In this work, we investigate the solvability and the approximate construction of solutions of certain types of regular non-linear boundary value problems for systems of ordinary differential equations on a compact interval. For this purpose, we construct analytically a uniformly convergent parametrised sequence of functions depending on the properties of the concrete boundary conditions and non-linearities in the given systems. The value of the parameter introduced artificially into the scheme is to be determined by solving a certain system of algebraic or transcendental equations. The text is divided into 10 sections. Sections 1 and 2 contain the notation used in what follows and provide a short introduction. In Section 3, the successive approximation techniques are treated for the investigation of periodic solutions of non-autonomous periodic systems. In Section 4, we apply the method for the study the periodic solutions of autonomous systems by using the appropriate reduction to a non-autonomous system. In Section 5, we establish conditions under which a system of non-linear non-autonomous ordinary differential equations has a family of solutions that are periodic with a common period and possess a certain symmetry property. Sections 6 and 7 deal with the investigation of non-linear two-point boundary value conditions by using a parametrisation that leads one to a family of problem with linear two-point conditions considered together with certain additional algebraic or transcendental equations with respect to certain parameters. In Section 8, we use the parametrisation approach to study some three-point non-linear boundary value problem which, as a result, can be investigated through auxiliary two-point problems. Most of theoretical results are illustrated by examples. Section 9 contains some historical remarks concerning the development and application of the method. Finally, in Section 10, we give several exercises concerning the successive approximation technique under consideration.

Successive Approximation Techniques in Non-Linear Boundary Value Problems for Ordinary Differential Equations

For a constructive analysis of the periodic boundary value problem for systems of non-linear non-autonomous ordinary differential equations, a numerical-analytic approach is developed, which allows one to both study the solvability and construct approximations to the solution. An interval halving technique, by using which one can weaken significantly the conditions required to guarantee the convergence, is introduced. The main assumption on the equation is that the non-linearity is locally Lipschitzian.

/pdf/constructive-analysis-of-periodic-solutions-with-interval-p32er8n4xk.pdf

Constructive analysis of periodic solutions with interval halving

We show how a suitable interval halving and parametrisation technique can help to es- sentially improve the sufficient convergence condition for the periodic successive approximations dealing with periodic solutions of nonlinear non-autonomous systems of ordinary differential equations.

/pdf/periodic-successive-approximations-and-interval-halving-3actr3pudc.pdf

Periodic successive approximations and interval halving

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples. 2000 Mathematics Subject Classification: 34B15; 65L10.

/pdf/numerical-analytic-technique-for-investigation-of-solutions-d3xk2218ce.pdf

Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

We obtain some results concerning the investigation of the solutions of three-point Cauchy-Nicoletti type boundary value problems for a certain class of linear functional differ- ential equations. We show that it is useful to reduce the given problem to the parametrized two-point boundary value problem for a suitably perturbed system containing some artificially introduced parameters both in the constructed inhomogeneous two-point boundary conditions and in the modified functional differential equations. To study the transformed parametrized two-point problem, we use a method which is based upon special type of successive approximations constructed in an analytic form. We prove the uniform convergence of these approximations to the parametrized limit function. Our technique leads to a certain system of algebraic equations with respect to the introduced parameters whose solutions provide those numerical values of the parameters that correspond to the solutions of the given three-point boundary value problem.

/pdf/on-a-cauchy-nicoletti-type-three-point-boundary-value-lwd0w12ppa.pdf

On a cauchy-nicoletti type three-point boundary value problem for linear differential equations with argument deviations

We establish conditions necessary for the existence of non-con- stant periodic solutions of non-linear autonomous dierence equations with Lipschitzian right-hand sides.

/pdf/on-periodic-solutions-of-autonomous-difference-equations-1nf5yf31i2.pdf

On periodic solutions of autonomous difference equations

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

A. N. Ronto

Author Tools

Chat about Author

Papers

On periodic solutions of autonomous difference equations