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

Abstract

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

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 -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 is nondecreasing with respect to . Clearly, if , 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 -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:where are the nonnegative constants satisfying , , , , and is a positive continuous function on , , may be singular at , , is a real sequence, and represents the impulsive term, where and denotes the right-hand limit of at .

In addition, and satisfy the following conditions:  for some , and there exists , such that , i.e., .  is a real sequence with , and denoted by

Clearly, is continuous on . Denote by . From , we have

Furthermore, for any .

The main results of this work are summarized as follows:(i)The necessary and sufficient conditions on and are established for the existence of -positive solutions of the BVP (1) (Theorems 1 and 2)(ii)We assume that the nonlinear term satisfies , which implies that is nondecreasing with respect to and nonincreasing with respect to (Remark 2). But 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 (Theorem 3 and 4).

The rest of this study is organized as follows. Some preliminaries and notations are presented in Section 2. 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

Then, with . The BVP (1) is rewritten into the BVP:

Lemma 1. Let the assumptions and hold. Then, is a solution of BVP (2) on if and only if is a solution of BVP (1) on .

Proof. (Necessity). Let be a solution of the BVP (2) on . Then, on each interval , is absolutely continuous. For , we haveSo,When , we haveIt follows thatObviously, satisfies the boundary conditions. Then, is a solution of the BVP (1) on .
(Sufficiency). Let be a solution of the BVP (1) on . Then, and , which followsWhen , since and , we haveDirect calculation shows that satisfies all boundary conditions. Then, is a solution of the BVP (2) on .

Definition 1. A function is called a positive solution of the BVP (2) if satisfies all the equations in (2) and for all . If is the positive solution of the BVP (2) and , namely, and exist, then is called a -positive solution of the BVP (2).
Throughout this study, the following assumptions on are needed.
for any , and there exist constants , such that, for every ,for any and .
.

Remark 1. The condition implies, for every ,for any .

Remark 2. If satisfies , then for any , is nondecreasing with respect to and nonincreasing with respect to .
Denote by the Banach space of all continuous functions on equipped with the norm . Define an operator bywhere is Green’s function of the BVP (2) with , andwhere .

Remark 3. By the definition of , is a positive solution of the BVP (2) if and only if is a positive fixed point of .

Lemma 2. (See [10]). Green’s function of the BVP (2) satisfieswhere

For any , let and . Then, . Define a cone by

Then, is a closed convex cone in .

Lemma 3. Let the assumptions – hold. Then, the operator , defined by (3), is well defined.

Proof. Assume that – hold. For fixed with for any , choosing a constant satisfying for , we haveSo, for , by (17), we haveThis implies that the operator 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 be a Banach space, and be a cone in . Assume that and are two bounded and open subsets of with , . Ifis a completely continuous operator such that either(i) and , or(ii) and ,then has at least one fixed point in .

3. Main Results

In this section, we establish necessary and sufficient conditions for the existence of -positive solutions of the BVP (1) The proof is based on Lemma 4.

Theorem 1. Let the assumptions – hold. Then, the BVP (1) has at least one positive solution if and only if

Proof. (Necessity). Let be a positive solution of the BVP (1) By Lemma 1, is a positive solution of the BVP (2). So, and exist and are finite.
Since for , it follows that is nonincreasing on and . Hence, by (2.1) of [6], there exists , such thatThen,Let . Then,Hence, on the one hand, by Remark 1, we haveOn the other hand, if for , thenBy (15) and Lemma 1, for . So, for , which contradicts the positivity of . Hence, there exists , such thatBy (2.1) of [6], there exists , such that for . Then,Let . Then,Hence, by and Remark 1, we have the following: Case (1): if , then Case (2): if , choose a small neighborhood of , such that for . So,Consequently, under both cases, we have(Sufficiency). At first, we prove that is a completely continuous operator. In fact, on the one hand, for any , by (15) and (17), we haveThen,On the other hand, since , there exists a constant , such that for any . Hence,Consequently, by and Remark 1, we haveLet . Then,Hence, . By Ascoli-Arzela’s theorem, one can prove that is completely continuous.
Second, we prove that there exists a constant , such thatwhere and .
For with , we have for any . Hence, by , we haveSo,where .
If , set ; when , we haveIf , set ; when , we haveThird, we prove that there exists a constant , such thatwhere and .
For with for , we haveIt follows thatHence, by the definition of operator and cone , for any , we havewhere and .
If , setting , . For , we haveIf , setting , . For , we haveBy Lemma 4, has at least one fixed point satisfying . Hence, , and it is a positive solution of the BVP (2).
Finally, we prove . Since , there exists a constant , such that for . Then, by Remark 1, one hasHence, is absolutely integrable on . So, and exist and are finite. Then, , and it is a positive solution of the BVP (2). By Lemma 1, belongs to , and it is a positive solution of the BVP (1).

Theorem 2. Assume that and satisfy the conditions , , , and , and there exist constants , such that for every ,for any .

Then, the BVP (1) has at least one positive solution if and only if (23) holds.

Proof. If the condition is replaced by , 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 toIf satisfies , then is nondecreasing with respect to and nonincreasing with respect to for every .

Remark 5. In [10], the authors assumed that must be nondecreasing with respect to . In the present work, we assume that is nondecreasing with respect to and nonincreasing with respect to . But is not monotonous with respect to both and . Therefore, our results extend the ones of [10].

Remark 6. If , the results in Theorems 1 and 2 are still true.

4. Examples

Example 1. Consider the second-order impulsive boundary value problem (IBVP):where satisfy .

Theorem 3. The IBVP (7) has at least one positive solution if and only ifwhere

Proof. Let . Obviously, and . Then, the condition holds with . Since , it follows that the condition holds with and . Then, , and for , we havewhich implies thatClearly, . Hence, the conditions and are satisfied. Therefore, by Theorem 1, the IBVP (7) has at least one positive solution if and only if (55) holds.

Example 2. Consider the second-order impulsive boundary value problem (IBVP):where satisfy .

Theorem 4. The IBVP (10) has at least one positive solution if and only ifwhere is given in (56).

Proof. Let . Obviously, and . Then, for , we havewhich implies thatThe remain proof is similar to the one of Theorem 3, and we omit the details here.

Remark 7. If or , it is clear that does not have monotonicity with respect to both . Hence, the results in [10] are not valid to Examples 1 and 2.

5. Conclusions

This study deals with the existence of -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 -positive solutions of the BVP (1) Two examples are given at last to illustrate the application of the obtained abstract results.

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest.