Boundary Value Problems
Volume 2010 (2010), Article ID 875675, 17 pages
doi:10.1155/2010/875675
Research Article

On Second-Order Differential Equations with Nonhomogeneous Φ -Laplacian

Mariella Cecchi,1 Zuzana Došlá,2 and Mauro Marini1

1Department of Electronics and Telecommunications, University of Florence, Via S. Marta 3, 50139 Firenze, Italy
2Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 60000 Brno, Czech Republic

Received 23 September 2009; Accepted 18 January 2010

Academic Editor: Ivan T. Kiguradze

Copyright © 2010 Mariella Cecchi et al. 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

Equation with general nonhomogeneous Φ -Laplacian, including classical and singular Φ -Laplacian, is investigated. Necessary and sufficient conditions for the existence of nonoscillatory solutions satisfying certain asymptotic boundary conditions are given and discrepancies between the general and classical Φ are illustrated as well.

1. Introduction

The aim of this paper is to investigate asymptotic properties for second-order nonlinear differential equation

𝑥 𝑎 ( 𝑡 ) Φ + 𝑏 ( 𝑡 ) 𝐹 ( 𝑥 ) = 0 , 𝑡 ≥ 𝑡 0 , ( 1 . 1 ) where

(i) Φ is an increasing odd homeomorphismus, Φ ∶ ( − 𝜌 , 𝜌 ) → ( − 𝜎 , 𝜎 ) such that Φ ( 0 ) = 0 and 0 < 𝜌 ≤ ∞ , 0 < 𝜎 ≤ ∞ ;(ii) 𝐹 is a real continuous increasing function on ℝ such that 𝐹 ( 𝑢 ) 𝑢 > 0 for 𝑢 ≠ 0 ;(iii) 𝑎 , 𝑏 are positive continuous functions for 𝑡 ≥ 𝑡 0 such that ∞ 𝑡 0 𝑏 ( 𝑡 ) 𝑑 𝑡 < ∞ , ( 1 . 2 ) and, whenever 𝜎 < ∞ , i n f { 𝜎 𝑎 ( 𝑡 ) } > 0 f o r a n y 𝑡 ≥ 𝑡 0 . ( 1 . 3 )

For sake of simplicity we will assume also that 𝐹 is an odd function.

Equation (1.1) is called equation with general Φ -Laplacian because D o m Φ and/or I m Φ are possibly bounded and it is not required that Φ satisfies the homogeneity property

Φ ( 𝑢 ) Φ ( 𝑣 ) = Φ ( 𝑢 𝑣 ) f o r a n y 𝑢 , 𝑣 ∈ D o m Φ . ( 1 . 4 ) Obviously, (1.4) holds for equations with the classical Φ -Laplacian, that is, for

Φ 𝑝 ( 𝑢 ) = | 𝑢 | 𝑝 − 2 𝑢 ( 𝑝 > 1 ) . ( 1 . 5 ) Prototypes of Φ , for which (1.4) does not hold, are the function Φ 𝐶 ∶ ℝ → ( − 1 , 1 )

Φ 𝐶 𝑢 ( 𝑢 ) = √ 1 + | 𝑢 | 2 , ( 1 . 6 ) which determines the curvature operator d i v ( Φ 𝐶 ( ∇ 𝑢 ) ) , and the function Φ 𝑅 ∶ ( − 1 , 1 ) → ℝ

Φ 𝑅 𝑢 ( 𝑢 ) = √ 1 − | 𝑢 | 2 , ( 1 . 7 ) which determines the relativity operator d i v ( Φ 𝑅 ( ∇ 𝑢 ) ) . Boundary value problems on compact intervals for equations of type (1.1) are widely studied. The classical Φ -Laplacian is examined in [1, 2], see also references therein; the cases of the curvature or the relativity operator are considered in [3–5]; finally, equations of type (1.1) with nonhomogeneous Φ -Laplacian defined in the whole ℝ are studied in [6]. As claimed in [6, page 25], the lack of the homogeneity property of Φ causes some difficulties for this study.

Oscillatory and asymptotic properties for (1.1) with classical Φ -Laplacian have attracted attention of many mathematicians in the last two decades; see, for example, [7–17] and references therein. Other papers deal with the qualitative behavior of solutions of systems of the form

𝑥 = 𝐴 ( 𝑡 ) 𝑓 1 𝑦 ( 𝑦 ) , = − 𝐵 ( 𝑡 ) 𝑓 2 ( 𝑥 ) , ( 1 . 8 ) see, for example, [18–21]. Since the homogeneity property (1.4) can fail, (1.1) is not equivalent with system (1.8) and so oscillatory and asymptotic properties of (1.1) with general Φ -Laplacian cannot be obtained, in general, from results concerning (1.8).

The aim of this paper is to consider (1.1) with general Φ -Laplacian and to study the existence of all possible types of nonoscillatory solutions of (1.1) and their mutual coexistence. We show that the lack of the homogeneity property of Φ can produce several new phenomena in asymptotic behavior of solutions of (1.1). The discrepancies on the asymptotic properties of solutions of (1.1) with classical and general Φ -Laplacian are presented and illustrated by some examples, as well.

Our main tools for solving the asymptotic boundary value problems are based on topological methods in locally convex spaces and integral inequalities.

We close the introduction by noticing that (1.1) covers a large class of second-order ordinary differential equations which arise in the study of radially symmetric solutions of partial differential equations of the type

d i v ( 𝐺 ( ∇ 𝑢 ) ) + 𝐵 ( | 𝑥 | ) 𝐹 ( 𝑢 ) = 0 , 𝑥 ∈ 𝐸 , ( 1 . 9 ) where 𝐺 ∶ ℝ 𝑛 → ℝ 𝑛 is continuous homeomorphismus, 𝑥 = ( 𝑥 1 , … 𝑥 𝑛 ) ∈ ℝ 𝑛 , 𝑛 ≥ 2 , ∇ 𝑢 = ( 𝐷 1 𝑢 , … , 𝐷 𝑛 𝑢 ) , 𝐷 𝑖 = 𝜕 / 𝜕 𝑥 𝑖 , 𝑖 = 1 , … , 𝑛 , | 𝑥 | = ∑ 𝑛 𝑖 = 1 𝑥 2 𝑖 , 𝐸 = { 𝑥 ∈ 𝑅 𝑛 ∶ | 𝑥 | ≥ 𝑐 } , 𝑐 > 0 . Denote 𝑟 = | 𝑥 | and 𝑑 𝑢 / 𝑑 𝑟 = 𝑢 𝑟 the radial derivative of 𝑢 . If there exists an odd function 𝑔 ∶ ℝ → ℝ such that

𝐺 𝑥 ( 𝑥 ) = 𝑟 𝑔 ( 𝑟 ) 𝑥 ∈ 𝐸 , ( 1 . 1 0 ) then a direct computation shows that 𝑢 is a radially symmetric solution of (1.9) if and only if the function 𝑦 = 𝑦 ( 𝑟 ) = 𝑢 ( | 𝑥 | ) is a solution of the ordinary differential equation

𝑟 𝑛 − 1 𝑔 𝑦 + 𝑟 𝑛 − 1 𝐵 ( 𝑟 ) 𝐹 ( 𝑦 ) = 0 , ( 𝑟 ≥ 𝑐 ) . ( 1 . 1 1 )

2. Homogeneity Property of Φ

We start by discussing the homogeneity property (1.4) and the consequences when it fails. To this aim, consider the functional equation

𝑋 ( 𝑢 ) 𝑋 ( 𝑣 ) = 𝑋 ( 𝑢 𝑣 ) . ( 2 . 1 ) The following holds.

Proposition 2.1. Any continuous and increasing solution 𝑋 of (2.1) has the form 𝑋 ( 𝑢 ) = | 𝑢 | 𝜇 𝑢 for some 𝜇 > − 1 .

Proof. Denote for 𝑢 > 0 𝑌 ( 𝑢 ) = l n 𝑋 ( 𝑒 𝑢 ) . ( 2 . 2 ) Then (2.2) transforms (2.1) into the Cauchy functional equation , 𝑌 ( 𝑢 + 𝑣 ) = 𝑌 ( 𝑢 ) + 𝑌 ( 𝑣 ) ( 2 . 3 ) whose continuous solutions are of the form 𝑌 ( 𝑢 ) = 𝜆 𝑢 , 𝜆 ∈ ℝ , see, for example, [22]. From here we have 𝑋 ( 𝑢 ) = 𝑢 𝜆 for 𝑢 > 0 and, because 𝑋 is increasing, it results 𝜆 > 0 . Moreover, the continuability of 𝑋 at 0 gives 𝑋 ( 0 ) = 0 . If 𝑢 < 0 , then from (2.1) we have 𝑋 ( 𝑢 ) = 𝑋 ( − 1 ) ( − 𝑢 ) 𝜆 . Because 𝑋 is increasing, we have 𝑋 ( − 1 ) < 0 and from the fact 𝑋 2 ( − 1 ) = 1 it follows 𝑋 ( − 1 ) = − 1 . Consequently, 𝑋 ( 𝑢 ) = | 𝑢 | 𝜆 s g n ( 𝑢 ) , where 𝜆 > 0 , for 𝑢 ∈ ℝ .

Let Φ ∗ be the inverse function to Φ and

Λ = 𝑡 ≥ 𝑡 0 ( 0 , 𝜎 𝑎 ( 𝑡 ) ) . ( 2 . 4 ) In view of (1.3), Λ is a nonempty interval and, if 𝜎 = ∞ , then Λ = ( 0 , ∞ ) .

As we will show later, a crucial role for the behavior of nonoscillatory solutions of (1.1) is played by the integral

𝐼 𝜆 = ∞ 𝑡 0 Φ ∗ 𝜆 𝑎 ( 𝑡 ) 𝑑 𝑡 , 𝜆 ∈ Λ . ( 2 . 5 ) If Φ satisfies the homogeneity property (1.4), then 𝐼 𝜆 is either divergent or convergent for any 𝜆 > 0 . If (1.4) does not hold, the convergence of (2.5) can depend on the choice of 𝜆 , as the following examples illustrate.

Example 2.2. Consider the continuous odd function Φ ∶ ( − 1 , 1 ) → ℝ given by Φ ( u ) = − ( l o g 𝑢 ) − 1 i f 0 < 𝑢 < 1 . ( 2 . 6 ) Thus Φ ∗ ( 𝑤 ) = 𝑒 − 1 / 𝑤 i f 0 < 𝑤 < ∞ . ( 2 . 7 ) Clearly, Λ = ( 0 , ∞ ) . Setting 𝑎 ( 𝑡 ) = l o g 𝑡 o n [ 2 , ∞ ) , ( 2 . 8 ) we have Φ ∗ 𝜆 = 1 𝑎 ( 𝑡 ) 𝑡 1 / 𝜆 , ( 2 . 9 ) and so (2.5) converges for 𝜆 < 1 and diverges for 𝜆 ≥ 1 .

Example 2.3. Consider the continuous odd function Φ ∶ ℝ → ( − 𝜎 , 𝜎 ) , 𝜎 = 1 + 𝑒 , given by ⎧ ⎪ ⎨ ⎪ ⎩ Φ ( 𝑢 ) = − ( l o g 𝑢 ) − 1 i f 1 0 < 𝑢 < 𝑒 , ( 1 + 𝑒 ) 𝑢 ( 𝑢 + 1 ) i f 1 𝑒 ≤ 𝑢 < ∞ . ( 2 . 1 0 ) Thus Φ ∗ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑒 ( 𝑤 ) = − 1 / 𝑤 i f 𝑤 0 < 𝑤 < 1 , ( 1 + 𝑒 − 𝑤 ) i f 1 ≤ 𝑤 < 1 + 𝑒 . ( 2 . 1 1 ) Setting 𝑎 ( 𝑡 ) = l o g 𝑡 o n 𝑒 4 , , ∞ ( 2 . 1 2 ) we have Λ = ( 0 , 𝜆 ) , where 𝜆 = 4 ( 1 + 𝑒 ) . For 𝜆 ∈ Λ we get (2.9) and thus the same conclusion as in Example 2.2 holds.

Observe that in the above examples the change of the convergence of 𝐼 𝜆 depends only on the behavior of Φ ∗ near zero and thus they can be modified in order to include functions Φ with Dom Φ = Im Φ = ℝ .

We close this section by recalling that in the study of oscillatory properties of (1.1) with the classical Φ -Laplacian it is often assumed

∞ 𝑡 0 Φ ∗ 𝑝 1 𝑎 ( 𝑡 ) 𝑑 𝑡 = ∞ . ( 2 . 1 3 ) In this case the operator 𝐿 1 𝑥 ≡ ( 𝑎 ( 𝑡 ) Φ 𝑝 ( 𝑥 ) ) is said to be in the canonical form and it can be reduced by the transformation of the independent variable ∫ 𝑠 = 𝑡 1 / Φ ∗ 𝑝 ( 𝑎 ( 𝜏 ) ) 𝑑 𝜏 to the operator with 𝑎 ( 𝑡 ) ≡ 1 . In particular, the Sturm-Liouville operator 𝐿 2 𝑥 ≡ ( 𝑎 ( 𝑡 ) 𝑥 ) in the canonical form can be transformed to the binomial operator 𝐿 𝑥 ≡ 𝑑 2 𝑥 / 𝑑 𝑠 2 . The lack of the homogeneity property (1.4) makes this approach impossible for a general Φ .

3. Unbounded Solutions

Throughout this paper, by solution of (1.1), we mean a function 𝑥 which is continuously differentiable together with its quasiderivative 𝑥 [ 1 ]

𝑥 [ 1 ] 𝑥 ( 𝑡 ) = 𝑎 ( 𝑡 ) Φ , ( 𝑡 ) ( 3 . 1 ) on some ray [ 𝑡 𝑥 , ∞ ) , where 𝑡 𝑥 ≥ 𝑡 0 , and satisfies (1.1) for 𝑡 > 𝑡 𝑥 . As usual, a solution 𝑥 of (1.1), defined on some neighborhood of infinity, is said to be nonoscillatory if 𝑥 ( 𝑡 ) ≠ 0 for any large 𝑡 , and oscillatory otherwise.

Since we assume that 𝐹 is an odd function, we will restrict our attention only to eventually positive solutions of (1.1) and we denote by 𝕊 the set of these solutions.

Let 𝑥 ∈ 𝕊 : we say 𝑥 ∈ 𝕄 + or 𝕄 − , if 𝑥 is eventually increasing or eventually decreasing. If 𝑥 is eventually positive, then 𝑥 [ 1 ] is decreasing for large 𝑡 . If 𝑥 [ 1 ] becomes negative for 𝑡 ≥ 𝑇 , because we can suppose also 𝑥 ( 𝑡 ) > 0 for 𝑡 ≥ 𝑇 , integrating (1.1) we obtain

𝑥 [ 1 ] ( 𝑡 ) ≥ 𝑥 [ 1 ] ( 𝑇 ) − ℎ 𝑡 𝑇 𝑏 ( 𝑠 ) 𝑑 𝑠 , ( 3 . 2 ) where ℎ = s u p 𝑡 ≥ 𝑇 𝐹 ( 𝑥 ( 𝑡 ) ) . Hence, (1.2) gives that 𝑥 [ 1 ] is bounded.

Unbounded solutions of (1.1) are in the class 𝕄 + and can be a priori divided into the subclasses:

𝕄 + ∞ , ℓ = 𝑥 ∈ 𝕄 + ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ∞ , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝑑 𝑥 , 0 < 𝑑 𝑥 , 𝕄 < ∞ + ∞ , 0 = 𝑥 ∈ 𝕄 + ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ∞ , l i m 𝑡 → ∞ 𝑥 [ 1 ] , ( 𝑡 ) = 0 ( 3 . 3 ) while bounded solutions of (1.1) can be a priori divided into the subclasses

𝕄 + ℓ , 0 = 𝑥 ∈ 𝕄 + ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ℓ 𝑥 , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 0 , 0 < ℓ 𝑥 , 𝕄 < ∞ + ℓ , ℓ = 𝑥 ∈ 𝕄 + ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ℓ 𝑥 , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝑑 𝑥 , 0 < ℓ 𝑥 < ∞ , 0 < 𝑑 𝑥 , 𝕄 < ∞ − ℓ , ℓ = 𝑥 ∈ 𝕄 − ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ℓ 𝑥 , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝑑 𝑥 , 0 < ℓ 𝑥 < ∞ , − ∞ < 𝑑 𝑥 , 𝕄 < 0 − 0 , ℓ = 𝑥 ∈ 𝕄 − ∶ l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = 0 , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝑑 𝑥 , − ∞ < 𝑑 x . < 0 ( 3 . 4 )

In the sequel, we give necessary and sufficient conditions for the existence of unbounded solutions of (1.1). Let 𝐼 𝜆 be defined by (2.5) and set

𝐾 𝜆 = ∞ 𝑡 0 𝑏 ( 𝑡 ) 𝐹 𝑡 𝑡 0 Φ ∗ 𝜆 𝐽 𝑎 ( 𝑠 ) 𝑑 𝑠 𝑑 𝑡 , 𝜇 = ∞ 𝑡 0 Φ ∗ 𝜇 1 𝑎 ( 𝑡 ) ∞ 𝑡 𝑏 ( 𝑠 ) 𝑑 𝑠 𝑑 𝑡 , ( 3 . 5 ) where 𝜆 ∈ Λ and 𝜇 > 0 . The following holds.

Theorem 3.1. ( i 1 ) If there exist positive constants 𝜆 , 𝐿 ∈ Λ such that 𝐿 < 𝜆 , 𝐼 𝐿 = ∞ , 𝐾 𝜆 < ∞ , ( 3 . 6 ) then there exist solutions of (1.1) in 𝕄 + ∞ , ℓ satisfying l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ∞ , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝐿 . ( 3 . 7 )
( i 2 ) Let 𝐼 𝜆 = ∞ for any 𝜆 ∈ Λ . If for some 𝜆 ∈ Λ and 𝜇 ∈ I m 𝐹 𝐾 𝜆 < ∞ , 𝐽 𝜇 = ∞ , ( 3 . 8 ) then there exist solutions of (1.1) in 𝕄 + ∞ , 0 .

Proof. Claim ( i 1 ). Denote 𝐴 𝑡 ; 𝑡 0 = 𝑡 𝑡 0 Φ ∗ 𝜆 𝑎 ( 𝑠 ) 𝑑 𝑠 . ( 3 . 9 ) Obviously, l i m 𝑡 → ∞ 𝐴 ( 𝑡 ; 𝑡 0 ) = ∞ . Let 𝜀 > 0 be such that 𝐿 + 𝜀 ≤ 𝜆 . Fixed 𝐻 > 0 , choose 𝑡 1 > 𝑡 0 large so that ∞ 𝑡 1 𝐴 𝑏 ( 𝑡 ) 𝐹 𝑡 ; 𝑡 0 𝑡 𝑑 𝑡 < 𝜀 , 𝐴 1 ; 𝑡 0 ≥ 𝐻 + 𝜀 . ( 3 . 1 0 ) Denote with 𝐶 [ 𝑡 1 , ∞ ) the Fréchet space of all continuous functions on [ 𝑡 1 , ∞ ) endowed with the topology of uniform convergence on compact subintervals of [ 𝑡 1 , ∞ ) and consider the set Ω ⊂ 𝐶 [ 𝑡 1 , ∞ ) given by 𝑡 Ω = 𝑢 ∈ 𝐶 1 , ∞ ∶ 𝐻 + 𝜀 ≤ 𝑢 ( 𝑡 ) ≤ 𝐻 + 𝜀 + 𝐴 𝑡 ; 𝑡 1 . ( 3 . 1 1 ) Define in Ω the operator 𝑇 as follows: 𝑇 ( 𝑢 ) ( 𝑡 ) = 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 1 𝑎 ( 𝑠 ) 𝐿 + ∞ 𝑠 𝑏 ( 𝜏 ) 𝐹 ( 𝑢 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 1 2 ) Obviously, 𝑇 ( 𝑢 ) ( 𝑡 ) ≥ 𝐻 + 𝜀 . From (3.10) we have for 𝑠 ≥ 𝑡 1 ∞ 𝑠 𝑏 ( 𝜏 ) 𝐹 ( 𝑢 ( 𝜏 ) ) 𝑑 𝜏 ≤ ∞ 𝑠 𝑏 ( 𝜏 ) 𝐹 𝐻 + 𝜀 + 𝐴 𝜏 ; 𝑡 1 𝑑 𝜏 ≤ ∞ 𝑡 1 𝐴 𝑏 ( 𝜏 ) 𝐹 𝜏 ; 𝑡 0 𝑑 𝜏 < 𝜀 , ( 3 . 1 3 ) and so, because 𝐿 + 𝜀 ≤ 𝜆 , 𝑇 ( 𝑢 ) ( 𝑡 ) ≤ 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 𝐿 + 𝜀 𝑎 ( 𝑠 ) 𝑑 𝑠 ≤ 𝐻 + 𝜀 + 𝐴 𝑡 ; 𝑡 1 , ( 3 . 1 4 ) that is, 𝑇 maps Ω into itself. Let us show that 𝑇 ( Ω ) is relatively compact, that is, 𝑇 ( Ω ) consists of functions equibounded and equicontinuous on every compact interval of [ 𝑡 1 , ∞ ) . Because 𝑇 ( Ω ) ⊂ Ω , the equiboundedness follows. Moreover, in view of the above estimates, for any 𝑢 ∈ Ω we have 𝑑 0 < 𝑑 𝑡 𝑇 ( 𝑢 ) ( 𝑡 ) ≤ Φ ∗ 𝐿 + 𝜀 , 𝑎 ( 𝑡 ) ( 3 . 1 5 ) which proves the equicontinuity of the elements of 𝑇 ( Ω ) . Now we prove the continuity of 𝑇 in Ω . Let { 𝑢 𝑛 } , 𝑛 ∈ ℕ , be a sequence in Ω which uniformly converges on every compact interval of [ 𝑡 1 , ∞ ) to 𝑢 ∈ Ω . Because 𝑇 ( Ω ) is relatively compact, the sequence { 𝑇 ( 𝑢 𝑛 ) } admits a subsequence { 𝑇 ( 𝑢 𝑛 𝑗 ) } converging, in the topology of 𝐶 [ 𝑡 1 , ∞ ) , to 𝑧 𝑢 ∈ 𝑇 ( Ω ) . Because 𝑡 𝑡 1 Φ ∗ 1 𝑎 ( 𝑠 ) 𝐿 + ∞ 𝑠 𝑢 𝑏 ( 𝜏 ) 𝐹 𝑛 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 ≤ 𝑡 𝑡 1 Φ ∗ 𝐿 + 𝜀 𝑎 ( 𝑠 ) 𝑑 𝑠 , ( 3 . 1 6 ) from the Lebesgue dominated convergence theorem, the sequence { 𝑇 ( 𝑢 𝑛 𝑗 ) ( 𝑡 ) } pointwise converges to 𝑇 ( 𝑢 ) ( 𝑡 ) . In view of the uniqueness of the limit, 𝑇 ( 𝑢 ) = 𝑧 𝑢 is the only cluster point of the compact sequence { 𝑇 ( 𝑢 𝑛 ) } , that is, the continuity of 𝑇 in the topology of 𝐶 [ 𝑡 1 , ∞ ) . Hence, by the Tychonov fixed point theorem there exists a solution 𝑥 of the integral equation 𝑥 ( 𝑡 ) = 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 1 𝑎 ( 𝑠 ) 𝐿 + ∞ 𝑠 𝑏 ( 𝜏 ) 𝐹 ( 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 1 7 ) Clearly, 𝑥 is a solution of (1.1). Using (3.6) and 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 𝐿 𝑎 ( 𝑠 ) < 𝑥 ( 𝑡 ) ≤ 𝑡 𝑡 0 Φ ∗ 𝜆 , 𝑎 ( 𝑠 ) ( 3 . 1 8 ) we get l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = ∞ , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝐿 , and the Claim ( 𝑖 1 ) is proved.
Claim ( i 2 ). Let { 𝐿 𝑛 } 𝑛 ≥ 1 be a decreasing sequence such that l i m 𝑛 𝐿 𝑛 = 0 , 0 < 𝐿 1 < 𝜆 . ( 3 . 1 9 ) Choose 𝜀 > 0 such that 𝐿 1 + 𝜀 < 𝜆 . Since { 𝐿 𝑛 } is decreasing and 𝜆 ∈ Λ , we have 𝐿 𝑛 ∈ Λ and 𝐿 𝑛 + 𝜀 < 𝜆 . Fixed 𝐻 such that 𝐻 > 𝐹 − 1 ( 𝜇 ) , using the argument of the claim ( i 1 ) , for any 𝑛 ≥ 1 there exists 𝑥 𝑛 ∈ 𝕄 + ∞ , ℓ such that l i m 𝑡 → ∞ 𝑥 𝑛 [ 1 ] ( 𝑡 ) = 𝐿 𝑛 . In virtue of the proof of claim ( i 1 ) , we have 𝑥 𝑛 ∈ Ω and 𝑥 𝑛 ( 𝑡 ) = 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 1 𝐿 𝑎 ( 𝑠 ) 𝑛 + ∞ 𝑠 𝑥 𝑏 ( 𝜏 ) 𝐹 𝑛 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 2 0 ) Since 𝑇 ( Ω ) is compact (in the topology of 𝐶 [ 𝑡 1 , ∞ ) ), there exists a subsequence of { 𝑥 𝑛 } converging to 𝑥 in any compact interval of 𝐶 [ 𝑡 1 , ∞ ) . For sake of simplicity, let { 𝑥 𝑛 } be such a sequence, that is, l i m 𝑛 𝑥 𝑛 = 𝑥 . Because 𝑡 𝑡 1 Φ ∗ 1 𝐿 𝑎 ( 𝑠 ) 𝑛 + ∞ 𝑠 𝑥 𝑏 ( 𝜏 ) 𝐹 𝑛 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 ≤ 𝑡 𝑡 1 Φ ∗ 𝜆 𝑎 ( 𝑠 ) 𝑑 𝑠 , ( 3 . 2 1 ) using the Lebesgue dominated convergence theorem, it results for 𝑡 ≥ 𝑡 1 𝑥 ( 𝑡 ) = 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 1 𝑎 ( 𝑠 ) ∞ 𝑠 𝑏 ( 𝜏 ) 𝐹 ( 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 2 2 ) Hence 𝑥 is a solution of (1.1) and l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 0 . Because 𝑥 ( 𝑡 ) ≥ 𝐻 + 𝜀 and 𝐹 , Φ are increasing, we have 𝐹 ( 𝑥 ( 𝑡 ) ) ≥ 𝐹 ( 𝐻 + 𝜀 ) > 𝜇 and 𝑥 ( 𝑡 ) ≥ 𝐻 + 𝜀 + 𝑡 𝑡 1 Φ ∗ 𝜇 𝑎 ( 𝑠 ) ∞ 𝑠 𝑏 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 2 3 ) Since 𝐽 𝜇 = ∞ , the solution 𝑥 is unbounded and the proof is complete.

Remark 3.2. If 𝐼 𝜆 = ∞ for any 𝜆 ∈ Λ and there exists 𝜆 ∈ Λ such that 𝐾 𝜆 < ∞ , then (1.1) has solutions in 𝕄 + ∞ , ℓ satisfying l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝐿 for any 𝐿 < 𝜆 . In some particular situations, the existence of solutions 𝑥 ∈ 𝕄 + ∞ , ℓ , satisfying the limit case l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝜆 , is examined in [23]. See also Example 3.4 below .

The following result is the partial converse of Theorem 3.1.

Theorem 3.3. If there exists a nonoscillatory unbounded solution 𝑥 of (1.1) such that l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝐿 ≥ 0 , then 𝐼 𝜆 = ∞ for any 𝜆 ∈ Λ , 𝜆 > 𝐿 , and ∞ 𝑡 0 𝜇 𝑏 ( 𝑡 ) 𝐹 𝑡 𝑡 0 Φ ∗ 𝐿 𝑎 ( 𝑠 ) 𝑑 𝑠 𝑑 𝑡 < ∞ ( 3 . 2 4 ) for any 𝜇 , 0 < 𝜇 < 1 .

Proof. Let 𝜀 > 0 be such that 𝐿 < 𝜆 − 𝜀 . Without loss of generality, we can suppose 𝑥 ( 𝑡 ) > 0 , 0 < 𝑥 [ 1 ] ( 𝑡 ) < 𝐿 + 𝜀 for 𝑡 ≥ 𝑇 . Thus 𝑥 ( 𝑡 ) < 𝑥 ( 𝑇 ) + 𝑡 𝑇 Φ ∗ 𝐿 + 𝜀 𝑎 ( 𝑠 ) 𝑑 𝑠 < 𝑥 ( 𝑇 ) + 𝑡 𝑇 Φ ∗ 𝜆 𝑎 ( 𝑠 ) 𝑑 𝑠 . ( 3 . 2 5 ) Since 𝑥 is unbounded, the first assertion follows.
Clearly, (3.24) holds for 𝐿 = 0 . Now, assume 𝐿 > 0 . Since 𝑥 [ 1 ] is decreasing for 𝑡 ≥ 𝑇 , we have 𝑥 ( 𝑡 ) ≥ 𝑡 𝑇 Φ ∗ 𝐿 𝑎 ( 𝑠 ) 𝑑 𝑠 . ( 3 . 2 6 ) Integrating (1.1), we obtain 𝑥 [ 1 ] ( 𝑡 ) ≥ 𝐿 + ∞ 𝑡 𝑏 ( 𝑠 ) 𝐹 𝑠 𝑇 Φ ∗ 𝐿 𝑎 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 2 7 ) If 𝐼 𝐿 < ∞ , from (1.2) inequality (3.24) follows. If 𝐼 𝐿 = ∞ , using l'Hopital rule, we have l i m 𝑡 → ∞ ∫ 𝑡 𝑇 Φ ∗ 𝐿 𝑎 − 1 ( 𝑠 ) 𝑑 𝑠 ∫ 𝑡 𝑡 0 Φ ∗ 𝐿 𝑎 − 1 ( 𝑠 ) 𝑑 𝑠 = l i m t → ∞ Φ ∗ 𝐿 𝑎 − 1 ( 𝑡 ) Φ ∗ 𝐿 𝑎 − 1 ( 𝑡 ) = 1 . ( 3 . 2 8 ) Thus we have for large 𝑠 , say 𝑠 ≥ 𝑇 , 𝑠 𝑇 Φ ∗ 𝐿 𝑎 ( 𝜏 ) 𝑑 𝜏 ≥ 𝜇 𝑠 𝑡 0 Φ ∗ 𝐿 𝑎 ( 𝜏 ) 𝑑 𝜏 . ( 3 . 2 9 ) From here, inequality (3.27) yields 𝑥 [ 1 ] ( 𝑡 ) ≥ ∞ 𝑡 𝜇 𝑏 ( 𝑠 ) 𝐹 𝑠 𝑡 0 Φ ∗ 𝐿 𝑎 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 . ( 3 . 3 0 ) Since 𝑥 [ 1 ] is bounded, (3.24) again follows.

The following example illustrates Theorem 3.1 and a possible discrepancy between equations with nonhomogeneous Φ -Laplacian and ones with Φ 𝑝 . More precisely, if 𝑥 , 𝑦 ∈ 𝕄 + ∞ , ℓ , then it may happen

l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) = 0 , ( 3 . 3 1 ) while, when Φ is the classical Φ -Laplacian, in view of the l'Hopital rule and the homogeneity property (1.4), the limit in (3.31) is finite and different from zero, that is, solutions in the class 𝕄 + ∞ , ℓ have the same growth at infinity when Φ = Φ 𝑝 .

Example 3.4. Consider the equation 𝑡 + 1 𝑡 Φ 𝐶 ( 𝑥 ) + 3 9 4 1 2 𝑡 2 √ 𝑡 2 𝐹 + 𝑡 ( 𝑥 ) = 0 , 𝑡 ≥ 1 , ( 3 . 3 2 ) where Φ 𝐶 is given by (1.6) and 𝐹 ( 𝑢 ) = | 𝑢 | 2 / 3 s g n 𝑢 . We have Λ = ( 0 , 1 ] and for 𝜆 ∈ Λ Φ ∗ 𝐶 𝜆 𝑡 = 𝑡 + 1 𝜆 𝑡 1 − 𝜆 2 𝑡 2 . + 2 𝑡 + 1 ( 3 . 3 3 ) Obviously, 𝐼 𝜆 = ∞ for any 𝜆 ∈ ( 0 , 1 ] . One can verify that 𝑦 ( 𝑡 ) = ( 2 / 3 ) 𝑡 3 / 2 is a solution of (3.32) in the class 𝕄 + ∞ , ℓ and l i m 𝑡 → ∞ 𝑦 [ 1 ] ( 𝑡 ) = 1 . Because 𝐾 𝜆 < ∞ for 𝜆 ∈ ( 0 , 1 ] , from Theorem 3.1, (3.32) has solutions 𝑥 in 𝕄 + ∞ , ℓ satisfying l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 𝐿 for any 𝐿 ∈ ( 0 , 1 ) . Since l i m 𝑡 → ∞ 𝑎 ( 𝑡 ) = 1 , we get l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = Φ ∗ ( 𝐿 ) > 0 and so (3.31) holds.

4. Bounded Solutions

Here we study the existence of bounded solutions of (1.1). The following holds.

Theorem 4.1. ( i 1 ) If there exists a solution 𝑥 of (1.1) in the class 𝕄 + ℓ , 0 such that l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = 𝐿 , then 𝐽 𝜇 < ∞ for any 𝜇 , 0 < 𝜇 < 𝐹 ( 𝐿 ) .
( i 2 ) If there exists a positive constant 𝜇 ∈ I m 𝐹 such that 𝐽 𝜇 < ∞ , then (1.1) has solutions satisfying l i m 𝑡 → ∞ 𝑥 ( 𝑡 ) = 𝐿 , l i m 𝑡 → ∞ 𝑥 [ 1 ] ( 𝑡 ) = 0 , ( 4 . 1 ) where 𝐿 = 𝐹 − 1 ( 𝜇 ) .

Proof. Claim ( i 1 ). Let 𝐿 𝜀 such that 𝐿 > 𝐿 𝜀 > 𝐹 − 1 ( 𝜇 ) . We can suppose, without loss of generality, 𝑥 ( 𝑡 ) > 𝐿 𝜀 , 𝑥 [ 1 ] ( 𝑡 ) > 0 for any 𝑡 ≥ 𝑇 ≥ 𝑡 0 . We have for 𝑡 ≥ 𝑇 𝑥 ( 𝑡 ) = Φ ∗ 1 𝑎 ( 𝑡 ) ∞ 𝑡 𝑏 ( 𝜏 ) 𝐹 ( 𝑥 ( 𝜏 ) ) 𝑑 𝜏 ( 4 . 2 ) or 𝐿 − 𝑥 ( 𝑡 ) ≥ ∞ 𝑡 Φ ∗ 1 𝑎 ( 𝑠 ) ∞ 𝑠 𝐿 𝑏 ( 𝜏 ) 𝐹 𝜀 ≥ 𝑑 𝜏 ∞ 𝑡 Φ ∗ 𝜇 𝑎 ( 𝑠 ) ∞ 𝑠 𝑏 ( 𝜏 ) 𝑑 𝜏 ( 4 . 3 ) which gives the assertion.
Claim ( i 2 ). The assertion follows by applying the Tychonov fixed point theorem to the operator 𝑇 given by 𝑇 ( 𝑢 ) ( 𝑡 ) = 𝐿 − ∞ 𝑡 Φ ∗ 1 𝑎 ( 𝑡 ) ∞ 𝑡 𝑏 ( 𝜏 ) 𝐹 ( 𝑢 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 ( 4 . 4 ) in the set Ω ⊂ 𝐶 [ 𝑡 1 , ∞ ) 𝑡 Ω = 𝑢 ∈ 𝐶 1 ∶ 1 , ∞ 2 , 𝐿 ≤ 𝑢 ( 𝑡 ) ≤ 𝐿 ( 4 . 5 ) where ∞ 𝑡 1 Φ ∗ 𝜇 𝑎 (