Showing papers in "Fixed Point Theory in 2021"

On an equation characterizing multi-cubic mappings and its stability and hyperstability

Abstract: In this paper, we introduce $n$-variables mappings which are cubic in each variable. We show that such mappings satisfy a functional equation. The main purpose is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-cubic mappings. As a consequence, we prove that a multi-cubic functional equation can be hyperstable.

On the local convergence of higher order methods in Banach spaces

Abstract: We study the local convergence analysis of two higher-order methods using Hölder continuity condition on the first Fréchet derivative to solve nonlinear equations in Banach spaces. Hölder continuous first derivative is used to extend the applicability of the method on such problems for which Lipschitz condition fails. Also, this convergence analysis generalizes the local convergence analysis based on Lipschitz continuity condition. Our analysis provides the radius of convergence ball and error bounds along with the uniqueness of the solution. Numerical examples like Hammerstein integral equation and a system of nonlinear equations are solved to verify our theoretical results.

Degree theory for discontinuous operators

Abstract: We introduce a new definition of topological degree for a meaningful class of operators which need not be continuous. Subsequently, we derive a number of fixed point theorems for such operators. As an application, we deduce a new existence result for first-order ODEs with discontinuous nonlinearities.

Fixed and periodic point results in cone b-metric spaces over Banach algebras; a survey

Abstract: In this paper, we consider the concept of cone b-metric spaces over Banach algebras and obtain some fixed point results for various definitions of contractive mappings. Moreover, we discuss about the property P and the property Q of fixed point problems. Our results are significant, since we omit the assumptions of normality of cones under which can be generalized and unified a number of recently announced results in the existing literature. In particular, we refer to the results of Huang et al. [H. Huang, G. Deng, S. Radenović, Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity. (2018), in press].

Boundary value problems for fractional-order differential inclusions in banach spaces with nondensely defined operators

Abstract: We consider a nonlocal boundary value problem for a semilinear differential inclusion of a fractional order in a Banach space assuming that its linear part is a non-densely defined HilleYosida operator. We apply the theory of integrated semigroups, fractional calculus and the fixed point theory of condensing multivalued maps to obtain a general existence principle (Theorem 3.2). Theorem 3.3 gives an example of a concrete realization of this result. Some important particular cases including a nonlocal Cauchy problem, periodic and anti-periodic boundary value problems are presented.

Hölder-type spaces, singular operators, and fixed point theorems

Abstract: In this note, we give a sufficient condition for the existence of Hölder-type solutions to a class of fractional initial value problems involving Caputo derivatives. Since imposing (classical or general) global Lipschitz conditions on the nonlinear operators involved leads to degeneracy phenomena, the main emphasis is put on local Lipschitz conditions or fixed point principles of Schauder and Darbo type. To this end, we study continuity and boundedness conditions for linear RiemannLiouville operators and nonlinear Nemytskij operators in Hölder spaces of integral type which have much better properties than classical Hölder spaces.

Existence of three weak solutions for a class of discrete problems driven by p-Laplacian operator

Abstract: In this paper, by using a theorem based on variational method which was recently proved by Ricceri, we establish the existence of three weak solutions for a class of p-Laplacian discrete problems. Remarks and examples are provided to illustrate our result.

Generalized Leray–Schauder nonlinear alternatives for general classes of maps

Abstract: New Leray–Schauder nonlinear alternatives are presented. These coincidence type results are established for set–valued maps.

Essential weakly Mönch type maps and continuation theory

Abstract: In this paper we present a variety of Leray–Schauder type continuation theorems for general classes of weakly Mönch type maps.

Fixed points and sets of multivalued contractions: an advanced survey with some new results

Abstract: The existence of fixed points and, in particular, coupled fixed points is investigated for multivalued contractions in complete metric spaces. Multivalued coupled fractals are furthermore explored as coupled fixed points of certain induced operators in hyperspaces, i.e. as coupled compact subsets of the original spaces. The structure of fixed point sets is considered in terms of absolute retracts. We also formulate a continuation principle for multivalued contractions as a nonlinear alternative based on the topological essentiality. Two illustrative examples about coupled multivalued fractals are supplied.