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1
Content available Karakostas fixed point theorem and semilinear neutral differential equations with impulses and nonlocal conditions
Leiva Hugo, Riera Lenin, Lalvay Sebastián
Journal of Mathematics and Applications
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2022
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Vol. 45
107--128
EN
This paper is concerned with the existence and uniqueness of solutions for a semilinear neutral differential equation with impulses and nonlocal conditions. First, we assume that the nonlinear terms are locally Lipschitz, and to achieve the existence of solutions, Karakostas Fixed Point Theorem is applied. After that, under some additional conditions, the uniqueness is proved as well. Next, assuming some bound on the non-linear terms the global existence is proved by applying a generalization of Gronwall inequality for impulsive differential equations. Then, we suppose stronger hypotheses on the nonlinear functions, such as globally Lipschitz conditions, that allow us to appy Banach Fixed Point Theorem to prove the existence and uniqueness of solutions. Finally, we present an example as an application of our method.
2
Content available Conditional stability estimate for an ill-posed elliptic equation by using nonlocal conditions
Benrabah Abderafik
Journal of Applied Mathematics and Computational Mechanics
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2022
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Vol. 21, nr 3
5--17
EN
We consider an ill-posed linear homogeneous fourth-order elliptic equation. We show that the problem is ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the given data. We propose a regularization method via nonlocal conditions and under some a priori bound assumptions different estimates for the regularized solution are obtained. Numerical examples for a rectangle domain show the effectiveness of the new method in providing highly accurate numerical solutions as the noise level tends to zero.
3
Content available remote On a first-order differential system with initial and nonlocal boundary conditions
Ngoc Le Thi Phuong, Long Nguyen Thanh
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
277--296
EN
This paper is devoted to the existence of solutions and the multiplicity of positive solutions of an initial-boundary value problem for a nonlinear first-order differential system with nonlocal conditions. The main tool is the fixed-point theorem in which we construct the novel representation of the associated Green’s functions with useful properties and define a cone in the Banach space suitably. Some examples are also given to demonstrate the validity of the main results.
4
Content available Controllability of time varying semilinear non-instantaneous impulsive systems with delay, and nonlocal conditions
Cabada Dalia, Garcia Katherine, Guevara Cristi, Leiva Hugo
Archives of Control Sciences
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2022
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Vol. 32, No. 2
335--357
EN
In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.
5
Content available remote On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces
Metwali Mohamed M. A.
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
86--94
EN
The existence of a.e. monotonic solutions for functional quadratic Hammerstein integral equations with the perturbation term is discussed in Orlicz spaces. We utilize the strategy of measure of noncompactness related to the Darbo fixed point principle. As an application, we discuss the presence of solution of the initial value problem with nonlocal conditions.
6
Content available remote Continuous dependence of mild solutions on initial nonlocal data, of the nonlocal semilinear functional-differential evolution Cauchy problems of the first and second order
Bednarz A., Byszewski L.
Technical Transactions
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2018
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Vol. 115, iss. 5
141--148
EN
The aim of the paper is to prove two theorems on the continuous dependence of mild solutions, on initial nonlocal data, of the nonlocal semilinear functional-differential evolution Cauchy problems of the first and second orders. The paper is based on publications [1–10] and is a generalization of paper [5].
PL
W artykule udowodniono dwa twierdzenia o ciągłej zależności rozwiązań całkowych od nielokalnych warunków początkowych, semiliniowych funkcjonalno-różniczkowych zagadnień ewolucyjnych Cauchy’ego pierwszego i drugiego rzędu. Artykuł bazuje na publikacjach [1–10] i jest uogólnieniem publikacji [5].
7
Content available remote Continuous dependence of mild solutions, on initial nonlocal data, of the nonlocal semilinear evolution Cauchy problems
Byszewski L.
Technical Transactions
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2017
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Vol. 9(114)
151--158
EN
The aim of the paper is to prove two theorems on continuous dependence of mild solutions, on initial nonlocal data, of the nonlocal semilinear evolution Cauchy problems. For this purpose, the method of semigroups and the theory of cosine family in Banach spaces are applied. The paper is based on publications [1–6] and is a generalization of paper [3].
PL
W artykule udowodniono dwa twierdzenia o ciągłej zależności rozwiązań całkowych od nielokalnych warunków początkowych, semiliniowych nielokalnych zagadnień Cauchy’ego. W tym celu zastosowano metodę półgrup i teorię rodziny cosinus w przestrzeniach Banacha. Artykuł bazuje na publikacjach [1–6] i jest pewnym uogólnieniem publikacji [3].
8
Content available remote Solvability of a Coupled System of Fractional Differential Equations with Nonlocal and Integral Boundary Conditions
Ahmad B., Ntouyas S. K., Alsaedi A., Hobiny A.
Fundamenta Informaticae
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2017
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Vol. 151, nr 1/4
91--108
EN
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with nonlocal and integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. The results are explained with the aid of examples. The case of nonlocal strip conditions is also discussed.
9
Content available Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions
Ntouyas S. K.
Opuscula Mathematica
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2013
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Vol. 33, no. 1
117-138
EN
This paper studies the boundary value problem of nonlinear fractional differential equations and inclusions of order q ∈ (1, 2] with nonlocal and integral boundary conditions. Some new existence and uniqueness results are obtained by using fixed point theorems.
10
Content available remote On nonlocal evolution problem for the equation of the first order
Byszewski L., Winiarska T.
Czasopismo Techniczne. Nauki Podstawowe
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2013
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R. 110, z. 1-NP
19--26
EN
The aim of the paper is to prove theorems about the existence and uniqueness of mild and classical solutions of a nonlocal semilinear functional-differential evolution Cauchy problem. The method of semigroups, the Banach fixed-point theorem and theorems (see [2]) about the existence and uniqueness of the classical solutions of the first-order differential evolution problems in a not necessarily reflexive Banach space are used to prove the existence and uniqueness of the solutions of the problems considered. The results obtained are based on publications [1–6].
PL
W artykule udowodniono twierdzenia o istnieniu i jednoznaczności rozwiązań całkowych i klasycznych nielokalnego semiliniowego funkcjonalno-różniczkowego ewolucyjnego zagadnienia Cauchy’ego. W tym celu zastosowano metodę półgrup, twierdzenie Banacha o punkcie stałym i twierdzenia ([2]) o istnieniu i jednoznaczności klasycznych rozwiązań ewolucyjnych zagadnień różniczkowych pierwszego rzędu w niekoniecznie refleksywnej przestrzeni Banacha. Artykuł bazuje na publikacjach [1‒6].
11
Content available remote On abstract nonlocal Cauchy problem
Bednarz A., Byszewski L.
Czasopismo Techniczne. Nauki Podstawowe
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2013
|
R. 110, z. 1-NP
11--17
EN
In this paper, we investigate the existence and uniqueness of the classical solution to an abstract nonlocal Cauchy problem. For this purpose, we apply a notion of mild solution and the Banach contraction theorem.
PL
W artykule zbadano istnienie i jednoznaczność klasycznego rozwiązania abstrakcyjnego nielokalnego zagadnienia Cauchy’ego. W tym celu zastosowano rozwiązanie całkowe i twierdzenie Banacha o kontrakcji.
12
Content available remote Continuous dependence of mild solutions, on initial nonlocal data, of the nonlocal evolution Cauchy problems
Byszewski L., Winiarska T.
Czasopismo Techniczne. Nauki Podstawowe
|
2013
|
R. 110, z. 1-NP
27--32
EN
The aim of the paper is to prove two theorems on continuous dependence of mild solutions, on initial nonlocal data, of the nonlocal Cauchy problems. For this purpose, the method of semigroups and the theory of cosine family in Banach spaces are applied. The paper is based on publications [1–5].
PL
W artykule udowodniono dwa twierdzenia o ciągłej zależności rozwiązań całkowych od nielokalnych warunków początkowych, nielokalnych zagadnień Cauchy’ego. W tym celu zastosowano metodę półgrup i teorię rodziny cosinus w przestrzeniach Banacha. Artykuł bazuje na publikacjach [1‒5].
13
Content available A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions
Wang J., Yan X., Zhang X. H., Wang T. M, Li X. Z.
Opuscula Mathematica
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2011
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Vol. 31, no. 1
119-135
EN
In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: [formula]. Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.
14
Content available Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces
Wang J., Yang Y., Wei W.
Opuscula Mathematica
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2010
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Vol. 30, no. 3
361-381
EN
In this paper, we study the existence and uniqueness of the PC-mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions. By means of the generalized Ascoli-Arzela Theorem given by us and the fixed point theorem, some existence and uniqueness results are obtained. Finally, an example is given to illustrate the theory.
15
Content available Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control
Wang J., Wei W., Yang Y.
Opuscula Mathematica
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2010
|
Vol. 30, no. 2
217-234
EN
In this paper, a class of fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is considered. Using a contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. The existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is also presented.
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