Convergence of solutions of perturbed nonlinear differential equations

Summary We use the nonlinear variation of parameters formula to investigate the convergence of the solutions of nonlinear perturbed systems of differential equations. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 9 ottobre 1971.     

On uniform estimates for positive solutions of nonlinear differential equations

The following differential equation is considered in this paper:

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

     

Some asymptotic properties of the solutions of perturbed nonlinear differential equations

Summary The behavior of the solutions of a class of perturbed nonlinear second order differential equations is studied where the perturbation term may be unbounded in each of its arguments. Upper and lower bounds for solutions and a nonoscillation theorem are included in the results.Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

Unbounded positive solutions of higher-order nonlinear functional differential equations

Classification schemes for positive solutions of a class of higher-order nonlinear functional differential equations are given in terms of their asymptotic behavior, and necessary as well as sufficient conditions for the existence of these solutions are also provided.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

Periodic solutions for perturbed differential equations

A variant of the Alekseev variation of constants integral equation is obtained relating the solutions of systems of the form $\text{x}\text{?}\text{= f(t, x, λ)}$ and $\text{y}\text{?}\text{= f(t, y, ψ(t, y)) + g(t, y)}$. For the case when f, g, and ψ have period P in t several theorems are given for the existence of periodic solutions extending known results when f is linear in x and does not depend on the parameter m-vector λ. Comparison with an older technique gives hypotheses where the method above is advantageous for establishing periodic solutions. An example is given for constructing limit cycles of autonomous second-order systems.     

Existence for positive solutions of second-order neutral nonlinear differential equations

We consider the nonlinear neutral differential equations. This work contains some sufficient conditions for the existence of a positive solution which is bounded with exponential functions. The case when the solution converges to zero is also treated.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

On continuous approximate solutions of nonlinear differential equations

Two recently developed methods for continuous approximate solution of equations involving differential operators—the Adomian (decomposition) method and the Sarafyan method—are compared.     

On solutions of third order nonlinear differential equations

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.