International Journal of Differential Equations

Volume 2014 (2014), Article ID 564914, 8 pages

http://dx.doi.org/10.1155/2014/564914

Research Article

Resonant Problems by Quasilinearization

Nadezhda Sveikate

Faculty of Natural Sciences and Mathematics, Daugavpils University, Parādes Street 1, Daugavpils 5401, Latvia

Received 29 January 2014; Accepted 6 March 2014; Published 22 April 2014

Academic Editor: Ioannis G. Stratis

Copyright © 2014 Nadezhda Sveikate. 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.

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domain and if solutions of the quasilinear problem are in , then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

1. Introduction

Two-point nonlinear boundary value problems often appear in applications. Consider the following one. Let a nonlinear equation

be given together with the Dirichlet boundary conditions

The left hand side of the equation is the second order linear form

where and generally are continuous functions given in the interval . The right side is continuous. A solution is a function in . Even equation of the form

and related boundary value problems are investigated insufficiently. The classical result says that the problem (4) and (2) is solvable if is a bounded function. Otherwise various cases are possible. To reduce the problem to that with bounded right hand side the method of lower and upper functions can be used. This method is well developed and related descriptions and other information can be found in the books [1–4]. This method cannot be applied to equations which exhibit oscillatory behavior however. The lower and upper functions in case of the existence of a solution exist but coincide with a solution.

As to the problem (1) and (2) with a bounded function , a similar result holds; namely, the problem (1) and (2) is solvable if the homogeneous problem

has only the trivial solution.

If this is not true the existence of a solution cannot be proved. There are examples, for instance, ,  and  showing that a solution does not exist. This type of problems are called resonant problems.

There is an intensive literature on resonant problems. We mention several papers [5–13].

To treat resonant problems, various approaches were used. We focused on the quasilinearization method [14, 15], which consists in reducing the resonant problem to nonresonant one using suitable estimations of expected solutions.

Below we give a description of the quasilinearization method.

We consider the Dirichlet boundary value problem

where and . The linear part of differential equation (6) is resonant with respect to given boundary conditions (7).

Definition 1. The linear part of differential equation (6) is called a nonresonant with respect to the boundary conditions (7) if the homogeneous problem

has only the trivial solution. Otherwise, if the homogeneous problem (8) has nontrivial solutions, then the linear part is called resonant.

For instance, is nonresonant, but is resonant in ; is nonresonant with respect to the boundary conditions , if , ; that is, the coefficient belongs to one of the intervals

These intervals are called nonresonant intervals.

The classical result states that the problem (6) and (7) is solvable if linear part is nonresonant and is continuous and bounded (the Picard theorem [3]). Therefore it is desirable to obtain the conditions for the boundary value problem (6) and (7) to be solvable for resonant linear parts.

In [16] the author considered the resonant boundary value problem (6) and (7) if and formulated the theorem that the problem (6) and (7) is solvable if

where , , is an integer, and .

In one of the most popular articles on resonant problems [10] the authors considered the boundary value problem

Here , is a constant, is a continuous function, and the limits

exist and are finite.

In [10] the authors formulated the theorem that problem (11) has at least one solution if the inequalities

hold. This theorem is known as the Landesman-Lazer condition. Boundary value problem (11) has also been studied by Lazer [11], Alonso and Ortega [6], Ahmad [5], and Cesari and Kannan [7].

The function in the Landesman-Lazer condition is of the specific form . And it is an actual question to find the conditions ensuring that the problem (6) and (7) is solvable if .

Our main result in this paper is to get conditions which guarantee that the resonant boundary value problem is solvable and we do not use the Landesman-Lazer condition.

We use the quasilinearization approach. This approach was developed in [14, 15]. Using this approach we reduce (6) to a quasilinear one of the form

where a function is continuous and bounded by a constant and the linear part is nonresonant yet with respect to the given boundary conditions (7) that means that the respective homogeneous problem

has only the trivial solution.

If such reduction is possible then according to Conti's theorem [17] the modified problem (14) and (7) is solvable.

If a solution of the modified quasilinear problem (14) and (7) is located in the domain , where both (6) and (14) are equivalent, then the original problem (6) and (7) at resonance has a solution.

Our paper consists of the introduction, four sections, conclusions and references.

Definitions of the type of a solution to Dirichlet boundary value problems are given in Section 2. The general result and related auxiliary results for quasilinear problems are stated.

In Section 3 we describe quasilinearization process and formulate and prove the theorem for resonant boundary value problem (6) and (7) to be solvable.

In Section 4 the idea of quasilinearization is applied to the investigation of the Emden-Fowler type resonant boundary problem.

In Section 5 we consider an example.

2. Auxiliary Results

We consider quasilinear problem (14) and (7).

Theorem 2 (see [17]). If in (14) is a continuous and bounded function and the homogeneous problem (15) has only the trivial solution, then the problem (14) and (7) is solvable.

The solution of quasilinear problem (14) and (7) can be written in the integral form

Respectively,

where is the Green function for the respective homogeneous problem (15).

If , then from (16) and (17) it follows that

where and are bounds for and , respectively.

For instance, if the linear part is then Green's function for Dirichlet problem (15) in the interval is given by

and satisfies the estimates

Definition 3 (see [14, 15]). We will say that the linear part is -nonresonant with respect to the boundary conditions (7), if a solution of the Cauchy problem

has exactly zeros in the interval and .

Definition 4 (see [14, 15]). We will say that is an -type solution of the problem (6) and (7) (resp., (14) and (7)) if for small enough the difference has exactly zeros in and , where is a solution of (6) (resp., (14)), which satisfies the initial conditions

Theorem 5 (see [15], Theorem 2.1). Quasilinear problem (14) and (7) with an -nonresonant linear part has an -type solution.

The proof can be found in [14, 15].

Definition 6. Let (6) and (14), where the linear part is nonresonant with respect to the boundary conditions (7) in the interval , be equivalent in the domain

in the sense that any solution of (6) with a graph in is also a solution of (14) and vice versa. Suppose that any solution of the quasilinear problem (14) and (7) satisfies the estimates

We will say then that the problem (6) and (7) allows for quasilinearization with respect to a domain and a linear part .

The following results in [14, 15] form a basis for application of the quasilinearization process for proving the existence of multiple solutions.

Theorem 7 (see [14, 15]). If the problem (6) and (7) allows for quasilinearization with respect to some domain and some -nonresonant linear part , then it has an -type solution.

Theorem 8 (see [14, 15]). Suppose that the problem (6) and (7) allows for quasilinearization with respect to and -nonresonant linear part , and, at the same time, it allows for quasilinearization with respect to a domain

and -nonresonant linear part , where . Then the problem (6) and (7) has at least 2 solutions of different types.

Corollary 9. Suppose that the problem (6) and (7) allows for quasilinearization with respect to essentially different (in the sense of Definition 3) linear parts and domains of the form (23). Then it has at least different solutions.

3. Preliminary Results

We consider the resonant problem (6) and (7), where is continuous function and the linear part is resonant.

We use quasilinearization process as follows.

(1)First we modify the equation adding a linear part so that the resulting linear part is not resonant yet

(2)We choose constants and and truncate right side:

where

(3)We check the inequalities

where and are the estimates of Green's function and its derivative associated with the linear part in (27); .

Theorem 10. Suppose that , and as above can be found such that the inequalities

are fulfilled.

Then the problem (6) and (7) has a solution such that    and .

Proof. Apply (26) and (27). From Theorem 2 the boundary value problem (27) and (7) is solvable and has a solution , which can be written in the integral form using Green's function.

The inequalities (30) and (31) are fulfilled. This means that

For these values of and the original equation (6) and the modified equation (27) are equivalent:

It follows that is also a solution of the original problem (6) and (7).

Remark 11. If the right side in (6) does not depend on , then can be set to and therefore only inequality (30) should be verified.

4. Application: Emden-Fowler Type Equation

Consider the Emden-Fowler type resonant problem

Theorem 12. Suppose that

and inequality

holds for some , , where is the root of the equation

Then there exists an -type solution of the problem (34) and (35).

Proof. Let us consider instead of (34) the equivalent one

The linear part is nonresonant with respect to the boundary conditions (35) if , where is an integer. We wish to make the right side in (39) bounded. Denote

The function is odd in for fixed . Consider it for nonnegative values of . There exists point of local extremum (it is either a point of maximum in case of or a point of minimum in case of ):

Figure 1 illustrates the case of for fixed .

We can calculate the value of the function at the point of maximum . Set

Choose such that

The value of is computed by solving the equation

or, equivalently, that of

with respect to for any fixed . Computation gives that

where a constant is described in (38). Set

Let us consider the quasilinear equation

where .

The modified quasilinear equation (48) is equivalent to the given equation (34) in the respective domains

The problem (48) and (35) is solvable and solutions can be written in integral form

where is the Green function to the respective homogeneous problem

It is given by

and satisfies the estimate

It follows from (50) that

If inequality

holds, then a solution of the quasilinear problem (48) and (35) satisfies the estimate

and the original problem (34) and (35) allows for quasilinearization with respect to the domain and the linear part . It follows from Theorem 5 that if the linear part is -nonresonant, then the problem (34) and (35) has an -type solution.

Then should satisfy the inequalities

or

where .

Consider inequality (55) and assume that satisfies the estimates (36). If , then

but in the case of we have

Hence inequality (55) reduces to (37).

Figure 1: Existence of a number .

We have computed results (see Table 1) for , various , which show that some satisfy inequality (37). It may happen that several fall into the same nonresonance interval . In Table 1 we select only one for any respective nonresonance interval (this prevents an error when estimating the number of solutions of different types). For instance, in the case , two values of , namely, and , fall into the same nonresonance interval . We show only in Table 1. Intervals of nonresonance are given in the third column of Table 1.

Table 1: Results of calculations for the boundary value problem , .

Nonresonant intervals

Example 13. For instance, we consider the resonant equation

with boundary conditions (35).

Function is odd. And for we can rewrite

Rewrite (62) equivalently:

The linear part in (63) is no more resonant with respect to (35).

We would like to make the function bounded and still continuous. The function is an odd function with a maximum at (cf. Figure 2). Define . Solve the equation for . The solution is .

Define the truncated function

In this example it is

The function is continuous and bounded by the number (cf. Figure 3). Therefore the problem

has a solution . Let us show that    and hence is also a solution of the problem (61) and (35).

A solution of (66) satisfies the integral equation

where is the Green function for the problem

It follows from (67) that

where , holds.

Then,

satisfies, since

This means that the problem (61) allows for quasilinearization with respect to the linear part , where and one can only propose that there exists 1-type solution.

The problem

allows for three essentially different quasilinearizations with , and . Then there exist at least 3 solutions of different types. First, it has the trivial solution, which is a 1-type solution. Figure 4 illustrates the 2-type and 3-type solutions of the boundary value problem (72).

Figure 2: Graph of function .

Figure 3: Graph of truncated function , (65).

(a)

(b)

Figure 4: (a) 2-type solution and (b) 3-type solution of the Emden-Fowler type boundary value problem (72).

5. Example: Equation

We consider the boundary value problem

Similar to the previous example we rewrite the equation equivalently:

The function is odd and we would like to make this function bounded and still continuous. The function has a maximum at . Define . Solve the equation for . The solution is .

We define the truncated function

which is bounded and continuous. The quasilinear boundary value problem

has a solution . We can write it through Green's function, where the estimate of Green's function is

Therefore,

This means that for this example the problem (73) allows for quasilinearization with respect to the linear part , where . Using quasilinearization process we can say that resonant boundary problem (73) is solvable.

6. Conclusions

We show that the resonant boundary value problem can be studied by using a quasilinearization process. The respective cases are considered, when the differential equations of resonant type with bounded or unbounded right side function allow for quasilinearization. As an application the conditions for solvability of the Emden-Fowler type resonant boundary value problem are given. Two examples are considered in detail showing the quasilinearization approach in action. By using quasilinearization process with different linear parts we can state the existence of different solutions of a problem thus obtaining multiplicity of results.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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