Existence of Solutions for Two-Point Integral Boundary Value Problems with Impulses

In this paper, we investigate the existence of at least one solution and at least two nonnegative solutions of impulsive differential equations with the two-point integral boundary conditions. We employ the recent fixed point theorems for the sum of two operators on Banach spaces. The applicability of the results is illustrated through an example.


Introduction
The theory of conventional differential equations is, no doubt, one of the versatile tools to model and study various real-world physical phenomena. There are numerous evolution processes that experience perturbations and undergo sudden changes in their states. These changes are relatively short-time as compared to the overall duration of the whole process. Such processes can be found in various fields of science and technology. For the description of applications in biology, medicine, population dynamics, economics, neural networks, readers can refer [23]. It is seen that conventional differential equations are inadequate to describe such phenomena. Therefore, it is natural to consider the governing differential equations along with their impulse effects. These equations are known as impulsive differential equations. Inspired by numerous applications, the theory of impulsive differential equations has been studied intensively and over the last three decades, it has been an active research area producing an extensive portfolio of results. This can be witnessed by the following published works by different authors [2-4, 13, 14, 21-23]. It is not an overemphasize that this theory is much richer than the corresponding differential equations. In consequence, it creates an important branch of nonlinear analysis.
On the other hand, due to its wide applicability in many actual phenomena, it is essential to include suitable conditions with differential equations. This often leads to the study of the initial and boundary value problems together with integral equations. Such problems form the basis of mathematical modelling of several dynamic phenomena. When the boundary of the physical process is not available for measurements, nonlocal conditions in a multi-point form may be imposed as an additional information, sufficient for the solvability. Keeping this in mind, equations with multipoint integral boundary conditions play an important and special role. They include two, three, multi-point, and nonlocal boundary conditions as special cases. Various differential equations with several types of nonlocal conditions have been studied extensively and a large number of papers are devoted on this study, see [5, 6, 10-12, 18-20, 24] to mention a few.
Ashyralyev and Sharifov [1] studied the existence and uniqueness of solution for the system of nonlinear differential equations of the type subject to impulsive conditions and two-point integral boundary condition  [16] also studied the existence and uniqueness of solution for the system of nonlinear differential equations of the type with two-point integral boundary conditions of the form (1.5) where A, B ∈ R n×n are given matrices such that det(A + T 0 m(s)ds B) = 0, and m, f , g : [0, T ] × R n are given functions which satisfy certain conditions. Mardanov et al. [16] used the same tools and obtained the similar results of [1].
In this paper, we investigate for the existence of at least one solution and at least two nonnegative solutions of the following boundary value problem (BVP) . . , p}, and the nonlinear functions involved satisfy the following hypotheses.
where v := (v 1 , . . . , v n ), |v| = v 2 1 + · · · + v 2 n , p j ≥ 0, and a 1 j , a 2 j : [0, T ] → [0, ∞) are continuous functions such that 0 ≤ a 1 j , a 2 j ≤ D, j ∈ {1, . . . , n}, for some positive constant D. (H 2 ) I k : R n → R n are continuous functions, I k := (I k1 , . . . , I kn ) such that where q j ≥ 0, a 1k j and a 2k j are constants such that 0 ≤ a 1k j , a 2k j ≤ D, (H 4 ) g : [0, T ] × R n → R n is a continuous function, g := (g 1 , . . . , g n ) such that Mardanov and Sharofov [15] investigated BVP (1.6) for the existence of unique solution and of at least one solution in the following cases.  8) where N 1 and N 2 are positive constants. Note that if then f and g do not satisfy (1.7) and (1.8). Thus, the results in this paper can be considered as complimentary results to the results in [15]. The plan of this paper is as follows. In the next section, we recall some notations, definitions, and auxiliary results that we need throughout this paper. In Sect. 3, we prove our main results about the existence and multiplicity of solutions for the problem (1.6) by using recent fixed point theorems for the sum of two operators T + S on Banach spaces, firstly by considering T linear and (I − S) be compact, secondly by taking this sum such that T is an expansive operator and S is completely continuous one. An example is given in Sect. 4 in order to illustrate our obtained results. In Sect. 5, a concluding remarks are given.

Preliminary Results
In this section, we will give some preliminary material needed to prove our main results. First, we recall some notations and definitions that we need throughout this paper. For To prove the existence of at least one solution to BVP (1.6), we will use the following fixed point theorem for a sum of two operators.
Theorem 2.1 will be used to prove Theorem 3.8 and its proof can be found in [8]. In the sequel, we are concerned with the existence of multiple nonnegative fixed points for the sum of an expansive mapping and a completely continuous one. So let us recall the definitions from the available literature.
. Let X and Y be real Banach spaces. A mapping T : X → Y is said to be expansive if there exists a constant h > 1 such that Definition 2.4 [9, Definition 7.7 in §12]. A closed convex set P in a real Banach space E is said to be cone if 1. αu ∈ P for any α ≥ 0 and for any u ∈ P, 2. u, −u ∈ P implies u = 0.

Remark 2.5 Every cone P defines a partial ordering ≤ in E defined by
In the sequel, P will refer to a cone in a Banach space (E, . ), is a subset of P, U is a bounded open subset of P, and P * =P\{0}. Assume that S : U → E is a completely continuous mapping and T : → E is an expansive one with constant h > 1. Now, we present a multiple fixed point theorem which will be used to ensure the existence of at least two nonnegative solutions to BVP (1.6). The proof of this theorem is based upon a recent fixed point index developed in [7].
and there exists u 0 ∈ P * such that the following conditions hold: Then, T + S has at least two nonzero fixed points u 1 , u 2 ∈ P such that

Main Results
We will start with the following useful lemmas, which give an integral representation of a solution of BVP (1.6).

Lemma 3.1 If x ∈ E is a solution of the problem
then it satisfies the integral equation

2)
and the conversely.
Proof Firstly, we will note that the solution of the equation is given by In particular, we have Now, we consider the problem For its solution, we have the representation Assume that the solution of the problem for some k ∈ {1, . . . , p − 1}, is given by Then Now, we consider the problem For its solution, we have the following representation Thus, the representation (3.2) holds. Now, assume that x ∈ E satisfies (3.2). Fix k ∈ {1, . . . , p} arbitrarily and let t ∈ (t k , t k+1 ]. Then and Hence, i.e., x satisfies (3.1). This completes the proof.

Lemma 3.2 If x ∈ E is a solution to the BVP (1.6)
, then x satisfies the integral equation

3)
and the conversely.

Proof
Suppose that x is a solution of the BVP (1.6). We rewrite the first equation of (1.6) in the form and using Lemma 3.1, we have that x satisfies (3.2). Hence, and From this, the boundary condition in (1.6) becomes

Now, substituting this value in (3.2), we get (3.3). On the other hand, if x ∈ E is a solution to the integral equation (3.3)
, then by direct computations, we see that x will be also a solution of BVP (1.6). This completes the proof.
Before moving further, for x ∈ E, we define the operator S 1 : E → R n as

Remark 3.3 If
x ∈ E satisfies the equation S 1 x = 0, then it is a solution to the BVP (1.6).

Lemma 3.4 Suppose that the hypotheses (H 1 )-(H 5 ) hold. Then for any x ∈ E with
x ≤ D, the following inequality holds.
where S 1 is defined in (3.4).
Proof In view of (H 1 ), (H 2 ), and (H 4 ), we write Then and subsequently, we obtain S 1 [x] ≤ D 1 . This completes the proof. Now, we introduce a new hypothesis and an operator as follows. (H 6 ) Suppose C > 0 is a constant such that C D 1 < D. For x ∈ E, define the operator S 2 : E → R n as where S 1 is defined as (3.4).

Remark 3.5 If
x ∈ E satisfies the equation for arbitrary constant Q, then x is a solution to the problem (1.6). Really, differentiating (3.6) with respect to t, we get

Lemma 3.6 Suppose that (H 1 )-(H 5 ) hold. Let x ∈ E be such that x ≤ D. Then
Proof The assertion follows directly from Lemma 3.4.

Existence of at Least One Solution
Let X be the set of all equicontinuous families in E and define Let also, ε ∈ 0, 1 2 . For x ∈ E, we define two operators T , S : E → R n as follows, where S 2 is defined in (3.5). Note that any fixed point of the operator T + S is a solution to the BVP (1.6). Proof Using definition of the operator S, we have This completes the proof. Now, we are in a position to state and prove our main result of this section which is as follows.

Theorem 3.8 Suppose that (H 1 )-(H 5 ) and (H 6 ) hold. Then the BVP (1.6) has at least one solution in E.
Proof Note that the operators S 1 , S 2 : E → R n defined in (3.4) and (3.5) respectively, are continuous operators. We consider two operators T and S defined in (3.8) and (3.9) respectively. In view of Lemma 3.7, it follows that the operator I − S : X → X and it is continuous. Since the continuous maps of equicontinuous families forms equicontinuous families, we conclude that the image set (I − S)(X ) resides in a compact subset of E. Suppose that there is an element x ∈ ∂ X and λ ∈ 0, 1 This, using definition of operator S, we write as That is, Whereupon we have the following But from Lemma 3.7, we have ((1 + ε)I − S)[x] < εD. Hence, we arrived at a contradiction. Therefore, our assumption that (3.10) is not correct. From here, we get {u ∈ X : u = λ(I − S) [u], u = D} = ∅ for any λ ∈ 0, 1 .
Thus, all conditions of Theorem 2.1 hold. Therefore, the operator T + S has a fixed point in E which is a solution of BVP (1.6). Hence, BVP (1.6) has at least one solution. This completes the proof.

Existence of at Least Two Nonnegative Solutions
Next, before proving the result concerning multiple nonnegative solutions of BVP (1.6), we introduce one more hypothesis as follows.
(H 7 ) Let m > 0 be large enough and r , L, D be positive constants that satisfy the following inequalities.
(i) r < L < D, Let ε > 0. For x ∈ E, we define two operators T 1 , S 3 : E → R n as follows, where S 2 is defined in (3.5). Note that both T 1 and S 3 are continuous operators on E. Further, any fixed point x ∈ E of the operator T 1 + S 3 is solution to the BVP (1.6).
Our main result in this section is as follows. Proof Let P := {x ∈ X : x ≥ 0} be a cone in E. First we define some subsets of P which are used in the proof.
Now, we consider two operators T 1 and S 3 defined in (3.11) and (3.12) respectively, and employ Theorem 2.6. The proof will be given in the following steps.
Step 1. For x, y ∈ , from (3.11) we see that whereupon the operator T 1 : → R n is an expansive with a constant 1 + mε > 1.
Step 2. For x ∈ U 3 , from (3.12) we see that This means that S 3 (U 3 ) is uniformly bounded. Now, since S 3 : U 3 → R n is continuous, it follows that S 3 (U 3 ) is equicontinuous. Consequently, the operator S 3 : U 3 → R n is completely continuous.

Concluding Remarks
This paper explores the existence of at least one solution and at least two nonnegative solutions for the differential equations with impulses and two-point integral boundary conditions by employing recent fixed point theorems for the sum of two operators on Banach spaces. The results of this paper are essentially new in the sense that they are considered in the context of impulsive conditions and two-point integral boundary conditions. The boundary condition taken in this work is more general and includes various others as special cases. The results obtained in this paper can be further investigated for higher-order differential equations with impulses. We also believe that other qualitative properties like data dependence, stability, controllability, and oscillations can be studied in the forthcoming papers.