scholarly journals On the solution of Steinhaus functional equation using weakly Picard operators

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 69-79
Author(s):  
Vasile Berinde
Keyword(s):  
Functional Equation ◽  
Convergent Series ◽  
Existence Results ◽  
Divergent Series ◽  
Picard Operator ◽  
Weakly Picard Operator

In this paper we obtain existence results regarding the solutions g of a Steinhaus type functional equation of the form g(x)+ g(f(x))= F(x), under the significantly weaker assumption that f is a weakly Picard operator. The solutions are given in terms of sums of either convergent series or divergent series but summable by some method of summability.

scholarly journals VII. A theory of asymptotic series

1912 ◽  
Vol 211 (471-483) ◽  
pp. 279-313 ◽  
Keyword(s):  
Power Series ◽  
Mathematical Analysis ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Consistent Theory

In their efforts to place mathematical analysis on the firm est possible foundations, Abel and Cauchy found it necessary to banish non-convergent series from their work ; from that time until a quarter of a century ago the theory o f divergent series was, in general, neglected by mathematicians. A consistent theory of divergent series was, however, developed by Poincaré in 1886, and, ten years later, Borel enunciated his theory of summability in connection with oscillating series. So far as diverging power series are concerned, the theory of Borel is more precise than that of Poincaré.

scholarly journals Existence and Ulam-Hyers stability results for multivalued coincidence problems

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 965-976 ◽  
Author(s):  
Oana Mleşniţe ◽  
Adrian Petruşel
Keyword(s):  
Fixed Point ◽  
Coincidence Point ◽  
Operator Technique ◽  
Picard Operator ◽  
Multivalued Operators ◽  
Weakly Picard Operator ◽  
Stability Results

In this paper, we will present some existence and Ulam-Hyers stability results for fixed point and coincidence point problems with multivalued operators using the weakly Picard operator technique in spaces endowed with vector metrics.

scholarly journals Properties of the solutions of those equations for which the Krasnoselskii iteration converges

2012 ◽  
Vol 28 (2) ◽  
pp. 329-336
Author(s):  
IOAN A. RUS ◽  
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Data Dependence ◽  
Picard Operator ◽  
Nonempty Convex Subset ◽  
Krasnoselskii Iteration ◽  
Nonempty Convex ◽  
Weakly Picard Operator

Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.

scholarly journals Ulam-Hyers stability for partial differential equations

2012 ◽  
Vol 21 (1) ◽  
pp. 73-78
Author(s):  
VASILE L. LAZAR ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Operator Technique ◽  
Partial Differential ◽  
Picard Operator ◽  
Weakly Picard Operator ◽  
Stability Results

Using the weakly Picard operator technique, we will present some Ulam-Hyers stability results for some partial differential equations.

scholarly journals The Products of Hyperreal Series and the Limitations of Cauchy Products

2021 ◽  
Vol 3 (2) ◽  
pp. 34-36
Author(s):  
Jonathan Bartlett
Keyword(s):  
Convergent Series ◽  
Divergent Series ◽  
Alternative Approaches

Cauchy products are used to take the products of convergent series. Here, we show the limitations of this approach in divergent series, including those that can be analyzed through the BGN method. Alternative approaches and formulas for divergent series are suggested, as well as their benefits and drawbacks.

Sign in / Sign up

Export Citation Format

Share Document