Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Infinitely many homoclinic solutions for a class of damped vibration problems

In this paper, we deal with the existence of infinitely many homoclinic solutions for the damped vibration problems where A is an antisymmetry N × N constant matrix, we establish some new existence results to guarantee that the above system has infinitely many homoclinic solutions under more relaxed assumptions on W(t,x), which satisfies a kind of new subquadratic condition by using fountain theorem. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.     

Quasiperiodic solutions of higher dimensional Duffing’s equations via the KAM theorem

In this paper, we consider the higher dimensional second order differential equations of the formẍ + ∇F(x,t) = 0,x ∈R ⁿ with a class of weakly coupled potentials F( x, t ), periodically depending on t. We prove the existence of infinitely many quasi-periodic solutions for such equations via the KAM theorem.     

Differentiability in optimization theory 1

This paper deals with necessary conditions for optimization problems with infinitely many inequality constraints assuming various differentiability conditions. By introducing a second topology N on a topological vector space we define generalized versions of differentiability and tangential cones. Different choices of N lead to Gâteaux-, Hadamaed- and weak differentiability with corresponding tangential cones. The general concept is used to derive necessary conditions for local optimal points in form of inequalities and generalized multiplier rules, Special versions of these theorems are obtained for different differentiability assumptions by choosing properly. An application to approximation theory is given.     

The existence of quasiperiodic solutions for coupled Duffing-type equations

In this paper, we prove the existence of infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations via KAM theorem. Moreover, the set of quasiperiodic solutions is of infinitely Lebesgue measure in the phase space.     

Infinitely many solutions for a class of discrete non-linear boundary value problems

The existence of infinitely many solutions for a discrete non-linear Dirichlet problem involving the p-Laplacian, under appropriate oscillating behaviours of the non-linear term, is established. The approach is based on the critical point theory.     

Multiple Solutions for a Differential Inclusion Problem with Nonhomogeneous Boundary Conditions

In this paper, we use a mountain pass theorem with Cerami type conditions for locally Lipschitz functions to investigate the existence of at least one nontrivial solution for a differential inclusion problem involving the p-Laplacian and with nonlinear and nonsmooth boundary conditions. Moreover, by a symmetric version of the mountain pass theorem, we prove the existence of infinitely many solutions.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

Global Ill‐Posedness of the Isentropic System of Gas Dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p (ρ) = ρ² and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1‐dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. © 2015 Wiley Periodicals, Inc.     

薛定谔方程及薛定谔-麦克斯韦方程的多解

不作周期性和对称性的假设,也没有Ambrosetti-Rabinowitz增长控制条件,我们得到了一类超线性薛定谔方程在全空间中无穷多解的存在性结果.同时,得到了一类超线性薛定谔-麦克斯韦方程无穷多解的存在性结果.     

Infinitely many bounded solutions for the p-Laplacian with nonlinear boundary conditions

By applying a method due to Saint Raymond, we prove the existence of infinitely many weak solutions for a quasilinear elliptic partial differential equation, involving the p-Laplacian operator, coupled with a nonlinear boundary condition. Our main assumption is a suitable oscillatory behaviour of the nonlinearity either at infinity or at zero.     

Homoclinic solutions for fourth order differential equations with superlinear nonlinearities

In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions.     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

Homoclinics for a Hamiltonian with wells at different levels

We consider a Hamiltonian equation of the form (HS) for which V has two distinct non-degenerate maxima at different levels: 0 is a local maximum and is an absolute maximum. Under standard non-degeneracy conditions on V, our main result is that there is a solution of (HS) homoclinic to 0. Then, supposing that another geometric condition holds, we show the existence of infinitely many solutions of (HS) homoclinic to 0 that are distinguished from one another by the number of times and regions where the solutions stay away from 0. As a corollary, we show that if there is a solution of (HS) homoclinic to , then there are infinitely many solutions of (HS) homoclinic to 0, distinguished by the number and position of intersections with 1/2.

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Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

Hyperfinite construction of G-expectation

The hyperfinite G-expectation is a nonstandard discrete analogue of G-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time G-expectation operator is defined as a hyperfinite G-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time G-expectation. We develop the basic theory for hyperfinite G-expectations and prove an existence theorem for liftings of (continuous-time) G-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the G-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of G-expectations, Stoch. Process. Appl. 122(2) (2012), pp. 664–675]).     

一类超线性椭圆方程的无穷多解

本文研究了一类超线性椭圆方程,这里的非线性项是奇的.我们不需要假设Ambrosetti-Rabinowitz条件,得到了无穷多个大能量解的存在性.我们的结论推广了邹文明最近的结果。     

Some Results on Subelliptic Equations

In this paper, we consider the principal on R^n, and we find a comparison principle for such semi-linear sub-elliptic equation in the whole R^n and infinitely many positive solutions of the problem. eigenvalue problem for Hormander＇s Laplacian principal eigenvalues. We also study a related prove that, under a suitable condition, we have     

Some Boolean algebras with finitely many distinguished ideals II

We describe the countably saturated models and prime models (up to isomorphism) of the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.