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.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(x, t) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).     

Strong commutativity preserving maps on Lie ideals

Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ²=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .     

Automatic Controls for Nonlinear Dynamical Systems with Lipschitzian Trajectories

We would like to investigate on the solution to the automatic control problem given by the differential equation y′(t) = f(t, y(t), w(t)) for a given initial function x in the initial domain D(x, ω, Y) for almost all t in the interval I, with controls given by w(t) = g(t, y(t), T(y)(t)), where T is a nonanticipating and Lipschitzian operator. The result will be generalized for a dynamical system y′(t) = f(t, y(t), T(y), u(t)).     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Solving second order initial value problems by a hybrid multistep method without predictors

A three-step seventh order hybrid linear multistep method (HLMM) with three non-step points is proposed for the direct solution of the special second order initial value problems (IVPs) of the form y″ = f(x, y) with an extension to y″ = f(x, y, y′). The main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures.The stability properties of the methods are discussed by expressing them as a one-step method in higher dimension. The methods are then applied in block form as simultaneous numerical integrators over non-overlapping intervals. Numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature.     

Existence and uniqueness of positive periodic solutions of functional differential equations

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y_λ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.     

On singular integrals depending on three parameters

In this paper we will prove the pointwise convergence of L(f; x, y, λ) to f(x₀, y₀), as (x, y, λ) tends to (x₀, y₀, λ₀) in the space L_2π, by the three parameter family of singular operators. In contrast to previous works, the kernel function is radial.     

Ill-posed problems with unbounded operators

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au=f is solvable, and ‖fδ−f‖?δ, then the following results are provided: Problem Fδ(u):=‖Au−fδ^‖2+α‖u^‖2 has a unique global minimizer uα,δ for any fδ, uα,δ=A^*−1(AA^*+αI)fδ. There is a function α=α(δ), limδ→0α(δ)=0 such that limδ→0‖uα(δ),δ−y‖=0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are given. Dynamical Systems Method (DSM) is justified for Eq. (1).     

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.     

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.     

Nonuniform nonresonance at the first eigenvalue for singular boundary value problems with sign changing nonlinearities

New nonresonant results are presented for the boundary value problem y″+f(t,y,y′)=0, 0<t<1 with Dirichlet boundary data. Our nonlinearity may be singular in its dependent variable and is allowed to change sign.     

Uniqueness results for the one-dimensional p-Laplacian

This paper is concerned with positive solutions of the boundary value problem ^′(|y^′|^p−2y^′)+f(y)=0, y(−b)=0=y(b) where p>1, b is a positive parameter. Assume that f is continuous on (0,+∞), changes sign from nonpositive to positive, and f(y)/yp−1 is nondecreasing in the interval of f>0. The uniqueness results are proved using a time-mapping analysis.     

Diagonalization method for a vector boundary problem of singular perturbation type

The nonlinear boundary value problem ?y″ + f(t, ?, y, y′) = 0, y(0, ?) = α(?), y(1, ?) = β(?), where ? > 0 is a small parameter and y, f are scalar functions, has been studied extensively. However, for n-dimensional vector functions y, f the problem seems open. Here we study this vector boundary problem and obtain results which are analogous to those for the scalar case. The approach in this paper is to transform the appropriate differential equation into a canonical or diagonalized system of two first-order equations.     

A uniqueness and Existence theorem for a singular third-order boundary value Problem on [0, ∞)

It is proved that the singular third-order boundary value problem y‴ = f(y), y(0) = 0, y(+∞) = 1, y′(+∞) = y″(+∞) = 0, has a unique solution. Here f(y) = (1 − y)^λg(y), λ > 0, g(y) is positive and continuous on (0, 1]. The problem arises in the study of draining and coating flows.     

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.     

Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the form

y^″(x)=f(x,y(x),(y○λ)(x),y^′(x))