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This paper establishes a theory of calculus for fuzzy-number-valued functions whose range is embedded in a specific Banach subspace of fuzzy numbers. These spaces are generated by strongly linearly independent (SLI) fuzzy numbers whose operations are induced by an isomorphism with R n . We use the notion of Frechet differentiability and Riemann integrability for these functions and present a fundamental theorem of calculus. Lastly, we develop a theory of fuzzy differential equations and provide an existence and uniqueness theorem.

In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are obtained to confirm the validity of the proposed method.

The purpose of this paper is to establish an existence and uniqueness theorem for the nonlocal fourth-order nonlinear beam differential equations with a parameter $ \begin{equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0 subject to the integral boundary conditions: $ \begin{equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, \end{equation*} $ where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $

In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model.

In this paper, the authors establish the existence and uniqueness theorem of Lp (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of Lp solutions for one-dimensional BDSDEs is also proved. These conclusions unify and generalize some known results.

This article is concerned with the study of the unique solvability of a time-nonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

In this paper, we investigate mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence and uniqueness theorem are obta...

Based on the effects of white noise and colored noise, we propose a stochastic Holling-III predator–prey model in an impulsive polluted environment. Firstly, we prove an existence and uniqueness theorem of the presented model. Secondly, we establish sufficient criteria of extinction, nonpersistence in mean, and weak persistence in mean for both prey and predator species. Thirdly, with the aid of Lyapunov functions, we prove that this system is ergodic and has a unique stationary distribution under certain conditions. Finally, we verify the theoretical results by performing some numerical simulations.

The aim of this paper is to study a comprehensive dynamics system called fractional fuzzy differential variational inequality, which is composed of a nonlinear fractional differential equation with Atangana-Baleanu fractional derivative and a time-dependent fuzzy variational inequality. We explore an existence and uniqueness theorem for the dynamics system under consideration. Our proof is based on F-KKM theorem, Banach fixed point principle, and the theory of monotone operators.

We introduce a type of Geraghty contractions in aJS-metric spaceX,calledα,D-proximal generalized Geraghty mappings. By using the triangular-α,D-proximal admissible property, we obtain the existence and uniqueness theorem of best proximity coincidence points for these mappings together with some corollaries and illustrative examples. As an application, we give a best proximity coincidence point result inXendowed with a binary relation.

This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.

Boundary-value problems for a third-order hyperbolic equation that is degenerate inside a domain and contains the Aller operator in the principal part, are examined. The existence and uniqueness theorem for solutions of the problem is proved.

Stoilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z -> z(k) and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps. (C) 2019 Elsevier GmbH. All rights reserved.

We consider a nonstationary Prandtl-type system of equations that describes the behavior of a boundary layer of a viscous incompressible fluid in the modification of O. A. Ladyzhenskaya. We prove an existence and uniqueness theorem both in Cartesian coordinates and in terms of the Crocco variables.

A quasi-stationary Darwin approximation for the system of Maxwell’s equations in inhomogeneous conducting media is studied. An existence and uniqueness theorem for the initial-boundary value problem for the resulting system of differential equations is proved. Estimates of the proximity between the solutions of the quasi-stationary problem under consideration and the corresponding nonstationary problem, depending on the characteristic values of the data, are presented.

No previous study has involved uncertain delay differential equations with jump. In this paper, we consider the uncertain delay differential equations with V-jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, we give the equivalent integral equation. Next, we establish an existence and uniqueness theorem of solution to the differential equations we proposed in the finite domain and the infinite domain, respectively. Once more, the concept of stability for uncertain delay differential equations with V-jump is proposed. In addition, the sufficient condition for stability theorem is derived. To judge existence, uniqueness, and stability briefly, we provide some examples in the end.

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise comparison between Berwald, Cartan and Chern connections is presented., LaTeX file, 14 pages

In this article the definition of strong solution of Ito-Skorokhod stochastic dynamic systems of random structure with external disturbances and all prehistory is presented, important inequalities, which are used to prove Existence and Uniqueness theorems are proved. Global Existence and Uniqueness theorem is proved.

This article studies a nonlocal inverse boundary-value problem for a two-dimensional second-order parabolic equation in a rectangular domain. The purpose of the article is to determine the unknown coefficient and the solution of the considered problem. To investigate the solvability of the inverse problem, we transform the original problem into some auxiliary problem with trivial boundary conditions. Using the contraction mappings principle, existence and uniqueness of the solution of an equivalent problem are proved. Further, using the equivalency, the existence and uniqueness theorem of the classical solution of the original problem is obtained.

For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

In this paper, a priori estimate for a linear third pseudoparabolic operator with bound is established, and applying the above result, the existence and uniqueness theorem of solutions for a class of nonlinear pseudoparabolic equations is obtained with the help of the homeomorphism method and the initial value method. Furthermore, an existence and uniqueness theorem of the semilinear equation is obtained as a corollary.

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

This paper studies conformable stochastic functional differential equations of neutral type. Firstly, the existence and uniqueness theorem of a solution is established. Secondly, the moment estimation and exponential stability results are given. Thirdly, the Ulam type stability in mean square is discussed. Finally, two examples are given to illustrate our results.

In this chapter, the first approach is made to establishing the convergence of scaled random sums, considering independent sequences. The classic Lindeberg–Lévy, Khinchine, Lindeberg–Feller, and Liapunov theorems are proved. The main focus is on the treatment of heterogeneous summands, applying the Lindeberg condition, and extensions are given to allow trending (growing or shrinking) variances. The final sections review cases of the central limit theorem under non-standard conditions and α-stable convergence.

This article investigates existence, uniqueness and stability solutions of new fractional Volterra integro-differential equations system with fractional boundary conditions by using the existence and uniqueness theorem. Theorems on existence and uniqueness of solution are established under some necessary and sufficient conditions on compact space. A simple example of application of the main results of this article is presented.

In this paper, we consider fractional integro-differential equations that involve convergent series on their right hand sides. We prove new existence and uniqueness theorem using Banach contraction...

In this paper, by using the Banach fixed point theorem, we prove the existence and uniqueness theorem of a fractional Volterra integral equation in the space of Lebesgue integrable 𝐿1(𝑅+) on unbounded interval [0,∞).

This paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of symmetrical martingale solutions (SM-solutions, for short) is introduced for BDSVIEs. And the existence and uniqueness theorem for BDSVIEs in the sense of SM-solutions is established., Accept by Computers and Mathematics with Applications

The mixed problem for nonlinear strongly dissipative wave equations with acoustic transmission conditions is considered. An existence and uniqueness theorem for local solutions is proved using the Faedo–Galerkin approximations, the compactness method, and the fixed point theorem. The existence of global solutions of this problem is also proved.

The purpose of this work is to present the dynamics of a fractional SIR model with generalized incidence rate using two differential derivatives, that are the Caputo and the AtanganaBaleanu. Firstly, we formulate the proposed model in Caputo sense and carried out the basic mathematical analysis such as positivity, basic reproduction number, local and global dynamics of the disease free and endemic case. The disease free equilibrium is locally and globally asymptotically stable when the basic reproduction number is less than 1. We also show that the SIR fractional model is locally and globally asymptotically stable at endemic state when the basic reproduction number greater than unity. Further, to analyze the dynamics of Caputo model we provide some numerical results by considering various incidence rates and with different fractional order parameters. Then, we reformulate the same model using the Atangana-Baleanu operator having nonsingular and non-local kernel. We prove the existence and uniqueness of the Atangana-Baleanu SIR model using PicardLindelof approach. Furthermore, we present an iterative scheme of the model and provide graphical results for various values of the fractional order α. From the graphical interpretation we conclude that the Atangana-Baleanu derivative is more prominent and provides biologically more feasible results than Caputo operator.

In this paper, we consider a free boundary problem related to ductal carcinoma in situ (DCIS) model. Existence and uniqueness theorem is proved by using Banach fixed point theorem. For computation ...

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by \begin{document}$ G $\end{document} -Brownian motion (VNSDDEGs in short) under global Caratheodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in \begin{document}$ L^p $\end{document} -sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.

We consider mixed problems for nonlinear equations of magnetoelasticity. Our main result in the three-dimensional case is the proof of an existence and uniqueness theorem; uniqueness is established under some extra restrictions on the smoothness of solutions. We also manage to prove the existence and uniqueness of a weak solution to the problem in the two-dimensional case; uniqueness is established without any additional a priori assumptions on the smoothness of solutions.

In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrodinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x ∈ Ω = [0,L], 0 ≤ t u (x,0) = u0 (x) ∈ H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here α is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrodinger equation under initial condition and Robin inhomogeneous boundary condition.

In this article, a numerical technique based on a combination of the Picard iteration method and hat basis functions to solve nonlinear stochastic Ito-Volterra integral equations with m-dimensional Brownian motion is proposed. The existence and uniqueness theorem for the solution of this class of Ito-Volterra integral equations is proved. Also, convergence analysis of the suggested method is investigated in details. Finally, some numerical examples are provided to demonstrate the accuracy of the proposed method and guarantee the theoretical results.

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.

We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a given pair of Galois- and reductive-group- representations can be identified as local Langlands correspondents. The results, which can be used to study local-global compatibility questions along eigenvarieties, are largely analogous to those of Emerton, Helm, and Moss on the local Langlands correspondence over local rings of mixed characteristic. We apply the theory to the interpolation of local zeta integrals and of $L$- and $\varepsilon$-factors., 28 pages. Several corrections and improvements (main results unchanged)

In the multidimensional case, we study the problem with initial and boundary conditions for the equation of vibrations of a beam with one end clamped and the other hinged. An existence and uniqueness theorem is proved for the posed problem in Sobolev classes. A solution of the problem under consideration is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem for which the eigenvalues are determined as the roots of a transcendental equation and the system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem about the uniqueness of a solution to the posed initial-boundary value problem is stated.