thanks to the continuity of f(t, α(t), ·) and f(t, β(t), ·) on the compact set E(t).
Now we introduce the main result of this article.
Theorem 1 Assume that the following conditions hold:
(H1) (Lower and upper solutions) There exist , with α ≤ β on I, which are a lower and an upper solution for problem (1).
(H2) (Carathéodory conditions)
(H2) - (a) For all x, y ∈ [mint∈Iα(t), maxt∈Iβ(t)] the function f(·,x,y) is measurable and for a.a. t ∈ I0, all x ∈ [α(t), β(t)] and all y ∈ E(t) (as defined in Definition 1) the functions f(t, ·, y) and f(t, x, •) are continuous.
(H2) - (b) For all the function τ(·, γ) is measurable and for a.a. t ∈ I0 the operator τ(t, ·) is continuous in (equipped with it usual topology of uniform convergence).
(H2) - (c) The nonlinear operator is continuous.
Then problem (1) has maximal and minimal solutions in [α, β].
It is an elementary matter to check that T is a completely continuous opera-tor from into itself (one has to take Remark 1 into account). Therefore, Schauder's Theorem ensures that T has a nonempty and compact set of fixed points in , which are exactly the solutions of problem (4).
a contradiction with (5).
Now, if is such that x ≥ x* on I then we have and, by (6), . So we conclude that which, along with x ≥ x*, implies that x = x* on I. Hence x* is a maximal element of . In the same way, we can prove that x* is a minimal element.
These definitions are not adequate to ensure the existence of solutions of (1) between given lower and upper solutions, as we show in the following example.
and then x(t) < α(t) for all t ∈ (0,1]. Hence (9) has no solution at all between α and β.
Remark 2 Notice that inequalities (2) and (3) imply (7) and (8), so lower and upper solutions in the sense of Definition 1 are lower and upper solutions in the usual sense, but the converse is false in general.
Definition 1 is probably the best possible for (1) because it reduces to some definitions that one can find in the literature in connection with particular cases of (1). Indeed, when the function τ does not depend on the second variable then for all t ∈ I0 we have E(t) = [α(τ(t)), β(τ(t))] in Definition 1. Therefore, if f is nondecreasing with respect to its third variable, then Definition 1 and the usual definition of lower and upper solutions are the same (we will use this fact in the proof of Theorem 2). If, in turn, f is nonincreasing with respect to its third variable, then Definition 1 coincides with the usual definition of coupled lower and upper solutions (see for example ).
In general, in the conditions of Theorem 1 we cannot expect problem (1) to have the extremal solutions in [α, β] (that is, the greatest and the least solutions in [α, β ]). This is justified by the following example.
Moreover, , so α and β are, respectively, a lower and an upper solution for (10), and then condition (H1) of Theorem 1 is fulfilled. As conditions (H2) and (H3) are also satisfied (take, for example, ψ ≡ 1) we deduce that problem (1) has maximal and minimal solutions in [α, β]. However we will show that this problem does not have the extremal solutions in [α, β].
The family x x (t) = λ cos t, t ∈ I0, with λ ∈ [-1,1], defines a set of solutions of problem (10) such that α ≤ x λ ≤ β for each λ ∈ [-1,1]. Notice that the zero solution is neither the least nor the greatest solution of (10) in [α, β]. Now let be an arbitrary solution of problem (10) and let us prove that is neither the least nor the greatest solution of (10) in [α, β]. First, if changes sign in I0 then cannot be an extremal solution of problem (10) because it cannot be compared with the solution x ≡ 0. If, on the other hand, in I0 then the differential equation yields a.e. on I0, which implies, along with the initial condition , that for all t ∈ I0. Reasoning in the same way, we can prove that in I0 implies . Hence, problem (10) does not have extremal solutions in [α, β].
The previous example notwithstanding, existence of extremal solutions for problem (1) between given lower and upper solutions can be proven under a few more assumptions. Specifically, we have the following extremality result.
If (11) satisfies all the conditions in Theorem 1 and, moreover, f is nondecreasing with respect to its third variable and Λ is nondecreasing in [α, β], then problem (11) has the extremal solutions in [α, β].
Proof. Theorem 1 guarantees that problem (11) has a nonempty set of solutions between α and β. We will show that this set of solutions is, in fact, a directed set, and then we can conclude that it has the extremal elements by virtue of [9, Theorem 1.2].
According to Remark 2, the lower solution α and the upper solution β satisfy, respectively, inequalities (7) and (8) and, conversely, if α and β satisfy (7) and (8) then they are lower and upper solutions in the sense of Definition 1.
We also have in I- because Λ is nondecreasing, so is a lower solution for problem (11). Theorem 1 ensures now that (11) has at least one solution .
Analogous arguments show that the set of solutions of (11) in [α, β] is downwards directed and, therefore, it is a directed set.
Next we show the applicability of Theorem 2.
Hence α and β are lower and upper solutions for problem (12) by virtue of Remark 2.
so Theorem 2 can be applied.
3 Construction of lower and upper solutions
where and .
In particular, problem (13) has maximal and minimal solutions between α and β, and this does not depend on the choice of τ.
So we deduce from (24) and (25) that α and β are lower and upper solutions for problem (13).
satisfies all the conditions in Proposition 1 for every compact interval I0. So the corresponding problem (13) has at least one solution for any choice of and .
We use now the ideas of Proposition 1 to construct lower and upper solutions for the general problem (1).
Then there exist such that α and β defined as in (17), (18) are lower and upper solutions for problem (1) with Λ = 0, and this does not depend on the choice of τ.
Therefore, α and β are lower and upper solutions for problem (1).
where γ ≥ 0, L > 0, and g is a nonnegative Carathéodory function.
so in particular conditions (26) and (27) hold. As conditions (28)-(33) also hold (see Example 4) we obtain that there exist such that α and β defined as in (17), (18) are lower and upper solutions for problem (34) for any choice of τ. In particular, if there exists ψ ∈ L1(I 0) such that for a.a. t ∈ I0 and all x ∈ [α(t), β(t)] we have g(t, x) ≤ ψ(t), then problem (34) has maximal and minimal solutions between α and β.
Remark 3 Notice that the lower and upper solutions obtained both in Propositions 1 and 2 satisfy a slightly stronger condition than the one required in Definition 1.
This study was partially supported by the FEDER and Ministerio de Edu-cación y Ciencia, Spain, project MTM2010-15314.
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