scholarly journals Stability of Picard operators under operator perturbations

2018 ◽  
Vol 56 (2) ◽  
pp. 3-12
Author(s):  
Adrian Petruşel ◽  
Ioan A. Rus
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Picard Operator

AbstractIn this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:(a) f is a Picard operator with its unique fixed point x *f;(b) there exists η > 0 such that d(f(x), g(x)) ≤ η, for every x ∈ M.The problem consists in estimating d(gn(x), x*f), for x ∈ M and n ∈ 𝕅*.II. Let B be a Banach space and f, g : B → B be two operators. We suppose that f is a Picard operator. The problem is to find sufficient conditions which guarantee that f + g is a Picard operator.

scholarly journals Suzuki ψF-contractions and some fixed point results

2018 ◽  
Vol 34 (1) ◽  
pp. 93-102
Author(s):  
NICOLAE-ADRIAN SECELEAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Continuity Point ◽  
Picard Operator ◽  
Nonlinear Anal ◽  
New Type

The purpose of this paper is to combine and extend some recent fixed point results of Suzuki, T., [A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317] and Secelean, N. A. & Wardowski, D., [ψF-contractions: not necessarily nonexpansive Picard operators, Results Math., 70 (2016), 415–431]. The continuity and the completeness conditions are replaced by orbitally continuity and orbitally completeness respectively. It is given an illustrative example of a Picard operator on a non complete metric space which is neither nonexpansive nor expansive and has a unique continuity point.

scholarly journals Switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives

2021 ◽  
Vol 6 (12) ◽  
pp. 13092-13118
Author(s):  
Rizwan Rizwan ◽  
◽  
Jung Rye Lee ◽  
Choonkil Park ◽  
Akbar Zada ◽  
...  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Langevin Equations ◽  
Fixed Point Approach ◽  
Proposed Model ◽  
Mixed Derivatives ◽  
Rassias Stability

<abstract><p>In this paper, we consider switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of generalized Diaz-Margolis's fixed point approach, over generalized complete metric space. We give an example which supports our main result.</p></abstract>

scholarly journals SOLVABILITY OF THE CAUCHY PROBLEM FOR EQUATIONS WITH RIEMANN–LIOUVILLE’S FRACTIONAL DERIVATIVES

2018 ◽  
Vol 62 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Petr P. Zabreiko ◽  
Svetlana V. Ponomareva
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Unique Solution ◽  
Metric Space ◽  
Fractional Derivatives ◽  
Volterra Equation ◽  
Complete Metric Space ◽  
The Cauchy Problem ◽  
The Right

In this article we study the solvability of the analogue of the Cauchy problem for ordinary differential equations with Riemann–Liouville’s fractional derivatives with a nonlinear restriction on the right-hand side of functions in certain spaces. The conditions for solvability of the problem under consideration in given function spaces, as well as the conditions for existence of a unique solution are given. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

scholarly journals Fixed points and stability in nonlinear neutral integro-differential equations with variable delay

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 781-795
Author(s):  
Imene Soualhia ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Initial Function ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Stability Theorem ◽  
Variable Delay ◽  
Necessary And Sufficient

The nonlinear neutral integro-differential equation x'(t) = -?t,t-?(t) a (t,s) g(x(s))ds+c(t)x'(t-?(t)), with variable delay ?(t) ? 0 is investigated. We find suitable conditions for ?, a, c and g so that for a given continuous initial function ? mapping P for the above equation can be defined on a carefully chosen complete metric space S0? in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient conditions. The obtained theorem improves and generalizes previous results due to Burton [6], Becker and Burton [5] and Jin and Luo [16].

scholarly journals Generalizations of the Converse of the Contraction Mapping Principle

1966 ◽  
Vol 18 ◽  
pp. 1095-1104 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contraction Mapping Principle ◽  
Converse Statement ◽  
Natural Formulation

This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).

scholarly journals Diametrically contractive mappings

2004 ◽  
Vol 70 (3) ◽  
pp. 463-468 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Compact Subset ◽  
Metric Space ◽  
Complete Metric Space ◽  
Contractive Mapping ◽  
Weakly Compact ◽  
Contractive Mappings

A contractive mapping on a complete metric space may fail to have a fixed point. Diametrically contractive mappings are introduced and it is shown that a diametrically contractive self-mapping of a weakly compact subset of a Banach space always has a fixed point.

scholarly journals Fixed point theorems for non-self maps I

1994 ◽  
Vol 17 (4) ◽  
pp. 713-716 ◽  
Author(s):  
Troy L. Hicks ◽  
Linda Marie Saliga
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Supposef:C→XwhereCis a closed subset ofX. Necessary and sufficient conditions are given forfto have a fixed point. All results hold whenXis complete metric space. Several results hold in a much more general setting.

scholarly journals A discussion on a Pata type contraction via iterate at a point

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1061-1066
Author(s):  
Erdal Karapınar ◽  
Andreea Fulga ◽  
Vladimir Rakocevic
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Continuity Assumption ◽  
The Given ◽  
Type Contraction

In this paper, we introduce the notion of Pata type contraction at a point in the context of a complete metric space. We observe that such contractions possesses unique fixed point without continuity assumption on the given mapping. Thus, is extended the original results of Pata. We also provide an example to illustrate its validity.

scholarly journals Fixed point theorem with contractive mapping on C*-Algebra valued G-Metric Space

2021 ◽  
Vol 2106 (1) ◽  
pp. 012015
Author(s):  
A Wijaya ◽  
N Hariadi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contractive Mapping ◽  
C Algebra ◽  
Metric Function ◽  
Interesting Theorem

Abstract Banach-Caccioppoli Fixed Point Theorem is an interesting theorem in metric space theory. This theorem states that if T : X → X is a contractive mapping on complete metric space, then T has a unique fixed point. In 2018, the notion of C *-algebra valued G-metric space was introduced by Congcong Shen, Lining Jiang, and Zhenhua Ma. The C *-algebra valued G-metric space is a generalization of the G-metric space and the C*-algebra valued metric space, meanwhile the G-metric space and the C *-algebra valued metric space itself is a generalization of known metric space. The G-metric generalized the domain of metric from X × X into X × X × X, the C *-algebra valued metric generalized the codomain from real number into C *-algebra, and the C *-algebra valued G-metric space generalized both the domain and the codomain. In C *-algebra valued G-metric space, there is one theorem that is similar to the Banach-Caccioppoli Fixed Point Theorem, called by fixed point theorem with contractive mapping on C *-algebra valued G-metric space. This theorem is already proven by Congcong Shen, Lining Jiang, Zhenhua Ma (2018). In this paper, we discuss another new proof of this theorem by using the metric function d(x, y) = max{G(x, x, y),G(y, x, x)}.

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