Existence and Uniqueness Results for a Fully Third-Order Boundary Value Problem

The boundary value problems of the third-order ordinary differential equation have many practical application backgrounds and their some special cases have been studied by many authors. However, few scholars have studied the boundary value problems of the complete third-order differential equations u′′′(t) = f (t,u(t),u′(t),u′′(t)). In this paper, we discuss the existence and uniqueness of solutions and positive solutions of the fully third-order ordinary differential equation on [0,1] with the boundary condition u(0) = u′(1) = u′′(1) = 0. Under some inequality conditions on nonlinearity f some new existence and uniqueness results of solutions and positive solutions are obtained.

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Existence and Uniqueness of Nested Regular Quadrilateral Central Configurations

Two cases of the nested configurations in R3 consisting of two regular quadrilaterals are discussed. One case of them do not form central configuration, the other case can be central configuration. In the second case the existence and uniqueness of the central configuration are studied. If the configuration is a central configuration, then all masses of outside layer are equivalent, similar to the masses of inside layer. At the same time the following relation between r(the ratio of the sizes) and mass ratio b = m/m must be satisfied in which the masses at outside layer are not less than the masses at inside layer, and the solution of this kind of central configuration is unique for the given ratio (6) of masses.

New Developments on Existence and Uniqueness of Solutions to Some Models in Atmospheric Dynamics

This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two-and three-dimensional quasi-geostrophic models and three-dimensional balanced model. The main aim of this paper is to introduce some results about the global and local (with respect to time) existence of solutions given by the authors in recent years, but others' important contributions and the literature on this subject are also quoted. We discuss briefly the relationships among the existence and uniqueness, physical instability and computational instability. In the appendixes, some key mathematical techniques in obtaining our results are presented, which are of vital importance to other problems in geophysical fluid dynamics as well.

Existence and uniqueness of a class of double pyramidal central configurations in six-body problems

Under the necessary conditions for a double pyramidal central configuration with a diamond base to exist in the real number space, the existence and uniqueness of such configurations were studied by employing combinedly the algebraic method and numerical calculation. It is found that there exists a planar curl triangle region G in a square Q such that any point in G and given by the ratio of the two diagonal lengths of the diamond base and the ratio of one diagonal length of the base to the height of the double pyramid configuration determines a unique double pyramid central configuration, while all points in Q-G have no referance to any central configuration.

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR GRADIENT SYSTEMS IN FINITE DIMENSIONAL SPACES

This paper deals with an abstract periodic gradient system in which the gradient is taken with respect to a variable metric. We obtain an existence and uniqueness result via the application of a global inverse theorem.

On existence and uniqueness of solutions to uncertain backward stochastic differential equations

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.

LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS

In this article,we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions.The initial vacuum is allowed.

Existence and uniqueness for a class of double pyramidal central configurations with a concave pentagonal base

Based on some necessary conditions for double pyramidal central configurations with a concave pentagonal base, for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with a concave pentagonal base in 7-body problems are proved and the range of the ratio between radius and half-height is obtained, within which the 7 bodies involved form a central configuration or form uniquely a central configuration.

Existence and uniqueness for a class of nine-bodies central configurations

On some necessary conditions for double pyramidal central configurations with concave heptagon for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with concave heptagon base for nine-body problems is proved in this paper, and the range of the ratio cr of the circularity radius of the heptagon to the half-height of the double pyramidal central configuration involved in this class configurations is obtained, which is in the interval （√3/3,1.099 600 679）, and the configuration involved in the bodies with any σ∈ （√3/3, 1.099 600 679） can form a central configuration which is a uniquely central configuration is proved.

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.

Existence and Uniqueness of Positive Periodic Solutions for First-order Functional Differential Equations

In this paper, we prove the existence and uniqueness of positive periodic solutions for first-order functional differential equation y＇（t） = α（t）y（t） ＋ f（t, y（t -τ-（t））） ＋ g（t）by using two fixed point theorems of general α-concave operators and homogeneous operators in ordered Banach spaces.

Existence and Uniqueness Theorem for the Cauchy Problem of Fuzzy Differential Equations under Non-Lipschitz Conditions

Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and uniqueness theorem for the Cauchy problem of fuzzy differential equations under non-Lipschitz conditions The comparison principles and the existence and uniqueness theorems of this paper generalize many well-known results up to now

THE EXISTENCE AND UNIQUENESS OF THE SCREW TENSOR FOR THE FINITE DISPLACEMENT OF A RIGIDBODY

The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a conseguence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

Existence and uniqueness of solutions for a hierarchical system of two age-structured populations

We propose a class of new hierarchical model for the evolution of two interacting age-structured populations,which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the interactions such as competition,cooperation and predator-prey relation.Based upon a group of natural conditions,the existence and uniqueness of solutions on infinite time interval are proved by means of fixed point and extension principle,and the continuous dependence of the solution on the initial age distribution is established.These results lay a sound basis for the investigation of stability,controllability and variable optimal control problems.

Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation

In this paper,we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions.Two definitions,i.e.,Riemann-Liouville definition and Caputo one,of the fractional derivative are considered in parallel.In both cases,we establish the well-posedness of the weak solution.Moveover,based on the proposed weak formulation,we construct an efficient spectral method for numerical approximations of the weak solution.The main contribution of this work are threefold:First,a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed.We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem,and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems.Secondly,we show that in the case of Riemann-Liouville definition,the well-posedness of the space-time fractional diffusion equation does not require any initial conditions.This contrasts with the case of Caputo definition,in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness.Finally,thanks to the weak formulation,we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem.

EXISTENCE AND UNIQUENESS OF STRONG PERIODIC SOLUTION OF THE EVOLUTION SYSTEM DESCRIBING GEOPHYSICAL FLOW-PART I: IN BOUNDED DOMAINS

The existence and uniqueness of a strong periodic solution of the evolution system describing geophysical flow in bounded domains of RN (N = 3, 4) are proven if external forces are periodic in time and sufficiently small.