A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations

Abstract

The aim of this paper is to extend the results of Harjani and Sadarangani and some other authors and to prove a new fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions. Our theorem can be used to investigate a large class of nonlinear problems.As an application, we discuss the existence of a solution for a periodic boundary value problem.

Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions

“…Also, we provide an example that supports our main result where previous results in literature are not applicable. Our results generalize and improve the main result of Yan et al [17] and several well-known results given by some authors in partially metric spaces. Moreover, we present a new extension of multidimensional fixed point theorems in metric spaces endowed with a transitive relation.…”

“…We note that Yan et al's result in [17] (Theorem 1.3) is not applicable in the above example since is not an altering distance function. This implies that the Banach contraction principle is not also applicable in the above example.…”

“…It is fascinating to point out that we use the result in Theorem 3.4 to derive a criterion for the existence of fixed points in some cases wherein several results contained in [17,[20][21][22] cannot guarantee the existence of fixed points. Indeed, the main results of Yan et al in [17] (Theorem 1.3) are not applicable in the following cases: -.X; / is not a partially ordered set; -is not an altering distance function; -is not continuous, the main results of Rus in [20] are not applicable in the following cases: -.X; / is not a partially ordered set (or quasi-ordered set); -T does not satisfy the contractive condition with the control function ' 2ˆ, the main results of Soleimani Rad et al in [21] are not applicable in the following cases: -T does not satisfy the contractive condition with several contractive constant, the main results of Su et al in [22] are not applicable in the following cases: -T does not satisfy the contractive condition in metric space without the transitive relation; -T does not satisfy the contractive condition with two control functions and (see the detail in [22]). Now, we give an example to illustrate utility of Theorem 3.4.…”

“…Figure 1). In 2012, Yan et al [17] discussed some results on existence and uniqueness of a fixed point in partially ordered metric spaces by using the concept of an altering distance function as follows.…”

Generalized C Ψ β - rational contraction and fixed point theorem with application to second order deferential equation

“…If we take a 2 = a 3 = 0, a 1 = 1 in Theorem 2.2, we obtain following result of Yan et al [5] satisfying weaker type of C …”