According to fractional calculus theory and Sadovskii's fixed point theorem, we establish sufficient conditions for controllability of the fractional integro-differential equation with state-dependent delay. An example is provided to illustrate the theory.

This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certain limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.

In this paper, we establish some composition theorems of $\mu $-pseudo almost automorphic functions via measure theory, then derive sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to fractional differential equations with Caputo derivatives.

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min \{ r, 1 \}$, respectively. We show that the order of convergence in the iterated modified projection method is $4$ if $ r = 0$ and is $ 2 r + 3$ if $ r \geq 1$. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples.

We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system of integral equations which do not involve the square of the integral operator. A multiscale collocation method is then applied to solve the system. A truncation strategy for the matrices appearing in the resulting discrete linear system is proposed to design a fast numerical solver for the system of integral equations. A quadrature method is used to compute the entries of the resulting matrices. We estimate the computational cost of the numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method for image restoration.