Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions

We present new sufficient conditions for the existence of a continuous utility function for an arbitrary binary relation on a topological space. Such conditions are basically obtained by using both the concept of a weakly continuous binary relation on a topological space and the concept of a countable network weight. In particular, we are concerned with suitable topological notions which generalize the concept of compactness and do not imply second countability or local compactness.hereditarily Lindeloef space; weakly continuous binary relation; countable network weight; hemicompactness; submetrizability

Extremal Solutions for a Class of Functional Differential Equations

We study, in Carathéodory assumptions, existence, continuation and continuous dependence of extremal solutions for an abstract and rather general class of hereditary differential equations. By some examples we prove that, unlike the nonfunctional case, solved Cauchy problems for hereditary differential equations may not have local extremal solutions

Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions

We present new sufficient conditions for the existence of a continuous utility function for an arbitrary binary relation on a topological space. Such conditions are basically obtained by using both the concept of a weakly continuous binary relation on a topological space and the concept of a countable network weight. In particular, we are concerned with suitable topological notions which generalize the concept of compactness and do not imply second countability or local compactness

Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions

We present new sufficient conditions for the existence of a continuous utility function for an arbitrary binary relation on a topological space. Such conditions are basically obtained by using both the concept of a weakly continuous binary relation on a topological space and the concept of a countable network weight. In particular, we are concerned with suitable topological notions which generalize the concept of compactness and do not imply second countability or local compactness

Normally preordered spaces and continuous multi-utilities

[EN] We study regular, normal and perfectly normal preorders by referring to suitable assumptions concerning the preorder and the topology of the space. We also present conditions for the existence of a countable continuous multi-utility representation, hence a Richter-Peleg multi-utility representation, by assuming the existence of a countable net weight.This research was carried out within the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni, Istituto Nazionale di Alta Matematica (Italy).Bosi, G.; Caterino, A.; Ceppitelli, R. (2016). Normally preordered spaces and continuous multi-utilities. Applied General Topology. 17(1):71-81. doi:10.4995/agt.2016.4561.SWORD7181171Alcantud, J. C. R., Bosi, G., & Zuanon, M. (2015). Richter–Peleg multi-utility representations of preorders. Theory and Decision, 80(3), 443-450. doi:10.1007/s11238-015-9506-zAlcantud, J. C. R., Bosi, G., Campión, M. J., Candeal, J. C., Induráin, E., & Rodríguez-Palmero, C. (2007). Continuous Utility Functions Through Scales. Theory and Decision, 64(4), 479-494. doi:10.1007/s11238-007-9025-7Bosi, G., & Herden, G. (2012). Continuous multi-utility representations of preorders. Journal of Mathematical Economics, 48(4), 212-218. doi:10.1016/j.jmateco.2012.05.001Bosi, G., & Herden, G. (2016). On continuous multi-utility representations of semi-closed and closed preorders. Mathematical Social Sciences, 79, 20-29. doi:10.1016/j.mathsocsci.2015.10.006Bosi, G., & Isler, R. (2000). Separation axioms in topological preordered spaces and the existence of continuous order-preserving functions. Applied General Topology, 1(1), 93. doi:10.4995/agt.2000.3026Bosi, G., & Mehta, G. B. (2002). Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. Journal of Mathematical Economics, 38(3), 311-328. doi:10.1016/s0304-4068(02)00058-7Caterino, A., & Ceppitelli, R. (2015). Jointly continuous utility functions on submetrizable kω-spaces. Topology and its Applications, 190, 109-118. doi:10.1016/j.topol.2015.04.012Caterino, A., Ceppitelli, R., & Holá, Ľ. (2013). Some generalizations of Backʼs Theorem. Topology and its Applications, 160(18), 2386-2395. doi:10.1016/j.topol.2013.07.033Caterino, A., Ceppitelli, R., & Maccarino, F. (2009). Continuous utility functions on submetrizable hemicompact k-spaces. Applied General Topology, 10(2), 187-195. doi:10.4995/agt.2009.1732Evren, Ö., & Ok, E. A. (2011). On the multi-utility representation of preference relations. Journal of Mathematical Economics, 47(4-5), 554-563. doi:10.1016/j.jmateco.2011.07.003K�nzi, H.-P. A. (1990). Completely regular ordered spaces. Order, 7(3), 283-293. doi:10.1007/bf00418656Minguzzi, E. (2011). Normally Preordered Spaces and Utilities. Order, 30(1), 137-150. doi:10.1007/s11083-011-9230-

Error bounds for solutions of systems of quasilinear hyperbolic partial differential equations in bicharacteristic form

The purpose of this paper is to obtain a theoretical estimate of the effect of a nonlinear perturbation on the solution of a boundary value problem for quasilinear hyperbolic systems in bicharacteristic canonic form with the same boundary data which L. Cesari introduced in [9], [10]. The estimate is given by the difference between the solutions of the linear and the perturbated systems

Continuous utility functions on submetrizable hemicompact k-spaces

Some theorems concerning the existence of continuous utility functions for closed preorders on submetrizable hemicompact k-spaces are proved. These spaces are precisely the inductive limits of increasing sequences of metric compact subspaces and in general are neither metrizable nor locally compact. These results generalize some well known theorems due to Levin

Preference orders and continuous representations

In this paper we prove some general theorems on the existence of continuous order-preserving functions on topological spaces with a continuous preorder. We use the concepts of network and netweight to prove new continuous representation theorems and we establish our main results for topological spaces that are countable unions of subspaces. Some results in the literature on pathconnected, locally connected and separably connected spaces are shown to be consequences of the general theorems proved in the paper. Finally, we prove a continuous representation theorem for hereditarily separable spaces