scholarly journals Applications of the Bielecki renorming technique

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Mihály Bessenyei ◽  
Zsolt Páles
Keyword(s):  
Functional Analysis ◽  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Existence And Uniqueness Theorem ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle ◽  
Elegant Proof

AbstractThe renorming technique allows one to apply the Banach Contraction Principle for maps which are not contractions with respect to the original metric. This method was invented by Bielecki and manifested in an extremely elegant proof of the Global Existence and Uniqueness Theorem for ODEs. The present paper provides further extensions and applications of Bielecki’s method to problems stemming from functional analysis and from the theory of functional equations.

A Global Existence and Uniqueness Theorem for Ordinary Differential Equations

1976 ◽  
Vol 19 (1) ◽  
pp. 105-107 ◽  
Author(s):  
W. Derrick ◽  
L. Janos
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Contraction Principle ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.

A Global Existence and Uniqueness Theorem for Ordinary Differential Equations of Generalized Order

1978 ◽  
Vol 21 (3) ◽  
pp. 267-271 ◽  
Author(s):  
Ahmed Z. Al-Abedeen ◽  
H. L. Arora
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle ◽  
Generalized Order

AbstractWe extend the Picard's theorem to ordinary differential equation of generalized order α, 0 ≤ α ≤ l, and prove a global existence and uniqueness theorem by using the Banach contraction principle.

scholarly journals UNIQUENESS RESULTS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

2013 ◽  
Vol 10 (03) ◽  
pp. 461-494 ◽  
Author(s):  
AHMAD EL HAJJ ◽  
RÉGIS MONNEAU
Keyword(s):  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Spatial Dimension ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of Solutions ◽  
Continuous Solutions ◽  
Uniqueness Results ◽  
Global Existence And Uniqueness

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.

scholarly journals Ulam Stability of n-th Order Delay Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3029
Author(s):  
Shuyi Wang ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Mathematical Induction ◽  
Ulam Stability ◽  
Existence And Uniqueness Theorem ◽  
Integro Differential Equation ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

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