On the construction of a nonnegative solution for one class of Urysohn-type nonlinear integral equations on a semiaxis

We investigate one class of Urysohn-type nonlinear integral equations with noncompact operator. It is assumed that a Wiener–Hopf–Hankel-type linear integral operator is a local minorant for the initial Urysohn operator. We prove an alternative theorem on the existence of positive solutions and investigate the asymptotic behavior of the obtained solutions at infinity.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On some bessel-function integral equations

Summary Two propositions of Bremekamp in the field of integral equations are revisited. Their common base is shown to be an elegant reciprocal relation between a pair of integral equations of Volterra type. Note dedicated toDario Graffi on the occasion of his seventieth birthday Entrata in Redazione il 7 marzo 1975.     

Asymptotic behavior of some nonlinear Volterra integral equations

Many people have looked at nonlinear Volterra integral equations of convolution type because they arise in a natural way from heat radiation problems. In this paper the author looks at such problems for systems of equations and looks at conditions for asymptotically constant solutions. Some theorems for comparing solutions are also derived. The theorems are applied to several integral equations derived from heat radiation problems.     

Existence of solutions for some nonlinear integral equations

In this study, we present an existence of solutions for some nonlinear functional- integral equations which include many key integral and functional equations that appear in nonlinear analysis and its applications. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aims in Banach algebra.     

On the positive almost periodic type solutions for some nonlinear delay integral equations

In this paper, we discuss the existence of positive almost periodic type solutions for some nonlinear delay integral equations, by constructing a new fixed point theorem in the cone. Some known results are extended.     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].     

Positive solvability of systems of nonlinear Hammerstein integral equations

The paper deals with the existence of positive continuous solutions to systems of nonlinear Hammerstein integral equations. The main tool used in the proofs is fixed point index theory in a cone. The results obtained here are essentially different from existing ones in the literature.     

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

In recent years, many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations. In this paper, the first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including, KdV, Kaup–Boussinesq and Wu–Zhang systems, analytically. By means of this method, some exact solutions for these systems of equations are formally obtained. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.     

A note on blow-up solutions to some nonlinear Volterra integral equations

In this paper we investigate certain nonlinear Volterra integral equations with the power nonlinearity. The basic results provide a criterion involving the kernel and the nonlinearity for the existence of the nontrivial and blow-up solution.     

On the approximation of some nonlinear equations

We approximate the solutions of a polynomial equation in a Banach space using the solutions of finite rank polynomial equations. Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Symmetry of solutions to some systems of integral equations

In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.

     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.