arXiv subjects
Publications and source records attributed to Maher Berzig.
In his study of Cantor-type intersection theorems and fixed points of almost affine mappings, Jachymski \cite{Jachymski} posed four open questions. In this note, we provide an affirmative answer to the fourth question.
We introduce the concept of $b$-suprametric spaces and establish a fixed point result for mappings satisfying a nonlinear contraction in such spaces. The obtained result generalizes a fixed point theorem of Czerwik and a recent result of the author.
We introduce the concept of shifting distance functions, and we establish a new fixed point theorem which generalizes the Banach contraction principle. An example is provided to illustrate our result.
In this paper, we introduce the notion of $α$--contractive mapping of Meir--Keeler type in complete metric spaces and prove new theorems which assure the existence, uniqueness and iterative approximation of the fixed point for this type of contraction. The presented theorems extend, generalize and improve several existing results in literature. To validate our results, we establish the existence and uniqueness of solution to a class of third order two point boundary value problems.
We consider the generalized continuous-time Lyapunov equation: $$ A^*XB + B^*XA =-Q, $$ where $Q$ is an $N\times N$ Hermitian positive definite matrix and $A,B$ are arbitrary $N\times N$ matrices. Under some conditions, using the coupled fixed point theorem of Bhaskar and Lakshmikantham, we establish the existence and uniqueness of Hermitian positive definite solution for such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists.