Existence Theorems Of Solutions Research Articles

General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator

In this paper, we consider the general mean-field backward doubly stochastic differential equations (mean-field BDSDEs) whose generator f can be discontinuous in y. We prove the existence theorem of solutions under stochastic linear growth conditions and also obtain the related comparison theorem. Naturally, we present those results under the linear growth condition, which is a special case of the stochastic condition. Finally, a financial claim sale problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance.

On the Fredholm Property and Solvability of a System of Integral Equations in the Transmission Problem for the Helmholtz Equation

A scalar three-dimensional boundary value problem of wave diffraction for the Helmholtz equation with transmission conditions that assume the presence of an infinitely thin material at the media interface is considered. Uniqueness and existence theorems for solutions are proved. The original problem is reduced to a system of integral equations over the media interface. Calculation formulas for the system of linear algebraic equations obtained after applying the collocation method and numerical results for solving the problem when the domain is a ball with certain transmission conditions are given.

The existence theory of solution in Sobolev space for fractional differential equations

We study the solution theory in Sobolev space for fractional differential equations of Caputo type: CDxαu(x)=f(x,u),0<α<1, with given initial value. Under proper conditions, we establish the existence theorem of solution in space Wp1(0,b). We emphasize that our existence theorem has no any continuity requirement on f(x,u) with respect to x∈[0,b]. The uniqueness condition is also given. Several examples are provided to support our theoretical analysis.

Mathematical Model of Human Capital Dynamics

A mathematical description of household economic behavior is studied. On the one hand, households are consumers that seek to maximize the discounted utility function in an imperfect market of savings and consumer loans. On the other hand, households are workers in the labor market; they receive a wage and seek to enhance their skills to receive a higher wage. An increase in the level of worker’s skill is achieved via investment in human capital. In this paper, a mathematical model of the worker’s behavior in the labor market is represented in the form of an infinite-horizon optimal control problem. A solution existence theorem is proved, and necessary optimality conditions are obtained in the form of Pontryagin’s maximum principle. The model is identified using Russian statistical data for various social layers.

Variational approach to instantaneous and noninstantaneous impulsive system of differential equations

In this paper, the existence and multiplicity of solutions for a coupled system of differential equations with instantaneous and noninstantaneous impulses are studied. By the virtue of variational methods, some new existence theorems of solutions are obtained. In addition, two examples are given to demonstrate our main results.

The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Affine Periodic Boundary Value Conditions

The solution to a sequential fractional differential equation with affine periodic boundary value conditions is investigated in this paper. The existence theorem of solution is established by means of the Leray–Schauder fixed point theorem and Krasnoselskii fixed point theorem. What is more, the uniqueness theorem of solution is demonstrated via Banach contraction mapping principle. In order to illustrate the main results, two examples are listed.

Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control

In this paper, we investigate a nonlinear and nonsmooth dynamics system (NNDS, for short) involving two multi-valued maps which are a convex subdifferential operator and a generalized subdifferential operator in the sense of Clarke, respectively. Under general assumptions, by using a surjectivity theorem for multi-valued mappings combined with the theory of nonsmooth analysis and arguments on pseudomonotone operators, the existence of a solution to (NNDS) is proved. Then, an optimal control problem governed by (NNDS) is introduced, and a solvability result for the optimal control problem is established. Moreover, we study a nonlinear feedback control system driven by (NNDS) and an u.s.c. multi-valued feedback map, and employ the Kakutani-Ky Fan fixed point theorem to obtain an existence theorem of solutions for the feedback control problem. Finally, we deliver a convergence result in the sense of Kuratowski describing the changes in the set of solutions for the feedback control problem as the initial data x0 is perturbed in Hilbert space H.

Solutions of system of generalized vector quasi-equilibrium problems in locally FC-uniform spaces

In this paper, we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally FC-uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally FC-uniform spaces. These theorems are new and generalize some known results in the literature.

A stochastic nonlinear differential propagation model for underwater acoustic propagation: Theory and solution

• The perturb-coefficient nonlinear propagation equation is derived from the continuity equation, the Euler’s equation and the adiabatic equation. • The physical elements are divided into two types, and the location parameter to present the location-affected elements is designed. The stochastic parameter is designed to model the random-occur physical elements. The small parameters are used to demonstrate the weakly-nonlinear theory. • The initial and boundary conditions are analyzed. The solution existence of the SNDP model is proved. • The operator splitting procedure is proposed to solve the model numerically. The principle of underwater acoustic signal propagation is of vital importance to realize the “digital ocean”. However, underwater circumstances are becoming more complex and multi-factorial because of raising human activities, changing climate, to name a few. For this study, we formulate a mathematical model to describe the complex variation of underwater propagating acoustic signals, and the solving method are presented. Firstly, the perturb-coefficient nonlinear propagation equation is derived based on hydrodynamics and the adiabatic relation between pressure and density. Secondly, physical elements are divided into two types, intrinsic and extrinsic. The expression of the two types are combined with the perturb-coefficient nonlinear propagation equation by location and stochastic parameters to obtain the stochastic nonlinear differential propagation model. Thirdly, initial and boundary conditions are analyzed. The existence theorem for solutions is proved. Finally, the operator splitting procedure is proposed to obtain the solution of the model. Two simulations demonstrate that this model is effective and can be used in multiple circumstances.

RESULT OF LOCAL EXISTENCE OF SOLUTIONS OF NONLOCAL VISCOELASTIC SYSTEM WITH RESPECT TO THE NONLINEARITY OF SOURCE TERMS

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

Existence theorems of solution of fractional differential equations with nonlocal conditions

This paper establishes a study on some important latest innovations in the existence of mild solution of semilinear for differential and fractional differential equations subject to nonlocal initial conditions. To apply this, the study uses Hausdorff measure of non-compactness and fixed point theorems. A wider applicability of these techniques are based on their reliability and reduction in the size of the mathematical work.

Solution existence theorems for finite horizon optimal economic growth problems

The solution existence of finite horizon optimal economic growth problems is studied by invoking Filippov's Existence Theorem for optimal control problems from the monograph of Cesari L. [Optimization theory and applications. New York: Springer-Verlag; 1983]. Our results are obtained not only for general problems but also for typical ones, where the production function is given by either the AK function or the Cobb–Douglas one, while the utility function can be in a linear or power form. Some open questions and conjectures about the regularity of global solutions of finite horizon optimal economic growth problems are formulated in this paper.

Adaptive synchronization for fractional stochastic neural network with delay

Under the Brownian motion environment, adaptive synchronization is mainly studied in this paper for fractional-order stochastic neural networks (FSNNs) with time delays and discontinuous activation functions. Firstly, an existence theorem of solutions is established and global solutions of FNNs are obtained under the definition of Filippov solution by using the fixed-point theorem for a condensing map. Secondly, an adaptive controller is designed to ensure the synchronization between FNNs and the corresponding fractional-order FSNNs. Finally, a numerical example is given to illustrate the given results.

Обратные задачи для двумерного уравнения теплопроводности по отысканию правой части

Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables. Preliminarily, we explicitly construct a solution to the direct initial-boundary-value problem. We prove the uniqueness of the solution to direct and inverse problems, making use of the completeness property of the system of eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. We also establish existence theorems for solutions of inverse problems and construct solutions explicitly.

Existence theorems of solution of fractional differential equations with nonlocal conditions

This paper establishes a study on some important latest innovations in the existence of mild solution of semilinear for differential and fractional differential equations subject to nonlocal initial conditions. To apply this, the study uses Hausdorff measure of non-compactness and fixed point theorems. A wider applicability of these techniques are based on their reliability and reduction in the size of the mathematical work.

Collectively fixed point theorems in noncompact abstract convex spaces with applications

&lt;abstract&gt;&lt;p&gt;In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.&lt;/p&gt;&lt;/abstract&gt;

On the solution existence for prox-regular perturbed sweeping processes

In the setting adopted by Edmond and Thibault [Mathematical Programming 104 (2005), 347--373], we study a class of perturbed sweeping processes. Under suitable assumptions, we obtain two solution existence theorems for perturbed sweeping processes with the constraint sets being prox-regular sublevel sets. The results are applied to analyzing the behavior of some concrete mechanical sweeping processes, which appear for the first time in this paper.

Set-Valued Symmetric Generalized Strong Vector Quasi-Equilibrium Problems with Variable Ordering Structures

In this paper, two types of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures are discussed. By using the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping, some existence theorems of solutions are established under suitable assumptions of cone-continuity and cone-convexity for the equilibrium mappings. Moreover, the results of compactness for solution sets are proven. As applications, some existence results of strong saddle points are obtained. The main results obtained in this paper unify and improve some recent works in the literature.

Mixed boundary–transmission problems for composite layered elastic structures

We investigate mixed type boundary–transmission problems of the generalized thermo‐electro‐magneto‐elasticity theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary–transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide.

Optimal economic growth problems with high values of total factor productivity

ABSTRACT This paper solves a question raised in the paper of Huong, Yao, and Yen [Optimal processes in a parametric optimal economic growth model. Taiwanese J Math. 2020. Available from: https://doi.org/10.11650/tjm/200203] about optimal economic growth problems with production functions and utility functions being all in the form of AK functions. By using a solution existence theorem from the paper of Huong [Solution existence theorems for finite horizon optimal economic growth problems. Preprint arXiv:2001.03298v2. Submitted] and a maximum principle from the book of Vinter [Optimal control. Boston (MA): Birkhäuser; 2000], we prove that the problem in question has a unique solution and give a comprehensive synthesis of the optimal processes. Our results show that if the value of total factor productivity is enough high and the planning time is short, then expanding the production facility does not lead to a higher total consumption satisfaction of the society. Meanwhile, if the value of total factor productivity is enough high and the planning time is relatively long, then the highest total consumption satisfaction of the society is attained only if the largest expansion of the production facility is made until a special time.