Solving Systems of Linear Integral Equations via Fixed Point Theory in Extended Parametric Sb‐Metric Spaces

IN THE PAPER, arbitrary 2-D BVP of termoelasticity in a wedge-shaped - layered region are reduced to special systems of singular integral equations with fixed point singularities. For this purpose, the Fourier and Mellin integral transforms of the solutions in the layered and wedge-shaped parts of the domain are fitted together along the common interface. This interface is characterized by the conditions of given discontinuities of displacements and tractions. The theory, developed by the author elsewhere, is applied to investigate the systems obtained. The results, concerning existence and properties of the solutions are presented depending on the exterior boundary conditions. The numerical method applied to solve the systems of equations is justified.

System of Volterra integral equations of the first kind: the system of Volterra linear integral equations of the first kind in the interval stability of solutions of Volterra integral equations of the second kind in the semiaxis the system of Volterra integral equations of the first kind in the semiaxis the system of Volterra linear integral equations of the first kind in the space of square-integrable functions. The system of Volterra linear integral equations of the first kind of the convolution type and commutative ring: the system of Volterra integral equations of the first kind of the convolution type commutative ring and a system of equations in it application to inverse problems. The system of Volterra integral equations of the first kind with two independent variables: the system of two-dimensional Volterra linear integral equations of the first kind the system of Volterra nonlinear two-dimensional integral equations of the first kind. Volterra operator equations: Volterra operator equations of the first kind in the scales of Hilbert spaces Volterra operator equations of the first kind in the scales of Banach spaces Volterra operator equations of the first kind with commuting kernels on uniqueness of solution of Volterra operator equations nonlinear Volterra operator equations of the first kind. Problems of integral geometry: the problem of integral geometry in the three dimensional band the problem of integral geometry in a two-dimensional band the problem of integral geometry with spatial curves the equations connected with problems of integral geometry.

In Chap. 6, on superlinear systems of Hammerstein integral equations and applications, we use the Leray–Schauder degree to obtain new results on the existence of solutions, and apply them to two-point boundary problems of systems of equations. We also are concerned with the existence of (component-wise) positive solutions for a semilinear elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone K 1×K 2, which is the Cartesian product of two cones in the space \(C(\overline{\Omega})\), and computing the fixed point index in K 1×K 2, we establish the existence of positive solutions for the system.KeywordsFixed Point IndexSemilinear Elliptic SystemsHammerstein Integral EquationsLeray-Schauder DegreeSublinearThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper diffraction of P-Waves in an infinite orthotropic strip containing two coplanar Griffith cracks is analyzed. Using Fourier transform the problem is reduced to a system of dual integral equation. Abel’s transform technique is used to further reduce the system of dual integral equation to a Fredholm integral equation of second kind and finally the integral equation is solved numerically to find the Stress Intensity Factor (SIF). Graphical representation has been made to show the effect of material orthotropy.

Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the "height" of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

The mathematical literature contains a good description of a scalar integral equation with a degenerate kernel (1), in which all of the written functions are scalar quantities. We are not aware of publications in which systems of integral equations of type (1) with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from a scalar case to a vector one. For example, in A.L. Kalashnikov’s text book “Methods for Approximate Solution of Integral Equations of the Second Kind,” systems of equations with degenerate kernels are briefly described, in which the products of scalar rather than matrix functions play the role of degenerate kernels. However, as the simplest examples show, the generalization of the ideas of the scalar case for the case of integral systems with kernels represented by the sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. In our opinion, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, has not been described previously. Bearing in mind the great capabilities of the theory of integral equations in applied problems, we considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in a multidimensional case and to implement this scheme in the Maple math software. It should be noted that only scalar integral equations are solved in Maple using the intsolve procedure. Since we could not find a similar procedure for solving systems of integral equations, the have developed our own procedure.

In this paper, we are concerned with existence of positive solutions for systems of nonlinear Hammerstein integral equations, in which one nonlinear term is superlinear and the other is sublinear. The discussion is based on the product formula of fixed point index on product cone and fixed point index theory in cones. As applications, we consider existence of positive solutions for systems of second-order ordinary differential equations with different boundary conditions.

This paper is concerned with the existence and uniqueness of positive solutions for a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution to the system are established. Moreover, an iterative scheme is constructed for approximating the solution. The case of quadratic system of fractional integral equations is also considered.

Mathematical modeling for many problems in different disciplines, such as engineering, chemistry, physics and biology leads to integral equation, or system of integral equations. It’s the reason of great interest for solving these equations. There are some analytical and numerical methods for solving Volterra integral equations, but extension of these methods to systems of such integral equations is not easy to employ. Adomian decomposition method, well address in [1,2] has been used to solved some of these systems such as systems of differential equations, systems of integral equations and even systems of integro-differential equation [3,4,5]. Applying this method needs some computations which is sometimes boring, having a program to do all computations would be interesting and helpful. In this article a maple program is prepared to solve a system of Volterra integral equations of the second kind, linear or non-linear.

Lagrangian coordinates in free boundary problems for parabolic equations

This paper presents some topics of the theory of infinite systems of differential and integral equations. Our considerations focus on showing the symmetries that can be encountered in the theory of nonlinear differential and integral equations from the viewpoint of initial conditions, such as the symmetry of the behaviour of solutions of differential equations with respect to initial conditions, the symmetry of the behaviour of solutions in +∞ and −∞ and some other essential properties of solutions of differential and integral equations. First of all, we describe the fundamental facts connected with the theory of infinite systems of both differential and integral equations. Particular attention is paid to the location of infinite systems of the mentioned equations in a suitable Banach space. Indeed, we define the spaces in question and describe the basic properties of those spaces. Next, we discuss conditions imposed on terms of equations of the considered infinite systems that guarantee the existence of solutions of those systems and allow us to obtain some essential information on those solutions. Moreover, after the description of the current state of investigations concerning the theory of infinite systems of differential and integral equations, we formulate a few open problems concerning the mentioned systems of equations.

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

This study aims to establish common fixed point theorems for a pair of compatible self-mappings within the framework of extended parametric Sb-metric spaces. To support our assertions, we provide corollaries and examples accompanied with graphical representations. Moreover, we leverage our principal outcome to guarantee the existence of a common solution to a system of integral equations.

There is considered a nonlinear integro-differential equation in partial derivatives of the fourth order, the existence and uniqueness of a local solution are proved, and the conditions for the existence of a local solution are determined. The proof is carried out using the method of an additional argument. Recently, this method has been used to reduce nonlinear differential equations in partial derivatives of high order into integral equations and systems of integral equations. This method was developed by Kyrgyz scientists and used to solve nonlinear differential equations in partial derivatives of the first order. Currently, the method of the additional argument is being developed for various new classes of nonlinear partial differential equations of higher order and systems of nonlinear partial differential equations. The nonlinear partial differential equation of the fourth order discussed in the article is presented in operator form, with the method of the additional argument applied consecutively several times. As a result, an integral equation is derived, and the existence and uniqueness of the solution to this integral equation are determined using the contraction mapping principle. The unknown function in the integral equation contains an additional argument, and by equating this additional argument to time, a local solution to the original initial value problem can be obtained. The results presented in the article can be applied in the study of solutions to other nonlinear differential and integro-differential equations of higher order.

Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane