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

Abstract

We consider boundary value problems for scalar differential equation [superscript]x[variant prime][variant prime][/superscript] +λf(x)=0, x(0)=0, x(1)=0, where f(x) is a seventh-degree polynomial and λ 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.

A. Kirichuka 1 and F. Sadyrbaev 2

Academic Editor:Jaume Gine

1, Daugavpils University, 13 Vienibas 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; 18 October 2015

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.

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 problem [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the first zero on the right of a solution of the Cauchy problem [figure omitted; refer to PDF]

The paper [11] discusses the cases of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 equation [figure omitted; refer to PDF] If [figure omitted; refer to PDF] is a solution of (3) with the initial conditions [figure omitted; refer to PDF] then we denote by [figure omitted; refer to PDF] the first zero function (time-map) for Cauchy problem (3), (4).

Consider the problem with a parameter [figure omitted; refer to PDF] and denote the first zero function [figure omitted; refer to PDF] .

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

3. Nonlinearity with Seventh-Degree Polynomial

In the sequel, we consider the problem with [figure omitted; refer to PDF] the seventh-degree polynomial as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with the Dirichlet conditions [figure omitted; refer to PDF]

Let function [figure omitted; refer to PDF] be the primitive function of [figure omitted; refer to PDF] [figure omitted; refer to PDF] and let the conditions [figure omitted; refer to PDF] be fulfilled.

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

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": [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Consider equation [figure omitted; refer to PDF]

Suppose that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are initial values such that trajectories [figure omitted; refer to PDF] enter the saddle points (as in Figure 1).

Figure 1: The phase portrait of (14).

[figure omitted; refer to PDF]

Part of the phase portrait for (14) has the form shown in Figure 1.

We will look now for solutions of the Dirichlet problem (14), (10) choosing the initial conditions in one of the four intervals separately [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

We note that phase portraits of (8) and (14) are equivalent (critical points are the same and trajectories are in one-to-one correspondence due to formula (7)). If we change [figure omitted; refer to PDF] then the initial values for trajectories entering saddle points change also as [figure omitted; refer to PDF] [figure omitted; refer to PDF] .

Proposition 1.

For any [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the time-map function for (14).

Proof.

Let [figure omitted; refer to PDF] be a solution which goes to the saddle point at [figure omitted; refer to PDF] . Consider solutions [figure omitted; refer to PDF] for [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the time needed for a point to move along phase trajectory from point [figure omitted; refer to PDF] to point [figure omitted; refer to PDF] . [figure omitted; refer to PDF] tends to zero as [figure omitted; refer to PDF] goes to zero (since [figure omitted; refer to PDF] is not a critical point) and [figure omitted; refer to PDF] tends to [figure omitted; refer to PDF] as [figure omitted; refer to PDF] goes to [figure omitted; refer to PDF] . By continuity of [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Proposition 2.

For the first zero function [figure omitted; refer to PDF] of (14) in each of the intervals [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , respectively.

Proof.

Consider solutions [figure omitted; refer to PDF] for [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the time needed for a point to move along phase trajectory from point [figure omitted; refer to PDF] to point [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] tends to [figure omitted; refer to PDF] as [figure omitted; refer to PDF] goes to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tends to [figure omitted; refer to PDF] as [figure omitted; refer to PDF] goes to [figure omitted; refer to PDF] . Function [figure omitted; refer to PDF] is continuous since the right side of (14) is a polynomial and there is continuous dependence of solutions on initial data. Therefore, there is a minimum of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] at some [figure omitted; refer to PDF] and, in this case also, the time function has a "U" shaped graph, similarly to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Remark 3.

The graph of the time-map function has U-shaped segments and the number of positive solutions may be one, two, or more solutions depending on whether the graph of [figure omitted; refer to PDF] has one, two, or more intersections with the unity level (the unity refers to the length of the interval).

Proposition 4.

There exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] the Dirichlet problem (8), (10) has at least seven solutions.

Proof.

It follows from Propositions 1 and 2 and formula (6), where [figure omitted; refer to PDF] is the time-map function for equation [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is given in (9).

Remark 5.

The relative positions of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , do not change if [figure omitted; refer to PDF] tends to [figure omitted; refer to PDF] . This follows from (6).

Remark 6.

Precise number of solutions of problem (8), (10) depends on convexity of time-map function [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] and this can be checked using criteria for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as in [13].

4. Examples

4.1. Example 1

Consider equation [figure omitted; refer to PDF] The phase portrait of (15) has 7 critical points; 3 of them are points of type "center" and 4 are points of type "saddle": [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] (see Figure 2).

Figure 2: The phase portrait of (15), [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Then [figure omitted; refer to PDF] The condition [figure omitted; refer to PDF] is fulfilled (see Figure 3).

Figure 3: Graphs of the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (dashed).

[figure omitted; refer to PDF]

Next we bring all the time-maps together and as a result get the bifurcation diagram in Figure 4.

Figure 4: The bifurcation diagram for Example [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

The graphs of [figure omitted; refer to PDF] for seven values of [figure omitted; refer to PDF] are depicted in Figures 5-18.

Figure 5: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 6: The solution of problem (15), (10), [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 7: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 8: Two solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 9: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 10: Three solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 11: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 12: Four solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 13: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 14: Five solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 15: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 16: Six solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 17: The graph of the function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 18: Seven solutions of problem (15), (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

4.2. Example 2

Consider equation [figure omitted; refer to PDF] The 012B phase portrait of (18) has 7 critical points; 3 of them are the points of type "center" and 4 are points of type "saddle": [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The structure of the phase portrait for (18) is the same as that for (15).

Then [figure omitted; refer to PDF]

The condition for the saddle points [figure omitted; refer to PDF] is fulfilled also.

If we look at the bifurcation diagram in Figure 19, we see that there is no such value [figure omitted; refer to PDF] where at some [figure omitted; refer to PDF] the Dirichlet problem (18), (10) has seven solution because the difference between minimum values of the time-map function is large enough.

Figure 19: The bifurcation diagram for Example [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

5. Conclusions

For a polynomial of the type (9) but with arbitrary large odd number of zeros [figure omitted; refer to PDF] in presence of the condition [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] are the zeros) there always exists large enough [figure omitted; refer to PDF] such that the respective boundary value problem (1) has at least [figure omitted; refer to PDF] positive solutions.

If [figure omitted; refer to PDF] is not large then in presence of the condition [figure omitted; refer to PDF] the number of solutions of the boundary value problem (1) depends on the relative positions of minima of the first zero function (time-map) [figure omitted; refer to PDF] . These relative positions may be regulated by the choice of parameters [figure omitted; refer to PDF] .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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