# Quasilinearization Technique forΦ-Laplacian Type Equations

Authors: Inara Yermachenko; Felix Sadyrbaev
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

An equationd/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 of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.


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## Body

## 1. Introduction

Consider theΦ-Laplacian type equation
(1.1)ddtΦ(t,x′)+f(t,x)=0,t∈I:=[a,b],
where f∈C(I×ℝ,ℝ) is Lipschitz function with respect to x, Φ∈C(I×ℝ,ℝ) is Lipschitz and monotone function with respect to x', together with the boundary conditions
(1.2)Φ(a,x′(a))=0,x(b)=0.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
(1.3)x′′+f(t,x)=0.
The equation (1.1) can also be interpreted as the Euler equation for the functional
(1.4)J(x)=∫01‍(Ψ(t,x′)-F(t,x))dt,
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 denotey=Φ(t,x') and rewrite (1.1) as a two-dimensional differential system of the form
(1.5)x′=Φ-1(t,y),y′=-f(t,x)
and apply the quasilinearization process described in [4–7]. Namely, we reduce the system (1.5) to a quasilinear one of the form
(1.6)x′-ky=Fk(t,y),y′+kx=Hk(t,x),
so that both systems (1.5) and (1.6) are equivalent in some domain Ωk={(t,x,y):a≤t≤b,|x|≤Nx,|y|≤Ny} and moreover the extracted linear part (LX)(t):=(x'-kyy'+kx) is nonresonant with respect to the boundary conditions
(1.7)y(a)=0,x(b)=0.If any solution of the quasilinear problem (1.6), (1.7) satisfies the inequalities |x(t)|≤Nx,|y(t)|≤Nyforallt∈[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 Ωk, 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 Section2 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 Fk,Hk are continuous, bounded (i.e., there exists a positive constant K such that |Fk|<K and |Hk|<K for all values of arguments) and satisfy the Lipschitz conditions in y and x, respectively. Consider also the relevant homogeneous system
(2.1)x′-ky=0,y′+kx=0.Definition 2.1.
A linear part(LX)(t):=(x'-kyy'+kx) 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 ofk let us introduce polar coordinates as
(2.2)x(t)=r(t)sinφ(t),y(t)=r(t)cosφ(t).
Then the angular function φ(t) for (2.1) satisfies φ'(t)=k. Suppose that k>0, then φ(t) is monotonically increasing and the boundary conditions (1.7) take the form φ(a)=π/2,φ(b)=πn,n∈ℕ.Definition 2.2.
One says that a linear part(LX)(t) in (2.1) isi-nonresonant with respect to the boundary conditions (1.7) if the angular function φ(t), defined by the initial condition φ(a)=π/2, takes exactlyitimes(i=0,1,…) values of the form πn in the interval (a,b) andφ(b)≠πn,n∈ℕ.The linear part(LX)(t) in (2.1) is nonresonant with respect to the boundary conditions (1.7) if the coefficient k>0 satisfies cosk(b-a)≠0, this means that k belongs to a certain interval of
(2.3)(0,π2(b-a)),(π2(b-a),3π2(b-a)),…,((2i-1)π2(b-a),(2i+1)π2(b-a)),…,i=1,2,3,….So the linear part (LX)(t) is 0-nonresonant with respect to the boundary conditions (1.7) if 0<k<(π/2(b-a)) and it is i-nonresonant with respect to the boundary conditions mentioned above if
(2.4)k∈((2i-1)π2(b-a),(2i+1)π2(b-a)),i∈ℕ.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.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
(2.5)x(t;δ)-ξ(t)=ρ(t)sinΘ(t;δ),y(t;δ)-η(t)=ρ(t)cosΘ(t;δ),
where Θ(a;δ)=π/2 and ρ(a)=δ.Definition 2.4.
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
(2.6)π2<Θ(b;δ)<πifi=0,iπ<Θ(b;δ)<(i+1)πifi≠0.


## 3. Results for Quasilinear Systems

Consider the quasilinear system (1.6), where the linear part (LX)(t):=(x'-kyy'+kx) is nonresonant with respect to the boundary conditions (1.7) and functions Fk,Hk 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 C21([a,b]×ℝ2,ℝ).Lemma 3.1.
A set𝕊 of solutions to the problem (1.6), (1.7) is nonempty and compact in C21([a,b]×ℝ2,ℝ).Proof.
The problem (1.6), (1.7) has a solution if the right sides Fk and Hk 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 cosk(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 solutionXmax=(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 Xmin=(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}.Proof.
A set𝕊1={(x(a),y(a))∈R2:(x,y)∈𝕊} is the image of a continuous map M:C21([a,b])→R2 defined by M(x,y)=(x(a),y(a)). Since 𝕊 is compact 𝕊1 is compact also. Moreover 𝕊1 is compact in a straight line y=0. Thus 𝕊1 is bounded and closed and therefore there exist the maximal and the minimal elements. The case of Xmax=Xmin corresponds to a unique solution of the BVP (1.6), (1.7).Lemma 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.Proof.
Consider the neighboring solution(x(t;δ),y(t;δ)) (see Definition 2.3). Notice that both (ξ(t),η(t)) and (x(t;δ),y(t;δ)) are solutions of (1.6) and Θ(a;δ)=π/2. The normalized functions u=(1/δ)(x(t;δ)-ξ(t)) and v=(1/δ)(y(t;δ)-η(t)) satisfy the system
(3.1)u′-kv=1δ[Fk(t,y(t;δ))-Fk(t,η(t))],v′+ku=1δ[Hk(t,x(t;δ))-Hk(t,ξ(t))].
The right sides in (3.1) tend to zero uniformly in t∈[a,b] as δ→+∞ since Fk and Hk 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]. As a consequence, Θ(b;δ) takes exactly i times values of the form πn together with φ(b).The main theorem follows.Theorem 3.4.
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 ani type solution.Proof.
Consider a solutionXmax=(ξ*,η*), mentioned in Lemma 3.2 and neighboring solutions (x(t;δ),y(t;δ)) (see Definition 2.3). We claim that Xmax is an i type solution to the problem. Suppose that this is not true. According to (2.6) there are two possibilities.
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 Xmax=(ξ*,η*) is not maximal in the sense of Lemma 3.2 This case is ruled out.
Case  2. Θ(b;δ)∉[iπ,(i+1)π]. Then there exists small positive δ1 such that jπ≤Θ(b;δ1)≤(j+1)π, where j≠i. By Lemma 3.3 exists δ2 such that for all δ≥δ2Θ(b;δ) satisfies
(3.2)iπ<Θ(b;δ)<(i+1)π.
Since Θ(b;δ) is continuous then there exists δ*∈[δ1,δ2] such that Θ(b;δ*)=πn. It follows again that (x(t;δ*),y(t;δ*)) is a solution of the BVP (1.6), (1.7). Therefore x(a;δ*)-ξ*(a)=δ*>0 and Xmax is not a maximal solution. The obtained contradiction completes the proof.


## 4. Application

Consider the differential equation(4.1)ddtΦ(t,x′)+f(t,x)=0,
whereΦ(t,x')=r(t)|x′|1/psgnx',f(t,x)=q(t)|x|psgnx,t∈I:=[a,b], p>1, r,q∈C(I;(0,+∞)) together with the boundary conditions
(4.2)Φ(a,x′(a))=0,x(b)=0,
which in polar coordinates take the form φ(a)=π/2,φ(b)=πn,n∈ℕ.It is worth mentioning that the problem of minimizing the functional(4.3)J(x)=∫ab‍[pr(t)|x′|((1+p)/p)-q(t)|x|p+1]dt,
with respect to the class of curves joining an arbitrary point of the line t=a with a given point (b,0) leads just to the boundary value problem (4.1), (4.2) [10][Ch. 1, Sec. 6].DenoteΦ(t,x')=y, then obtain a two-dimensional differential system
(4.4)x'=(r(t))-p|y|psgny,y'=-q(t)|x|psgnx,
together with the boundary conditions
(4.5)y(a)=0,x(b)=0.
The obtained system (4.4) is equivalent to a system
(4.6)x'-ky=(r(t))-p|y|psgny-ky,y'+kx=kx-q(t)|x|psgnx,
where the coefficient k>0 satisfies cosk(b-a)≠0. This means if coefficient k∈(((2i-1)π/2(b-a)),((2i+1)π/2(b-a))),i∈ℕ then extracted linear part (LX)(t) in (4.6) is i-nonresonant with respect to the boundary conditions (4.5).DenoteUk(t,y):=(r(t))-p|y|psgny-ky.FunctionUk(t,y) is odd in y for fixed t=t*. We calculate the value of this function at the point of local extremum y0. Set
(4.7)my(t*)=|Uk(t*,y0)|=(kp)p/(p-1)(p-1)(r(t*))p/(p-1).
Choose ny(t*) such that|y|≤ny(t*)⇒|Uk(t*,y)|≤my(t*). Computation gives that
(4.8)ny(t*)=k1/(p-1)(r(t*))p/(p-1)γ,
where a constant γ is a root of the equation γp=γ+(p-1)p(p/(1-p)).Similarly we transform the functionVk(t,x):=kx-q(t)|x|psgnx and instead of the functions Uk(t,y), Vk(t,x) consider
(4.9)Uk(t,y)^:=Uk(t,Δ(-Ny,y,Ny)),Vk(t,x)^:=Vk(t,Δ(-Nx,x,Nx)),
where the truncation function Δ is given by
(4.10)Δ(x,y,z)={x,y<xy,x≤y≤zz,y>z,Ny=min{ny(t):t∈[a,b]} and Nx=min{nx(t):t∈[a,b]}, besides
(4.11)sup|Uk(t,y)^|=My=max{my(t):t∈[a,b]},sup|Vk(t,x)^|=Mx=max{mx(t):t∈[a,b]}.The nonlinear system (4.6) and the quasilinear one,
(4.12)x′-ky=Uk(t,y)^,y′+kx=Vk(t,x)^,
are equivalent in a domain
(4.13)Ωk={(t,x,y):a≤t≤b,|x(t)|≤Nx,|y(t)|≤Ny}.The modified quasilinear problem (4.12), (4.5) is solvable if k belongs to one from the intervals mentioned above. The respective solution (xk(t),yk(t)) can be written in the integral form
(4.14)xk(t)=∫ab‍(Gk11(t,s)Uk(s,y(s))^+Gk12(t,s)Vk(s,x(s))^)ds,yk(t)=∫ab‍(Gk21(t,s)Uk(s,y(s))^+Gk22(t,s)Vk(s,x(s))^)ds,
where Gkij(t,s)(i,j=1,2) are the elements of the Green’s matrix to the respective homogeneous problem
(4.15)x'-ky=0,y'+kx=0,y(a)=0,x(b)=0.
Then
(4.16)|xk(t)|≤(b-a)(Γ11(k)·My+Γ12(k)·Mx),|yk(t)|≤(b-a)(Γ21(k)·My+Γ22(k)·Mx),
where Γij(k)(i,j=1,2) are the best estimates (which are known precisely) of the respective elements Gkij(t,s) of the Green’s matrix.If the inequalities(4.17)(b-a)(Γ11(k)·My+Γ12(k)·Mx)<Nx,(b-a)(Γ21(k)·My+Γ22(k)·Mx)<Ny
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
(4.18)𝔾k(t,s)={1∆(-sin(k(s-a))sin(k(b-t))-cos(k(s-a))sin(k(b-t))sin(k(s-a))cos(k(b-t))cos(k(s-a))cos(k(b-t)))ifa≤s≤t≤b,1∆(-cos(k(t-a))cos(k(b-s))-cos(k(t-a))sin(k(b-s))sin(k(t-a))cos(k(b-s))sin(k(t-a))sin(k(b-s)))ifa≤t<s≤b,
where ∆=cosk(b-a), therefore
(4.19)|Gkij(t,s)|≤1|cosk(b-a)|=:Γk,(i,j=1,2).Suppose that0<r1≤r(t)≤r2 and 0<q1≤q(t)≤q2forallt∈[a,b].Taking into consideration the expressions forMy, Mx, Ny, Nx, γ, and the estimate Γk we obtain that both inequalities in (4.17) hold if the following inequality is fulfilled
(4.20)k(b-a)|cosk(b-a)|·pp/(1-p)·(p-1)·(r2p/(p-1)+q11/(1-p))<A·γ,
where A=min{r1p/(p-1),q21/(1-p)}.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 functionsr(t) and q(t) and parameter p there are 4 different possible cases. Denote:
(4.21)μ=1r2p·q2,ifr1-p<q2,r2-p<q1μ=r1p·q1,ifr1-p>q2,r2-p>q1μ=(r1r2)p,ifr1-p>q2,r2-p<q1μ=q1q2,ifr1-p<q2,r2-p>q1,
then inequality (4.20) is fulfilled if the following inequality holds
(4.22)2k(b-a)|cosk(b-a)|·pp/(1-p)·(p-1)·μ1/(1-p)<γ.
The following theorem is valid.Theorem 4.1.
Suppose that functionsr(t)andq(t)in the Φ-Laplacian type equation (4.1) are such that 0<r1≤r(t)≤r2 and 0<q1≤q(t)≤q2forallt∈[a,b]. If there exists some number k∈(((2i-1)π/2(b-a)),((2i+1)π/2(b-a))), i∈ℕ, which satisfies the inequality
(4.23)2k(b-a)|cosk(b-a)|·pp/(1-p)·(p-1)·μ1/(1-p)<γ,
where γ is a root of the equation γp=γ+(p-1)·pp/(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).Corollary 4.2.
If there exist numberskj∈(((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).Denote:τi is a root of the equation τ=-cotτ, which belongs to the interval (((2i-1)π/2),πi), i∈ℕ. If the inequality
(4.24)2τi·pp/(1-p)·(p-1)·μ1/(1-p)<γ
holds then (4.23) is fulfilled also. The results of calculations are provided in Table 1. For certain values of p and μ this table shows which numbers ki-1 of the form ki-1=(τi/(b-a)), i∈ℕ satisfy the inequality (4.23). The subscript of number k in Table 1 indicates that nonlinear problem under consideration has a solution of definite type, for instance, k0,k1 show that there exist 0 type and 1 type solutions.Table 1
Results of calculations for the problem (4.1), (4.2).
p
γ
μ
k
i
3
/
2
1.2509
μ
≥
0.8390
k
0
;
k
1
4
/
3
1.2703
μ
≥
0.7903
k
0
;
k
1
5
/
4
1.2813
μ
≥
0.7852
k
0
;
k
1
μ
≥
0.9437
k
0
;
k
1
;
k
2
μ
≥
0.7907
k
0
;
k
1
6
/
5
1.2884
μ
≥
0.9161
k
0
;
k
1
;
k
2
μ
≥
0.9949
k
0
;
k
1
;
k
2
;
k
3
μ
≥
0.7991
k
0
;
k
1
7
/
6
1.2933
μ
≥
0.9034
k
0
;
k
1
;
k
2
μ
≥
0.9702
k
0
;
k
1
;
k
2
;
k
3


### 4.1. Example

Consider the problem(4.25)ddt(|x′|5/6sgnx')+0.04(cosπt+25)|x|6/5sgnx=0,x′(0)=0,x(1)=0,
which is a special case of the problem (4.1), (4.2)with p=6/5, r(t)≡1, and q(t)=0.04(cosπt+25).For allt∈[0,1]0.96≤q(t)≤1.04, since q2>1=r1-pandq1<1=r2-p, then μ=q1/q2, μ≈0.923.In accordance with calculations (see Table1) 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).Different type solutions of the problem (4.25).
(a)
ξ
0
(
t
)
≡
0
(b)
ξ
1
′
(
0
)
=
0,ξ1(0)=10.77
(c)
ξ
2
′
(
0
)
=
0
,
ξ
2
(
0
)
=
2733.569Phase portraits of the differences between solutionξi (i=0,1,2) of (4.25) and respective neighboring solution in the interval t∈[0,1].
(a)
forξ0 if δ=0.2
(b)
forξ1 if δ=0.1
(c)
forξ2 if δ=0.1


## 4.1. Example

Consider the problem(4.25)ddt(|x′|5/6sgnx')+0.04(cosπt+25)|x|6/5sgnx=0,x′(0)=0,x(1)=0,
which is a special case of the problem (4.1), (4.2)with p=6/5, r(t)≡1, and q(t)=0.04(cosπt+25).For allt∈[0,1]0.96≤q(t)≤1.04, since q2>1=r1-pandq1<1=r2-p, then μ=q1/q2, μ≈0.923.In accordance with calculations (see Table1) 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).Different type solutions of the problem (4.25).
(a)
ξ
0
(
t
)
≡
0
(b)
ξ
1
′
(
0
)
=
0,ξ1(0)=10.77
(c)
ξ
2
′
(
0
)
=
0
,
ξ
2
(
0
)
=
2733.569Phase portraits of the differences between solutionξi (i=0,1,2) of (4.25) and respective neighboring solution in the interval t∈[0,1].
(a)
forξ0 if δ=0.2
(b)
forξ1 if δ=0.1
(c)
forξ2 if δ=0.1



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