Singular Differential Systems Research Articles

Asymptotic Stability of Caputo Fractional Singular Differential Systems with Multiple Delays

Asymptotic Stability of Caputo Fractional Singular Differential Systems with Multiple Delays

Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters

By using the method of reducing the order of a derivative, the higher-order fractional differential equation is transformed into the lower-order fractional differential equation and combined with the mixed monotone operator, a unique positive solution is obtained in this paper for a singular p-Laplacian boundary value system with the Riemann–Stieltjes integral boundary conditions. This equation system is very wide because there are many parameters, which can be changeable in the equation system in this paper, and the nonlinearity is allowed to be singular in regard to not only the time variable but also the space variable. Moreover, the unique positive solution that we obtained in this paper is dependent on λ, and an iterative sequence and convergence rate are given, which are important for practical application. An example is given to demonstrate the application of our main results.

The rapid convergence for nonlinear singular differential systems with "maxima''

The rapid convergence for nonlinear singular differential systems with "maxima''

Bifurcation analysis for a singular differential system with two parameters via to topological degree theory

Based on the relation between Leray–Schauder degree and a pair of strict lower and upper solutions, we focus on the bifurcation analysis for a singular differential system with two parameters, explicit bifurcation points for relative parameters are obtained by using the property of solution for the akin systems and topological degree theory.

Multiplicity Results for Positive Solutions to Differential Systems of Singular Coupled Integral Boundary Value Problems

By constructing a special cone and using a fixed-point theorem in cone, this paper investigates the existence of multiple solutions of coupled integral boundary value problems for a nonlinear singular differential system.

Existence of positive solutions for a singular semipositone differential system with nonlocal boundary conditions

In this paper we consider the existence of at least one positive solution to a class of singular semipositone coupled system of nonlocal boundary value problems. We show that the system possesses at least one positive solution by using fixed point index theory. We remark that to some extent our systems and results generalize and extend some previous works.

Existence of Solutions of a New Class of Impulsive Initial Value Problems of Singular Nonlinear Fractional Differential Systems

Abstract: Sufficient conditions are given for the existence of solutions of impulsive boundary value problems for singular nonlinear fractional differential systems. We allow the nonlinearities p ( t ) f ( t , y ) $$p(t)f(t,y)$$ and q ( t ) g ( t , x ) $$q(t)g(t,x)$$ in fractional differential equations to be singular at t = 0 $$t\!=\!0$$ . Both f $$f$$ and g $$g$$ may be super-linear and sub-linear. The analysis relies on some well-known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. A conclusion section is given at the end of the paper.

Spectral Analysis for a Singular Differential System with Integral Boundary Conditions

In this paper, by constructing a cone K1 × K2 in the Cartesian product space C[0, 1] × C[0, 1], and using spectral analysis of the relevant linear operator for the corresponding differential system, some properties of the first eigenvalue corresponding to the relevant linear operator are obtained, and the fixed-point index of nonlinear operator in the K1 × K2 is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.

Interval state estimation for singular differential equation systems with delays

Interval state estimation for singular differential equation systems with delays

Variational properties of the solutions of singular second-order differential equations and systems

This paper is devoted to the existence and variational characterization of the weak solutions of the Dirichlet boundary value problem for singular second-order ordinary differential equations and systems. The solution appears as a minimizer of the energy functional associated with the equation, and in the case of systems, as a Nash-type equilibrium of the set of energy functionals. The results are connected with the recent abstract fixed point theory due to the second author and with its application, given by the first author, to semilinear operator problems of Michlin type.

Existence of unbounded positive solutions of boundary value problems for differential systems on whole lines

This paper is concerned with integral type boundary value problems of second order singular differential systems with quasi-Laplacian operators on whole lines. A Banach space and a nonlinear completely continuous operator are defined. By using the Banach space and the nonlinear operator, together with the Schauder?s fixed point theorem, sufficient conditions to guarantee the existence of at least one unbounded positive solution are established. Finally, we present a concrete example to illustrate the efficiency of the main theorem.

Solvability of a boundary value problem for singular multi-term fractional differential system with impulse effects

In this article, we first of all convert the boundary value problems for impulsive fractional differential equations to integral equations. Second, we construct a weighted function space and prove the completely continuous property of a nonlinear operator. Finally, we establish existence results for solutions of a boundary value problem for a nonlinear impulsive fractional differential system. Our analysis relies on the well-known Schauder fixed point theorem. An example is given to illustrate main results.

A periodic boundary value problem for nonlinear singular differential systems with ‘maxima’

In this paper, by using the method of quasilinearization to discuss the periodic boundary value problem for a nonlinear singular differential system with ‘maxima’, we obtain monotone iterative sequences of approximate solutions which converge uniformly and quadratically to the solution of the nonlinear singular differential system with ‘maxima’.

Periodic solutions for second order singular differential systems with parameters

In this paper we consider the existence of periodic solutions of one-parameter and two-parameter families of second order singular differential equations.

Existence of unbounded solutions of boundary value problems for singular differential systems on whole line

Motivated by (Kyoung and Yong-Hoon in Sci. China Math. 53:967-984, 2010) and (Chen and Zhang in Sci. China Math. 54:959-972, 2011), this paper is concerned with a boundary value problem of singular second-order differential systems with quasi-Laplacian operators on the whole line. By constructing a completely continuous nonlinear operator and using a fixed point theorem, sufficient conditions guaranteeing the existence of at least one unbounded solution are established. The methods used are standard, however, their exposition in the framework of such a kind of problems is new and skillful. Three concrete examples are given to illustrate the main theorem.

The Applications of Algebraic Methods on Stable Analysis for General Differential Dynamical Systems with Multidelays

The distribution of purely imaginary eigenvalues and stabilities of generally singular or neutral differential dynamical systems with multidelays are discussed. Choosing delays as parameters, firstly with commensurate case, we find new algebraic criteria to determine the distribution of purely imaginary eigenvalues by using matrix pencil, linear operator, matrix polynomial eigenvalues problem, and the Kronecker product. Additionally, we get practical checkable conditions to verdict the asymptotic stability and Hopf bifurcation of differential dynamical systems. At last, with more general case, the incommensurate, we mainly study critical delays when the system appears purely imaginary eigenvalue.

Symmetric positive solutions for second-order singular differential systems with multi-point coupled integral boundary conditions

Symmetric positive solutions for second-order singular differential systems with multi-point coupled integral boundary conditions

Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions

In this paper, we investigate the four-point coupled boundary value problem of nonlinear semipositone Hadamard fractional differential equations $$\begin{aligned} D^\alpha u(t)\,+\,\lambda f(t,u(t),v(t))\!=\!0, \quad D^\beta v(t)\,+\,\lambda g(t,u(t),v(t))\!=\!0, \quad t\!\in \!(1,e),\quad \!\!\lambda \!>\!0,\\ u^{(j)}(1)\!=\!v^{(j)}(1)\!=\!0,\quad 0\!\le \! j\le n-2, \quad u(e)\!=\!av(\xi ),\quad v(e)\!=\!bu(\eta ), \quad \xi ,\eta \in (1,e), \end{aligned}$$ where \(\lambda ,a,b\) are three parameters with \(0<ab(\log \eta )^{\alpha -1}(\log \xi )^{\beta -1}<1\), \(\alpha ,\beta \in (n-1,n]\) are two real numbers and \(n\ge 3\), \(D^\alpha , D^\beta \) are the Hadamard fractional derivative of fractional order, and \(f,g\) are continuous and may be singular at \(t=0\) and \(t=1\). We firstly give the corresponding Green’s function for the boundary value problem and some of its properties. Moreover, by applying Guo-Krasnoselskii fixed point theorems, we derive an interval of \(\lambda \) such that any \(\lambda \) lying in this interval, the singular boundary value problem has at least one positive solution. As applications, two interesting examples are presented to illustrate the main results.

Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters

Abstract In this paper, we study the existence of a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. By using the properties of the Green’s function and the Guo-Krasnosel’skii fixed point theorem, we obtain some existence results of positive solutions under some conditions concerning the nonlinear functions. The method of this paper is a unified method for establishing the existence of positive solutions for a large number of nonlinear differential equations with coupled boundary conditions. In the end, examples are given to demonstrate the validity of our main results. MSC:34B16, 34B18.

Addendum to ‘(Singular) state models and (singular) LTID systems’

In this note, a construction of state representations of singular linear time-invariant differential systems is described directly in terms of trajectories.