Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy~ and the results also hold for some nonlinear equations with the form x＇（t） = f（t,x（t）） ＋ λg（t,x（t））, where f,g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.     

On the Cauchy problem for an Emden-Fowler equation

We consider the equation y″ = P(x)x ^a y ^σ , σ < 0, and prove the unique solvability of the Cauchy problem y(0) = 0, y′(0) = λ.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

A nonlinear three-point boundary value problem

We consider the differential equation ?(py′)′ + qy + λay + μby + f(x, y, y′) = 0, x? (α, γ) subject to the boundary conditions cos(α₁) y(α) ? sin(α₁) y′(α) = 0cos(β₁) y(β) ? sin(β₁) y′(β) = 0 β? (α, γ)cos(γ₁) y(γ) ? sin(γ₁) y′(γ) = 0. The functions p, g, a, b, and f are well-behaved functions of x; f is smooth and of “higher order” in y and y′; the scalars λ and μ are eigenparameters. With mild restrictions on a and b it is known that the linearized problem, f ≡ 0, has eigensolutions, ($\text{λ}{}^1\text{, μ}{}^1\text{, ψ}{}^1$). In this paper we use an Implicit Function Theorem argument to establish the existence of a local branch of solutions, bifurcating from ($\text{λ}{}^1\text{, μ}{}^1\text{, 0}$), to the above nonlinear two-parameter eigenvalue problem.     

Minimum norm solutions of single stiff linear analytic differential equations

The $\text{l}$₂-norm of the infinite vector of the terms of the Taylor series of an analytic function is used to measure the “unsmoothness” of the function. The sets of solutions to the scalar differential equations y′(t) = λy(t) + f(t) and y′(t) = q(t)y(t) + f(t) are analyzed with respect to this norm. A number of results on the particular solution with minimum norm are given.     

Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl m_α(λ) function for the differential equation d²y/dx²+[λ−q(x)]y=0. The test potential q(x)=x², for which the relevant m_α(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the m_α(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m_α(λ) function for these potentials to illustrate the computational effectiveness of the method used.     

L∞ bounds for solutions of parametrized elliptic equations

This generalizes earlier results (T. I. Seidman, Indiana Univ. Math. J.30 (1981), 305–311) for ?Δu = λf(u). For the family of equations (su1) Au = g(u, λ) with appropriate boundary conditions the object is to construct from g and the boundary conditions a function η(λ, r) such that a bound y(λ) on ∥u∥_∞ can be obtained by solving the ODE: y′(λ) = η(λ, y) with y(λ₀) = B(λ₀) = bound at λ = λ₀.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

More bounds for eigenvalues

A method introduced by Leighton [J. Math. Anal. Appl.35, 381–388 (1971)] for bounding eigenvalues has been extended to include problems of the form ?y″ + p(x) y = λy, when p(x) ? 0 on [0, 1]. The boundary conditions are the general homogeneous conditions y(0) ? ay′(0) = 0 = y(1) + by′(1), where 0 ? a, b ? ∞. Upper and lower bounds for the eigenvalues of these problems are obtained, and these bounds may be made as close together as desired, thereby allowing λ to be estimated precisely.     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

On the Baire Classification of Positive Characteristic Exponents in the Perron Effect of Change of Their Values

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ?ⁿ {0}.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

On the oscillation of a second-order delay equation

The oscillatory nature of two equations (r(t) y′(t))′ + p₁(t)y(t) = f(t), (r(t) y′(t))′ + p₂(t) y(t ? τ(t))= 0, is compared when positive functions p₁ and p₂ are not “too close” or “too far apart.” Then the main theorem states that if h(t) is eventually negative and a twice continuously differentiable function which satisfies (r(t) h′(t))′ + p₁(t) h(t) ? 0, then this inequality is necessary and sufficient for every bounded solution of (r(t) y′(t))′ + p₂(t) y(t ? τ(t)) = 0 to be nonoscillatory.     

Second order linear O.D.E. and Riccati equations

We prove that approximate solutions of the Riccati equation ?′ + ?² = a(x) yield asymptotic solutions y = e^∝x?(s)ds of the second order linear equation y″ = a(x)y. We show that the iterative scheme ?₀ = a, ?_{n + 1}² = a ? ?_n′ leads to asymptotic solutions of the cited linear equation in many interesting cases.     

The Hukuhara-Kneser property for parabolic systems with nonlinear boundary conditions

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.