Open mappings: The case for a new direction in fixed point theory

Abstract

Classical fixed point theorems often begin with the assumption that we have a mapping P of a closed convex set in a Banach space G into itself. It then adds a number of conditions which will ensure that there is at least one fixed point in the set G. We continue two earlier studies in which we now propose to stop the process after we have mapped G not only into itself, but into its interior. We then study what we may deduce from this alone.

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

“…We should note that for t ∈ 0, b m and x ∈ M 2 , it follows that x = x * 1 is a fixed point. Then, according to (13), we can derive that…”

“…Burton and Purnaras [13] suggested a new direction in fixed point theory such that in classical fixed point theorems, one has a mapping P of a closed convex set in a Banach space into itself. Instead of this case, in Burton and Purnaras [13], for the map P defined by the right-hand side of IE (8), a closed bounded convex nonempty set P is constructed such that P : G → G 0 , where G 0 is the interior of P.…”

Global Existence and Uniqueness of Solutions of Integral Equations with Multiple Variable Delays and Integro Differential Equations: Progressive Contractions

“…As for some other fixed point results, applications of fixed point methods, etc., one can find several interesting results in the papers of Abbas and Benchohra [12], Banaś and Rzepka [13], Becker et al [5], Burton and Purnaras [14][15][16], Burton and Zhang [17], Chauhan et al [18], Ilea and Otrocol [10], Khan et al ([19]), Petruşel et al ([20,21]), ), the books of Burton [6], Smart [25], and the references therein. On the other hand, recently, Assari et al [26] and Assari and Dehghan [27] presented a numerical method for solving logarithmic Fredholm integral equations, which occur as a reformulation of twodimensional Helmholtz equations over the unit circle with the Robin boundary conditions, and a computational scheme to solve nonlinear logarithmic singular boundary integral equations, which arise from boundary value problems of Laplace equations with nonlinear Robin boundary conditions, respectively.…”