Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of $S$-derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.

In this paper, the coupled Schrödinger-Boussinesq equations (SBE) will be solved by the sech, tanh, csch, and the modified simplest equation method (MSEM). We obtain exact solutions of the nonlinear for bright, dark, and singular 1-soliton solution. Kerr law nonlinearity media are studied. Results have proven that modified simple equation method does not produce the soliton solution in general case. Solutions may find practical applications and will be important for the conservation laws for dispersive optical solitons.

The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.

The one-dimensional advection-diffusion-reaction equation is a mathematical model describing transport and diffusion problems such as pollutants and suspended matter in a stream or canal. If the pollutant concentration at the discharge point is not uniform, then numerical methods and data analysis techniques were introduced. In this research, a numerical simulation of the one-dimensional water-quality model in a stream is proposed. The governing equation is advection-diffusion-reaction equation with nonuniform boundary condition functions. The approximated pollutant concentrations are obtained by a Saulyev finite difference technique. The boundary condition functions due to nonuniform pollutant concentrations at the discharge point are defined by the quadratic interpolation technique. The approximated solutions to the model are verified by a comparison with the analytical solution. The proposed numerical technique worked very well to give dependable and accurate solutions to these kinds of several real-world applications.

We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed.

We obtain in this article a solution of sequential differential equation involving the Hadamard fractional derivative and focusing the orders in the intervals $(\mathrm{1,2})$ and $(\mathrm{2,3})$. Firstly, we obtain the solution of the linear equations using variation of parameter technique, and next we investigate the existence theorems of the corresponding nonlinear types using some fixed-point theorems. Finally, some examples are given to explain the theorems.

This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.

Air pollutant levels in Bangkok are generally high in street tunnels. They are particularly elevated in almost closed street tunnels such as an area under the Bangkok sky train platform with high traffic volume where dispersion is limited. There are no air quality measurement stations in the vicinity, while the human population is high. In this research, the numerical simulation is used to measure the air pollutant levels. The three-dimensional air pollution measurement model in a heavy traffic area under the Bangkok sky train platform is proposed. The finite difference techniques are employed to approximate the modelled solutions. The vehicle air pollutant emission due to the high traffic volume is mathematically assumed by the pollutant sources term. The simulation is also considered in averaged and moving pollutant sources due to manner vehicle emission. The proposed approximated air pollutant concentration indicators can be replaced by user required gaseous pollutants indices such as NOx, SO2, CO, and PM2.5.

In 2016, some inequalities of the Ostrowski type for functions (of two variables) differentiable on the coordinates were established. In this paper, we extend these results to an arbitrary time scale by means of a parameter $\lambda \in \left[\mathrm{0,1}\right]$. The aforementioned results are regained for the case when the time scale $\mathbb{T}=\mathbb{R}$. Besides extension, our results are employed to the continuous and discrete calculus to get some new inequalities in this direction.

For each $x_{\mathrm{0}}\in [\mathrm{0,2}\pi )$ and $k\in \mathbf{N}$, we obtain some existence theorems of periodic solutions to the two-point boundary value problem $u^{\mathrm{\prime }\mathrm{\prime }}(x)+k^{\mathrm{2}}u(x-x_{\mathrm{0}})+g(x,u(x-x_{\mathrm{0}}))=h(x)$ in $(\mathrm{0},\mathrm{2}\pi )$ with $u(\mathrm{0})-u(\mathrm{2}\pi )=u^{\mathrm{\prime }}(\mathrm{0})-u^{\mathrm{\prime }}(\mathrm{2}\pi )=\mathrm{0}$ when $g:(\mathrm{0,2}\pi )\times \mathbf{R}\to \mathbf{R}$ is a Caratheodory function which grows linearly in $u$ as $\left|u\right|\to \mathrm{\infty }$, and $h\in L^{\mathrm{1}}(\mathrm{0,2}\pi )$ may satisfy a generalized Landesman-Lazer condition for all $v\in N(L)\backslash \{\mathrm{0}\}$. Here $N(L)$ denotes the subspace of $L^{\mathrm{1}}(\mathrm{0,2}\pi )$ spanned by $\sin kx$ and $\cos kx$, , $g_{\beta }^{+}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to \mathrm{\infty }}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta })$, and $g_{\beta }^{-}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to -\mathrm{\infty }}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta })$.

We consider convergence acceleration of the modified Fourier expansions by rational trigonometric corrections which lead to modified-trigonometric-rational approximations. The rational corrections contain some unknown parameters and determination of their optimal values for improved pointwise convergence is the main goal of this paper. The goal was accomplished by deriving the exact constants of the asymptotic errors of the approximations with further elimination of the corresponding main terms by appropriate selection of those parameters. Numerical experiments outline the convergence improvement of the optimal rational approximations compared to the expansions by the modified Fourier basis.

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

The Riemann solutions of a deposition model are shown. A singular flux-function limit of the obtained Riemann solutions is considered. As a result, it is shown that the Riemann solutions of the deposition model just converge to the Riemann solutions of the limit system, the scalar conservation law with a linear flux function involving discontinuous coefficient. Especially, for some initial data, the two-shock Riemann solution of the deposition model tends to the delta-shock Riemann solution of the limit system; by contrast, for some initial data, the two-rarefaction-wave Riemann solution of the deposition model tends to the vacuum Riemann solution of the limit system. Some numerical results exhibiting the formation processes of delta-shocks and vacuum states are presented.

We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular $C^{\mathrm{1}}$ Hermite data-sets. Each such data-set is satisfied by two scaled Enneper surfaces, and we can obtain four interpolants on each surface. Examples of these interpolants were found to be better, in terms of bending energy and arc-length, than those obtained using a previous PH-preserving mapping.

Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a $C_{\mathrm{0}}$-quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.

The aim of this paper is to study various properties of Mittag-Leffler (M-L) function. Here we establish two theorems which give the image of this M-L function under the generalized fractional integral operators involving Fox’s $H$-function as kernel. Corresponding assertions in terms of Euler, Mellin, Laplace, Whittaker, and $K$-transforms are also presented. On account of general nature of M-L function a number of results involving special functions can be obtained merely by giving particular values for the parameters.

By a rotational system, we mean a closed subset $X$ of the circle, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, together with a continuous transformation $f:X\to X$ with the requirements that the dynamical system $(X,f)$ be minimal and that $f$ respect the standard orientation of $\mathbb{T}$. We show that infinite rotational systems $(X,f)$, with the property that map $f$ has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, $F:\mathbb{T}\to \mathbb{T}$. Because our main result makes no explicit mention of a global transformation on $\mathbb{T}$, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation $F:\mathbb{T}\to \mathbb{T}$ with finite preimages. In particular, there are no explicit conditions on the degree of $F$. We then give a development of known results in the case where $F\left(\theta \right)=d·\theta \mathrm{mod}\mathrm{1}$ for an integer $d>\mathrm{1}$. The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.

We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.

An efficient iteration method is introduced and used for solving a type of system of nonlinear Volterra integro-differential equations. The scheme is based on a combination of the spectral collocation technique and the parametric iteration method. This method is easy to implement and requires no tedious computational work. Some numerical examples are presented to show the validity and efficiency of the proposed method in comparison with the corresponding exact solutions.

Through a modification on the parameters associated with generating function of the $q$-extensions for the Apostol type polynomials of order $\alpha$ and level $m$, we obtain some new results related to a unified presentation of the $q$-analog of the generalized Apostol type polynomials of order $\alpha$ and level $m$. In addition, we introduce some algebraic and differential properties for the $q$-analog of the generalized Apostol type polynomials of order $\alpha$ and level $m$ and the relation of these with the $q$-Stirling numbers of the second kind, the generalized $q$-Bernoulli polynomials of level $m$, the generalized $q$-Apostol type Bernoulli polynomials, the generalized $q$-Apostol type Euler polynomials, the generalized $q$-Apostol type Genocchi polynomials of order $\alpha$ and level $m$, and the $q$-Bernstein polynomials.

We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^{*}$. Let $T:X\supseteq D(T)\to {\mathrm{2}}^{X^{*}}$ be a maximal monotone operator and $C:X\supseteq D(C)\to X^{*}$ be bounded and continuous with $D(T)\subseteq D(C)$. The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the type $T+C$ provided that $C$ is compact or $T$ is of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition on $T+C$. The operator $C$ is neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.

Let $\mathrm{\Omega }$ be a smoothly bounded pseudoconvex domain in ${\mathbb{C}}^{\mathrm{3}}$ and assume that where $z_{\mathrm{0}}\in b\mathrm{\Omega }$, the boundary of $\mathrm{\Omega }$. Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” $C_b(z_{\mathrm{0}},{\mathrm{\delta }}_{\mathrm{0}})\subset \mathrm{\Omega }\cup \left\{z_{\mathrm{0}}\right\}$.

Vaccine-induced protection is substantial to control, prevent, and reduce the spread of infectious diseases and to get rid of infectious diseases. In this paper, we propose an SVEIR epidemic model with age-dependent vaccination, latency, and infection. The model also considers that the waning vaccine-induced immunity depends on vaccination age and the vaccinated individuals fall back to the susceptible class after losing immunity. The model is a coupled system of (hyperbolic) partial differential equations with ordinary differential equations. The global dynamics of the model is established through construction of appropriate Lyapunov functionals and application of Lasalle’s invariance principle. As a result, the global stability of the infection-free equilibrium and endemic equilibrium is obtained and is fully determined by the basic reproduction number ${\mathfrak{R}}_{\mathrm{0}}$.

This study developed a new method of hypothesis testing of model conformity between truncated spline nonparametric regression influenced by spatial heterogeneity and truncated spline nonparametric regression. This hypothesis test aims to determine the most appropriate model used in the analysis of spatial data. The test statistic for model conformity hypothesis testing was constructed based on the likelihood ratio of the parameter set under H₀ whose components consisted of parameters that were not influenced by the geographical factor and the set under the population parameter whose components consisted of parameters influenced by the geographical factor. We have proven the distribution of test statistics $V$ and verified that each of the numerators and denominators in the statistic test $V$ followed a distribution of ${\chi }^{\mathrm{2}}$. Since there was a symmetric and idempotent matrix S, it could be proved that ${\tilde{\mathrm{Y}}}^{\mathrm{T}}S \tilde{\mathrm{Y}}/{\sigma }^{\mathrm{2}}~{\chi }_{\left(n-lm-\mathrm{1}\right)}^{\mathrm{2}}$. Matrix $D\left(u_i,v_i\right)$ was positive semidefinite and contained weighting matrix $\mathbf{W}\left(u_i,v_i\right)$ which had different values in every location; therefore matrix $D\left(u_i,v_i\right)$ was not idempotent. If ${\tilde{\mathrm{Y}}}^{\mathrm{T}}D\left(u_i,v_i\right)\tilde{\mathrm{Y}}\ge \mathrm{0}$ and $D\left(u_i,v_i\right)$ was not idempotent and also $\tilde{\mathrm{Y}}$ was a $N\left(\mathit{0},\mathit{I}\right)$ distributed random vector, then there were constants $k$ and $r$; hence ${\tilde{\mathrm{Y}}}^{\mathrm{T}}D\left(u_i,v_i\right)\tilde{\mathrm{Y}}~k{\chi }_r^{\mathrm{2}}$; therefore it was concluded that test statistic $V$ followed an F distribution. The modeling is implemented to find factors that influence the unemployment rate in 38 areas in Java in Indonesia.

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points $F(T)$ of a multivalued (or single-valued) $k-$strictly pseudocontractive-type mapping $T$ and the set of solutions $EP(F)$ of an equilibrium problem for a bifunction $F$ in a real Hilbert space $H$. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence $\{K_n{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of closed convex subsets of $H$ from an arbitrary $x_{\mathrm{0}}\in H$ and a sequence $\{x_n{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of the metric projections of $x_{\mathrm{0}}$ into $K_n$. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

In this paper, we consider a general symmetric diffusion semigroup ${\left\{T_{t}f\right\}}_{t\ge \mathrm{0}}$ on a topological space $X$ with a positive $\sigma$-finite measure, given, for $t>\mathrm{0}$, by an integral kernel operator: $T_{t}f(x)\triangleq {\int }_X\mathrm{‍}{\rho }_t(x,y)f(y)dy$. As one of the contributions of our paper, we define a diffusion distance whose specification follows naturally from imposing a reasonable Lipschitz condition on diffused versions of arbitrary bounded functions. We next show that the mild assumption we make, that balls of positive radius have positive measure, is equivalent to a similar, and an even milder looking, geometric demand. In the main part of the paper, we establish that local convergence of $T_{t}f$ to $f$ is equivalent to local equicontinuity (in $t$) of the family ${\left\{T_{t}f\right\}}_{t\ge \mathrm{0}}$. As a corollary of our main result, we show that, for $t_{\mathrm{0}}>\mathrm{0}$, $T_{t+t_{\mathrm{0}}}f$ converges locally to $T_{t_{\mathrm{0}}}f$, as $t$ converges to ${\mathrm{0}}^{+}$. In the Appendix, we show that for very general metrics $\mathcal{D}$ on $X$, not necessarily arising from diffusion, ${\int }_X\mathrm{‍}{\rho }_t(x,y)\mathcal{D}(x,y)dy\to \mathrm{0}\text{a.e.}$, as $t\to {\mathrm{0}}^{+}.$ R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly in $x$, in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded function $f$ being Lipschitz, and the rate of convergence of $T_{t}f$ to $f$, as $t\to {\mathrm{0}}^{+}$. We do not make such an assumption in the present work.

In this paper we introduce the notion of proximal $\rho$-normal structure of pair of $\rho$-admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.

We are concerned with some existence and attractivity results of a coupled fractional Riemann–Liouville–Volterra–Stieltjes multidelay partial integral system. We prove the existence of solutions using Schauder’s fixed point theorem; then we show that the solutions are uniformly globally attractive.

In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated $C_{\mathrm{0}}$-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.

In this paper, we are concerned with the solution of the third-order nonlinear differential equation $f^{″\mathrm{\prime }}+f{f}^{″}+\beta f^{\mathrm{\prime }}(f^{\mathrm{\prime }}-\mathrm{1})=\mathrm{0}$, satisfying the boundary conditions $f(\mathrm{0})=a\in \mathbb{R}$, , and $f^{\mathrm{\prime }}(t)\to \lambda$, as $t\to +\mathrm{\infty }$, where $\lambda \in \{\mathrm{0,1}\}$ and The problem arises in the study of the opposing mixed convection approximation in a porous medium. We prove the existence, nonexistence, and the sign of convex and convex-concave solutions of the problem above according to the mixed convection parameter and the temperature parameter .

We consider the Dirichlet initial boundary value problem ${\partial }_t{u}^{m(x)}-\mathrm{div}\left({\left|\nabla u\right|}^{p\left(x,t\right)-\mathrm{2}}\nabla u\right)=a\left(x,t\right)u^{q(x,t)}$, where the exponents $p(x,t)>\mathrm{1}$, $q(x,t)>\mathrm{0}$, and $m(x)>\mathrm{0}$ are given functions. We assume that $a(x,t)$ is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if , then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when , we get the positivity of solutions for large $t$. In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that and either $q(x,t)=m(x)$ or $a(x,t)\le \mathrm{0}$ a.e. Finally, we give a uniform localization of the support of solutions for all $t>\mathrm{0}$, in the case where a.e. and .

In this paper, some new results are obtained for the even order neutral delay difference equation $\mathrm{\Delta }\left(a_n{\mathrm{\Delta }}^{m-\mathrm{1}}\left(x_n+p_n{x}_{n-k}^{\alpha }\right)\right)+q_n{x}_{n-\mathcal{l}}^{\beta }=\mathrm{0}$, where $m\ge \mathrm{2}$ is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.

In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function.

In this paper, the notion of $\mathcal{L}$-contractions is introduced and a new fixed point theorem for such contractions is established.

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series.