Continuums of solutions for integral inclusions on the half line

In this paper we show that the solution set of certain Volterra inclusions on the half line is a continuum.     

урысон非线性积分方程的正解

分别在k(x,y,u)具有关于u的非线性控制和关于u的变系数线性上控之下，研究урысон非线性积分方程的正解和非负解。     

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.     

On monotonic solutions of an integral equation of Volterra type

Using a technique associated with measures of noncompactness we prove the existence of nondecreasing solutions to integral equations of Volterra type in C[0,1].     

Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space

We consider positive solutions of $\varDelta u=0$ in $\mathbf{R}_+^n$ , $\partial _{\nu }u=u^q$ on $\partial \mathbf{R}_+^n$ , where $n\ge 3$ and $q>n/(n-2)$ . We investigate the qualitative property of positive $x_n$ -axial symmetric solutions. In particular, we are concerned with the asymptotic expansion and the intersection property of positive $x_n$ -axial symmetric solutions.     

The existence of positive almost periodic type solutions for some neutral nonlinear integral equation

This paper presents the conditions of existence of positive almost periodic type solutions for some neutral nonlinear integral equation, by using a fixed point theorem in the cone. Some known results are extended.     

Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

On solvability of one integral equation on half line with Chebyshev polynomial nonlinearity

We consider an integral equation on half-line with Chebyshev polynomial nonlinearity, arising in dynamic theory of universe and p-adic string theory.We prove existence of the positive and monotonically increasing continuous solution in class of essentially bounded functions on half-line. We also found two sided estimates for obtained solution, as well as the limit of solution at infinity (Theorem 2.1). We prove uniqueness of a solution in the certain class of functions (Theorem 2.2). We generalize the results for more general integral equation with “double” nonlinearity (Theorem 2.3). At the end we give some examples of functions, describing nonlinearity. Using suggested constructive solution method, we present some results of numerical calculations, having direct application in cosmology.     

On monotonic solutions of an integral equation of Volterra type with supremum

Using a technique associated with measures of noncompactness, we prove the existence of nondecreasing solutions of an integral equation of Volterra type in C[0,1].     

Classification of solutions for an integral equation

Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x₀ ? ?ⁿ. This solves an open problem posed by Lieb 12 . The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well‐known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.     

On the numerical solutions of integral equation of mixed type

Sufficient conditions for oscillation of all solutions of a class of second-order quasilinear delay differential equations with fixed moments of impulse effect are found.