Volume 27, Issue 1

The dynamics of short intense electromagnetic pulses propagating in a relativistic pair plasma is governed by a nonlinear Schrödinger equation with a new type of focusing-defocusing saturable nonlinearity. In this context, we provide an existence theory for ring-profiled optical vortex solitons. We prove the existence of both saddle point and minimum type solutions. Via a constrained minimization approach, we prove the existence of solutions where the photon number may be prescribed, and we get the nonexistence of small-photon-number solutions. We also use the constrained minimization to compute the soliton’s profile as a function of the photon number and other relevant parameters.

A method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

Trapezoidal inequalities for functions of diverse nature are useful in numerical computations. The authors have proved an identity for a generalized integral operator via a differentiable function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in the recent decades. Various special cases have been identified. Some applications of presented results have been analyzed.

This article investigates a nonlinear fractional Caputo–Langevin equation Dβ⁢(Dα+λ)⁢x⁢(t)=f⁢(t,x⁢(t)),0<t<1, 0<α≤1, 1<β≤2,D^{\beta}(D^{\alpha}+\lambda)x(t)=f(t,x(t)),\quad 0<t<1,\,0<\alpha\leq 1,\,1<% \beta\leq 2, subject to the multi-point boundary conditions x⁢(0)=0,𝒟2⁢α⁢x⁢(1)+λ⁢𝒟α⁢x⁢(1)=0,x⁢(1)=∫0ηx⁢(τ)⁢𝑑τ for some ⁢0<η<1,x(0)=0,\qquad\mathcal{D}^{2\alpha}x(1)+\lambda\mathcal{D^{\alpha}}x(1)=0,% \qquad x(1)=\int_{0}^{\eta}x(\tau)\,d\tau\quad\text{for some }0<\eta<1, where Dα{D^{\alpha}} is the Caputo fractional derivative of order α, f:[0,1]×ℝ→ℝ{f:[0,1]\times\mathbb{R}\to\mathbb{R}} is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

We firstly establish inequalities for functions whose high degree derivatives are convex via an equality which was presented previously. Then we derive inequalities for functions whose high-order derivatives are absolutely continuous by using the same equality. In addition, we examine connections between inequalities obtained in earlier works and our results. Finally, some estimates of composite quadrature rules are given.

We provide the optimal bounds for the sine and hyperbolic tangent means in terms of various weighted means of the arithmetic and the contraharmonic means.

The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb’s law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution.

In this paper, we introduce and investigate three new general classes of bi-starlike and bi-convex functions of Ma–Minda type defined by the Sǎlǎgean integro-differential operator. Bounds of the first three coefficients |a2|{\lvert a_{2}\rvert}, |a3|{\lvert a_{3}\rvert} and |a4|{\lvert a_{4}\rvert} are given.

This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.

Very general univariate and multivariate Caputo ψ-fractional integral inequalities of Poincaré, Sobolev and Hilbert–Pachpatte types are presented. Estimates are with respect to ∥⋅∥p{\lVert\,\cdot\,\rVert_{p}}, 1≤p<∞{1\leq p<\infty}. Applications are given.

The aim of this paper is to present new results on the existence of solutions for a class of the boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H -functions and the incomplete H¯{\overline{H}}-functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H -functions (IHF) that paved the way to a natural extension and decomposition of H -function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯δ{\mathsf{P}_{\delta}}-transform) of this family of incomplete H -functions. Further, we give several interesting new and known results which are special cases our key results.

In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.