Solvability of Boundary Value Problems for Singular Quasi-Laplacian Differential Equations on the Whole Line∗

This paper is concerned with some integral type boundary value problems associated to second order singular differential equations with quasi-Laplacian on the whole line. The emphasis is put on the one-dimensional p-Laplacian term [Φ(ρ(t)a(t, x(t), x′(t))x′(t))]′ involving a nonnegative function ρ that may be singular at t = 0 and such that ∫ 0 −∞ ds ρ(s) = ∫ +∞ 0 ds ρ(s) = +∞. A Banach space and a nonlinear completely continuous operator are defined in this paper. By using the Schauder’s fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established. An example is presented to illustrate the main theorem.


Introduction
The multi-point boundary-value problems for linear second order ordinary differential equations (ODEs) were initiated by Il'in and Moiseev [15]. Since then, more general nonlinear multi-point boundary-value problems (BVPs) were studied by several authors, see the paper [8,9,10,19], the text books [1,13,14], the survey papers [11,18] and the references therein. However, the study of the existence of solutions of differential equations on the whole real line with nonlinear differential operators does not seem to be sufficiently developed [5].
Differential equations governed by nonlinear differential operators have been widely studied. In this setting the most investigated operator is the classical p-Laplacian, that is Φ p (x) = |x| p−2 x with p > 1, which, in recent years, has been Y. Liu generalized to other types of differential operators, that preserve the monotonicity of the p-Laplacian, but are not homogeneous. These more general operators, which are usually referred to as Φ-Laplacian (or quasi-Laplacian), are involved in some models, e.g. in non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity and theory of capillary surfaces. The related nonlinear differential equation has the form where Φ : R → R is an increasing homeomorphism such that Φ(0) = 0. More recently, equations involving other types of differential operators have been studied from a different point of view arising from other types of models, e.g. reaction diffusion equations with non-constant diffusivity and porous media equations. This leads to consider nonlinear differential operators of the type [a(t, x, x )Φ(x )] , where a is a positive continuous function. For a comprehensive bibliography on this subject, see e.g. [11,16,18].
In [17], the authors study a class of BVPs for the second order nonlinear ordinary differential equations on the whole line. Two theorems have been proved. The first one is established by the use of the Schauder theorem and concerns the existence of solutions, while the second one deals with the existence and uniqueness of solutions and is derived by the Banach contraction principle.
In [5], Bianconi  where Φ is a monotone function which generalizes the one-dimensional p-Laplacian operator. A criterion for the existence and non-existence of solutions of BVP (1.1) is established. In [2,4], Avramescu and Vladimirescu study the following boundary value problem 2) is obtained. In [3], Avramescu and Vladimirescu study the following boundary value problem under some adequate hypothesises and using the Bohnenblust-Karlin fixed point theorem, the existence of solutions of BVP (1.3) is established.

Solvability of BVPs for Singular Quasi-Laplacian Differential Equations 425
Cabada and Cid [6] prove the solvability of the boundary value problem on the whole line where f is a continuous function, Φ : (−a, a) → R is a homeomorphism with a ∈ (0, +∞), i.e., Φ is singular. Calamai [7] and Marcelli, Papalini [17] discuss the solvability of the following strongly nonlinear BVP: where α < β, Φ is a general increasing homeomorphism with bounded domain (singular Φ-Laplacian), a is a positive continuous function and f is a Caratheodory nonlinear function. Conditions for the existence and nonexistence of heteroclinic solutions in terms of the behavior of y → f (t, x, y) and y → Φ(y) as y → 0, and of t → f (t, x, y) as |t| → +∞ are established. The approach is based on fixed point techniques suitably combined to the method of upper and lower solutions. Motivated by the mentioned papers, we consider the more general BVP for a second order singular differential equation on the whole line with quasi-Laplacian operator • a : R × R × R → (0, +∞) is continuous and satisfies that there exist constants m > 0, M > 0 such that and for each r > 0, |x|, |y| ≤ r imply that a(t, (1 + τ (t))x, y/ρ(t)) → a ±∞ uniformly as t → ±∞.
• f , g, h defined on R 3 are nonnegative Caratheodory functions.
By a solution of BVP (1.7) we mean a function x ∈ C 1 (R) such that belongs to W 1,1 (R) and all equations in (1.7) are satisfied. The purpose is to establish sufficient conditions for the existence of at least one solution of BVP (1.7). The results in this paper generalize and improve some known ones since the quasi-Laplacian term [Φ(ρ(t)a(t, x(t), x (t))x (t))] involves the nonnegative function ρ that may satisfy ρ(0) = 0.
The remainder of this paper is organized as follows: the preliminary results are given in Section 2, the main results are presented in Section 3. An example is presented in Section 4 to illustrate the prototype of the main theorem.

Preliminary Results
In this section, we present some background definitions in Banach spaces and state an important fixed point theorem. The preliminary results are given too.
Let X be a Banach space. An operator T ; X → X is completely continuous if it is continuous and maps bounded sets into relatively compact sets. Definition 1. f : R×R×R → R is called a Carathédory function if it satisfies (i) t → f (t, (1 + τ (t))x, y/ρ(t)) is measurable for any x, y ∈ R, (ii) (x, y) → f (t, (1 + τ (t))x, y/ρ(t)) is continuous for a.e. t ∈ R, (iii) for each r > 0, there exists nonnegative function φ r ∈ L 1 (R) such that |u|, |v| ≤ r implies For x ∈ X, define the norm of x by One can prove that X is a Banach space with the norm x for x ∈ X.

Lemma 2.
Suppose that x ∈ X. Denote

Furthermore, it holds that
where M is defined in Section 1.
Proof. Since x ∈ X, f , h are Caratheodory functions, then It is easy to see that G(c) is strictly increasing on R. We find that where A x satisfies (2.1) and (2.4) Lemma 3. The following properties hold : It is easy to see that (2.5) holds. .

Y. Liu
(iii) It is easy to see that x ∈ X is a solution of BVP (1.7) if and only if x is a fixed point of T in X.
(iv) The following five steps are needed (Steps 1-2 imply that T : X → X is continuous and Steps 3-5 imply that T maps bounded sets into relatively compact sets). It follows that T : X → X is completely continuous.

Solvability of BVPs for Singular Quasi-Laplacian Differential Equations 431
Suppose that {A xn } does not converge to A x0 . Then there exist two subse- Let k → ∞, using Lebesgue's dominated convergence theorem, the above equality implies Step 2. We show that T is continuous on X.
Since A x is continuous, then B x is continuous too. From the continuity of A x and B x , and since f , g, h are Caratheodory functions, the result follows.
To prove that T maps bounded sets into relatively compact sets, we must prove that T D is relative compact. Recall W ⊂ X is relatively compact if (i) it is bounded, x ∈ W } are equi-continuous on any closed subinterval of (−∞, +∞), Hence we must do the following three steps.
Step 3. We show that T maps bounded subsets into bounded sets. Let D ⊆ X be a given bounded set. Then, there exists Similarly we have Therefore, On the other hand, we have Then Step 4. Let D be a bounded subset of X. We prove that both with t 1 ≤ t 2 and x ∈ X , since f, g, h are Caratheodory functions, then there exists φ M0 ∈ L 1 (R) such that (2.7) and (2.8) hold. One sees that (2.6) holds.

Solvability of BVPs for Singular Quasi-Laplacian Differential Equations 435
Case 2. −K ≤ t 1 ≤ t 2 ≤ 0. We have similarly that From Cases 1-3, we get Then there exists σ 3 > 0 such that Step 5. Let D be a bounded subset of X. We show that both { T x 1+τ (t) : x ∈ D} and {ρ(T x) : x ∈ D} are equi-convergent at +∞ and −∞ respectively.
It is easy to know that there exists T 1 > 0 such that t > T 1 implies Similarly to Step 4, we can get that uniformly as t → +∞, we know that Then there exists T 2 > 0 such that Then It is easy to see that there exists T 4 > T 3 such that From Steps 3-5, we see that T maps bounded sets into relatively compact sets. Therefore, the operator T : X → X is completely continuous. The proof is complete.

Main Theorems
In this section, the main results on the existence of solutions of BVP (1.7) are established.

Theorem 1. Suppose that there exist nonnegative functions
Then BVP (1.7) has at least one solution.
Proof. We will apply Lemma 1 to prove this theorem. Let X and T be defined in Section 2. From Lemma 3, T : X → X is a completely continuous operator. Let

Mρ(s) ds
Choose By the definition of T , together with (2.4), we get , t ≥ 0, , t ≥ 0, , t ≤ 0, It follows that It follows that

Then (3.2) and (3.3) imply that
a contradiction to (3.1). So T (∂Ω) ⊂ Ω. Thus Lemma 1 implies that the operator T has at least one fixed point in Ω. So BVP (1.7) has at least one solution.
Corollary 1. Suppose that there exists r > 0 such that where x, y ∈ [−r, r]. Then BVP (1.7) has at least one solution.
Proof. From Lemma 3, T : X → X is a completely continuous operator. Now we define Ω = {x ∈ X : x < r}. For any x ∈ ∂Ω, x = r. So By the assumptions, similarly to the proof of Theorem 1, we get Mρ(s) ds So T x ≤ x for all x ∈ ∂Ω. Similar to the process in Theorem 1, the result follows. The proof is complete.
Then BVP (1.7) has at least one solution.
Proof. Let Then, there exists r > 0, such that By Corollary 1, BVP (1.7) has at least one solution. The proof is complete.

An Example
Now, we present an example to illustrate Theorem 1. where λ ∈ R is a constant, Φ(x) = |x| 2 x is a one-dimensional p-Lapalcian. Then BVP (4.1) has at least one solution if
By direct computation, we get It follows from Theorem 1 that BVP (4.1) has at least one solution if