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In this paper, we present new variants of some known fixed point theorems and new fixed point results for cyclic operators on ordered sets, on distance spaces, and on ordered distance spaces. Some results in terms of the fixed point structures are also presented. Our results include existence, uniqueness, convergence of successive approximations, well posedness of fixed point problems and the Ostrowsky property. We also identify some of the main research directions regarding cyclic operators and formulate several open problems for each of them. [ABSTRACT FROM AUTHOR]

Let (X , ·) be a Hilbert space and T : X → X be a decreasing operator. Under a metric condition involving the convex combination of x and T(x), we will prove some fixed point theorems which generalize and complement several results in the theory of nonlinear operators. Our results are closely related to the admissible perturbations approach in fixed point theory. [ABSTRACT FROM AUTHOR]

The main aim of this article is to investigate some new sufficient conditions of the existence and uniqueness results of ψ -Hilfer fractional functional differential equations with nonlocal condition and finite delay. Further, Ulam–Hyers–Mittag-Leffler stability of solutions to a proposed problem is discussed. Krasnoselskii fixed point theorem, Picard operator method and Gronwall's inequality lemma are the main tools of our analysis. An illustrative example is provided in support of the results obtained. [ABSTRACT FROM AUTHOR]

This article investigates the existence and uniqueness of solutions to the second-order Volterra integrodifferential equations with nonlocal and boundary conditions through its equivalent integral equations and fixed point of Banach. Furthermore, utilizing the Picard operator theory, we obtain the dependence of solutions on the initial nonlocal data and on functions involved on the right-hand side of the equations. [ABSTRACT FROM AUTHOR]

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach's fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation. [ABSTRACT FROM AUTHOR]

Using the fixed point method and the weakly Picard operator technique, we establish some abstract Ulam-Hyers stability results for operatorial equations in noncommutative metric spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide some sufficient conditions under which a self-mapping T defined on a non-empty set X endowed with some convergence property is a Picard operator. A relevant example showing that such a mapping T on a non-metrizable space is a Picard operator is given. Our results can be used to obtain some known fixed point theorems on generalized metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider, discuss, improve, and complement recent fixed points results for mappings satisfying cyclical contractive conditions established by Pacurar and Rus (Nonlinear Anal. 72:1181-1187, 2010) and Chandok and Postolache (Fixed Point Theory Appl. 2013:28, 2013). By using a new lemma we get much shorter and nicer proofs of some results with the new concept of mappings. [ABSTRACT FROM AUTHOR]