Nonlinear Analysis: Modelling and Control
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Prescribed-time practical scaled consensus of multiagent systems via time-based generator approach
Full text linkIn this paper, we consider the leaderless and leader-following practical scaled consensus problems of multiagent systems (MASs). To achieve leaderless practical scaled consensus, a distributed control protocol is introduced that incorporates with an innovative time-based generator (TBG). Under this protocol, all agents achieve practical scaled consensus within the prescribed-time frame while providing a precise estimation of the practical error. To fulfill practical requirements, we devise two leader-following scaled consensus protocols for both directed detail-balanced graphs and general directed networks. Furthermore, a comprehensive analysis for the convergence of MASs is given by employing the Lyapunov stability theory. Finally, the effectiveness and feasibility of the proposed theoretical results are verified
Boundary value problems for third-order differential equations involving singular Phi-Laplacian operators
Full text linkWe study strongly nonlinear, third-order differential equations of type (Φ(k(t)u\u27\u27(t)))\u27 = f(t, u(t), u\u27(t), u\u27\u27(t)), a.e. t ∈ J, where Φ is the singular Φ-Laplacian operator. That is, Φ : (–r, r) -> R, r > 0, is a generic strictly increasing homeomorphism with bounded domain, which generalizes the relativistic operator Φ(u) := u (r2 – u2)–1/2. Moreover, k is a nonnegative continuous function, which can vanish on a set of zero measure, so such equations can be singular, and f is a general Carathédory function. For these equations, we investigate boundary value problems both in compact intervals (when J = [a; b]) and in a half-line (with J = [a;+∞)), and we prove existence results under mild assumptions. Our approach is based on fixed point techniques
A class of nonlinear double-phase Dirichlet fractional differential equations
Full text linkIn this paper, we study the existence of positive solutions for a new class of double-phase Dirichlet fractional differential equations with singular and superlinear terms. By applying the Nehari manifold method we show that for all small values of the parameter τ > 0, the considered equation has at least two positive solutions
Codimension-two bifurcation analysis of a discrete predator–prey system with fear effect and Allee effect
Full text linkIn this paper, we study the dynamic behavior of a discrete predator–prey model with fear effect and Allee effect by theoretical analysis and numerical simulation. Firstly, the existence and stability of the equilibrium points of the model are proved. Secondly, the existence of codimension-2 bifurcations (1 : 2, 1 : 3, and 1 : 4 strong resonances) in the case of two parameters is verified by bifurcation theory. In order to illustrate the complexity of the dynamic behavior of the model in the two-parameter space, we simulate the bifurcation diagrams, phase diagrams, maximum Lyapunov exponent diagrams, and isoperiodic diagram, and we verify the influence of model parameters on the population size
A discontinuous nonlinear singular elliptic problem with the fractional rho-Laplacian
Full text linkIn this paper, we use the topological degree method, based on the abstract Hammerstein equation, to investigate the existence of weak solutions for a certain class of elliptic Dirichlet boundary value problems. These problems involve the fractional ρ-Laplacian operator and involve discontinuous nonlinearities in the framework of fractional Sobolev spaces
A revisit to tail risk measures in the presence of bivariate regularly varying tailed insurance and financial risks
Full text linkConsider a discrete-time insurance risk model in which the one-period insurance and financial risks are assumed to be independent and identically distributed random pairs, but a strong dependence structure is allowed to exist between each pair. Recently, Q. Tang and Y. Yang employed a framework of bivariate regular variation to model the heavy tails and the dependence of the insurance and financial risks, and they also established an asymptotic formula for the finite-time ruin probability [Interplay of insurance and financial risks in a stochastic environment, Scand. Actuar. J., 2019(5):432–451, 2019]. In this paper, by adopting a different approach, we study the asymptotic behavior of some tail risk measures for the aggregate discounted net loss, including the tail probability and the conditional loss-based tail expectation. We show both analytically and numerically how the heavy tailedness and the dependence of each pair of insurance and financial risks affect the tail risk measures
Background risk model in presence of heavy tails under dependence
Full text linkIn this paper, we examine two problems on applied probability, which are directly connected with the dependence in presence of heavy tails. The first problem is related to max-sum equivalence of the randomly weighted sums in bivariate setup. Introducing a new dependence, called generalized tail asymptotic independence, we establish the bivariate max-sum equivalence under a rather general dependence structure when the primary random variables follow distributions from the intersection of the dominatedly varying and the long-tailed distributions. Based on this max-sum equivalence, we provide a result about the asymptotic behavior of two kinds of ruin probabilities over a finite-time horizon in a bivariate renewal risk model with constant interest rate. The second problem is related to the asymptotic behavior of the tail distortion risk measure in a static portfolio called background risk model. In opposite to other approaches on this topic, we use a general enough assumption that is based on multivariate regular variation
Eigenvalue problems for a k-Hessian-type equation
Full text linkIn this work, we focus on the eigenvalue problem for a class of k-Hessian-type equations. Under some suitable assumptions, we first determine the intervals of the parameter for the existence of nontrivial radial solutions. To this aim, we apply the eigenvalue theory and Jensen inequality. Finally, the behavior of the solutions with respect to the parameter is analyzed via Guo’s fixed point theorem
Positive solutions for a Hadamard-type fractional-order three-point boundary value problem on the half-line
Full text linkIn this paper, we study a Hadamard-type fractional-order three-point boundary value problem on the half-line. Under some growth conditions concerning the spectral radius of the relevant linear operator, the existence and multiplicity of positive solutions is obtained using a fixed-point method. Our results improve and generalize some results in the literature
On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals
No full textIn the present research, we investigate the existence and uniqueness of solutions for a system of (k; χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann–Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. Examples are also constructed to illustrate the obtained results