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1

Existence, uniqueness and parameter perturbation analysis results of a fractional integro-differential boundary problem

Zhang Nan, Zhang Lingling, Ngungu Mercy, Adeniji Adejimi, Addai Emmanuel

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 4

art. no. e145938

EN

In the formulation, the existence, uniqueness and stability of solutions and parameter perturbation analysis to Riemann-Liouville fractional differential equations with integro-differential boundary conditions are discussed by the properties of Green’s function and cone theory. First, some theorems have been established from standard fixed point theorems in a proper Banach space to guarantee the existence and uniqueness of positive solution. Moreover, we discuss the Hyers-Ulam stability and parameter perturbation analysis, which examines the stability of solutions in the presence of small changes in the equation main parameters, that is, the derivative order η, the integral order β of the boundary condition, the boundary parameter ξ , and the boundary value τ. As an application, we present a concrete example to demonstrate the accuracy and usefulness of the proposed work. By using numerical simulation, we obtain the figure of unique solution and change trend figure of the unique solution with small disturbances to occur in different kinds of parameters.

2

Numerical and theoretical analysis of an awareness COVID-19 epidemic model via generalized Atangana-Baleanu fractional derivative

Baba Isa Abdullahi, Ahmed Idris, Al-Mdallal Qasem M, Jarad Fahd, Yunusa Salisu

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 1

7--18

EN

In this paper, a COVID-19 Awareness model in the setting of a generalized fractional Atangana-Baleanu derivative is proposed. The existence and uniqueness of a solution of the proposed fractional-order model are investigated under the techniques of fixed point theorems. In addition, we perform the predictor-corrector method to find its numeric solutions and present the graphs of the various solutions using different values of the parameters embodied in the derivative.

3

Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations

Basti Bilal, Arioua Yacine, Benhamidouche Nouredine

Journal of Mathematics and Applications

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2019

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Vol. 42

35--61

EN

The present paper deals with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with Katugampola fractional derivative. The main results are proved by means of Guo-Krasnoselskii and Banach fixed point theorems. For applications purposes, some examples are provided to demonstrate the usefulness of our main results.

4

On nonlinear neutral Liouville-Caputo-type fractional differential equations with Riemann-Liouville integral boundary conditions

Ahmad Bashir, Ntouyas Sotiris K., Alsaedi Ahmed

Journal of Applied Analysis

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2019

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Vol. 25, nr 2

119--130

EN

This paper studies neutral Liouville-Caputo-type fractional differential equations and inclusions supplemented with nonlocal Riemann-Liouville-type integral boundary conditions. Sadovskii’s fixed point theorem is applied to establish the existence result for the single-valued case, while the multivalued case is investigated by using nonlinear alternative for contractive maps. Examples are constructed to illustrate the main results. The case of nonlinear nonlocal boundary conditions is also discussed.

5

Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

Benchohra Mouffak, Bouriah Soufyane, Nieto Juan J.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

437--450

EN

In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

6

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Zada Akbar, Waheed Hira, Alzabut Jehad, Wang Xiaoming

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

296--335

EN

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

7

On the existence of solutions of a perturbed functional integral equation in the space of Lebesgue integrable functions on R+

Al Sayed W., Darwish M. A.

Journal of Mathematics and Applications

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2018

|

Vol. 41

19--27

EN

In this paper, we investigate and study the existence of solutions for perturbed functional integral equations of convolution type using Darbo's fixed point theorem, which is associated with the measure of noncompactness in the space of Lebesgue integrable functions on R+. Finally, we offer an example to demonstrate that our abstract result is applicable.

8

Uwarunkowania bezpieczeństwa rozwoju społeczności

Załoga W.

Studia Bezpieczeństwa Narodowego

|

2016

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R. 6, Nr 10

341--358

PL

Treść artykułu odnosi się do zagadnień z obszaru szeroko pojętego bezpieczeństwa oraz bezpieczeństwa społecznego w ujęciu regionalnym i globalnym z uwzględnieniem rozwoju społeczności. W artykule zaprezentowano również główne uwarunkowania w aspekcie bezpieczeństwa mające wpływ na rozwój społeczności. Wynikają one z natury świata i życia społecznego, miejsca narodu na ziemi, historii, tradycji, charakteru współczesnych zagrożeń oraz stanu organizacji bezpieczeństwa międzynarodowego. Wybór tak szerokiego spektrum czynników wpływających na bezpieczeństwo wynika m.in. z potrzeby uniknięcia jego błędnego rozpatrywania, dokonywanego pod wpływem aktualnie modnego i powszechnie uznanego typu zagrożenia, które zwykle czasowo dominuje w ocenach i organizacji bezpieczeństwa. Treść artykułu odnosząca się do uwarunkowań bezpieczeństwa opiera się na zmienności, nieprzewidywalności środowiska w ujęciu regionalnym i globalnym i sprowadza się do trzech podstawowych tez: konieczności traktowania bezpieczeństwa narodowego jako procesu podlegającego nieustannym zmianom, w którym spoczywanie na laurach w jakimś okresie stanu bezpieczeństwa prowadzi nieuchronnie do tragedii; konieczności traktowania wszelkich hipotez, przepowiedni i proroctw, oznajmujących status quo lub odnoszących się do przyszłości, nie tylko jako nieprawidłowych, nieodpowiedzialnych, ale i groźnych dla bezpieczeństwa narodu; wpływająca na zmienność, nieprzewidywalność i zaskoczenie w odniesieniu do bezpieczeństwa narodowego, to konieczność jego kreacji, tworzenia, a nie tylko reagowania na zaistniałe zagrożenia. W niniejszej publikacji Bezpieczeństwo przedstawione w ogólnym znaczeniu odnosi się głównie do zagadnień obejmujących zaspokojenie takich potrzeb, jak: istnienie, przetrwanie, ciągłość, tożsamość, niezależność, spokój, posiadanie i pewność rozwoju. Bezpieczeństwo w ogólnym tego słowa znaczeniu, ma charakter podmiotowy, a będąc naczelną potrzebą człowieka i grup społecznych, jest zarazem podstawową potrzebą państwa i systemów międzynarodowych. Zaakcentowane zostały zagadnienia dotyczące zagrożeń społecznych wynikających z rosnącego bezrobocia globalnego i lokalnego, ubóstwa, wojny, braku perspektyw poprawy warunków życia, oraz zagrożeń spowodowanych masową, nielegalną migracją ludności, grożącą trwałą destabilizacją społeczeństwa tych krajów. Artykuł poświęcony jest również zagadnieniom dotyczącym organizacji bezpieczeństwa w ramach współpracy Unii Europejskiej. Państwa członkowskie Unii Europejskiej prowadząc wspólną politykę zagraniczną w ramach zagwarantowania bezpieczeństwa, znacząco zwiększają rolę międzynarodową integrującej się Europy. Zaakcentowano rolę państw członkowskich Unii Europejskiej w systemie bezpieczeństwa. Bezpieczeństwo to szerokie pojęcie, a zapotrzebowanie na nie przychodzi wraz z narodzinami człowieka oraz nieodłącznie towarzyszy ludziom przez całe życie, niezależnie od stanu i stopnia czy kierunku aktywności. Ze względu na swoją istotę i społeczne znaczenie, wraz z postępem ludzkości, staje się coraz bardziej złożonym zagadnieniem. Przez swoją istotę bezpieczeństwo jest problemem zarówno z punktu widzenia indywidualnych osób, jak też całych społeczności.

EN

Contents of the article refer to the broad area of issues determining the social security at the regional and global levels, taking into account the community development. The key considerations are made in terms of analyzing security problems in the area of vital importance for State security, at the holistic level. In face of particular observations and analyses, author defines factors corresponding to the secure social development. The determinates are the result of the world conditions that the State is submerged in, and particular factors of social life tradition, nation’s significance for common culture, history, tradition, and also what predestines the evolvement of a character of contemporary threats to the State and international security organizations. Author choses the widest spectrum of factors affecting security and creating potential threats. This is done understanding that some very particular notions of security, as they may be fashionable by universal recognition, may not always be the corretconsideration when discussing different dimensions of security and State’s social development and security organizations of significance. In this sense author, understand that the referring topic of the article on conditioning of general and particular security interactions of different fields is based on a changeability, and unpredictability of the security environment in the regional and global areas. Hence, what should be considered rests in three specific points of interest: the need to treat national security as the process being subjected to ceaseless changes, in which there are some periods of relative security, but in long run may lead to unavoidable tragedy, if we forget about various threats; the need to treat all ad hoc hypotheses, the “prophecies” of forecast and predictions, indicative of the status quo or referring to the future, not only as irresponsible, but also potentially dangerous to the State security; the need of influencing the changeability factor rationally, the unpredictable and potential threats have to be considered in reference to national security, However, the secure environment is rests on creation, what also implies that existing threats cannot be ignored. Author argues that importance of the State security and its overall significance rests mainly on fundament that includes meeting specific needs, such as: that of being, of surviving, of continuity, of identity, of independence, of peace, of possession and the certainty of development. As it is also noticed, the security in its generic term refers to subjective character of human being interaction within a community or social group, however it also is reflected in a similar fashion within the international system. The problems of threats may grow out of social threats, or be the result of a global economic system, unemployment, war, lack of prospects for development, poverty, mass or other type of illegal immigration, and other threats that may destabilize society of different countries. Keeping the above in mind, the author also considers the issues pertaining to the organizational security and co-operational frameworks within the European Union. Of course, the membership in this large organization is partially a guarantee of security, considerably as important as the common foreign policy in view of international system and the integrating role of Europe. In this sense also a particular role of the member states of the European Union, as an addition to common security system is considered. The broad spectrum of security demand very carefull allocation of ways and means protecting different needs, including the social needs, of a State. Yet, on the account of complex nature of State’s existence, with security as the most significant element of social progress and the progress of humankind, this is an intricate issue. The essential characteristics of issue described can be observed, however, the point of view of an individual person, as well as that of an entire community.

9

On some qualitative properties of mild solutions of nonlocal semilinear functional differential equations

Jain R. S., Dhakne M. B.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

854--865

EN

In the present paper, we investigate the qualitative properties such as existence, uniqueness and continuous dependence on initial data of mild solutions of first and second order nonlocal semilinear functional differential equations with delay in Banach spaces. Our analysis is based on semigroup theory and modified version of Banach contraction theorem.

10

Podstawy poznawcze obrony powietrznej

Glen A.

Zeszyty Naukowe AON

|

2012

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nr 4(89)

43-83

PL

W artykule zawarto opis podstaw aksjologicznych, ontologicznych i epistemologicznych zjawiska obrony powietrznej (OP). Scharakteryzowano system wartości OP, wyjaśniono, jak OP istnieje w sferze realnej i konceptualnej rzeczywistości. Zdefiniowano OP w szerokim rozumieniu, jako część bezpieczeństwa powietrznego państwa, i wąskim – jako część walki zbrojnej. Określono sposób rozumienia całości, kompleksu i systemu OP, a także misji, domeny i celu OP. Wyjaśniono istotę OP ze względu na jej podmiot, przedmiot, rodzaj działań i charakter środków napadu powietrznego.

EN

The article includes a description of the axiological, ontological and epistemological basis of the phenomena of air defence (AD). It characterises the value system of AD, and explains how AD exists in the physical and conceptual reality. AD is defined in a broad sense as part of the air security of the state and in a narrow sense as part of the armed struggle. It specifies the mode of understanding of the whole complex system of AD, as well as the mission, domain and goal of AD. The essence of AD is explained with regard to its subject, object, the type of actions and the nature of an air attack.

11

Positive realization of fractional discrete-time linear systems with delays

Sajewski Ł.

Pomiary Automatyka Robotyka

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2012

|

R. 16, nr 2

323-327

EN

The positive realization problem for single-input single output fractional discrete-time linear systems with delays in state vector and input is formulated and a method for finding a positive realization of a given proper transfer function is proposed. Sufficient conditions for the existence ot a positive realization of this class of linear systems are established. A procedure for computation of a positive realization is proposed and illustrated by a numerical example.

PL

Sformułowane zostanie zadanie realizacji dodatniej dla klasy dyskretnych układów liniowych niecałkowitego rzędu z opóźnieniami w wektorze stanu i wymuszeniu. Zaproponowana zostanie metoda wyznaczania realizacji dodatniej dla danej transmitancji właściwej układu o jednym wejściu i jednym wyjściu. Podane zostaną warunki wystarczające istnienia realizacji dodatniej dla rozpatrywanej klasy liniowych układów z opóźnieniami. Przedstawiona zostanie procedura wyznaczania realizacji dodatniej zilustrowana przykładem numerycznym.

12

Determination of the set of Metzler matrices for given stable polynomials

Kaczorek T.

Pomiary Automatyka Kontrola

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2012

|

R. 58, nr 5

407-412

EN

The problem of determination of the set of Metzler matrices for given stable polynomials is formulated and partly solved. For stable polynomial of the second degree there exists a set of Metzler matrices if and only if the polynomial has only real negatives zeros. If the stable polynomial has only real negative zeros then the set of corresponding Metzler matrices is given by the set of lower or upper triangular matrices with diagonal entries equal to the negative real zeros and any nonnegative off-diagonal entries. Sufficient condition are establish for the existence of the set of Metzler matrices for stable polynomials with a real negative zeros and the complex conjugate zeros).

PL

W artykule sformułowani i częściowo rozwiązano problem wyznaczania zbioru macierzy Metzlera dla danych stabilnych wielomianów. Wykazano, ze dla stabilnych wielomianów stopnia drugiego istnieje zbiór macierzy Metzlera wtedy i tylko wtedy, gdy wielomian ten ma tylko ujemne pierwiastki rzeczywiste. Jeżeli stabilny wielomian dowolnego stopnia ma tylko pierwiastki rzeczywiste, to odpowiadający jemu zbiór macierzy Metzlera jest dany zbiorem macierzy dolno lub górno-trójkątnych z elementami na głównej przekątnej równych ujemnym zerom tego wielomianu oraz nieujemnymi elementami poza główną przekątną. Warunkiem koniecznym na to, aby dla danego stabilnego wielomianu istniał zbiór macierzy Metzlera jest posiadanie przez ten wielomian co najmniej dwóch zer przeczystych. Podano warunki dostateczne na istnienie zbioru macierzy Metzlera dla danych stabilnych wielomianów z ujemnymi zerami rzeczywistymi i zespolonymi parami sprzężonymi.

13

Existence and determination of the set of Metzler matrices for given stable polynomials

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2012

|

Vol. 22, no. 2

389-399

EN

The problem of the existence and determination of the set of Metzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples.

14

Some applications of dominated convergence theorems to a higher order singular boundary value problem

Yao Q.

Journal of Applied Analysis

|

2010

|

Vol. 16, nr 1

15-30

EN

The existence of positive solution is considered for a singular higher-order boundary value problem, where the nonlinear term is a strong Carathéodory function. Two new existence theorems are proved by applying the Lebesgue dominated convergence theorem, the Fatou lemma and the Krasnosel'skii fixed point theorem of cone expansion or cone compression type.

15

Positive partial realization problem for linear discrete-time systems

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2007

|

Vol. 17, no 2

165-171

EN

A partial realization problem for positive linear discrete-time systems is addressed. Sufficient conditions for the existence of its solution are established. A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed. The procedure is illustrated by four numerical examples.

16

A realization problem for positive continuous-time systems with reduced numbers of delays

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2006

|

Vol. 16, no 3

325-331

EN

A realization problem for positive, continuous-time linear systems with reduced numbers of delays in state and in control is formulated and solved. Sufficient conditions for the existence of positive realizations with reduced numbers of delays of a given proper transfer function are established. A procedure for the computation of positive realizations with reduced numbers of delays is presented and illustrated by an example.

17

How to prove existence in shape optimization

Bucur D.

Control and Cybernetics

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2005

|

Vol. 34, no 1

103-116

EN

This paper deals with the existence question in optimal design. We present, a general variational technique for proving existence, and give several examples concerning functionals of eigenvalues and of energy type. In particular, we show how the isoperi-metric problem for the Dirichlet eigenvalues of an elliptic operator of general order fits into this frame.

18

On the connective Cahn-Hilliard equation

Liu C.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

|

Vol. 53, no 3

299-314

EN

The author studies the convective Cahn-Hilliard equation. Some results on the existence of classical solutions and asymptotic behavior of solutions are established. The instability of the traveling waves is also discussed.

19

Existence in optimal control problems of certain Fredholm integral equations

Roubicek T., Schmidt W.

Control and Cybernetics

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2001

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Vol. 30, no 3

303-322

EN

Existence of an optimal control for certain systems governed by nonlinear Fredholm integral equations that are of a Hammerstein type wlith respect to the control is proved under convexity of the orientor field by using a relaxed problem. Then, this convexity assumption is put away by finer analysis of the maximum principle. Illustrative examples are presented.