International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 634612 | 22 pages | https://doi.org/10.1155/2011/634612

Multiple Positive Solutions for First-Order Impulsive Integrodifferential Equations on the Half Line in Banach Spaces

Academic Editor: A. M. El-Sayed
Received16 May 2011
Accepted03 Jul 2011
Published14 Aug 2011

Abstract

The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

1. Introduction

Let 𝐸 be a real Banach space and 𝑃 a cone in 𝐸 which defines a partial ordering in 𝐸 by π‘₯≀𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that πœƒβ‰€π‘₯≀𝑦 implies ||π‘₯||≀𝑁||𝑦||, where πœƒ denotes the zero element of 𝐸 and the smallest 𝑁 is called the normal constant of 𝑃. If π‘₯≀𝑦 and π‘₯≠𝑦, we write π‘₯<𝑦. For details on cone theory, see [1].

In paper [2], we considered the infinite boundary value problem (IBVP) for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line in 𝐸: π‘’ξ…ž||(𝑑)=𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)),βˆ€π‘‘βˆˆπ½β€²,Δ𝑒𝑑=π‘‘π‘˜=πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ(π‘˜=1,2,3,…),𝑒(∞)=𝛽𝑒(0),(1.1) where 𝐽=[0,∞), 0<𝑑1<β‹―<π‘‘π‘˜<…,π‘‘π‘˜β†’βˆž, 𝐽′=𝐽⧡{𝑑1,…,π‘‘π‘˜,…}, π‘“βˆˆπΆ[𝐽×𝑃×𝑃×𝑃,𝑃], πΌπ‘˜βˆˆπΆ[𝑃,𝑃] (π‘˜=1,2,3,…), 𝛽>1, 𝑒(∞)=limπ‘‘β†’βˆžπ‘’(𝑑), and (ξ€œπ‘‡π‘’)(𝑑)=𝑑0ξ€œπΎ(𝑑,𝑠)𝑒(𝑠)𝑑𝑠,(𝑆𝑒)(𝑑)=∞0𝐻(𝑑,𝑠)𝑒(𝑠)𝑑𝑠,(1.2)𝐾∈𝐢[𝐷,𝑅+], 𝐷={(𝑑,𝑠)βˆˆπ½Γ—π½βˆΆπ‘‘β‰₯𝑠}, 𝐻∈𝐢[𝐽×𝐽,𝑅+],𝑅+ denotes the set of all nonnegative numbers. Δ𝑒|𝑑=π‘‘π‘˜ denotes the jump of 𝑒(𝑑) at 𝑑=π‘‘π‘˜, that is, ||Δ𝑒𝑑=π‘‘π‘˜ξ€·π‘‘=𝑒+π‘˜ξ€Έξ€·π‘‘βˆ’π‘’βˆ’π‘˜ξ€Έ,(1.3) where 𝑒(𝑑+π‘˜) and 𝑒(π‘‘βˆ’π‘˜) represent the right and left limits of 𝑒(𝑑) at 𝑑=π‘‘π‘˜, respectively. By using the fixed point index theory, we discussed the multiple positive solutions of IBVP(1.1). But the discussion dealt with sublinear equations, that is, we assume that there exists π‘βˆˆπΆ[𝐽,𝑅+]∩𝐿[𝐽,𝑅+] such that (‖𝑓𝑑,𝑒,𝑣,𝑀)‖𝑐(𝑑)(‖𝑒‖+‖𝑣‖+‖𝑀‖)⟢0as𝑒,𝑣,π‘€βˆˆπ‘ƒ,‖𝑒‖+‖𝑣‖+β€–π‘€β€–βŸΆβˆž(1.4) uniformly for π‘‘βˆˆπ½ (see condition (𝐻5) in [2]).

Now, in this paper, we discuss the multiple positive solutions of an infinite three-point boundary value problem (which includes IBVP(1.1) as a special case) for superlinear case by means of different method, that is, by using the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [3] (see also [1]), and the key point is to introduce a new cone 𝑄.

Consider the infinite three-point boundary value problem for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line in 𝐸: π‘’ξ…ž||(𝑑)=𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)),βˆ€π‘‘βˆˆπ½β€²,Δ𝑒𝑑=π‘‘π‘˜=πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ(π‘˜=1,2,3,…),𝑒(∞)=𝛾𝑒(πœ‚)+𝛽𝑒(0),(1.5) where 0≀𝛾<1,𝛽+𝛾>1, and π‘‘π‘šβˆ’1<πœ‚β‰€π‘‘π‘š (for some π‘š). It is clear that IBVP(1.5) includes IBVP(1.1) as a special case when 𝛾=0.

Let PC[𝐽,𝐸]={π‘’βˆΆπ‘’ is a map from 𝐽 into 𝐸 such that 𝑒(𝑑) is continuous at π‘‘β‰ π‘‘π‘˜, left continuous at 𝑑=π‘‘π‘˜, and 𝑒(𝑑+π‘˜) exists, π‘˜=1,2,3,…} and BPC[𝐽,𝐸]={π‘’βˆˆPC[𝐽,𝐸]∢supπ‘‘βˆˆπ½||𝑒(𝑑)||<∞}. It is clear that BPC[𝐽,𝐸] is a Banach space with norm ‖𝑒‖𝐡=supπ‘‘βˆˆπ½β€–π‘’(𝑑)β€–.(1.6) Let BPC[𝐽,𝑃]={π‘’βˆˆBPC[𝐽,𝐸]βˆΆπ‘’(𝑑)β‰₯πœƒ,βˆ€π‘‘βˆˆπ½} and 𝑄={π‘’βˆˆBPC[𝐽,𝑃]βˆΆπ‘’(𝑑)β‰₯π›½βˆ’1(1βˆ’π›Ύ)𝑒(𝑠),βˆ€π‘‘,π‘ βˆˆπ½}. Obviously, BPC[𝐽,𝑃] and 𝑄 are two cones in space BPC[𝐽,𝐸] and π‘„βŠ‚BPC[𝐽,𝑃]. π‘’βˆˆBPC[𝐽,𝑃]∩𝐢1[𝐽′,𝐸] is called a positive solution of IBVP(1.5) if 𝑒(𝑑)>πœƒ for π‘‘βˆˆπ½ and 𝑒(𝑑) satisfies (1.5).

2. Several Lemmas

Let us list some conditions.(𝐻1)supπ‘‘βˆˆπ½βˆ«π‘‘0𝐾(𝑑,𝑠)𝑑𝑠<∞,supπ‘‘βˆˆπ½βˆ«βˆž0𝐻(𝑑,𝑠)𝑑𝑠<∞, and limπ‘‘ξ…žβ†’π‘‘ξ€œβˆž0||π»ξ€·π‘‘ξ…žξ€Έ||,π‘ βˆ’π»(𝑑,𝑠)𝑑𝑠=0,βˆ€π‘‘βˆˆπ½.(2.1) In this case, let π‘˜βˆ—=supπ‘‘βˆˆπ½ξ€œπ‘‘0𝐾(𝑑,𝑠)𝑑𝑠,β„Žβˆ—=supπ‘‘βˆˆπ½ξ€œβˆž0𝐻(𝑑,𝑠)𝑑𝑠.(2.2)(𝐻2)There exist π‘ŽβˆˆπΆ[𝐽,𝑅+] and π‘”βˆˆπΆ[𝑅+×𝑅+×𝑅+,𝑅+] such that π‘Žβ€–π‘“(𝑑,𝑒,𝑣,𝑀)β€–β‰€π‘Ž(𝑑)𝑔(‖𝑒‖,‖𝑣‖,‖𝑀‖),βˆ€π‘‘βˆˆπ½,𝑒,𝑣,π‘€βˆˆπ‘ƒ,βˆ—=ξ€œβˆž0π‘Ž(𝑑)𝑑𝑑<∞.(2.3)(𝐻3)There exist π›Ύπ‘˜β‰₯0 (π‘˜=1,2,3,…) and 𝐹∈𝐢[𝑅+,𝑅+] such that β€–β€–πΌπ‘˜β€–β€–(𝑒)β‰€π›Ύπ‘˜π›ΎπΉ(‖𝑒‖),βˆ€π‘’βˆˆπ‘ƒ(π‘˜=1,2,3,…),βˆ—=βˆžξ“π‘˜=1π›Ύπ‘˜<∞.(2.4)(𝐻4)For any π‘‘βˆˆπ½ and π‘Ÿ>0,𝑓(𝑑,π‘ƒπ‘Ÿ,π‘ƒπ‘Ÿ,π‘ƒπ‘Ÿ)={𝑓(𝑑,𝑒,𝑣,𝑀)βˆΆπ‘’,𝑣,π‘€βˆˆπ‘ƒπ‘Ÿ} and πΌπ‘˜(π‘ƒπ‘Ÿ)={πΌπ‘˜(𝑒)βˆΆπ‘’βˆˆπ‘ƒπ‘Ÿ} (π‘˜=1,2,3,…) are relatively compact in 𝐸, where π‘ƒπ‘Ÿ={π‘’βˆˆπ‘ƒβˆΆ||𝑒||β‰€π‘Ÿ}.

Remark 2.1. Obviously, condition (𝐻4) is satisfied automatically when 𝐸 is finite dimensional.

Remark 2.2. It is clear that if condition (𝐻1) is satisfied, then the operators 𝑇 and 𝑆 defined by (1.2) are bounded linear operators from BPC[𝐽,𝐸] into BPC[𝐽,𝐸] and ||𝑇||β‰€π‘˜βˆ—,||𝑆||β‰€β„Žβˆ—; moreover, we have 𝑇(BPC[𝐽,𝑃])βŠ‚BPC[𝐽,𝑃] and 𝑆(BPC[𝐽,𝑃])βŠ‚BPC[𝐽,𝑃].

We shall reduce IBVP(1.5) to an impulsive integral equation. To this end, we consider the operator 𝐴 defined by 1(𝐴𝑒)(𝑑)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έπ‘‘0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+0<π‘‘π‘˜<π‘‘πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½.(2.5) In what follows, we write 𝐽1=[0,𝑑1],π½π‘˜=(π‘‘π‘˜βˆ’1,π‘‘π‘˜] (π‘˜=2,3,4,…).

Lemma 2.3. If conditions (𝐻1)–(𝐻4) are satisfied, then operator 𝐴 defined by (2.5) is a completely continuous (i.e., continuous and compact) operator from 𝐡𝑃𝐢[𝐽,𝑃] into 𝑄.

Proof. Let π‘Ÿ>0 be given. Let π‘€π‘Ÿξ€½π‘”=max(π‘₯,𝑦,𝑧)∢0≀π‘₯β‰€π‘Ÿ,0β‰€π‘¦β‰€π‘˜βˆ—π‘Ÿ,0β‰€π‘§β‰€β„Žβˆ—π‘Ÿξ€Ύπ‘,(2.6)π‘Ÿ=max{𝐹(π‘₯)∢0≀π‘₯β‰€π‘Ÿ}.(2.7) For π‘’βˆˆBPC[𝐽,𝑃],||𝑒||π΅β‰€π‘Ÿ, we see that by virtue of condition (𝐻2) and (2.6), ‖𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))β€–β‰€π‘€π‘Ÿπ‘Ž(𝑑),βˆ€π‘‘βˆˆπ½,(2.8) which implies the convergence of the infinite integral ξ€œβˆž0β€–β€–β€–ξ€œπ‘“(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))𝑑𝑑,(2.9)∞0β€–β€–β€–β‰€ξ€œπ‘“(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))π‘‘π‘‘βˆž0‖𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))β€–π‘‘π‘‘β‰€π‘€π‘Ÿπ‘Žβˆ—.(2.10) On the other hand, condition (𝐻3) and (2.7) imply the convergence of the infinite series βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜,β€–β€–β€–β€–ξ€Έξ€Έ(2.11)βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–β€–β€–β‰€ξ€Έξ€Έβˆžξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξ€Έξ€Έβ‰€π‘π‘Ÿπ›Ύβˆ—.(2.12) It follows from (2.5), (2.10), and (2.12) that ||||||||≀1(𝐴𝑒)(𝑑)ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚β€–Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))‖𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+‖𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))β€–π‘‘π‘ βˆžξ“π‘˜=π‘šβ€–β€–πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–+ξ€Έξ€Έ(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξƒ°+ξ€œξ€Έξ€Έπ‘‘0‖𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))‖𝑑𝑠+0<π‘‘π‘˜<π‘‘β€–β€–πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–β‰€1ξ€Έξ€Έξƒ―ξ€œπ›½+π›Ύβˆ’1∞0‖𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))‖𝑑𝑠+βˆžξ“π‘˜=1||||πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜||||ξƒ°+ξ€œξ€Έξ€Έβˆž0‖𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))‖𝑑𝑠+βˆžξ“π‘˜=1||||πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜||||=𝛽+π›Ύξƒ―ξ€œπ›½+π›Ύβˆ’1∞0‖𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))‖𝑑𝑠+βˆžξ“π‘˜=1||||πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜||||≀𝛽+𝛾𝑀𝛽+π›Ύβˆ’1π‘Ÿπ‘Žβˆ—+π‘π‘Ÿπ›Ύβˆ—ξ€Έ,βˆ€π‘‘βˆˆπ½,(2.13) which implies that π΄π‘’βˆˆBPC[𝐽,𝑃] and ‖𝐴𝑒‖𝐡≀𝛽+𝛾𝑀𝛽+π›Ύβˆ’1π‘Ÿπ‘Žβˆ—+π‘π‘Ÿπ›Ύβˆ—ξ€Έ.(2.14) Moreover, by (2.5), we have 1(𝐴𝑒)(𝑑)β‰₯ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°,ξ€Έξ€Έβˆ€π‘‘βˆˆπ½,(2.15)1(𝐴𝑒)(𝑑)β‰€ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜+ξ€Έξ€Έ(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έβˆž0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½.(2.16) It is clear that ξ€œβˆž0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β‰€1ξ€Έξ€Έξƒ―ξ€œ1βˆ’π›Ύβˆžπœ‚ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°,ξ€Έξ€Έ(2.17) so, (2.16) and (2.17) imply ξ‚»1(𝐴𝑒)(𝑑)≀+1𝛽+π›Ύβˆ’1ξ‚ΌΓ—ξƒ―ξ€œ1βˆ’π›Ύβˆžπœ‚ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜+ξ€Έξ€Έ(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½.(2.18) It follows from (2.15) and (2.18) that 1(𝐴𝑒)(𝑑)β‰₯ξ‚΅1𝛽+π›Ύβˆ’1+1𝛽+π›Ύβˆ’1ξ‚Ά1βˆ’π›Ύβˆ’1(𝐴𝑒)(𝑠)=π›½βˆ’1(1βˆ’π›Ύ)(𝐴𝑒)(𝑠),βˆ€π‘‘,π‘ βˆˆπ½.(2.19) Hence, π΄π‘’βˆˆπ‘„. That is, 𝐴 maps BPC[𝐽,𝑃] into 𝑄.
Now, we are going to show that 𝐴 is continuous. Let 𝑒𝑛,π‘’βˆˆBPC[𝐽,𝑃],||π‘’π‘›βˆ’π‘’||𝐡→0 (π‘›β†’βˆž). Then π‘Ÿ=sup𝑛||𝑒𝑛||𝐡<∞ and ||𝑒||π΅β‰€π‘Ÿ. Similar to (2.14), it is easy to get β€–β€–π΄π‘’π‘›βˆ’π΄π‘’β€–β€–π΅β‰€π›½+π›Ύξƒ―ξ€œπ›½+π›Ύβˆ’1∞0‖‖𝑓𝑠,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛(𝑠)βˆ’π‘“π‘ ,𝑇𝑒(𝑠),𝑒𝑆(𝑠),𝑒‖‖+(𝑠)π‘‘π‘ βˆžξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξƒ°ξ€Έξ€Έ(𝑛=1,2,3,…).(2.20) It is clear that 𝑓𝑑,𝑒𝑛(𝑑),𝑇𝑒𝑛(𝑑),𝑆𝑒𝑛(𝑑)βŸΆπ‘“π‘‘,𝑒𝑇(𝑑),𝑒𝑆(𝑑),𝑒(𝑑)asπ‘›βŸΆβˆž,βˆ€π‘‘βˆˆπ½.(2.21) Moreover, we see from (2.8) that ‖‖𝑓𝑑,𝑒𝑛(𝑑),𝑇𝑒𝑛(𝑑),𝑆𝑒𝑛(𝑑)βˆ’π‘“π‘‘,𝑒𝑇(𝑑),𝑒𝑆(𝑑),𝑒‖‖(𝑑)≀2π‘€π‘Ÿξ€Ίπ‘Ž(𝑑)=𝜎(𝑑),βˆ€π‘‘βˆˆπ½(𝑛=1,2,3,…);𝜎∈𝐿𝐽,𝑅+ξ€».(2.22) It follows from (2.21), (2.22) and the dominated convergence theorem that limπ‘›β†’βˆžξ€œβˆž0‖‖𝑓𝑑,𝑒𝑛(𝑑),𝑇𝑒𝑛(𝑑),𝑆𝑒𝑛(𝑑)βˆ’π‘“π‘‘,𝑇𝑒(𝑑),𝑒𝑆(𝑑),𝑒‖‖(𝑑)𝑑𝑑=0.(2.23) On the other hand, for any πœ–>0, we can choose a positive integer 𝑗 such that π‘π‘Ÿβˆžξ“π‘˜=𝑗+1π›Ύπ‘˜<πœ–.(2.24) And then, choose a positive integer 𝑛0 such that π‘—ξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξ€Έξ€Έ<πœ–,βˆ€π‘›>𝑛0.(2.25) From (2.24) and (2.25), we get βˆžξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξ€Έξ€Έ<πœ–+2π‘π‘Ÿβˆžξ“π‘˜=𝑗+1π›Ύπ‘˜<3πœ–,βˆ€π‘›>𝑛0,(2.26) hence limβˆžπ‘›β†’βˆžξ“π‘˜=1β€–β€–πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξ€Έξ€Έ=0.(2.27) It follows from (2.20), (2.23), and (2.51) that ||π΄π‘’π‘›βˆ’π΄π‘’||𝐡→0asπ‘›β†’βˆž, and the continuity of 𝐴 is proved.
Finally, we prove that 𝐴 is compact. Let 𝑉={𝑒𝑛}βŠ‚BPC[𝐽,𝑃] be bounded and ||𝑒𝑛||π΅β‰€π‘Ÿ (𝑛=1,2,3,…). Consider 𝐽𝑖=(π‘‘π‘–βˆ’1,𝑑𝑖] for any fixed 𝑖. By (2.5) and (2.8), we have β€–β€–ξ€·π΄π‘’π‘›π‘‘ξ€Έξ€·ξ…žξ€Έβˆ’ξ€·π΄π‘’π‘›ξ€Έ(β€–β€–β‰€ξ€œπ‘‘)π‘‘π‘‘ξ…žβ€–β€–π‘“ξ€·π‘ ,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛(‖‖𝑠)π‘‘π‘ β‰€π‘€π‘Ÿξ€œπ‘‘β€²π‘‘π‘Ž(𝑠)𝑑𝑠,βˆ€π‘‘,π‘‘ξ…žβˆˆπ½π‘–,π‘‘ξ…ž>𝑑(𝑛=1,2,3,…),(2.28) which implies that the functions {𝑀𝑛(𝑑)} (𝑛=1,2,3,…) defined by 𝑀𝑛(⎧βŽͺ⎨βŽͺβŽ©ξ€·π‘‘)=𝐴𝑒𝑛(𝑑),βˆ€π‘‘βˆˆπ½π‘–=ξ€·π‘‘π‘–βˆ’1,𝑑𝑖,𝐴𝑒𝑛𝑑+π‘–βˆ’1ξ€Έ,βˆ€π‘‘=π‘‘π‘–βˆ’1(𝑛=1,2,3,…)(2.29) ((𝐴𝑒𝑛)(𝑑+π‘–βˆ’1) denotes the right limit of (𝐴𝑒𝑛)(𝑑) at 𝑑=π‘‘π‘–βˆ’1) are equicontinuous on 𝐽𝑖=[π‘‘π‘–βˆ’1,𝑑𝑖]. On the other hand, for any πœ–>0, choose a sufficiently large 𝜏>πœ‚ and a sufficiently large positive integer 𝑗>π‘š such that π‘€π‘Ÿξ€œβˆžπœπ‘Ž(𝑠)𝑑𝑠<πœ–,π‘π‘Ÿβˆžξ“π‘˜=𝑗+1π›Ύπ‘˜<πœ–.(2.30) We have, by (2.29), (2.5), (2.8), (2.30), and condition (𝐻3), 𝑀𝑛=1(𝑑)ξƒ―ξ€œπ›½+π›Ύβˆ’1πœπœ‚π‘“ξ€·π‘ ,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛+ξ€œ(𝑠)π‘‘π‘ βˆžπœπ‘“ξ€·π‘ ,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),π‘†π‘’π‘›ξ€Έξ€ΈΓ—ξ€œ(𝑠)𝑑𝑠+(1βˆ’π›Ύ)πœ‚0𝑓𝑠,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛(𝑠)𝑑𝑠+π‘—ξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜+ξ€Έξ€Έβˆžξ“π‘˜=𝑗+1πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έπ‘‘0𝑓𝑠,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛+(𝑠)π‘‘π‘ π‘–βˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½π‘–(β€–β€–β€–ξ€œπ‘›=1,2,3,…),(2.31)βˆžπœπ‘“ξ€·π‘ ,𝑒𝑛(𝑠),𝑇𝑒𝑛(𝑠),𝑆𝑒𝑛‖‖‖‖‖‖‖(𝑠)𝑑𝑠<πœ–(𝑛=1,2,3,…),(2.32)βˆžξ“π‘˜=𝑗+1πΌπ‘˜ξ€·π‘’π‘›ξ€·π‘‘π‘˜β€–β€–β€–β€–ξ€Έξ€Έ<πœ–(𝑛=1,2,3,…).(2.33) It follows from (2.31), (2.32), (2.33), (2.8), and [4, Theorem 1.2.3] that 1𝛼(π‘Š(𝑑))≀2ξ€œπ›½+π›Ύβˆ’1πœπœ‚ξ€œπ›Ό(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))𝑑𝑠+2πœ–+2(1βˆ’π›Ύ)πœ‚0+𝛼(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))π‘‘π‘ π‘—ξ“π‘˜=π‘šπ›Όξ€·πΌπ‘˜ξ€·π‘‰ξ€·π‘‘π‘˜ξ€Έξ€Έξ€Έ+2πœ–+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1π›Όξ€·πΌπ‘˜ξ€·π‘‰ξ€·π‘‘π‘˜ξƒ°ξ€œξ€Έξ€Έξ€Έ+2𝑑0𝛼(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))𝑑𝑠+π‘–βˆ’1ξ“π‘˜=1π›Όξ€·πΌπ‘˜ξ€·π‘‰ξ€·π‘‘π‘˜ξ€Έξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½π‘–,(2.34) where π‘Š(𝑑)={𝑀𝑛(𝑑)βˆΆπ‘›=1,2,3,…}, 𝑉(𝑠)={𝑒𝑛(𝑠)βˆΆπ‘›=1,2,3,…}, (𝑇𝑉)(𝑠)={(𝑇𝑒𝑛)(𝑠)βˆΆπ‘›=1,2,3,…}, (𝑆𝑉)(𝑠)={(𝑆𝑒𝑛)(𝑠)βˆΆπ‘›=1,2,3,…} and 𝛼(π‘ˆ) denotes the Kuratowski measure of noncompactness of bounded set π‘ˆβŠ‚πΈ (see [4, Section 1.2]). Since 𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)βŠ‚π‘ƒπ‘Ÿβˆ— for π‘ βˆˆπ½, where π‘Ÿβˆ—=max{π‘Ÿ,π‘˜βˆ—π‘Ÿ,β„Žβˆ—π‘Ÿ}, we see that, by condition (𝐻4), 𝛼𝐼𝛼(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))=0,βˆ€π‘‘βˆˆπ½,(2.35)π‘˜ξ€·π‘‰ξ€·π‘‘π‘˜ξ€Έξ€Έξ€Έ=0(π‘˜=1,2,3,…).(2.36) It follows from (2.34)–(2.36) that 𝛼(π‘Š(𝑑))≀4πœ–π›½+π›Ύβˆ’1,βˆ€π‘‘βˆˆπ½π‘–,(2.37) which implies by virtue of the arbitrariness of πœ– that 𝛼(π‘Š(𝑑))=0 for π‘‘βˆˆπ½π‘–.
By Ascoli-Arzela theorem (see [4, Theorem 1.2.5]), we conclude that π‘Š={π‘€π‘›βˆΆπ‘›=1,2,3,…} is relatively compact in 𝐢[𝐽𝑖,𝐸]; hence, {𝑀𝑛(𝑑)} has a subsequence which is convergent uniformly on 𝐽𝑖, so, {(𝐴𝑒𝑛(𝑑)} has a subsequence which is convergent uniformly on 𝐽𝑖. Since 𝑖 may be any positive integer, so, by diagonal method, we can choose a subsequence {(𝐴𝑒𝑛𝑖)(𝑑)} of {(𝐴𝑒𝑛)(𝑑)} such that {(𝐴𝑒𝑛𝑖)(𝑑)} is convergent uniformly on each π½π‘˜ (π‘˜=1,2,3,…). Let limπ‘–β†’βˆžξ€·π΄π‘’π‘›π‘–ξ€Έ(𝑑)=𝑣(𝑑),βˆ€π‘‘βˆˆπ½.(2.38) It is clear that π‘£βˆˆPC[𝐽,𝑃]. By (2.14), we have ‖‖𝐴𝑒𝑛𝑖‖‖𝐡≀𝛽+𝛾𝑀𝛽+π›Ύβˆ’1π‘Ÿπ‘Žβˆ—+π‘π‘Ÿπ›Ύβˆ—ξ€Έ,(𝑖=1,2,3,…),(2.39) which implies that π‘£βˆˆBPC[𝐽,𝑃] and ‖𝑣‖𝐡≀𝛽+𝛾𝑀𝛽+π›Ύβˆ’1π‘Ÿπ‘Žβˆ—+π‘π‘Ÿπ›Ύβˆ—ξ€Έ.(2.40) Let πœ–>0 be arbitrarily given and choose a sufficiently large positive number 𝜏 such that π‘€π‘Ÿξ€œβˆžπœπ‘Ž(𝑠)𝑑𝑠+π‘π‘Ÿξ“π‘‘π‘˜β‰₯πœπ›Ύπ‘˜<πœ–.(2.41) For any 𝜏<𝑑<∞, we have, by (2.5), 𝐴𝑒𝑛𝑖(𝑑)βˆ’π΄π‘’π‘›π‘–ξ€Έ(ξ€œπœ)=π‘‘πœπ‘“ξ€·π‘ ,𝑒𝑛𝑖(𝑠),𝑇𝑒𝑛𝑖(𝑠),𝑆𝑒𝑛𝑖(ξ€Έ+𝑠)π‘‘π‘ πœβ‰€π‘‘π‘˜<π‘‘πΌπ‘˜ξ€·π‘’π‘›π‘–ξ€Έ(𝑑),(𝑖=1,2,3,…),(2.42) which implies by virtue of (2.8), condition (𝐻3) and (2.41) that ‖‖𝐴𝑒𝑛𝑖(𝑑)βˆ’π΄π‘’π‘›π‘–ξ€Έ(β€–β€–πœ)β‰€π‘€π‘Ÿξ€œπ‘‘πœπ‘Ž(𝑠)𝑑𝑠+π‘π‘Ÿξ“πœβ‰€π‘‘π‘˜<π‘‘π›Ύπ‘˜<πœ–,(𝑖=1,2,3,…).(2.43) Letting π‘–β†’βˆž in (2.43), we get ‖𝑣(𝑑)βˆ’π‘£(𝜏)β€–β‰€πœ–,βˆ€π‘‘>𝜏.(2.44) On the other hand, since {(𝐴𝑒𝑛𝑖)(𝑑)} converges uniformly to 𝑣(𝑑) on [0,𝜏] as π‘–β†’βˆž, there exists a positive integer 𝑖0 such that ‖‖𝐴𝑒𝑛𝑖‖‖[](𝑑)βˆ’π‘£(𝑑)<πœ–,βˆ€π‘‘βˆˆ0,𝜏,𝑖>𝑖0.(2.45) It follows from (2.43)–(2.45) that ‖‖𝐴𝑒𝑛𝑖‖‖≀‖‖(𝑑)βˆ’π‘£(𝑑)𝐴𝑒𝑛𝑖(𝑑)βˆ’π΄π‘’π‘›π‘–ξ€Έβ€–β€–+β€–β€–ξ€·(𝜏)𝐴𝑒𝑛𝑖‖‖‖(𝜏)βˆ’π‘£(𝜏)+‖𝑣(𝜏)βˆ’π‘£(𝑑)<3πœ–,βˆ€π‘‘>𝜏,𝑖>𝑖0.(2.46) By (2.45) and (2.46), we have β€–β€–π΄π‘’π‘›π‘–β€–β€–βˆ’π‘£π΅β‰€3πœ–,βˆ€π‘–>𝑖0,(2.47) hence ||π΄π‘’π‘›π‘–βˆ’π‘£||𝐡→0 as π‘–β†’βˆž, and the compactness of 𝐴 is proved.

Lemma 2.4. Let conditions (𝐻1)–(𝐻4) be satisfied. Then π‘’βˆˆπ΅π‘ƒπΆ[𝐽,𝑃]∩𝐢1[𝐽′,𝐸] is a solution of IBVP(1.5) if and only if π‘’βˆˆπ‘„ is a solution of the following impulsive integral equation: 1𝑒(𝑑)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έπ‘‘0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+0<π‘‘π‘˜<π‘‘πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½.(2.48) that is, 𝑒 is a fixed point of operator 𝐴 defined by (2.5) in 𝑄.

Proof. For π‘’βˆˆPC[𝐽,𝐸]∩𝐢1[𝐽′,𝐸], it is easy to get the following formula: ξ€œπ‘’(𝑑)=𝑒(0)+𝑑0π‘’ξ…ž(𝑠)𝑑𝑠+0<π‘‘π‘˜<𝑑𝑒𝑑+π‘˜ξ€Έξ€·π‘‘βˆ’π‘’π‘˜ξ€Έξ€»,βˆ€π‘‘βˆˆπ½.(2.49)
Let π‘’βˆˆBPC[𝐽,𝑃]∩𝐢1[𝐽′,𝐸] be a solution of IBVP(1.5). By (1.5) and (2.49), we have ξ€œπ‘’(𝑑)=𝑒(0)+𝑑0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+0<π‘‘π‘˜<π‘‘πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ,βˆ€π‘‘βˆˆπ½.(2.50) We have shown in the proof of Lemma 2.3 that the infinite integral (2.9) and the infinite series (2.11) are convergent, so, by taking limits as π‘‘β†’βˆž in both sides of (2.50), we get ξ€œπ‘’(∞)=𝑒(0)+∞0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ.(2.51) On the other hand, by (1.5) and (2.50), we have ξ€œπ‘’(∞)=𝛾𝑒(πœ‚)+𝛽𝑒(0),(2.52)𝑒(πœ‚)=𝑒(0)+πœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜.ξ€Έξ€Έ(2.53) It follows from (2.51)–(2.53) that 1𝑒(0)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°,ξ€Έξ€Έ(2.54) and, substituting it into (2.50), we see that 𝑒(𝑑) satisfies (2.48), that is, 𝑒=𝐴𝑒. Since π΄π‘’βˆˆπ‘„ by virtue of Lemma 2.3, we conclude that π‘’βˆˆπ‘„.
Conversely, assume that π‘’βˆˆπ‘„ is a solution of (2.48). We have, by (2.48), 1𝑒(0)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°,1ξ€Έξ€Έ(2.55)𝑒(πœ‚)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚Γ—ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έπœ‚0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜.ξ€Έξ€Έ(2.56) Moreover, by taking limits as π‘‘β†’βˆž in (2.33), we see that 𝑒(∞) exists and 1𝑒(∞)=ξƒ―ξ€œπ›½+π›Ύβˆ’1βˆžπœ‚ξ€œπ‘“(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+(1βˆ’π›Ύ)πœ‚0+𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))π‘‘π‘ βˆžξ“π‘˜=π‘šπΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ+(1βˆ’π›Ύ)π‘šβˆ’1ξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒ°+ξ€œξ€Έξ€Έβˆž0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜.ξ€Έξ€Έ(2.57) It follows from (2.55)–(2.57) that 𝛾𝑒(πœ‚)+𝛽𝑒(0)=𝑒(∞).(2.58) On the other hand, direct differentiation of (2.48) gives π‘’ξ…ž(𝑑)=𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)),βˆ€π‘‘βˆˆπ½β€²,(2.59) and, it is clear, by (2.48), ||Δ𝑒𝑑=π‘‘π‘˜=πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ(π‘˜=1,2,3,…).(2.60) Hence, π‘’βˆˆπΆ1[𝐽′,𝐸] and 𝑒(𝑑) satisfies (1.5).

Corollary 2.5. Let cone 𝑃 be normal. If 𝑒 is a fixed point of operator 𝐴 defined by (1.5) in 𝑄 and ||𝑒||𝐡>0, then 𝑒(𝑑)>πœƒ for π‘‘βˆˆπ½, so, 𝑒 is a positive solution of IBVP(1.5).

Proof. For π‘’βˆˆπ‘„, we have 𝑒(𝑑)β‰₯π›½βˆ’1(1βˆ’π›Ύ)𝑒(𝑠)β‰₯πœƒ,βˆ€π‘‘,π‘ βˆˆπ½,(2.61) so, ‖𝑒(𝑑)β€–β‰₯π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)‖𝑒‖𝐡,βˆ€π‘‘βˆˆπ½,(2.62) where 𝑁 denotes the normal constant of 𝑃. Since ||𝑒||𝐡>0, (2.61) and (2.62) imply that 𝑒(𝑑)>πœƒ for π‘‘βˆˆπ½.

Lemma 2.6 (Fixed point theorem of cone expansion and compression with norm type, see [3, Theorem  3] or [1, Theorem  2.3.4]). Let 𝑃 be a cone in real Banach space 𝐸 and Ξ©1,Ξ©2 two bounded open sets in 𝐸 such that πœƒβˆˆΞ©1, Ξ©1βŠ‚Ξ©2, where πœƒ denotes the zero element of 𝐸 and Ξ©2 denotes the closure of Ξ©2. Let operator π΄βˆΆπ‘ƒβˆ©(Ξ©2⧡Ω1)→𝑃 be completely continuous. Suppose that one of the following two conditions is satisfied: (a)‖𝐴π‘₯‖≀‖π‘₯β€–,βˆ€π‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©1;‖𝐴π‘₯β€–β‰₯β€–π‘₯β€–,βˆ€π‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©2,(2.63) where πœ•Ξ©π‘– denotes the boundary of Ω𝑖 (𝑖=1,2).(b)‖𝐴π‘₯β€–β‰₯β€–π‘₯β€–,βˆ€π‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©1,‖𝐴π‘₯‖≀‖π‘₯β€–,βˆ€π‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©2.(2.64)Then 𝐴 has at least one fixed point in π‘ƒβˆ©(Ξ©2⧡Ω1).

3. Main Theorems

Let us list more conditions.(𝐻5) There exist 𝑒0βˆˆπ‘ƒβ§΅{πœƒ},π‘βˆˆπΆ[𝐽,𝑅+], and 𝜏∈𝐢[𝑃,𝑅+] such that 𝑓(𝑑,𝑒,𝑣,𝑀)β‰₯𝑏(𝑑)𝜏(𝑒)𝑒0,βˆ€π‘‘βˆˆπ½,𝑒,𝑣,π‘€βˆˆπ‘ƒ,𝜏(𝑒)π‘β€–π‘’β€–βŸΆβˆžasπ‘’βˆˆπ‘ƒ,β€–π‘’β€–βŸΆβˆž,βˆ—=ξ€œβˆž0𝑏(𝑑)𝑑𝑑<∞.(3.1)

Remark 3.1. Condition (𝐻5) means that 𝑓(𝑑,𝑒,𝑣,𝑀) is superlinear with respect to 𝑒.(𝐻6) There exist 𝑒1βˆˆπ‘ƒβ§΅{πœƒ},π‘βˆˆπΆ[𝐽,𝑅+], and 𝜎∈𝐢[𝑃,𝑅+] such that 𝑓(𝑑,𝑒,𝑣,𝑀)β‰₯𝑐(𝑑)𝜎(𝑒)𝑒1,βˆ€π‘‘βˆˆπ½,𝑒,𝑣,π‘€βˆˆπ‘ƒ,𝜎(𝑒)π‘β€–π‘’β€–βŸΆβˆžasπ‘’βˆˆπ‘ƒ,β€–π‘’β€–βŸΆ0,βˆ—=ξ€œβˆž0𝑐(𝑑)𝑑𝑑<∞.(3.2)

Theorem 3.2. Let cone 𝑃 be normal and conditions (𝐻1)–(𝐻6) satisfied. Assume that there exists a πœ‰>0 such that 𝑁(𝛽+𝛾)𝑀𝛽+π›Ύβˆ’1πœ‰π‘Žβˆ—+π‘πœ‰π›Ύβˆ—ξ€Έ<πœ‰,(3.3) where π‘€πœ‰ξ€½π‘”=max(π‘₯,𝑦,𝑧)∢0≀π‘₯β‰€πœ‰,0β‰€π‘¦β‰€π‘˜βˆ—πœ‰,0β‰€π‘§β‰€β„Žβˆ—πœ‰ξ€Ύ,π‘πœ‰=max{𝐹(π‘₯)∢0≀π‘₯β‰€πœ‰}.(3.4) (for 𝑔(π‘₯,𝑦,𝑧),𝐹(π‘₯),π‘Žβˆ— and π›Ύβˆ—, see conditions (𝐻2) and (𝐻3)). Then IBVP(1.5) has at least two positive solutions π‘’βˆ—,π‘’βˆ—βˆ—βˆˆπ‘„βˆ©πΆ1[𝐽′,𝐸] such that 0<||π‘’βˆ—||𝐡<πœ‰<||π‘’βˆ—βˆ—||𝐡.

Proof. By Lemmas 2.3, 2.4, and Corollary 2.5, operator 𝐴 defined by (2.5) is completely continuous from 𝑄 into 𝑄, and we need to prove that 𝐴 has two fixed points π‘’βˆ— and π‘’βˆ—βˆ— in 𝑄 such that 0<||π‘’βˆ—||𝐡<πœ‰<||π‘’βˆ—βˆ—||𝐡.
By condition (𝐻5), there exists an π‘Ÿ1>0 such that 𝜏(𝑒)β‰₯𝛽(𝛽+π›Ύβˆ’1)𝑁2(1βˆ’π›Ύ)2π‘βˆ—β€–β€–π‘’0‖‖‖𝑒‖,βˆ€π‘’βˆˆπ‘ƒ,‖𝑒‖β‰₯π‘Ÿ1,(3.5) where 𝑁 denotes the normal constant of 𝑃, so, 𝑓(𝑑,𝑒,𝑣,𝑀)β‰₯𝛽(𝛽+π›Ύβˆ’1)𝑁2‖𝑒‖(1βˆ’π›Ύ)2π‘βˆ—β€–β€–π‘’0‖‖𝑏(𝑑)𝑒0|||||,βˆ€π‘‘βˆˆπ½,𝑒,𝑣,π‘€βˆˆπ‘ƒ,𝑒|β‰₯π‘Ÿ1.(3.6) Choose π‘Ÿ2ξ€½>max𝑁𝛽(1βˆ’π›Ύ)βˆ’1π‘Ÿ1ξ€Ύ,πœ‰.(3.7) For π‘’βˆˆπ‘„,||𝑒||𝐡=π‘Ÿ2; we have by (2.62) and (3.7), ‖𝑒(𝑑)β€–β‰₯π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)‖𝑒‖𝐡=π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)π‘Ÿ2>π‘Ÿ1,βˆ€π‘‘βˆˆπ½,(3.8) so, (2.5), (3.8), (3.6), and (2.62) imply (𝐴𝑒)(𝑑)β‰₯1βˆ’π›Ύξ‚΅ξ€œπ›½+π›Ύβˆ’1∞0ξ‚Άβ‰₯𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠𝛽𝑁2(1βˆ’π›Ύ)π‘βˆ—β€–β€–π‘’0β€–β€–ξ‚΅ξ€œβˆž0𝑒‖𝑒(𝑠)‖𝑏(𝑠)𝑑𝑠0β‰₯π‘β€–π‘’β€–π΅π‘βˆ—β€–β€–π‘’0β€–β€–ξ‚΅ξ€œβˆž0𝑒𝑏(𝑠)𝑑𝑠0=𝑁‖𝑒‖𝐡‖‖𝑒0‖‖𝑒0,βˆ€π‘‘βˆˆπ½,(3.9) and consequently, ||||||||𝐴𝑒𝐡β‰₯|||||𝑒|𝐡||||,βˆ€π‘’βˆˆπ‘„,|𝑒|𝐡=π‘Ÿ2.(3.10) Similarly, by condition (𝐻6), there exists π‘Ÿ3>0 such that 𝜎(𝑒)β‰₯𝛽(𝛽+π›Ύβˆ’1)𝑁2(1βˆ’π›Ύ)2π‘βˆ—||||𝑒1|||||||||||||𝑒|,βˆ€π‘’βˆˆπ‘ƒ,0<|𝑒|<π‘Ÿ3,(3.11) so, 𝑓(𝑑,𝑒,𝑣,𝑀)β‰₯𝛽(𝛽+π›Ύβˆ’1)𝑁2|||||𝑒|(1βˆ’π›Ύ)2π‘βˆ—||||𝑒1||||𝑐(𝑑)𝑒1||||,βˆ€π‘‘βˆˆπ½,𝑒,𝑣,π‘€βˆˆπ‘ƒ,0<|𝑒|<π‘Ÿ3.(3.12) Choose 0<π‘Ÿ4ξ€½π‘Ÿ<min3ξ€Ύ,πœ‰.(3.13) For π‘’βˆˆπ‘„,||𝑒||𝐡=π‘Ÿ4, we have by (3.13) and (2.62), π‘Ÿ3>||||𝑒||||(𝑑)β‰₯π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)‖𝑒‖𝐡=π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)π‘Ÿ4>0,(3.14) so, similar to (3.9), we get by (2.5), (3.12), and (3.14) (𝐴𝑒)(𝑑)β‰₯1βˆ’π›Ύξ‚΅ξ€œπ›½+π›Ύβˆ’1∞0ξ‚Άβ‰₯𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠𝛽𝑁2(1βˆ’π›Ύ)π‘βˆ—||||𝑒1||||ξ‚΅ξ€œβˆž0||||||||𝑒𝑒(𝑠)𝑐(𝑠)𝑑𝑠1β‰₯π‘β€–π‘’β€–π΅π‘βˆ—||||𝑒1||||ξ‚΅ξ€œβˆž0𝑐𝑒(𝑠)𝑑𝑠1=𝑁‖𝑒‖𝐡||||𝑒1||||𝑒1,βˆ€π‘‘βˆˆπ½;(3.15) hence ‖𝐴𝑒‖𝐡β‰₯‖𝑒‖𝐡||||,βˆ€π‘’βˆˆπ‘„,|𝑒|=π‘Ÿ4.(3.16) On the other hand, for π‘’βˆˆπ‘„, ||𝑒||𝐡=πœ‰, by condition (𝐻2), condition (𝐻3), (3.4), we have ‖𝑓(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))β€–β‰€π‘€πœ‰β€–β€–πΌπ‘Ž(𝑑),βˆ€π‘‘βˆˆπ½,(3.17)π‘˜ξ€·π‘’ξ€·π‘‘π‘˜β€–β€–ξ€Έξ€Έβ‰€π‘πœ‰π›Ύπ‘˜(π‘˜=1,2,3,…).(3.18) It is clear that (𝐴𝑒)(𝑑)≀𝛽+π›Ύξƒ©ξ€œπ›½+π›Ύβˆ’1∞0𝑓(𝑠,𝑒(𝑠),(𝑇𝑒)(𝑠),(𝑆𝑒)(𝑠))𝑑𝑠+βˆžξ“π‘˜=1πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξƒͺξ€Έξ€Έβˆ€π‘‘βˆˆπ½.(3.19) It follows from (3.17)–(3.19) that ‖𝐴𝑒‖𝐡≀𝑁(𝛽+𝛾)𝑀𝛽+π›Ύβˆ’1πœ‰π‘Žβˆ—+π‘πœ‰π›Ύβˆ—ξ€Έ.(3.20) Thus, (3.20) and (3.3) imply ‖𝐴𝑒‖𝐡<‖𝑒‖𝐡,βˆ€π‘’βˆˆπ‘„,‖𝑒‖𝐡=πœ‰.(3.21) From (3.7) and (3.13), we know 0<π‘Ÿ4<πœ‰<π‘Ÿ2; hence, (3.10), (3.16), (3.21), and Lemma 2.6 imply that 𝐴 has two fixed points π‘’βˆ—,π‘’βˆ—βˆ—βˆˆπ‘„ such that π‘Ÿ4<||π‘’βˆ—||𝐡<πœ‰<||π‘’βˆ—βˆ—||𝐡<π‘Ÿ2. The proof is complete.

Theorem 3.3. Let cone 𝑃 be normal and conditions (𝐻1)–(𝐻5) satisfied. Assume that 𝑔(π‘₯,𝑦,𝑧)π‘₯+𝑦+π‘§βŸΆ0asπ‘₯+𝑦+π‘§βŸΆ0+,𝐹(π‘₯)π‘₯⟢0asπ‘₯⟢0+.(3.22) (for 𝑔(π‘₯,𝑦,𝑧) and 𝐹(π‘₯), see conditions (𝐻2) and (𝐻3)). Then IBVP(1.5) has at least one positive solution π‘’βˆ—βˆˆπ‘„βˆ©πΆ1[𝐽′,𝐸].

Proof. As in the proof of Theorem 3.2, we can choose π‘Ÿ2>0 such that (3.10) holds (in this case, we only choose π‘Ÿ2>𝑁𝛽(1βˆ’π›Ύ)βˆ’1π‘Ÿ1 instead of (3.7)). On the other hand, by (3.22), there exists π‘Ÿ5>0 such that 𝑔(π‘₯,𝑦,𝑧)β‰€πœ–0(π‘₯+𝑦+𝑧),βˆ€0<π‘₯+𝑦+𝑧<π‘Ÿ5,𝐹(π‘₯)β‰€πœ–0π‘₯,βˆ€0<π‘₯<π‘Ÿ5,(3.23) where πœ–0=𝛽+π›Ύβˆ’1𝑁(𝛽+𝛾)ξ€Ίξ€·1+π‘˜βˆ—+β„Žβˆ—ξ€Έπ‘Žβˆ—+π›Ύβˆ—ξ€».(3.24) Choose 0<π‘Ÿ6ξ‚†π‘Ÿ<min51+π‘˜βˆ—+β„Žβˆ—,π‘Ÿ2.(3.25) For π‘’βˆˆπ‘„,||𝑒||𝐡=π‘Ÿ6, we have by (2.62) and (3.25), 0<π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)π‘Ÿ6≀||||𝑒||||(𝑑)β‰€π‘Ÿ6<π‘Ÿ5,βˆ€π‘‘βˆˆπ½,0<π‘βˆ’1π›½βˆ’1(1βˆ’π›Ύ)π‘Ÿ6≀||||||||+||||||||+||||||||≀𝑒(𝑑)(𝑇𝑒)(𝑑)(𝑆𝑒)(𝑑)1+π‘˜βˆ—+β„Žβˆ—ξ€Έπ‘Ÿ6<π‘Ÿ5,βˆ€π‘‘βˆˆπ½,(3.26) so, (3.23) imply 𝑔||||𝑒||||,||||||||,||||||||ξ€Έ(𝑑)(𝑇𝑒)(𝑑)(𝑆𝑒)(𝑑)β‰€πœ–0ξ€·||||𝑒||||+||||||||+||||||||ξ€Έ(𝑑)(𝑇𝑒)(𝑑)(𝑆𝑒)(𝑑)β‰€πœ–0ξ€·1+π‘˜βˆ—+β„Žβˆ—ξ€Έπ‘Ÿ6𝐹||||𝑒𝑑,βˆ€π‘‘βˆˆπ½,π‘˜ξ€Έ||||ξ€Έβ‰€πœ–0||||π‘’ξ€·π‘‘π‘˜ξ€Έ||||β‰€πœ–0π‘Ÿ6,(π‘˜=1,2,3,…).(3.27) It follows from (3.19), condition (𝐻2), condition (𝐻3), (3.27), and (3.24) that ||||||||≀(𝐴𝑒)(𝑑)𝑁(𝛽+𝛾)ξƒ―πœ–π›½+π›Ύβˆ’10ξ€·1+π‘˜βˆ—+β„Žβˆ—ξ€Έπ‘Ÿ6ξ€œβˆž0π‘Ž(𝑠)𝑑𝑠+πœ–0π‘Ÿ6βˆžξ“π‘˜=1π›Ύπ‘˜ξƒ°=𝑁(𝛽+𝛾)πœ–0π‘Ÿ6𝛽+π›Ύβˆ’1ξ€½ξ€·1+π‘˜βˆ—+β„Žβˆ—ξ€Έπ‘Žβˆ—+π›Ύβˆ—ξ€Ύ=π‘Ÿ6,βˆ€π‘‘βˆˆπ½,(3.28) and consequently, ‖𝐴𝑒‖𝐡≀‖𝑒‖𝐡,βˆ€π‘’βˆˆπ‘„,‖𝑒‖𝐡=π‘Ÿ6.(3.29) Since 0<π‘Ÿ6<π‘Ÿ2 by virtue of (3.25), we conclude from (3.10), (3.29), and Lemma 2.6 that 𝐴 has a fixed point π‘’βˆ—βˆˆπ‘„ such that π‘Ÿ6≀||π‘’βˆ—||π΅β‰€π‘Ÿ2. The theorem is proved.

Example 3.4. Consider the infinite system of scalar first-order impulsive integrodifferential equations of mixed type on the half line: π‘’ξ…žπ‘›1(𝑑)=8𝑛2π‘’βˆ’5π‘‘βŽ›βŽœβŽœβŽξƒ¬π‘’π‘›+1(𝑑)+βˆžξ“π‘š=1π‘’π‘šξƒ­(𝑑)2+ξ„Άξ„΅ξ„΅βŽ·3𝑒2𝑛(𝑑)+βˆžξ“π‘š=1π‘’π‘šβŽžβŽŸβŽŸβŽ +1(𝑑)9𝑛3π‘’βˆ’6π‘‘ξƒ―ξ‚΅ξ€œπ‘‘0π‘’βˆ’(𝑑+1)𝑠𝑒𝑛(𝑠)𝑑𝑠2+ξ‚΅ξ€œβˆž0𝑒𝑛+2(𝑠)𝑑𝑠(1+𝑑+𝑠)2ξ‚Ά3ξƒ°,βˆ€0≀𝑑<∞,π‘‘β‰ π‘˜(π‘˜=1,2,3,…;𝑛=1,2,3,…),Δ𝑒𝑛||𝑑=π‘˜=16𝑛23βˆ’π‘˜ξ‚€ξ€Ίπ‘’π‘›ξ€»(π‘˜)2+𝑒𝑛+2ξ€»(π‘˜)21,(π‘˜=1,2,3,…;𝑛=1,2,3,…),𝑒(∞)=2𝑒𝑛92+6𝑒𝑛(0),(𝑛=1,2,3,…).(3.30) Evidently, 𝑒𝑛(𝑑)≑0(𝑛=1,2,3,…) is the trivial solution of infinite system (3.30).

Conclusion. Infinite system (3.30) has at least two positive solutions {π‘’βˆ—π‘›(𝑑)} (𝑛=1,2,3,…) and {π‘’π‘›βˆ—βˆ—(𝑑)} (𝑛=1,2,3,…) such that 0<inf∞0≀𝑑<βˆžξ“π‘›=1π‘’βˆ—π‘›(𝑑)≀sup∞0≀𝑑<βˆžξ“π‘›=1π‘’βˆ—π‘›(𝑑)<1<sup∞0≀𝑑<βˆžξ“π‘›=1π‘’π‘›βˆ—βˆ—(𝑑),inf∞0≀𝑑<βˆžξ“π‘›=1π‘’π‘›βˆ—βˆ—(𝑑)>0.(3.31)

Proof. Let 𝐸=𝑙1={𝑒=(𝑒1,…,π‘’π‘›βˆ‘,…)βˆΆπ‘›=1∞|𝑒𝑛|<∞} with norm βˆ‘||𝑒||=βˆžπ‘›=1|𝑒𝑛| and 𝑃=(𝑒1,…,𝑒𝑛,…)βˆΆπ‘’π‘›β‰₯0,𝑛=1,2,3,…}. Then 𝑃 is a normal cone in 𝐸 with normal constant 𝑁=1, and infinite system (3.30) can be regarded as an infinite three-point boundary value problem of form (1.5). In this situation, 𝑒=(𝑒1,…,𝑒𝑛,…), 𝑣=(𝑣1,…,𝑣𝑛,…), 𝑀=(𝑀1,…,𝑀𝑛,…), π‘‘π‘˜=π‘˜(π‘˜=1,2,3,…), 𝐾(𝑑,𝑠)=π‘’βˆ’(𝑑+1)𝑠, 𝐻(𝑑,𝑠)=(1+𝑑+𝑠)βˆ’2, πœ‚=9/2, 𝛾=1/2, 𝛽=6, 𝑓=(𝑓1,…,𝑓𝑛,…), and πΌπ‘˜=(πΌπ‘˜1,…,πΌπ‘˜π‘›β€¦), in which 𝑓𝑛1(𝑑,𝑒,𝑣,𝑀)=8𝑛2π‘’βˆ’5π‘‘βŽ›βŽœβŽœβŽξƒ¬π‘’π‘›+1+βˆžξ“π‘š=1π‘’π‘šξƒ­2+ξ„Άξ„΅ξ„΅βŽ·3𝑒2𝑛+βˆžξ“π‘š=1π‘’π‘šβŽžβŽŸβŽŸβŽ +19𝑛3π‘’βˆ’6𝑑𝑣2𝑛+𝑀3𝑛+2ξ€Έ,[πΌβˆ€π‘‘βˆˆπ½=0,∞),𝑒,𝑣,π‘€βˆˆπ‘ƒ(𝑛=1,2,3,…),(3.32)π‘˜π‘›(1𝑒)=6𝑛23βˆ’π‘˜ξ€·π‘’2𝑛+𝑒22𝑛+1ξ€Έ,βˆ€π‘’βˆˆπ‘ƒ(π‘˜=1,2,3,…;𝑛=1,2,3,…).(3.33) It is easy to see that π‘“βˆˆπΆ[𝐽×𝑃×𝑃×𝑃,𝑃], πΌπ‘˜βˆˆπΆ[𝑃,𝑃] (