On a first-order differential system with initial and nonlocal boundary conditions

1 Introduction

In this paper, we consider the following nonlinear system

associated with the initial and nonlocal boundary conditions

where f , g : [ 0 , T ] × R n × R n → R n , h : [ 0 , T ] → R are given functions and u 0 ∈ R n ; μ j , T j ( j = 1 , N ¯ ) are given constants, with 0 < T 1 < T 2 < ⋯ < T N = T .

Nonlocal initial and boundary value problems with integral boundary conditions or multipoint boundary value problems for differential equations have been extensively studied due to their numerous applications in various fields of science and applied mathematics; see [1,2, 3,4,5, 6,7,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22] and the references given therein.

In [4], Bolojan-Nica et al. proved the solvability of a system of N first-order differential equations of the form

subject to the coupled nonlocal conditions

where α i j : C ( [ 0 , 1 ] ) → R were linear continuous functionals. As noted in [4], α i j could be written in a form involving Stieltjes integrals, namely,

for some functions of bounded variation A i j ( i , j = 1 , … , n ). On the other hand, the formula (1.5) covered, as special cases, initial conditions of the type:

where a k ∈ R and 0 ≤ t 1 < t 2 < ⋯ < t m ≤ 1 were given points. When all the functionals α i j had discrete expressions, (1.4) gave a multipoint condition. The approach there was based on the fixed-point theorems of Perov, Schauder, and Schaefer and on a vector method for treating systems, which used matrices with spectral radius less than one.

In [3], by extending the use of techniques in [4] to nonlocal conditions given by nonlinear functionals, O. Bolojan et al. proved the existence results of solutions to an initial value problem for a nonlinear first-order differential system subject to nonlinear nonlocal initial conditions of the form

where f 1 , f 2 : [ 0 , 1 ] × R 2 → R were L 1 -Carathéodory functions, α , β : C ( [ 0 , 1 ] ) × C ( [ 0 , 1 ] ) → R were nonlinear continuous functionals, and the solution ( x , y ) was sought in W 1 , 1 ( 0 , 1 ; R 2 ) . That problem was studied by using the fixed-point principles by Perov, Schauder, and Leray-Schauder, together with the technique that used convergent matrices and vector norms.

In [16], by applying the Banach’s fixed-point theorem and the Schaefer’s fixed-point theorem, Mardanov et al. proved the existence and uniqueness theorems for the system of ordinary differential equations with three-point boundary conditions as follows:

where A , B , and C were constant square matrices of order n such that det ( A + B + C ) ≠ 0 , f : [ 0 , T ] × R n → R n was a given function, d ∈ R n was a given vector, t 1 satisfied the condition of 0 < t 1 < T , and y : [ 0 , T ] → R n was unknown.

In [9], Han considered the second-order three-point boundary value problem in the form

with η ∈ ( 0 , 1 ) . By means of the fixed-point theorem in cones, the existence and multiplicity of positive solutions were proved.

In [20], Truong et al. studied the following m -point boundary value problem:

where m ≥ 3 , η j ∈ ( 0 , 1 ) , and α j ≥ 0 , for all j = 1 , m − 2 ¯ such that ∑ j = 1 m − 2 α j < 1 . By applying well-known Guo-Krasnoselskii’s fixed-point theorem and applying the monotone iterative technique, the results obtained in [20] were the existence and multiplicity of positive solutions. Furthermore, the compactness of the set of positive solutions was proved.

In [5], Boucherif applied the fixed-point theorem in a cone to study the existence of positive solutions for the problem given by

where f : [ 0 , 1 ] × R → R was continuous, g 0 , g 1 : [ 0 , 1 ] → [ 0 , + ∞ ) were continuous and positive, and c and d were nonnegative real parameters.

In [1], Agarwal et al. formulated existence results for solutions to discrete equations that approximate three-point boundary value problems for second-order ordinary differential equations. The proofs of these results were finished based on extending the notion of discrete compatibility, which was a degree-based relationship between the given boundary conditions and the lower or upper solutions chosen, to three-point boundary conditions. On the other hand, the invariance of the degree under the homotopy of the degree theory was also applied in the aforementioned proofs.

In [11], Henderson and Luca investigated the following multipoint boundary value problem for the system of nonlinear higher-order ordinary differential equations of the type

with the multipoint boundary conditions

Under sufficient assumptions on f and g , the authors proved the existence and multiplicity of positive solutions of the aforementioned problem by applying the fixed-point index theory.

Motivated by the aforementioned works, by applying standard fixed-point theorems, we study the initial and multipoint boundary problems (1.1) and (1.2).

This paper is organized as follows. In Section 2, we present some preliminaries. Here, the Green function is established for Prob. (1.1) and (1.2) such that the considered problem is reduced to the equivalent integral system. Section 3 is devoted to the existence and uniqueness of solutions based on the fixed-point theorems of Banach and Krasnoselskii. In Section 4, we prove the existence of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone. In Section 5, we observe some multiplicity results for positive solutions. Finally, a remark is also given in Section 6 for a system of multiple differential equations. Furthermore, various examples (Examples 3.1, 3.2, and 4.1) are also presented to demonstrate the validity of the main results. The methods used for proofs are standard, but their applications in this framework of the system (1.1) and (1.2) with nonlocal conditions are new, where we construct the novel representation of the Green’s functions with useful properties and define a cone suitably. We note more that properties of the associated Green’s function are of fundamental importance in many of our arguments, and based on these properties, we establish sufficient conditions for which solutions/positive solutions exist.

2 Preliminaries

Throughout this paper, x i denotes the ith component of x ∈ R n , ∣ x ∣ 1 = ∣ x 1 ∣ + ⋯ + ∣ x n ∣ denotes the norm of x = ( x 1 , … , x n ) ∈ R n , C ( [ 0 , T ] ; R n ) denotes the Banach space of all continuous functions u : [ 0 , T ] ⟶ R n endowed with the usual maximum norm ∥ u ∥ C ( [ 0 , T ] ; R n ) = max 0 ≤ t ≤ T ∣ u ( t ) ∣ 1 ; C ( [ 0 , T ] ) ≡ C ( [ 0 , T ] ; R ) , and L 1 ( 0 , T ) denotes the set of all integrable functions from ( 0 , T ) to R . Denote the Banach space of continuously differentiable functions u : [ 0 , T ] ⟶ R n by C 1 ( [ 0 , T ] ; R n ) endowed with the usual norm as follows:

In addition, for all x = ( x 1 , … , x n ) ∈ R n , we say x ≥ 0 (or x ∈ R + n ) if and only if x i ≥ 0 , for all i = 1 , … , n ; x > 0 if and only if x ≥ 0 , and x ≠ 0 ; and we say x ≥ y if and only if x − y ≥ 0 , for all x , y ∈ R n . It is clear to see that if one has x ≥ y ≥ 0 then ∣ x ∣ 1 ≥ ∣ y ∣ 1 ; therefore, ∣ x ∣ 1 = ∣ x − y ∣ 1 + ∣ y ∣ 1 . Moreover, if u ∈ L 1 ( 0 , T ; R n ) , u ( t ) ≥ 0 , ∀ t ∈ [ 0 , T ] , and then, we have

Now, for the convenience of the reader, we recall the following fixed-point theorems.

Next, we construct an expression for solutions of Prob. (1.1) and (1.2), with f , g ∈ C ( [ 0 , T ] × R n × R n ; R n ) , where C ( [ 0 , T ] × R n × R n ; R n ) is the space of all continuous functions mapping from [ 0 , T ] × R n × R n to R n . For details, in the following, we shall construct the Green function for Prob. (1.1) and (1.2) such that the considered problem is reduced to an equivalent integral system.

For each α , β > 0 , we denote f α ( t , u , v ) = f ( t , u , v ) + α u and g β ( t , u , v ) = g ( t , u , v ) + β v .

The second integral equation of (2.1) can be written in the following form:

where G ( t , s ) is the Green’s function for Prob. (1.1) and (1.2), and this function is defined as follows:

We have a property of the Green’s function G ( t , s ) as follows.

If the sign of μ j and h ( t ) can not be determined, we note more that

where

3 Existence and uniqueness

In this section, we shall prove two existence results (Theorems 3.1 and 3.5) for Prob. (1.1) and (1.2), where h ∈ C ( [ 0 , T ] ) and f , g ∈ C ( [ 0 , T ] × R n × R n ; R n ) . Based on the Banach’s fixed-point theorem, Theorem 3.1 gives a sufficient condition for the unique existence of a solution. Under weaker conditions, Theorem 3.5 gives a sufficient condition for the existence, and this is proved by using the Krasnoselskii’s fixed-point theorem.

We consider the Banach space X = C ( [ 0 , T ] ; R n ) × C ( [ 0 , T ] ; R n ) equipped with the norm

and we define an operator P : X ⟶ X as follows:

where

We make the following assumptions:

1. h ∈ C ( [ 0 , T ] ) ;

2. 0 < ∫ 0 T ∣ h ( t ) ∣ d t + ∑ j = 1 N ∣ μ j ∣ < 1 ;

3. There exists a positive function L f ∈ L 1 ( 0 , T ) such that

   (3.3) ∣ f ( t , u , v ) − f ( t , u ¯ , v ¯ ) ∣ 1 ≤ L f ( t ) ( ∣ u − u ¯ ∣ 1 + ∣ v − v ¯ ∣ 1 ) ,

   for all ( t , u , v ) , ( t , u ¯ , v ¯ ) ∈ [ 0 , T ] × R n × R n ;

4. There exists a positive function L g ∈ L 1 ( 0 , T ) such that

   (3.4) ∣ g ( t , u , v ) − g ( t , u ¯ , v ¯ ) ∣ 1 ≤ L g ( t ) ( ∣ u − u ¯ ∣ 1 + ∣ v − v ¯ ∣ 1 ) ,

   for all ( t , u , v ) , ( t , u ¯ , v ¯ ) ∈ [ 0 , T ] × R n × R n .

In what follows, under weaker conditions, the second result is given without the Lipschitzian condition on g as in ( H 3 ) . The main tool is the Krasnoselskii’s fixed-point theorem. We make the assumption ( H 4 ) as below.

( H ¯ 4 ) g : [ 0 , T ] × R n × R n → R n is a continuous function, and there exist two positive functions g ˆ 1 , g ˆ 2 ∈ L 1 ( 0 , T ) such that

We now define two operators U , C : X → X as follows: