Existence of travelling wave solutions for a model of tumour invasion

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Existence of multiple positive solutions for N-laplacian in a bounded domain in R-N

Let Ohm be a bounded domain in IRN, N greater than or equal to 2, lambda > 0, q is an element of (0, N - 1) and alpha is an element of (1, N/N-1 In this article we show the existence of at least two positive solutions for the following quasilinear elliptic problem with an exponential type nonlinearity:

On existence of periodic solutions for stable interval plants with odd, sector type nonlinearities

In several systems, the physical parameters of the system vary over time or operating points. A popular way of representing such plants with structured or parametric uncertainties is by means of interval polynomials. However, ensuring the stability of such systems is a robust control problem. Fortunately, Kharitonov's theorem enables the analysis of such interval plants and also provides tools for design of robust controllers in such cases. The present paper considers one such case, where the interval plant is connected with a timeinvariant, static, odd, sector type nonlinearity in its feedback path. This paper provides necessary conditions for the existence of self sustaining periodic oscillations in such interval plants, and indicates a possible design algorithm to avoid such periodic solutions or limit cycles. The describing function technique is used to approximate the nonlinearity and subsequently arrive at the results. Furthermore, the value set approach, along with Mikhailov conditions, are resorted to in providing graphical techniques for the derivation of the conditions and subsequent design algorithm of the controller.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations

In this paper, we give sufficient conditions for the uniform boundedness and uniform ultimate boundedness of solutions of a class of retarded functional differential equations with impulse effects acting on variable times. We employ the theory of generalized ordinary differential equations to obtain our results. As an example, we investigate the boundedness of the solution of a circulating fuel nuclear reactor model.

Existence and concentration of solutions for a class of biharmonic equations

Some superlinear fourth order elliptic equations are considered. A family of solutions is proved to exist and to concentrate at a point in the limit. The proof relies on variational methods and makes use of a weak version of the Ambrosetti-Rabinowitz condition. The existence and concentration of solutions are related to a suitable truncated equation. (C) 2012 Elsevier Inc. All rights reserved.

Continuous Dependence of Solutions of Quasidifferential Equations with Non-Fixed Time of Impulses

In this article on quasidifferential equation with non-fixed time of impulses we consider the continuous dependence of the solutions on the initial conditions as well as the mappings defined by these equations. We prove general theorems for quasidifferential equations from which follows corresponding results for differential equations, differential inclusion and equations with Hukuhara derivative.

Existence and Uniqueness of Solutions for Partial Differential-Functional Equations of the First Order with Deviating Argument of the Derivative of Unknown Function

We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09

Existence of Homoclinic Solutions for Nonlinear Second-Order Problems

In this work, we consider the second-order discontinuous equation in the real line, u′′(t)−ku(t)=f(t,u(t),u′(t)),a.e.t∈R, with k>0 and f:R3→R an L1 -Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.

Sufficient conditions for the existence of heteroclinic solutions for Phi-Laplacian differential equations

In this paper we consider the second order discontinuous equation in the real line, (a(t)φ(u′(t)))′ = f(t,u(t),u′(t)), a.e.t∈R, u(-∞) = ν⁻, u(+∞)=ν⁺, with φ an increasing homeomorphism such that φ(0)=0 and φ(R)=R, a∈C(R,R\{0})∩C¹(R,R) with a(t)>0, or a(t)<0, for t∈R, f:R³→R a L¹-Carathéodory function and ν⁻,ν⁺∈R such that ν⁻<ν⁺. We point out that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities φ and f. Moreover, as far as we know, this result is even new when φ(y)=y, that is, for equation (a(t)u′(t))′=f(t,u(t),u′(t)), a.e.t∈R.

Random existence of charge ordered stripes and its influence on the magnetotransport properties of La0.6Sr0.4MnO3 perovskite substituted with diamagnetic ions at Mn sublattice

Phase-singular solid solutions of La0.6Sr0.4Mn1-yMeyO3 (0 <= y <= 0.3) [Me=Li1+, Mg2+, Al3+, Ti4+, Nb5+, Mo6+ or W6+] [LSMey] perovskite of rhombohedral symmetry (space group: R (3) over barc) have been prepared wherein the valence of the diamagnetic substituent at Mn site ranged from 1 to 6. With increasing y-content in LSMey, the metal-insulator (TM-I) transition in resistivity-temperature rho(T) curves shifted to low temperatures. The magnetization studies M(H) as well as the M(T) indicated two groups for LSMey. (1) Group A with Me=Mg, Al, Ti, or Nb which are paramagnetic insulators (PIs) at room temperature with low values of M (< 0.5 mu(B)/Mn); the magnetic transition [ferromagnetic insulator (FMI)-PI] temperature (T-C) shifts to low temperatures and nearly coincides with that of TM-I and the maximum magnetoresistance (MR) of similar to 50% prevails near T-C (approximate to TM-I). (2) Group-B samples with Me=Li, Mo, or W which are FMIs with M-s=3.3-3.58 mu(B)/Mn and marginal reduction in T-C similar to 350 K as compared to the undoped LSMO (T-C similar to 378 K). The latter samples show large temperature differences Delta T=T-c-TM-I, reaching up to similar to 288 K. The maximum MR (similar to 60%) prevails at low temperatures corresponding to the M-I transition TM-I rather than around T-C. High resolution lattice images as well as microscopy analysis revealed the prevalence of inhomogeneous phase mixtures of randomly distributed charge ordered-insulating (COI) bistripes (similar to 3-5 nm width) within FMI charge-disordered regions, yet maintaining crystallographically single phase with no secondary precipitate formation. The averaged ionic radius < r(B)>, valency, or charge/radius ratio < CRR > cannot be correlated with that of large Delta T; hence cannot be used to parametrize the discrepancy between T-C and TM-I. The M-I transition is controlled by the charge conduction within the electronically heterogeneous mixtures (COI bistripes+FMI charge disordered); large MR at TM-I suggests that the spin-ordered FM-insulating regions assist the charge transport, whereas the T-C is associated with the bulk spin ordered regions corresponding to the FMI phase of higher volume fraction of which anchors the T-C to higher temperatures. The present analysis showed that the double-exchange model alone cannot account for the wide bifurcation of the magnetic and electric transitions, contributions from the charge as well as lattice degrees of freedom to be separated from spin/orbital ordering. The heterogeneous phase mixtures (COI+FMI) cannot be treated as of granular composite behavior. (c) 2008 American Institute of Physics.