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In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at t = 0 and an integral boundary condition at t = 1 . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result. [ABSTRACT FROM AUTHOR]

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertex-deleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes.

Let T be a linear tree, and let S (T) denote the set of real symmetric matrices whose graph is T. If U (T) is the minimum number of eigenvalues with multiplicity 1 among matrices in S (T) and T ′ is a linear tree resulting from the addition of one vertex to T , we show that | U (T ′) − U (T) | ⩽ 1. We also determine the exact set of possible values of U (T) − U (T ′) , depending upon the manner in which the vertex is added to T to get T ′. These results are then used to give a new bound for U (T) , the diameter bound, and to improve an existing bound, 2 + D 2 (T). [ABSTRACT FROM AUTHOR]

We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for singular kernels. The first part of the paper presents results concerning the existence of a principal eigenfunction. Then we present various necessary and/or sufficient conditions for the maximum principle to hold, and use these to characterize the simplicity of the principal eigenvalue. [ABSTRACT FROM AUTHOR]

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of static bifurcation theory for a class of degenerate boundary value problems for semilinear elliptic differential operators of second-order which includes as particular cases the Dirichlet and Robin problems. The purpose of this paper is to generalize some results of Szulkin (Nonlinear Anal TMA 6:95–116, 1982) to the degenerate case. We study the behavior of nontrivial solution branches for bounded nonlinear terms, and prove that the branches turn back towards the first eigenvalue of the linearized degenerate boundary value problem. Our proof is based on global static bifurcation theory for positive mappings in ordered Banach spaces, with the special emphasis on the Semenov approximation in Chemistry. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the L p theory of pseudo-differential operators. [ABSTRACT FROM AUTHOR]

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The second one consists in the deformation of the boundary of the domain itself, and requires more technical properties of the bilinear form of the variational problem. [ABSTRACT FROM AUTHOR]

A large interval of random perturbation is presented in this study. Under the derived interval, the probability that the simple eigenvalue of original symmetric matrix is still simple is not less than a given confidence level. Numerical examples are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed two-species predator-prey model with general functional response under the homogeneous Neumann boundary condition. We discuss the stability of the trivial and semi-trivial solutions and obtain the spatially nonhomogeneous bifurcation solutions stemming from the semi-trivial trivial solutions (θ a , 0) and (0 , θ b). Besides, the stability and some results of Hopf bifurcation at the spatially nonhomogeneous bifurcation steady-state solutions are investigated by analyzing the distribution of the eigenvalues. The method we applied here is mainly based on spectral analysis, comparison principle, Lyapunov-Schmidt reduction, and bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper we consider a spectral problem for ordinary differential equations of fourth order with spectral parameter in the boundary conditions. This problem describes the bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, on the right end of which a mass is concentrated and on the left end a tracking force acts. We investigate the location of eigenvalues on the real axis, we study the structure of root spaces and oscillation properties of eigenfunctions and we obtain sufficient conditions for the subsystems of root functions of this problem to form a basis in L p , 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

We investigate the simplicity of the lowest eigenvalue for an eigenvalue problem involving the Finsler p-Laplace operator and a nonlocal term on a bounded domain subject to the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

The simplified Jacobi-Davidson (JD) method is a variant of the JD method without subspace acceleration. If the correction equation is solved approximately, the inexact simplified JD method is obtained. In this paper, we present a new convergence analysis of the inexact simplified JD method. The analysis applies to polynomial eigenvalue problems with simple eigenpairs. We first establish a relationship between the solution of the correction equation and the residual of the approximate eigenpair. From this relationship, we find the difference of two adjacent approximate eigenvalues bounded in terms of the residual norm of the approximate eigenpair. Then we prove the convergence of the inexact simplified JD method in terms of the residual norm of the approximate eigenpair. Depending on how accurately we solve the correction equation, the convergence rate of the inexact simplified JD may take several different forms. Numerical experiments confirm the convergence analysis. [ABSTRACT FROM AUTHOR]

In this paper, by transforming the symmetric random perturbed matrix into a special form and using the result from Li and Li (2005) [7] , we first present an improved random perturbation interval. Under the random perturbation in the interval, the simple eigenvalue of the original matrix is still simple. Then, we extend the result to the case of several eigenvalues. Finally, numerical experiments are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This note deals with a problem on approximation of a matrix tuple by a finite family of diagonalizable matrices with simple eigenvalues. In addition, for a given tuple of matrix functions, it is required that the product of their values at those diagonalizable matrices has a simple spectrum. We solve this problem making use of topological properties of the full matrix algebra. [ABSTRACT FROM AUTHOR]