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This paper deals with the existence of nontrivial nonnegative solutions of Schrödinger–Hardy systems driven by two possibly different fractional ℘ -Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators. [ABSTRACT FROM AUTHOR]

Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on , which exhibit conformal invariance of exponential type. In this way we intend to complete the work that we undertook in Li et al. (2011) , where we introduced the family of elementary conformal tensors in order to describe all fully nonlinear differential operators of any integer order on which are conformally invariant of degree . Examples of the differential operators that we study in this paper are those related to the –curvature equation on and to the Gauss equation on . [Copyright &y& Elsevier]

Abstract: In this paper, existence results are derived for the unique, smallest, greatest, minimal, and/or maximal solutions of finite systems of nonlinear distributional Cauchy problems. The dependence of solutions on the data is also studied. Applications to systems of distributional differential equations with impulses and to higher-order distributional Cauchy problems are presented. The main tools are fixed point results in function spaces and a primitive integral of distributions defined in the paper, including recently introduced regulated and continuous primitive integrals on compact intervals. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the impulsive effects on the stability of the zero solution of the linear neutral differential equations with variable delays. By transforming the equations into ones without impulses and using fixed point theory, some sufficient conditions for asymptotic stability and exponential stability of the zero solution are obtained. The paper extends and improves results on sufficient conditions obtained by Jin and Luo (2008) , and Ardjouni and Djoudi (2011) , which is shown clearly in Example 1. This paper also shows that the impulse intensity and the impulse time both influence the decay rate of the convergence to zero of the solutions. Finally, two examples are given to show applications of some results obtained. [Copyright &y& Elsevier]

Abstract: We study the Cauchy problem for the nonlinear heat equation in the sub critical case of . In the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper Escobedo and Kavian (1987)  as uniformly with respect to , where is a positive solution of equation which decays rapidly at infinity: . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327–e2330] and many others. [Copyright &y& Elsevier]

Abstract: The problem of how the dynamics of the smooth gene networks is related to the simplified dynamics of the Boolean networks is studied. The emphasis is put on the gene regulatory networks with delay. Asymptotic analysis which is applied in the paper goes back to Tikhonov’s theory of singular perturbed differential equations and a modified algorithm of reducing delay equations to ordinary differential equations. A number of illustrative examples complements the theory which is offered in the paper. [Copyright &y& Elsevier]

Abstract: This paper investigates the boundary control problem of the viscosity Degasperis–Procesi equation in [0, 1]. Results of the existence of a solution to the viscosity Degasperis–Procesi equation in a short time interval under the boundary condition, are given in the paper. It proves the global exponential stability under the given boundary condition in , , , space. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of multiple solutions for weighted -Laplacian equation Dirichlet problems. Several sub-super-solution type results have been given via Leray–Schauder’s degree method. In particular, we give a counterexample which means that in paper [J. Henderson, H.B. Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000) 443–454] is not an open subset of . [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of positive solutions for a class of nonlinear second-order four-point boundary value problems with alternating coefficient. Our approach relies on the Krasnosel’skii fixed point theorem. The result of this paper is new and extent the previously known result. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and uniqueness of a nontrivial solution for the following third-order -Laplacian -point eigenvalue problems on time scales: where is -Laplacian operator, i.e., is a parameter, . We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. The conditions we used in the paper are different from those in [Z.C. Hao, L. Debnath, On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. Math. Lett. 18 (2005) 543–553; D. Jiang, R.P. Agarwal, A uniqueness and existence theorem for a singular third-order boundary value problem on , Appl. Math. Lett. 15 (2002) 445–451; S.H. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl. 323 (2006) 413–425; Z. Liu, J.S. Ume, D.R. Anderson, S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, J. Math. Anal. Appl. 334 (2007) 299–313; Q. Yao, Positive solutions to a special singular second-order boundary value problem, Math. Comput. Model. (in press); Q. Yao, Successive iteration of positive solution for a discontinuous third-order boundary value problem, Comput. Math. Appl. 53 (2007) 741–749]. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of nonlinear parabolic problems in divergence form whose solutions, without appropriate data restrictions, might blow up at some finite time. The purpose of this paper is to establish conditions on the data sufficient to guarantee blow-up of solution at some finite time , conditions to ensure that the solution remains bounded as well as conditions to derive some explicit exponential decay bounds for the solution and its derivatives. [Copyright &y& Elsevier]

Abstract: In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected. [Copyright &y& Elsevier]

Abstract: This paper concerns the existence of positive solutions to a singular differential equation with the boundary conditions where , and on . By theories of ordinary differential equations, Bertsch and Ughi [M. Bertsch, M. Ughi, Positivity properties of viscosity solutions of a degenerate parabolic equation, Nonlinear Anal. 14 (1990) 571–592] proved that when ( is a positive integer) and , the problem admits a decreasing positive solution. In this paper, we show, by the classical method of elliptical regularization, that if , then the above problem admits at least a positive solution which is not decreasing. As a by-product of the results, the problem with and admits at least two positive solutions if . [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear periodic systems driven by the ordinary -Laplacian and having a nonsmooth potential function. Under minimal and natural hypotheses on the nonsmooth potential and using variational methods based on the nonsmooth critical point theory we prove four existence theorems and a multiplicity theorem. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (i.e. it is unbounded both above and below). [Copyright &y& Elsevier]

Abstract: This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zolésio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field. [Copyright &y& Elsevier]

Abstract: In this paper we study stability of delay impulsive systems on time scales. By using Lyapunov functional and analysis method, some stability and instability criteria for the delay system with impulses at fixed times on time scales are obtained. Two examples are presented to illustrate the efficiency of results proposed in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472]. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial–boundary value problem for the linear wave equation where , are given constants and , , are given functions, and the unknown function and the unknown boundary value satisfy the following nonlinear integral equation: where , , , are given constants and , are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1.1)–(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of , Part 2 is devoted the study of the asymptotic behavior of the solution as . Finally, in Part 3 we obtain an asymptotic expansion of the solution of the problem (1.1)–(1.5) up to order in four small parameters , , , . [Copyright &y& Elsevier]

Abstract: In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)]. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial-boundary value problem for the linear wave equationwhere K, are given constants and , , , are given functions, the unknown function and the unknown boundary value satisfy the following linear integral equationwhere g, k, , are given functions. The paper consists of four parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1)–(2). The proof is based on a Galerkin type approximation associated to various energy estimates-type bounds, weak-convergence and compactness arguments. In Part 2, it is devoted to the study of the regularity of the solution , with respect to the functions , , f, g, k, . In Part 3 we prove that the solution , of this problem is stable with respect to the data , , , , f, g, k, . Finally, in Part 4 we obtain an asymptotic expansion of the solution , of this problem up to order in , , for , sufficiently small. [Copyright &y& Elsevier]

Abstract: This paper deals with a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments by employing a fixed point theorem in cones. Easily verifiable sufficient criteria are established. The paper extends some previous results and reports some new results about impulsive functional differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we establish an existence theorem of two local minima for functionals on reflexive Banach space. As a consequence, we obtain an existence theorem of three critical points which improves the conclusion of a result in a recent paper of G.Bonanno concerning some remarks on a three-critical-points theorem established by B. Ricceri. [Copyright &y& Elsevier]

Abstract: This paper provides existence results for the nonlinear impulsive periodic boundary value problemwhere and . The basic assumption is the existence of lower/upper functions associated with the problem. Here, we generalize and extend the existence results of our previous papers. [Copyright &y& Elsevier]

In this paper we consider a general differential equation of the form x˙=f(x) with f∈C1(Rn,Rn) and n⩾2. Borg, Hartman, Leonov and others have studied sufficient conditions for the existence, uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction. They used a certain contraction property of the flow with respect to the Euclidian or a Riemannian metric. In this paper we also prove sufficient conditions including upper bounds for the Floquet exponents of the periodic orbit. Moreover, we show the necessity of these conditions using Floquet theory and a Lyapunov function. [Copyright &y& Elsevier]

In this paper we use the linear theory developed in Cano-Casanova and Lo´pez-Go´mez (J. Differential Equations 178 (2002) 123) to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the boundary operator. Besides working with mixed non-classical boundary conditions, this paper considers problems where bifurcation from infinity occurs. Treating this kind of problems is far from easy, since a complete knowledge of the set of values of the parameters where they exhibit a positive solution is imperative. Our results are far from immediate, as a consequence of the vanishing weight functions involved in our general setting. Now a days, the analysis of these general problems, where the coefficients are allowed to vary and eventually vanishing or changing sign, is focusing a great deal of attention. [Copyright &y& Elsevier]

The current paper concerns with traveling wave solutions to time periodic two-dimensional lattice differential equations of bistable type. Employing the so-called ‘vanishing viscosity’ approach, we obtain the existence of monotone traveling wave solutions connecting the two spatially homogeneous periodic solutions. Uniqueness of the wave speed and uniqueness of continuous wave front are shown by repeatedly utilizing comparison arguments. The existence and uniqueness results of the current paper together with the stability results proved in an author''s earlier paper generalize those in the time-independent case. [Copyright &y& Elsevier]

In this paper we consider the second order nonlinear elliptic system div [ A ( | x | , | u | 2 , | ∇ u | 2 ) ∇ u ] + B ( | x | , | u | 2 , | ∇ u | 2 ) u = [ cof ∇ u ] ∇ P , where the unknown vector field u satisfies the incompressibility constraint det ∇ u = 1 a.e. along with suitable boundary conditions and P = P ( x ) is an a priori unknown hydrostatic pressure field. Here, A = A ( r , s , ξ ) and B = B ( r , s , ξ ) are sufficiently regular scalar functions satisfying natural structural properties. Most notably in the case of a finite symmetric annulus we prove the existence of a countably infinite scale of topologically distinct twisting solutions to the system in all even dimensions. In sharp contrast in odd dimensions the only twisting solution is the map u ≡ x . We study a related class of systems by introducing the novel notion of a discriminant . Using this a complete and explicit characterisation of all twisting solutions for n ≥ 2 is given and a curious dichotomy in the behaviour of the system and its solutions captured and analysed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following linear system of second order differential equations (0.1) u ′ ′ + A (t) u = 0 , 0 ≤ t < ∞ where, for each t , A (t) is an n × n matrix with real components, and positive with respect to the usual cone K in R n. Conditions are provided in order that the first conjugate point T of t = 0 , i.e. the smallest T such that the above equation has a nontrivial solution u : 0 , T → K satisfying the boundary conditions u (0) = 0 = u (T) will be a bifurcation point for higher order perturbations of the equation. The paper is mainly motivated by results in Ahmad and Lazer (1997, 1980); Ahmad and Salazar (1981) and Schmitt (1975); Schmitt and Smith (1978). Some additional new consequences are discussed. [ABSTRACT FROM AUTHOR]

That a superposition of fundamental solutions to the p -Laplace Equation is p -superharmonic–even in the non-linear cases p > 2 –has been known since M. Crandall and J. Zhang published their paper Another Way to Say Harmonic in 2003. We give a simple proof and extend the result by means of an explicit formula for the p -Laplacian of the superposition. [ABSTRACT FROM AUTHOR]

In this paper we consider the global existence for the 3D generalized Hall-MHD equations with fractional dissipative terms ( − Δ ) α u and ( − Δ ) α b under small initial data in the setting of Sobolev norms with lower regularity. For the global existence we enlarge the range of dissipative exponents α = β from ( 1 , 7 6 ] to ( 1 , 3 2 ) , which established in a recent work. In addition, the long time behavior and rates of decay for both the solutions and higher derivatives in different Sobolev spaces are obtained by using the Fourier splitting method, which extends the previous work by Chae and Schonbek. [ABSTRACT FROM AUTHOR]

In this paper, we firstly derive the blow-up criterion and the lower bound of the maximal time of existence to the modified b-family of equations. The new blow-up mechanism for the solutions with certain initial profiles and various parameters is then described in detail. [ABSTRACT FROM AUTHOR]

This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑ i = 1 n ∂ ∂ x i a i x , u , D u = b x , u , D u under p , q − growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u , other than on its gradient D u and on the x variable. [ABSTRACT FROM AUTHOR]

In this paper a family of multi-dimensional fractional differential operators T α and their corresponding pseudo-differential equations over p -adic fields are investigated. The test function class D ( Q p n ) and distribution class D ′ ( Q p n ) are invariant under the actions of these operators. The p -adic Laplacian Δ p and a fundamental solution of the Laplace equation are constructed. We study the spectral properties of the Laplacian Δ p , and obtain an orthonormal basis of the eigen-functions of this operator in L 2 ( Q p n ) . Furthermore, the Cauchy problems for the wave and heat equations on the p -adic fields related to Δ p are also studied. [ABSTRACT FROM AUTHOR]

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary and evolution problems, and steady/ergodic large time behavior for the associated evolution problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the smooth solutions u ( x , t ) , with suitable decay at infinity, of the Euler equations on R 2 × ( 0 , ∞ ) . We assume that the initial vorticity ω 0 = ∇ × u 0 has a compact support which leads by Marchioro (1994, 1996) to a consequence that the support of the vorticity ω ( t ) for any finite time is a bounded set. Then, we will show that the decay(growth) rate, in L p ( R 2 ) spaces for 2 < p ≤ ∞ (for 1 < p < 2 ), of the smooth solutions of the Euler equations on R 2 cannot be faster than t − 1 3 + 2 3 p as t → ∞ . We will see that the L p -norm of the solution u ( x , t ) is equivalent to the L q -norm for any 1 < q < p ≤ ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Dirac-harmonic equation for differential forms and prove some basic norm inequalities, including the Caccioppoli-type inequality, Poincaré-type inequality and the weak reverse Hölder inequality for the solutions of this kind of differential equations. [ABSTRACT FROM AUTHOR]

The purpose of this review paper is two-fold. First, we point out one more time the importance and usefulness of the S. Lie theory of symmetries of differential equations. In particular some illustrative examples are presented. Then, we discuss the Noetherian approach to Rellich–Pohozaev type identities. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to develop applications of advanced tools of first-order and second-order variational analysis and generalized differentiation to the fundamental notion of full stability of local minimizers of general classes of constrained optimization and minimax problems. In particular, we derive second-order characterizations of full stability and investigate its relationships with other notions of stability for parameterized conic programs and minimax problems. Furthermore, the developed variational approach allows us to largely unify and provide new self-contained proofs of some quite recent results in this direction for problems of constrained optimization with C 2 -smooth data. [ABSTRACT FROM AUTHOR]

The FitzHugh–Nagumo model is a reaction–diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues. One of the model parameters is the ratio ϵ of two time scales, which takes values between 0.001 and 0.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at ϵ = 0 , it has been shown that the FitzHugh–Nagumo equation admits a stable traveling pulse solution for sufficiently small ϵ > 0 . Here we prove the existence of such a solution for ϵ = 0.01 , both for circular axons and axons of infinite length. This is in many ways a completely different mathematical problem. In particular, it is non-perturbative and requires new types of estimates. Some of these estimates are verified with the aid of a computer. The methods developed in this paper should apply to many other problems involving homoclinic orbits, including the FitzHugh–Nagumo equation for a wide range of other parameter values. [ABSTRACT FROM AUTHOR]

The paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients, considered in a non-convex domain: { x ′ ( t ) + H ( t , x ( t ) ) ∂ − φ ( x ( t ) ) ∋ g ( t , x ( t ) ) , t ≥ 0 , x ( 0 ) = x 0 , where ∂ − φ stands for the Fréchet subdifferential of the semiconvex function φ and the matrix application x ↦ H ( ⋅ , x ) is a Lipschitz mapping. The presence of the oblique reflection brought by the term H ∂ − φ leads to the use of different techniques comparing to the cases of standard reflection in non-convex domains or oblique reflection in convex domains. The last section of the article is focused on the qualitative analysis of a non-convex Skorohod problem, with generalized reflection and, as applications, stochastic variational inequalities driven by oblique Fréchet subgradients are addressed. [ABSTRACT FROM AUTHOR]

In this paper we study a Hénon-like equation (see Eq. (1)), where the nonlinearity f (t) is not homogeneous (i.e., it is not a power). By minimization on the Nehari manifold, we prove that for large values of the parameter α there is a breaking of symmetry and non radial solutions appear. This holds for sub- and super-critical growth of the nonlinearity f . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using the remote start or dissipative method, we investigate ergodicity for several kinds of functional stochastic equations including functional stochastic differential equations (SDEs) with variable delays, neutral functional SDEs, functional SDEs driven by jump processes, and semi-linear functional stochastic partial differential equations (SPDEs). Using the ergodicity derived, we then treat a couple of applications in stochastic approximation and optimization problems. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with weighted pseudo almost automorphic behavior of hyperbolic semilinear integro-differential equations in intermediate Banach spaces, where the nonlinear perturbation is weighted pseudo almost automorphic type or weighted Stepanov-like pseudo almost automorphic type. As applications, some interesting examples are presented to illustrate the main findings. [Copyright &y& Elsevier]

This paper is devoted to the study of spatial propagation dynamics of species in locally spatially inhomogeneous patchy environments or media. For a lattice differential equation with monostable nonlinearity in a discrete homogeneous media, it is well-known that there exists a minimal wave speed such that a traveling front exists if and only if the wave speed is not slower than this minimal wave speed. We shall show that strongly localized spatial inhomogeneous patchy environments may prevent the existence of transition fronts (generalized traveling fronts). Transition fronts may exist in weakly localized spatial inhomogeneous patchy environments but only in a finite range of speeds, which implies that it is plausible to obtain a maximal wave speed of existence of transition fronts. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider nonlocal boundary value problems for systems of second-order differential equations with dependence on the first-order derivatives and deviating arguments. By using a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems. We investigate our problem both for delayed and advanced arguments and also for the case when . In all cases, arguments can change the character on , so, in some subinterval of , they can be delayed in and advanced in . Some remarks concern also the case when differential equations do not depend on the first-order derivatives. Examples illustrate some results. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the Ulam–Hyers stability of first order differential systems with nonlocal initial conditions on compact intervals (in ) and on noncompact intervals (in , with suitable weight function ). The novelty consists in the simultaneous use of vectorial norms, convergent to zero matrices, Sobolev spaces for studying the Ulam–Hyers stability of equations with nonlocal conditions on noncompact intervals. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of -asymptotically -periodicity in the Stepanov sense. We study some properties of the space consisting of -asymptotically -periodic functions, and we use this notion to establish the existence of -asymptotically -periodic solutions to linear and semi-linear first-order abstract differential equations. In particular, we characterize the strongly continuous semigroups of bounded linear operators defined on reflexive spaces which are -asymptotically -periodic as those semigroups which are the direct sum of an -periodic and a strongly stable semigroup. [Copyright &y& Elsevier]