Abstract and Applied Analysis

Volume 2015, Article ID 302185, 6 pages

http://dx.doi.org/10.1155/2015/302185

## Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis

^{1}Daugavpils University, 13 Vienības Street, Daugavpils LV-5401, Latvia^{2}Institute of Mathematics and Computer Science of University of Latvia, Raina Bulvaris 29, Riga LV-1469, Latvia

Received 1 July 2015; Accepted 28 September 2015

Academic Editor: Jaume Giné

Copyright © 2015 A. Kirichuka and F. Sadyrbaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider boundary value problems for scalar differential equation , , , where is a seventh-degree polynomial and is a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.

#### 1. Introduction

Nonlinear boundary value problems for ordinary differential equations still form rapidly developed branch of classical analysis. The traditional issues like existence of solutions, uniqueness, and continuous dependence on boundary data are discussed in a number of classical and modern sources [1, 2]. Less studied are complicated problems of the number of solutions as well as of their dependence on parameters. Of special value are results on existence of positive solutions due to multiple applications. We mention here the works [3–7], where two-point boundary value problems with parameters were considered for the second-order ordinary differential equations. The problem of finding multiple positive solutions was treated in [8]. Nonlinearities of polynomial type were considered in [9]. The time-map technique was applied for investigation of similar problems in [10].

Our goal in this note is to demonstrate how elementary phase plane analysis combined with evaluations of time-map functions can provide the researchers with significant information on the number and properties of solutions. We have chosen problemwhere is a seventh-degree polynomial. Our technique is based on a phase plane analysis. To find positive solutions we will use the first zero function (the so called time-map function). By the first zero function we mean the mapping , where is the first zero on the right of a solution of the Cauchy problem

The paper [11] discusses the cases of being third- and fifth-degree polynomials. It was observed that problem (1) may have, respectively, three or five positive solutions. We focus on the case of being seventh-degree polynomial.

In Section 2, we provide basic facts about first zero functions. In Section 3, we formulate proposition about number of positive solutions for the Dirichlet boundary value problem. The example in Section 4 provides the detailed description of the respective time-map functions, solutions, and bifurcation diagrams. In the final section, we summarize the results and make conclusions.

#### 2. Basic Facts about Time-Map Function

Consider differential equationIf is a solution of (3) with the initial conditions then we denote by the first zero function (time-map) for Cauchy problem (3), (4).

Consider the problem with a parameterand denote the first zero function .

The relation between these two time-map functions was established previously [11, 12]:If is a solution of the Cauchy problem (3), (4), thensolves the initial value problem (5). For details consult [12].

#### 3. Nonlinearity with Seventh-Degree Polynomial

In the sequel, we consider the problem with the seventh-degree polynomial as follows:wherewith the Dirichlet conditions

Let function be the primitive function of and let the conditionsbe fulfilled.

Let us consider phase plane for (8). The value of the first zero function at is the time needed to move from a point to the first intersection point with the -axis. Denote by a solution of the Cauchy problem (5). A set of such that solves the Dirichlet problem (8), (10) will be called* a solution curve*. All such satisfy the equality

The phase portrait for (8) has 7 critical points; 3 of them are the points of type “center" and 4 are points of type “saddle”: , , , and .

Consider equation

Suppose that , , , and are initial values such that trajectories enter the saddle points (as in Figure 1).