Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Biles, Daniel C.; Schechter, Eric

doi:10.1090/s0002-9939-00-05584-2

Cited by 13 publications

(7 citation statements)

References 29 publications

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“…We also find in [4] a proof which uses Perron's method, see [16], a refined version of Peano's own proof in [14]. It is fair to acknowledge that probably the best application of Perron's method to (1.1) in dimension n = 1 is due to Goodman [7], who even allowed f to be discontinuous with respect to the independent variable and whose approach has proven efficient in more general settings, see [1,2,8,12].…”

Section: Introductionmentioning

confidence: 80%

“…The reader is referred to [1,2,8,9,19,20] and references therein for more information on quasimonotone systems.…”

Section: A Finer Results In Dimension Onementioning

confidence: 99%

“…2. Theorem 4.1 and Corollary 4.3 can be extended to quasimonotone systems, see [1,2,9]. Theorem 4.1 for systems even works when the lower and upper solutions are not ordered, see [13], and f may be discontinuous or singular, see [1].…”

Section: The Power Of Lower and Upper Solutions: Existence For Nonloc...mentioning

confidence: 99%

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Peano's Existence Theorem revisited

Pouso

2012

Preprint

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We present new proofs to four versions of Peano's Existence Theorem for ordinary differential equations and systems. We hope to have gained readability with respect to other usual proofs. We also intend to highlight some ideas due to Peano which are still being used today but in specialized contexts: it appears that the lower and upper solutions method has one of its oldest roots in Peano's paper of 1886."Le dimostrazioni finora date dell'esistenza degli integrali delle equazioni differenziali lasciano a desiderare sotto l'aspetto della semplicità."

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Section: Introductionmentioning

confidence: 80%

“…The reader is referred to [1,2,8,9,19,20] and references therein for more information on quasimonotone systems.…”

Section: A Finer Results In Dimension Onementioning

confidence: 99%

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Peano's Existence Theorem revisited

Pouso

2012

Preprint

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“…A very interesting feature of (H 3 ) − (b) is that it allows downwards jump discontinuities with respect to x(t), something explicitly avoided just a few years ago in the literature on existence of Carathéodory solutions for discontinuous first-order differential equations, see [3], [5], [17], [18]. In particular, (H 3 ) − (b) does not imply continuity with respect to x(t) when combined with (H 6 ), so f and τ can be discontinuous with respect to t and to x(t) in Theorems 3.2 and 3.4.…”

Section: Discussion About the Assumptionsmentioning

confidence: 99%

Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments

Figueroa

Pouso

2011

Nonlinear Analysis: Theory, Methods & Applications

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We establish a coupled fixed points theorem for a meaningful class of mixed monotone multivalued operators and then we use it to derive some results on existence of quasisolutions and solutions to first-order functional differential equations with statedependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays.Primary classification number: 34K05. * Partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.Differential equations with deviating arguments are being intensively studied, and we can quote recent papers as [11], [13], [21]. This paper considers the general situation where deviations depend on the unknown, which is interesting from the viewpoints of both theory and applications. Indeed, the particular case of differential equations with state-dependent delays is receiving a lot of attention, see for instance [10], [19], [24], [25], [26]. We also refer readers to the survey by Hartung et al. [16] and references therein, for mathematical models which use state-dependent delays, and for basic and qualitative theory on this type of problems.Our starting point is the work by Jankowski in [20], who studied the problem( 1.2) for continuous f and τ and monotonicity with respect to spatial variables. In that paper, the author uses a monotone method in presence of lower and upper solutions in order to obtain the existence of extremal quasisolutions for problem (1.2) and then he applies a maximum principle to guarantee that, under certain Lipschitz conditions, those extremal quasisolutions are the same function, thus proving that problem (1.2) has a unique solution between the given lower and upper solutions. The author also assumes in [20] that f ≥ 0 on the compact subset of I + ×R delimited by the graphs of the lower and the upper solutions.It is the aim of the present work to state and prove a new result on coupled fixed points for multivalued operators and then apply it to (1.1), showing in this way that the existence results in [20] hold valid in more general settings, namely, 1. In the case τ (t, x(t), x) = τ (t, x(t)) the nonlinearity f has to be neither nonnegative nor monotone with respect to x(τ ) between assumed lower and upper solutions; 2. Delay differential equations are included in the framework of (1.1);3. Neither f nor τ need be continuous in any of their arguments for the existence of quasisolutions, and they can be discontinuous with respect to the t and the x(t) variables for the existence of solutions;4. The right-hand side in the differential equation features a nondecreasing functional dependence in its fourth variable which makes nonincreasingness with respect to the third variable be less stringent, as we show in Section 3.4. Moreover, it provides us with a unified framework for the study of many other types of functional differential eq...

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“…Although (ii) * is commonplace in the current literature of discontinuous differential equations (see [2][3][4]), it is not a completely satisfactory assumption. First, despite the fact that everyone agrees that measurability is quite a weak condition, it is easy to find elementary examples of solvable Cauchy problems satisfying (ii) and (iii) * , but not (ii) * (see [14]).…”

Section: Conditions For Superpositional Measurabilitymentioning

confidence: 99%