Unique Continuous Solution Research Articles

Semipositone elliptic equations in unbounded domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document}

We investigate a class of semipositone nonlinear elliptic boundary value problems in unbounded domains D of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document}, n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\geq 3$\\end{document}. Under some mild-assumptions, we obtain the existence of a unique positive continuous solution and provide information about its global behavior.

Integral-algebraic equations with variable limits of integration. The case when the lower limit is at

The paper is devoted to the study and numerical solution of linear systems of integral equations with an identically singular matrix in the principal part (called integral-algebraic equations). The systems studied in the paper are fundamentally different from those previously studied in that the upper and lower limits of integration are variable. These systems of equations will be referred to as integral-algebraic equations with variable integration limits. The paper develops and justifies an algorithm for a numerical solution of the first order of accuracy and establishes conditions for the existence of a unique continuous solution to the given problem. There are numerical computations provided for model examples.

Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.

Existence Results for a Class of Fractional Differential Beam Type Equations

Fractional differential beam type equations are considered. By using an efficient approach, we prove the existence and uniqueness of continuous solutions. An iterative scheme for approximating the solution is given. Some examples are presented.

Backward reachability approach to state-constrained stochastic optimal control problem for jump-diffusion models

AbstractIn this paper, we consider the stochastic optimal control problem for jump-diffusion models with state constraints. In general, the value function of such problems is the discontinuous viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation since the regularity cannot be guaranteed at the boundary of the state constraint. By adapting the stochastic target theory, we obtain an equivalent representation of the original value function by means of the backward reachable set. We then show that this backward reachable can be characterized by the zero-level set of the auxiliary value function for the unconstrained stochastic control problem, which includes two additional unbounded controls as a consequence of the martingale representation theorem. We prove that the auxiliary value function is the unique continuous viscosity solution of the associated HJB equation, which is the second-order nonlinear partial integro-differential equation (PIDE). Our paper provides an explicit way to characterize the original (possibly discontinuous) value function as a zero-level set of the continuous solution of the auxiliary HJB equation. The proof of the existence and uniqueness requires a new technique due to the unbounded control sets and the presence of the singularity of the corresponding Lévy measure in the nonlocal operator of the HJB equation.

Optimal sizing of the sediment replenishment capacity based on robust ergodic control of subordinator-driven dynamics

A countermeasure against sediment depletion in rivers is sand replenishment. The cost-efficient sizing of its capacity is critical for sustainable environmental management. However, only temporally discrete replenishment is possible, and sediment transport is intermittent and uncertain. In this paper, we approach this new design problem using a robust ergodic control. First, a stochastic differential equation driven by a (tempered) stable subordinator describes the sediment storage dynamics. Sand replenishment is considered an impulse control, wherein observation/replenishment chances arrive at Poisson jump times. The subordinator is ambiguous since the decision-maker does not precisely know its Lévy measure. Using a dynamic risk measure that balances replenishment cost, disutility triggered by sediment depletion, and penalization of ambiguity, we derived the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation to determine the optimal storage capacity. We obtained that the HJBI equation has an analytical solution when a stable subordinator is used. Furthermore, we prove it admits a unique continuous viscosity solution and that it is optimal and applied the proposed model to real data.

The Equivalence of Definitions of the Natural Logarithm Function

Summary There are four definitions of the natural logarithm function commonly found in analysis texts: the inverse of the natural exponential, an integral, a limit, and the unique continuous solution of a functional equation. This article provides historical and pedagogical comments about each and establishes the equivalence of these four definitions.

Nonnegative solutions for a class of semipositone nonlinear elliptic equations in bounded domains of R^{n}

In this paper, we obtain sufficient conditions for the existence of a unique nonnegative continuous solution of semipositone semilinear elliptic problem in bounded domains of \(\mathbb{R}^n\) (\(n\geq 2\)). The global behavior of this solution is also given.

On an irreversible investment problem with two-factor uncertainty

We consider a real options model for the optimal irreversible investment problem of a profit-maximizing company. The company has the opportunity to invest in a production plant capable of producing two products, of which the prices follow two independent geometric Brownian motions. After paying a constant sunk investment cost, the company sells the products on the market and thus receives a continuous stochastic revenue flow. This investment problem is set as a two-dimensional optimal stopping problem. We find that the optimal investment decision is triggered by a convex curve, which we characterize as the unique continuous solution to a nonlinear integral equation. Furthermore, we provide analytical and numerical comparative statics results of the dependency of the project's value and investment decision with respect to the model's parameters.

Exponential convergence of a proximal projection neural network for mixed variational inequalities and applications

This paper proposes a novel proximal projection neural network (PPNN) to deal with mixed variational inequalities. It is shown that the PPNN has a unique continuous solution under the condition of Lipschitz continuity and that the trajectories of the PPNN converge to the unique equilibrium solution exponentially under some mild conditions. In addition, we study the influence of different parameters on the convergence rate. Furthermore, the proposed PPNN is applied in solving nonlinear complementarity problems, min–max problems, sparse recovery problems and classification and feature selection problems. Finally, numerical and experimental examples are presented to validate the effectiveness of the proposed neurodynamic network.

Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel

In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.

Discrete approximation of the viscous HJ equation

We consider a stochastic discretization of the stationary viscous Hamilton-Jacobi equation on the flat d–dimensional torus \(\mathbb {T}^d\) associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on \(\mathbb {T}^d\), up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton–Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.

A proximal neurodynamic model for solving inverse mixed variational inequalities

This paper proposes a proximal neurodynamic model (PNDM) for solving inverse mixed variational inequalities (IMVIs) based on the proximal operator. It is shown that the PNDM has a unique continuous solution under the condition of Lipschitz continuity (L-continuity). It is also shown that the equilibrium point of the proposed PNDM is asymptotically stable or exponentially stable under some mild conditions. Finally, three numerical examples are presented to illustrate effectiveness of the proposed PNDM.

Existence and Uniqueness of Continuous Solution for a Non-local Coupled System Modeling the Dynamics of Dislocation Densities

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Based on a new gradient entropy estimate in $$L \log L$$ space, we prove the global existence of a continuous solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity. A comparison principle with respect to time is used for proving uniqueness of the solution for the local problem.

Existence and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions

We deal with the following Riemann–Liouville fractional nonlinear boundary value problem: {Dαv(x)+f(x,v(x))=0,2<α≤3,x∈(0,1),v(0)=v′(0)=v(1)=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\mathcal{D}^{\\alpha }v(x)+f(x,v(x))=0, & 2< \\alpha \\leq 3, x\\in (0,1), \\\\ v(0)=v^{\\prime }(0)=v(1)=0. \\end{cases} $$\\end{document} Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying |v(x)|≤cxα−1(1−x)for all x∈[0,1] and some c>0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl\\vert v(x) \\bigr\\vert \\leq cx^{\\alpha -1}(1-x)\\quad\\text{for all }x \\in [ 0,1]\\text{ and some }c>0. $$\\end{document} Our results improve those obtained by Zou and He (Appl. Math. Lett. 74:68–73, 2017).

Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients

&lt;p style='text-indent:20px;'&gt;The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.&lt;/p&gt;

Unique Continuation of Solutions of Second-Order Quasi-Elliptic Equations

Unique Continuation of Solutions of Second-Order Quasi-Elliptic Equations

Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

Optimal Market Making under Partial Information with General Intensities

ABSTRACT Starting from the Avellaneda–Stoikov framework, we consider a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes but also on unobservable factors modelled by a hidden Markov chain. We tackle this stochastic control problem under partial information with a model that unifies and generalizes many existing ones under full information, combining several risk metrics and constraints, and using general decreasing intensity functionals. We use stochastic filtering, control and piecewise-deterministic Markov processes theory, to reduce the dimensionality of the problem and characterize the reduced value function as the unique continuous viscosity solution of its dynamic programming equation. We then solve the analogous full information problem and compare the results numerically through a concrete example. We show that the optimal full information spreads are biased when the exact market regime is unknown, and the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observed order flow volatility. This effect becomes higher, the longer the waiting time in between orders.

The Generalized Pomeron Functional Equation

This paper investigates the linear functional equation with constant coefficients φt=κφλt+ft, where both κ&gt;0 and 1&gt;λ&gt;0 are constants, f is a given continuous function on ℝ, and φ:ℝ⟶ℝ is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ&gt;1, then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0&lt;κ&lt;1, then the equation has a unique continuous solution; and (iii) if κ=1, then the equation has a continuous solution depending on a single parameter φ0 under a suitable condition on f.