We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.

We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: $L_j=\partial /\partial t_j+(\partial \varphi /\partial t_j)(t,A)A$, $j=\mathrm{1,2},\dots ,n$, where $A:D(A)\subset H\to H$ is a self-adjoint linear operator, positive with $\mathrm{0}\in \rho (A)$, in a Hilbert space $H$, and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of $A^{-\mathrm{1}}$ with coefficients in $C^{\mathrm{\infty }}(\mathrm{\Omega })$, $\mathrm{\Omega }$ being an open set of ${\mathbb{R}}^n$, for any $n\in \mathbb{N}$, different from what happens in the work of Hounie (1979) who studies the problem only in the case $n=\mathrm{1}$. We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem $t$′$(s)=-\nabla \mathrm{R}\mathrm{e}\mathrm{}{\varphi }_{\mathrm{0}}(t(s))$, $s\ge \mathrm{0}$, $t(\mathrm{0})=t_{\mathrm{0}}\in \mathrm{\Omega },{\varphi }_{\mathrm{0}}:\mathrm{\Omega }\to \mathbb{C}$ being the first coefficient of $\varphi (t,A)$. Besides, to get over the problem out of the elliptic region, that is, in the points $t$^∗ $\in \mathrm{\Omega }$ such that $\nabla \mathrm{R}\mathrm{e}{\varphi }_{\mathrm{0}}(t$^∗$)$ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator $A=\mathrm{1}-\mathrm{\Delta }:H^{\mathrm{2}}({\mathbb{R}}^N)\subset L^{\mathrm{2}}({\mathbb{R}}^N)\to L^{\mathrm{2}}({\mathbb{R}}^N)$.

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error $\inf \{d(x,y):y\in T(x)\}$, and hence the existence of a consummate approximate solution to the equation $T(X)=x$.

We consider two-dimensional predator-prey system with Beddington-DeAngelis type functional response on periodic time scales in shifts. For this special case we try to find under which conditions the system has ${\delta }_{\pm }$-periodic solution.

A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.

We discuss the Hankel determinants $H_2(n)=a_n{a}_{n+2}-{\left(a_{n+1}\right)}^2$ for typically real functions, that is, analytic functions which satisfy the condition $\mathrm{Im} z \mathrm{Im}f(z)\ge 0$ in the unit disk Δ. Main results are concerned with $H_2(2)$ and $H_2(3)$. The sharp upper and lower bounds are given. In general case, for $n\ge 4$, the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green’s functions, and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.

We have generalized the notion of statistical boundedness by introducing the concept of $f$-statistical boundedness for scalar sequences where $f$ is an unbounded modulus. It is shown that bounded sequences are precisely those sequences which are $f$-statistically bounded for every unbounded modulus $f$. A decomposition theorem for $f$-statistical convergence for vector valued sequences and a structure theorem for $f$-statistical boundedness have also been established.

We have introduced and studied a new concept of $f$-lacunary statistical convergence, where $f$ is an unbounded modulus. It is shown that, under certain conditions on a modulus $f$, the concepts of lacunary strong convergence with respect to a modulus $f$ and $f$-lacunary statistical convergence are equivalent on bounded sequences. We further characterize those $\theta$ for which $S_{\theta }^f=S^f$, where $S_{\theta }^f$ and $S^f$ denote the sets of all $f$-lacunary statistically convergent sequences and $f$-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods of $f$-statistical convergence is given. Finally, we give an $S_{\theta }^f$-analog of the Cauchy criterion for convergence and a Tauberian theorem for $S_{\theta }^f$-convergence is also proved.

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.

A potential estimate type approach is used in order to obtain some a priori bounds for the solutions of certain classes of Dirichlet problems associated with nondivergence structure elliptic equations.

We construct a continuous function $f:[\mathrm{0,1}]\to \mathbb{R}$ such that $f$ possesses $N^{-\mathrm{1}}$-property, but $f$ does not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property $(N)$ does not hold for $N^{-\mathrm{1}}$-property. Some relevant properties are presented.

This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

The aim of this paper is to connect the zeros of polynomials in two variables with the eigenvalues of a self-adjoint operator. This is done by use of a functional-analytic method. The polynomials in two variables are assumed to satisfy a five-term recurrence relation, similar to the three-term recurrence relation that the classical orthogonal polynomials satisfy.

The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operator $A$ and maximal monotone operators $B$ with $D(B)\subset H$: $x_{n+\mathrm{1}}={\alpha }_{n}f(x_n)+{\gamma }_n{x}_n+{\delta }_n(I+r_{n}B{)}^{-\mathrm{1}}(I-r_{n}A)x_n+e_n$, for $n=\mathrm{1,2},\dots ,$ for given $x_{\mathrm{1}}$ in a real Hilbert space $H$, where $({\alpha }_n)$, $({\gamma }_n)$, and $({\delta }_n)$ are sequences in $(\mathrm{0,1})$ with ${\alpha }_n+{\gamma }_n+{\delta }_n=\mathrm{1}$ for all $n\ge \mathrm{1}$, $(e_n)$ denotes the error sequence, and $f:H\to H$ is a contraction. The algorithm is known to converge under the following assumptions on ${\delta }_n$ and $e_n$: (i) $({\delta }_n)$ is bounded below away from 0 and above away from 1 and (ii) $(e_n)$ is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i) $({\delta }_n)$ is bounded below away from 0 and above away from 3/2 and (ii) $(e_n)$ is square summable in norm; and we still obtain strong convergence results.

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

This paper presents a complete stochastic solution represented by the first probability density function for random first-order linear difference equations. The study is based on Random Variable Transformation method. The obtained results are given in terms of the probability density functions of the data, namely, initial condition, forcing term, and diffusion coefficient. To conduct the study, all possible cases regarding statistical dependence of the random input parameters are considered. A complete collection of illustrative examples covering all the possible scenarios is provided.

We derive a new class of linear multistep methods (LMMs) via the interpolation and collocation technique. We discuss the use of these methods as boundary value methods and block unification methods for the numerical approximation of the general second-order initial and boundary value problems. The convergence of these families of methods is also established. Several test problems are given to show a computational comparison of these methods in terms of accuracy and the computational efficiency.

We study extension of $p$-trigonometric functions ${\mathrm{s}\mathrm{i}\mathrm{n}}_p$ and ${\mathrm{c}\mathrm{o}\mathrm{s}}_p$ and of $p$-hyperbolic functions ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_p$ and ${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_p$ to complex domain. Our aim is to answer the question under what conditions on $p$ these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, $\mathrm{s}\mathrm{i}\mathrm{n}(z)=-i·\sinh \left(i·z\right)$. In particular, we prove in the paper that for $p=\mathrm{6,10,14},\dots$ the $p$-trigonometric and $p$-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for $p$-trigonometric and $p$-hyperbolic functions.

We introduce the unified biunivalent function class ${\mathcal{M}}_{q,\sigma }^{\delta ,\lambda }(\gamma ,\phi )$ defined based on quasi-subordination and obtained the coefficient estimates for Taylor-Maclaurin coefficients $|a_{\mathrm{2}}|$ and $|a_{\mathrm{3}}|$. Several related classes of functions are also considered and connections to earlier known and new results are established.

We generalize the results of Krasniqi 2012 and Wei and Yu 2012 to the case of $r$-differences.

Two classes of generalized convex functions in the sense of Beckenbach are considered. For both classes, we show that the existence of support curves implies their generalized convexity and obtain an extremum property of these functions. Furthermore, we establish Hadamard’s inequality for them.

This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes.

We propose a new method for the specific nonlinear and nonconvex global optimization problem by using a linear relaxation technique. To simplify the specific nonlinear and nonconvex optimization problem, we transform the problem to the lower linear relaxation form, and we solve the linear relaxation optimization problem by the Branch and Bound Algorithm. Under some reasonable assumptions, the global convergence of the algorithm is certified for the problem. Numerical results show that this method is more efficient than the previous methods.

In this article we study the nonlinear Robin boundary-value problem $-{\mathrm{\Delta }}_{p(x)}u=f(x,u)\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }$, $|\nabla u{|}^{p\left(x\right)-\mathrm{2}}(\partial u/\partial \nu )+\beta (x){\left|u\right|}^{p(x)-\mathrm{2}}u=\mathrm{0}$ on $\partial \mathrm{\Omega }$. Using the variational method, under appropriate assumptions on $f$, we obtain results on existence and multiplicity of solutions.

We study the one-dimensional forced harmonic oscillator with relativistic effects. Under some conditions of the parameters, the existence of a unique stable periodic solution is proved which is of twist type. The results depend on a Twist Theorem for nonlinear Hill’s equations which is established and proved here.

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

This paper describes the studied effects of thermal radiation and chemical reaction on unsteady MHD non-Newtonian (obeying Walter’s B model) fluid in porous medium. The resulting problems are solved numerically. Graphical results for various interesting parameters are presented. Also the effects of the different parameters on the skin-friction and the heat fluxes are obtained and discussed numerically.

We investigate some results about mean-inequalities involving a large number of bivariate means. As application, we derive a lot of inequalities between four or more means among the standard means known in the literature.

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^{⁎}$. Let $T:X\supseteq D(T)\to {\mathrm{2}}^{X^{⁎}}$ and $A:X\supseteq D(A)\to {\mathrm{2}}^{X^{⁎}}$ be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for $T+A$ under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition $\stackrel{\circ }{D(T)}\cap D(A)\ne \mathrm{\varnothing }$ and Browder and Hess who used the quasiboundedness of $T$ and condition $\mathrm{0}\in D(T)\cap D(A)$. In particular, the maximality of $T+\partial \varphi$ is proved provided that $\stackrel{\circ }{D(T)}\cap D(\varphi )\ne \mathrm{\varnothing }$, where $\varphi :X\to (-\mathrm{\infty },\mathrm{\infty }]$ is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.

We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.

This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving the $p$-Laplacian and the $p$-pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned operators.

We study a class of semilinear nonlocal elliptic systems posed on settings without compact Sobolev embedding. By employing critical point theory and concentration estimates, we prove the existence of infinitely many solutions for values of the dimension $N$, where $N>\mathrm{6}s,$ provided

We have considered a multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we studied a Fritz-John type necessary condition. Then, we introduced two constraint qualifications and derive the weak and strong Karush-Kuhn-Tucker (KKT in brief) types necessary conditions for an efficient solution of the considered problem. Finally an extension of a Caristi-Ferrara-Stefanescu result for the ($\mathrm{\Phi },\rho$)-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in terms of Clark subdifferential.

In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature.

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases and

By making use of the concept of $q$-calculus, various types of generalized starlike functions of order $\alpha$ were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of $q$-starlike functions of order $\alpha$. We also introduce and study a new subclass of $q$-starlike functions of order $\alpha$. Moreover, we give some properties of those $q$-starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.

Let $l^{\mathrm{2}}=L^{\mathrm{2}}\left(\mathbb{N},\mu \right)$, where $\mathbb{N}$ is set of all positive integers and $\mu$ is the counting measure whose $\sigma$-algebra is the power set of $\mathbb{N}$. In this paper, we obtain necessary and sufficient conditions for a weighted composition operator to be antinormal on the Hilbert space $l^{\mathrm{2}}$. We also determine a class of antinormal weighted composition operators on Hardy space $H^{\mathrm{2}}\left(\mathbb{D}\right)$.

Suppose $C$ is a cone contained in real vector space $V$. When does $V$ contain a hyperplane $H$ that intersects each of the 0-rays in $C\setminus \{\mathrm{0}\}$ exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question. We also present an example of a salient, closed Banach space cone $C$ for which there does not exist a hyperplane that intersects each 0-ray in $C\setminus \{\mathrm{0}\}$ exactly once.

This paper investigates the two-sided first exit problem for a jump process having jumps with rational Laplace transform. The corresponding boundary value problem is solved to obtain an explicit formula for the first passage functional. Also, we derive the distribution of the first passage time to two-sided barriers and the value at the first passage time.

We study quasi-hyperbolicity of the delay semigroup associated with the equation $u^{\mathrm{\prime }}(t)=Bu(t)+\mathrm{\Phi }{u}_t$, where $u_t$ is the history function and $(B,D(B))$ is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup.

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.

This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of $\mathbb{R}$. The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field.

We show that the dual ${\left(B_{p\left(·\right)}^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\mathrm{\Omega }\right)\right)}^{\mathrm{\prime }}$ of the variable exponent Hörmander space $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to the Hörmander space $B_{\mathrm{\infty }}^c(\mathrm{\Omega })$ (when the exponent $p(·)$ satisfies the conditions , the Hardy-Littlewood maximal operator $M$ is bounded on $L_{p(·)/p_{\mathrm{0}}}$ for some and $\mathrm{\Omega }$ is an open set in ${\mathbb{R}}^n$) and that the Fréchet envelope of $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is the space $B_{\mathrm{1}}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$. Our proofs rely heavily on the properties of the Banach envelopes of the $p_{\mathrm{0}}$-Banach local spaces of $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for $p(·)\equiv p$, , are also given (e.g., all quasi-Banach subspace of $B_p^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to a subspace of $l_p$, or $l_{\mathrm{\infty }}$ is not isomorphic to a complemented subspace of the Shapiro space $h_{p^{-}}$). Finally, some questions are proposed.

The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations.

We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.

We continue the study of discrete anisotropic equations and we will provide new multiplicity results of the solutions for a discrete anisotropic equation. We investigate the existence of infinitely many solutions for a perturbed discrete anisotropic boundary value problem. The approach is based on variational methods and critical point theory.

An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.