Fixed point theorems with generalized altering distance functions in partially ordered metric spaces via w-distances and applications

Abstract

In this paper, we establish some fixed point theorems with T-contractions and w-distances in partially ordered metric spaces. The main tool used in our proof is a generalized altering distance function. Our results can be applied directly to study multidimensional fixed point which covers the concepts of coupled, tripled, quadruple fixed point etc. Moreover, a Fredholm integral equation and an initial-value problem for partial differential equation of parabolic type are also discussed.

Fixed point results for generalized contractive mappings involving altering distance functions on complete quasi-metric spaces and applications

“…Now we prove that (3.13) of [25] is not satisfied. Indeed, assume that there exist a generalized altering distance function ψ and a Jachymski function φ with φ(t) < ψ(t) for all t > 0 such that…”

“…Chaipunya et al [7] considered the initial value problem (5.4) when g(x, t, u, u x ) = g(x, t, v, v x ) ≡ 0. Very recently, in [25], we discussed the corresponding initial-value problem of single parabolic equation for (5.4). Unlike to Theorem 4.2 of [25], it should be noted that we do not require f and g to have the mixed monotone properties.…”

“…Motivated by the works in [16,25], we study the existence of solutions for the integral equations in the following system…”

Fixed-Point Theorems for α-Admissible Mappings with w-Distance and Applications to Nonlinear Integral Equations

“…In 1996, Kada et al [3] introduced the notion of w-distance and elaborated with the help of examples, that the concept of w-distance is general than that of metric on a nonempty set. Several researchers [4][5][6][7] used this notion to obtain some fixed point results not only in usual metric spaces but also in other spaces.…”

“…Motivated by the results in [1][2][3][4][5][6][7][8][9][10], in this paper we prove the existence and uniqueness of fixed points for α-admissible mappings satisfying contractive inequality of integral type via w-distance in complete metric spaces, which are used to study solvability of nonlinear Fredholm and Volterra integral equations. Our results generalize eorems 1-5 and two examples are given.…”