Mixed Monotone Iterative Technique and Boundary Value Problem for a Class of Impulsive Integro-Differential Equations

In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.

MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF LINE

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SUPERLINEAR INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF LINE

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

INTEGRAL BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

By developing a comparison result and using the monotone iterative technique, we obtain the existence of the minimal and the maximal solutions to an integral boundary value problem for first order impulsive integro-differential equations.

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).

PERIODIC BOUNDARY VALUE PROBLEMS FOR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES

This paper investigates periodic boundary value problem for first order nonlinear impulsive integro-differelltial equations of mixed type in a Banach space. By establishing a comparison result, criteria on the existence of maximal and minimal solutions are obtained.

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.

Initial Value Problems for Second Order Impulsive Integro-Differential Equation in Banach Spaces

By an established comparison result and using the upper and lower solutions,one sufficient condition of existence of minimal and maximal solutions to initial value problem for second order impulsive integro-differential equation in Banach spaces is obtained and the related results are essentially improved.At the same time, another sufficient condition of existence of minimal and maximal solutions based on the Kuratowski measure of noncompactness is given.

Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

ON BOUNDEDNESS OF SOLUTIONS OFIMPULSIVE INTEGRO-DIFFERENTIAL SYSTEMSWITH FIXED MOMENTS OF IMPULSE EFFECTS

Several boundedness criteria for the impulsive integro-differential systems with fixed moments of impulse effects are established, employing the method of Lyapunov functions and Razumikhin technique.

Oscillation Theorem of Systems of Quasilinear Impulsive Delay Hyperbolic Equations

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

Solving Fractional Integro-Differential Equations by Using Sumudu Transform Method and Hermite Spectral Collocation Method

In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

CONTINUOUS DEPENDENCE OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS ON THE INITIAL CONDITIONS AND BARRIER CURVES

The basic objects of investigation in this article are nonlinear impulsive dif- ferential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier curves. For such type of equations, suf- ficient conditions are found under which the solutions are continuously dependent on the perturbations with respect to the initial conditions and barrier curves. The results are applied to a mathematical model of population dynamics.

STABILITY RESULTS OF RANDOM IMPULSIVE SEMILINEAR DIFFERENTIAL EQUATIONS

In this paper, we study the existence, uniqueness, continuous dependence, Ulam stabilities and exponential stability of random impulsive semilineax differential equations under sufficient condition. The results are obtained by using the contraction mapping principle. Finally an example is given to illustrate the applications of the abstract results.

OSCILLATION CRITERIA OF NEUTRAL TYPE IMPULSIVE HYPERBOLIC EQUATIONS

In this paper, oscillatory properties of all solutions for neutral type impulsive hyperbolic equations with several delays under the Robin boundary condition are investigated and several new sufficient conditions for oscillation are presented.

ON MONOTONE ITERATIVE METHOD FOR INITIAL VALUE PROBLEMS OF NONLINEAR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.

Asymptotic stability for impulsive functional differential equations

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

NON-INSTANTANEOUS IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAY AND PRACTICAL STABILITY

Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay,depending on both the time and the state variable.The case when the lower limit of the Caputo fractional derivative is fixed at the initial time,and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied.Practical stability properties,based on the modified Razumikhin method are investigated.Several examples are given in this paper to illustrate the results.