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The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p (x) -Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč's version of the celebrated Díaz–Saá's inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each "positive bump" of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed. [ABSTRACT FROM AUTHOR]

In this work, we propose a method to enhance the convergence of the Born series. The Born series is widely used in scattering theory, but its convergence is only guaranteed under certain restrictive conditions that limit the cases where this formulation can be applied. The proposed method, based on modifying the singularities of the resolvent operator by a change of variables, accelerates the convergence of the series or even achieves convergence in otherwise divergent cases. Due to its low computational cost and ease of implementation, the method preserves the advantages of the Born series, while it extends its range of applications. It can also be applied to a variety of methods relying on the same mathematical principles as the Born series. Numerical examples are provided to show that a significant improvement can be achieved with simple techniques and with limited knowledge about the operator spectrum. [ABSTRACT FROM AUTHOR]

Visceral leishmaniasis in dogs is believed to have an impact on the prevalence of the disease in human populations. Here, we continue the analysis of the nested immuno-epidemiological model of visceral leishmaniasis in dogs, including a proof of well-posedness using functional analytical methods. Once well-posedness is established, we continue stability analysis of the endemic equilibria and provide necessary and sufficient conditions for the presence of backward bifurcation, and prove the instability of the lower endemic equilibrium in the presence of backward bifurcation. Lastly, we provide a number of simulations of the model using a number of control strategies. Control measures currently in use attempt to reduce the parasite load in the host, reduce the vector population, reduce the vector biting rate, and remove infected hosts. We examine various combinations of these strategies and conclude that a strategy combining culling infected dogs and removing vectors from the population by means such as insecticide will be the most effective. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we give a new short proof of the local quadratic convergence of the Dominant Pole Spectrum Eigensolver (DPSE). Also, we introduce here the Diagonal Dominant Pole Spectrum Eigensolver (DDPSE), another fixed-point method that computes several eigenvalues of a matrix A at a time, which also has local quadratic convergence. From results of some experiments with a large power system model, it is shown that DDPSE can also be used in small-signal stability studies to compute dominant poles of a transfer function of the type c T (A − s I) − 1 b , where s ∈ C , b and c are vectors, by its own or combined with DPSE. Besides DDPSE is also effective in finding low damped modes of a large scale power system model. [ABSTRACT FROM AUTHOR]

This article investigates nonlinear, second-order ordinary differential equations subject to various two-point boundary conditions. A condition is introduced that ensures a priori bounds on the derivatives of solutions to the problem. In particular, quadratic growth conditions on the right-hand side of the differential equation are not employed. The ideas are then applied to ensure the existence of at least one solution. The main tools involve differential inequalities and fixed-point methods.

Quasi-Newton acceleration is an interesting tool to improve the performance of numerical methods based on the fixed-point paradigm. In this paper, a briefly description of the compartmental model employed for a typical task in perfusion imaging is presented. Next, a quasi-Newton strategy for accelerating the convergence of fixed-point iterations is analyzed. For that, classical secant updates are considered. Finally, the quasi-Newton strategy is applied on the practical problem of represent the kinetic behavior of a PET (Positron Emission Tomography) tracer during cardiac perfusion. The performance of the method when applied to real data problems is illustrated numerically. [ABSTRACT FROM AUTHOR]

We study the ridge regression ( L 2 regularized least squares) problem and its dual, which is also a ridge regression problem. We observe that the optimality conditions describing the primal and dual optimal solutions can be formulated in several different but equivalent ways. The optimality conditions we identify form a linear system involving a structured matrix depending on a single relaxation parameter which we introduce for regularization purposes. This leads to the idea of studying and comparing, in theory and practice, the performance of the fixed point method applied to these reformulations. We compute the optimal relaxation parameters and uncover interesting connections between the complexity bounds of the variants of the fixed point scheme we consider. These connections follow from a close link between the spectral properties of the associated matrices. For instance, some reformulations involve purely imaginary eigenvalues; some involve real eigenvalues and others have all eigenvalues on the complex circle. We show that the deterministic Quartz method—which is a special case of the randomized dual coordinate ascent method with arbitrary sampling recently developed by Qu, Richtárik and Zhang—can be cast in our framework, and achieves the best rate in theory and in numerical experiments among the fixed point methods we study. Remarkably, the method achieves an accelerated convergence rate. Numerical experiments indicate that our main algorithm is competitive with the conjugate gradient method. [ABSTRACT FROM AUTHOR]

Continued advances in computational power and methods have enabled image-based biomechanical modeling to become an important tool in basic science, diagnostic and therapeutic medicine, and medical device design. One of the many challenges of this approach, however, is identification of a stress-free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs. Fortunately, iterative methods have been proposed to solve this inverse problem, among them Sellier’s method. This method is particularly appealing because it is easy to implement, convergences reasonably fast, and can be coupled to nearly any finite element package. By means of several practical examples, however, we demonstrate that in its original formulation Sellier’s method is not optimally fast and may not converge for problems with large deformations. Fortunately, we can also show that a simple, inexpensive augmentation of Sellier’s method based on Aitken’s delta-squared process can not only ensure convergence but also significantly accelerate the method. [ABSTRACT FROM AUTHOR]

From the fixed-point methods and the properties of theμ-pseudo-almost automorphic functions, the existence and uniqueness of measure pseudo-almost automorphic solution are obtained for differential equations involving reflection of the argument. [ABSTRACT FROM AUTHOR]

This article investigates the existence of solutions to first-order nonlinear boundary-value problems (BVPs) involving systems of ordinary differential equations and two-point boundary conditions. Some sufficient conditions are presented that will ensure solvability. The main tools employed are novel differential inequalities and fixed-point methods.

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one-dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the existence of solutions to boundary value problems (BVPs) involving systems of nonlinear first-order ordinary differential equations and two-point, anti-periodic boundary conditions. A new existence result for the BVPs above is obtained by using fixed-point theory. [Copyright &y& Elsevier]

Abstract. We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth, and thus finding its global solution is in principle a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes for solving this problem. The first algorithm is a very simple explicit fixed-point-based formula, and the second is based on solving at each iteration a nonlinear least squares problem, which can be solved globally and efficiently after transforming it into an equivalent quadratic minimization problem with a single quadratic constraint. We show that the nonsmoothness of the problem can be avoided by choosing a specific "good" starting point for both algorithms, and we prove the convergence of the two schemes to stationary points. We present empirical results that support the underlying theoretical analysis and suggest that, despite of its nonconvexity, the ML problem can effectively be solved globally using the devised schemes. [ABSTRACT FROM AUTHOR]

Abstract: This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented. [Copyright &y& Elsevier]

Abstract: This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution. [Copyright &y& Elsevier]

This article investigates the existence of solutions to first-order nonlinear boundary-value problems (BVPs) involving systems of ordinary differential equations and two-point boundary conditions. Some sufficient conditions are presented that will ensure solvability. The main tools employed are novel differential inequalities and fixed-point methods. [ABSTRACT FROM AUTHOR]

From the very beginning. Complex Analysis is closely connected with partial differential equations. The article surveys present generalizations of these connexions to higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we give a formal derivation of several systems of equations for injection moulding. This is done starting from the basic equations for nonisothermal, non-Newtonian flows in a three-dimensional domain. We derive systems for both (T0, p0) and (T1, p1) in the presence of body forces and sources. We find that body forces and sources have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”. Our formulation includes not only the pressure gradient, but also body forces and sources, which play the role of a nonlinearity. Later, we prove the existence of weak solutions to certain boundary value problems and initial-boundary value problems associated with the resulting equations for (T0,p0) but in a more general mathematical setting. [Copyright &y& Elsevier]

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brézis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set. [ABSTRACT FROM AUTHOR]