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Let T be a periodic time scale. The purpose of this paper is to use Krasnoselskii's fixed point theorem to prove the existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale. We invert these equations to construct a sum of a contraction and a compact map which is suitable for applying the Krasnoselskii's theorem. The results obtained here extend the work of Candan c1.

The paper addresses a nonlinear heat equation (the porous medium equation) in the case of a power-law dependence of the heat conductivity coefficient on temperature. The equation is used for describing high-temperature processes, filtration of gases and fluids, groundwater infiltration, migration of biological populations, etc. The heat waves (waves of filtration) with a finite velocity of propagation over a cold background form an important class of solutions to the equation under consideration. A special boundary value problem having solutions of such type is studied. The boundary condition of the problem is given on a sufficiently smooth closed curve with variable geometry. The new theorem of existence and uniqueness of the analytic solution is proved.

In this paper, we consider a non-instantaneous impulsive system represented by second order nonlinear differential equation with deviated argument in a Banach space X. We used the strongly continuous cosine family of linear operators and Banach fixed point method to study the existence and uniqueness of the solution of the non-instantaneous impulsive system. Also, we study the existence and uniqueness of the solution of the nonlocal problem and stability of the non-instantaneous impulsive system. Finally, we give examples to illustrate the application of these abstract results. Keywords: Non-instantaneous impulses, Deviated argument, Banach fixed point theorem

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

The purpose of this paper is to introduce and investigate a new system of global fractional-order interval implicit projection neural networks. An existence and uniqueness theorem of the equilibrium point for the system of global fractional-order interval implicit projection neural networks is obtained under some suitable assumptions. Moreover, Mittag–Leffler stability for the system of global fractional-order interval implicit projection neural networks is also proved. Finally, two numerical examples are given to illustrate the validity of our results.

We consider the class of boundary value problems for elliptic, parabolic, and hyperbolic equations in cylinders separated by a two-layer film into two half-cylinders. We prove the existence and uniqueness theorem and express the solutions in terms of solutions to analogous classical problems in cylinders without films.

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

In this paper, we study the existence and uniqueness theorem for L p ( 1 p ≤ 2 ) solutions to a class of infinite time interval backward doubly stochastic differential equations (BDSDEs for short) under monotonicity and general increasing conditions. Furthermore, we obtain the comparison theorem for 1-dimensional infinite time interval BDSDEs in L p ( 1 p ≤ 2 ) .

A nonlocal boundary-value problem for a linear ordinary differential equation with fractional discretely distributed differentiation operator is considered. The existence and uniqueness theorem for the solution of this problem is proved.

The fact that every combinator has a fixed point is at the heart of the λ -calculus as a model of computation. We consider several aspects of such phenomenon; our specific, perhaps eccentric, point of view focuses on problems and results that we consider worthy of further investigations. We first consider the relation with self application, in comparison with the opposite view, which stresses the role of coding, unifying the first and the second fixed point theorems. Then, we consider the relation with the diagonal argument, a relation which is at the origin of the fixed point theorem itself. We also review the Recursion Theorem, which is considered a recursion theoretic version of the fixed point theorem. We end considering systems of equations which are related to fixed points.

We present a generalized formulation of sweeping process where the behavior of the solution is prescribed at the jump points of the driving moving set. An existence and uniqueness theorem for such formulation is proved. As a consequence we derive a formulation and an existence/uniqueness theorem for sweeping processes driven by an arbitrary $${\textit{BV}}$$ moving set, whose evolution is not necessarily right continuous. Applications to the play operator of elastoplasticity are also shown.

We prove Liouville-type theorem for semilinear parabolic system of the form \begin{document}$u_t-\Delta u =a_{11}u^{p}+a_{12} u^rv^{s+1}$\end{document} , \begin{document}$v_t-\Delta v =a_{21} u^{r+1}v^{s}+a_{22}v^{p}$\end{document} where \begin{document}$r, s>0$\end{document} , \begin{document}$p=r+s+1$\end{document} . The real matrix \begin{document}$A=(a_{ij})$\end{document} satisfies conditions \begin{document}$ a_{12}, a_{21}\geq 0$\end{document} and \begin{document}$a_{11}, a_{22}>0$\end{document} . This paper is a continuation of Phan-Souplet (Math. Ann., 366,1561-1585,2016) where the authors considered the special case \begin{document}$s=r$\end{document} for the system of \begin{document}$m$\end{document} components. Our tool for the proof of Liouville-type theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34,525–598 1981) and Bidaut-Veron (Equations aux derivees partielles et applications. Elsevier, Paris, pp 189–198,1998).

S.G. Pyatkov, M.A. Verzhbitskii Yugra State University, Khanty-Mansyisk, Russian Federation E-mail: s_pyakov@ugrasu.ru. С.Г. Пятков, М.А. Вержбицкий Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: s_pyakov@ugrasu.ru In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions. Рассматривается обратная задача об определении вместе с решением начально-краевой задачи для параболического уравнения второго порядка неизвестных функций, входящих в граничное условие Дирихле. Задачи такого вида об определении граничных данных возникают в самых различных задачах математической физики: управление процессами теплообмена и проектирование тепловой защиты, диагностика и идентификация теплопередачи в сверхзвуковых гетерогенных потоках, идентификация и моделирование теплопереноса в теплозащитных материалах и покрытиях, моделирование свойств и тепловых режимов многоразовой тепловой защиты аэрокосмических аппаратов, исследование композиционных материалов и т. п. В качестве условий переопределения берутся интегралы от решения по пространственной области с весами. Проблема сводится к операторному уравнению типа Вольтерра. Мы устанавливаем теорему существования и единственности решений этой обратной задачи в пространствах Соболева. Решение является регулярным, т. е. все обобщенные производные, входящие в уравнение существуют и суммируемы с некоторой степенью. Доказательство основано на теореме о неподвижной точке и последовательном доказательстве разрешимости на малых промежутках времени. Приведена также оценка устойчивости решений. Полученные условия разрешимости близки к необходимым условиям. Публикация подготовлена в результате проведения научного исследования за счет средств гранта на развитие научных школ с участием молодых ученых федерального государственного бюджетного образовательного учреждения высшего образования «Югорский государственный университет»

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

Talking about a black hole, one has in mind the process of unlimited self-compression of gravitating matter with a mass greater than critical. With a mass greater than the critical one, the elasticity of neutron matter cannot withstand gravitational compression. However, compression cannot be unlimited, because with increasing pressure, neutrons turn into some other “more elementary” particles. These can be bosons of the Standard Model of elementary particles. The wave function of the condensate of neutral bosons at zero temperature is a scalar field. If instead of the constraint det gik < 0 we use a weaker condition of regularity (all invariants of the metric tensor gik are finite), then there is a regular static spherically symmetric solution to Klein-Gordon and Einstein equations, claiming to describe the state to which the gravitational collapse leads. With no restriction on total mass. In this solution, the metric component grr changes its sign twice: g rr (r) = 0 at r=rg and r=rh > rg . Between these two gravitational radii the signature of the metric tensor gik is (+, +, -, -). Gravitational radius rg inside the gravitating body ensures regularity in the center. Within the framework of the phenomenological model “λψ4 ”, relying on the existence and uniqueness theorem, the main properties of a collapsed black hole are determined. At r = rg a regular solution to Klein-Gordon and Einstein equations exists, but it is not a unique one. Gravitational radius rg is the branch point at which, among all possible continuous solutions, we have to choose a proper one, corresponding to the problem under consideration. We are interested in solutions that correspond to a finite mass of a black hole. It turns out that the density value of bosons is constant at r < rg. It depends only on the elasticity of a condensate, and does not depend on the total mass. The energy-momentum tensor at r ⩽ rg corresponds to the ultra relativistic equation of state p = ɛ/3. In addition to the discrete spectrum of static solutions with a mass less than the critical one (where grr < 0 does not change sign), there is a continuous spectrum of equilibrium states with grr(r) changing sign twice, and with no restriction on mass. Among the states of continuous spectrum, the maximum possible density of bosons depends on the mass of the condensate and on the rest mass of bosons. The rest energy of massive Standard Model bosons is about 100 GeV. In this case, for the black hole in the center of our Milky Way galaxy, the maximum possible density of particles should not exceed 3 × 1081 cm-3.

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L 2 stability of the LDG scheme and optimal L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.

In this work, we give sufficient conditions for the existence of a mild solution for some impulsive integro-differential equations in Banach spaces. We study the existence without assuming the Lipschitz condition on the nonlinear term f. The compactness on the C0-semigroup(T(t))t≥0 in a Banach space is not needed. We use Hausdorff’s measure of noncompactness, resolvent operators and Darbo’s fixed point Theorem to obtain the main result of this work. Keywords: Impulsive integro-differential equations, Mild solutions, Sadovskii’s fixed point Theorem, Resolvent operators, Noncompactness measures

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time existence and uniqueness theorem for a $G$-invariant solution $g(t)$ satisfying the boundary condition $\mathop{\mathrm{II}}(g(t))=F(t,g_{\partial M}(t))$ and the initial condition $g(0)=\hat g$. Here, $\mathop{\mathrm{II}}(g(t))$ is the second fundamental form of $\partial M$, $g_{\partial M}$ is the metric induced on $\partial M$ by $g(t)$, $F$ is a smooth map and $\hat g$ is a metric on $M$. Second, we study Perelman's $\mathcal F$-functional on $M$. Our results show, roughly speaking, that $\mathcal F$ is non-decreasing on a $G$-invariant solution to the modified Ricci flow, provided that this solution satisfies boundary conditions inspired by the 2012 paper of Gianniotis., Comment: 18 pages

In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions.

We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential equations in which the latter are determined on the boundary. This system describes a model of bimolecular surface reaction between carbon monoxide and nitrous oxide running on supported rhodium in the case of slow desorption of the products.

This paper proposes an accelerated iterative procedure for a nonlinear fourth order elliptic equation with nonlocal boundary conditions. First, an existence and uniqueness theorem is proved for the fourth order elliptic equation via the accelerated iterative procedure. To solve this problem numerically, a finite difference based numerical scheme is also developed in view of the main theorem. Theoretically, the monotone property as well as the convergence analysis are proved for both the continuous and discretized cases. The main result also supplements several algorithms for computing the solution of the fourth order elliptic integro-partial differential equation. The proposed scheme not only accelerates the scheme in the literature but also provides a greater flexibility in choosing the initial guess. The efficacy of the proposed scheme is demonstrated through a comparative numerical study with the recent literature. The numerical simulation confirms the theoretical claims too.

In 1995, Palfy and Szabo stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Palfy and Szabo is identified as the Reidemeister condition.

We study the following fractional Yamabe-type equation: { A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. Here Ω is a regular bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , and A s {A_{s}} , s ∈ ( 0 , 1 ) {s\in(0,1)} , represents the fractional Laplacian operator ( - Δ ) s {(-\Delta)^{s}} in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.

The generalized commutator of a list of k real matrices is defined as a multilinear skew-symmetric function and the linear operator on the vector space is defined by . The Amitsur–Levitzki ...

Let $A$ be a Banach algebra, and let $\mathcal{X}$ be a left Banach $A$ -module. In this paper, using the notation of point multipliers on left Banach modules, we introduce a certain type of spectrum for the elements of $\mathcal{X}$ and we also introduce a certain subset of $\mathcal{X}$ which behaves as the set of invertible elements of a commutative unital Banach algebra. Among other things, we use these sets to give some Gleason–Kahane–Żelazko theorems for left Banach $A$ -modules.

In this paper, a generalized fixed point theorem and its results are established in the concept of multiplicative distance which was introduced by Agamirza et.al [3] to improve the non-Newtonian calculus. Our results include some existing results in the concept of multiplicative metric space.

Chemical self-assembly has been considered one of the most important scientific problems in the 21st century; however, because the process of self-assembly is very complex, there is currently little mathematic theory describing it. Based on some assumptions of the classic hard sphere model, this paper presents a novel multi-agent system for chemical self-assembly that can be formulated as a group of stochastic differential equations. In consideration of this model, an existence and uniqueness theorem of the solution is presented. An optimal problem is proposed by taking the temperature as the control input and choosing the Hamiltonian as the optimal object, and a numerical solution for this optimal problem is also developed. Simulations show that the proposed control scheme can greatly improve the product of self-assembly.

The tool of uncertain fractional differential equations (UFDEs) is devoted to describing the behavior of complex systems with memory effects in the uncertain environment. In this paper, we mainly investigate the Caputo-Hadamard UFDEs. First, the definition of Caputo-Hadamard UFDE is proposed and the analytical solution to a linear Caputo-Hadamard UFDE is provided. Then, an existence and uniqueness theorem of solution to Caputo-Hadamard UFDE is studied.

Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting cases to the electrical circuit. Moreover, we investigate Ulam–Hyers stability of Caputo type fractional Langevin differential equation. At the end, we provide an example to verify our main results.

In this paper, we study the following backward stochastic differential equations driven by G -Brownian motion ( G -BSDEs in short) Y t = ξ + ∫ t T f ( s , Y s , Z s ) d s + ∫ t T g ( s , Y s , Z s ) d 〈 B 〉 s − ∫ t T Z s d B s − ( K T − K t ) with a kind of non-Lipschitz coefficients. An existence and uniqueness theorem is established.

The main aim of this article is to establish analogues of Landau’s theorem for solutions to the$\overline{\unicode[STIX]{x2202}}$-equation in Dirichlet-type spaces.

In this paper, we consider the Cauchy problem of a fluid-particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro-micro decomposition.

The nonlinear versions of Sturm-Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.

We use suitably Page’s theorem to get effective results for interesting problems, by avoiding the ineffective Siegel’s theorem.

In this article, we study existence and uniqueness of a class of highly non-linear boundary value problem of fractional order differential equations. The concerned problem is investigated by means of classical fixed point theorem for the mentioned requirements. The Ulam–Hyer’s stability is also established for the class of fractional differential equations. Appropriate example is also provided which demonstrate the applicability of our results.

The Valiron–Mohon’ko theorem plays an important role in the theory of complex differential equations. In this paper, a difference analogue of the Valiron–Mohon’ko theorem is established, which can be used to get the characteristic functions on irreducible rational functions of f(z) and its shifts. Using our results and some properties of periodic functions, the proof of [10, Theorem 1.1] can be organized in a short way.

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ2×2 and Julia set of one dimensional complex dynamic system.

In this paper, we study sufficient conditions for the existence of positive solutions to a class of fractional differential equations of arbitrary order. Our solutions are placed in the space of Lipschitz functions and, perhaps, this is a part of the originality of the paper. For our study, we use a recent result about the relative compactness in Holder spaces and the classical Schauder fixed point theorem.

We study the existence of solutions continuously depending on a parameter for higher-order nonlinear ordinary differential equations with linear boundary conditions. In particular, we prove a theorem of Fredholm type providing tests for the unique solvability of this problem.

In this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields., Dans cette note, nous prouvons le théorème du flot tubulaire pour les champs vectoriels Lipschitz à divergence nulle.

In this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem.

Summary A well-known result on pathwise uniqueness of the solution of stochastic differential equations in R is the Yamada-Watanabe theorem. We have extended this result by replacing the Lipschitz assumption on the drift coefficient by much weaker assumption of semi-monotonicity.