Existence of approximate solution to abstract nonlocal Cauchy problem

The aim of the paper is to prove a theorem about the existence of an approximate solution to an abstract nonlinear nonlocal Cauchy problem in a Banach space. The right-hand side of the nonlocal condition belongs to a locally closed subset of a Banach space. The paper is a continuation of papers [1], [2] and generalizes some results from [3].

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Uniqueness criterion for solution of abstract nonlocal Cauchy problem

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s104895339300005x ◽

1993 ◽

Vol 6 (1) ◽

pp. 49-54 ◽

Cited By ~ 8

Author(s):

L. Byszewski

Keyword(s):

Cauchy Problem ◽

Nonlocal Condition ◽

Nonlocal Problem ◽

Dissipative Operator ◽

Uniqueness Criterion ◽

Nonlocal Cauchy Problem

The aim of the paper is to prove an uniqueness criterion for a solution of an abstract nonlocal Cauchy problem. A dissipative operator in the nonlocal problem and an arbitrary functional in the nonlocal condition are considered. The paper is a continuation of papers [1]-[3] and generalizes some results from [4].

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Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term $$F=F(Du, D_xDu)$$ F = F ( D u , D x D u ) on the Right-Hand Side Does not Depend on u Explicitly

Series in Contemporary Mathematics - Nonlinear Wave Equations ◽

10.1007/978-3-662-55725-9_13 ◽

2017 ◽

pp. 303-317

Author(s):

Tatsien Li ◽

Yi Zhou

Keyword(s):

Cauchy Problem ◽

Lower Bound ◽

Life Span ◽

Nonlinear Term ◽

Classical Solutions ◽

Right Hand ◽

The Cauchy Problem ◽

The Right

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

Doklady of the National Academy of Sciences of Belarus ◽

10.29235/1561-8323-2018-62-4-391-397 ◽

2018 ◽

Vol 62 (4) ◽

pp. 391-397 ◽

Cited By ~ 1

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.

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Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term $$F=F(u,Du, D_xDu)$$ F = F ( u , D u , D x D u ) on the Right-Hand Side Depends on u Explicitly

Series in Contemporary Mathematics - Nonlinear Wave Equations ◽

10.1007/978-3-662-55725-9_14 ◽

2017 ◽

pp. 319-361

Author(s):

Tatsien Li ◽

Yi Zhou

Keyword(s):

Cauchy Problem ◽

Lower Bound ◽

Life Span ◽

Nonlinear Term ◽

Classical Solutions ◽

Right Hand ◽

The Cauchy Problem ◽

The Right

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Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method

Journal of Mathematics ◽

10.1155/2017/8213932 ◽

2017 ◽

Vol 2017 ◽

pp. 1-6

Author(s):

K. Issa ◽

F. Salehi

Keyword(s):

Approximate Solution ◽

Galerkin Method ◽

Integrodifferential Equation ◽

Numerical Results ◽

Integrodifferential Equations ◽

Right Hand ◽

The Right

In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting equation using Chebyshev-Galerkin method. Details of the method are presented and some numerical results along with absolute errors are given to clarify the method. Where necessary, we made comparison with the results obtained previously in the literature. The results obtained reveal the accuracy of the method presented in this study.

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Existence of solutions of nonlinear integrodifferential equation with nonlocal condition

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s104895339700035x ◽

1997 ◽

Vol 10 (3) ◽

pp. 279-288 ◽

Cited By ~ 7

Author(s):

K. Balachandran ◽

M. Chandrasekaran

Keyword(s):

Cauchy Problem ◽

Integrodifferential Equation ◽

Existence And Uniqueness ◽

Contraction Mapping ◽

Existence Of Solutions ◽

Nonlocal Condition ◽

Global Solutions ◽

Contraction Mapping Principle ◽

Local And Global Solutions ◽

Nonlocal Cauchy Problem

In this paper we prove the existence and uniqueness of local and global solutions of a nonlocal Cauchy problem for a class of integrodifferential equation. The method of semigroups and the contraction mapping principle are used to establish the results.

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Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.41.2010.786 ◽

2010 ◽

Vol 41 (4) ◽

pp. 361-374

Author(s):

H. L. Tidke ◽

M. B. Dhakne

Keyword(s):

Cauchy Problem ◽

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Integrodifferential Equation ◽

Existence And Uniqueness ◽

Nonlocal Condition ◽

Strong Solutions ◽

Integrodifferential Equations ◽

Nonlocal Cauchy Problem

The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.

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ON THE SELF-REGULARIZATION OF ILL-POSED PROBLEMS BY THE LEAST ERROR PROJECTION METHOD

Mathematical Modelling and Analysis ◽

10.3846/13926292.2014.923944 ◽

2014 ◽

Vol 19 (3) ◽

pp. 299-308 ◽

Cited By ~ 4

Author(s):

Alina Ganina ◽

Uno Hamarik ◽

Urve Kangro

Keyword(s):

Approximate Solution ◽

Projection Method ◽

Regularization Method ◽

The Self ◽

Noise Levels ◽

Right Hand ◽

Ill Posed ◽

The Right

We consider linear ill-posed problems where both the operator and the right hand side are given approximately. For approximate solution of this equation we use the least error projection method. This method occurs to be a regularization method if the dimension of the projected equation is chosen properly depending on the noise levels of the operator and the right hand side. We formulate the monotone error rule for choice of the dimension of the projected equation and prove the regularization properties.

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CONTINUOUS DEPENDENCE ON A PARAMETER OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202593000242 ◽

1993 ◽

Vol 03 (04) ◽

pp. 477-483

Author(s):

D.D. BAINOV ◽

S.I. KOSTADINOV ◽

NGUYEN VAN MINH ◽

P.P. ZABREIKO

Keyword(s):

Differential Equation ◽

Banach Space ◽

Differential Equations ◽

Small Parameter ◽

Continuous Dependence ◽

Impulsive Differential Equation ◽

Impulsive Differential Equations ◽

Right Hand ◽

Dependence On A Parameter ◽

The Right

Continuous dependence of the solutions of an impulsive differential equation on a small parameter is proved under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.

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