On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.     

Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coefficient of nonlinear term

As p-Laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Lu’s continuation theorem, which is an extension of Manásevich-Mawhin, we study the existence of periodic solutions for a Rayleigh type p-Laplacian equation

(φ_p(x^′(t)))^′+f(x^′(t))+g₁(x(t-τ₁(t,|x|_∞)))+β(t)g₂(x(t-τ₂(t,|x|_∞)))=e(t).     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space

Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an X-valued almost automorphic function and X is a finite dimensional Banach space.     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

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.     

Computer Algebra and Bifurcations

In this paper we will present the family of Newton algorithms. From the computer algebra point of view, the most basic of them is well known for the local analysis of plane algebraic curves f(x,y)=0 and consists in expanding y as Puiseux series in the variable x. A similar algorithm has been developped for multi-variate algebraic equations and for linear differential equations, using the same basic tools: a “regular” case, associated with a “simple” class of solutions, and a “simple” method of calculus of these solutions; a Newton polygon; changes of variable of type ramification; changes of unknown function of two types y=ct ^μ+? or y=exp?(c/t ^μ)?. Our purpose is first to define a “regular” case for nonlinear implicit differential equations f(t,y,y′)=0. We will then apply the result to an explicit differential equation with a parameter y′=f(y,α) in order to make a link between the expansions of the solutions obtained by our local analysis and the classical theory of bifurcations.     

Positive periodic solutions of nonautonomous functional differential systems

Considered is the periodic functional differential system with a parameter, x^′(t)=A(t,x(t))x(t)+λf(t,xt). Using the eigenvalue problems of completely continuous operators, we establish some criteria on the existence of positive periodic solutions. Moreover, we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions, which are compared with existing ones.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

Three positive solutions to a discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ³x(t − k) = f(x(t)), x(a) = Δx(t₂) = Δ²x(b + 1) = 0. Under various assumptions on f and the integers a, t₂, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.     

Periodic solutions for p-Laplacian neutral functional differential equation with deviating arguments

By using the theory of coincidence degree, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with deviating arguments such as (φp(x^′(t)−cx^′(t−σ)))+g(t,x(t−τ(t)))=p(t), a result on the existence of periodic solutions is obtained.     

Asymptotic behavior of a differential equation with distributed delays

In this paper, sufficient conditions are established for the asymptotical behavior of solutions of the delay differential equation

x^′(t)=F(t,xt)+G(t,xt)     

Three periodic solutions for a nonlinear first order functional differential equation

This paper is concerned with the existence of three positive T-periodic solutions of the first order functional differential equations of the form

x^′(t)=a(t)x(t)-λb(t)f(t,x(h(t))),     

Duality for a Class of Multiobjective Control Problems

In this paper, a class of multiobjective control problems is considered, where the objective and constraint functions involved are f(t, x(t), ?(t), y(t), z(t)) with x(t) ∈ Rⁿ, y(t) ∈ Rⁿ, and z(t) ∈ R^m, where x(t) and z(t) are the control variables and y(t) is the state variable. Under the assumption of invexity and its generalization, duality theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems.     

On oscillation of a Volterra integral equation with delay

The purpose of this paper is to obtain sufficient conditions for oscillation of all solutions of the equation x(t) = f(t) + ∝_a^t K(t, s, x(s), x(g(s))) ds to study the behaviour of its oscillatory solutions in a dependence on the distance between their consecutive zeros and to establish a theorem for localization of the zeros of its solutions.     

Boundary blow-up for differential equations with indefinite weight

We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0⁺ and t→1⁻. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1].