Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses

This paper deals with the existence of multiple periodic solutions for n-dimensional functional differential equations with impulses. By employing the Krasnoselskii fixed point theorem, we obtain some easily verifiable sufficient criteria which extend previous results.     

A new existence theory for positive periodic solutions to functional differential equations with impulse effects

The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results.     

Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain,

where f:[t₁,σ(t₂)]×[0,∞)×R→[0,∞) is continuous and p:[t₁,σ(t₂)]→[0,∞) a nonnegative function that is allowed to vanish on some subintervals of [t₁,σ(t₂)] of the measure chain. The method involves applications of a new fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699–707]. The emphasis is put on the nonlinear term f involved with the first order delta derivative x^Δ(t).     

Impulsive boundary value problem for nonlinear differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for the boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on Altman’s fixed point theorem and Leray-Schauder’s fixed point theorem.     

Existence and uniqueness of mild solutions for abstract delay fractional differential equations

In this paper, by means of solution operator approach and contraction mapping theorem, the existence and uniqueness of mild solutions for a class of abstract delay fractional differential equations are obtained.     

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.     

Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions

This paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order α∈(1,2]. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.     

Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Green’s function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

Multiple positive solutions for singular BVPs on the positive half-line

In this work, we are concerned with the existence of multiple positive solutions to a second-order nonlinear singular boundary value problem set on the positive half-line. We mainly use the Krasnozels’ki and Leggett–Williams fixed point theorems in cones to prove existence of one positive solution, two positive solutions and three positive solutions. The results complement, extend and correct some recent ones.     

Positive solutions of operator equations on ordered Banach spaces and applications

This paper is concerned with an operator equation on ordered Banach spaces. The existence and uniqueness of its’ positive solutions is obtained by using the properties of cones and monotone iterative technique. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for two classes of integral equations.     

The weighted continuous galerkin scheme for ordinary differential equations

We introduce the Weighted Continuous Galerkin Scheme for initial value ordinary differential equations. This is an extension of the Continuous Galerkin Scheme, having an extra parameter for the purpose of error reduction. We prove convergence in the L ₂ norm in the time variable in a new way, similar to (elliptic) finite element techniques. Using the optimal L ₂ estimates, we then prove max norm convergence. Numerical evidence for the effectiveness of the proposed scheme is presented.     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

On sign-changing solutions for a second-order integral boundary value problem

In this paper, we obtain the existence of sign-changing solutions for nonlinear second-order differential equations with integral boundary value conditions, by applying a new fixed point theorem in ordered Banach spaces, with the lattice structure derived by Liu and Sun. Our results improve on those in the literature.     

Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions

This paper studies the existence of solutions for nonlinear fractional differential equations and inclusions of order q∈(3,4] with anti-periodic boundary conditions. In the case of inclusion problem, the existence results are established for convex as well as nonconvex multivalued maps. Our results are based on some fixed point theorems, Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type. Some illustrative examples are discussed.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.