Linear integral volterra equation of the first kind in the banach space

10.12737/4716 ◽  
2014 ◽  
Vol 2 (4) ◽  
pp. 113-114
Author(s):  
I. Sapronov ◽  
P. Zyukin ◽  
V. Zenina
Keyword(s):  
Banach Space ◽  
Volterra Equation ◽  
Integral Volterra Equation ◽  
Linear Integral

LINEAR INTEGRAL VOLTERRA EQUATION OF THE FIRST KIND IN THE BANACH SPACE

2015 ◽  
Vol 4 (2) ◽  
pp. 8-13
Author(s):  
Зенина ◽  
V. Zenina ◽  
Сапронов ◽  
Ivan Sapronov ◽  
Уточкина ◽  
...  
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Volterra Equation ◽  
Polynomial Kernel ◽  
Integral Volterra Equation ◽  
Integrable Functions ◽  
Linear Integral ◽  
Space Of Integrable Functions

We construct solutions to a singular Volterra integral equation of the first kind with а polynomial kernel in the space of integrable functions whose values be-long to a Banach space.

scholarly journals Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Szymon Dudek ◽  
Leszek Olszowy
Keyword(s):  
Banach Space ◽  
Hausdorff Distance ◽  
Continuous Dependence ◽  
Volterra Equation ◽  
Fréchet Space ◽  
Frechet Space ◽  
Integral Volterra Equation ◽  
Quadratic Integral ◽  
Continuous Dependence Of Solutions

We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Fréchet space.

scholarly journals Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the SpacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)

2013 ◽  
Vol 2013 ◽  
pp. 1-20
Author(s):  
İsmet Özdemir ◽  
Ali M. Akhmedov ◽  
Ö. Faruk Temizer
Keyword(s):  
Banach Space ◽  
Numerical Methods ◽  
Banach Spaces ◽  
Integral Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Fredholm Integral Equations ◽  
Bounded Linear ◽  
Linear Integral ◽  
Linear Integral Operators

The spacesHα,δ,γ((a,b)×(a,b),ℝ)andHα,δ((a,b),ℝ)were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the setsHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)by taking an arbitrary Banach spaceXinstead ofℝ, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms∥·∥α,δ,γand∥·∥α,δ. Besides, the bounded linear integral operators on the spacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X), some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the formf(s)=ϕ(s)+λ∫abA(s,t)f(t)dt,f(s)=ϕ(s)+λ∫abA(t,s)f(t)dtandf(s,t)=ϕ(s,t)+λ∫abA(s,t)f(t,s)dtare investigated in these spaces by analytical methods.

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 To the solution of the singular inhomogeneous integral Volterra equation

2017 ◽  
Vol 86 (2) ◽  
pp. 20-31
Author(s):  
T.N. Bekjan ◽  
◽  
M.T. Jenaliyev ◽  
S.A. Iskakov ◽  
M.I. Ramazanov ◽  
...  
Keyword(s):  
Volterra Equation ◽  
Integral Volterra Equation

scholarly journals Interpolation rational integral fraction of nth order on a continuum set of nodes

2021 ◽  
pp. 101-105
Author(s):  
Ihor Demkiv ◽  
Yaroslav Baranetskyi ◽  
Halyna Berehova
Keyword(s):  
Integral Equation ◽  
Volterra Equation ◽  
Standard Form ◽  
Integral Volterra Equation ◽  
Substitution Rule ◽  
Rational Integral

The paper constructs and investigates an integral rational interpolant of the nth order on a continuum set of nodes, which is the ratio of a functional polynomial of the first degree to a functional polynomial of the (n-1)th degree. Subintegral kernels are determined from the corresponding continuum conditions. Additionally, we obtain an integral equation to determine the kernel of the numerator integral. This integral equation, using elementary transformations, is reduced to the standard form of the integral Volterra equation of the second kind. Substituting the obtained solution into expressions for the rest of the kernels, we obtain expressions for all kernels included in the integral rational interpolant. Then, in order for a rational functional of the nth order to be interpolation on continuous nodes, it is sufficient for this functional to satisfy the substitution rule. Note that the resulting interpolant preserves any rational functional of the obtained form.

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