Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

A nonlocal multi-point singular fractional integro-differential problem of Lane–Emden type

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane–Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

Symmetry of integral equation systems on bounded domains

In this paper, we investigate the symmetry of domains and solutions of integral equation systems on bounded domains. Under some natural integrability conditions, we prove that the domains are balls, all positive solutions of systems are radially symmetric and monotone decreasing with respect to the radius.     

广义KdV方程与广义Burgers方程的精确解

利用试探函数法和直接积分法构造广义KdV方程与广义Burgers方程的新的精确解.     

用试探方程法求Jaulent-Miodek方程的新的精确行波解

利用试探方程法将Jaulent-Miodek方程约化为初等积分的形式,进而求出了该方程的精确行波解,其中包括椭圆函数双周期解和有理函数解等新解.     

The Method of Moving Planes for Integral Equation in an Extremal Case

In this paper, we study the symmetry results and monotonicity of solutions for an integral equation $$u(x)=-c_N∫_{\mathbb{R}^N}e^{u(y)}log|x-y|dy$$ in an external case.     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

L_（loc）~p-solutions of Nonlinear Impulsive Volterra Integral Equation in Abstract Space

In this paper,we discuss Llocp-solutions of a kind of nonlinear impulsive Volterra integral equation and present an existence theorem of solutions in Banach space.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

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

The existence of solutions of a nonlinear quadratic Volterra integral equation is studied. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. Such an approach allows us to obtain a result on the existence of solutions of an equation in question which are uniformly locally attractive or asymptotically stable.     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

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.     

Qualitative properties of solutions of an integral equation associated with the Benjamin–Ono–Zakharov–Kuznetsov operator

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions.     

Integral boundary value problems for first order impulsive integro-differential equations of mixed type

This paper considers existence of solutions for a class of first order impulsive differential equation with integral boundary value conditions. We present a new comparison theorem and show that the monotone iterative technique coupled with lower and upper solutions is still valid. The results we obtained generalize and improve many recent results.     

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.     

POSITIVE WEIGHTED PSEUDO-ALMOST PERIODIC SOLUTIONS TO INFINITE DELAY INTEGRAL EQUATION

This paper studies the existence of almost periodic type solutions to the nonlinear infinite delay integral equation. By means of the fixed theorem on a suitable space with Hilbert’s projective metric, we establish some sufficient conditions for the existence of positive weighted pseudo-almost periodic solutions to the equation.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.