Measures of noncompactness and infinite systems of integral equations of Urysohn type in $L^\infty(\mathfrak{G})$

"In this article, applying the concept of measure of noncompactness, some fixed point theorems in the Fréchet space $L^\infty(\mathfrak{G})$ (where $\mathfrak{G}\subseteq \mathbb{R}^{\omega}$) have been proved. We handle our obtained consequences to inquiry the existence of solutions for infinite systems of Urysohn type integral equations. Our results extend some famous related results in the literature. Finally, to indicate the effectiveness of our results we present a genuine example."

A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type Ω -contractions in the setting of extended b -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations

This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton–Jacobi–Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the solution and prove the solvability of the corresponding ergodic problem.

Finite-size effects in field theory. Scaling behaviour

A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.

Transport of finite chains with free ends through thin channels

We describe the transport of a finite chain of [Formula: see text] identical particles in a thermal bath, through thin channels that forbid any crossing with a conceptually and technically simple method, that is neither restricted to the thermodynamic limit (infinite systems with finite density) nor to overdamped systems. We obtain analytically the mean squared displacement of each particle. Regardless of the damping, we identify a correlated regime for which chain transport is dominated by the correlations between individual particles. At large damping, the mean squared displacement evidences the typical single file behavior, with a time dependence that scales as [Formula: see text]. At small damping, the correlated regime is rather described by a diffusion-like behavior, with a diffusivity which is neither the individual particle diffusivity nor the Fickian diffusivity of the chain as a whole. We emphasize that, for a chain with free ends, the fluctuations of the chain ends are larger by a factor two than the fluctuations of its center. This effect is observed whatever the damping [Formula: see text], but the duration of this fluctuations enhancement is found to scale as [Formula: see text] for low damping and as [Formula: see text] for high damping. We discuss the relevance of this model to the transport of actual systems in confined geometries.

Electromagnetic field of the circular magnetic source in biconical section

The problem of axially-symmetric electromagnetic wave diffraction from the perfectly conducting biconical scatterer formed by the finite cone placed in the semi-infinite conical region is solved rigorously using the mode-matching and analytical regularization techniques. The problem is reduced to the infinite systems of linear algebraic equations (ISLAE) of the second kind. The obtained equations admit the reduction procedure and can be solved with a given accuracy for any geometrical parameters and frequency. The numerical examples of the solution are presented. The analysis of the source location influences on the far-field pattern for different geometrical parameters of the bicone is carried out.