Existence of solutions for singular boundary problems for higher order differential equations

We offer sufficient conditions for the existence of solutions for the boundary value problem $\begin{gathered} y(n) + f(t,y,y^1 ,...,y^{(n - 2)} = 0, 0< t< 1, n \geqslant 2 \\ y^{(i)} (0) = 0, 0 \leqslant i \leqslant n - 3 \\ y^{(n - 2)} (0) - \beta y^{(n - 1)} (0) = 0 \\ \gamma y^{(n - 2)} (1) + \delta y^{(n - 1)} (1) = 0 \\ \end{gathered} $ where α, β, γ and δ are constants satisfying αγ+αδ+βγ>0, β, δ≥0, β+α>0 and δ+γ>0.     

Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions

A bundle of differential operators

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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Darboux’s identity and its analogs

Let us choose a positive integern and denoteF(x, y)= , wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation

Norm of an Integral Operator Related to the Harmonic Bergman Projection

The norm of the operator

Asymptotic representations of solutions of two-term nonautonomous nth-order ordinary differential equations with exponential nonlinearity

We obtain necessary and sufficient conditions for the existence of a certain class of solutions of the differential equation $$ (|y^{(n - 1)} |^{\lambda - 1} y^{(n - 1)} )' = \alpha _0 p(t)e^{\sigma y} $$ , where α ₀ ∈ {?1, 1}, σ, λ ∈ R \ {0}, and p: [a, ω[→]0,+∞[(?∞ < a < ω ≤ + ∞) is a continuously differentiable function. We also establish asymptotic representations of such solutions.     

Uniform estimates for positive solutions to quasi-linear differential equations of even order

The existence of uniform estimates for positive solutions with the same domain to the even-order differential equation

Weakly nonlinear solutions of equationP 1 2

Using the isomonodromic deformation method, we study the equation P ₁ ² , $$\frac{1}{{10}}y^{(4)} + y''y + \frac{1}{2}(y')^2 + y^3 = x$$ , which is the first higher equation in the hierarchy of the first Painlevé equation. We construct the asymptotics of its weakly nonlinear solutions for x→∞ along the Stokes rays, and those of the real regular solutions for x→±∞. Bibliography: 11 titles.     

Solvability of a higher-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance

Birkhoff regularity in terms of the growth of the norm for the Green function

We consider the ordinary differential operator L generated on [0, 1] by the differential expression

Eine Verschärfung der Abschätzung des Restes Taylorscher Näherungspolynome

I begin with a new short proof of: (I) LetP(t) inR ^d be a function oft havingn continuous derivatives fora≤t≤x. ThenP(x)∈ convK, where $$K = \left\{ {\sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}} P^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}P^{(n)} (t),a \leqslant t \leqslant x} \right\}.$$ for applying (I) let bef(t) a real function such that the point ((t?a) ⁿ⁺¹,f(t)) fulfills the conditions of (I). Then (I) gives a sharper estimate of then ^th remainder term off(x) than the Lagrange remainder formula. Iff( ⁿ )(t) is also convex ina≤t≤x, thenf(x)∈[c,d], where $$\begin{gathered} c = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}f^{(n)} \left( {\frac{{na + x}}{{n + 1}}} \right)} , \hfill \\ d = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}} \frac{{nf^{(n)} (a) + f^{(n)} (x)}}{{n + 1}}. \hfill \\ \end{gathered} $$      

Solutions of Sturm-Liouville type multi-point boundary value problems for higher-order differential equations

The existence of solutions of the following multi-point boundary value problem $\left\{ \begin{gathered} x^{(n)} (t) = f(t,x(t),x^\prime (t),...,x^{(n - 2)} (t)) + r(t),0 < t < 1, \\ x^{(i)} (\xi _i ) = 0 for i = 0,1,... ,n - 3, ( * ) \\ \alpha x^{(n - 2)} (0) = \beta x^{(n - 1)} (0) = \gamma x^{(n - 1)} (1) + \tau x^{(n - 1)} (1) = 0 \\ \end{gathered} \right.$ is studied. Sufficient conditions for the existence of at least one solution of BVP(*) are established. It is of interest that the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don’t apply the Green’s functions of the corresponding problem and the method to obtain a priori bounds of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.     

Asymptotic Formulas for the Solutions of Integro-Differential Equations

We study the problem of asymptotic integration of the linear integro-differential equation

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ where λ is a spectral parameter; p _j (x) ∈ L ₁(0, 1), j = 0, 1, 2, are complex-valued functions; α _s;l , s = 1, 2, 3, $l = \overline {0,s - 1} $ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α _3,2 + α _1,0 ≠ α _2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L _p (0, 1), 1 < p < ∞, when α _3,2+α _1,0 ≠ α _2,1, p _j (x) ∈ W ₁ ^j (0, 1), j = 1, 2, and p ₀(x) ∈ L ₁(0, 1); moreover, this basis is unconditional for p = 2.     

Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima-tion:then|f~(k)(x)-P_n~(k)(f,x)|=O(1)△_n~(q-k)(x)ωwhere P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodesX_nUY_n(see the definition of the next).     

Approximation by Nörlund means of double fourier series to continuous functions in two variables

We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions \(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and \(\tilde f^{(1,1)} (x,y)\) .     

图有特殊[a,b]-因子的度条件

设G是一个n阶图 .设 1 a 相似文献   

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form ■ where a_j(x) are meromorphic functions in the variable x for j = 0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.     

Oscillation of even order nonlinear functional differential equations with damping

This paper is concerned with a class of even order nonlinear damped differential equations

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.