International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 975760, 11 pages

doi:10.1155/2012/975760

Research Article

Quasilinearization Technique for -Laplacian Type Equations

Inara Yermachenko¹ and Felix Sadyrbaev1,2

¹Faculty of Natural Sciences and Mathematics, Daugavpils University, Daugavpils, LV-5400, Latvia

²Institute of Mathematics and Computer Science, University of Latvia, Riga, LV-1459, Latvia

Received 31 March 2012; Revised 8 August 2012; Accepted 4 September 2012

Academic Editor: Paolo Ricci

Copyright © 2012 Inara Yermachenko and Felix 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.

An equation is considered together with the boundary conditions , . This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.

1. Introduction

Consider the -Laplacian type equation

where is Lipschitz function with respect to , is Lipschitz and monotone function with respect to , together with the boundary conditions

This equation (even in a greater generality) was intensively studied in the last time ([1–3] and references therein). If then it reduces to

The equation (1.1) can also be interpreted as the Euler equation for the functional

where and .

Our aim is to obtain the multiplicity results. For this we denote and rewrite (1.1) as a two-dimensional differential system of the form

and apply the quasilinearization process described in [4–7]. Namely, we reduce the system (1.5) to a quasilinear one of the form

so that both systems (1.5) and (1.6) are equivalent in some domain and moreover the extracted linear part is nonresonant with respect to the boundary conditions

If any solution of the quasilinear problem (1.6), (1.7) satisfies the inequalities , then we say that the original problem for -Laplacian type equation (1.1), (1.2) allows for quasilinearization.

If a solution of the problem (1.6), (1.7) is located in , then this also solves the problem (1.5), (1.7) and therefore the respective solves the original problem (1.1), (1.2). Notice that the type of a solution to the problem (1.1), (1.2) is induced by oscillatory type of a solution to the quasilinear problem (1.6), (1.7), which, in turn, is defined by oscillatory properties of the extracted nonresonant linear part (see below).

If the original nonlinear problem allows for quasilinearization with respect to the linear parts with different types of nonresonance, then this problem is expected to have multiple solutions.

The paper is organized as follows. In Section 2 definitions are given. In Section 3 the main result is proved concerning the solvability of a quasilinear boundary value problem. Section 4 contains application of the main result and the quasilinearization technique for studying a nonlinear system; the numerical results are provided and a corresponding example was analyzed.

2. Definitions

Consider the quasilinear system (1.6), where functions are continuous, bounded (i.e., there exists a positive constant such that and for all values of arguments) and satisfy the Lipschitz conditions in and , respectively. Consider also the relevant homogeneous system

Definition 2.1. A linear part is nonresonant with respect to the boundary conditions (1.7) if the homogeneous problem (2.1), (1.7) has only the trivial solution.

In order to classify the linear parts for different values of let us introduce polar coordinates as

Then the angular function for (2.1) satisfies . Suppose that , then is monotonically increasing and the boundary conditions (1.7) take the form .

Definition 2.2. One says that a linear part in (2.1) is-nonresonant with respect to the boundary conditions (1.7) if the angular function , defined by the initial condition , takes exactlytimes values of the form in the interval and.

The linear part in (2.1) is nonresonant with respect to the boundary conditions (1.7) if the coefficient satisfies , this means that belongs to a certain interval of

.So the linear part is -nonresonant with respect to the boundary conditions (1.7) if and it is -nonresonant with respect to the boundary conditions mentioned above if

Let be a solution of the quasilinear problem (1.6), (1.7).

Definition 2.3. One says that is a neighboring solution of a solution , if solves the same system (1.6), satisfies the condition and there exists such that for all .

In order to classify solutions of the quasilinear problem under consideration introduce local polar coordinates for the difference between neighboring solution and investigated solution as

where and .

Definition 2.4. One says that is an type solution of the problem (1.6), (1.7), if there exists such that for any the angular function , defined by the initial condition , takes exactly values of the form in the interval and .

Remark 2.5. If is an type solution of (1.6), (1.7), then the angular function in (2.5) satisfies the inequalities

3. Results for Quasilinear Systems

Consider the quasilinear system (1.6), where the linear part is nonresonant with respect to the boundary conditions (1.7) and functions are continuous, bounded and satisfy the Lipschitz conditions with respect to and , respectively. By a solution we mean a two-dimensional vector function with continuously differentiable components an element of the space .

Lemma 3.1. A set of solutions to the problem (1.6), (1.7) is nonempty and compact in .

Proof. The problem (1.6), (1.7) has a solution if the right sides and are bounded. This can be proved by direct application of Schauder fixed point theorem and follows from the well-known results ([8, 9], for instance). The Existence Theorem of [9][Ch. 2, 2] when adapted for the problem (1.6), (1.7) says that this problem is solvable if the homogeneous one (2.1), (1.7) has only the trivial solution. This is the case since the nonresonance condition fulfils.

Compactness follows from the integral representation of a solution of the problem (1.6), (1.7) via the Green’s matrix (4.14) and standard evaluations in order to show that the Arzela-Ascoli criterium is satisfied.

Lemma 3.2. There exists a maximal solution of quasilinear problem (1.6), (1.7) with the property that and . Similarly there exists a minimal solution of (1.6), (1.7) with a property and .

Proof. A set is the image of a continuous map defined by . Since is compact is compact also. Moreover is compact in a straight line . Thus is bounded and closed and therefore there exist the maximal and the minimal elements. The case of corresponds to a unique solution of the BVP (1.6), (1.7).

Lemma 3.3. Suppose that the linear part in (1.6) is -nonresonant with respect to the boundary conditions (1.7). Let be any element of . Then the angular function introduced by (2.5) for large enough takes exactly times values of the form in the interval and .

Proof. Consider the neighboring solution (see Definition 2.3). Notice that both and are solutions of (1.6) and . The normalized functions and satisfy the system

The right sides in (3.1) tend to zero uniformly in as since and are bounded functions. The functions , tend to solutions , of the homogeneous equation (2.1), which satisfy the initial conditions , , where is the angular function for . Therefore as , uniformly in . As a consequence, takes exactly times values of the form together with .

The main theorem follows.

Theorem 3.4. If a linear part in the quasilinear system (1.6) is -nonresonant with respect to the boundary conditions (1.7), then the quasilinear problem (1.6), (1.7) has an type solution.

Proof. Consider a solution , mentioned in Lemma 3.2 and neighboring solutions (see Definition 2.3). We claim that is an type solution to the problem. Suppose that this is not true. According to (2.6) there are two possibilities.

Case  1. For any there exists such that (for some natural value of ). Therefore solves the BVP (1.6), (1.7) as well. Since by virtue of (2.5), that is, , a solution is not maximal in the sense of Lemma 3.2 This case is ruled out.

Case  2. . Then there exists small positive such that , where . By Lemma 3.3 exists such that for all satisfies

Since is continuous then there exists such that . It follows again that is a solution of the BVP (1.6), (1.7). Therefore and is not a maximal solution. The obtained contradiction completes the proof.

4. Application

Consider the differential equation

where,,, , together with the boundary conditions

which in polar coordinates take the form .

It is worth mentioning that the problem of minimizing the functional

with respect to the class of curves joining an arbitrary point of the line with a given point leads just to the boundary value problem (4.1), (4.2) [10][Ch. 1, Sec. 6].

Denote , then obtain a two-dimensional differential system

together with the boundary conditions

The obtained system (4.4) is equivalent to a system

where the coefficient satisfies . This means if coefficient then extracted linear part in (4.6) is -nonresonant with respect to the boundary conditions (4.5).

Denote .

Function is odd in for fixed . We calculate the value of this function at the point of local extremum . Set

Choose such that. Computation gives that

where a constant is a root of the equation .

Similarly we transform the function and instead of the functions , consider

where the truncation function is given by

and , besides

The nonlinear system (4.6) and the quasilinear one,

are equivalent in a domain

The modified quasilinear problem (4.12), (4.5) is solvable if belongs to one from the intervals mentioned above. The respective solution can be written in the integral form

where are the elements of the Green’s matrix to the respective homogeneous problem

Then

where are the best estimates (which are known precisely) of the respective elements of the Green’s matrix.

If the inequalities

hold then the nonlinear problem (4.6), (4.5) (or, equivalently, the original problem (4.1), (4.2)) allows for quasilinearization and therefore has a solution of definite type.

Since the Green’s matrix of the homogeneous linear problem (4.15) is given by

where , therefore

Suppose that and .

Taking into consideration the expressions for , , , , , and the estimate we obtain that both inequalities in (4.17) hold if the following inequality is fulfilled

where .

Thus a fulfilment of the inequality (4.20) is a sufficient condition for existence of a solution of definite oscillatory type to the problem (4.1), (4.2).

Depending on the functions and and parameter there are 4 different possible cases. Denote:

then inequality (4.20) is fulfilled if the following inequality holds

The following theorem is valid.

Theorem 4.1. Suppose that functionsandin the -Laplacian type equation (4.1) are such that and . If there exists some number , , which satisfies the inequality

where is a root of the equation and is number of the form (4.21), then there exists an type solution of the nonlinear problem (4.1), (4.2).

Corollary 4.2. If there exist numbers , , which satisfy the inequality (4.23), then there exist at least solutions of different types to the problem (4.1), (4.2).

Denote: is a root of the equation , which belongs to the interval , . If the inequality

holds then (4.23) is fulfilled also. The results of calculations are provided in Table 1. For certain values of and this table shows which numbers of the form , satisfy the inequality (4.23). The subscript of number in Table 1 indicates that nonlinear problem under consideration has a solution of definite type, for instance, show that there exist type and type solutions.

Table 1: Results of calculations for the problem (4.1), (4.2).

4.1. Example

Consider the problem

which is a special case of the problem (4.1), (4.2)with , , and .

For all , since , then , .

In accordance with calculations (see Table 1) there exist at least three different solutions of the problem (4.25) of type, type and type, respectively. We have computed them (see Figures 1 and 2).

(a)

(b) ,

(c)

Figure 1: Different type solutions of the problem (4.25).

(a) for  if

(b) for if

(c) for if

Figure 2: Phase portraits of the differences between solution () of (4.25) and respective neighboring solution in the interval .

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