In this article, we investigate the solution of a new class absolute value equation (NCAVE) A1x ? |B1x c| = d. Based on spectral radius condition, singular condition and row column W-property, some necessary sufficient conditions for unique solvability NCAVE are gained. Some results generalized (NGAVE) A1x?|B1x| d also obtained.

The unique solvability issues of the Cauchy problem with a fractional derivative is considered in case when coefficient at desired function continuous function. solution found an explicit form. uniqueness theorem proved. existence for to proved by reducing it Volterra equation second kind singularity kernel, and necessary sufficient conditions are obtained.

In this paper we study a potential problem with a nonlinear boundary condition. Using the Green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation. We shall give a brief discussion of the solvability of the integral equation. The aim of this paper, however, is to analyse the collocation method for finding ...

The two-point boundary value problem is considered for the system of nonlinear generalized ordinary differential equations with singularities on a non-closed interval. Singularity is understood in a sense of the vector-function corresponding to the system which belongs to the local Carathéodory class with respect to the matrix-function corresponding to the system. The general sufficient conditi...

We prove the unique solvability of boundary-value problem for a loaded third-order integrodifferential equation with parabolic-hyperbolic operator. By method integral equations, we existence and uniqueness solutions problems.

We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of ...

This work provides a complete analysis of eddy current problems, ranging from a proof of unique solvability to the analysis of a multiharmonic discretization technique. For proving existence and uniqueness, we use a Schur complement approach in order to combine the structurally different results for conducting and non-conducting regions. For solving the time-dependent problem, we take advantage...

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...