Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

A bijection is presented between (1): partitions with conditions f_j+f_j+1≤k−1 and f₁≤i−1, where f_j is the frequency of the part j in the partition, and (2): sets of k−1 ordered partitions (n⁽¹⁾,n⁽²⁾,…,n^(k−1)) such that and , where m_j is the number of parts in n^(j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k−1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.     

Similarity solutions for liquid metal systems near a sharply cornered conductive region

We study the boundary value problem (Pm,a): on (0,+∞), subject to the boundary conditions f(0)=a∈R, f^′(0)=−1 and f^′(+∞)=0. The problem arises in the study of similarity solutions for high frequency excitation of liquid metal systems in an antisymmetric magnetic field. We give a complete picture of solutions of (Pm,a) for the physical interesting case: m<−1 and a?0.     

An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)ⁿx^(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x^(Δ∇)i(a)=0 and x^(Δ∇)i(c)=βx^(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.     

Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

Exact multiplicity results for a class of two-point boundary value problems

This paper is concerned with the exact number of positive solutions for boundary value problems ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf^′(u) changes sign from negative to positive.     

Factorization of entire solutions of some algebraic differential equations

We investigate the factorization of entire solutions of the following algebraic differential equations:

bn(z)finjn(f^′)+bn−1(z)fin−1jn−1(f^′)+?+b₀(z)fi₀j₀(f^′)=b(z),     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

We consider the boundary value problems: (?p(x^′′(t)))+q(t)f(t,x(t),x(t−1),x^′(t))=0, ?p(s)=|s|^p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems.     

Basic Fourier series on a q-linear grid: Convergence theorems

For 0<q<1 define the symmetric q-linear operator acting on a suitable function f(x) by δf(x)=f(q1/2x)−f(q−1/2x). The q-linear initial value problem , f(0)=1, has two entire functions Cq(z) and Sq(z) as linearly independent solutions, which are orthogonal on a discrete set. Sufficient conditions for pointwise convergence and for uniform convergence of the corresponding Fourier expansion are given.     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

Normal Forms for Nonautonomous Differential Equations

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λ_j−∑ⁿ_i=1 ?_iλ_i≠0 for eigenvalues is generalized to the new nonresonance condition λ_j∩∑ⁿ_i=1 ?_iλ_i=∅ for spectral intervals.     

On an eigenvalue problem involving the one-dimensional p-Laplacian

We apply general results on operator equations in ordered spaces and properties of the principal eigenvalues for weighted semi-linear equations to prove the existence of a global continua of positive solutions and eigenvalue intervals to the problem (?(x′))′+λf(t,x,x′)=0 in (0,1), x(0)=x(1)=0, where ?(x)=|x|^p−2x, p>1, λ>0.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).