## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (191.996 recursos)

Journal of Integral Equations and Applications

2. #### Essential norm of a Volterra-type integral operator from Hardy spaces to some analytic function spaces

Zhou, Jizhen; Zhu, Xiangling
In this paper, we obtain some estimates of essential norm of the Volterra-type integral operator $T_g$, where $T_gf(z)=\int ^z_0f(\zeta )g'(\zeta )\,d\zeta ,$ from Hardy spaces to the BMOA space, Besov spaces, Berg\-man spaces and Bloch-type spaces.

3. #### Existence of solutions and controllability for impulsive fractional order damped systems

Liu, Zhenhai; Li, Xuemei
In this paper, we are concerned with the controllability of linear and nonlinear Caputo impulsive fractional order damped systems in Banach spaces. Our main purpose is to establish some necessary and sufficient conditions for controllability for this kind of impulsive control system by using Mittag-Leffler matrix functions and the Schauder fixed point theorem.

4. #### Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs

Goodrich, Christopher S.
We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation $y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )$ $\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,$ where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign...

5. #### Approximation of solutions to a delay equation with a random forcing term and non local conditions

Chaudhary, Renu; Pandey, Dwijendra N.
The existence and approximation of a solution to a delay equation with a random forcing term and non local conditions is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, the convergence of Faedo-Galerkin approximations of the solution is shown. An example is given which illustrates the results.

6. #### On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions

Bensidhoum, Fatima-Zahra; Dib, Hacen
The present work deals with some spectral properties of the problem ¶ \medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$, ¶ \vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip ¶ \noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda$ is a scalar parameter. ¶ First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the...

7. #### Application of measure of noncompactness to Volterra equations of convolution type

Alvarez, Edgardo; Lizama, Carlos
Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.

8. #### Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

Domínguez, Víctor; Lyon, Mark; Turc, Catalin
We present a comparison among the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously...

9. #### $C^\sigma ,\alpha$ estimates for concave, non-local parabolic equations with critical drift

Lara, Héctor Chang; Dávila, Gonzalo
Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a $C^{\sigma +\alpha }$ estimate in the spatial variable and $C^{1,\alpha }$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.

10. #### Boundary integral operator for the fractional Laplacian on the boundary of a bounded smooth domain

Chang, Tongkeun
We introduce the boundary integral operator induced from the fractional Laplace equation on the boundary of a bounded smooth domain. For~$\frac 12\lt \alpha \lt 1$, we show the bijectivity of the boundary integral operator~$S_{2\alpha }:L^p(\partial \Omega )\to H^{2\alpha -1}_p(\partial \Omega )$ for $1 \lt p \lt \infty$. As an application, we demonstrate the existence of the solution of the Dirichlet boundary value problem of the fractional Laplace equation.

11. #### Numerical methods for systems of nonlinear integro-parabolic equations of Volterra type

Boglaev, Igor
This paper deals with the numerical solution of systems of nonlinear integro-parabolic problems of Volterra type. The numerical approach is based on the method of upper and lower solutions. A monotone iterative method is constructed. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Construction of initial upper and lower solutions is discussed. Numerical experiments are presented.

12. #### A collocation method solving integral equation models for image restoration

Liu, Yuzhen; Shen, Lixin; Xu, Yuesheng; Yang, Hongqi
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...

13. #### Approximate solution of Urysohn integral equations with non-smooth kernels

Kulkarni, Rekha P.; Nidhin, T.J.
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 $r + 2 + \min... 14. #### Compositions of pseudo almost automorphic functions via measure theory and applications Fan, Zhenbin; Dong, Qixiang; Li, Gang In this paper, we establish some composition theorems of$\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. 15. #### Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering Bonnet, Marc 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... 16. #### Controllability of fractional integrodifferential equations with state-dependent delay Aissani, Khalida; Benchohra, Mouffak 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. 17. #### A mode III interface crack with surface strain gradient elasticity Wang, Xu; Schiavone, Peter We study the contribution of surface strain gradient elasticity to the anti-plane deformations of an elastically isotropic bimaterial containing a mode~III interface crack. The surface strain gradient elasticity is incorporated using an enriched version of the continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete semi-analytic solution valid everywhere in the solid (including at the crack tips) by reducing the boundary value problem to two coupled hyper-singular integro-differential equations which are solved numerically using Chebyshev polynomials and a collocation method. Our solution demonstrates that the presence of surface strain gradient elasticity on the crack faces leads to bounded... 18. #### The direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder Gintides, Drossos; Mindrinos, Leonidas In this paper we consider the direct scattering problem of obliquely incident time-harmonic electromagnetic plane waves by an infinitely long dielectric cylinder. We assume that the cylinder and the outer medium are homogeneous and isotropic. From the symmetry of the problem, Maxwell's equations are reduced to a system of two 2D Helmholtz equations in the cylinder and two 2D Helmholtz equations in the exterior domain where the fields are coupled on the boundary. We prove uniqueness and existence of this differential system by formulating an equivalent system of integral equations using the direct method. We transform this system into a... 19. #### On a nonlinear abstract Volterra equation Emmrich, Etienne; Vallet, Guy Existence of solutions is shown for equations of the type$Av + B( KGv,v) = f$, where$A$,$B$and$G$are possibly nonlinear operators acting on a Banach space$V$, and$K$is a Volterra operator of convolution type. The proof relies on the convergence of a suitable time discretization scheme. 20. #### Probabilistic regularization of Fredholm integral equations of the first kind Micheli, Enrico De; Viano, Giovanni Alberto The main purpose of this paper is to focus on various issues inherent to the regularization theory of Fredholm integral equations of the first kind. Particular attention is devoted to the probabilistic approach to regularization, and a regularizing algorithm based on statistical methods is then proposed and tested on examples. The information theory approach is studied from two different viewpoints: the first approach is the standard one based on probability theory; the second one is formulated, in analogy with communication theory, in terms of the$\varepsilon \$-capacity in the sense elaborated by Kolmogorov and his school. The classical problem of...

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