We introduce an analog of Fourier transform ${\mathcal{F}}_h^{\rho }$ in interior of light cone that commutes with the action of the Lorentz group. We describe some properties of ${\mathcal{F}}_h^{\rho }$, namely, its action on pseudoradial functions and functions being products of pseudoradial function and space hyperbolic harmonics. We prove that ${\mathcal{F}}_h^{\rho }$-transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.

We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding between $C(K)$ and $C\left(L\right)$ for 0-dimensional $K$ and $L$ such that the following requirement holds for all $h\ne 0$ and $f\ge 0$ in $C(K)$: if $0\le Th\le Tf$, then there are constants $a\ne 0$ and $b$ with $0\le a·h+b\le f$ and $a·h+b\ne 0$.

In 1961, Wang showed that if $A$ is the commutative $C^{\mathrm{*}}$-algebra $C_{\mathrm{0}}(X)$ with $X$ a locally compact Hausdorff space, then $M(C_{\mathrm{0}}(X))\cong C_b(X)$. Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing that $\text{H}\text{o}{\text{m}}_{C_{\mathrm{0}}(X,A)}(C_{\mathrm{0}}(X,E),C_{\mathrm{0}}(X,F))\simeq C_{s,b}(X,\text{H}\text{o}{\text{m}}_A(E,F)),$ where $E$ and $F$ are $p$-normed spaces which are also essential isometric left $A$-modules with $A$ being a certain commutative $F$-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator from $L^p\left(\mu \right)$ into the space $L^{q,\infty }\left(\mu \right)$. Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented.

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field ${ℂ}^{*}$ of $*$-complex numbers and the concept of $*$-metric. Also, we give the definitions and the basic important properties of $*$-boundedness and $*$-continuity. Later, we define the space $C_{*}(\mathrm{\Omega })$ of $*$-continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that $C_{*}(\mathrm{\Omega })$ is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between $C_{*}(\mathrm{\Omega })$ and the set of $C_{*}^{\mathrm{\prime }}(\mathrm{\Omega })$ $*$-differentiable functions.

In 2010, Agarwal et al. studied the existence of a one-dimensional fractional neutral functional differential equation. In this paper, we study an initial value problem for a class of $k$-dimensional systems of fractional neutral functional differential equations by using Krasnoselskii’s fixed point theorem. In fact, our main result generalizes their main result in a sense.

Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

Maddox defined the space $\ell (p)$ of the sequences $x=(x_k)$ such that , in Maddox, 1967. In the present paper, the Nörlund sequence space $N^t(p)$ of nonabsolute type is introduced and proved that the spaces $N^t(p)$ and $\ell (p)$ are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the space $N^t(p)$ are computed and the basis of the space $N^t(p)$ is constructed. The classes $(N^t(p):\mu )$ and $(\mu :N^t(p))$ of infinite matrices are characterized. Finally, some geometric properties of the space $N^t(p)$ are investigated.

We define and study Sobolev-type spaces $W_A^{s,p}\left({ℝ}_{+}\right)$ associated with singular second-order differential operator on $\left(0,\infty \right)$. Some properties are given; in particular we establish a compactness-type imbedding result which allows a Reillich-type theorem. Next, we introduce a generalized Weierstrass transform and, using the theory of reproducing kernels, some applications are given.

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.

We prove Amann type three solutions theorem for one dimensional p-Laplacian problems with a singular weight function. To prove this theorem, we define a strong upper and lower solutions and compute the Leray-Schauder degree on a newly established weighted solution space. As an application, we consider the combustion model and show the existence of three positive radial solutions on an exterior domain.

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for a class of $N$-Laplacian equations with critical growth and indefinite weight $-\text{div}\left({\left|\nabla u\right|}^{N-2}\nabla u\right)+V\left(x\right){\left|u\right|}^{N-2}u=\lambda \left({\left|u\right|}^{N-2}u/{\left|x\right|}^{\beta }\right)+\left(f\left(x,u\right)/{\left|x\right|}^{\beta }\right)+\mathrm{\varepsilon }h\left(x\right)$, $x\in {ℝ}^N$, $u\ne 0$, $x\in {ℝ}^N$, where , $V(x)$ is an indefinite weight, $f:{ℝ}^N\times ℝ\to ℝ$ behaves like $\text{exp}\left(\alpha {\left|u\right|}^{N/\left(N-1\right)}\right)$ and does not satisfy the Ambrosetti-Rabinowitz condition, and $h\in {(W^{\mathrm{1},N}({ℝ}^N))}^{\mathrm{*}}$.

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

We prove that every map satisfying the $g$-weakly C-contractive inequality in partial metric space has a unique coincidence point. Our results generalize several well-known existing results in the literature.