Existence and uniqueness results for a semilinear Black–Scholes type equation

This paper deals with the application of fixed point theorem to determine the source term of semilinear Black–Scholes type equation and thereby establish the existence and uniqueness of the solution. The proof mainly relies on the iteration method and the Schauder fixed point theorem.     

Comment on “Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet”

The article named above appeared recently in Applied Mathematics Letters and investigated a boundary value problem governing viscous flow over a nonlinearly stretching sheet. The authors of the work assert existence and (under certain restrictions) uniqueness of a solution to the problem for all relevant values of the parameter governing the stretching rate of the sheet. Unfortunately, several proofs presented in the article are incorrect. We will prove that for a range of parameter space the solution to the BVP is not unique. For these parameter values there are infinitely many solutions to the problem. The same incorrect analysis is reproduced in several other papers (see the references). Some of the claims of these papers are contradicted by established results on, for example, the Falkner–Skan problem.     

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.     

Existence and uniqueness of positive solutions to?m-point boundary value problem for?nonlinear fractional differential equation

In this paper, we consider the following nonlinear fractional m-point boundary value problem where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. By the properties of the Green function, the lower and upper solution method and fixed-point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As applications, examples are presented to illustrate the main results.     

Existence of a periodic solution for continuous and discrete periodic second-order equations with variable potentials

Green’s functions for new second-order periodic differential and difference equations with variable potentials are found, then used as kernels in integral operators to guarantee the existence of a positive periodic solution to continuous and discrete second-order periodic boundary value problems with periodic coefficient functions. A new version of the Leggett-Williams fixed point theorem is employed.     

Existence and Hyers–Ulam stability of solutions for a delayed hyperbolic partial differential equation

Periodica Mathematica Hungarica - In this paper, we first prove the existence and uniqueness of the solutions for a delayed hyperbolic partial differential equation by applying the progressive...     

Existence and uniqueness for a quasilinear hyperbolic equation with σ-finite Borel measures as initial conditions

The aim of this paper is to discuss the Cauchy problem for a quasilinear hyperbolic equation of the form

     

Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.     

Existence and uniqueness of periodic solutions for?a?kind of?Liénard equation with multiple deviating arguments

In this paper, a kind of Liénard equation with multiple deviating arguments of the form $$x''(t)+f(x(t))x'(t)+\sum_{k=1}^{n}g_{k}(t,x(t-\tau_{k}(t)))=p(t)$$ is considered. By using coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of periodic solutions of this equation. The obtained results are also valid and new for the problem discussed in the previous literature.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

Corrigendum to “Existence and uniqueness results for a nonlinear differential equation arising in MHD Falkner-Skan flow” [Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2272–2277]

We correct the hypothesis for which the existence and uniqueness theorems of Van Gorder and Vajravelu [Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2272–2277] hold. This correction modifies the range of parameters valid under the given theorems.     

Existence and iterative algorithm of solutions for a system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures.     

Existence and uniqueness of global weak solutions to a Cahn–Hilliard–Stokes–Darcy system for two phase incompressible flows in karstic geometry

We study the well-posedness of a coupled Cahn–Hilliard–Stokes–Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak–strong uniqueness property of the weak solutions is provided as well.     

An extension of Krugman’s core–periphery model to the case of a continuous domain: Existence and uniqueness of solutions of a system of nonlinear integral equations in spatial economics

We consider a model that is an extension of Krugman’s core–periphery model to the case of a bounded closed domain included in a Euclidean space. We can describe the relation of the density of workers, the density of nominal wages, and the density of real wages by the system of nonlinear integral equations of the model. If we obtain a solution of the system under the condition that the density of workers is given, then the solution is called a short-run equilibrium. In this paper we prove that this model has a short-run equilibrium, and we obtain a sufficient condition for its uniqueness. Moreover we obtain upper and lower estimates for short-run equilibria, and we construct a useful iteration scheme to numerically obtain short-run equilibria.     

A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)

$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$

where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).

     

A population explosion in an evolutionary game in spatial economics: Blow up radial solutions to the initial value problem for the replicator equation whose growth rate is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting

We consider a spatially continuous evolutionary game whose payoff is defined as the density of real wages that is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting. This evolutionary game is expressed by the initial value problem for the replicator equation whose growth rate contains an operator which acts on an unknown function that denotes the density of workers. We prove that this initial value problem has a unique global solution, and that if workers are distributed radially in space and concentrated in the neighborhood of a city at the initial time, then all workers will move toward the center of the city in such a way that the density of workers converges to the Dirac delta function in the sense of distribution. In the real world this result describes a population explosion caused by concentration of workers motivated by the disparity in real wages.     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

Necessary and Sufficient Conditions of Existence for Positive Solution to a Class of Quasilinear Elliptic Systems in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we establish the necessary and sufficient conditions of existence for a positive solution to a class of non-variational quasilinear elliptic systems in R ^N . The sufficient condition of existence result bases on the Mountain Pass Lemma and the sub-super solution methods, and the necessary condition is a consequence of a Picone’s identity. The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology and so on.