Volume 24, Issue 2

A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate scaling type function. For it, sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general multivariate specialized operators are presented fulfilling all the above properties; the higher order of multivariate approximation of these operators is also studied.

We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials -(p⁢x′)′⁢(t)-r⁢(t)⁢p⁢(t)⁢x′⁢(t)+q⁢(t)⁢x⁢(t)=f⁢(t,x⁢(t)){-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))}, where the functions p⁢(t)>0{p(t)>0}, q⁢(t){q(t)}, r⁢(t){r(t)} and f⁢(t,x){f(t,x)} are 𝒞2{\mathcal{C}^{2}} and T -periodic in the variable t .

In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 2013, 143–163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 2013, 11–28]. Examples are included to illustrate the validation of the results.

Let X be a normed space, U⊂X∖{0}{U\subset X\setminus\{0\}} a non-empty subset, and (G,+){(G,+)} a commutative group equipped with a complete ultrametric d that is invariant (i.e., d(x+z,y+z)=d(x,y{d(x+z,y+z)=d(x,y}) for x,y,z∈G{x,y,z\in G}). Under some weak natural assumptions on U and on the function γ:U3→[0,∞){\gamma\colon U^{3}\to[0,\infty)}, we study the new generalized hyperstability results when f:U→G{f\colon U\to G} satisfies the inequality d⁢(α⁢f⁢(x+yα+z),α⁢f⁢(z)+f⁢(y)+f⁢(x))≤γ⁢(x,y,z)d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)% \biggr{)}\leq\gamma(x,y,z) for all x,y,z∈U{x,y,z\in U}, where x+yα+z∈U{\frac{x+y}{\alpha}+z\in U} and α≥2{\alpha\geq 2} is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.

In this paper, we discuss the class of typically real functions that are convex in two directions. We determine the Koebe domain for this class and its different representations.

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

In this paper, we first study a composite relaxed resolvent operator and prove some of its properties. After that, we introduce a Yosida approximation operator based on the composite relaxed resolvent operator and demonstrate some properties of the Yosida approximation operator. Finally, we obtain the solution of a system of Yosida inclusions by applying these concepts. We construct a conjoin example in support of many concepts derived in this paper. Our concepts and results are new in the literature and can be used for further research.

The purpose of this paper is to deal with the multiple values and unicity problem of meromorphic mappings of ℂm{\mathbb{C}^{m}} into ℙn⁢(ℂ){\mathbb{P}^{n}(\mathbb{C})} sharing different families of moving hyperplanes with a small identity set. We prove several uniqueness theorems for meromorphic mappings, which extend and improve some recent results.

In the present paper, the notion of generalized (s,m){(s,m)}-preinvex Godunova–Levin function of second kind is introduced, and some new integral inequalities involving generalized (s,m){(s,m)}-preinvex Godunova–Levin functions of second kind along with beta function are given. By using a new identity for fractional integrals, some new estimates on generalizations of Hermite–Hadamard, Ostrowski and Simpson type inequalities for generalized (s,m){(s,m)}-preinvex Godunova–Levin functions of second kind via Riemann–Liouville fractional integral are established.

In this paper, sufficient optimality conditions are established for the multiobjective control problem using efficiency of higher order as a criterion for optimality. The ρ-type 1 invex functionals (taken in pair) of higher order are proposed for the continuous case. Existence of such functionals is confirmed by a number of examples. It is shown with the help of an example that this class is more general than the existing class of functionals. Weak and strong duality theorems are also derived for a mixed dual in order to relate efficient solutions of higher order for primal and dual problems.

The aim of this paper is to introduce a new concept of fuzzy δ- I -continuity between mixed fuzzy ideal topological spaces and investigate some properties of this mapping.