In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and W-property. Based on unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP).

The spectral radius condition \[\rho (\vert A^{-1} \vert\cdot \vert B \vert)<1\]for the unique solvability of generalized absolute value matrix equation (GAVME) \[AX + B \vert X \vert = D\] is provided. For some instances, our condition is superior to the earlier published singular values conditions \(\sigma_{\max}(\vert B \vert)<\sigma_{\min}(A)\) [M. Dehghan, 2020] and \(\sigma_{\max}(B)<\sigma_{\min}(A)\) [Kai Xie, 2021]. For the validity of our condition, we also provided an example.

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

In this article, we take the piecewise linear equation system \(x-W|x|=b\), which is also known by absolute value equation, where \(W\in {\mathbb R}^ {n\times n}\), \(b\in {\mathbb R}^{n}\) are given and to undetermined the value of \(x\in {\mathbb R}^{n}\). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. (small corrections operated in the pdf file on January 7, 2023)

In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvability of the system of non-linear integral equations. Finally, we give an example to validate the applicability of our results.