Authors:Yong Zhou Pages: 1 - 6 Abstract: Publication date: January 2018 Source:Applied Mathematics Letters, Volume 75 Author(s): Yong Zhou In this paper, we initiate the question of attractivity of solutions for fractional differential equations in abstract space. We establish sufficient conditions for the existence of globally attractive solutions for the Cauchy problems. Our results essentially reveal the characteristics of solutions for fractional differential equations with the Riemann–Liouville derivative.

Authors:Xiaoping Wang Pages: 7 - 12 Abstract: Publication date: January 2018 Source:Applied Mathematics Letters, Volume 75 Author(s): Xiaoping Wang In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system u ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × N and W : R × R N → R . Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim x → ∞ W ( t , x ) x 2 = ∞ , uniformly in t ∈ ( a , b ) for some − ∞ < a < b < + ∞ , instead of the common one lim x → ∞ W ( t , x ) x 2 = ∞ , uniformly in t ∈ R , which is essential to show the existence of nontrivial homoclinic solutions for the above system in all existing literature.

Authors:Vjacheslav Yurko Pages: 1 - 6 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Vjacheslav Yurko Inverse spectral problems are studied for the first order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and a constructive algorithm is provided for this class of inverse problems.

Authors:Qing Zhu; Zhan Zhou; Lin Wang Pages: 7 - 14 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Qing Zhu, Zhan Zhou, Lin Wang Deriving exact solutions of nonlinear evolution equations has recently received tremendous attention in mathematics and physics. In this paper, using the ( G ′ ∕ G ) -expansion method, we obtain some exact solutions for a coupled discrete nonlinear Schrödinger system with a saturation nonlinearity. These exact solutions include the hyperbolic function type, trigonometric function type and rational function type, of which the hyperbolic function type solutions may generate kink and anti-kink solitons.

Authors:Jianli Liu; Yaoli Fang Pages: 15 - 19 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Jianli Liu, Yaoli Fang In this short paper, we consider the formation of singularities for the solutions of the Euler–Poisson equations with damping with attractive forces in R n ( n ≥ 1 ) and the repulsive forces in R . Under the appropriate assumptions on initial data, we can obtain the divergence of the velocity will blowup in the finite time.

Authors:Jianhua Chen; Xianhua Tang; Bitao Cheng Pages: 20 - 26 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Jianhua Chen, Xianhua Tang, Bitao Cheng In this paper, we study the following generalized quasilinear Schrödinger equation − d i v ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) ∇ u 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , g ∈ C 1 ( R , R + ) , V ( x ) and f ( x , u ) are 1-periodic on x . By using a change of variable, we obtain the existence of ground states solutions. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the manifold by using the diagonal method.

Authors:Shuxia Pan Pages: 46 - 51 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Shuxia Pan This paper is concerned with the asymptotic spreading of a predator–prey system, which formulates that the predator invades the habitat of the prey. By constructing auxiliary equation, we obtain the speed of asymptotic spreading of the predator.

Authors:Xiaoyu Zeng; Yimin Zhang Pages: 52 - 59 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Xiaoyu Zeng, Yimin Zhang For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent p for its L 2 -normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical points for the functional on the L 2 -normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.

Authors:Yumei Zou; Guoping He Pages: 68 - 73 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Yumei Zou, Guoping He We are concerned with the uniqueness of solutions for the following nonlinear fractional boundary value problem: D p x ( t ) + f ( t , x ( t ) ) = 0 , 2 < p ≤ 3 , t ∈ ( 0 , 1 ) , x ( 0 ) = x ′ ( 0 ) = 0 , x ( 1 ) = 0 where D 0 + p denotes the standard Riemann–Liouville fractional derivative. Our analysis relies on the theory of linear operators and the ‖ ⋅ ‖ e norm.

Authors:M. Alvarez-Ramírez; M. Corbera; Josep M. Cors; A. García Pages: 74 - 78 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): M. Alvarez-Ramírez, M. Corbera, Josep M. Cors, A. García In this work, we study a one-parameter family of differential equations and the different scenarios that arise with the change of parameter. We remark that these are not bifurcations in the usual sense but a wider phenomenon related with changes of continuity or differentiability. We offer an alternative point of view for the study for the motion of a system of two particles which will always move in some fixed line, we take R for the position space. If we fix the center of mass at the origin, the system reduces to that of a single particle of unit mass in a central force field. We take the potential energy function U ( x ) = x β , where x is the position of the single particle and β is some positive real number.

Authors:Xinguang Zhang; Lishan Liu; Yonghong Wu; Yujun Cui Pages: 85 - 93 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Xinguang Zhang, Lishan Liu, Yonghong Wu, Yujun Cui In this paper, we study the entire blow-up solutions for a quasilinear p -Laplacian Schrödinger elliptic equation with a non-square diffusion term. By using the dual approach and some new iterative techniques, the difficulty due to the non-square diffusion term and the p -Laplacian operator is overcome and the nonexistence and existence of entire blow-up solutions are established.

Authors:L.D. Petković; M.S. Petković Pages: 94 - 101 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): L.D. Petković, M.S. Petković A new one-parameter family of simultaneous methods for the determination of all (simple or multiple) zeros of a polynomial is derived. The order of the basic family of simultaneous methods is four. Using suitable corrective approximations, the order of convergence of this family is increased even to six without any additional evaluations of the polynomial and its derivatives, which points to a high computational efficiency of the new family of root solvers. Comparison with the existing methods in regard to computational efficiency and numerical examples are also performed.

Authors:Li Chen; Heinz Siedentop Pages: 102 - 107 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Li Chen, Heinz Siedentop We prove a blow-up criterion for the solutions to the ν -dimensional Patlak–Keller–Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two, it is exactly Dolbeault’s and Perthame’s blow-up condition, i.e., blow-up occurs, if the total mass exceeds 8 π .

Authors:Didier Clamond Pages: 114 - 120 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Didier Clamond This short note is about the gauge condition for the velocity potential, the definitions of the Bernoulli constant and of the velocity speeds in the context of water waves. These definitions are often implicit and thus the source of confusion in the literature. This note aims at addressing this issue. The discussion is related to water waves because the confusion are frequent in this field, but it is relevant for more general problems in fluid mechanics.

Authors:Wensheng Yin; Yong Ren Pages: 121 - 126 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Wensheng Yin, Yong Ren The purpose of this paper is to discuss a class of stochastic differential equations with delay driven by G -Brownian motion ( G -SDDEs, in short). The asymptotical boundedness and exponential stability of the equations are obtained by means of G -Lyapunov function. An example is proposed to illustrate the theory obtained.

Authors:Honglv Ma; Chengkui Zhong Pages: 127 - 133 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Honglv Ma, Chengkui Zhong In this paper we obtain the existence of a global attractor for the Kirchhoff equations with strong nonlinear damping, which attracts every H 0 1 ( Ω ) × L 2 ( Ω ) -bounded set with respect to the H 0 1 ( Ω ) × H 0 1 ( Ω ) norm.

Authors:Xiaopeng Zhao; Jinde Cao Pages: 134 - 139 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Xiaopeng Zhao, Jinde Cao In this paper, by using Strauss’ inequality, we prove the existence of global mild solutions to the 3D magnetohydrodynamics- α equations with small initial data.

Authors:Hua Jin; Wenbin Liu Pages: 140 - 146 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Hua Jin, Wenbin Liu In this paper, for any dimension N > 2 s ( 0 < s < 1 ) , we study the fractional Kirchhoff equation a + b ∫ R N ( − Δ ) s 2 u 2 d x ( − Δ ) s u + u = f ( u ) in R N , with a critical nonlinearity, where ( − Δ ) s is the fractional Laplacian. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti–Rabinowitz condition when the parameter b is small. Moreover, we obtain the asymptotic behavior of solutions as b → 0 . The method we use and the result we get are applicable in any dimension N > 2 s . Our result improves the study made in the low dimension 2 s < N < 4 s .

Authors:Soon-Mo Jung; Jaiok Roh Pages: 147 - 153 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Soon-Mo Jung, Jaiok Roh In Jung and Roh (2017), we investigated some properties of approximate solutions of the second-order inhomogeneous linear differential equations. In this paper as an application of above paper we will study the stability of the time independent Schrödinger equation with a potential box of finite walls.

Authors:Alexander G. Ramm Pages: 154 - 160 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Alexander G. Ramm The Navier–Stokes (NS) problem consists of finding a vector-function v from the Navier–Stokes equations. The solution v to NS problem is defined in this paper as the solution to an integral equation. The kernel G of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term ( v ⋅ ∇ ) v . The kernel G is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions v with finite norm N 1 ( v ) = sup ξ ∈ R 3 , t ∈ [ 0 , T ] ( 1 + ξ ) ( v + ∇ v ) ≤ c ( ∗ ) , where T > 0 and C > 0 are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given.

Authors:Alina Dobrogowska; Grzegorz Jakimowicz Pages: 161 - 166 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Alina Dobrogowska, Grzegorz Jakimowicz The main purpose of this work is to apply the factorization method to difference equations. The factorization method offers the possibility of finding solutions of new classes of difference equations. We show how to integrate certain classes of second order difference equations. Finally, an example is presented to illustrate our main result.

Authors:Nejmeddine Chorfi; Vicenţiu D. Rădulescu Pages: 167 - 173 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Nejmeddine Chorfi, Vicenţiu D. Rădulescu This paper deals with the study of a nonlinear eigenvalue problem driven by a new class of non-homogeneous differential operators with variable exponent and involving a nonlinear term with variable growth. The framework in the present paper corresponds to the case of small perturbations of the nonlinear term. Combining variational arguments with energy estimates, we establish the existence of eigenvalues in a neighborhood of the origin. Our abstract setting includes several models described by nonhomogeneous differential operators, including the case of the capillarity equation.

Authors:Jun-Gang Wang; Yan Li; Yu-Hong Ran Pages: 174 - 180 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Jun-Gang Wang, Yan Li, Yu-Hong Ran In this paper we investigate the convergence of Chebyshev type regularization strategy combined with the Morozov discrepancy principle, which is an a posteriori parameter choice rule and independent of the a priori information of exact solution. Compared with the standard Tikhonov regularization method, the Chebyshev type regularization method has no saturation restriction so that its convergence order holds for any υ ≥ 1 2 , where p = 2 υ + 1 2 υ in the H o ̈ l d e r type estimate.

Authors:Qunfei Long; Jianqing Chen Pages: 181 - 186 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Qunfei Long, Jianqing Chen We discuss a class of quasi-linear pseudo-parabolic equation with nonlocal source u t − Δ u t − ∇ ⋅ ∇ u 2 q ∇ u = u p ( x , t ) ∫ Ω k ( x , y ) u p + 1 ( y , t ) d y x , y ∈ Ω , t ∈ ( 0 , T 0 ] , where q ≤ p and 0 < q < n − 2 2 . By establishing the criterions for blow-up, we determine the upper bounds for blow-up time under not only q < p and non-positive initial energy but also q = p and negative initial energy. The results show that the upper bound for blow-up time under q < p is different from it under q = p . Moreover, we also determine the lower bound for blow-up time.

Authors:Martin Stynes; José Luis Gracia Pages: 187 - 192 Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Martin Stynes, José Luis Gracia A simple and inexpensive preprocessing of an initial–boundary value problem with a Caputo time derivative is shown theoretically and numerically to yield an enhanced convergence rate for the L1 scheme. The same preprocessing can also be used with other methods for time-dependent problems.

Authors:Koya Sakakibara; Shigetoshi Yazaki Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Koya Sakakibara, Shigetoshi Yazaki Method of fundamental solutions (MFS) is a meshfree numerical solver for linear homogeneous partial differential equations, and it has been applied to several problems. It is well known that some numerical solutions by MFS do not satisfy invariance properties, which is unnatural from viewpoint of physics. In this paper, we propose a unified way to obtain invariant schemes for MFS. The key idea is to introduce scaling factors.

Authors:Liang Wang; Daqing Jiang Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Liang Wang, Daqing Jiang A model of the chemostat involving stochastic perturbation is considered. Instead of assuming the familiar Monod kinetics for nutrient uptake, a general class of functions is used which includes both monotone and non-monotone uptake functions. Using the stochastic Lyapunov analysis method, under restrictions on the intensity of the noise, we show the existence of a stationary distribution and the ergodicity of the stochastic system.

Authors:Marcio V. Ferreira; Jaime E. Muñoz Rivera; Amelie Rambaud; Octavio Vera Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Marcio V. Ferreira, Jaime E. Muñoz Rivera, Amélie Rambaud, Octavio Vera The paper establishes the exponential stability of a micropolar thermoelastic homogeneous and linear body with frictional dissipation and thermal conduction of hyperbolic type, allowing for second sound. Existence and uniqueness are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional.

Authors:Qun Liu; Daqing Jiang Pages: 8 - 15 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Qun Liu, Daqing Jiang In this paper, we analyze a stochastic SIR model with nonlinear perturbation. By the Lyapunov function method, we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model. Moreover, sufficient conditions for extinction of the disease are also obtained.

Authors:Chein-Shan Liu; Botong Li Pages: 49 - 55 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Chein-Shan Liu, Botong Li We consider an inverse problem for reconstructing the second-order Sturm–Liouville operator with the help of boundary data. The concept of energetic boundary functions, which satisfy the given boundary conditions and preserve the energy, is introduced for the first time. Then, we can derive a linear system to recover the unknown important coefficients of leading coefficient and potential function through a few iterations. Two examples are given to verify the iterative method.

Authors:P. Gera; D. Salac Pages: 56 - 61 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): P. Gera, D. Salac The Cahn–Hilliard system has been used to describe a wide number of phase separation processes, from co-polymer systems to lipid membranes. In this work the convergence properties of a closest-point based scheme are investigated. In place of solving the original fourth-order system directly, two coupled second-order systems are solved. The system is solved using an approximate Schur-decomposition as a preconditioner. The results indicate that with a sufficiently high-order time discretization the method only depends on the underlying spatial resolution.

Authors:Nguyen Huy Tuan; Doan Vuong Nguyen; Vo Van Au; Daniel Lesnic Pages: 69 - 77 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Nguyen Huy Tuan, Doan Vuong Nguyen, Vo Van Au, Daniel Lesnic We study for the first time the inverse backward problem for the strongly damped wave equation. First, we show that the problem is severely ill-posed in the sense of Hadamard. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.

Authors:Swetlana Giere; Volker John Pages: 78 - 83 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Swetlana Giere, Volker John This note proposes, analyzes, and studies numerically a regularization approach in the computation of the initial condition for reduced-order models (ROMs) of convection–diffusion equations. The aim of this approach consists in reducing significantly spurious oscillations in the ROM solutions.

Authors:Zhong Zheng; Feng-Lin Huang; Yu-Cheng Peng Pages: 91 - 97 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Zhong Zheng, Feng-Lin Huang, Yu-Cheng Peng In this paper, a double-step scale splitting (DSS) iteration method is proposed to solve a class of complex symmetric of linear equation. Unconditional convergence result is established under appropriate restrictions. Two reciprocal optimal iteration parameters and the corresponding optimal convergence factor are also determined. Numerical experiments on a few model problems are used to verify the effectiveness of the DSS iteration method.

Authors:Wen Zhang; Jian Zhang; Zhiming Luo Pages: 98 - 105 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Wen Zhang, Jian Zhang, Zhiming Luo This paper is concerned with the following fourth-order elliptic equation Δ 2 u − Δ u + V ( x ) u = K ( x ) f ( u ) + μ ξ ( x ) u p − 2 u , x ∈ R N , u ∈ H 2 ( R N ) , where Δ 2 ≔ Δ ( Δ ) is the biharmonic operator, N ≥ 5 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. By working in weighted Sobolev spaces and using a variational method, we prove that the above equation has two nontrivial solutions.

Authors:Toshikazu Kuniya Pages: 106 - 112 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Toshikazu Kuniya We are concerned with the numerical approximation of the basic reproduction number R 0 , which is the well-known epidemiological threshold value defined by the spectral radius of the next generation operator. For a class of age-structured epidemic models in infinite-dimensional spaces, R 0 has the abstract form and cannot be explicitly calculated in general. We discretize the linearized equation for the infective population into a system of ordinary differential equations in a finite n -dimensional space and obtain a corresponding threshold value R 0 , n , which can be explicitly calculated as the positive dominant eigenvalue of the next generation matrix. Under the compactness of the next generation operator, we show that R 0 , n → R 0 as n → + ∞ in terms of the spectral approximation theory.

Authors:Feifei Jing; Jian Li; Zhangxin Chen; Wenjing Yan Pages: 113 - 119 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Feifei Jing, Jian Li, Zhangxin Chen, Wenjing Yan This work aims at employing the discontinuous Galerkin (DG) methods for the incompressible flow with nonlinear leak boundary conditions of friction type, whose continuous variational problem is an inequality due to the subdifferential property of such boundary conditions. Error estimates are derived for the velocity and pressure in the corresponding norms, this work ends with numerical results to demonstrate the theoretically predicted convergence orders and reflect the leak or non-leak phenomena.

Authors:Mehdi Dehghan; Mostafa Abbaszadeh; Weihua Deng Pages: 120 - 127 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Mehdi Dehghan, Mostafa Abbaszadeh, Weihua Deng The main aim of the current paper is to develop a high-order numerical scheme to solve the space–time tempered fractional diffusion-wave equation. The convergence order of the proposed method is O ( τ 2 + h 4 ) . Also, we prove the unconditional stability and convergence of the developed method. The numerical results show the efficiency of the provided numerical scheme.

Authors:Fenglong Sun; Lishan Liu; Yonghong Wu Pages: 128 - 135 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Fenglong Sun, Lishan Liu, Yonghong Wu By introducing a subspace of H 2 ( Ω ) with constraints ∂ u ∂ n ∂ Ω = 0 and ∫ Ω u d x = 0 and using the Fountain Theorem, we obtain the existence of infinitely many sign-changing high energy solutions for a biharmonic equations with p -Laplacian and Neumann boundary condition.

Authors:Junchao Chen; Shundong Zhu Pages: 136 - 142 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Junchao Chen, Shundong Zhu The residual symmetry is derived for the negative-order Korteweg–de Vries equation from the truncated Painlevé expansion. This nonlocal symmetry is transformed into the Lie point symmetry and the finite symmetry transformation is presented. The multiple residual symmetries are constructed and localized by introducing new auxiliary variables, and then n th Bäcklund transformation in terms of determinant is provided. With the help of the consistent tanh expansion (CTE) method, the explicit soliton-cnoidal wave interaction solutions are obtained from the last consistent differential equation.

Authors:Xin-Yi Gao Pages: 143 - 149 Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Xin-Yi Gao Optical fiber communication system is one of the core supporting systems of the modern internet age, and studies on the ultrashort optical pulses are at the forefront of fiber optics, modern optics and optical engineering. Hereby, symbolic computation on the recently-proposed generalized higher-order variable-coefficient Hirota equation is performed, for certain ultrashort optical pulses propagating in a nonlinear inhomogeneous fiber. For the complex envelope function associated with the optical-pulse electric field in the fiber, an auto-Bäcklund transformation is worked out, along with a family of the analytic solutions. Both our Bäcklund transformation and analytic solutions depend on the optical-fiber variable coefficients which represent the effects of the first-order dispersion, second-order dispersion, third-order dispersion, Kerr nonlinearity, time delaying, phase modulation and gain/loss. Relevant constraints among those coefficients are also presented. We expect that the work could be of some use for the fiber-optics investigations.

Abstract: Publication date: December 2017 Source:Applied Mathematics Letters, Volume 74 Author(s): Hüseyin Koçak This study investigates the behaviour of blow-up and global similarity solutions for the nonlinear dispersion equation (NDE), u t = ( u n u ) x x x ± ( u p − 1 u ) x x in R × R + , n > 0 and p > n + 1 , and attempts to give some aspects analytically and numerically. One can easily see that the proposed NDE is more complicated and hence more difficult than the corresponding semilinear equation. We will particularly pay attention to the first critical exponent p = p 0 = n + 2 , which helps us to simplify the rescaled nonlinear equation and compare with semilinear ones studied in the literature.

Authors:Pituk Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Mihály Pituk A new sufficient condition is given for the oscillation of all solutions of a linear scalar delay differential equation with slowly varying and uniformly positive coefficient at infinity. Based on the celebrated Myshkis type necessary condition, we show that this new sufficient condition is sharp within the considered class of delay differential equations.

Authors:Liang Bai; Juan Nieto Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Liang Bai, Juan J. Nieto In this note we introduce the concept of a weak solution for a linear equation with not instantaneous impulses. We use the classical Lax–Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems.

Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): J. Džurina We establish new comparison theorem for deducing oscillation of noncanonical nonlinear differential equation r ( t ) u ′ ( t ) γ ′ + p ( t ) u γ ( τ ( t ) ) = 0 from that of suitable linear equation y ″ ( t ) + p ˜ ( t ) u ( τ ˜ ( t ) ) = 0 . Our results are based on Trench’s theory of canonical operators and comparison technique.

Authors:Pavel Abstract: Publication date: November 2017 Source:Applied Mathematics Letters, Volume 73 Author(s): Pavel Řehák We consider the n th order dynamic equations x Δ n + p 1 ( t ) x Δ n − 1 + ⋯ + p n ( t ) x = 0 and y Δ n + p 1 ( t ) y Δ n − 1 + ⋯ + p n ( t ) y = f ( t , y ( τ ( t ) ) ) on a time scale T , where τ is a composition of the forward jump operators, p i are real rd-continuous functions and f is a continuous function; T is assumed to be unbounded above. We establish conditions that guarantee asymptotic equivalence between some solutions of these equations. No restriction is placed on whether the solutions are oscillatory or nonoscillatory. Applications to second order Emden–Fowler type dynamic equations and Euler type dynamic equations are shown.