Quasilinearization Technique for Φ-Laplacian Type Equations

Abstract

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

1. Introduction

Consider the Φ-Laplacian type equation 𝑑Φ𝑑𝑡𝑡,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,ℝ) is Lipschitz and monotone function with respect to 𝑥′, together with the boundary conditions Φ𝑎,𝑥^′(𝑎)=0,𝑥(𝑏)=0. (1.2)

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. (1.3) The equation (1.1) can also be interpreted as the Euler equation for the functional 𝐽(𝑥)=¹₀Ψ𝑡,𝑥^′−𝐹(𝑡,𝑥)𝑑𝑡, (1.4) 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,𝑦𝑡,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,𝑦𝑡,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_𝑦forall𝑡∈[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.

2. 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^′𝑦−𝑘𝑦=0,_′+𝑘𝑥=0. (2.1)

Definition 2.1. A linear part (𝐿𝑋)(𝑡)∶=^{𝑥′−𝑘𝑦}_{𝑦′+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 of 𝑘 let us introduce polar coordinates as 𝑥(𝑡)=𝑟(𝑡)sin𝜑(𝑡),𝑦(𝑡)=𝑟(𝑡)cos𝜑(𝑡). (2.2) 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 𝜋0,2(),𝜋𝑏−𝑎2(),𝑏−𝑎3𝜋2()()𝜋𝑏−𝑎,…,2𝑖−12(),()𝜋𝑏−𝑎2𝑖+12(𝑏−𝑎),…, (2.3)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 ()𝜋𝑘∈2𝑖−12(),()𝜋𝑏−𝑎2𝑖+12(𝑏−𝑎),𝑖∈ℕ. (2.4)

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 𝑥(𝑡;𝛿)−𝜉(𝑡)=𝜌(𝑡)sinΘ(𝑡;𝛿),𝑦(𝑡;𝛿)−𝜂(𝑡)=𝜌(𝑡)cosΘ(𝑡;𝛿), (2.5) 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 𝜋2<Θ(𝑏;𝛿)<𝜋if()<()𝜋𝑖=0,𝑖𝜋<Θ𝑏;𝛿𝑖+1if𝑖≠0. (2.6)

3. 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 𝐶¹₂([𝑎,𝑏]×ℝ²,ℝ).

Lemma 3.1. A set 𝕊 of solutions to the problem (1.6), (1.7) is nonempty and compact in 𝐶¹₂([𝑎,𝑏]×ℝ²,ℝ).

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 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 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}.

Proof. A set 𝕊₁={(𝑥(𝑎),𝑦(𝑎))∈𝑅²∶(𝑥,𝑦)∈𝕊} is the image of a continuous map 𝑀∶𝐶¹₂([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 𝑋_max=𝑋_min 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 𝑢^′1−𝑘𝑣=𝛿𝐹_𝑘𝑡,𝑦(𝑡;𝛿)−𝐹_𝑘(𝑡,𝑣𝑡,𝜂)_′1+𝑘𝑢=𝛿𝐻_𝑘(𝑡,𝑥(𝑡;𝛿))−𝐻_𝑘(𝑡,𝜉(𝑡)). (3.1) The 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]. 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 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.
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.
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 𝑖𝜋<Θ(𝑏;𝛿)<(𝑖+1)𝜋. (3.2) 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 𝑋_max is not a maximal solution. The obtained contradiction completes the proof.

4. Application

Consider the differential equation 𝑑Φ𝑑𝑡𝑡,𝑥^′+𝑓(𝑡,𝑥)=0, (4.1) 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 Φ𝑎,𝑥^′(𝑎)=0,𝑥(𝑏)=0, (4.2) 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 𝐽(𝑥)=^𝑏_𝑎𝑝𝑟(𝑡)𝑥^{′((1+𝑝)/𝑝)}−𝑞(𝑡)|𝑥|^𝑝+1𝑑𝑡, (4.3) 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].

Denote Φ(𝑡,𝑥′)=𝑦, then obtain a two-dimensional differential system 𝑥′=(𝑟(𝑡))^−𝑝𝑦^𝑝(𝑡𝑥|sgn𝑦,𝑦′=−𝑞)|^𝑝sgn𝑥, (4.4) together with the boundary conditions 𝑦(𝑎)=0,𝑥(𝑏)=0. (4.5) The obtained system (4.4) is equivalent to a system 𝑥′−𝑘𝑦=(𝑟(𝑡))^−𝑝𝑦^𝑝(𝑡𝑥|sgn𝑦−𝑘𝑦,𝑦′+𝑘𝑥=𝑘𝑥−𝑞)|^𝑝sgn𝑥, (4.6) 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).

Denote 𝑈_𝑘(𝑡,𝑦)∶=(𝑟(𝑡))^−𝑝|𝑦|^𝑝sgn𝑦−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 𝑦₀. Set 𝑚_𝑦𝑡^∗=𝑈_𝑘𝑡^∗,𝑦₀=𝑘𝑝^{𝑝/(𝑝−1)}𝑝−1𝑟𝑡^{∗𝑝/(𝑝−1)}. (4.7) Choose 𝑛_𝑦(𝑡^∗) such that|𝑦|≤𝑛_𝑦(𝑡^∗)⇒|𝑈_𝑘(𝑡^∗,𝑦)|≤𝑚_𝑦(𝑡^∗). Computation gives that 𝑛_𝑦𝑡^∗=𝑘^1/(𝑝−1)𝑟𝑡^{∗𝑝/(𝑝−1)}𝛾, (4.8) 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|^𝑝sgn𝑥 and instead of the functions 𝑈_𝑘(𝑡,𝑦), 𝑉_𝑘(𝑡,𝑥) consider 𝑈_𝑘𝑡,𝑦∶=𝑈_𝑘𝑡,Δ−𝑁_𝑦,𝑦,𝑁_𝑦,𝑉_𝑘(𝑡,𝑥)∶=𝑉_𝑘𝑡,Δ−𝑁_𝑥,𝑥,𝑁_𝑥, (4.9) where the truncation function Δ is given by Δ𝑥,𝑦,𝑧=𝑥,𝑦<𝑥𝑦,𝑥≤𝑦≤𝑧𝑧,𝑦>𝑧, (4.10)𝑁_𝑦=min{𝑛_𝑦(𝑡)∶𝑡∈[𝑎,𝑏]} and 𝑁_𝑥=min{𝑛_𝑥(𝑡)∶𝑡∈[𝑎,𝑏]}, besides sup𝑈_𝑘𝑡,𝑦=𝑀_𝑦=max𝑚_𝑦(𝑡)[,𝑉∶𝑡∈𝑎,𝑏]sup_𝑘(𝑡,𝑥)=𝑀_𝑥=max𝑚_𝑥(𝑡)∶𝑡∈[𝑎,𝑏]. (4.11)

The nonlinear system (4.6) and the quasilinear one, 𝑥^′−𝑘𝑦=𝑈_𝑘,𝑦𝑡,𝑦_′+𝑘𝑥=𝑉_𝑘(𝑡,𝑥), (4.12) are equivalent in a domain Ω_𝑘=𝑡,𝑥,𝑦∶𝑎≤𝑡≤𝑏,|𝑥(𝑡)|≤𝑁_𝑥,𝑦(𝑡)≤𝑁_𝑦. (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 𝑥_𝑘(𝑡)=^𝑏_𝑎𝐺_𝑘¹¹(𝑡,𝑠)𝑈_𝑘𝑠,𝑦(𝑠)+𝐺_𝑘¹²(𝑡,𝑠)𝑉_𝑘((𝑠𝑦𝑠,𝑥))𝑑𝑠,_𝑘(𝑡)=^𝑏_𝑎𝐺_𝑘²¹(𝑡,𝑠)𝑈_𝑘𝑠,𝑦(𝑠)+𝐺_𝑘²²(𝑡,𝑠)𝑉_𝑘(𝑠,𝑥(𝑠))𝑑𝑠, (4.14) where 𝐺_𝑘^𝑖𝑗(𝑡,𝑠)(𝑖,𝑗=1,2) are the elements of the Green’s matrix to the respective homogeneous problem 𝑦(𝑎)(𝑏)𝑥′−𝑘𝑦=0,𝑦′+𝑘𝑥=0,=0,𝑥=0. (4.15) Then 𝑥_𝑘(𝑡)≤(𝑏−𝑎)Γ₁₁(𝑘)⋅𝑀_𝑦+Γ₁₂(𝑘)⋅𝑀_𝑥,𝑦_𝑘(𝑡)≤(𝑏−𝑎)Γ₂₁(𝑘)⋅𝑀_𝑦+Γ₂₂(𝑘)⋅𝑀_𝑥, (4.16) where Γ_𝑖𝑗(𝑘)(𝑖,𝑗=1,2) are the best estimates (which are known precisely) of the respective elements 𝐺_𝑘^𝑖𝑗(𝑡,𝑠) of the Green’s matrix.

If the inequalities (𝑏−𝑎)Γ₁₁(𝑘)⋅𝑀_𝑦+Γ₁₂(𝑘)⋅𝑀_𝑥<𝑁_𝑥,(𝑏−𝑎)Γ₂₁(𝑘)⋅𝑀_𝑦+Γ₂₂(𝑘)⋅𝑀_𝑥<𝑁_𝑦 (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 𝔾_𝑘()=1𝑡,𝑠▵(𝑘((𝑘((𝑘((𝑘((𝑘((𝑘((𝑘((𝑘(−sin𝑠−𝑎))sin𝑏−𝑡))−cos𝑠−𝑎))sin𝑏−𝑡))sin𝑠−𝑎))cos𝑏−𝑡))cos𝑠−𝑎))cos𝑏−𝑡))if1𝑎≤𝑠≤𝑡≤𝑏,▵(𝑘((𝑘((𝑘((𝑘((𝑘((𝑘((𝑘((𝑘(−cos𝑡−𝑎))cos𝑏−𝑠))−cos𝑡−𝑎))sin𝑏−𝑠))sin𝑡−𝑎))cos𝑏−𝑠))sin𝑡−𝑎))sin𝑏−𝑠))if𝑎≤𝑡<𝑠≤𝑏, (4.18) where ▵=cos𝑘(𝑏−𝑎), therefore 𝐺_𝑘^𝑖𝑗(≤1𝑡,𝑠)|cos𝑘(𝑏−𝑎)|=∶Γ_𝑘,𝑖,𝑗=1,2. (4.19)

Suppose that 0<𝑟₁≤𝑟(𝑡)≤𝑟₂ and 0<𝑞₁≤𝑞(𝑡)≤𝑞₂forall𝑡∈[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 𝑘(𝑏−𝑎)|cos𝑘(𝑏−𝑎)|⋅𝑝^{𝑝/(1−𝑝)}⋅𝑝−1⋅𝑟₂^{𝑝/(𝑝−1)}+𝑞₁^1/(1−𝑝)<𝐴⋅𝛾, (4.20) where 𝐴=min{𝑟₁^{𝑝/(𝑝−1)},𝑞₂^1/(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 functions 𝑟(𝑡) and 𝑞(𝑡) and parameter 𝑝 there are 4 different possible cases. Denote: 1𝜇=𝑟^𝑝₂⋅𝑞₂,if𝑟₁^−𝑝<𝑞₂,𝑟₂^−𝑝<𝑞₁𝜇=𝑟^𝑝₁⋅𝑞₁,if𝑟₁^−𝑝>𝑞₂,𝑟₂^−𝑝>𝑞₁𝑟𝜇=¹𝑟₂^𝑝,if𝑟₁^−𝑝>𝑞₂,𝑟₂^−𝑝<𝑞₁𝑞𝜇=¹𝑞₂,if𝑟₁^−𝑝<𝑞₂,𝑟₂^−𝑝>𝑞₁, (4.21) then inequality (4.20) is fulfilled if the following inequality holds 2𝑘(𝑏−𝑎)|cos𝑘(𝑏−𝑎)|⋅𝑝^{𝑝/(1−𝑝)}⋅𝑝−1⋅𝜇^1/(1−𝑝)<𝛾. (4.22) 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 2𝑘(𝑏−𝑎)|cos𝑘(𝑏−𝑎)|⋅𝑝^{𝑝/(1−𝑝)}⋅𝑝−1⋅𝜇^1/(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).

Corollary 4.2. If there exist numbers 𝑘_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).

Denote: 𝜏_𝑖 is a root of the equation 𝜏=−cot𝜏, which belongs to the interval (((2𝑖−1)𝜋/2),𝜋𝑖), 𝑖∈ℕ. If the inequality 2𝜏_𝑖⋅𝑝^{𝑝/(1−𝑝)}⋅𝑝−1⋅𝜇^1/(1−𝑝)<𝛾 (4.24) 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 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 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).

4.1. Example

Consider the problem 𝑑𝑥𝑑𝑡^′5/6sgn𝑥′+0.04(cos𝜋𝑡+25)|𝑥|^6/5sgn𝑥=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(cos𝜋𝑡+25).

For all 𝑡∈[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 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).

(a) 𝜉0(𝑡)≡0

(b) 𝜉′1(0)=0,𝜉1(0)=10.77

(c) 𝜉′2(0)=0,𝜉2(0)=2733.569

Figure 1 

Different type solutions of the problem (4.25).

(a) for 𝜉0 if 𝛿=0.2

(b) for 𝜉1 if 𝛿=0.1

(c) for 𝜉2 if 𝛿=0.1

Figure 2 

Phase portraits of the differences between solution 𝜉𝑖 (𝑖=0,1,2) of (4.25) and respective neighboring solution in the interval 𝑡∈[0,1].