Journal of Applied Mathematics

Journal of Applied Mathematics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 814103 | https://doi.org/10.1155/2011/814103

Guizhen Zhi, Yunrui Guo, Yan Wang, Qihu Zhang, "Existence of Solutions for a Class of Variable Exponent Integrodifferential System Boundary Value Problems", Journal of Applied Mathematics, vol. 2011, Article ID 814103, 40 pages, 2011. https://doi.org/10.1155/2011/814103

Existence of Solutions for a Class of Variable Exponent Integrodifferential System Boundary Value Problems

Academic Editor: Alexandre Carvalho
Received29 May 2011
Revised25 Sep 2011
Accepted02 Oct 2011
Published22 Nov 2011

Abstract

This paper investigates the existence of solutions for a class of variable exponent integrodifferential system with multipoint and integral boundary value condition in half line. When the nonlinearity term 𝑓 satisfies sub-(π‘βˆ’βˆ’1) growth condition or general growth condition, we give the existence of solutions and nonnegative solutions via Leray-Schauder degree at nonresonance, respectively. Moreover, the existence of solutions for the problem at resonance has been discussed.

1. Introduction

In this paper, we consider the existence of solutions for the following variable exponent integrodifferential systemβˆ’Ξ”π‘(𝑑)𝑒+𝛿𝑓𝑑,𝑒,(𝑀(𝑑))1/(𝑝(𝑑)βˆ’1)π‘’ξ…žξ€Έ,𝑆(𝑒),𝑇(𝑒)=0,π‘‘βˆˆ(0,+∞),(1.1) with the following nonlinear multipoint and integral boundary value conditionξ€œπ‘’(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1,lim𝑑→+∞||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž(𝑑)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘€ξ€·πœ‰π‘–ξ€Έ||π‘’ξ…ž||𝑝(πœ‰π‘–)βˆ’2π‘’ξ…žξ€·πœ‰π‘–ξ€Έ+𝑒2,(1.2) where π‘’βˆΆ[0,+∞)→ℝ𝑁;𝑆 and 𝑇 are linear operators defined by ξ€œπ‘†(𝑒)(𝑑)=𝑑0ξ€œπœ“(𝑠,𝑑)𝑒(𝑠)𝑑𝑠,𝑇(𝑒)(𝑑)=0+βˆžπœ’(𝑠,𝑑)𝑒(𝑠)𝑑𝑠,(1.3) where βˆ«πœ“βˆˆπΆ(𝐷,ℝ),πœ’βˆˆπΆ(𝐷,ℝ),𝐷={(𝑠,𝑑)∈[0,+∞)Γ—[0,+∞)};0+∞|πœ“(𝑠,𝑑)|𝑑𝑠 and ∫0+∞|πœ’(𝑠,𝑑)|𝑑𝑠 are uniformly bounded with 𝑑;π‘βˆˆπΆ([0,+∞),ℝ),𝑝(𝑑)>1,lim𝑑→+βˆžπ‘(𝑑) exists and lim𝑑→+βˆžπ‘(𝑑)>1;βˆ’Ξ”π‘(𝑑)π‘’βˆΆ=βˆ’(𝑀(𝑑)|π‘’ξ…ž|𝑝(𝑑)βˆ’2π‘’ξ…ž)ξ…ž is called the weighted 𝑝(𝑑)-Laplacian; π‘€βˆˆπΆ([0,+∞),ℝ) satisfies 0<𝑀(𝑑), for all π‘‘βˆˆ(0,+∞), and (𝑀(𝑑))βˆ’1/(𝑝(𝑑)βˆ’1)∈𝐿1(0,+∞);0<πœ‰1<β‹―<πœ‰π‘šβˆ’2<+∞,𝛼𝑖β‰₯0,(𝑖=1,…,π‘šβˆ’2) and βˆ‘0β‰€π‘šβˆ’2𝑖=1𝛼𝑖≀1;π‘’βˆˆπΏ1(0,+∞) is nonnegative, β€‰βˆ«πœŽ=0+βˆžπ‘’(𝑑)𝑑𝑑 and 𝜎∈[0,1];𝑒1,𝑒2βˆˆβ„π‘;𝛿 is a positive parameter.

If βˆ‘π‘šβˆ’2𝑖=1𝛼𝑖<1 and 𝜎<1, we say the problem is nonresonant; but if βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1] and 𝜎=1, we say the problem is resonant.

The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic. Many results have been obtained on these problems, for example, [1–23]. We refer to [3, 19, 23], for the applied background on these problems. If 𝑀(𝑑)≑1 and 𝑝(𝑑)≑𝑝 (a constant), βˆ’Ξ”π‘(𝑑) becomes the well-known 𝑝-Laplacian. If 𝑝(𝑑) is a general function, βˆ’Ξ”π‘(𝑑) represents a nonhomogeneity and possesses more nonlinearity, and thus βˆ’Ξ”π‘(𝑑) is more complicated than βˆ’Ξ”π‘. For example, if Ξ©βŠ‚β„π‘› is a bounded domain, the Rayleigh quotientπœ†π‘(π‘₯)=infπ‘’βˆˆπ‘Š01,𝑝(π‘₯)(Ξ©)⧡{0}∫Ω||||(1/𝑝(π‘₯))βˆ‡π‘’π‘(π‘₯)𝑑π‘₯∫Ω(1/𝑝(π‘₯))|𝑒|𝑝(π‘₯)𝑑π‘₯(1.4) is zero in general, and only under some special conditions πœ†π‘(π‘₯)>0 (see [9, 16–18]), but the fact that πœ†π‘>0 is very important in the study of 𝑝-Laplacian problems.

Integral boundary conditions for evolution problems have been applied variously in chemical engineering, thermoelasticity, underground water flow, and population dynamics. There are many papers on the differential equations with integral boundary value conditions, for example, [24–29]. On the existence of solutions for 𝑝(π‘₯)-Laplacian systems boundary value problems, we refer to [2, 4, 7, 8, 10–12, 20–22]. In [20], the present author deals with the existence and asymptotic behavior of solutions for (1.1) with the following linear boundary value conditions𝑒(0)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘’ξ€·πœ‰π‘–ξ€Έ+𝑒0,lim𝑑→+βˆžξ€œπ‘’(𝑑)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑,(1.5) when βˆ‘0β‰€π‘šβˆ’2𝑖=1𝛼𝑖<1 and ∫0≀0+βˆžπ‘’(𝑑)≀1. But results on the existence of solutions for variable exponent integrodifferential systems with nonlinear boundary value conditions are rare. In this paper, when 𝑝(𝑑) is a general function, we investigate the existence of solutions and nonnegative solutions for variable exponent integrodifferential systems with nonlinear multipoint and integral boundary value conditions, when the problem is at nonresonance. Moreover, we discuss the existence of solutions for the problem at resonance. Since the nonlinear multipoint boundary value condition is on the derivative of solution 𝑒, we meet more difficulties than [20].

Let 𝑁β‰₯1 and 𝐽=[0,+∞), the function 𝑓=(𝑓1,…,𝑓𝑁)βˆΆπ½Γ—β„π‘Γ—β„π‘Γ—β„π‘Γ—β„π‘β†’β„π‘ is assumed to be Caratheodory, by this we mean,(i)for almost every π‘‘βˆˆπ½, the function 𝑓(𝑑,β‹…,β‹…,β‹…,β‹…) is continuous;(ii)for each (π‘₯,𝑦,𝑧,𝑀)βˆˆβ„π‘Γ—β„π‘Γ—β„π‘Γ—β„π‘, the function 𝑓(β‹…,π‘₯,𝑦,𝑧,𝑀) is measurable on 𝐽;(iii)for each 𝑅>0, there is a π›½π‘…βˆˆπΏ1(𝐽,ℝ) such that, for almost every π‘‘βˆˆπ½ and every (π‘₯,𝑦,𝑧,𝑀)βˆˆβ„π‘Γ—β„π‘Γ—β„π‘Γ—β„π‘ with |π‘₯|≀𝑅,|𝑦|≀𝑅,|𝑧|≀𝑅,|𝑀|≀𝑅, one has||||𝑓(𝑑,π‘₯,𝑦,𝑧,𝑀)≀𝛽𝑅(𝑑).(1.6)

Throughout the paper, we denote||𝑒𝑀(0)ξ…ž||𝑝(0)βˆ’2π‘’ξ…ž(0)=lim𝑑→0+||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž||𝑒(𝑑),𝑀(+∞)ξ…ž||𝑝(+∞)βˆ’2π‘’ξ…ž(+∞)=lim𝑑→+∞||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž(𝑑).(1.7)

The inner product in ℝ𝑁 will be denoted by βŸ¨β‹…,β‹…βŸ©,|β‹…| will denote the absolute value and the Euclidean norm on ℝ𝑁. Let 𝐴𝐢(0,+∞) denote the space of absolutely continuous functions on the interval (0,+∞). For 𝑁β‰₯1, we set 𝐢=𝐢(𝐽,ℝ𝑁),𝐢1={π‘’βˆˆπΆβˆ£π‘’ξ…žβˆˆπΆ((0,+∞),ℝ𝑁),lim𝑑→0+𝑀(𝑑)1/(𝑝(𝑑)βˆ’1)π‘’ξ…ž(𝑑)exists}. For any 𝑒(𝑑)=(𝑒1(𝑑),…,𝑒𝑁(𝑑))∈𝐢, we denote |𝑒𝑖|0=supπ‘‘βˆˆ(0,+∞)|𝑒𝑖(𝑑)|,‖𝑒‖0βˆ‘=(𝑁𝑖=1|𝑒𝑖|20)1/2, and ‖𝑒‖1=‖𝑒‖0+β€–(𝑀(𝑑))1/(𝑝(𝑑)βˆ’1)π‘’ξ…žβ€–0. Spaces 𝐢 and 𝐢1 will be equipped with the norm β€–β‹…β€–0 and β€–β‹…β€–1, respectively. Then, (𝐢,β€–β‹…β€–0) and (𝐢1,β€–β‹…β€–1) are Banach spaces. Denote 𝐿1=𝐿1(𝐽,ℝ𝑁) with the norm ‖𝑒‖𝐿1βˆ‘=[𝑁𝑖=1(∫∞0|𝑒𝑖|𝑑𝑑)2]1/2.

We say a function π‘’βˆΆπ½β†’β„π‘ is a solution of (1.1) if π‘’βˆˆπΆ1 with 𝑀(𝑑)|π‘’ξ…ž|𝑝(𝑑)βˆ’2π‘’ξ…ž absolutely continuous on (0,+∞), which satisfies (1.1) a.e. on 𝐽.

In this paper, we always use 𝐢𝑖 to denote positive constants, if it cannot lead to confusion. Denote π‘§βˆ’=infπ‘‘βˆˆπ½π‘§(𝑑),𝑧+=supπ‘‘βˆˆπ½π‘§(𝑑),foranyπ‘§βˆˆπΆ(𝐽,ℝ).(1.8)

We say 𝑓 satisfies sub-(π‘βˆ’βˆ’1) growth condition, if 𝑓 satisfieslim||𝑦|||π‘₯|++|𝑧|+|𝑀|β†’+βˆžπ‘“(𝑑,π‘₯,𝑦,𝑧,𝑀)ξ€·||𝑦||ξ€Έ|π‘₯|++|𝑧|+|𝑀|π‘ž(𝑑)βˆ’1=0,forπ‘‘βˆˆπ½uniformly,(1.9) where π‘ž(𝑑)∈𝐢(𝐽,ℝ) and 1<π‘žβˆ’β‰€π‘ž+<π‘βˆ’. We say 𝑓 satisfies general growth condition, if 𝑓 does not satisfy sub-(π‘βˆ’βˆ’1) growth condition.

We will discuss the existence of solutions of (1.1)-(1.2) in the following three cases.Case (i): βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎∈[0,1);Case (ii): βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎=1;Case (iii): βˆ‘π‘šβˆ’2𝑖=1𝛼𝑖=1,𝜎=1.

This paper is divided into five sections. In the second section, we present some preliminary and give the operator equations which have the same solutions of (1.1)-(1.2) in the three cases, respectively. In the third section, we will discuss the existence of solutions of (1.1)-(1.2) when βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎∈[0,1), and we give the existence of nonnegative solutions. In the fourth section, we will discuss the existence of solutions of (1.1)-(1.2) when βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎=1. In the fifth section, we will discuss the existence of solutions of (1.1)-(1.2) when βˆ‘π‘šβˆ’2𝑖=1𝛼𝑖=1,𝜎=1.

2. Preliminary

For any (𝑑,π‘₯)βˆˆπ½Γ—β„π‘, denote πœ‘(𝑑,π‘₯)=|π‘₯|𝑝(𝑑)βˆ’2π‘₯. Obviously, πœ‘ has the following properties.

Lemma 2.1 (see [7]). πœ‘ is a continuous function and satisfies the following.(i)For any π‘‘βˆˆ[0,+∞), β€‰πœ‘(𝑑,β‹…) is strictly monotone, that is ξ«πœ‘ξ€·π‘‘,π‘₯1ξ€Έξ€·βˆ’πœ‘π‘‘,π‘₯2ξ€Έ,π‘₯1βˆ’π‘₯2>0,foranyπ‘₯1,π‘₯2βˆˆβ„π‘,π‘₯1β‰ x2.(2.1)(ii)There exists a function π›½βˆΆ[0,+∞)β†’[0,+∞),𝛽(𝑠)β†’+∞ as 𝑠→+∞, such that βŸ¨πœ‘(𝑑,π‘₯),π‘₯⟩β‰₯𝛽(|π‘₯|)|π‘₯|,βˆ€π‘₯βˆˆβ„π‘.(2.2)

It is well known that πœ‘(𝑑,β‹…) is a homeomorphism from ℝ𝑁 to ℝ𝑁 for any fixed π‘‘βˆˆ[0,+∞). For any π‘‘βˆˆπ½, denote by πœ‘βˆ’1(𝑑,β‹…) the inverse operator of πœ‘(𝑑,β‹…), then πœ‘βˆ’1(𝑑,π‘₯)=|π‘₯|(2βˆ’π‘(𝑑))/(𝑝(𝑑)βˆ’1)π‘₯,forπ‘₯βˆˆβ„π‘β§΅{0},πœ‘βˆ’1(𝑑,0)=0.(2.3)

It is clear that πœ‘βˆ’1(𝑑,β‹…) is continuous and sends bounded sets into bounded sets.

Let us now consider the following problem with boundary value condition  (1.2)𝑀(𝑑)πœ‘π‘‘,π‘’ξ…ž(𝑑)ξ€Έξ€Έξ…ž=𝑔(𝑑),π‘‘βˆˆ(0,+∞),(2.4) where π‘”βˆˆπΏ1.

If 𝑒 is a solution of (2.4) with (1.2), by integrating (2.4) from 0 to 𝑑, we find that 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…ž(𝑑)=𝑀(0)πœ‘0,π‘’ξ…ž(ξ€Έ+ξ€œ0)𝑑0𝑔(𝑠)𝑑𝑠.(2.5)

Define operator 𝐹∢𝐿1→𝐢 as ξ€œπΉ(𝑔)(𝑑)=𝑑0𝑔(𝑠)𝑑𝑠,βˆ€π‘‘βˆˆπ½,βˆ€π‘”βˆˆπΏ1.(2.6)

By solving for π‘’ξ…ž in (2.5) and integrating, we find thatξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1𝑀(0)πœ‘0,π‘’ξ…žξ€Έ(0)+𝐹(𝑔)ξ€Έξ€»ξ€Ύ(𝑑),π‘‘βˆˆπ½.(2.7)

In the following, we will give the operator equations which have the same solutions of (1.1)-(1.2) in three cases, respectively.

2.1. Case (i): βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎∈[0,1)

We denote 𝜌=𝑀(0)πœ‘(0,π‘’ξ…ž(0)) in (2.7). It is easy to see that 𝜌 is dependent on 𝑔(β‹…), then we find that ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1(𝜌+𝐹(𝑔))ξ€»ξ€Ύ(𝑑),π‘‘βˆˆπ½.(2.8)

The boundary value condition (1.2) implies that βˆ«π‘’(0)=0+βˆžξ‚†βˆ«π‘’(𝑑)𝑑0πœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1(𝜌+𝐹(𝑔)(π‘Ÿ))π‘‘π‘Ÿπ‘‘π‘‘+𝑒1βˆ’βˆ«1βˆ’πœŽ0+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1(ξ€»πœŒ+𝐹(𝑔)(π‘Ÿ))π‘‘π‘Ÿ,11βˆ’πœŽπœŒ=βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξƒ©π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0ξ€œπ‘”(𝑠)π‘‘π‘ βˆ’0+βˆžπ‘”(𝑠)𝑑𝑠+𝑒2ξƒͺ.(2.9)

For fixed β„ŽβˆˆπΏ1, we define 𝜌∢𝐿1→ℝ𝑁 as1𝜌(β„Ž)=βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξƒ©π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0ξ€œβ„Ž(𝑑)π‘‘π‘‘βˆ’0+βˆžβ„Ž(𝑑)𝑑𝑑+𝑒2ξƒͺ.(2.10)

Lemma 2.2. 𝜌∢𝐿1→ℝ𝑁 is continuous and sends bounded sets of 𝐿1 to bounded sets of ℝ𝑁. Moreover, ||||β‰€πœŒ(β„Ž)2π‘βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–β‹…ξ€·β€–β„Žβ€–πΏ1+||𝑒2||ξ€Έ.(2.11)

Proof. Since 𝜌(β‹…) consists of continuous operators, it is continuous. It is easy to see that ||||β‰€πœŒ(β„Ž)2π‘βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–β‹…ξ€·β€–β„Žβ€–πΏ1+||𝑒2||ξ€Έ.(2.12)
This completes the proof.

It is clear that 𝜌(β‹…) is continuous and sends bounded sets of 𝐿1 to bounded sets of ℝ𝑁, and hence it is a compact continuous mapping.

If 𝑒 is a solution of (2.4) with (1.2), we find that ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1∫(𝜌+𝐹(𝑔))ξ€»ξ€Ύ(𝑑),π‘‘βˆˆπ½,𝑒(0)=0+βˆžξ‚†βˆ«π‘’(𝑑)𝑑0πœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1(𝜌+𝐹(𝑔)(π‘Ÿ))π‘‘π‘Ÿπ‘‘π‘‘+𝑒1βˆ’βˆ«1βˆ’πœŽ0+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€»(𝜌+𝐹(𝑔)(π‘Ÿ))π‘‘π‘Ÿ.1βˆ’πœŽ(2.13)

We denote ξ€½πœ‘πΎ(β„Ž)(𝑑)∢=(πΎβˆ˜β„Ž)(𝑑)=πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1(𝜌(β„Ž)+𝐹(β„Ž))ξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ(0,+∞).(2.14)

We say a set π‘ˆβŠ‚πΏ1 be equi-integrable, if there exists a nonnegative πœŒβˆ—βˆˆπΏ1(𝐽,ℝ), such that ||||β„Ž(𝑑)β‰€πœŒβˆ—(𝑑)a.e.in𝐽,foranyβ„Žβˆˆπ‘ˆ.(2.15)

Lemma 2.3. The operator 𝐾 is continuous and sends equi-integrable sets in 𝐿1 to relatively compact sets in 𝐢1.

Proof. It is easy to check that 𝐾(β„Ž)(𝑑)∈𝐢1, for all β„ŽβˆˆπΏ1. Since (𝑀(𝑑))βˆ’1/(𝑝(𝑑)βˆ’1)∈𝐿1 and 𝐾(β„Ž)ξ…ž(𝑑)=πœ‘βˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€»[(𝜌(β„Ž)+𝐹(β„Ž)),βˆ€π‘‘βˆˆ0,+∞),(2.16) it is easy to check that 𝐾 is a continuous operator from 𝐿1 to 𝐢1.
Let now π‘ˆ be an equi-integrable set in 𝐿1, then there exists a nonnegative πœŒβˆ—βˆˆπΏ1(𝐽,ℝ), such that ||||β„Ž(𝑑)β‰€πœŒβˆ—(𝑑)a.e.in𝐽,foranyβ„Žβˆˆπ‘ˆ.(2.17)
We want to show that 𝐾(π‘ˆ)βŠ‚πΆ1 is a compact set.
Let {𝑒𝑛} be a sequence in 𝐾(π‘ˆ), then there exists a sequence {β„Žπ‘›}βŠ‚π‘ˆ such that 𝑒𝑛=𝐾(β„Žπ‘›). Since β„Žπ‘›(𝑑)=(β„Ž1𝑛(𝑑),…,β„Žπ‘π‘›(𝑑)), where β„Žπ‘–π‘›(𝑑)∈𝐿1(𝐽,ℝ)(𝑖=1,…,𝑁), we have ||β„Žπ‘–π‘›||≀||β„Ž(𝑑)𝑛||(𝑑),βˆ€π‘–=1,…,𝑁,(2.18) then for any 𝑑1,𝑑2∈𝐽 with 𝑑1<𝑑2, we have ||||ξ€œπ‘‘2𝑑1β„Žπ‘›||||=ξ„Άξ„΅ξ„΅βŽ·(𝑑)𝑑𝑑𝑁𝑖=1ξ‚΅ξ€œπ‘‘2𝑑1β„Žπ‘–π‘›ξ‚Ά(𝑑)𝑑𝑑2β‰€ξ„Άξ„΅ξ„΅βŽ·π‘ξ“π‘–=1ξ‚΅ξ€œπ‘‘2𝑑1||β„Žπ‘–π‘›||ξ‚Ά(𝑑)𝑑𝑑2β‰€ξ„Άξ„΅ξ„΅βŽ·π‘ξ“π‘–=1ξ‚΅ξ€œπ‘‘2𝑑1||β„Žπ‘›||ξ‚Ά(𝑑)𝑑𝑑2ξ€œβ‰€π‘π‘‘2𝑑1||β„Žπ‘›||(𝑑)𝑑𝑑,(2.19) which together with (2.17) implies ||πΉξ€·β„Žπ‘›π‘‘ξ€Έξ€·1ξ€Έξ€·β„Žβˆ’πΉπ‘›π‘‘ξ€Έξ€·2ξ€Έ||=||||ξ€œπ‘‘10β„Žπ‘›ξ€œ(𝑑)π‘‘π‘‘βˆ’π‘‘20β„Žπ‘›||||=||||ξ€œ(𝑑)𝑑𝑑𝑑2𝑑1β„Žπ‘›(||||ξ€œπ‘‘)𝑑𝑑≀𝑁𝑑2𝑑1||β„Žπ‘›(||ξ€œπ‘‘)𝑑𝑑≀𝑁𝑑2𝑑1πœŒβˆ—(𝑑)𝑑𝑑.(2.20)
Hence, the sequence {𝐹(β„Žπ‘›)} is uniformly bounded. According to the absolute continuity of Lebesgue integral, for any πœ€>0, there exists a 𝛿>0 such that if 0≀𝑑1βˆ’π‘‘2<𝛿, then we have ∫0≀𝑁𝑑2𝑑1πœŒβˆ—(𝑑)𝑑𝑑<πœ€. Thus, (2.20) means that {𝐹(β„Žπ‘›)} is equicontinuous.
Denote Ξ©π‘š=[0,π‘š]. Obviously, {𝐹(β„Žπ‘›)} is uniformly bounded and equicontinuous on Ξ©π‘š for π‘š=1,2,…. By Ascoli-Arzela Theorem, there exists a subsequence {𝐹(β„Žπ‘›(1))} of {𝐹(β„Žπ‘›)} being convergent in 𝐢(Ξ©1), we may assume 𝐹(β„Žπ‘›(1))→𝑣1(β‹…) in 𝐢(Ξ©1). Since {𝐹(β„Žπ‘›(1))} is uniformly bounded and equicontinuous on Ξ©2, there exists a subsequence {𝐹(β„Žπ‘›(2))} of {𝐹(β„Žπ‘›(1))} such that {𝐹(β„Žπ‘›(2))} is convergent in 𝐢(Ξ©2), we may assume 𝐹(β„Žπ‘›(2))→𝑣2(β‹…) in 𝐢(Ξ©2). Obviously, 𝑣2(𝑑)=𝑣1(𝑑), for any π‘‘βˆˆΞ©1. Repeating the process, we get a subsequence {𝐹(β„Žπ‘›(π‘š+1))} of {𝐹(β„Žπ‘›(π‘š))} such that {𝐹(β„Žπ‘›(π‘š+1))} is convergent in 𝐢(Ξ©π‘š+1), we may assume 𝐹(β„Žπ‘›(π‘š+1))β†’π‘£π‘š+1(β‹…) in 𝐢(Ξ©π‘š+1). Obviously, π‘£π‘š+1(𝑑)=π‘£π‘š(𝑑) for any π‘‘βˆˆΞ©π‘š. Select the diagonal element, we can see that {𝐹(β„Žπ‘›(𝑛))} is a subsequence of {𝐹(β„Žπ‘›)} which satisfies that {𝐹(β„Žπ‘›(𝑛))} is convergent in 𝐢(Ξ©π‘š)(π‘š=1,2,…) and 𝐹(β„Žπ‘›(𝑛))β†’π‘£π‘š(β‹…) in 𝐢(Ξ©π‘š)(π‘š=1,2,…). Thus, we get a function 𝑣 which is defined on [0,+∞) such that 𝑣(𝑑)=π‘£π‘š(𝑑) for any π‘‘βˆˆΞ©π‘š, and 𝐹(β„Žπ‘›(𝑛))→𝑣(β‹…) in 𝐢(Ξ©π‘š)(π‘š=1,2,…).
From (2.20), it is easy to see that for any 𝑛=1,2,…, limπ‘‘β†’βˆžπΉ(β„Žπ‘›)(𝑑) exists (we denote the limit by 𝐹(β„Žπ‘›(𝑛))(+∞)), and, for any πœ€>0, there exists an integer π‘…πœ€>0 such that βˆ«π‘…+βˆžπœ€πœŒβˆ—(𝑑)𝑑𝑑<πœ€/𝑁, and then |||πΉξ‚€β„Žπ‘›(𝑛)ξ‚ξ€·β„Ž(+∞)βˆ’πΉπ‘›ξ€Έ|||ξ€œ(𝑑)≀𝑁𝑅+βˆžπœ€πœŒβˆ—(𝑑)𝑑𝑑<πœ€,βˆ€π‘‘β‰₯π‘…πœ€,βˆ€π‘›=1,2,….(2.21)
Since {𝐹(β„Žπ‘›)} is uniformly bounded, then {𝐹(β„Žπ‘›(𝑛))(+∞)} is bounded. By choosing a subsequence, we may assume that limπ‘›β†’βˆžπΉξ‚€β„Žπ‘›(𝑛)(+∞)=𝑏.(2.22)
We claim that limπ‘‘β†’βˆžπ‘£(𝑑)=𝑏. In fact, for any 𝑑β‰₯π‘…πœ€, from (2.21), we have ||||≀|||ξ‚€β„Žπ‘£(𝑑)βˆ’π‘π‘£(𝑑)βˆ’πΉπ‘›(𝑛)|||+|||πΉξ‚€β„Ž(𝑑)𝑛(𝑛)ξ‚ξ‚€β„Ž(𝑑)βˆ’πΉπ‘›(𝑛)|||+|||πΉξ‚€β„Ž(+∞)𝑛(𝑛)|||≀|||ξ‚€β„Ž(+∞)βˆ’π‘π‘£(𝑑)βˆ’πΉπ‘›(𝑛)||||||πΉξ‚€β„Ž(𝑑)+πœ€+𝑛(𝑛)|||.(+∞)βˆ’π‘(2.23)
Since limπ‘›β†’βˆžπΉ(β„Žπ‘›(𝑛))(𝑑)=𝑣(𝑑) and limπ‘›β†’βˆžπΉ(β„Žπ‘›(𝑛))(+∞)=𝑏, letting π‘›β†’βˆž, the above inequality implies ||||𝑣(𝑑)βˆ’π‘β‰€πœ€,βˆ€π‘‘β‰₯π‘…πœ€.(2.24)
Thus, limπ‘‘β†’βˆžπ‘£(𝑑)=𝑏=limπ‘›β†’βˆžπΉξ‚€β„Žπ‘›(𝑛)(+∞).(2.25)
Next, we will prove that 𝐹(β„Žπ‘›(𝑛)) tend to 𝑣 uniformly.
Suppose 𝑑β‰₯π‘…πœ€. From (2.21) and (2.24), we have |||πΉξ‚€β„Žπ‘›(𝑛)|||≀|||πΉξ‚€β„Ž(𝑑)βˆ’π‘£(𝑑)𝑛(𝑛)ξ‚ξ‚€β„Ž(𝑑)βˆ’πΉπ‘›(𝑛)|||+|||πΉξ‚€β„Ž(+∞)𝑛(𝑛)|||+|||||||πΉξ‚€β„Ž(+∞)βˆ’π‘π‘βˆ’π‘£(𝑑)β‰€πœ€+𝑛(𝑛)||||||πΉξ‚€β„Ž(+∞)βˆ’π‘+πœ€=2πœ€+𝑛(𝑛)|||.(+∞)βˆ’π‘(2.26)
From (2.22), there exists a 𝑁1>0 such that |𝐹(β„Žπ‘›(𝑛))(+∞)βˆ’π‘|β‰€πœ€ for 𝑛β‰₯𝑁1. Thus, for any 𝑑β‰₯π‘…πœ€, |||πΉξ‚€β„Žπ‘›(𝑛)|||(𝑑)βˆ’π‘£(𝑑)≀3πœ€,βˆ€π‘›β‰₯𝑁1.(2.27)
Suppose π‘‘βˆˆ[0,π‘…πœ€]. Since 𝐹(β„Žπ‘›(𝑛))→𝑣 in 𝐢([0,π‘…πœ€]), there exists a 𝑁2>0 such that |||πΉξ‚€β„Žπ‘›(𝑛)|||(𝑑)βˆ’π‘£(𝑑)β‰€πœ€,βˆ€π‘›β‰₯𝑁2.(2.28)
Thus, |||πΉξ‚€β„Žπ‘›(𝑛)|||[(𝑑)βˆ’π‘£(𝑑)≀3πœ€,βˆ€π‘‘βˆˆ0,+∞),βˆ€π‘›β‰₯𝑁1+𝑁2.(2.29)
This means that 𝐹(β„Žπ‘›(𝑛)) tend to 𝑣 uniformly, that is, 𝐹(β„Žπ‘›(𝑛)) tend to 𝑣 in 𝐢.
According to the bounded continuous of the operator 𝜌, we can choose a subsequence of {𝜌(β„Žπ‘›)+𝐹(β„Žπ‘›)} (which we still denote {𝜌(β„Žπ‘›)+𝐹(β„Žπ‘›)}) which is convergent in 𝐢, then 𝑀(𝑑)πœ‘(𝑑,𝐾(β„Žπ‘›)ξ…ž(𝑑))=𝜌(β„Žπ‘›)+𝐹(β„Žπ‘›) is convergent in 𝐢.
Since πΎξ€·β„Žπ‘›ξ€Έξ€½πœ‘(𝑑)=πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒξ€·β„Žπ‘›ξ€Έξ€·β„Ž+𝐹𝑛[ξ€Έξ€Έξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ0,+∞),(2.30) it follows from the continuity of πœ‘βˆ’1 and the integrability of 𝑀(𝑑)βˆ’1/(𝑝(𝑑)βˆ’1) in 𝐿1 that 𝐾(β„Žπ‘›) is convergent in 𝐢. Thus, {𝑒𝑛} is convergent in 𝐢1. This completes the proof.

Let us define π‘ƒβˆΆπΆ1→𝐢1 as𝑃1(β„Ž)=ξ‚΅ξ€œ1βˆ’πœŽ0+βˆžπ‘’(𝑑)𝐾(β„Ž)(𝑑)π‘‘π‘‘βˆ’πΎ(β„Ž)(+∞)+𝑒1ξ‚Ά.(2.31) It is easy to see that 𝑃 is compact continuous.

Throughout the paper, we denote 𝑁𝑓(𝑒)∢[0,+∞)×𝐢1→𝐿1 the Nemytskii operator associated to 𝑓 defined by 𝑁𝑓(𝑒)(𝑑)=𝑓𝑑,𝑒(𝑑),(𝑀(𝑑))1/(𝑝(𝑑)βˆ’1)π‘’ξ…žξ€Έ(𝑑),𝑆(𝑒)(𝑑),𝑇(𝑒)(𝑑),a.e.on𝐽.(2.32)

Lemma 2.4. In the Case (i), 𝑒 is a solution of (1.1)-(1.2) if and only if 𝑒 is a solution of the following abstract equation: 𝑒=𝑃𝛿𝑁𝑓(𝑒)+𝐾𝛿𝑁𝑓(𝑒).(2.33)

Proof. If 𝑒 is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to 𝑑, we find that 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έξ€·(𝑑)=πœŒπ›Ώπ‘π‘“ξ€Έξ€·(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(𝑑),βˆ€π‘‘βˆˆ(0,+∞),(2.34) which implies that ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒξ€·π›Ώπ‘π‘“ξ€Έξ€·(𝑒)+𝐹𝛿𝑁𝑓[(𝑒)(𝑑)ξ€Έξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ0,+∞).(2.35)
From βˆ«π‘’(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1, we have βˆ«π‘’(0)=0+βˆžξ‚†βˆ«π‘’(𝑑)𝑑0πœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·ξ€·πœŒ+𝐹𝛿𝑁𝑓(𝑒)ξ€Έξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘+𝑒1βˆ’βˆ«1βˆ’πœŽ0+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·ξ€·πœŒ+𝐹𝛿𝑁𝑓(𝑒)ξ€Έξ€Έξ€»π‘‘π‘Ÿξ€·1βˆ’πœŽ=𝑃𝛿𝑁𝑓.(𝑒)(2.36)
So we have 𝑒=𝑃𝛿𝑁𝑓(𝑒)+𝐾𝛿𝑁𝑓(𝑒).(2.37)
Conversely, if 𝑒 is a solution of (2.33), we have 𝑒(0)=𝑃𝛿𝑁𝑓(𝑒)+𝐾𝛿𝑁𝑓(𝑒)(0)=𝑃𝛿𝑁𝑓=1(𝑒)ξ‚΅ξ€œ1βˆ’πœŽ0+βˆžπ‘’(𝑑)𝐾(β„Ž)(𝑑)π‘‘π‘‘βˆ’πΎ(β„Ž)(+∞)+𝑒1ξ‚Ά,(2.38) which implies that ξ€œ(1βˆ’πœŽ)𝑒(0)+𝐾(β„Ž)(+∞)=0+βˆžπ‘’(𝑑)(𝑒(𝑑)βˆ’π‘’(0))𝑑𝑑+𝑒1,(2.39) then π‘’ξ€œ(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1.(2.40)
From (2.33), we can obtain 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έξ€·(𝑑)=πœŒπ›Ώπ‘π‘“ξ€Έξ€·(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(𝑑).(2.41)
It follows from the condition of the mapping 𝜌 that 𝑀(+∞)πœ‘+∞,π‘’ξ…žξ€Έξ€·(+∞)=πœŒπ›Ώπ‘π‘“ξ€Έξ€·(𝑒)+𝐹𝛿𝑁𝑓=1(𝑒)(+∞)βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξƒ©π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0π›Ώπ‘π‘“ξ€œ(𝑒)(𝑑)π‘‘π‘‘βˆ’0+βˆžπ›Ώπ‘π‘“(𝑒)(𝑑)𝑑𝑑+𝑒2ξƒͺ+ξ€œ0+βˆžπ›Ώπ‘π‘“(𝑒)(𝑑)𝑑𝑑,(2.42) and then 1βˆ’π‘šβˆ’2𝑖=1𝛼𝑖ξƒͺ𝑀(+∞)πœ‘+∞,π‘’ξ…žξ€Έ=(+∞)π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0π›Ώπ‘π‘“ξ€œ(𝑒)(𝑑)π‘‘π‘‘βˆ’0+βˆžπ›Ώπ‘π‘“(𝑒)(𝑑)𝑑𝑑+𝑒2+1βˆ’π‘šβˆ’2𝑖=1𝛼𝑖ξƒͺξ€œ0+βˆžπ›Ώπ‘π‘“=(𝑒)(𝑑)π‘‘π‘‘π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0𝛿𝑁𝑓(𝑒)(𝑑)π‘‘π‘‘βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œ0+βˆžπ›Ώπ‘π‘“(𝑒)(𝑑)𝑑𝑑+𝑒2=π‘šβˆ’2𝑖=1π›Όπ‘–π‘€ξ€·πœ‰π‘–ξ€Έπœ‘ξ€·πœ‰π‘–,π‘’ξ…žξ€·πœ‰π‘–βˆ’ξ€Έξ€Έπ‘šβˆ’2𝑖=1𝛼𝑖𝑀(+∞)πœ‘+∞,π‘’ξ…ž(ξ€Έ+∞)+𝑒2,(2.43) thus lim𝑑→+∞||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž(𝑑)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘€ξ€·πœ‰π‘–ξ€Έ||π‘’ξ…ž||𝑝(πœ‰π‘–)βˆ’2π‘’ξ…žξ€·πœ‰π‘–ξ€Έ+𝑒2.(2.44)
From (2.40) and (2.44), we obtain (1.2).
From (2.41), we have 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έξ€Έξ…ž=𝛿𝑁𝑓(𝑒)(𝑑).(2.45)
Hence, 𝑒 is a solution of (1.1)-(1.2). This completes the proof.

2.2. Case (ii): βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎=1

We denote 𝜌1=𝑀(0)πœ‘(0,π‘’ξ…ž(0)) in (2.7). It is easy to see that 𝜌1 is dependent on 𝑔(β‹…), and we have ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒ1+𝐹(𝑔)ξ€Έξ€»ξ€Ύ(𝑑),π‘‘βˆˆπ½.(2.46)

The boundary value condition (1.2) implies that ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ1ξ‚Ό+𝐹(𝑔)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1𝜌=0,1=1βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξƒ©π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0ξ€œπ‘”(𝑑)π‘‘π‘‘βˆ’0+βˆžπ‘”(𝑑)𝑑𝑑+𝑒2ξƒͺ.(2.47)

For fixed β„ŽβˆˆπΏ1, we define 𝜌1∢𝐿1→ℝ𝑁 as𝜌11(β„Ž)=βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–ξƒ©π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰π‘–0ξ€œβ„Ž(𝑑)π‘‘π‘‘βˆ’0+βˆžβ„Ž(𝑑)𝑑𝑑+𝑒2ξƒͺ.(2.48)

Similar to Lemma 2.2, we have the following.

Lemma 2.5. 𝜌1∢𝐿1→ℝ𝑁 is continuous and sends bounded sets of 𝐿1 to bounded sets of ℝ𝑁. Moreover, ||𝜌1||≀(β„Ž)2π‘βˆ‘1βˆ’π‘šβˆ’2𝑖=1π›Όπ‘–β‹…ξ€·β€–β„Žβ€–πΏ1+||𝑒2||ξ€Έ.(2.49)

It is clear that 𝜌1(β‹…) is continuous and sends bounded sets of 𝐿1 to bounded sets of ℝ𝑁, and, hence, it is a compact continuous mapping.

Let us define𝑃1∢𝐢1⟢𝐢1,π‘’βŸΌπ‘’(0),Θ∢𝐿1βŸΆβ„π‘ξ€œ,β„ŽβŸΌ0+βˆžξ‚»ξ€œπ‘’(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ1(ξ‚Όβ„Ž)+𝐹(β„Ž)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1,𝐾1𝐾(β„Ž)(𝑑)∢=1ξ€Έξ€½πœ‘βˆ˜β„Ž(𝑑)=πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒ1[(β„Ž)+𝐹(β„Ž)ξ€Έξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ0,+∞).(2.50)

Lemma 2.6. The operator 𝐾1 is continuous and sends equi-integrable sets in 𝐿1 to relatively compact sets in 𝐢1.

Proof. Similar to the proof of Lemma 2.3, we omit it here.

Lemma 2.7. In Case (ii), 𝑒 is a solution of (1.1)-(1.2) if and only if 𝑒 is a solution of the following abstract equation: 𝑒=𝑃1𝑒+Ξ˜π›Ώπ‘π‘“ξ€Έ(𝑒)+𝐾1𝛿𝑁𝑓.(𝑒)(2.51)

Proof. If 𝑒 is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to 𝑑, we find that 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έ(𝑑)=𝜌1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(𝑑),βˆ€π‘‘βˆˆ(0,+∞),(2.52) which implies that ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒ1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓[(𝑒)(𝑑)ξ€Έξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ0,+∞).(2.53)
From 𝜎=1 and βˆ«π‘’(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1, we obtain ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(𝑑)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1ξ€·=Ξ˜π›Ώπ‘π‘“ξ€Έ(𝑒)=0,(2.54) then 𝑒=𝑃1𝑒+Ξ˜π›Ώπ‘π‘“ξ€Έ(𝑒)+𝐾1𝛿𝑁𝑓.(𝑒)(2.55)
Conversely, if 𝑒 is a solution of (2.51), then 𝑒(0)=𝑃1𝑒+Ξ˜π›Ώπ‘π‘“ξ€Έ(𝑒)+𝐾1𝛿𝑁𝑓(𝑒)(0)=𝑒(0)+Ξ˜π›Ώπ‘π‘“ξ€Έ.(𝑒)(2.56)
Thus, Θ(𝛿𝑁𝑓(𝑒))=0, and we have ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)0+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓=ξ€œ(𝑒)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑0πœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘+𝑒1,(2.57) then ξ€œ0+βˆžξ€œ{𝑒(𝑑)(𝑒(+∞)βˆ’π‘’(0))}𝑑𝑑=0+∞{𝑒(𝑑)(𝑒(𝑑)βˆ’π‘’(0))}𝑑𝑑+𝑒1.(2.58)
Thus, π‘’ξ€œ(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1.(2.59)
Similar to the proof of Lemma 2.4, we can have lim𝑑→+∞||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž(𝑑)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘€ξ€·πœ‰π‘–ξ€Έ||π‘’ξ…ž||𝑝(πœ‰π‘–)βˆ’2π‘’ξ…žξ€·πœ‰π‘–ξ€Έ+𝑒2.(2.60)
From (2.59) and (2.60), we obtain (1.2).
From (2.51), we have 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έ(𝑑)=𝜌1𝛿𝑁𝑓(𝑒)+𝐹𝛿𝑁𝑓(𝑒)(𝑑),(2.61) then 𝑀(𝑑)πœ‘π‘‘,π‘’ξ…žξ€Έξ€Έξ…ž=𝛿𝑁𝑓(𝑒)(𝑑).(2.62)
Hence, 𝑒 is a solution of (1.1)-(1.2). This completes the proof.

2.3. Case (iii): βˆ‘π‘šβˆ’2𝑖=1𝛼𝑖=1,𝜎=1

We denote 𝜌2=𝑀(0)πœ‘(0,π‘’ξ…ž(0)) in (2.7). It is easy to see that 𝜌2 is dependent on 𝑔(β‹…), then we find that ξ€½πœ‘π‘’(𝑑)=𝑒(0)+πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒ2+𝐹(𝑔)ξ€Έξ€»ξ€Ύ(𝑑),π‘‘βˆˆπ½.(2.63)

The boundary value condition (1.2) implies that ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ2ξ‚Ό+𝐹(𝑔)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1=0,π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰+βˆžπ‘–π‘”(𝑑)π‘‘π‘‘βˆ’π‘’2=0.(2.64)

For fixed β„ŽβˆˆπΏ1, we denote Ξ›β„Žξ€·πœŒ2ξ€Έ=ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ2ξ‚Ό+𝐹(β„Ž)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1.(2.65)

Throughout the paper, we denote 𝐸#=ξ€œ0+βˆžπ‘’ξ€œ(𝑑)𝑑+∞(𝑀(π‘Ÿ))βˆ’1/(𝑝(π‘Ÿ)βˆ’1)π‘‘π‘Ÿπ‘‘π‘‘.(2.66)

Lemma 2.8. The function Ξ›β„Ž(β‹…) has the following properties.(i)For any fixed β„ŽβˆˆπΏ1, the equation Ξ›β„Žξ€·πœŒ2ξ€Έ=0(2.67)has a unique solution 𝜌2(β„Ž)βˆˆβ„π‘.(ii)The function 𝜌2∢𝐿1→ℝ𝑁, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, ||𝜌2||𝐸(β„Ž)≀3𝑁#+1𝐸#𝑝+ξ‚ƒβ€–β„Žβ€–0||𝑒+2𝑁1||𝑝#βˆ’1ξ‚„,(2.68)where the notation 𝑀𝑝#βˆ’1 means 𝑀𝑝#βˆ’1=𝑀𝑝+βˆ’1M,𝑀>1,π‘βˆ’βˆ’1,𝑀≀1.(2.69)

Proof. (i) From Lemma 2.1, it is immediate that βŸ¨Ξ›β„Ž(π‘₯)βˆ’Ξ›β„Ž(𝑦),π‘₯βˆ’π‘¦βŸ©>0,forπ‘₯≠𝑦,(2.70) and, hence, if (2.67) has a solution, then it is unique.
Let 𝑑0=3𝑁((𝐸#+1)/𝐸#)𝑝+[β€–β„Žβ€–0+2𝑁|𝑒1|𝑝#βˆ’1]. Since(𝑀(𝑑))βˆ’1/(𝑝(𝑑)βˆ’1)∈𝐿1(0,+∞) and β„ŽβˆˆπΏ1, if |𝜌2|>𝑑0, it is easy to see that there exists an π‘–βˆˆ{1,…,𝑁} such that the 𝑖th component πœŒπ‘–2 of 𝜌2 satisfies |πœŒπ‘–2|β‰₯|𝜌2|/𝑁>3((𝐸#+1)/𝐸#)𝑝+[β€–β„Žβ€–0+2𝑁|𝑒1|𝑝#βˆ’1]. Thus, (πœŒπ‘–2+β„Žπ‘–(𝑑)) keeps sign on 𝐽 and ||πœŒπ‘–2+β„Žπ‘–||β‰₯||𝜌(𝑑)𝑖2||βˆ’β€–β„Žβ€–0β‰₯2||𝜌2||𝐸3𝑁>2#+1𝐸#𝑝+ξ‚ƒβ€–β„Žβ€–0||𝑒+2𝑁1||𝑝#βˆ’1ξ‚„,βˆ€π‘‘βˆˆπ½.(2.71)
Obviously, |𝜌2+β„Ž(𝑑)|≀4|𝜌2|/3≀2𝑁|πœŒπ‘–2+β„Žπ‘–(𝑑)|, then ||𝜌2||+β„Ž(𝑑)(2βˆ’π‘(𝑑))/(𝑝(𝑑)βˆ’1)||πœŒπ‘–2+β„Žπ‘–||>1(𝑑)||𝜌2𝑁𝑖2+β„Žπ‘–||(𝑑)1/(𝑝(𝑑)βˆ’1)>𝐸#+12𝑁𝐸#||𝑒1||,βˆ€π‘‘βˆˆπ½.(2.72)
Thus, the 𝑖th component Ξ›π‘–β„Ž(𝜌2) of Ξ›β„Ž(𝜌2) is nonzero and keeps sign, and then we have ξ€œ0+βˆžξ‚»π‘’ξ€œ(𝑑)𝑑+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€·πœŒ2ξ‚Ό+𝐹(β„Ž)(π‘Ÿ)ξ€Έξ€»π‘‘π‘Ÿπ‘‘π‘‘βˆ’π‘’1β‰ 0.(2.73)
Let us consider the equation πœ†Ξ›β„Žξ€·πœŒ2ξ€Έ+(1βˆ’πœ†)𝜌2[]=0,πœ†βˆˆ0,1.(2.74)
It is easy to see that all the solutions of (2.74) belong to 𝑏(𝑑0+1)={π‘₯βˆˆβ„π‘βˆ£|π‘₯|<𝑑0+1}. So, we have π‘‘π΅ξ€ΊΞ›β„Žξ€·πœŒ2𝑑,𝑏0ξ€Έξ€»+1,0=𝑑𝐡𝑑𝐼,𝑏0ξ€Έξ€»+1,0β‰ 0,(2.75) and it means the existence of solutions of Ξ›β„Ž(𝜌2)=0.
In this way, we define a function 𝜌2(β„Ž)∢𝐿1→ℝ𝑁, which satisfies Ξ›β„Žξ€·πœŒ2ξ€Έ(β„Ž)=0.(2.76)
(ii) By the proof of (i), we also obtain 𝜌2 sends bounded sets to bounded sets, and ||𝜌2||𝐸(β„Ž)≀3𝑁#+1𝐸#𝑝+ξ‚ƒβ€–β„Žβ€–0||𝑒+2𝑁1||𝑝#βˆ’1ξ‚„.(2.77)
It only remains to prove the continuity of 𝜌2. Let {𝑒𝑛} be a convergent sequence in 𝐿1 and 𝑒𝑛→𝑒 as 𝑛→+∞. Since {𝜌2(𝑒𝑛)} is a bounded sequence, then it contains a convergent subsequence {𝜌2(𝑒𝑛𝑗)}. Let 𝜌2(𝑒𝑛𝑗)β†’πœŒ0 as 𝑗→+∞. Since Λ𝑒𝑛𝑗(𝜌2(𝑒𝑛𝑗))=0, letting 𝑗→+∞, we have Λ𝑒(𝜌0)=0. From (i), we get 𝜌0=𝜌2(𝑒), it means that 𝜌2 is continuous. This completes the proof.

It is clear that 𝜌2(β‹…) is continuous and sends bounded sets of 𝐿1 to bounded sets of ℝ𝑁, and, hence, it is a compact continuous mapping.

Let us define𝑃2∢𝐢1⟢𝐢1,π‘’βŸΌπ‘’(0),(2.78)π‘„βˆΆπΏ1βŸΆβ„π‘,β„ŽβŸΌπ‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰+βˆžπ‘–β„Ž(𝑑)π‘‘π‘‘βˆ’π‘’2,𝑄(2.79)βˆ—βˆΆπΏ1⟢𝐿1,β„ŽβŸΌπœ(𝑑)π‘šβˆ’2𝑖=1π›Όπ‘–ξ€œπœ‰+βˆžπ‘–β„Ž(𝑑)π‘‘π‘‘βˆ’π‘’2ξƒͺ,(2.80) where 𝜏∈([0,+∞),ℝ) and satisfies βˆ‘0<𝜏(𝑑)<1,π‘‘βˆˆπ½,π‘šβˆ’2𝑖=1π›Όπ‘–βˆ«πœ‰+βˆžπ‘–πœ(𝑑)𝑑𝑑=1. We denote 𝐾2∢𝐿1→𝐢1 as 𝐾2ξ€½πœ‘(β„Ž)(𝑑)∢=πΉβˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€·πœŒ2ξ€·ξ€·πΌβˆ’π‘„βˆ—ξ€Έβ„Žξ€Έ+πΉξ€·ξ€·πΌβˆ’π‘„βˆ—ξ€Έβ„Ž[ξ€Έξ€Έξ€»ξ€Ύ(𝑑),βˆ€π‘‘βˆˆ0,+∞).(2.81)

Similar to Lemmas 2.3 and 2.7, we have the following

Lemma 2.9. The operator 𝐾2 is continuous and sends equi-integrable sets in 𝐿1 to relatively compact sets in 𝐢1.

Lemma 2.10. In Case (iii), 𝑒 is a solution of (1.1)-(1.2) if and only if 𝑒 is a solution of the following abstract equation: 𝑒=𝑃2𝑒+𝑄𝛿𝑁𝑓(𝑒)+𝐾2𝛿𝑁𝑓.(𝑒)(2.82)

3. Existence of Solutions in Case (i)

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2) when βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎∈[0,1). Moreover, we give the existence of nonnegative solutions.

Theorem 3.1. In Case (i), if 𝑓 satisfies sub-(π‘βˆ’βˆ’1) growth condition, then problem (1.1)-(1.2) has at least a solution for any fixed parameter 𝛿.

Proof. Denote Ψ𝑓(𝑒,πœ†)∢=𝑃(πœ†π›Ώπ‘π‘“(𝑒))+𝐾(πœ†π›Ώπ‘π‘“(𝑒)), where 𝑁𝑓(𝑒) is defined in (2.32). We know that (1.1)-(1.2) has the same solution of 𝑒=Ψ𝑓(𝑒,πœ†),(3.1) when πœ†=1.
It is easy to see that the operator 𝑃 is compact continuous. According to Lemmas 2.2 and 2.3, we can see that Ψ𝑓(β‹…,β‹…) is compact continuous from 𝐢1Γ—[0,1] to 𝐢1.
We claim that all the solutions of (3.1) are uniformly bounded for πœ†βˆˆ[0,1]. In fact, if it is false, we can find a sequence of solutions {(𝑒𝑛,πœ†π‘›)} for (3.1) such that ‖𝑒𝑛‖1β†’+∞ as 𝑛→+∞ and ‖𝑒𝑛‖1>1 for any 𝑛=1,2,….
From Lemma 2.2, we have ||πœŒξ€·πœ†π‘›π›Ώπ‘π‘“ξ€·π‘’π‘›||≀𝐢1‖‖𝑁𝑓𝑒𝑛‖‖𝐿1+||𝑒2||≀𝐢2β€–β€–π‘’π‘›β€–β€–π‘ž+1βˆ’1,(3.2) then we have ||πœŒξ€·πœ†π‘›π›Ώπ‘π‘“ξ€·π‘’π‘›ξ€·πœ†ξ€Έξ€Έ+𝐹𝑛𝛿𝑁𝑓𝑒𝑛||≀||πœŒξ€·πœ†ξ€Έξ€Έπ‘›π›Ώπ‘π‘“ξ€·π‘’π‘›||+||πΉξ€·πœ†ξ€Έξ€Έπ‘›π›Ώπ‘π‘“ξ€·π‘’π‘›||≀𝐢3β€–β€–π‘’π‘›β€–β€–π‘ž+1βˆ’1.(3.3)
From (3.1), we have ||𝑒𝑀(𝑑)ξ…žπ‘›||(𝑑)𝑝(𝑑)βˆ’2π‘’ξ…žπ‘›ξ€·πœ†(𝑑)=πœŒπ‘›π›Ώπ‘π‘“ξ€·π‘’π‘›ξ€·πœ†ξ€Έξ€Έ+𝐹𝑛𝛿𝑁𝑓𝑒𝑛,βˆ€π‘‘βˆˆπ½,(3.4) then ||𝑒𝑀(𝑑)ξ…žπ‘›||(𝑑)𝑝(𝑑)βˆ’1≀||πœŒξ€·πœ†π‘›π›Ώπ‘π‘“ξ€·π‘’π‘›||+||πΉξ€·πœ†ξ€Έξ€Έπ‘›π›Ώπ‘π‘“ξ€·π‘’π‘›||≀𝐢3β€–β€–π‘’π‘›β€–β€–π‘ž+1βˆ’1.(3.5)
Denote 𝛼=(π‘ž+βˆ’1)/(π‘βˆ’βˆ’1), from the above inequality, we have β€–β€–(𝑀(𝑑))1/(𝑝(𝑑)βˆ’1)π‘’ξ…žπ‘›β€–β€–(𝑑)0≀𝐢4‖‖𝑒𝑛‖‖𝛼1.(3.6)
It follows from (2.36) and (3.3) that ||𝑒𝑛(||0)≀𝐢5‖‖𝑒𝑛‖‖𝛼1π‘ž,where𝛼=+βˆ’1π‘βˆ’.βˆ’1(3.7)
For any 𝑗=1,…,𝑁, we have ||𝑒𝑗𝑛||=||||𝑒(𝑑)π‘—π‘›ξ€œ(0)+𝑑0ξ€·π‘’π‘—π‘›ξ€Έξ…ž||||≀||𝑒(π‘Ÿ)π‘‘π‘Ÿπ‘—π‘›(||+||||ξ€œ0)𝑑0(𝑀(π‘Ÿ))βˆ’1/(𝑝(π‘Ÿ)βˆ’1)supπ‘‘βˆˆ(0,+∞)|||(𝑀(𝑑))1/(𝑝(𝑑)βˆ’1)ξ€·π‘’π‘—π‘›ξ€Έξ…ž(|||||||≀𝐢𝑑)π‘‘π‘Ÿ6+𝐢4𝐸‖‖𝑒𝑛‖‖𝛼1≀𝐢7‖‖𝑒𝑛‖‖𝛼1,(3.8) which implies that ||𝑒𝑗𝑛||0≀𝐢8‖‖𝑒𝑛‖‖𝛼1,𝑗=1,…,𝑁,𝑛=1,2,….(3.9)
Thus, ‖‖𝑒𝑛‖‖0≀𝐢9‖‖𝑒𝑛‖‖𝛼1,𝑛=1,2,….(3.10)
It follows from (3.6) and (3.10) that {‖𝑒𝑛‖1} is bounded.
Thus, we can choose a large enough 𝑅0>0 such that all the solutions of (3.1) belong to 𝐡(𝑅0)={π‘’βˆˆπΆ1βˆ£β€–π‘’β€–1<𝑅0}. Thus, the Leray-Schauder degree 𝑑LS[πΌβˆ’Ξ¨π‘“(β‹…,πœ†),𝐡(𝑅0),0] is well defined for each πœ†βˆˆ[0,1], and 𝑑LSξ€ΊπΌβˆ’Ξ¨π‘“ξ€·π‘…(β‹…,1),𝐡0ξ€Έξ€»,0=𝑑LSξ€ΊπΌβˆ’Ξ¨π‘“ξ€·π‘…(β‹…,0),𝐡0ξ€Έξ€».,0(3.11)
Let 𝑒0=∫0+βˆžξ‚†βˆ«π‘’(𝑑)𝑑0πœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€»ξ‚‡βˆ«πœŒ(0)π‘‘π‘Ÿπ‘‘π‘‘βˆ’0+βˆžπœ‘βˆ’1ξ€Ίπ‘Ÿ,(𝑀(π‘Ÿ))βˆ’1ξ€»πœŒ(0)π‘‘π‘Ÿ+𝑒1+ξ€œ1βˆ’πœŽπ‘Ÿ0πœ‘βˆ’1𝑑,(𝑀(𝑑))βˆ’1ξ€»πœŒ(0)𝑑𝑑,(3.12) where 𝜌(0) is defined in (2.10), thus 𝑒0 is the unique solution of 𝑒=Ψ𝑓(𝑒,0).
It is easy to see that 𝑒 is a solution of 𝑒=Ψ𝑓(𝑒,0) if and only if 𝑒 is a solution of the following system βˆ’Ξ”π‘(𝑑)ξ€œπ‘’=0,π‘‘βˆˆ(0,+∞),𝑒(+∞)=0+βˆžπ‘’(𝑑)𝑒(𝑑)𝑑𝑑+𝑒1,lim𝑑→+∞||𝑒𝑀(𝑑)ξ…ž||𝑝(𝑑)βˆ’2π‘’ξ…ž(𝑑)=π‘šβˆ’2𝑖=1π›Όπ‘–π‘€ξ€·πœ‰π‘–ξ€Έ||π‘’ξ…ž||𝑝(πœ‰π‘–)βˆ’2π‘’ξ…žξ€·πœ‰π‘–ξ€Έ+𝑒2.(I)
Obviously, system (I ) possesses a unique solution 𝑒0. Note that 𝑒0∈𝐡(𝑅0), we have 𝑑LSξ€ΊπΌβˆ’Ξ¨π‘“ξ€·π‘…(β‹…,1),𝐡0ξ€Έξ€»,0=𝑑LSξ€ΊπΌβˆ’Ξ¨π‘“ξ€·π‘…(β‹…,0),𝐡0ξ€Έξ€»,0β‰ 0.(3.13)
Therefore, (1.1)-(1.2) has at least one solution when βˆ‘π‘šβˆ’2𝑖=1π›Όπ‘–βˆˆ[0,1),𝜎∈[0,1). This completes the proof.

Denote Ξ©