Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhemar Jean-Claude Barre de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in order that the considered system has global classical solutions. In order to prove our main results we use a new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.

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Existence of solutions for a class of impulsive Burgers equation

Georgiev Svetlin G., Hakem Ali

Статья научная

We study a class of impulsive Burgers equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. The arguments are based on recent theoretical results. Here we focus our attention on a class of Burgers equations and we investigate it for the existence of classical solutions. The Burgers equation can be used for modeling both traveling and standing nonlinear plane waves. The simplest model equation can describe the second-order nonlinear effects connected with the propagation of high-amplitude (finite-amplitude waves) plane waves and, in addition, the dissipative effects in real fluids. There are several approximate solutions to the Burgers equation. These solutions are always fixed to areas before and after the shock formation. For an area where the shock wave is forming no approximate solution has yet been found. Therefore, it is therefore necessary to solve the Burgers equation numerically in this area.

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Existence results for a Dirichlet boundary value problem involving the p(x)-Laplacian operator

Ait Hammou Mustapha

Статья научная

The aim of this paper is to establish the existence of weak solutions, in W1,p(x)0(Ω), for a Dirichlet boundary value problem involving the p(x)-Laplacian operator. Our technical approach is based on the Berkovits topological degree theory for a class of demicontinuous operators of generalized (S+) type. We also use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, and specially properties of p(x)-Laplacian operator. In order to use this theory, we will transform our problem into an abstract Hammerstein equation of the form v+S∘Tv=0 in the reflexive Banach space W-1,p′(x)(Ω) which is the dual space of W1,p(x)0(Ω). Note also that the problem can be seen as a nonlinear eigenvalue problem of the formAu=λu, where Au:=-Div(|∇u|p(x)-2∇u)-f(x,u). When this problem admits a non-zero weak solution u, λ is an eigenvalue of it and u is an associated eigenfunction.

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$Existence results for functional perturbed differential equations of fractional order with state-dependent delay in Banach spaces$

Existence results for functional perturbed differential equations of fractional order with state-dependent delay in Banach spaces

Helal Mohamed

Статья научная

In this paper, we provide sufficient conditions for the existence of solutions of initial value problem, for perturbed partial functional hyperbolic differential equations of fractional order involving Caputo fractional derivative with state-dependent delay by reducing the research to the search of the existence and the uniqueness of fixed points of appropriate operators. Our main result for this problem is based on a nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Banach spaces due to Burton and Kirk and a fractional version of Gronwall's inequality. We should observe the structure of the space and the properties of the operators to obtain existence results. To our knowledge, there are very few papers devoted to fractional differential equations with finite and/or infinite constant delay on bounded domains. Many other questions and issues can be investigated regarding the existence in the space of weighted continuous functions, the uniqueness, the structure of the solutions set and also whether or not the condition satisfied by the operators are optimal. This paper can be considered as a contribution in this setting case. Examples are given to illustrate this work.

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$Existence theorem for a fractal Sturm-Liouville problem$

Existence theorem for a fractal Sturm-Liouville problem

Allahverdiev B.P., Tuna H.

Статья научная

In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm-Lioville type problem is discussed, namely the fractal Sturm-Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of Fα-calculus defined on fractal subsets of real numbers are given, i.e., Fα-continuity, Fα-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm-Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm-Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.

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Exponential stability for a swelling porous-heat system with thermodiffusion effects and delay

Douib M., Zitouni S.

Статья научная

In the present work, we consider a one-dimensional swelling porous-heat system with single time-delay in a bounded domain under Dirichlet-Neumann boundary conditions subject to thermodiffusion effects and frictional damping to control the delay term. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous-heat. At first, we state and prove the well-posedness of the solution of the system by the semigroup approach using Lumer-Philips theorem under suitable assumption on the weight of the delay. Then, we show that the considered dissipation in which we depended on are strong enough to guarantee an exponential decay result by using the energy method that consists to construct an appropriate Lyapunov functional based on the multiplier technique, this result is obtained without the equal-speed requirement. Our result is new and an extension of many other works in this area.

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Extension of an almost f-algebra multiplication

Chil Elmiloud

Статья научная

It is proved that an almost f-algebra multiplication and a d-algebra multiplication defined on a majorizing vector sublattice of a Dedekind complete vector lattice can be extended to the whole vector lattice by using purely algebraic and order theoretical means.

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Functional calculus and Minkowski duality on vector lattices

Kusraev Anatoly G.

Статья научная

The paper extends homogeneous functional calculus on vector lattices. It is shown that the function of elements of a relatively uniformly complete vector lattice can naturally be defined if the positively homogeneous function is defined on some conic set and is continuous on some closed convex subcone. An interplay between Minkowski duality and homogeneous functional calculus leads to the envelope representation of abstract convex elements generated by the linear hull of a finite collection in a uniformly complete vector lattice.

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Functions with uniform sublevel sets on cones

Dastouri A., Ranjbari A.

Статья научная

Extended real-valued functions on a real vector space with uniform sublevel sets are important in optimization theory. Weidner studied these functions in [1]. In the present paper, we study the class of these functions, which coincides with the class of Gerstewitz functionals, on cones. These cone are not necessarily embeddable in vector spaces. Almost any Weidner's results are not true on cones without extra conditions. We show that the mentioned conditions are necessary, by nontrivial examples. Specially for element k from the cone P, we define k-directional closed subsets of the cone and prove some properties of them. For a subset A of the cone P, we characterize domain of the φA,k (function with uniform sublevel set) and show that this function is k-transitive. One of the important conditions for satisfying the results, is that k has the symmetric element in the cone. Also, we prove that, under some conditions, the class of Gerstewitz functionals coincides with the class of k-translative functions on P.

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Generalization of Eberlein's and sine's ergodic theorems to lr-nets

Emelyanov Eduard, Nazife Erkusan

Статья научная

The notion of LR-nets provides an appropriate setting for study of various ergodic theorems in Banach spaces. In the present paper, we prove Theorems 2.1, 3.1 which extend Eberlein's and Sine's ergodic theorems to LR-nets. Together with Theorem 1.1, these two theorems form the necessary background for further investigation of strongly convergent LR-nets. Theorem 2.1 is due to F. Rabiger, and was announced without a proof in [1].

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Generalization of the Ostrowski inequalities on time scales

Khan A.R., Mehmood F., Shaikh M.A.

Статья научная

The idea of time scales calculus’ theory was initiated and introduced by Hilger (1988) in his PhD thesis order to unify discret and continuous analysis and to expend the discrete and continous theories to cases ``in between''. Since then, mathematical research in this field has exceeded more than 1000 publications and a lot of applications in the fields of science, i.e., operations research, economics, physics, engineering, statistics, finance and biology. Ostrowski proved an inequality to estimate the absolute deviation of a differentiable function from its integral mean. This result was obtained by applying the Montgomery identity. In the present paper we derive a generalization of the Montgomery identity to the various time scale versions such as discrete case, continuous case and the case of quantum calculus, by obtaining this generalization of Montgomery identity we would prove our results about the generalization of the Ostrowski inequalities (without weighted case) to the several time scales such as discrete case, continuous case and the case of quantum calculus and recapture the several published results of different authors of various papers and thus unify corresponding discrete version and continuous version. Similarly we would also derive our results about the generalization of the Ostrowski inequalities (weighted case) to the different time scales such as discrete case and continuous case and recapture the different published results of several authors of various papers and thus unify corresponding discrete version and continuous version. Moreover, we would use our obtained results (without weighted case) to the case of quantum calculus.

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Gift of the teacher GP Akilov

Kutateladze Semen S.

Персоналии

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Grand Morrey type spaces

Samko Stefan G., Umarkhadzhiev Salaudin M.

Статья научная

The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets Ω with finite measure |Ω|

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$H-операторы в идеальных пространствах со смешанной квазинормой E (\ Omega)$

H-операторы в идеальных пространствах со смешанной квазинормой E (\ Omega)

Фетисов Валерий Георгиевич, Рандриананжа Р.Р.

Статья научная

Цель настоящей работы - с единой точки зрения рассмотреть поведение нелинейных операторов типа суперпозиции, интегральных операторов Гаммерштейна и Урысона в общих квазинормированных идеальных пространствах. Некоторые из нижеприведенных результатов анонсированы нами ранее в статье [1].

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$H\"Older type inequalities for orthosymmetric bilinear operators$

H\"Older type inequalities for orthosymmetric bilinear operators

Kusraev Anatoly G.

Статья научная

An interplay between squares of vector lattice and homogeneous functional calculus is considered and H\"older type inequalities for orthosymmetric bilinear operators are obtained.

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Hankel determinant of third kind for certain subclass of multivalent analytic functions

Vamshee Krishna D., Shalini D.

Статья научная

The objective of this paper is to obtain an upper bound (not sharp) to the third order Hankel determinant for certain subclass of multivalent (p-valent) analytic functions, defined in the open unit disc E. Using the Toeplitz determinants, we may estimate the Hankel determinant of third kind for the normalized multivalent analytic functions belongng to this subclass. But, using the technique adopted by Zaprawa [1], i.e., grouping the suitable terms in order to apply Lemmas due to Hayami [2], Livingston [3] and Pommerenke [4], we observe that, the bound estimated by the method adopted by Zaprawa is more refined than using upon applying the Toeplitz determinants.

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Hardy type inequalities in classical and grand Lebesgue spaces lp), 0

Ouardani Abderrahmane, Abdelkader Senouci

Статья научная

In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1), 0In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1), 0далее...

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Homogeneous functions of regular linear and bilinear operators

Kusraev Anatoly Georgievich

Статья научная

Using envelope representations explicit formulae for computing \widehat{\varphi}(T_1,...,T_N) for any finite sequence of regular linear or bilinear operators T_1,...,T_N on vector lattices are derived.

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I. I. Gordon who was an adressee of L. S. Pontryagin (introductory notes)

Gordon E.I.

Персоналии

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Increasing unions of Stein spaces with singularities

Alaoui Youssef

Статья научная

We show that if X is a Stein space and, if Ω⊂X is exhaustable by a sequence Ω1⊂Ω2⊂…⊂Ωn⊂… of open Stein subsets of X, then Ω is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for X=Cn and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When X has dimension 2, we prove that the same result follows if we assume only that Ω⊂⊂X is a domain of holomorphy in a Stein normal space. It is known, however, that if X is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets X1⊂X2⊂⋯⊂Xn⊂…, it does not follow in general that X is holomorphically-convex or holomorphically-separate (even if X has no singularities). One can even obtain 2-dimensional complex manifolds on which all holomorphic functions are constant.

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