# Existence ofC1-Positive Solutions for a Class of Second-Order Impulsive Differential Equations

Authors: Hong Li
Journal: Complexity
 (2022)

---

## Abstract

In this study, under some inequality conditions, necessary and sufficient conditions, using fixed-point theorem in cones, are established for the existence ofC1-positive solutions for a class of second-order impulsive differential equations. Two examples are given in the last section to illustrate the abstract results.


---

## Body

## 1. Introduction

The theory of differential equations with impulsive effects has an extensive application in realistic mathematical models. It has been used to describe many evolution processes, containing abrupt change, such as biological systems, population dynamics, and optimal control. Hence, in recent years, more and more attention has been paid on this topic. For the general theory of impulsive differential equations, one can see the monographs of Lakshmikantham et al. [1], Bainov and Simeonov [2], and Benchohra et al. [3]. There are also some studies focusing on impulsive differential equations. In [4], Ye investigated the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces by using the Hausdorff measure of noncompactness conditions. In [5], Hernández et al. concerned with the existence of solutions for partial neutral functional differential equations of first and second order with impulses by using fixed-point theorems. The existence of solutions for fractional differential equations have also been studied widely. In [6], Gu et al. studied the existence of positive solutions for impulsive fractional differential equations attached with integral boundary conditions via global bifurcation techniques. In [7], Benchohra and Seba demonstrated the existence and uniqueness of solutions for the initial value problem of fractional differential equations by utilizing fixed-point theorems.The existence of solutions for second-order differential equations, involving different boundary conditions, has been studied by many authors. Chu and Nieto in [8], utilizing the nonlinear alternative principle of Leray-Schauder type and Schauder’s fixed-point theorem, presented existence results of positive T-periodic solutions for second-order differential equations. In 2019, Ma and Zhang in [9] proved sharp conditions for the existence of positive solutions of second-order singular differential equation with integral boundary conditions. Recently, Zhang and Tian [10] established sharp conditions for the existence of positive solutions of second-order impulsive differential equations. But in their work, a key assumption is that the nonlinearity ft,x is nondecreasing with respect to x≥0. Clearly, if ft,x=t2x1/3+t3x−1/3, the results obtained in [9, 10] are not valid. The aim of this study is to extend the results in [10] to more general cases. We establish necessary and sufficient conditions for the existence of C1-positive solutions for a class of second-order impulsive differential equations. The results obtained in this study extend and improve some existing works.In the present work, we consider the boundary value problem (BVP for short) of second-order impulsive differential equation:(1)−x″t=τtft,xt,yt,t∈I,t≠tk,Δx|t=tk=δkxtk,k=1,2,…,n,αx0−βx′0=0,γx1+ωx′1=0,where α,β,γ,ω are the nonnegative constants satisfying α+βγ+αω>0, I=0,1, τ∈Lp0,11≤p<+∞, yt=ςtx−1t, and ς is a positive continuous function on I, f∈CI×0,+∞×0,+∞,0,+∞, ft,x,y may be singular at t=0,1, t0=0<t1<t2<⋯<tn<tn+1=1, δkk=1n is a real sequence, and Δx|t=tk=xtk+−xtkk=1,2,…,n represents the impulsive term, where xtk=xtk− and xtk+ denotes the right-hand limit of xt at t=tk.In addition,τ and δkk=1n satisfy the following conditions:A1τ∈Lp0,1 for some 1≤p<+∞, and there exists ξ>0, such that τt≥ξ, i.e., t∈I.A2δkk=1n is a real sequence with δk>−1,k=1,2,…,n, and denoted by(2)δt=∏0<tk<t1+δk,t∈I∗≔0,1.Clearly,δt is continuous on I∗. Denote by δM=maxt∈I∗δt,δm=mint∈I∗δt. From A2, we have(3)δt≥δm>0,∀t∈I∗,δt=1,∀t∈0,t1,(4)δ′t≡0,∀t∈I∗.Furthermore,δ−1t=∏0<tk<t1+δk−1 for any t∈I∗.The main results of this work are summarized as follows:(i)
The necessary and sufficient conditions onτ and f are established for the existence of C1-positive solutions of the BVP (1) (Theorems 1 and 2)(ii)
We assume that the nonlinear termft,x,y satisfies A3, which implies that ft,x,y is nondecreasing with respect to x and nonincreasing with respect to y (Remark 2). But ft,x,y does not have monotonicity in whole;(iii)
Correct examples are given in the last section to illustrate our abstract results, which show that the results obtained in this study contain some existing works (Theorem3 and 4).The rest of this study is organized as follows. Some preliminaries and notations are presented in Section2. Particularly, we transform the BVP (1) to a problem without impulse in this section. In Section 3, we prove the main results by using the fixed-point theorem of cone mapping and give some remarks. Examples are given in Section 4 to illustrate the abstract results.


## 2. Preliminaries

In this section, some preliminaries and notations, which are useful in the proof of the main results, are presented. In order to discuss the BVP (1) more clearly, we first transform the BVP (1) to a problem without impulse. Let(5)xt=δtut,∀t∈I∗.Then,yt=δtu−1t with ςt=δ2t. The BVP (1) is rewritten into the BVP:(6)−u″t=δ−1tτtft,δtut,δtu−1t,t∈I,αu0−βu′0=0,γδ1u1+ωδ1u′1=0.Lemma 1.
Let the assumptionsA1 and A2 hold. Then, ut is a solution of BVP (2) on I if and only if xt=δtut is a solution of BVP (1) on I.Proof.
(Necessity). Letut be a solution of the BVP (2) on I. Then, on each interval tk,tk+1,k=0,1,2,…,n, xt=δtut is absolutely continuous. For t≠tk, we have(7)x′t=δtu′t,x″t=δtu″t.
So,(8)−x″t=−δtu″t=τtft,δtut,δtu−1t=τtft,xt,yt.
Whent=tk, we have(9)xtk+=limt⟶tk+δtut=∏0<ti≤tk1+δiutk,xtk=xtk−=limt⟶tk−δtut=∏0<ti≤tk−11+δiutk.
It follows that(10)Δx|t=tk=xtk+−xtk=∏0<ti≤tk−11+δiδkutk=δkδtkutk=δkxtk.
Obviously,xt satisfies the boundary conditions. Then, xt is a solution of the BVP (1) on I.
(Sufficiency). Letxt be a solution of the BVP (1) on I. Then, ut=δ−1txt and u″t=δ−1tx″t, which follows(11)−δtu″t=−x″t=τtft,xt,yt=τtft,δtut,δtu−1t,∀t∈I.
Whent=tk, since xtk=xtk− and δ−1tk+=1/1+δkδ−1tk−, we have(12)utk+−utk−=δ−1tk+xtk+−δ−1tk−xtk−=δ−1tk+xtk−+δkxtk−−δ−1tk−xtk−=1+δkδ−1tk+xtk−−δ−1tk−xtk−=0.
Direct calculation shows thatut satisfies all boundary conditions. Then, ut is a solution of the BVP (2) on I.Definition 1.
A functionu∈C0,1∩C20,1 is called a positive solution of the BVP (2) if ut satisfies all the equations in (2) and ut>0 for all t∈I. If ut is the positive solution of the BVP (2) and u∈C10,1, namely, u′0+ and u′1− exist, then u is called a C1-positive solution of the BVP (2).
Throughout this study, the following assumptions onf are needed.
A3ft,δm,δM>0 for any t∈I, and there exist constants σ1≥σ2>1, such that, for every ℓ∈0,1,(13)ℓσ1ft,x,y≤ft,ℓx,ℓ−1y≤ℓσ2ft,x,y,for any t∈I and x,y∈0,+∞.
A4∫01δ−1tft,δM,δmdt<+∞.Remark 1.
The conditionA3 implies, for every ℓ≥1,(14)ℓσ2ft,x,y≤ft,ℓx,ℓ−1y≤ℓσ1ft,x,y,for any t∈I,x,y∈0,+∞.Remark 2.
Ifft,x,y satisfies A3, then for any t∈I, ft,x,y is nondecreasing with respect to x∈0,+∞ and nonincreasing with respect to y∈0,+∞.
Denote byE:=C0,1 the Banach space of all continuous functions on 0,1 equipped with the norm u=maxt∈I∗ut. Define an operator Q:E⟶E by(15)Qut=∫01Gt,sδ−1sτsfs,δsus,δsu−1sds,t∈I∗,where Gt,s is Green’s function of the BVP (2) with u″=0, and(16)Gt,s=1ργ+ω−γtβ+αs,0≤s≤t≤1,γ+ω−γsβ+αt,0≤t≤s≤1,where ρ=α+βγ+αω>0.Remark 3.
By the definition ofQ, u∈C0,1∩C20,1 is a positive solution of the BVP (2) if and only if u∈C0,1 is a positive fixed point of Q.Lemma 2.
(See [10]). Green’s function Gt,s of the BVP (2) satisfies(17)0<α∗≤qtGs,s≤Gt,s≤Gs,s≤β∗,∀t,s∈I∗,where(18)qt=minαt+βα+β,γ+ω−γtγ+ω,α∗=ωβρ,β∗=γ+ωβρ.For anyν∈0,1/2, let I0=ν,1−ν and η=mint∈I0qt. Then, η>0. Define a cone P by(19)P=u∈E:∃ζ≥1s.t.0<ut≤ζGt,t,∀t∈I∗,mint∈I0ut≥ηu.Then,P is a closed convex cone in E.Lemma 3.
Let the assumptionsA1–A4 hold. Then, the operator Q:P⟶E, defined by (3), is well defined.Proof.
Assume thatA1–A4 hold. For fixed u∈P with ut>0 for any t∈I∗, choosing a constant a∈0,1 satisfying 0<aut<1 for t∈I∗, we have(20)ft,δtut,δtu−1t≤1aσ1ft,aδtut,1aδtu−1t≤1aσ1autσ2ft,δt,δt≤aσ2−σ1uσ2ft,δM,δm.
So, fort∈I∗, by (17), we have(21)0<Qut=∫01Gt,sδ−1sτsfs,δsus,δsu−1sds≤aσ2−σ1uσ2β∗τp∫01δ−1sfs,δM,δmds<+∞.
This implies that the operatorQ:P⟶E is well defined.
To end this section, we state a fixed-point theorem of cone mapping, which is useful in the proof of our main results.Lemma 4.
(See [11]). Let E be a Banach space, and P⊂E be a cone in E. Assume that Ω1 and Ω2 are two bounded and open subsets of E with θ∈Ω1, Ω¯1⊂Ω2. If(22)Q:P∩Ω¯2∖Ω1⟶P,is a completely continuous operator such that either(i)
Qu≤u,∀u∈P∩∂Ω1 and Qu≥u,∀u∈P∩∂Ω2, or(ii)
Qu≥u,∀u∈P∩∂Ω1 and Qu≤u,∀u∈P∩∂Ω2,then Q has at least one fixed point in P∩Ω¯2∖Ω1.


## 3. Main Results

In this section, we establish necessary and sufficient conditions for the existence ofC1-positive solutions of the BVP (1) The proof is based on Lemma 4.Theorem 1.
Let the assumptionsA1–A4 hold. Then, the BVP (1) has at least one positive solution x∈C10,1 if and only if(23)0<∫01δ−1sτsfs,δsGs,s,δsG−1s,sds<+∞.Proof.
(Necessity). Letx∈C10,1 be a positive solution of the BVP (1) By Lemma 1, u∈C10,1 is a positive solution of the BVP (2). So, u′0 and u′1 exist and are finite.
Sinceu″t≤0 for t∈I∗, it follows that u′t is nonincreasing on I∗ and u′0≥u′1. Hence, by (2.1) of [6], there exists b>0, such that(24)us≥bGs,s,s∈I∗.
Then,(25)u−1s≤1bG−1s,s,s∈I∗.
Letℓ=minb,1. Then,(26)δsus≥ℓδsGs,s,δsu−1s≤1ℓδsG−1s,s,s∈I∗.
Hence, on the one hand, by Remark1, we have(27)∫01δ−1sτsfs,δsGs,s,δsG−1s,sds≤∫01δ−1sτsfs,1ℓδsus,ℓδsu−1sds≤1ℓσ1∫01δ−1sτsfs,δsus,δsu−1sds=1ℓσ1∫01−u″sds=1ℓσ1u′0−u′1<+∞.
On the other hand, iffs,δsus,δsu−1s≡0 for s∈I∗, then(28)δ−1sτsfs,δsus,δsu−1s≡0.
By (15) and Lemma 1, ut≡0 for t∈I∗. So, xt≡0 for t∈I∗, which contradicts the positivity of x. Hence, there exists t∗∈0,1, such that(29)ft∗,δt∗ut∗,δt∗u−1t∗>0.
By (2.1) of [6], there exists c>0, such that us≤cGs,s for s∈I∗. Then,(30)u−1s≥1cG−1s,s,s∈I∗.
Letℓ1=maxc,1. Then,(31)δt∗ut∗≤ℓ1δt∗Gt∗,t∗,δt∗u−1t∗≥1ℓ1δt∗G−1t∗,t∗.
Hence, byA1 and Remark 1, we have the following:Case (1): ifτt∗>0, then(32)0<δ−1t∗τt∗ft∗,δt∗ut∗,δt∗u−1t∗≤δ−1t∗τt∗ft∗,ℓ1δt∗Gt∗,t∗,1ℓ1δt∗G−1t∗,t∗≤ℓ1σ1δ−1t∗τt∗ft∗,δt∗Gt∗,t∗,δt∗G−1t∗,t∗.Case (2): ifτt∗=0, choose a small neighborhood a1,b1 of t∗, such that τs≡0 for s∈a1,b1⊂I. So,(33)∫a1b1δ−1sτsfs,δsGs,s,δsG−1s,sds>0.
Consequently, under both cases, we have(34)∫01δ−1sτsfs,δsGs,s,δsG−1s,sds>0.
(Sufficiency). At first, we prove thatQ:P⟶P is a completely continuous operator. In fact, on the one hand, for any u∈P, by (15) and (17), we have(35)Qut≥∫01qtGs,sδ−1sτsfs,δsus,δsu−1sds.
Then,(36)mint∈I0Qut≥mint∈I0qt∫01Gs,sδ−1sτsfs,δsus,δsu−1sds≥ηQu,∀t∈I∗.
On the other hand, sinceu∈P, there exists a constant ζ≥1, such that us≤ζGs,s for any s∈I∗. Hence,(37)δsus≤ζδsGs,s,δsu−1s≥1ζδsG−1s,s,s∈I∗.
Consequently, byA3 and Remark 1, we have(38)Qut≤∫01Gt,sδ−1sτsfs,ζδsGs,s,1ζδsG−1s,sds≤ζσ1∫01Gt,sδ−1sτsfs,δsGs,s,δsG−1s,sds≤ζσ1Gt,t∫01δ−1sτsfs,δsGs,s,δsG−1s,sds,∀t∈I∗.
Letζ∗=maxζσ1∫01δ−1sτsfs,δsGs,s,δsG−1s,sds,1. Then,(39)Qut≤ζ∗Gt,t,∀t∈I∗.
Hence,QP⊂P. By Ascoli-Arzela’s theorem, one can prove that Q:P⟶P is completely continuous.
Second, we prove that there exists a constantr>0, such that(40)Qu≤u,∀u∈P∩∂Ωr,where Ωr=u∈E:u<r and ∂Ωr=u∈E:u=r.
Foru∈P with u≤1, we have ut≤u≤1 for any t∈I∗. Hence, by A3, we have(41)ft,δtut,δtu−1t≤uσ2tft,δt,δt≤uσ2ft,δM,δm,∀t∈I∗.
So,(42)Qu≤maxt∈I∗∫01Gt,sδ−1sτsuσ2ft,δM,δmds≤β∗τpuσ2∫01δ−1sft,δM,δmds=Auσ2,where A=β∗τp∫01δ−1sft,δM,δmds.
IfA>1, set r=1A1/σ2−1<1; when u∈P∩∂Ωr, we have(43)Qu≤Auσ2=A1−σ2/σ2−1=r=u.
IfA≤1, set r=1; when u∈P∩∂Ωr, we have(44)Qu≤Auσ2=A≤1=r=u.
Third, we prove that there exists a constantR>r, such that(45)Qu≥u,∀u∈P∩∂ΩR,where ΩR=u∈E:u<R and ∂ΩR=u∈E:‖u‖=R.
Foru∈P with ut≥1 for t∈I∗, we have(46)δtut≥δm,δtu−1t≤δt≤δM.
It follows that(47)ft,δtut,δtu−1t≥uσ2tft,δt,δt≥qσ2tuσ2ft,δm,δM,∀t∈I∗.
Hence, by the definition of operatorQ and cone P, for any t∈I∗, we have(48)Qut≥α∗ξ∫01δ−1sqσ2suσ2fs,δs,δsds≥α∗ξuσ2∫ν1−νδ−1sqσ2sfs,δs,δsds≥uσ2α∗ξησ2∫ν1−νδ−1sfs,δm,δMds=Buσ2,where B=α∗ξησ2∫ν1−νδ−1sfs,δm,δMds and η=mint∈I0qt.
IfB<1, setting R=1/B1/σ2−1, R>1≥r. For u∈P∩∂ΩR, we have(49)Qu≥Qut≥Buσ2=B1−σ2/σ2−1=R=u.
IfB≥1, setting R=B+1, R>1≥r. For u∈P∩∂ΩR, we have(50)Qu≥Qut≥Buσ2≥Bu≥u.
By Lemma4, Q has at least one fixed point u∗∈P∩Ω¯R∖Ωr satisfying 0<r≤u∗≤R. Hence, mint∈I0u∗t≥ηu∗>0, and it is a positive solution of the BVP (2).
Finally, we proveu∗∈C10,1. Since u∗∈P, there exists a constant ζ1≥1, such that u∗s≤ζ1Gs,s for s∈I∗. Then, by Remark 1, one has(51)∫01u∗″sds=∫01δ−1sτsfs,δsu∗s,δsu∗−1sds≤∫01δ−1sτsfs,ζ1δsGs,s,1ζ1δsG−1s,sds≤ζ1σ1∫01δ−1sτsfs,δsGs,s,δsG−1s,sds<+∞.
Hence,u∗ is absolutely integrable on I∗. So, u∗′0+ and u∗′1− exist and are finite. Then, u∗∈C10,1, and it is a positive solution of the BVP (2). By Lemma 1, x∗t=δtu∗t,∀t∈I∗ belongs to C10,1, and it is a positive solution of the BVP (1).Theorem 2.
Assume thatτ and f satisfy the conditions A1, A2, A4, and A5ft,δm,δM>0,t∈I, and there exist constants 0<σ3≤σ4<1, such that for every ℓ∈0,1,(52)ℓσ4ft,x,y≤ft,ℓx,ℓ−1y≤ℓσ3ft,x,y,for any t∈I,x,y∈0,+∞.Then, the BVP (1) has at least one positive solution x∈C10,1 if and only if (23) holds.Proof.
If the conditionA3 is replaced by A5, we can also obtain the conclusion of Lemma 3. The proof of this theorem is the same as the one of Theorem 1. So, we omit the details here.Remark 4.
Condition (52) is equivalent to(53)ℓσ3ft,x,y≤ft,ℓx,ℓ−1y≤ℓσ4ft,x,y,ℓ≥1.
Ifft,x,y satisfies A5, then ft,x,y is nondecreasing with respect to x and nonincreasing with respect to y for every t∈I.Remark 5.
In [10], the authors assumed that ft,x must be nondecreasing with respect to x. In the present work, we assume that ft,x,y is nondecreasing with respect to x and nonincreasing with respect to y. But ft,x,y is not monotonous with respect to both x and y. Therefore, our results extend the ones of [10].Remark 6.
Ifτ∈L∞0,1, the results in Theorems 1 and 2 are still true.


## 4. Examples

Example 1.
Consider the second-order impulsive boundary value problem (IBVP):(54)−x″t=3+sintet2x3+etx−3,t∈0,1,t≠tk,Δx|t=tk=xtk,k=1,2,…,n,αx0−βx′0=0,γx1+ωx′1=0,where α,β,γ,ω≥0 satisfy α+βγ+αω>0.Theorem 3.
The IBVP (7) has at least one positive solution x∈C10,1 if and only if(55)0<∫013+sinsfs,2nGs,s,G−1s,sds<+∞,where(56)Gs,s=γ+ω−γsβ+αsα+βγ+αω,0≤s≤1.Proof.
Letft,x,y=et2x3+ety−3. Obviously, ft,x,x=et2x3+etx−3 and τt=3+sint. Then, the condition A1 holds with ξ=2. Since δk=1, it follows that the condition A2 holds with δt=Π0<tk<t2 and δm=1,δM=2n. Then, ft,δm,δM=et2+2−3net>0,∀t∈0,1, and for ℓ∈0,1, we have(57)ft,ℓx,ℓ−1y=et2ℓx3+etℓ−1y−3=ℓ3et2x3+ℓ4ety−3,which implies that(58)ℓ5ft,x,y≤ft,ℓx,ℓ−1y≤ℓ2ft,x,y,∀t∈0,1.
Clearly,∫0123net2+etdt<+∞. Hence, the conditions A3 and A4 are satisfied. Therefore, by Theorem 1, the IBVP (7) has at least one positive solution x∈C10,1 if and only if (55) holds.Example 2.
Consider the second-order impulsive boundary value problem (IBVP):(59)−x″t=2−cost2t2x1/3+t3x−1/3,t∈0,1,t≠tk,Δx|t=tk=xtk,k=1,2,…,n,αx0−βx′0=0,γx1+ωx′1=0,where α,β,γ,ω≥0 satisfy α+βγ+αω>0.Theorem 4.
The IBVP (10) has at least one positive solution x∈C10,1 if and only if(60)0<∫012−coss2fs,2nGs,s,G−1s,sds<+∞,where Gs,s is given in (56).Proof.
Letft,x,y=t2x1/3+t3y−1/3. Obviously, ft,x,x=t2x1/3+t3x−1/3 and τt=2−cost2. Then, for ℓ∈0,1, we have(61)ft,ℓx,ℓ−1y=t2ℓx1/3+t3ℓ−1y−1/3=ℓ1/3t2x1/3+ℓ1/4t3y−1/3,which implies that(62)ℓ1/2ft,x,y≤ft,ℓx,ℓ−1y≤ℓ1/5ft,x,y,∀t∈0,1.
The remain proof is similar to the one of Theorem3, and we omit the details here.Remark 7.
Ifft,x,x=t2x1/3+t3x−1/3 or ft,x,x=t2x1/3+t3x−1/3, it is clear that Ft,x=ft,x,x does not have monotonicity with respect to both x. Hence, the results in [10] are not valid to Examples 1 and 2.


## 5. Conclusions

This study deals with the existence ofC1-positive solutions for a class of second-order impulsive differential equations. By using the fixed-point theorem of cone mapping, we establish the necessary and sufficient conditions for the existence of C1-positive solutions of the BVP (1) Two examples are given at last to illustrate the application of the obtained abstract results.



---

*Source: 9521849-2022-01-25.xml*