We consider the nonlinear eigenvalue problem $u^{\mathrm{″}}(t)+\lambda f(u(t))=0$, $u(t)>0$, $t\in I=:(-\mathrm{1,1})$, $u(1)=u(-1)=0$, where $f(u)$ is a cubic-like nonlinear term and $\lambda >0$ is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions on $f(u)$, there exist exactly three bifurcation branches $\lambda ={\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$), and these curves are parameterized by the maximum norm $\xi$ of the solution $u_{\lambda }$ corresponding to $\lambda$. In this paper, we establish the precise global structures for ${\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$), which can be applied to the inverse bifurcation problems. The precise local structures for ${\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$) are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solution $u_2(\lambda ,t)$, which corresponds to $\lambda ={\lambda }_2(\xi )$, as $\lambda \to \mathrm{\infty }$.

We study boundary value problems for $q$-difference equations and inclusions with nonlocal and integral boundary conditions which have different quantum numbers. Some new existence and uniqueness results are obtained by using fixed point theorems. Examples are given to illustrate the results.

We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equations $P_{\mathrm{1}},\dots ,P_{\mathrm{6}}$.

We apply the averaging theory of first and second order to a class of generalized Kukles polynomial differential systems to study the maximum number of limit cycles of these systems.

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form: $X^{(4)}+F(\dot{X},\ddot{X})\stackrel{⃛}{X}+\mathrm{\Phi }(\ddot{X})+G(\dot{X}(t-r))+H(X(t-r))=0.$ By constructing a Lyapunov functional, we obtained the result of stability.

The inverse problem by the Weyl matrix is studied for the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval. We construct special fundamental systems of solutions for this equation and prove the uniqueness theorem of the inverse problem.

Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance.

We introduce and study two subclasses of multivalent functions denoted by ${\mathcal{M}}_{p,n}^{m,\alpha ,\beta ,\sigma }({\lambda }_1;{\lambda }_2)$ and ${\mathcal{N}}_{p,n}^{m,\alpha ,\beta ,\sigma }(\mu ,\delta ;\gamma )$. Further, by using the method of differential subordination, certain inclusion relations between the two subclasses aforementioned are given. Moreover, several consequences of the main results are also discussed.

We will investigate the decay estimate of the energy of the global solutions to the p-Laplacian wave equation with dissipation of the form $u_{tt}-\text{div}\mathrm{ }({\left|{\nabla }_{x}u\right|}^{p-2}{\nabla }_{x}u)+\sigma (t)(u_t-\text{div}\mathrm{ }({\left|{\nabla }_x{u}_t\right|}^{m-2}{\nabla }_x{u}_t))=0$ under suitable assumptions on the positive function $\sigma$. For this end we use the multiplier method combined with nonlinear integral inequalities given by Martinez; the proof is based on the construction of a special weight function that depends on the behavior of $\sigma$.

Self-similar blow-up solutions for the generalized deterministic KPZ equation $u_t=u_{xx}+|u_x{|}^q$ with $q>2$ are considered. The asymptotic behavior of self-similar solutions is studied.

This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equations $x_{n+1}=A+x_{n-1}^p/y_n^p\text{and}{y}_{n+1}=A+y_{n-1}^p/x_n^p$, $n=\mathrm{0,1},\dots$, where $A\in (0,\infty )$, $p\in [1,\infty )$, $x_i\in (0,\infty )$, and $y_i\in (0,\infty )$, $i=-\mathrm{1,0}$.

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential system $\stackrel{.}{x}=-y+\epsilon P(x,y,z)+h_{\mathrm{1}}(t)$, $\stackrel{.}{y}=x+\epsilon Q(x,y,z)+h_{\mathrm{2}}(t)$, and $\stackrel{.}{z}=az+\epsilon R(x,y,z)+h_{\mathrm{3}}(t)$, where $P$, $Q$, and $R$ are polynomials in the variables $x$, $y$, and $z$ of degree $n,$$h_i(t)=h_i(t+\mathrm{2}\pi )$ with $i=\mathrm{1,2},3$ being periodic functions, $a$ is a real number, and $\epsilon$ is a small parameter.

The behaviour of solutions for certain third-order nonlinear differential equations with multiple deviating arguments is considered. By employing Lyapunov’s second method, a complete Lyapunov functional is constructed and used to establish sufficient conditions that guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results not only are new but also include many outstanding results in the literature. Finally, the correctness and effectiveness of the results are justified with examples.

We combine the Adomian decomposition method (ADM) and Adomian’s asymptotic decomposition method (AADM) for solving Riccati equations. We investigate the approximate global solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from Adomian’s asymptotic decomposition method for Riccati equations and in such cases when we do not find any region of overlap between the obtained approximate solutions by the two proposed methods, we connect the two approximations by the Padé approximant of the near-field approximation. We illustrate the efficiency of the technique for several specific examples of the Riccati equation for which the exact solution is known in advance.

Mild solutions generated by a $\left(a, k\right)$-regularized family to fractional stochastic relaxation equations are studied. The main objective is to establish the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to linear and semilinear case of these equations. Under different hypotheses, some new theorems concerning the main objective are derived.

We consider nonlinear impulsive differential equations with ψ-exponential and ψ-ordinary dichotomous linear part in a Banach space. By the help of Banach’s fixed-point principle sufficient conditions are found for the existence of ψ-bounded solutions of these equations on $\mathbb{R}$ and ${\mathbb{R}}_{+}$.

We consider a family with respect to a small parameter of nonlinear boundary-value problems as well as the corresponding spectral problems in a domain perforated periodically along a part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain.

We establish existence and uniqueness of solutions to the Cauchy problem associated with a new one-dimensional weakly-nonlinear, weakly-dispersive system which arises as an asymptotical approximation of the full potential theory equations for modelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth, taking into account the effect of surface tension. Furthermore, numerical schemes of spectral type are introduced for approximating the evolution in time of solutions of this system and its travelling wave solutions, in both the periodic and nonperiodic case.

We consider the time-fractional derivative in the Caputo sense of order $\alpha \in (0, 1)$. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in ${\mathbb{R}}^{+}$, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when $\alpha \nearrow 1$ of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.

The present paper treats three concepts of nonuniform polynomial trichotomies for noninvertible evolution operators acting on Banach spaces. The connections between these concepts are established through numerous examples and counterexamples for systems defined on the Banach space of square-summable sequences.

This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.

Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasi-uniform and locally bounded meshes. These avoid the ill-conditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arc-length monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes.

We consider the equation $u^{″}=P(z)u+F(z)\hspace{0.17em}\hspace{0.17em}(z\in \mathbb{C})$, where $P(z)$ is a polynomial and $F(z)$ is an entire function. Let $z_k(u)\hspace{0.17em}\hspace{0.17em}(k=\mathrm{1,2},\dots )$ be the zeros of a solution $u(z)$ to that equation. Lower estimates for the products ${\prod }_{k=\mathrm{1}}^j|z_k(u)|\hspace{0.17em}\hspace{0.17em}(j=\mathrm{1,2},\dots )$ are derived. In particular, they give us a bound for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.

We give a characterization of the control for ill-posed problems of oscillating solutions. More precisely, we study the control of Cauchy elliptic problems via a regularization approach which generates incomplete information. We obtain a singular optimality system characterizing the no-regret control for the Cauchy problem.

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy $g$. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive $h$ (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter $h$ goes to zero.

We study in this paper nonlinear anisotropic problems with Robin boundary conditions. We prove, by using the technic of monotone operators in Banach spaces, the existence of a sequence of weak solutions of approximation problems associated with the anisotropic Robin boundary value problem. For the existence and uniqueness of entropy solutions, we prove that the sequence of weak solutions converges to a measurable function which is the entropy solution of the anisotropic Robin boundary value problem.

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

We investigate the existence and iteration of positive solutions for the following third-order $p$-Laplacian dynamic equations on time scales: $({\varphi }_p(u^{\mathrm{\Delta }\mathrm{\Delta }}(t)){)}^{\nabla }+q(t)f(t,u(t),u^{\mathrm{\Delta }\mathrm{\Delta }}(t))=\mathrm{0},\hspace{0.17em}\hspace{0.17em}t\in [a,b],$$\alpha u(\rho (a))-\beta u^{\mathrm{\Delta }}(\rho (a))=\mathrm{0},\hspace{0.17em}\hspace{0.17em}\gamma u(b)+\delta u^{\mathrm{\Delta }}(b)=\mathrm{0},\hspace{0.17em}\hspace{0.17em}{u}^{\mathrm{\Delta }\mathrm{\Delta }}(\rho (a))=\mathrm{0},$ where ${\varphi }_p(s)$ is $p$-Laplacian operator; that is, ${\varphi }_p(s)={\left|s\right|}^{p-\mathrm{2}}s,\hspace{0.17em}\hspace{0.17em}p>\mathrm{1},\hspace{0.17em}\hspace{0.17em}{\varphi }_p^{-\mathrm{1}}={\varphi }_q$, and $\mathrm{1}/p+\mathrm{1}/q=\mathrm{1}.$ By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

This work is devoted to the study of a general class of anisotropic problems involving $\overrightarrow{p}(·)$-Laplace operator. Based on the variational method, we establish the existence of a nontrivial solution without Ambrosetti-Rabinowitz type conditions.

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as $L$ must be greater than the critical patch size $L_c$. It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function.