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

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 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that 𝜃≤𝑥≤𝑦 implies ||𝑥||≤𝑁||𝑦||, where 𝜃 denotes the zero element of 𝐸 and the smallest 𝑁 is called the normal constant of 𝑃. If 𝑥≤𝑦 and 𝑥≠𝑦, we write 𝑥<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: 𝑢^(𝑡Δ)𝑢=||𝑓_{𝑡=𝑡𝑘}(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)),∀𝑡∈𝐽′,=𝐼_𝑘𝑢𝑢(∞𝑡)_𝑘=𝛽𝑢((𝑘0)=,1,2,3,…),(1.1) where 𝐽=[0,∞), 0<𝑡₁<⋯<𝑡_𝑘<…,𝑡_𝑘→∞, 𝐽′=𝐽⧵{𝑡₁,…,𝑡_𝑘,…}, 𝑓∈𝐶[𝐽×𝑃×𝑃×𝑃,𝑃], 𝐼_𝑘∈𝐶[𝑃,𝑃] (𝑘=1,2,3,…), 𝛽>1, 𝑢(∞)=lim_𝑡→∞𝑢(𝑡), and (𝑇𝑢)(𝑡)=^𝑡0𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,(𝑆𝑢)(𝑡)=^∞0𝐻(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,(1.2)𝐾∈𝐶[𝐷,𝑅₊], 𝐷={(𝑡,𝑠)∈𝐽×𝐽∶𝑡≥𝑠}, 𝐻∈𝐶[𝐽×𝐽,𝑅₊],𝑅₊ denotes the set of all nonnegative numbers. Δ𝑢|_{𝑡=𝑡𝑘} denotes the jump of 𝑢(𝑡) at 𝑡=𝑡_𝑘, that is, Δ𝑢||_{𝑡=𝑡𝑘}=𝑢𝑡^+𝑘−𝑢𝑡^−𝑘,(1.3) where 𝑢(𝑡^+𝑘) and 𝑢(𝑡^−𝑘) represent the right and left limits of 𝑢(𝑡) at 𝑡=𝑡_𝑘, 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 𝑐∈𝐶[𝐽,𝑅₊]∩𝐿[𝐽,𝑅₊] such that ‖𝑓(𝑡,𝑢,𝑣,𝑤)‖𝑐(𝑡)(‖𝑢‖+‖𝑣‖+‖𝑤‖)⟶0as𝑢,𝑣,𝑤∈𝑃,‖𝑢‖+‖𝑣‖+‖𝑤‖⟶∞(1.4) uniformly for 𝑡∈𝐽 (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: 𝑢^(𝑡Δ)𝑢=||𝑓_{𝑡=𝑡𝑘}(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)),∀𝑡∈𝐽′,=𝑢(𝐼∞_𝑘)𝑢=𝑡𝛾_𝑘𝑢(𝜂)+(𝑘𝛽=𝑢(10,)2,,3,…),(1.5) where 0≤𝛾<1,𝛽+𝛾>1, and 𝑡_𝑚−1<𝜂≤𝑡_𝑚 (for some 𝑚). It is clear that IBVP(1.5) includes IBVP(1.1) as a special case when 𝛾=0.

Let PC[𝐽,𝐸]={𝑢∶𝑢 is a map from 𝐽 into 𝐸 such that 𝑢(𝑡) is continuous at 𝑡≠𝑡_𝑘, left continuous at 𝑡=𝑡_𝑘, and 𝑢(𝑡^+𝑘) exists, 𝑘=1,2,3,…} and BPC[𝐽,𝐸]={𝑢∈PC[𝐽,𝐸]∶sup_𝑡∈𝐽||𝑢(𝑡)||<∞}. It is clear that BPC[𝐽,𝐸] is a Banach space with norm ‖𝑢‖_𝐵=sup_𝑡∈𝐽‖𝑢(𝑡)‖.(1.6) Let BPC[𝐽,𝑃]={𝑢∈BPC[𝐽,𝐸]∶𝑢(𝑡)≥𝜃,∀𝑡∈𝐽} and 𝑄={𝑢∈BPC[𝐽,𝑃]∶𝑢(𝑡)≥𝛽⁻¹(1−𝛾)𝑢(𝑠),∀𝑡,𝑠∈𝐽}. Obviously, BPC[𝐽,𝑃] and 𝑄 are two cones in space BPC[𝐽,𝐸] and 𝑄⊂BPC[𝐽,𝑃]. 𝑢∈BPC[𝐽,𝑃]∩𝐶¹[𝐽′,𝐸] is called a positive solution of IBVP(1.5) if 𝑢(𝑡)>𝜃 for 𝑡∈𝐽 and 𝑢(𝑡) satisfies (1.5).

2. Several Lemmas

Let us list some conditions.(𝐻₁)sup_𝑡∈𝐽∫^𝑡0𝐾(𝑡,𝑠)𝑑𝑠<∞,sup_𝑡∈𝐽∫^∞0𝐻(𝑡,𝑠)𝑑𝑠<∞, and lim_{𝑡→𝑡}^∞0||𝐻𝑡^,𝑠−𝐻(𝑡,𝑠)||𝑑𝑠=0,∀𝑡∈𝐽.(2.1) In this case, let 𝑘^∗=sup_𝑡∈𝐽^𝑡0𝐾(𝑡,𝑠)𝑑𝑠,ℎ^∗=sup_𝑡∈𝐽^∞0𝐻(𝑡,𝑠)𝑑𝑠.(2.2)(𝐻₂)There exist 𝑎∈𝐶[𝐽,𝑅₊] and 𝑔∈𝐶[𝑅₊×𝑅₊×𝑅₊,𝑅₊] such that ‖𝑓(𝑡,𝑢,𝑣,𝑤)‖≤𝑎(𝑡)𝑎𝑔(‖𝑢‖,‖𝑣‖,‖𝑤‖),∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,_∗=^∞0𝑎(𝑡)𝑑𝑡<∞.(2.3)(𝐻₃)There exist 𝛾_𝑘≥0 (𝑘=1,2,3,…) and 𝐹∈𝐶[𝑅₊,𝑅₊] such that ‖‖𝐼_𝑘(𝑢)‖‖≤𝛾_𝑘𝐹(‖𝛾𝑢‖),∀𝑢∈𝑃(𝑘=1,2,3,…),_∗=^∞_𝑘=1𝛾_𝑘<∞.(2.4)(𝐻₄)For any 𝑡∈𝐽 and 𝑟>0,𝑓(𝑡,𝑃_𝑟,𝑃_𝑟,𝑃_𝑟)={𝑓(𝑡,𝑢,𝑣,𝑤)∶𝑢,𝑣,𝑤∈𝑃_𝑟} and 𝐼_𝑘(𝑃_𝑟)={𝐼_𝑘(𝑢)∶𝑢∈𝑃_𝑟} (𝑘=1,2,3,…) are relatively compact in 𝐸, where 𝑃_𝑟={𝑢∈𝑃∶||𝑢||≤𝑟}.

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||≤𝑘^∗,||𝑆||≤ℎ^∗; moreover, we have 𝑇(BPC[𝐽,𝑃])⊂BPC[𝐽,𝑃] and 𝑆(BPC[𝐽,𝑃])⊂BPC[𝐽,𝑃].

We shall reduce IBVP(1.5) to an impulsive integral equation. To this end, we consider the operator 𝐴 defined by (𝐴𝑢)(𝑡)=1𝛽+𝛾−1^∞𝜂×𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)_𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1+𝐼_𝑘𝑢𝑡_𝑘_𝑡0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡_𝑘,∀𝑡∈𝐽.(2.5) In what follows, we write 𝐽₁=[0,𝑡₁],𝐽_𝑘=(𝑡_𝑘−1,𝑡_𝑘] (𝑘=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 𝐵𝑃𝐶[𝐽,𝑃] into 𝑄.

Proof. Let 𝑟>0 be given. Let 𝑀_𝑟=max𝑔(𝑥𝑁,𝑦,𝑧)∶0≤𝑥≤𝑟,0≤𝑦≤𝑘^∗𝑟,0≤𝑧≤ℎ^∗𝑟,(2.6)_𝑟=max{𝐹(𝑥)∶0≤𝑥≤𝑟}.(2.7) For 𝑢∈BPC[𝐽,𝑃],||𝑢||_𝐵≤𝑟, we see that by virtue of condition (𝐻₂) and (2.6), ‖𝑓(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))‖≤𝑀_𝑟𝑎(𝑡),∀𝑡∈𝐽,(2.8) which implies the convergence of the infinite integral ^∞0‖‖‖𝑓(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))𝑑𝑡,(2.9)_∞0𝑓(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))𝑑𝑡‖‖‖≤^∞0‖𝑓(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))‖𝑑𝑡≤𝑀_𝑟𝑎^∗.(2.10) On the other hand, condition (𝐻₃) and (2.7) imply the convergence of the infinite series ^∞_𝑘=1‖‖‖‖𝐼_𝑘𝑢𝑡_𝑘,(2.11)_∞_𝑘=1𝐼_𝑘𝑢𝑡_𝑘‖‖‖‖≤^∞_𝑘=1‖‖𝐼_𝑘𝑢𝑡_𝑘‖‖≤𝑁_𝑟𝛾^∗.(2.12) It follows from (2.5), (2.10), and (2.12) that ||||(𝐴𝑢)(𝑡)||||≤1𝛽+𝛾−1^∞𝜂×‖𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠+(1−𝛾)_𝜂0+‖𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠_∞𝑘_=𝑚‖‖𝐼_𝑘𝑢𝑡_𝑘‖‖+(1−𝛾)^𝑚−1_𝑘=1+‖‖𝐼_𝑘𝑢𝑡_𝑘_𝑡0‖‖≤‖1𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠+_{0<𝑡𝑘<𝑡}‖‖𝐼_𝑘𝑢𝑡_𝑘‖‖𝛽+𝛾−1^∞0‖𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠+^∞_𝑘=1+||||𝐼_𝑘𝑢𝑡_𝑘_∞0||||‖𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠+^∞_𝑘=1=𝛽+𝛾||||𝐼_𝑘𝑢𝑡_𝑘||||𝛽+𝛾−1^∞0‖𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))‖𝑑𝑠+^∞_𝑘=1≤𝛽+𝛾||||𝐼_𝑘𝑢𝑡_𝑘||||𝛽+𝛾−1𝑀^𝑟𝑎^∗+𝑁_𝑟𝛾^∗,∀𝑡∈𝐽,(2.13) which implies that 𝐴𝑢∈BPC[𝐽,𝑃] and ‖𝐴𝑢‖_𝐵≤𝛽+𝛾𝛽+𝛾−1𝑀^𝑟𝑎^∗+𝑁_𝑟𝛾^∗.(2.14) Moreover, by (2.5), we have (𝐴𝑢)(𝑡)≥1𝛽+𝛾−1^∞𝜂×𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)_𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1∀𝑡∈𝐽,𝐼_𝑘𝑢𝑡_𝑘,(2.15)(𝐴𝑢)(𝑡)≤1𝛽+𝛾−1^∞𝜂×𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)_𝜂0+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1+𝐼_𝑘𝑢𝑡_𝑘_∞0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘=1𝐼_𝑘𝑢𝑡_𝑘,∀𝑡∈𝐽.(2.16) It is clear that ^∞0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘=1≤1𝐼_𝑘𝑢𝑡_𝑘1−𝛾^∞𝜂+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)^𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1𝐼_𝑘𝑢𝑡_𝑘,(2.17) so, (2.16) and (2.17) imply (𝐴𝑢)(𝑡)≤1×𝛽+𝛾−1+1−1𝛾_∞𝜂+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)^𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1𝐼_𝑘𝑢𝑡_𝑘,∀𝑡∈𝐽.(2.18) It follows from (2.15) and (2.18) that (𝐴𝑢)(𝑡)≥1𝛽+𝛾−1𝛽+1𝛾−1+1−1𝛾⁻¹(𝐴𝑢)(𝑠)=𝛽⁻¹(1−𝛾)(𝐴𝑢)(𝑠),∀𝑡,𝑠∈𝐽.(2.19) Hence, 𝐴𝑢∈𝑄. That is, 𝐴 maps BPC[𝐽,𝑃] into 𝑄.
Now, we are going to show that 𝐴 is continuous. Let 𝑢_𝑛,𝑢∈BPC[𝐽,𝑃],||𝑢_𝑛−𝑢||_𝐵→0 (𝑛→∞). Then 𝑟=sup_𝑛||𝑢_𝑛||_𝐵<∞ and ||𝑢||_𝐵≤𝑟. Similar to (2.14), it is easy to get ‖‖𝐴𝑢_𝑛−𝐴𝑢‖‖_𝐵≤𝛽+𝛾𝛽+𝛾−1^∞0+‖‖𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)−𝑓𝑠,𝑢(𝑠),𝑇𝑢(𝑠),𝑆𝑢(𝑠)𝑑𝑠_∞‖‖_𝑘=1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖(𝑛=1,2,3,…).(2.20) It is clear that 𝑓𝑡,𝑢_𝑛(𝑡),𝑇𝑢_𝑛(𝑡),𝑆𝑢_𝑛(𝑡)⟶𝑓𝑡,𝑢(𝑡),𝑇𝑢(𝑡),𝑆𝑢(𝑡)as𝑛⟶∞,∀𝑡∈𝐽.(2.21) Moreover, we see from (2.8) that ‖‖𝑓≤𝑡,𝑢2_𝑛𝑀(𝑡_𝑟),𝑇𝑢_𝑛(𝑡),𝑆𝑢_𝑛(𝑡)−𝑓𝑡,𝑢(𝑡),𝑇𝑢(𝑡),𝑆𝑢(𝑡)‖‖𝑎(𝑡)=𝜎(𝑡),∀𝑡∈𝐽(𝑛=1,2,3,…);𝜎∈𝐿𝐽,𝑅₊.(2.22) It follows from (2.21), (2.22) and the dominated convergence theorem that lim_𝑛→∞^∞0‖‖𝑓𝑡,𝑢_𝑛(𝑡),𝑇𝑢_𝑛(𝑡),𝑆𝑢_𝑛(𝑡)−𝑓𝑡,𝑢(𝑡),𝑇𝑢(𝑡),𝑆𝑢(𝑡)‖‖𝑑𝑡=0.(2.23) On the other hand, for any 𝜖>0, we can choose a positive integer 𝑗 such that 𝑁_𝑟∞𝑘_=𝑗+1𝛾_𝑘<𝜖.(2.24) And then, choose a positive integer 𝑛₀ such that ^𝑗_𝑘=1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖<𝜖,∀𝑛>𝑛₀.(2.25) From (2.24) and (2.25), we get ^∞_𝑘=1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖<𝜖+2𝑁_𝑟∞𝑘_=𝑗+1𝛾_𝑘<3𝜖,∀𝑛>𝑛₀,(2.26) hence lim_{∞𝑛→∞}_𝑘=1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖=0.(2.27) It follows from (2.20), (2.23), and (2.51) that ||𝐴𝑢_𝑛−𝐴𝑢||_𝐵→0as𝑛→∞, and the continuity of 𝐴 is proved.
Finally, we prove that 𝐴 is compact. Let 𝑉={𝑢_𝑛}⊂BPC[𝐽,𝑃] be bounded and ||𝑢_𝑛||_𝐵≤𝑟 (𝑛=1,2,3,…). Consider 𝐽_𝑖=(𝑡_𝑖−1,𝑡_𝑖] for any fixed 𝑖. By (2.5) and (2.8), we have ‖‖𝐴𝑢_𝑛𝑡^−𝐴𝑢_𝑛(𝑡)‖‖≤≤𝑀^𝑡𝑡_𝑟‖‖𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)‖‖𝑑𝑠^𝑡′_𝑡𝑎(𝑠)𝑑𝑠,∀𝑡,𝑡^∈𝐽_𝑖,𝑡^>𝑡(𝑛=1,2,3,…),(2.28) which implies that the functions {𝑤_𝑛(𝑡)} (𝑛=1,2,3,…) defined by 𝑤_𝑛(𝑡)=⎧⎪⎨⎪⎩𝐴𝑢^𝑛𝐴𝑢_𝑛(𝑡),∀𝑡∈𝐽_𝑖=𝑡_𝑖−1,𝑡_𝑖,𝑡^+𝑖−1,∀𝑡=𝑡_𝑖−1(𝑛=1,2,3,…)(2.29) ((𝐴𝑢_𝑛)(𝑡^+𝑖−1) denotes the right limit of (𝐴𝑢_𝑛)(𝑡) at 𝑡=𝑡_𝑖−1) are equicontinuous on 𝐽_𝑖=[𝑡_𝑖−1,𝑡_𝑖]. On the other hand, for any 𝜖>0, choose a sufficiently large 𝜏>𝜂 and a sufficiently large positive integer 𝑗>𝑚 such that 𝑀_𝑟^∞𝜏𝑎(𝑠)𝑑𝑠<𝜖,𝑁_𝑟∞𝑘_=𝑗+1𝛾_𝑘<𝜖.(2.30) We have, by (2.29), (2.5), (2.8), (2.30), and condition (𝐻₃), 𝑤_𝑛=(𝑡)1𝛽+𝛾−1^𝜏𝜂+𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)𝑑𝑠_∞𝜏×𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)𝑑𝑠+(1−𝛾)_𝜂0𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)𝑑𝑠+^𝑗_𝑘_=𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘+^∞_𝑘_=𝑗+1+(1−𝛾)_𝑚−1𝐼_𝑘𝑢_𝑛𝑡_𝑘_𝑘=1+𝐼_𝑘𝑢_𝑛𝑡_𝑘_𝑡0+𝑓𝑠,𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)𝑑𝑠_𝑖−1_𝑘=1𝐼_𝑘‖‖‖𝑢_𝑛𝑡_𝑘,∀𝑡∈𝐽_𝑖(𝑛=1,2,3,…),(2.31)_∞𝜏𝑓𝑠,‖‖‖‖𝑢_𝑛(𝑠),𝑇𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)𝑑𝑠‖‖‖<𝜖(𝑛=1,2,3,…),(2.32)_∞𝑘_=𝑗+1𝐼_𝑘𝑢_𝑛𝑡_𝑘‖‖‖‖<𝜖(𝑛=1,2,3,…).(2.33) It follows from (2.31), (2.32), (2.33), (2.8), and [4, Theorem 1.2.3] that 𝛼(𝑊(𝑡))≤1𝛽+𝛾−12^𝜏𝜂+2(1−𝛼𝛾()𝑓_𝜂0(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))𝑑𝑠+2𝜖+𝛼(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))𝑑𝑠_𝑗𝑘_=𝑚𝛼𝐼_𝑘𝑉𝑡_𝑘+2𝜖+(1−𝛾)^𝑚−1_𝑘=1+2_𝑡0𝛼𝐼_𝑘𝑉𝑡_𝑘𝛼(𝑓(𝑠,𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))𝑑𝑠+^𝑖−1_𝑘=1𝛼𝐼_𝑘𝑉𝑡_𝑘,∀𝑡∈𝐽_𝑖,(2.34) where 𝑊(𝑡)={𝑤_𝑛(𝑡)∶𝑛=1,2,3,…}, 𝑉(𝑠)={𝑢_𝑛(𝑠)∶𝑛=1,2,3,…}, (𝑇𝑉)(𝑠)={(𝑇𝑢_𝑛)(𝑠)∶𝑛=1,2,3,…}, (𝑆𝑉)(𝑠)={(𝑆𝑢_𝑛)(𝑠)∶𝑛=1,2,3,…} and 𝛼(𝑈) denotes the Kuratowski measure of noncompactness of bounded set 𝑈⊂𝐸 (see [4, Section 1.2]). Since 𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)⊂𝑃_𝑟∗ for 𝑠∈𝐽, where 𝑟^∗=max{𝑟,𝑘^∗𝑟,ℎ^∗𝑟}, we see that, by condition (𝐻₄), 𝛼(𝑓𝛼(𝑠,𝐼𝑉(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)))=0,∀𝑡∈𝐽,(2.35)_𝑘𝑉𝑡_𝑘=0(𝑘=1,2,3,…).(2.36) It follows from (2.34)–(2.36) that 𝛼(𝑊(𝑡))≤4𝜖𝛽+𝛾−1,∀𝑡∈𝐽_𝑖,(2.37) which implies by virtue of the arbitrariness of 𝜖 that 𝛼(𝑊(𝑡))=0 for 𝑡∈𝐽_𝑖.
By Ascoli-Arzela theorem (see [4, Theorem 1.2.5]), we conclude that 𝑊={𝑤_𝑛∶𝑛=1,2,3,…} is relatively compact in 𝐶[𝐽_𝑖,𝐸]; hence, {𝑤_𝑛(𝑡)} has a subsequence which is convergent uniformly on 𝐽_𝑖, so, {(𝐴𝑢_𝑛(𝑡)} has a subsequence which is convergent uniformly on 𝐽_𝑖. Since 𝑖 may be any positive integer, so, by diagonal method, we can choose a subsequence {(𝐴𝑢_𝑛𝑖)(𝑡)} of {(𝐴𝑢_𝑛)(𝑡)} such that {(𝐴𝑢_𝑛𝑖)(𝑡)} is convergent uniformly on each 𝐽_𝑘 (𝑘=1,2,3,…). Let lim_𝑖→∞𝐴𝑢_𝑛𝑖(𝑡)=𝑣(𝑡),∀𝑡∈𝐽.(2.38) It is clear that 𝑣∈PC[𝐽,𝑃]. By (2.14), we have ‖‖𝐴𝑢_𝑛𝑖‖‖_𝐵≤𝛽+𝛾𝛽+𝛾−1𝑀^𝑟𝑎^∗+𝑁_𝑟𝛾^∗,(𝑖=1,2,3,…),(2.39) which implies that 𝑣∈BPC[𝐽,𝑃] and ‖𝑣‖_𝐵≤𝛽+𝛾𝛽+𝛾−1𝑀^𝑟𝑎^∗+𝑁_𝑟𝛾^∗.(2.40) Let 𝜖>0 be arbitrarily given and choose a sufficiently large positive number 𝜏 such that 𝑀_𝑟^∞𝜏𝑎(𝑠)𝑑𝑠+𝑁_𝑟𝑡_𝑘≥𝜏𝛾_𝑘<𝜖.(2.41) For any 𝜏<𝑡<∞, we have, by (2.5), 𝐴𝑢_𝑛𝑖(𝑡)−𝐴𝑢_𝑛𝑖(𝜏)=+^𝑡𝜏𝑓𝑠,𝑢_𝑛𝑖(𝑠),𝑇𝑢_𝑛𝑖(𝑠),𝑆𝑢_𝑛𝑖(𝑠)𝑑𝑠_{𝜏≤𝑡𝑘<𝑡}𝐼_𝑘𝑢_𝑛𝑖(𝑡),(𝑖=1,2,3,…),(2.42) which implies by virtue of (2.8), condition (𝐻₃) and (2.41) that ‖‖𝐴𝑢_𝑛𝑖(𝑡)−𝐴𝑢_𝑛𝑖(𝜏)‖‖≤𝑀_𝑟^𝑡𝜏𝑎(𝑠)𝑑𝑠+𝑁_𝑟𝜏≤_{𝑡𝑘<𝑡}𝛾_𝑘<𝜖,(𝑖=1,2,3,…).(2.43) Letting 𝑖→∞ in (2.43), we get ‖𝑣(𝑡)−𝑣(𝜏)‖≤𝜖,∀𝑡>𝜏.(2.44) On the other hand, since {(𝐴𝑢_𝑛𝑖)(𝑡)} converges uniformly to 𝑣(𝑡) on [0,𝜏] as 𝑖→∞, there exists a positive integer 𝑖₀ such that ‖‖𝐴𝑢_𝑛𝑖(𝑡)−𝑣(𝑡)‖‖<𝜖,∀𝑡∈[0,𝜏],𝑖>𝑖₀.(2.45) It follows from (2.43)–(2.45) that ‖‖𝐴𝑢_𝑛𝑖(𝑡)−𝑣(𝑡)‖‖≤<3‖‖𝜖,𝐴𝑢∀𝑡_𝑛𝑖>(𝑡𝜏),−𝑖>𝐴𝑢𝑖_𝑛𝑖0(𝜏)‖‖+‖‖𝐴𝑢_𝑛𝑖(𝜏)−𝑣(𝜏)‖‖+‖𝑣(𝜏)−𝑣(𝑡)‖.(2.46) By (2.45) and (2.46), we have ‖‖𝐴𝑢_𝑛𝑖−𝑣‖‖_𝐵≤3𝜖,∀𝑖>𝑖₀,(2.47) hence ||𝐴𝑢_𝑛𝑖−𝑣||_𝐵→0 as 𝑖→∞, and the compactness of 𝐴 is proved.

Lemma 2.4. Let conditions (𝐻₁)–(𝐻₄) be satisfied. Then 𝑢∈𝐵𝑃𝐶[𝐽,𝑃]∩𝐶¹[𝐽′,𝐸] is a solution of IBVP(1.5) if and only if 𝑢∈𝑄 is a solution of the following impulsive integral equation: 𝑢(𝑡)=1𝛽+𝛾−1^∞𝜂×𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)_𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1+𝐼_𝑘𝑢𝑡_𝑘_𝑡0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡_𝑘,∀𝑡∈𝐽.(2.48) that is, 𝑢 is a fixed point of operator 𝐴 defined by (2.5) in 𝑄.

Proof. For 𝑢∈PC[𝐽,𝐸]∩𝐶¹[𝐽′,𝐸], it is easy to get the following formula: 𝑢(𝑡)=𝑢(0)+^𝑡0𝑢^(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑢𝑡^+𝑘−𝑢𝑡_𝑘,∀𝑡∈𝐽.(2.49)
Let 𝑢∈BPC[𝐽,𝑃]∩𝐶¹[𝐽′,𝐸] be a solution of IBVP(1.5). By (1.5) and (2.49), we have 𝑢(𝑡)=𝑢(0)+^𝑡0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡_𝑘,∀𝑡∈𝐽.(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 𝑡→∞ in both sides of (2.50), we get 𝑢(∞)=𝑢(0)+^∞0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘=1𝐼_𝑘𝑢𝑡_𝑘.(2.51) On the other hand, by (1.5) and (2.50), we have 𝑢(𝜂)=𝑢(0)+_𝜂0𝑢(∞)=𝛾𝑢(𝜂)+𝛽𝑢(0),(2.52)𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^𝑚−1_𝑘=1𝐼_𝑘𝑢𝑡_𝑘.(2.53) It follows from (2.51)–(2.53) that 𝑢(0)=1𝛽+𝛾−1^∞𝜂+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)^𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1𝐼_𝑘𝑢𝑡_𝑘,(2.54) and, substituting it into (2.50), we see that 𝑢(𝑡) satisfies (2.48), that is, 𝑢=𝐴𝑢. Since 𝐴𝑢∈𝑄 by virtue of Lemma 2.3, we conclude that 𝑢∈𝑄.
Conversely, assume that 𝑢∈𝑄 is a solution of (2.48). We have, by (2.48), 𝑢(0)=1𝛽+𝛾−1^∞𝜂+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)^𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1𝑢(𝜂)=1𝐼_𝑘𝑢𝑡_𝑘,(2.55)𝛽+𝛾−1^∞𝜂×𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)_𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1+𝐼_𝑘𝑢𝑡_𝑘_𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^𝑚−1_𝑘=1𝐼_𝑘𝑢𝑡_𝑘.(2.56) Moreover, by taking limits as 𝑡→∞ in (2.33), we see that 𝑢(∞) exists and 𝑢(∞)=1𝛽+𝛾−1^∞𝜂+𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+(1−𝛾)^𝜂0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠_∞𝑘_=𝑚𝐼_𝑘𝑢𝑡_𝑘+(1−𝛾)^𝑚−1_𝑘=1+𝐼_𝑘𝑢𝑡_𝑘_∞0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘=1𝐼_𝑘𝑢𝑡_𝑘.(2.57) It follows from (2.55)–(2.57) that 𝛾𝑢(𝜂)+𝛽𝑢(0)=𝑢(∞).(2.58) On the other hand, direct differentiation of (2.48) gives 𝑢^(𝑡)=𝑓(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)),∀𝑡∈𝐽′,(2.59) and, it is clear, by (2.48), Δ𝑢||_{𝑡=𝑡𝑘}=𝐼_𝑘𝑢𝑡_𝑘(𝑘=1,2,3,…).(2.60) Hence, 𝑢∈𝐶¹[𝐽′,𝐸] and 𝑢(𝑡) satisfies (1.5).

Corollary 2.5. Let cone 𝑃 be normal. If 𝑢 is a fixed point of operator 𝐴 defined by (1.5) in 𝑄 and ||𝑢||_𝐵>0, then 𝑢(𝑡)>𝜃 for 𝑡∈𝐽, so, 𝑢 is a positive solution of IBVP(1.5).

Proof. For 𝑢∈𝑄, we have 𝑢(𝑡)≥𝛽⁻¹(1−𝛾)𝑢(𝑠)≥𝜃,∀𝑡,𝑠∈𝐽,(2.61) so, ‖𝑢(𝑡)‖≥𝑁⁻¹𝛽⁻¹(1−𝛾)‖𝑢‖_𝐵,∀𝑡∈𝐽,(2.62) where 𝑁 denotes the normal constant of 𝑃. Since ||𝑢||_𝐵>0, (2.61) and (2.62) imply that 𝑢(𝑡)>𝜃 for 𝑡∈𝐽.

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 Ω₁,Ω₂ two bounded open sets in 𝐸 such that 𝜃∈Ω₁, Ω₁⊂Ω₂, where 𝜃 denotes the zero element of 𝐸 and Ω₂ denotes the closure of Ω₂. Let operator 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 be completely continuous. Suppose that one of the following two conditions is satisfied: (a)‖𝐴𝑥‖≤‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₁;‖𝐴𝑥‖≥‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₂,(2.63) where 𝜕Ω_𝑖 denotes the boundary of Ω_𝑖 (𝑖=1,2).(b)‖𝐴𝑥‖≥‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₁,‖𝐴𝑥‖≤‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₂.(2.64)Then 𝐴 has at least one fixed point in 𝑃∩(Ω₂⧵Ω₁).

3. Main Theorems

Let us list more conditions.(𝐻₅) There exist 𝑢₀∈𝑃⧵{𝜃},𝑏∈𝐶[𝐽,𝑅₊], and 𝜏∈𝐶[𝑃,𝑅₊] such that 𝑓(𝑡,𝑢,𝜏𝑣(,𝑢𝑤))≥𝑏(𝑡)𝜏(𝑢)𝑢₀,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,‖𝑢‖⟶𝑏∞as𝑢∈𝑃,‖𝑢‖⟶∞,_∗=^∞0𝑏(𝑡)𝑑𝑡<∞.(3.1)

Remark 3.1. Condition (𝐻₅) means that 𝑓(𝑡,𝑢,𝑣,𝑤) is superlinear with respect to 𝑢.(𝐻₆) There exist 𝑢₁∈𝑃⧵{𝜃},𝑐∈𝐶[𝐽,𝑅₊], and 𝜎∈𝐶[𝑃,𝑅₊] such that 𝑓(𝑡,𝑢,𝜎𝑣,(𝑢𝑤))≥𝑐(𝑡)𝜎(𝑢)𝑢₁,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,‖𝑢‖⟶𝑐∞as𝑢∈𝑃,‖𝑢‖⟶0,_∗=^∞0𝑐(𝑡)𝑑𝑡<∞.(3.2)

Theorem 3.2. Let cone 𝑃 be normal and conditions (𝐻₁)–(𝐻₆) satisfied. Assume that there exists a 𝜉>0 such that 𝑁(𝛽+𝛾)𝛽+𝛾−1𝑀^𝜉𝑎^∗+𝑁_𝜉𝛾^∗<𝜉,(3.3) where 𝑀_𝜉=max𝑔(𝑥𝑁,𝑦_𝜉,𝑧)∶0≤𝑥≤𝜉,0≤𝑦≤𝑘^∗𝜉,0≤𝑧≤ℎ^∗𝜉,=max{𝐹(𝑥)∶0≤𝑥≤𝜉}.(3.4) (for 𝑔(𝑥,𝑦,𝑧),𝐹(𝑥),𝑎^∗ and 𝛾^∗, see conditions (𝐻₂) and (𝐻₃)). Then IBVP(1.5) has at least two positive solutions 𝑢^∗,𝑢^∗∗∈𝑄∩𝐶¹[𝐽′,𝐸] such that 0<||𝑢^∗||_𝐵<𝜉<||𝑢^∗∗||_𝐵.

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 𝑢^∗ and 𝑢^∗∗ in 𝑄 such that 0<||𝑢^∗||_𝐵<𝜉<||𝑢^∗∗||_𝐵.
By condition (𝐻₅), there exists an 𝑟₁>0 such that 𝜏(𝑢)≥𝛽(𝛽+𝛾−1)𝑁²(1−𝛾)²𝑏^∗‖‖𝑢₀‖‖‖𝑢‖,∀𝑢∈𝑃,‖𝑢‖≥𝑟¹,(3.5) where 𝑁 denotes the normal constant of 𝑃, so, 𝑓(𝑡,𝑢,𝑣,𝑤)≥𝛽(𝛽+𝛾−1)𝑁²‖𝑢‖(1−𝛾)²𝑏^∗‖‖𝑢₀‖‖𝑏(𝑡)𝑢⁰,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,|||𝑢|||≥𝑟₁.(3.6) Choose 𝑟₂>max𝑁𝛽(1−𝛾)⁻¹𝑟₁,𝜉.(3.7) For 𝑢∈𝑄,||𝑢||_𝐵=𝑟₂; we have by (2.62) and (3.7), ‖𝑢(𝑡)‖≥𝑁⁻¹𝛽⁻¹(1−𝛾)‖𝑢‖_𝐵=𝑁⁻¹𝛽⁻¹(1−𝛾)𝑟₂>𝑟₁,∀𝑡∈𝐽,(3.8) so, (2.5), (3.8), (3.6), and (2.62) imply (𝐴𝑢)(𝑡)≥1−𝛾𝛽+𝛾−1^∞0≥𝛽𝑁₂𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠(1−𝛾)𝑏^∗‖‖𝑢₀‖‖^∞0‖𝑢(𝑠)‖𝑏(𝑠)𝑑𝑠𝑢₀≥𝑁‖𝑢‖^𝐵𝑏^∗‖‖𝑢₀‖‖^∞0𝑏(𝑠)𝑑𝑠𝑢₀=𝑁‖𝑢‖^𝐵‖‖𝑢₀‖‖𝑢⁰,∀𝑡∈𝐽,(3.9) and consequently, ||||𝐴𝑢||||_𝐵≥|||𝑢|||_𝐵,∀𝑢∈𝑄,|||𝑢|||_𝐵=𝑟₂.(3.10) Similarly, by condition (𝐻₆), there exists 𝑟₃>0 such that 𝜎(𝑢)≥𝛽(𝛽+𝛾−1)𝑁²(1−𝛾)²𝑐^∗||||𝑢₁|||||||𝑢|||,∀𝑢∈𝑃,0<|||𝑢|||<𝑟³,(3.11) so, 𝑓(𝑡,𝑢,𝑣,𝑤)≥𝛽(𝛽+𝛾−1)𝑁²|||𝑢|||(1−𝛾)²𝑐^∗||||𝑢₁||||𝑐(𝑡)𝑢¹,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,0<|||𝑢|||<𝑟₃.(3.12) Choose 0<𝑟₄<min𝑟₃,𝜉.(3.13) For 𝑢∈𝑄,||𝑢||_𝐵=𝑟₄, we have by (3.13) and (2.62), 𝑟₃>||||𝑢(𝑡)||||≥𝑁⁻¹𝛽⁻¹(1−𝛾)‖𝑢‖_𝐵=𝑁⁻¹𝛽⁻¹(1−𝛾)𝑟₄>0,(3.14) so, similar to (3.9), we get by (2.5), (3.12), and (3.14) (𝐴𝑢)(𝑡)≥1−𝛾𝛽+𝛾−1^∞0≥𝛽𝑁₂𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠(1−𝛾)𝑐^∗||||𝑢₁||||^∞0||||𝑢(𝑠)||||𝑐(𝑠)𝑑𝑠𝑢₁≥𝑁‖𝑢‖^𝐵𝑐^∗||||𝑢₁||||^∞0𝑐(𝑠)𝑑𝑠𝑢₁=𝑁‖𝑢‖^𝐵||||𝑢₁||||𝑢¹,∀𝑡∈𝐽;(3.15) hence ‖𝐴𝑢‖_𝐵≥‖𝑢‖_𝐵,∀𝑢∈𝑄,|||𝑢|||=𝑟₄.(3.16) On the other hand, for 𝑢∈𝑄, ||𝑢||_𝐵=𝜉, by condition (𝐻₂), condition (𝐻₃), (3.4), we have ‖𝑓(𝑡‖‖,𝑢𝐼(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))‖≤𝑀_𝜉𝑎(𝑡),∀𝑡∈𝐽,(3.17)_𝑘𝑢𝑡_𝑘‖‖≤𝑁_𝜉𝛾_𝑘(𝑘=1,2,3,…).(3.18) It is clear that (𝐴𝑢)(𝑡)≤𝛽+𝛾𝛽+𝛾−1^∞0𝑓(𝑠,𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠+^∞_𝑘=1𝐼_𝑘𝑢𝑡_𝑘∀𝑡∈𝐽.(3.19) It follows from (3.17)–(3.19) that ‖𝐴𝑢‖_𝐵≤𝑁(𝛽+𝛾)𝛽+𝛾−1𝑀^𝜉𝑎^∗+𝑁_𝜉𝛾^∗.(3.20) Thus, (3.20) and (3.3) imply ‖𝐴𝑢‖_𝐵<‖𝑢‖_𝐵,∀𝑢∈𝑄,‖𝑢‖_𝐵=𝜉.(3.21) From (3.7) and (3.13), we know 0<𝑟₄<𝜉<𝑟₂; hence, (3.10), (3.16), (3.21), and Lemma 2.6 imply that 𝐴 has two fixed points 𝑢^∗,𝑢^∗∗∈𝑄 such that 𝑟₄<||𝑢^∗||_𝐵<𝜉<||𝑢^∗∗||_𝐵<𝑟₂. The proof is complete.

Theorem 3.3. Let cone 𝑃 be normal and conditions (𝐻₁)–(𝐻₅) satisfied. Assume that 𝑔(𝑥,𝑦,𝑧)𝑥+𝑦+𝑧⟶0as𝑥+𝑦+𝑧⟶0⁺𝐹(𝑥),𝑥⟶0as𝑥⟶0⁺.(3.22) (for 𝑔(𝑥,𝑦,𝑧) and 𝐹(𝑥), see conditions (𝐻₂) and (𝐻₃)). Then IBVP(1.5) has at least one positive solution 𝑢^∗∈𝑄∩𝐶¹[𝐽′,𝐸].

Proof. As in the proof of Theorem 3.2, we can choose 𝑟₂>0 such that (3.10) holds (in this case, we only choose 𝑟₂>𝑁𝛽(1−𝛾)⁻¹𝑟₁ instead of (3.7)). On the other hand, by (3.22), there exists 𝑟₅>0 such that 𝑔(𝑥,𝑦,𝑧)≤𝐹𝜖₀(𝑥(𝑥)≤+𝜖𝑦₀+𝑧),∀0<𝑥+𝑦+𝑧<𝑟₅,𝑥,∀0<𝑥<𝑟₅,(3.23) where 𝜖₀=𝛽+𝛾−1𝑁(𝛽+𝛾)1+𝑘^∗+ℎ^∗𝑎^∗+𝛾^∗.(3.24) Choose 0<𝑟₆<min𝑟⁵1+𝑘^∗+ℎ^∗,𝑟₂.(3.25) For 𝑢∈𝑄,||𝑢||_𝐵=𝑟₆, we have by (2.62) and (3.25), 0<𝑁₋₁0<𝑁⁻¹𝛽⁻¹(1−𝛾)𝑟₆≤||||𝑢(𝑡)||||≤𝑟₆<𝑟₅,∀𝑡∈𝐽,𝛽⁻¹(1−𝛾)𝑟₆≤||||𝑢(𝑡)||||+||||(𝑇𝑢)(𝑡)||||+||||(𝑆𝑢)(𝑡)||||≤1+𝑘^∗+ℎ^∗𝑟₆<𝑟₅,∀𝑡∈𝐽,(3.26) so, (3.23) imply 𝑔||||𝑢(𝑡)||||,||||(𝑇𝑢)(𝑡)||||,||||(𝑆𝑢)(𝑡)||||≤≤𝜖𝜖₀₀||||𝑢(𝑡)||||+||||(𝑇𝑢)(𝑡)||||+||||(𝑆𝑢)(𝑡)||||𝐹||||𝑢𝑡1+𝑘^∗+ℎ^∗𝑟₆,∀𝑡∈𝐽,_𝑘||||≤𝜖₀||||𝑢𝑡_𝑘||||≤𝜖₀𝑟₆,(𝑘=1,2,3,…).(3.27) It follows from (3.19), condition (𝐻₂), condition (𝐻₃), (3.27), and (3.24) that ||||(𝐴𝑢)(𝑡)||||≤𝑁(𝛽+𝛾)𝛽+𝛾−1𝜖⁰1+𝑘^∗+ℎ^∗𝑟₆^∞0𝑎(𝑠)𝑑𝑠+𝜖₀𝑟_6∞_𝑘=1𝛾_𝑘=𝑁(𝛽+𝛾)𝜖₀𝑟₆𝛽+𝛾−11+𝑘^∗+ℎ^∗𝑎^∗+𝛾^∗=𝑟₆,∀𝑡∈𝐽,(3.28) and consequently, ‖𝐴𝑢‖_𝐵≤‖𝑢‖_𝐵,∀𝑢∈𝑄,‖𝑢‖_𝐵=𝑟₆.(3.29) Since 0<𝑟₆<𝑟₂ by virtue of (3.25), we conclude from (3.10), (3.29), and Lemma 2.6 that 𝐴 has a fixed point 𝑢^∗∈𝑄 such that 𝑟₆≤||𝑢^∗||_𝐵≤𝑟₂. 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: 𝑢^𝑛(𝑡)=18𝑛²𝑒^−5𝑡⎛⎜⎜⎝𝑢_𝑛+1(𝑡)+^∞_𝑚₌₁𝑢_𝑚(𝑡)²+⎷3𝑢_2𝑛(𝑡)+^∞_𝑚₌₁+1𝑢_𝑚(𝑡)⎞⎟⎟⎠9𝑛³𝑒^−6𝑡^𝑡0𝑒^{−(𝑡+1)𝑠}𝑢_𝑛(𝑠)𝑑𝑠²+^∞0𝑢_𝑛+2(𝑠)𝑑𝑠Δ𝑢_𝑛∀0≤𝑡<∞,𝑡≠𝑘(𝑘=1,2,3,…;(𝑛1=+1𝑡,+2,𝑠3),²…³),,||_𝑡=𝑘=16𝑛²𝑢(∞)=13^−𝑘𝑢^𝑛(𝑘)²+𝑢_𝑛+2(𝑘)²,(𝑘=1,2,3,…;𝑛=1,2,3,…),2𝑢^𝑛29+6𝑢^𝑛(0),(𝑛=1,2,3,…).(3.30) Evidently, 𝑢_𝑛(𝑡)≡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 {𝑢^∗𝑛(𝑡)} (𝑛=1,2,3,…) and {𝑢^𝑛∗∗(𝑡)} (𝑛=1,2,3,…) such that 0<inf_{∞0≤𝑡<∞}_𝑛=1𝑢^∗𝑛(𝑡)≤sup_{∞0≤𝑡<∞}_𝑛=1𝑢^∗𝑛(𝑡)<1<sup_{∞0≤𝑡<∞}_𝑛=1𝑢^𝑛∗∗(𝑡),inf_{∞0≤𝑡<∞}_𝑛=1𝑢^𝑛∗∗(𝑡)>0.(3.31)

Proof. Let 𝐸=𝑙¹={𝑢=(𝑢₁,…,𝑢_𝑛,…)∶∑_𝑛=1∞|𝑢_𝑛|<∞} with norm ||𝑢||=∑^∞𝑛=1|𝑢_𝑛| and 𝑃=(𝑢₁,…,𝑢_𝑛,…)∶𝑢_𝑛≥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, 𝑢=(𝑢₁,…,𝑢_𝑛,…), 𝑣=(𝑣₁,…,𝑣_𝑛,…), 𝑤=(𝑤₁,…,𝑤_𝑛,…), 𝑡_𝑘=𝑘(𝑘=1,2,3,…), 𝐾(𝑡,𝑠)=𝑒^{−(𝑡+1)𝑠}, 𝐻(𝑡,𝑠)=(1+𝑡+𝑠)⁻², 𝜂=9/2, 𝛾=1/2, 𝛽=6, 𝑓=(𝑓₁,…,𝑓_𝑛,…), and 𝐼_𝑘=(𝐼_𝑘1,…,𝐼_𝑘𝑛…), in which 𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)=18𝑛²𝑒^−5𝑡⎛⎜⎜⎝𝑢_𝑛+1+^∞_𝑚₌₁𝑢_𝑚²+⎷3𝑢_2𝑛+^∞_𝑚₌₁𝑢_𝑚⎞⎟⎟⎠+19𝑛³𝐼∀𝑡∈𝐽=[0,∞),𝑢,𝑣,𝑤∈𝑃(𝑛𝑒=^−6𝑡1,𝑣2^2𝑛,+3,𝑤…)^3𝑛+2,,(3.32)_𝑘𝑛(𝑢)=16𝑛²3^−𝑘𝑢^2𝑛+𝑢^22𝑛+1,∀𝑢∈𝑃(𝑘=1,2,3,…;𝑛=1,2,3,…).(3.33) It is easy to see that 𝑓∈𝐶[𝐽×𝑃×𝑃×𝑃,𝑃], 𝐼_𝑘∈𝐶[𝑃,𝑃] (𝑘=1,2,3,…), and condition (𝐻₁) is satisfied and 𝑘^∗≤1,ℎ^∗≤1. We have, by (3.32), 0≤≤𝑓1_𝑛(𝑡,𝑢,𝑣,𝑤)8𝑛²𝑒^−5𝑡2|||𝑢|||²+4|||𝑢|||+19𝑛³≤1𝑒^−6𝑡‖𝑣‖²+‖𝑤‖³𝑛²𝑒^−5𝑡12‖𝑢‖²+14|||𝑢|||+19‖𝑣‖²+19‖𝑤‖³,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃(𝑛=1,2,3,…),(3.34) so, observing the inequality ∑^∞𝑛=1(1/𝑛²)<2, we get ‖𝑓(𝑡,𝑢,𝑣,𝑤)‖=^∞_𝑛=1𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)≤𝑒^−5𝑡‖𝑢‖²+12√‖𝑢‖+29‖𝑣‖²+2∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃,9‖𝑤‖³,(3.35) which implies that condition (𝐻₂) is satisfied for 𝑎(𝑡)=𝑒^−5𝑡(^∗=1/5) and 𝑔(𝑥,𝑦,𝑧)=𝑥²+12√𝑥+29𝑦²+29𝑧³.(3.36) By (3.33), we have 0≤𝐼_𝑘𝑛(𝑢)≤16𝑛²3^−𝑘‖𝑢‖²,∀𝑢∈𝑃(𝑘=1,2,3,…;𝑛=1,2,3,…),(3.37) so, condition (𝐻₃) is satisfied for 𝛾_𝑘=3^−𝑘−1(𝛾^∗=1/6) and 𝐹(𝑥)=𝑥².(3.38) On the other hand, (3.32) implies 𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)≥18𝑛²𝑓𝑒^−5𝑡‖𝑢‖²,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃(𝑛=1,2,3,…),_𝑛(𝑡,𝑢,𝑣,𝑤)≥18𝑛²𝑒^−5𝑡√‖𝑢‖,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃(𝑛=1,2,3,…),(3.39) so, we see that condition (𝐻₅) is satisfied for 𝑏(𝑡)=(1/8)𝑒^−5𝑡(𝑏^∗=1/40), 𝜏(𝑢)=||𝑢||² and 𝑢₀=(1,…,1/𝑛²,…), and condition (𝐻₆) is satisfied for 𝑐(𝑡)=(1/8)𝑒^−5𝑡(𝑐^∗=1/40), 𝜎(𝑢)=√||𝑢||, and let 𝑢₁=(1,…,1/𝑛²,…). Now, we check that condition (𝐻₄) is satisfied. Let 𝑡∈𝐽 and 𝑟>0 be fixed, and {𝑧^(𝑚)} be any sequence in 𝑓(𝑡,𝑃_𝑟,𝑃_𝑟,𝑃_𝑟), where 𝑧^(𝑚)=(𝑧^1(𝑚),…,𝑧^𝑛(𝑚),…). Then, we have, by (3.34), 0≤𝑧^𝑛(𝑚)≤1𝑛²1118𝑟²+14√𝑟+19𝑟³,(𝑛,𝑚=1,2,3,…).(3.40) So, {𝑧^𝑛(𝑚)} is bounded, and, by diagonal method, we can choose a subsequence {𝑚_𝑖}⊂{𝑚} such that 𝑧^(𝑚𝑖)⟶𝑧_𝑛as𝑖⟶∞(𝑛=1,2,3,…),(3.41) which implies by virtue of (3.40) that 0≤𝑧_𝑛≤1𝑛²1118𝑟²+14√𝑟+19𝑟³,(𝑛=1,2,3,…).(3.42) Consequently, 𝑧=(𝑧₁,…,𝑧_𝑛,…)∈𝑙¹=𝐸. Let 𝜖>0 be given. Choose a positive integer 𝑛₀ such that ⎛⎜⎜⎝^∞_𝑛=_𝑛0+11𝑛²⎞⎟⎟⎠1118𝑟²+14√𝑟+19𝑟³<3𝜖.(3.43) By (3.41), we see that there exists a positive integer 𝑖₀ such that |||𝑧_{(𝑚𝑖)𝑛}−𝑧_𝑛|||<𝜖3𝑛₀,∀𝑖>𝑖₀𝑛=1,2,…,𝑛₀.(3.44) It follows from (3.40)–(3.44) that ‖‖𝑧^(𝑚𝑖)−𝑧‖‖=^∞_𝑛=1|||𝑧_{(𝑚𝑖)𝑛}−𝑧_𝑛|||≤^𝑛0_𝑛=1|||𝑧_{(𝑚𝑖)𝑛}−𝑧_𝑛|||+^∞_𝑛=_𝑛0+1+_∞|||𝑧_{(𝑚𝑖)𝑛}|||_𝑛=_𝑛0+1||𝑧_𝑛||<𝜖3+3𝜖+3𝜖=𝜖,∀𝑖>𝑖⁰.(3.45) Thus, we have proved that 𝑓(𝑡,𝑃_𝑟,𝑃_𝑟,𝑃_𝑟) is relatively compact in 𝐸. Similarly, by using (3.37), we can prove that 𝐼_𝑘(𝑃_𝑟) is relatively compact in 𝐸. Hence, condition (𝐻₄) is satisfied. Finally, it is easy to check that inequality (3.3) is satisfied for 𝜉=1 (in this case, 𝑀_𝜉≤17/36 and 𝑁_𝜉=1). Hence, our conclusion follows from Theorem 3.2.

Example 3.5. Consider the infinite system of scalar first-order impulsive integrodifferential equations of mixed type on the half line: 𝑢^𝑛(𝑡)=1𝑛³(1+𝑡)³𝑢^𝑛(𝑡)+2𝑢_𝑛+1(𝑡)+^∞_𝑚₌₁𝑢_𝑚(𝑡)³+1𝑛⁴(1+𝑡)⁴^𝑡0𝑢_2𝑛(𝑠)𝑑𝑠1+𝑡𝑠+𝑠²⁴+1𝑛⁵(1+𝑡)⁵^∞0Δ𝑢_𝑛∀𝑒0^−𝑠≤sin𝑡²<(𝑡∞−,𝑠𝑡)≠𝑢2_3𝑛𝑘(𝑠)(𝑑𝑘𝑠=⁵1,,2,3,…;𝑛=1,2,3,…),∣_𝑡=2𝑘=1𝑛²𝑒^−𝑘𝑢_𝑛(2𝑘)³+1𝑛³4𝑢_𝑛2^−𝑘𝑢_2𝑛+1(2𝑘)⁴,(𝑘=1,2,3,…;𝑛=1,2,3,…),(∞)=3𝑢_𝑛(7)+2𝑢_𝑛(0),(𝑛=1,2,3,…).(3.46) Evidently, 𝑢_𝑛(𝑡)≡0(𝑛=1,2,3,…) is the trivial solution of infinite system (3.46).

Conclusion. Infinite system (3.46) has at least one positive solution {𝑢^∗𝑛(𝑡)} (𝑛=1,2,3,…) such that inf_{∞0≤𝑡<∞}_𝑛=1𝑢^∗𝑛(𝑡)>0.(3.47)

Proof. Let 𝐸=𝑙¹=({𝑢=(𝑢₁,…,𝑢_𝑛,…)∶∑^∞𝑛=1|𝑢_𝑛|<∞} with norm ||𝑢||=∑^∞𝑛=1|𝑢_𝑛| and 𝑃={𝑢=(𝑢₁,…,𝑢_𝑛,…)∈𝑙¹∶𝑢_𝑛≥0,𝑛=1,2,3,…}. Then 𝑃 is a normal cone in 𝐸 with normal constant 𝑁=1, and infinite system (3.46) can be regarded as an infinite three-point boundary value problem of form (1.5) in 𝐸. In this situation, 𝑢=(𝑢₁,…,𝑢_𝑛,…), 𝑣=(𝑣₁,…,𝑣_𝑛,…), 𝑤=(𝑤₁,…,𝑤_𝑛,…), 𝑡_𝑘=2𝑘 (𝑘=1,2,3,…), 𝐾(𝑡,𝑠)=(1+𝑡𝑠+𝑠²)⁻¹, 𝐻(𝑡,𝑠)=𝑒^−𝑠sin²(𝑡−𝑠), 𝜂=7, 𝛾=3/4, 𝛽=1/2, 𝑓=(𝑓₁,…,𝑓_𝑛,…), and 𝐼_𝑘=(𝐼_𝑘1,…,𝐼_𝑘𝑛,…), in which 𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)=1𝑛³(1+𝑡)⁻³𝑢^𝑛+2𝑢_𝑛+1+^∞_𝑚₌₁𝑢_𝑚³+1𝑛⁴(1+𝑡)⁻⁴𝑣^42𝑛+1𝑛⁵𝐼∀𝑡∈𝐽=[0,∞),𝑢,𝑣,𝑤∈𝑃(𝑛=1,(12,+3,𝑡…)⁻⁵)𝑤,^53𝑛_𝑘𝑛,(𝑢)=1𝑛²𝑒^−𝑘𝑢^3𝑛+1𝑛³2^−𝑘𝑢^42𝑛+1,(𝑘=1,2,3,…;𝑛=1,2,3,…).(3.48) It is clear that 𝑓∈𝐶[𝐽×𝑃×𝑃×𝑃,𝑃],𝐼_𝑘∈𝐶[𝑃,𝑃] (𝑘=1,2,3,…), and condition (𝐻₁) is satisfied and 𝑘^∗≤𝜋/2,ℎ^∗≤1. We have 0≤𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)≤1𝑛³0≤𝐼_𝑘𝑛(1+𝑡)⁻³(3‖𝑢‖)³+‖𝑣‖⁴+‖𝑤‖⁵,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃(𝑛=1,2,3,…),(𝑢)≤1𝑛²2^−𝑘‖𝑢‖³+‖𝑢‖⁴,∀𝑢∈𝑃(𝑘=1,2,3,…,𝑛=1,2,3,…),(3.49) so, condition (𝐻₂) is satisfied for 𝑎(𝑡)=(1+𝑡)⁻³(𝑎^∗=(1/2)) and 𝑔(𝑥,𝑦,𝑧)=54𝑥³+2𝑦⁴+2𝑧⁵,(3.50) and (𝐻₃) is satisfied for 𝛾_𝑘=2^−𝑘(𝛾^∗=1) and 𝐹(𝑥)=2𝑥³+2𝑥⁴.(3.51) From 𝑓_𝑛(𝑡,𝑢,𝑣,𝑤)≥1𝑛³(1+𝑡)⁻³‖𝑢‖³,∀𝑡∈𝐽,𝑢,𝑣,𝑤∈𝑃(𝑛=1,2,3,…),(3.52) we see that condition (𝐻₅) is satisfied for 𝑏(𝑡)=(1+𝑡)⁻³(𝑏^∗=1/2),𝜏(𝑢)=||𝑢||³, and 𝑢₀=(1,…,1/𝑛³,…). Moreover, it is clear that (3.22) are satisfied. Similar to the discussion in Example 3.4, we can prove that 𝑓(𝑡,𝑃_𝑟,𝑃_𝑟,𝑃_𝑟) and 𝐼_𝑘(𝑃_𝑟) (for fixed 𝑡∈𝐽 and 𝑟>0;𝑘=1,2,3,…) are relatively compact in 𝐸=𝑙¹; so, condition (𝐻₄) is satisfied. Hence, our conclusion follows from Theorem 3.3.

Acknowledgment

This paper is supported by the National Nature Science Foundation of China (no. 10671167).