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We investigate the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half line ℝ+ ≔ (0, ∞). Inspired by the relationship between a micropolar fluid model and Navier-Stokes equations, we prove that the composite wave consisting of the transonic boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave and the 3-rarefaction wave for the inflow problem on the micropolar fluid model is time-asymptotically stable under some smallness conditions. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

We obtain new Kolmogorov-type sharp inequalities estimating the norm of the Marchaud fractional derivative $$ {\left\Vert {D}_{-}^kf\right\Vert}_{\infty } $$ of a function f defined on the positive half line in terms of ‖f‖p, 1

We show that, under the long-tailedness of the densities of normalized Levy measures, the densities of infinitely divisible distributions on the half-line are subexponential if and only if the densities of their normalized Levy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half-line are subexponential if and only if their normalized Levy measures are locally subexponential.

In the first part of this paper, we pose and solve a new version of Skorokhod problem with jumps, reflected in the half line $$\mathbb {R}^{+}$$ and associated to a right limited and left limited function. We show the existence and uniqueness of the solution for this problem. Then, we apply our result to prove existence and uniqueness of solutions of reflected stochastic differential equations driven by optional semimartingales on an unusual probability space.

The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line. We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3 × 3 matrix Riemann-Hilbert problem formulated in the complex k-plane. The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k) and S(k) that depend on the initial data and boundary values, respectively. Then, applying nonlinear steepest descent techniques to the associated 3 × 3 matrix-valued Riemann-Hilbert problem, we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.

This paper deals with a fourth-order nonlinear parabolic equation with logarithmic nonlinearity coming from the modeling of epitaxial growth. First, by establishing a new infinite time blow-up condition which is independent of the mountain-pass level, we show the solution can be extended over time (the whole half line) and then blows up at $$\infty $$ ; second, we prove that the solution can blow up at $$\infty $$ with arbitrary initial energy using this new infinite time blow-up condition; thirdly, some numerical simulations are presented to verify and illustrate the theoretical results. The results of this paper complete and extend the previous studies on this type of model.

We show that the half-line m functions associated with the vector-valued discrete Schrodinger operators are the elements in the Siegel upper half space. We introduce a metric on the space of m functions associated to these operators. Then, we show that the action of transfer matrices on these m functions is distance decreasing.

We introduce a new system of Jacobi rational functions, which is mutually orthogonal with respect to the weight function $$x^\beta $$ on the half line. Some properties and approximation results on Jacobi rational functions are established. The Jacobi rational spectral methods are proposed for various elliptic boundary value problems on unbounded domains and their convergence is proved. Numerical results demonstrate the spectral accuracy of these approaches.

An initial boundary-value problem for the Hirota equation on the half-line, 0 0, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert (RH) problem in the complex k-plane. This RH problem has explicit (x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x, 0) = q0(x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g0(t), qx(0, t) = g1(t) and qxx(0, t) = g2(t). The spectral functions satisfy an algebraic global relation which characterizes, say, g2(t) in terms of {q0(x), g0(t), g1(t)}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

In this paper we are interested in the existence of solutions for the following boundary value problem at resonance on the half-line where $$f:\mathbb {R}^{+}\times \mathbb {R}^{2}\rightarrow \mathbb {R}$$ is q-Caratheodory, $$ g:\mathbb {R}^{+}\rightarrow \mathbb {R}^{+} $$ is a nondecreasing function with $$g(0)=0$$ under the resonance condition $$g(\infty )= 1$$ . The coincidence degree theory of Mawhin [17] is used to establish the existence of at least one solution for the posed problem.

In the last years the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. This turns out to be a rather challenging problem, mainly when studied in low Sobolev regularity. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view.

We investigate a class of semipositone fractional integral boundary value problem on the half-line. Using the fixed point index theorems in a cone, we obtain the existence result of positive solutions.

By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with $\operatorname{dim}\operatorname{Ker}L = 1$ dim Ker L = 1 . And an example is given to show that our result here is valid.

We establish necessary and sufficient conditions for a system of positive numbers $$ {M}_{k_1},{M}_{k_2},{M}_{k_3},{M}_{k_4},0={k}_1

In this paper we consider a second-order singular differential operator $$ \Delta _{\alpha ,n}u = u'' + ((2\alpha +1)/x)u'-\frac{4n(\alpha +n)}{x^{2}}u, \alpha >-1/2, n\in {\mathbb {N}}$$ on the half line which generalize the Bessel operator $$ \Delta _{\alpha }u = u'' + ((2\alpha +1)/x)u', \alpha >-1/2 $$ . A generalized integral transform $$ T_{\varphi }^{\alpha ,n}$$ associated with $$\Delta _{\alpha ,n}$$ is studied in $$ L_{\alpha ,n}^{2}$$ and we have established a relation between $$ T_{\varphi }^{\alpha ,n}$$ its adjoint and generalized Bessel Plancherel transform $$\phi _{\alpha ,n}$$ associated with $$\Delta _{\alpha ,n}.$$ We derive new results concerning the relation between $$\phi _{\alpha ,n}$$ , and the generalized Riemann-Liouville transform, generalized Weyl transform, generalized Sonine transform and a Hankel potential type transform associated with $$ \Delta _{\alpha ,n}.$$

In this article, we prove under eigenvalue criteria, existence result for nodal solutions to the boundary value problem posed on the positive half-line: $$\begin{aligned} \left\{ \begin{array}{l} -u^{\prime \prime }(t)+q(t)u(t)=u(t)f(t,u(t))\quad t>0, \\ u(0)=\lim _{t\rightarrow +\infty }u(t)=0, \end{array} \right. \end{aligned}$$ where $$q\in C\left( \mathbb {\mathbb {R} }^{+},\mathbb {\mathbb {R}}^{+}\right) $$ may be unbounded from above and $$ f:{\mathbb {R}}^{+}\times \mathbb {R} \rightarrow \mathbb {R} $$ is a continuous function.

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation $$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$$ on the half line (−∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u0(x) = u(x, 0) and the boundary data g0(t) = u(0, t), g1(t) = ux(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

This work considers two types of a second order impulsive coupled system of differential equations with generalized jump conditions in half-line. For both problems it will be presented two localization results with different monotonicity assumptions on the nonlinearities and on the impulsive conditions. The arguments apply lower and upper solutions method combined with Nagumo type condition and truncature techniques. Last section contains an applications of one of this impulsive problems to logging timber by helicopter.

We study a Nikol’skii type inequality for even entire functions of given exponential type between the uniform norm on the half-line [0,∞) and the norm (∫ 0 ∞ |f(x)| q x2α+1dx)1/q of the space L q ((0,∞), x2α+1) with the Bessel weight for 1 ≤ q −1/2. An extremal function is characterized. In particular, we prove that the uniform norm of an extremal function is attained only at the end point x = 0 of the half-line. To prove these results, we use the Bessel generalized translation.

Inverse dynamic, spectral, quantum, and acoustic scattering problems for the Schrodinger operator on a half-line are considered. The goal of the paper is to establish the relationship between different types of inverse data for these problems. The central objects, which serve as a source for all formulas, are the kernels of the so-called connecting operators and the corresponding Krein equations.

We discuss a variational approach to abstract doubly nonlinear evolution systems defined on the time half line (Formula presented.). This relies on the minimization of weighted energy-dissipation (WED) functionals, namely a family of (Formula presented.)-dependent functionals defined over entire trajectories. We prove WED functionals admit minimizers and that the corresponding Euler-Lagrange system, which is indeed an elliptic-in-time regularization of the original problem, is strongly solvable. Such WED minimizers converge, up to subsequences, to a solution of the doubly nonlinear system as (Formula presented.). The analysis relies on a specific estimate on WED minimizers, which is specifically tailored to the unbounded time interval case. In particular, previous results on the bounded time interval are extended and generalized. Applications of the theory to classes of nonlinear PDEs are also presented.

In this paper, the existence of solutions for a fourth-order boundary value problem with nonlinear derivative dependence on the half-line is studied which is based on critical point theory and iterative methods.

An explicit formula for the wave operators associated with Schrodinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional pseudo-differential operators of order zero. As an application, we give a topological interpretation for Levinson’s theorem, which relates the scattering phase shift and the number of bound states of the system.

In this paper, we consider a new fractional differential system on an unbounded domain $$\begin{aligned} &D^{\alpha }u(t)+\varphi \bigl(t,v(t),D^{\gamma _{1}}v(t)\bigr)=0, \quad t \in [0,+ \infty ), \alpha \in (2,3], \\ &D^{\beta }v(t)+\psi \bigl(t,u(t),D^{\gamma _{2}}u(t)\bigr)=0, \quad t\in [0,+\infty ), \beta \in (2,3], \end{aligned}$$ subject to the conditions $$\begin{aligned} &I^{3-\alpha }u(t)|_{t=0}=0, \quad\quad D^{\alpha -2}u(t)|_{t=0}= \int _{0}^{h} g _{1}(s)u(s)\,ds, \quad \quad D^{\alpha -1}u(+\infty )=Mu(\xi )+a, \\ &I^{3-\beta }v(t)|_{t=0}=0, \quad\quad D^{\beta -2}v(t)|_{t=0}= \int _{0}^{h} g _{2}(s)v(s)\,ds, \quad \quad D^{\beta -1}v(+\infty )=Nv(\eta )+b. \end{aligned}$$ The nonlinear terms φ and ψ are dependent on the fractional derivative of lower order $\gamma _{i}\in (0,1)$ , $i=1,2$ , which creates additional complexity to verify the existence of solutions. Moreover, a proper choice of Banach space allows the solutions to be defined on the half-line. From some standard fixed point theorems, sufficient conditions for the existence and uniqueness of solutions to boundary value problems are developed. Finally, the main result is applied to an illustrative example.

It is shown that all the transmission eigenvalues (including multiplicity) of the Schrodinger operator on the half line can not uniquely determine the potential if the certain constant $$\gamma $$ is unknown, which gives an answer to the open question raised by Aktosun et al. (Inverse Probl 30:075001, 2014).

We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrodinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a no...

It has been conjectured that the defocusing nonlinear Schr\"odinger (NLS) equation on the half-line does not admit solitons. We give a proof of this conjecture., Comment: 5 pages

In this paper, we introduce the fractional wavelet transformations (FrWT) involving Hankel–Clifford integral transformation (HClIT) on the positive half line and studied some of its basic properties. Also we obtain Parseval’s relation and an inversion formula. Examples of fractional powers of Hankel–Clifford integral transformation (FrHClIT) and FrWT are given. Then, we introduce the concept of fractional wavelet packet transformations FrBWPT and FrWPIT, and investigate their properties.

We study the existence of positive solutions to boundary-value problems for two systems of two secondorder nonlinear three-point ϕ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces.

This paper is concerned with the large-time behavior of solutions to an initial-boundary-value problem for full compressible Navier-Stokes equations on the half line (0,∞), which is named impermeable wall problem. It is shown that the 3-rarefaction wave is stable under partially large initial perturbation if the adiabatic exponent γ is close to 1. Here partially large initial perturbation means that the perturbation of absolute temperature is small, while the perturbations of specific volume and velocity can be large. The proof is given by the elementary energy method.

We study a class of second-order superlinear equations. New oscillations criteria are established by using a general class of parameter functions in the averaging techniques. We extend and improve the oscillation criteria of several authors. One of our results is based on the information for the whole half line and the other is based on the information on a sequence subintervals of the whole half line.

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function \(\varPsi \) is the Bernstein function of a self-decomposable probability law \(\pi \) on the positive half-axis if and only if alternating sums of \(\varPsi \) satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that \(\varPsi \) is a self-decomposable Bernstein function if and only if a related d-variate function \(C_{\psi ,d}\), \(\psi :=\exp (-\varPsi )\), is a d-variate copula for each \(d \ge 2\). A canonical stochastic construction is presented, in which \(\pi \) (respectively \(\varPsi \)) determines the probability law of an exchangeable sequence of random variables \(\{U_k\}_{k\in {\mathbb {N}}}\) such that \((U_1,\ldots ,U_d) \sim C_{\psi ,d}\) for each \(d \ge 2\). The random variables \(\{U_k\}_{k\in {\mathbb {N}}},\) are i.i.d. conditioned on an increasing Sato process whose law is characterized by \(\varPsi \). The probability law of \(\{U_k\}_{k \in {\mathbb {N}}}\) is studied in quite some detail.

In this paper, we deal with the following nonlinear fractional differential problem in the half-line $${\mathbb{R}^{+}=(0,+ \infty)}$$ $$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$ where $${m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}$$ , the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in $${\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}$$ satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schauder fixed point theorem. Some examples illustrating our main result are also given.

In this paper we study the Bessel process R_t^{(��)} with index ��\neq 0 starting from x>0 and killed when it reaches a positive level a, where x>a>0. We provide sharp estimates of the transition probability density p_a^{(��)}(t,x,y) for the whole range of space parameters x,y>a and every t>0., 18 pages

In this paper, we consider the solvability of a third-order three-point boundary value problem on a half-line of the form: $$\begin{aligned} {\left\{ \begin{array}{ll} x^{\prime \prime \prime }(t)=f\left( t,x(t),{x^{\prime }}(t),{x^{\prime \prime }}(t)\right) ,\quad 0< t

In this paper, we study a boundary value problem for a coupled differential system of fractional order on the half-line. The differential operator is taken in the Riemann–Liouville sense and the nonlinear terms involve the fractional derivative of the unknown functions. Applying the Schauder fixed point theorem, we prove the existence of infinitely many positive unbounded solutions of the fractional differential system. Also, we give examples to illustrate our main result.

In this paper, we consider a class of infinite-point boundary value problems of fractional differential equations on the infinite interval $[0,+\infty)$ with a disturbance parameter. By using the method of upper and lower solutions, fixed point index theory and some fixed point theorems, the existence, multiplicity and nonexistence for the positive solution of the boundary value problem are obtained, respectively. The impact of the disturbance parameters on the existence of positive solutions is also given. Finally, some examples are presented to illustrate the wide range of potential applications of our main results.

The aim of the paper is twofold. At first, it is an erratum to: Gabor G., Grudzka A.: Structure of the solution set to impulsive functional differential inclusions on the half-line. NoDEA 19, 609–627 (2012). We fill a little gap in a proof of Theorem 3.5 by a change of a compactness type condition, and give a discussion on assumptions sufficient to obtain an Rδ-structure of solution sets. Simultaneously, we improve the main theorems by the use of modified proof-arguments. Namely, the assumptions on impulse maps can be weakened.

We obtain new sharp Kolmogorov-type inequalities for the fractional derivatives of functions defined on the half line.

In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is an RW allowing jumps to the left at a maximal size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.

In this paper, existence and iteration of positive solutions for a class of high-order fractional differential equations with integral conditions on a half-line is investigated. By means of a monotone iterative technique, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.

To understand the spreading of invasive and native species, in this paper we consider the diffusive competition models with a free boundary in the heterogeneous time-periodic environments, in which the variable intrinsic growth rates of these two species change signs and may be very negative in some large regions. We study the spreading–vanishing dichotomy, long-time dynamical behavior of solution, sharp criteria for spreading and vanishing, and estimates of the asymptotic spreading speed of the free boundary. Moreover, we establish the existence of positive solutions to a T-periodic boundary value problem of the diffusive competition system with sign-changing growth rates in the half line.

We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e., where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces. Доведено iснування iнтегральних (стiйких, нестiйких, центральних) многовидiв для розв’язкiв напiвлiнiйного iнтегрального рiвняння у випадку, коли сiм’я еволюцiй (U(t,s))tleqs має експоненцiальну трихотомiю на пiвосi або на всiй осi, а нелiнiйний збурюючий член f задовольняє φ-лiпшицевi умови, тобто належить до деяких класiв допустимих просторiв функцiй. Наш основний метод базується на методах Ляпунова – Перрона, процедурах перемасштабування та технiцi застосування допустимостi просторiв функцiй.

By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.

We consider the existence of multiple positive solutions of the singular boundary value problem for the Sturm-Liouville equation with impulses on the half line. By applying the fixed-point index theorem of a cone map, existence and multiplicity results about positive solutions are obtained. Our results improve and generalize some well-known results. An example is presented to demonstrate applications of our main results.

In this paper, we consider the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. By using a variational method and a variant fountain theorem, we obtain some new criteria to guarantee that the impulsive problem has at least one solution and infinitely many solutions respectively. We extend and improve some recent results.

In this paper, we construct the global weak solutions to the initial-boundary problem for the Navier–Stokes system with capillarity in the half space \({\mathbb{R}_+^1}\). The result extends Eugene Tsyganov’s existence theorem which considered the problem in the finite region published in J. Differential Equaions 245:3936–3955, 2008.