Quasilinearization Technique for Φ-Laplacian Type Equations

Copyright q 2012 I. Yermachenko 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. An equation d/dt Φ t, x′ f t, x 0 is considered together with the boundary conditions Φ a, x′ a 0, x b 0. This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence ofmultiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.


Introduction
This equation even in a greater generality was intensively studied in the last time 1-3 and references therein .If Φ t, x x then it reduces to x f t, x 0.

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The equation 1.1 can also be interpreted as the Euler equation for the functional where Φ t, x ∂Ψ t, x /∂x and f t, x ∂F t, x /∂x .Our aim is to obtain the multiplicity results.For this we denote y Φ t, x and rewrite 1.1 as a two-dimensional differential system of the form x Φ −1 t, y , y −f t, x 1.5 and apply the quasilinearization process described in 4-7 .Namely, we reduce the system 1.5 to a quasilinear one of the form x − ky F k t, y , y kx H k t, x , 1.6 so that both systems 1.5 and 1.6 are equivalent in some domain Ω k { t, x, y : a ≤ t ≤ b, |x| ≤ N x , |y| ≤ N y } and moreover the extracted linear part LX t : x −ky y kx is nonresonant with respect to the boundary conditions y a 0, x b 0. 1.7 If any solution of the quasilinear problem 1.6 , 1.7 satisfies the inequalities |x t | ≤ N x , |y t | ≤ N y for all t ∈ a, b , then we say that the original problem for Φ-Laplacian type equation 1.1 , 1.2 allows for quasilinearization.
If a solution x t , y t of the problem 1.6 , 1.7 is located in Ω k , then this x t , y t also solves the problem 1.5 , 1.7 and therefore the respective x t solves the original problem 1.1 , 1.2 .Notice that the type of a solution x t to the problem 1.1 , 1.2 is induced by oscillatory type of a solution x t , y t to the quasilinear problem 1.6 , 1.7 , which, in turn, is defined by oscillatory properties of the extracted nonresonant linear part LX t 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.

Definitions
Consider the quasilinear system 1.6 , where functions F k , H k are continuous, bounded i.e., there exists a positive constant K such that |F k | < K and |H k | < K for all values of arguments and satisfy the Lipschitz conditions in y and x, respectively.Consider also the relevant homogeneous system x − ky 0, y kx 0.
Let ξ t , η t be a solution of the quasilinear problem 1.6 , 1.7 .
Definition 2.3.One says that x t; δ , y t; δ is a neighboring solution of a solution ξ t , η t , if x t; δ , y t; δ solves the same system 1.6 , satisfies the condition y a; δ 0 and there exists ε > 0 such that for all δ ∈ 0, ε x a; δ ξ a δ.

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In order to classify solutions of the quasilinear problem under consideration introduce local polar coordinates for the difference between neighboring solution x t; δ , y t; δ and investigated solution ξ t , η t as 2.6

Results for Quasilinear Systems
Consider the quasilinear system 1.6 , where the linear part LX t : x −ky y kx is nonresonant with respect to the boundary conditions 1.7 and functions F k , H k are continuous, bounded and satisfy the Lipschitz conditions with respect to y and x, respectively.By a solution we mean a two-dimensional vector function x, y with continuously differentiable components an element of the space Proof.The problem 1.6 , 1.7 has a solution if the right sides F k and H k 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 cos k b − a / 0 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 X max
x * , y * of quasilinear problem 1.6 , 1.7 with the property that y * a 0 and x * a max{x a : x, y ∈ S, y a 0}.Similarly there exists a minimal solution X min x * , y * of 1.6 , 1.7 with a property y * a 0 and x * a min{x a : x, y ∈ S, y a 0}.
x a , y a .Since S is compact S 1 is compact also.Moreover S 1 is compact in a straight line y 0. Thus S 1 is bounded and closed and therefore there exist the maximal and the minimal elements.The case of X max X min corresponds to a unique solution of the BVP 1.6 , 1.7 .Proof.Consider a solution X max ξ * , η * , mentioned in Lemma 3.2 and neighboring solutions x t; δ , y t; δ see Definition 2.3 .We claim that X max is an i type solution to the problem.Suppose that this is not true.According to 2.6 there are two possibilities.

Lemma
Case 1.For any ε > 0 there exists 0 < δ < ε such that Θ b; δ πn for some natural value of n .Therefore x t; δ , y t; δ solves the BVP 1.6 , 1.7 as well.Since x a; δ − ξ * a δ > 0 by virtue of 2.5 , that is, x a; δ > ξ * a , a solution X max ξ * , η * is not maximal in the sense of Lemma 3.2 This case is ruled out.

Application
Consider the differential equation  where a constant γ is a root of the equation γ p γ p − 1 p p/ 1−p .Similarly we transform the function V k t, x : kx − q t |x| p sgn x and instead of the functions U k t, y , V k t, x consider where the truncation function Δ is given by

4.11
The nonlinear system 4.6 and the quasilinear one, x − ky U k t, y , y kx V k t, x , 4.12 are equivalent in a domain

4.13
The modified quasilinear problem 4.12 , 4.5 is solvable if k belongs to one from the intervals mentioned above.The respective solution x k t , y k t can be written in the integral form

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where G ij k t, s i, j 1, 2 are the elements of the Green's matrix to the respective homogeneous problem x − ky 0, y kx 0, y a 0, x b 0.

4.15
Then where Γ ij k i, j 1, 2 are the best estimates which are known precisely of the respective elements 4.17 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

4.19
Suppose that 0 < r 1 ≤ r t ≤ r 2 and 0 < q 1 ≤ q t ≤ q 2 for all t ∈ a, b .Taking into consideration the expressions for M y , M x , N y , N x , γ, and the estimate Γ k we obtain that both inequalities in 4.17 hold if the following inequality is fulfilled where A min{r International Journal of Mathematics and Mathematical Sciences 9 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 r t and q t and parameter p there are 4 different possible cases.Denote:

4.21
then inequality 4.20 is fulfilled if the following inequality holds

4.22
The following theorem is valid.
Theorem 4.1.Suppose that functions r t and q t in the Φ-Laplacian type equation 4.1 are such that 0 < r 1 ≤ r t ≤ r 2 and 0 < q 1 ≤ q t ≤ q 2 for all t ∈ a, b .If there exists some number  1 indicates that nonlinear problem under consideration has a solution of definite type, for instance, k 0 , k 1 show that there exist 0 type and 1 type solutions.

Example
Consider the problem d dt x 5/6 sgn x 0.04 cos πt 25 |x| 6/5 sgn x 0, x 0 0, x 1 0, 4.25 which is a special case of the problem 4.1 , 4.2 with p 6/5, r t ≡ 1, and q t 0.04 cos πt 25 .For all t ∈ 0, 1 0.96 ≤ q t ≤ 1.04, since q 2 > 1 r In accordance with calculations see Table 1 there exist at least three different solutions of the problem 4.25 of 0 type, 1 type and 2 type, respectively.We have computed them see Figures 1 and 2 .
Consider the Φ-Laplacian type equationd dt Φ t, x f t, x 0, t ∈ I : a, b , 1.1 where f ∈ C I × R, R is Lipschitz function with respect to x, Φ ∈ C I × R, Ris Lipschitz and monotone function with respect to x , together with the boundary conditions

Theorem 3 . 4 .
As a consequence, Θ b; δ takes exactly i times values of the form πn together with ϕ b .The main theorem follows.If a linear part LX t in the quasilinear system 1.6 is i-nonresonant with respect to the boundary conditions 1.7 , then the quasilinear problem 1.6 , 1.7 has an i type solution.

Figure 1 :
Figure 1: Different type solutions of the problem 4.25 .
One says that ξ t , η t is an i type solution of the problem 1.6 , 1.7 , if there exists ε > 0 such that for any δ ∈ 0, ε the angular function Θ t; δ , defined by the initial condition Θ a; δ π/2, takes exactly i values of the form πn in the interval a, b and Θ b; δ / πn, n ∈ N.Remark 2.5.If ξ t , η t is an i type solution of 1.6 , 1.7 , then the angular function Θ t; δ in 2.5 satisfies the inequalities 3.3.Suppose that the linear part LX t in 1.6 is i-nonresonant with respect to the boundary conditions 1.7 .Let ξ t , η t be any element of S. Then the angular function Θ t; δ introduced by 2.5 for large enough δ takes exactly i times values of the form πn, n ∈ N in the interval a, b and Θ b; δ / πn.
k t, x t; δ − H k t, ξ t .3.1The right sides in 3.1 tend to zero uniformly in t ∈ a, b as δ → ∞ since F k and H k are bounded functions.The functions u t , v t tend to solutions x t , y t of the homogeneous equation 2.1 , which satisfy the initial conditions ϕ a π/2, x a 1, where ϕ t is the angular function for x t , y t .Therefore Θ t; δ → ϕ t as δ → ∞, uniformly in t ∈ a, b .
extracted linear part LX t in 4.6 is inonresonant with respect to the boundary conditions 4.5 .Denote U k t, y : r t −p |y| p sgn y − ky.Function U k t, y is odd in y for fixed t t * .We calculate the value of this function at the point of local extremum y 0 .Set Choose n y t * such that |y| ≤ n y t * ⇒ |U k t * , y | ≤ m y t * .Computation gives that * U k t * , y 0 k p p/ p−1 p − 1 r t * p/ p−1 .

Corollary 4.2. If
there exist numbersk j ∈ 2j−1 π/2 b−a , 2j 1 π/2 b−a , j 1, 2, . ..,n, which satisfy the inequality 4.23 , then there exist at least n solutions of different types to the problem 4.1 , 4.2 .The results of calculations are provided in Table 1.For certain values of p and μ this table shows which numbers k i−1 of the form k i−1 τ i / b − a , i ∈ N satisfy the inequality 4.23 .The subscript of number k in Table 1−p < γ, 4.23 where γ is a root of the equation γ p γ p − 1 • p p/ 1−p and μ is number of the form 4.21 , then there exists an i type solution of the nonlinear problem 4.1 , 4.2 .

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