scholarly journals Equilibria and Systemic Risk in Saturated Networks

2021 ◽  
Author(s):  
Leonardo Massai ◽  
Giacomo Como ◽  
Fabio Fagnani
Keyword(s):  
Systemic Risk ◽  
Nash Equilibria ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Linear Functions ◽  
Shock Propagation ◽  
Fundamental Study ◽  
Network Games ◽  
Bifurcation Phenomenon

We undertake a fundamental study of network equilibria modeled as solutions of fixed-point equations for monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations. It is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. This model also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness, encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals Blow-Up Analysis for the Periodic Two-Component μ-Hunter-Saxton System

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yunxi Guo ◽  
Tingjian Xiong
Keyword(s):  
Transport Equation ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Blow Up Analysis ◽  
Classic Method ◽  
Two Component ◽  
Localization Analysis ◽  
Spatially Periodic ◽  
Periodic Setting

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.

scholarly journals Wave Breaking Phenomenon for DGH Equation with Strong Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhengguang Guo ◽  
Min Zhao
Keyword(s):  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Dissipative Term

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.

Complete blow-up for quasilinear degenerate parabolic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 877-908 ◽  
Author(s):  
R. Suzuki
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Image Position

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

scholarly journals Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lingling Zhang ◽  
Hui Wang
Keyword(s):  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Gradient Term ◽  
Nonlinear Parabolic Problems ◽  
Nonlinear Parabolic ◽  
Robin Boundary ◽  
Upper Estimate

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:(b(u))t=∇·(h(t)k(x)a(u)∇u)+f(x,u,|∇u|2,t), inD×(0,T),(∂u/∂n)+γu=0, on∂D×(0,T),u(x,0)=u0(x)>0, inD¯, whereD⊂RN  (N≥2)is a bounded domain with smooth boundary∂D. Under some appropriate assumption on the functionsf,h,k,b, andaand initial valueu0, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

CONTINUOUS PIECEWISE LINEAR FUNCTIONS

2005 ◽  
Vol 10 (1) ◽  
pp. 77-99 ◽  
Author(s):  
CHARALAMBOS D. ALIPRANTIS ◽  
DAVID HARRIS ◽  
RABEE TOURKY
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Linear Functions ◽  
Dimensional Euclidean Space ◽  
Piecewise Linear Functions ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Mathematical Background ◽  
Linear Regressions ◽  
Special Case

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

scholarly journals Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Coupled System ◽  
Initial Energy ◽  
Evolution System ◽  
High Energies ◽  
Second Order In Time

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

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