Integral Equations Of Fractional Order Research Articles

Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra

Abstract We present an existence theorem for at least one continuous solution for a nonlinear quadratic functional integral equation of fractional order. Also, a general quadratic integral of fractional order will be considered.

Existence and stability of nonlinear, fractional order Riemann-Liouville Volterra-Stieltjes multi-delay integral equations

We study the existence and the stability of solutions for Riemann-Liouville, Volterra-Stieltjes quadratic delay integral equations of fractional order. Our results are obtained by applying some fixed point theorems.

Picard and Adomian decomposition methods for a quadratic integral equation of fractional order

We study two analytical methods: the classical method of successive approximations (Picard method) (Curtain and Pritchard in Functional analysis in modern applied mathematics, Academic press, London, 1977) and Adomian method which gives the solution as a series (see Adomian in Stochastic system, Academic press, New York, 1983; Adomian in Nonlinear stochastic operator equations. Academic press, San Diego, 1986; Adomian in Nonlinear stochastic systems: theory and applications to physics. Kluwer, Dordrecht, 1989; Adomian et al. in J Math Comput 23:17–23, 1992; Abbaoui and Cherruault in Comput Math Appl 28:103–109, 1994; Adomian in Solving frontier problems of physics: the decomposition method. Kluwer, Dordrecht, 1995; Cherruault in Kybernetes 18:31–38, 1989; Cherruault et al. in Int J Biomed Comput 38:89–93, 1995). The existence and uniqueness of the solution and the convergence will be discussed for each method and some examples will be studied.

Numerical Solution of Constant-Q Fractional Order Integral Modeling

In this paper fractional order integral model is set up about quality factor. Complex rectangular and Chebyshev polynomial are utilized to approximate the fractional order integral equation. Moreover, plane wave is used to calculate quality factor-Q. The numerical results show that Chebyshev polynomial approximation is more suitable to describe integral modeling of quality factor.

On Attractivity and Positivity of Solutions for Functional Integral Equations of Fractional Order

We investigate a class of functional integral equations of fractional order given byx(t)=q(t)+f1(t,x(α1(t)),x(α2(t)))+(f2(t,x(β1(t)),x(β2(t)))/Γ(α))×∫0t(t−s)α−1f3(t,s,x(γ1(s)),x(γ2(s)))ds: sufficient conditions for the existence, global attractivity, and ultimate positivity of solutions of the equations are derived. The main tools include the techniques of measures of noncompactness and a recent measure theoretic fixed point theorem of Dhage. Our investigations are placed in the Banach space of continuous and bounded real-valued functions defined on unbounded intervals. Moreover, two examples are given to illustrate our results.

Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order

AbstractIn this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order with three parameters. Moreover, we also present the uniqueness and another existence results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained results.

Existence results for nonlinear quadratic functional integral equations of fractional orders

Existence results for nonlinear quadratic functional integral equations of fractional orders

Existence and attractivity for fractional order integral equations in Frechet spaces

In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of RiemannLiouville fractional order, by using an extension of the Burton-Kirk fixed point theorem in the case of a Frechet space.

Existence and stability for fractional order integral equations with multiple time delay in Fr´echet spaces

In this paper, we present some results concerning the existence of solutions for a system of integral equations of Riemann-Liouville fractional order with multiple time delay in Fr´ echet spaces, we use an extension of the Burton-Kirk fixed point theorem. Also we investigate the stability of solutions of this system.

Existence results for coupled systems of quadratic integral equations of fractional orders

We are concerning here with the existence of at least one positive continuous solution of coupled systems of quadratic integral equations of fractional orders. Then an existence theorem for a coupled system of Cauchy problems will be proved.

The technique of Volterra–Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders

Our study in this paper concerns the infinite systems of nonlinear integral equations of Volterra–Stieltjes type. We show that some assumptions imposed on terms of the mentioned systems guarantee the solvability of those systems in the space C(I,c0) with I=[0,1] and c0 being the classical sequence space. As a particular and very important case of the investigated systems of nonlinear Volterra–Stieltjes integral equations we obtain infinite systems of nonlinear integral equations of fractional orders. Thus, the approach applied in the paper enables us to obtain rather general results concerning the existence of solutions of infinite systems of nonlinear fractional integral equations.

A new approach to the theory of functional integral equations of fractional order

The aim of the paper is to present a new approach to the theory of functional integral equations of fractional order. That approach depends on converting of the mentioned equations to the form of functional integral equations of Volterra–Stieltjes type. It turns out that the study of functional integral equations of Volterra–Stieltjes type is more convenient and effective than the study of functional integral equations of fractional order. An example illustrating our approach is also discussed.

On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order

On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order

On local attractivity of solutions of a functional integral equation of fractional order with deviating arguments

In this paper, we study the existence of solutions of a nonlinear functional integral equation of fractional order with deviating arguments. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive. An example is provided to illustrate the theory.

Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity

A functional integral equation of fractional order is investigated in the space of real functions which are defined, continuous and bounded on the real half-axis. Using the technique of measures of noncompactness and the fixed point theorem of Darbo type we prove that the equation in question has solutions in the mentioned function space. Moreover, a suitable choice of a measure of noncompactness enables us to prove that those solutions tend to limits at infinity. An example illustrating our result is also given.

On existence and asymptotic behaviour of solutions of a fractional integral equation

We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions, defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.

On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order

We study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. The mentioned equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. We will show that solutions of the investigated integral equation are locally attractive.

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112–119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.