Solvability of convolution type integro-differential equations on ℝ

The paper looks for the solutions of integro-differential equations of the form

$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $

in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L ₁ (?), 0 ≤ k ∈ L ₁ (?), ∫_? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.

     

Existence theorems for nonlinear differential equations having trichotomy in banach spaces

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation

$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$

(P)

where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f ^d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ _t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay

$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$

(Q)

has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

     

Subdivision schemes with polynomially decaying masks

In this paper we investigate the L ₂-solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix, which plays a vital role for characterizations of wavelets and biorthogonal wavelets with infinite support. A vector refinement equation with polynomially decaying masks and a general dilation matrix is the form:

$ \phi(x)=\sum_{\alpha\in\Bbb Z^s}a(\alpha)\medspace\phi(Mx-\alpha),\quad x\in\Bbb R^s, $

where the vector of functions \(\phi=(\phi_{1},\cdots,\phi_{r})^{T}\) is in \((L_{2}(\Bbb R^s))^{r},\) \(a:=(a(\alpha))_{\alpha\in\Bbb Z^s}\) is a polynomially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that \(\lim_{n\to\infty}M^{-n}=0.\) The corresponding cascade operator on \((L_2(\Bbb R^s))^r\) is given by:

$ Q_{a}f(x):=\sum_{\alpha\in\Bbb Z^s}a(\alpha)f(Mx-\alpha),\quad x\in\Bbb R^s, \quad f=(f_1,...,f_r)^T\in (L_2(\Bbb R^s))^r. $

The iterative scheme \((Q_a^nf)_{n=1,2,\cdots,}\) is called vector cascade algorithm. In this paper we give a complete characterization of convergence of the sequence \((Q_a^nf)_{n=1,2\cdots}\) in L ₂-norm. Some properties of the transition operator restricted to a certain linear space are discussed. As an application of convergence, we also obtain a characterization of smoothness of solutions of refinement equation mentioned above for the case r?=?1.

     

Existence of solutions for a critical fractional Kirchhoff type problem in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity:

$${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$

where (?Δ) ^s is the fractional Laplacian operator with 0 < s < 1, 2 _s * = 2N/(N ? 2s), N > 2s, p ∈ (1, 2 _s *), θ ∈ [1, 2 _s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

     

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form

$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$

When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s?1 in D u, 2 ≤ s ≤ n, and asserts local well-posedness in L ^q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

     

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

We present a variation-of-constants formula for functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t + f\left( {y_t,t} \right),\;y_{t_0}= \varphi $$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t \(t \mapsto f\left( {y_t,t} \right)\) is Kurzweil integrable with t in an interval of ?, for each regulated function y. This means that t \(t \mapsto f\left( {y_t,t} \right)\) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J.Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x} \right],x\left( {{t_0}} \right) = \tilde x$$

and the solutions of the perturbed Cauchy problem

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x + F\left( {x,t} \right)} \right],x\left( {{t_0}} \right) = \tilde x$$

Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t,\;y_{t_0} = \varphi$$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.

     

Hölder estimates for viscosity solutions of equations of fractional <Emphasis Type="Italic">p</Emphasis>-Laplace type

We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation

$$PV {\int_{\mathbb{R}^{n}}\frac{|u(x)-u(x + y)|^{p-2}(u(x) - u(x + y))}{|y|^{n+sp}}\,dy = 0},$$

where \({s \in (0,1)}\) and \({p > 2}\) or \({1/(1-s) < p < 2}\). Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.

     

Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation

We study the asymptotic behavior as t → ∞ of the solution of the initial-boundary value problem for the nonlinear integro-differential equation

$$\frac{{\partial U}}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {a\left( {\mathop \smallint \limits_0^t \left( {\frac{{\partial U}}{{\partial x}}} \right)^2 d\tau } \right)\frac{{\partial U}}{{\partial x}}} \right],$$

where a(S) = (1 + S) ^p , 0 < p ≤ 1. We consider problems with homogeneous boundary conditions as well as with a nonhomogeneous boundary condition on part of the boundary. The orders of convergence are established.

     

Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent

In this paper, we consider the following problem

$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta u\left( x \right) + u\left( x \right) = \lambda \left( {{u^p}\left( x \right) + h\left( x \right)} \right),\;x \in {\mathbb{R}^N},} \\ {u\left( x \right) \in {H^1}\left( {{\mathbb{R}^N}} \right),\;u\left( x \right) \succ 0,\;x \in {\mathbb{R}^N},\;} \end{array}} \right.\;\left( * \right)$$

, where λ > 0 is a parameter, p = (N+2)/(N?2). We will prove that there exists a positive constant 0 < λ* < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ*), no solution for λ > λ*, a unique solution for λ = λ*. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ*) and 3 ≤ N ≤ 5. For N ≥ 6, under some monotonicity conditions of h we show that there exists a constant 0 < λ** < λ* such that problem (*) possesses a unique solution for λ ∈ (0, λ**).

     

Quasi-Periodic Solutions for Non-Autonomous mKdV Equation

In this paper, we consider the non-autonomous modified Korteweg-de Vries (mKdV) equation

$${u_t} = {u_{xxx}} - 6f\left( {\omega t} \right){u^2}{u_x},x \in \mathbb{R}/2\pi \mathbb{Z}$$

, where f(ωt) is real analytic and quasi-periodic in t with frequency vector ω = (ω₁,ω₂, · · ·; ω _m ). Basing on an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, we obtain the existence of Cantor families of smooth quasi-periodic solutions.

     

Oscillation of third order differential equation with damping term

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term

$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$

We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

A modulation invariant Carleson embedding theorem outside local <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

Yen Do and Christoph Thiele developed a theory of Carleson embeddings in outer L_p spaces for the wave packet transform

$${F_\phi }(f)(u,t,\eta ) = \int {f(x){e^{i\eta (u - x)}}\phi \left( {\frac{{u - x}}{t}} \right)} \frac{{dx}}{t},(u,t,\eta ) \in R \times (0,\infty ) \times R$$

of functions f ∈ L_p(R) in the range 2 ≤ p ≤ ∞, referred to as local L². In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of L^p estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.

     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness

We consider a bulk charge potential of the form

$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$

where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as

$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$

where the function γ_ij(x) does not exceed in absolute value a certain quantity of the order of h², the surface integral is understood in the sense of Hadamard finite value, and the n_i(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.

     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation

$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 ? (α/N) < p < (2N ? α)/(N?2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$

where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

     

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

On a property of the reducibility exponent of linear differential systems

We establish that the reducibility exponent (Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system

$$\dot x = A(t)x, x \in \mathbb{R}^n , t \geqslant 0$$

, with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1_A+Q) with an arbitrary piecewise continuous perturbation Q satisfying the condition \(\overline {\lim } _{t \to + \infty } t^{ - 1} \ln \left\| {Q(t)} \right\| \leqslant - \sigma \) is reducible to the original system (1 _A ) by some Lyapunov transformation.