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

Abstract

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...

The Method of Coupled Fixed Points and Coupled Quasisolutions When Working with ODE’s with Arguments of Bounded Variation

“…Moreover, there exists a very extensive literature about the existence of extremal solutions for problem (P γ1,γ2 ) for discontinuous F γ1,γ2 . The reader is referred to [5], [6], [10], [11] and references therein for some results of this type. Notice that although most of these references deal with initial value problems, these results can easily be adapted for final value problems.…”

“…In the paper [5] we proved a new result on the existence of coupled fixed points for multivalued operators, and then we used it to guarantee the existence of coupled quasisolutions and solutions to a certain first-order ordinary differential equation with state-dependent delay. In that paper, the nonlinearity was allowed to have both nondecreasing and nonincreasing arguments and the existence of solutions was obtained under strong Lipschitz conditions.…”