On solutions of functional-integral equations of Urysohn type on an unbounded interval

In this paper we establish the existence of solutions of functional-integral and quadratic Urysohn integral on the interval . The technique of proving applied in this paper is based on the concept of measure of noncompactness and the fixed point theorem. Some new results are given.     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval

In this paper, we study the existence of mild solutions for initial value problems for semilinear Volterra integrodifferential equations in a Banach space. The arguments are based on the concept of measure of noncompactness in Fréchet space and the Tikhonov fixed point theorem.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Eigenfunctions for singular fully nonlinear equations in unbounded domains

In this paper, we prove the existence of a generalized eigenvalue and a corresponding eigenfunction for fully nonlinear elliptic operators singular or degenerate, homogeneous of degree 1+α, α > −1 in unbounded domains of IR ^N . The main tool will be Harnack’s inequality.     

Fractional order differential equations on an unbounded domain

We are concerned with the existence of bounded solutions of a boundary value problem on an unbounded domain for differential equations involving the Caputo fractional derivative. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Singular viscosity solutions to fully nonlinear elliptic equations

We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.     

Iterative methods for solving nonlinear equations with finitely many roots in an interval

In this paper we consider a nonlinear equation f(x)=0$f\left(x\right)=0$ having finitely many roots in a bounded interval. Based on the so-called numerical integration method [B.I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008) 691–699] without any initial guess, we propose iterative methods to obtain all the roots of the nonlinear equation. In the result, an algorithm to find all of the simple roots and multiple ones as well as the extrema of f(x)$f\left(x\right)$ is developed. Moreover, criteria for distinguishing zeros and extrema are included in the algorithm. Availability of the proposed method is demonstrated by some numerical examples.     

Note on the solvability of equations involving unbounded linear and quasibounded nonlinear operators

Under the assumption of a vanishing magnetic field (curl A = 0), a transformation of variables is exhibited which uncouples the Maxwell-Dirac equations. It is then shown that the Cauchy problem in two space dimensions has a unique solution for C^∞ data with compact support.