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Abstract
In this paper, we consider the following perturbed fractional Hamiltonian systems
where \(\alpha \in (1/2,1],\,\,L \in C(\mathbb{R},{\mathbb{R}^{N \times N}})\) is symmetric and not necessarily required to be positive definite, \(W \in {C^1}(\mathbb{R}\times {\mathbb{R}^N},\mathbb{R})\) is locally subquadratic and locally even near the origin, and perturbed term \(G \in {C^1}(\mathbb{R} \times {\mathbb{R}^N},\mathbb{R})\) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results.
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In this work, we consider the existence of solution to the following fractional advection–dispersion equation $$\begin{aligned} -\frac{d}{dt} \left( p {_{-\infty }}I_{t}^{\beta }(u'(t)) + q\; {_{t}}I_{\infty }^{\beta }(u'(t))\right) + b(t)u = f(t, u(t)),\;\;t\in \mathbb {R}\end{aligned}$$ (0.1)where \(\beta \in (0,1)\), \(_{-\infty }I_{t}^{\beta }\) and \(_{t}I_{\infty }^{\beta }\) denote left and right Liouville–Weyl fractional integrals of order \(\beta \) respectively, \(0<p=1-q<1\), \(f:\mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) and \(b:\mathbb {R} \rightarrow \mathbb {R}^{+}\) are continuous functions. Due to the general assumption on the constant p and q, the problem (0.1) does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem (0.1) under suitable assumptions.
In this artile, we prove the existene and multipliity of nontrivial solutions for the nonperiodi perturbed frational Hamiltonian systems (formula presented), where α ∈ (1/2, 1], λ > 0 is a parameter, t ∈ ℝ, x ∈ ℝN,−∞Dtα andtDα∞ are left and right Liouville-Weyl frational derivatives of order α on the whole axis R respetively, the matrix L(t) is not neessary positive definite for all t ∈ R nor oerive, W ∈ C¹(R×ℝN, ℝ) and f ∈ L²(ℝ, ℝN)\{0} small enough. Replaing the Ambrosetti-Rabinowitz Condition by general superquadrati assumptions, we establish the existene and multipliity results for the above system. Some examples are also given to illustrate our results.
In this paper we are concerned with the existence of solutions for the following perturbed fractional Hamiltonian systems: (Formula presented.) , (Formula presented.) (PFHS), where (Formula presented.) , (Formula presented.) , (Formula presented.) , (Formula presented.) is a symmetric and positive definite matrix for all (Formula presented.) , (Formula presented.) , and (Formula presented.) is the gradient of (Formula presented.) at u, (Formula presented.) and belongs to (Formula presented.). The novelty of this paper is that, assuming (Formula presented.) is bounded in the sense that there are constants (Formula presented.) such that (Formula presented.) for all (Formula presented.) and (Formula presented.) satisfies the Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, (Formula presented.) is sufficiently small in (Formula presented.) , we show that (PFHS) possesses at least two nontrivial solutions. Recent results are generalized and significantly improved.
In this paper, we study the existence of infinitely many homoclinic solutions of the perturbed second-order Hamiltonian system
$$-\ddot{u}(t)+L(t)u=W_u(t,u(t))+G_u(t,u(t)),$$where \({L(t)}\) and \({W(t,u)}\) are neither autonomous nor periodic in \({t}\). Under the assumptions that \({W(t,u)}\) is indefinite in sign and only locally superquadratic as \({|u|\to +\infty}\) and \({G(t,u)}\) is not even in \({u}\), we prove the existence of infinitely many homoclinic solutions in spite of the lack of the symmetry of this problem by Bolle’s perturbation method in critical point theory. Our results generalize some known results and are even new in the symmetric case.
In this article we study the existence of infinitely many homoclinic
solutions for a class of second-order Hamiltonian systems
$$
\ddot{u}-L(t)u+W_u(t,u)=0, \quad \forall t\in\mathbb{R},
$$
where L is not required to be either uniformly positive
definite or coercive, and W is superquadratic at infinity in u
but does not need to satisfy the Ambrosetti-Rabinowitz superquadratic condition.
This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.
In this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form
tD∞α(-∞Dtαu(t))+L(t)u(t)=∇W(t,u(t)),
where α∈(12,1), t∈ℝ, u∈ℝn, L∈C(ℝ,ℝn2) is a symmetric and positive definite matrix for all t∈ℝ, W∈C1(ℝ×ℝn,ℝ) and ∇W(t,u) is the gradient of W(t,u) at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants 0<τ1<τ2<∞ such that τ1|u|2≤(L(t)u,u)≤τ2|u|2 for all (t,u)∈ℝ×ℝn and W(t,u) is of the form (a(t)/(p+1))|u|p+1 such that a∈L∞(ℝ,ℝ) can change its sign and 0<p<1 is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.
In this paper, by the critical point theory, the boundary value problem is discussed for a fractional differential equation containing the left and right fractional derivative operators, and various criteria on the existence of solutions are obtained. To the authors' knowledge, this is the first time, the existence of solutions to the fractional boundary value problem is dealt with by using critical point theory.
In this work we prove the existence of solution for a class of perturbed
fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00}
-{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla
W(t,u(t)) = f(t), \end{eqnarray} where $\alpha \in (1/2, 1)$, $t\in
\mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is
a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in
C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$ and $\nabla W$ is the
gradient of $W$ at $u$. The novelty of this paper is that, assuming $L$ is
coercive at infinity we show that (\ref{eq00}) at least has one nontrivial
solution.
In this work we want to prove the existence of solution for a class of
fractional Hamiltonian systems given by
{eqnarray*}_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = &
\nabla W(t,u(t)) u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}) {eqnarray*}
In this article, we study the existence of solutions for the fractional Hamiltonian system $$\displaylines{ {}_tD_\infty^\alpha(_{-\infty}D_t^\alpha u(t))+L(t)u(t)=\nabla W(t,u(t)),\cr u\in H^\alpha(\mathbb{R},\mathbb{R}^N), }$$ where \( {}_tD_\infty^\alpha\) and \(_{-\infty}D_t^\alpha\) are the Liouville-Weyl fractional derivatives of order \(1/2<\alpha<1\), \(L\in C(\mathbb{R},\mathbb{R}^{N\times N})\) is a symmetric matrix-valued function, which is unnecessarily required to be coercive, and \(W\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})\) satisfies some kind of local superquadratic conditions, which is rather weaker than the usual Ambrosetti-Rabinowitz condition. For more information see https://ejde.math.txstate.edu/Volumes/2020/29/abstr.html
In this paper, we consider the following class of fractional Hamiltonian systems $$\begin{aligned} \begin{aligned}&_xD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{x}u(x))+L(x)u(x)=\nabla W(x,u(x)) + f(x),\quad x\in \mathbb {R}\\&\quad u\in H^{\alpha }(\mathbb {R},\mathbb {R}^N), \end{aligned} \end{aligned}$$where \(\alpha \in (1/2,1)\), \(L\in C(\mathbb {R},\mathbb {R}^{N^2})\) is a symmetric positive definite matrix, \(W\in C^1(\mathbb {R}\times \mathbb {R}^N,\mathbb {R})\) is superquadratic and even in u. By using Bolle’s perturbation method in critical point theory, we prove the existence of infinitely many solutions in spite of the lack of the symmetry of this problem. Moreover, we study the spectral properties of the operator \({_{x}}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{x})+L(x)\).
In this paper we study solutions of the following nonperiodic fractional Hamiltonian systems: -tD∞α(-∞Dtαx(t))-L(t).x(t)+∇W(t,x(t))=0,x∈Hα(R,RN),where α∈12,1,t∈R,x∈RN,-∞Dtα and tD∞α are the left and right Liouville-Weyl fractional derivatives of order α on the whole axis R respectively, L:R⟶R2N and W:R×RN⟶R are suitable functions. Applying a new Fountain Theorem established by W. Zou, we prove the existence of infinitely many solutions for the above system in the case where the matrix L(t) is not required to be either uniformly positive definite or coercive, and W∈C1(R×RN,R) is superquadratic at infinity in x but does not needed to satisfy the Ambrosetti-Rabinowitz condition.
In this paper, we mainly consider the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems u¨−L(t)u+Wu(t,u)=0, where L(t) is not necessarily positive definite and the growth rate of potential function W can be in (1, 3/2). Using the variant fountain theorem, we obtain the existence of infinitely many homoclinic solutions for the second-order Hamiltonian systems.
In this paper, we establish two existence theorems for multiple solutions for the following fractional Hamiltonian system
where
In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: (FHS) where α ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ∇ W is the gradient of W at u. The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.
A systematic treatment is presented of the principal mathematical
problems in nonrelativistic quantum mechanics that are associated with
the study of the Schroedinger equation. In particular, attention is
given to the spectral theory of one-dimensional and multidimensional
Schroedinger operators, scattering theory, and the method of functional
integrals. The discussion also covers operator symbols, operators in
Hilbert space, Sobolev spaces, and elliptic equations.
In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.
In this paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order Hamiltonian system ü(t)−L(t)u(t)+∇W(t,u(t))=0, where L(t) and W(t,x) are neither autonomous nor periodic in t. Under the assumption that W(t,x) is indefinite sign and subquadratic as |x|→+∞ and L(t) is a N×N real symmetric positive definite matrices for all t∈R, we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory.
In this paper we prove the existence of infinitely many distinct $T$-periodic solutions for the perturbed second order Hamiltonian system $\ddot{q} + V'(q) = f(t)$ under the conditions that $V: R^N \rightarrow R$ is continuously differentiable and superquadratic, and that $f$ is square integrable and $T$-periodic. In the proof we use the minimax method of the calculus of variation combining with a priori estimates on minimax values of the corresponding functionals.
In this paper we study the following nonperiodic second order Hamiltonian system−u¨(t)+L(t)u(t)=∇uR(t,u),∀t∈R, where L(t) may not be uniformly positive definite for all t∈R, and R(t,u) is asymptotically quadratic in u as |u|→∞. By applying a generalized linking theorem in Bartsch and Ding (2006) [8], we obtain the existence and multiplicity of homoclinic orbits for the above system.
: Some existence results for T periodic solutions of (0.1) were presented in (1) for superquadratic Hamiltonian systems using finite dimensional minimax arguments together with estimates suitable to pass to a limit. An improved existence mechanism was introduced in (2) and applied to some of the super-quadratic problems of (1) as well as to several subquadratic cases. It will show here that these problems possess not only one T periodic solutions z sub 1 but infinitely many distinct subharmonic solutions z sub k. A word of caution must be entered at this point. Although z sub k has period kT, it may not be the case that z sub k has minimal (i.e. primitive) period kT. Indeed simple examples show that there may be an upper bound on the minimal period of z sub k.
In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
: The existence of various kinds of connecting orbits is established for a certain Hamiltonian system as well as its time dependent analogue. For the autonomous case, our main assumption is that V has a global maximum, e.g. at X = O and we find a various kinds of orbits terminating at O. For the time dependent case V has a local but not global maximum at X = O and we find a homoclinic orbit emanating from and terminating at O. Keywords: Periodic solution.
We consider the following boundary value problem:x+∇V(x)=0,x(0)=x0,x(T)=x1, whereT>0,x0,x1∈nare given. We prove that this problem has infinitely many solutions providedVis even and superquadratic.
*In the title of this book (monograph), please immediately correct the word "Fractinal" to read "Fractional". Thanks!*
A new critical point theorem and its applications on second-order perturbed hamiltonian systems
- Y Liu
- F Guo
- Y Luo
Multiple critical points of perturbed symmetric functionals
- P H Rabinowitz