Existence of C 1 -Positive Solutions for a Class of Second-Order Impulsive Differential Equations

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Introduction
e 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].ere 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.e 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.
e 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 secondorder singular differential equation with integral boundary conditions.Recently, Zhang and Tian [10] established sharp conditions for the existence of positive solutions of secondorder impulsive differential equations.But in their work, a key assumption is that the nonlinearity f(t, x) is nondecreasing with respect to x ≥ 0. Clearly, if f(t, x) � t 2 x 1/3 + t 3 x − 1/3 , the results obtained in [9,10] are not valid.e 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 C 1 -positive solutions for a class of second-order impulsive differential equations.e 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 α, β, c, ω are the nonnegative constants satisfying , and ς is a positive continuous function on I, f ∈ C(I × (0, +∞) × (0, +∞), [0, +∞)), f(t, x, y) may be singular at t � 0, 1, In addition, τ and δ k   n k�1 satisfy the following conditions: Clearly, δ(t) is continuous on I * .Denote by δ M � max t∈I * δ(t), δ m � min t∈I * δ(t).From (A2), we have Furthermore, δ − 1 (t) �  0<t k <t (1 + δ k ) − 1 for any t ∈ I * .e main results of this work are summarized as follows: (i) e necessary and sufficient conditions on τ and f are established for the existence of C 1 -positive solutions of the BVP (1) ( eorems 1 and 2) (ii) We assume that the nonlinear term f(t, x, y) satisfies (A3), which implies that f(t, x, y) is nondecreasing with respect to x and nonincreasing with respect to y (Remark 2).But f(t, 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 ( eorem 3 and 4).
e 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.

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 en, y(t) � δ(t)u − 1 (t) with ς(t) � δ 2 (t).e BVP (1) is rewritten into the BVP:

Lemma 1. Let the assumptions (A1) and (A2) hold. en, u(t) is a solution of BVP (2) on I if and only if x(t) � δ(t)u(t) is a solution of BVP (1) on I.
Proof.(Necessity).Let u(t) be a solution of the BVP (2) on I. en, on each interval So, When t � t k , we have It follows that Obviously, x(t) satisfies the boundary conditions.en, x(t) is a solution of the BVP (1) on I.
2 Complexity (Sufficiency).Let x(t) be a solution of the BVP (1) on Direct calculation shows that u(t) satisfies all boundary conditions.en, u(t) is a solution of the BVP (2) on I.
roughout this study, the following assumptions on f are needed.
Proof.Assume that (A1)-(A4) hold.For fixed u ∈ P with u(t) > 0 for any t ∈ I * , choosing a constant a ∈ (0, 1) satisfying 0 < au(t) < 1 for t ∈ I * , we have So, for t ∈ I * , by (17), we have is implies that the operator Q: 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.

Main Results
In this section, we establish necessary and sufficient conditions for the existence of C 1 -positive solutions of the BVP (1) e proof is based on Lemma 4.
Second, we prove that there exists a constant r > 0, such that where For u ∈ P with ‖u‖ ≤ 1, we have u(t) ≤ ‖u‖ ≤ 1 for any t ∈ I * .Hence, by (A3), we have where ird, we prove that there exists a constant R > r, such that where For u ∈ P with u(t) ≥ 1 for t ∈ I * , we have It follows that Hence, by the definition of operator Q and cone P, for any t ∈ I * , we have By Lemma 4, Q has at least one fixed point u * ∈ P ∩ (Ω R ∖Ω r ) satisfying 0 < r ≤ ‖u * ‖ ≤ R. Hence, min t∈I 0 u * (t) ≥ η‖u * ‖ > 0, and it is a positive solution of the BVP (2).
Proof.If the condition (A3) is replaced by (A5), we can also obtain the conclusion of Lemma 3. e proof of this theorem is the same as the one of eorem 1.So, we omit the details here.

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Remark 4. Condition (52) is equivalent to If f(t, x, y) satisfies (A5), then f(t, 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 f(t, x) must be nondecreasing with respect to x.In the present work, we assume that f(t, x, y) is nondecreasing with respect to x and nonincreasing with respect to y.But f(t, x, y) is not monotonous with respect to both x and y. erefore, our results extend the ones of [10].
], the results in eorems 1 and 2 are still true.

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