1. Introduction and preliminaries

As is known, integro-differential equations find many applications in various mathematical problems, see Cordunean's book [1], Guo et al.'s book [2] and references therein for details. For the recent developments involving existence of solutions to BVPs for integro-differential equations and impulsive integro-differential equations we can refer to [3–17]. So far the main method appeared in the references to guarantee the existence of solutions is the method of upper and low solutions. Motivated by the ideas in the recent works [18, 19], we come up with a new approach to ensure the existence of at least one solution for certain family of first-order nonlinear integro-differential equations with periodic boundary value conditions or antiperiodic boundary value conditions. Our methods involve new differential inequalities and the classical fixed-point theory.

This paper mainly considers the existence of solutions for the following first-order nonlinear integro-differential system with periodic boundary value conditions.

(1.1)

and first-order integro-differential system with "non-periodic" conditions.

(1.2)

where (Kx)(t) denotes

with k i (t, s) : [0, 1] × [0, 1] → [0, +∞) continuous for i = 1, 2, ⋯, n; A and B are n × n matrices with real valued elements, θ is the zero vector in ℝ n . For A = (a ij ) n × n , we denote ||A|| by . In what follows, we assume function f : [0, 1] × ℝ n × ℝ n → ℝ n is continuous, and det (A + B) ≠ 0.

Noticing that det (A+B) ≠ 0, conditions Ax(0)+Bx(1) = θ do not include the periodic conditions x(0) = x(1). Furthermore, if A = B = I, where I denotes n × n identity matrix, then Ax(0)+Bx(1) = θ reduces to the so-called "anti-periodic" conditions x(0) = -x(1). The authors of [20–24] consider this kind of "anti-periodic" conditions for differential equations or impulsive differential equations. To the best of our knowledge it is the first article to deal with integro-differential equations with "anti-periodic" conditions so far.

We are also concerned with the following BPVP of integro-differential equations of mixed type:

(1.3)

where function f : [0, 1] × ℝ n × ℝ n × ℝ n → ℝ n is continuous, (Lx) (t) denotes

with l i (t, s) : [0, 1] × [0, 1] → ℝ, i = 1, 2, ⋯, n being continuous.

This article is organized as follows. In Sect. 1 we give some preliminaries. Section 2 presents some existence theorems for PVPs (1.1), (1.3) and a couple of examples to illustrate how our newly developed results work. In Sect. 3 we focus on the existence of solutions for (1.2) and also an example is given.

In what follows, if x, y ∈ ℝ n , then 〈x, y〉 denotes the usual inner product and ||x|| denotes the Euclidean norm of x on ℝ n . Let

with the norm

The following well-known fixed-point theorem will be used in the proof of Theorem 3.3.

Theorem 1.1 (Schaefer)[25]. Let X be a normed space with H : X → X a compact mapping. If the set

is bounded, then H has at least one fixed-point.

2. Existence results for periodic conditions

To begin with, we consider the following periodic boundary value problem

(2.1)

where g : [0, 1] × ℝ n × ℝ n → ℝ n and m : [0, 1] → ℝ are both continuous functions, with m having no zeros in [0, 1].

Lemma 2.1. The BVP (2.1) is equivalent to the integral equation

Proof. The result can be obtained by direct computation.

Theorem 2.1. Let g and m be as in Lemma 2.1. Assume that there exist constants R > 0, α ≥ 0 such that

(2.2)

and

(2.3)

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (2.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. Let C = C([0, 1], R n ) and Ω = {x(t) ∈ C, ||x(t)|| C < R}. Define an operator by

(2.4)

for all t ∈ [0, 1].

Since g is continuous, see that T is also a continuous map. It is easy to verify the operator T is compact by the Arzela-Ascoli theorem. Indeed, for the ball Ω,

(2.5)

implies

Define H λ = I - λT, λ ∈ [0, 1], where I is the identity. So if (2.5) is true, then from the homotopy principle of Schauder degree [[25], Chap.4.], we have

Therefore, it follows from the non-zero property of Leray-Schauder degree that H 1(x) = x - Tx = 0 has at least one x ∈ Ω.

Now our problem is reduced to prove that (2.5) is true. Observe that the family of problems

(2.6)

is equivalent to the family of PBVPs

(2.7)

Consider the function r(t) = ||x(t)||2, t ∈ [0, 1], where x(t) is a solution of (2.7). Then r(t) is differentiable and we have by the product rule

Denote

Let x be a solution of (2.6) with We now show that x ∉ ∂Ω. From (2.5) and (2.3) we have, for each t ∈ [0, 1] and each λ ∈ [0, 1],

Then it follows from (2.2) that x ∉ ∂ Ω. Thus, (2.5) is true and the proof is completed.

Corollary 2.1. Let g and m be as in Lemma 2.1 with m(t) < 0, t ∈ [0, 1]. If there exist constants R > 0, α ≥ 0 such that

and

(2.8)

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}, then PBVP (2.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. Multiply both sides of (2.8) by λ ∈ [0, 1] to obtain

It completes the proof.

Now consider the existence of solutions of PBVP (1.1). It is easy to see (1.1) is equivalent to the PBVP

(2.9)

Theorem 2.2. If there exist constants R > 0, α ≥ 0 such that

(2.10)

and

(2.11)

where M(R) is a positive constant dependent on R, B R = {x ∈ ℝ n , ||x|| ≤ R}, then PBVP (1.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. Consider the PVPB (2.9), which is of the form (2.1) with m(t) ≡ - 1 and g(t, x, (Kx)(t)) = f (t, x, (Kx)(t)) - x. Clearly,

So, (2.2) reduces to (2.10). Besides, (2.3) reduces to (2.11). Hence the result follows from Theorem 2.1.

Corollary 2.2. Assume there are constants R > 0, α ≥ 0 such that

(2.12)

and

(2.13)

where M (R) is a positive constant dependent on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (1.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. Multiply both sides of (2.13) by λ ∈ [0, 1] to obtain

Considering that

we have (2.11) is true if (2.13) is true. Then the proof is completed.

Now an example is provided to show how our theorems work.

Example 2.1 1. Consider the following PBVP with n = 2.

(2.14)
Figure 1
figure 1

Figure of Example 2.1

Full size image

Let us show (2.14) has at least one solution (x(t), y(t))⊤ with , ∀t ∈ [0, 1].

It is clear that (2.14) has no constant solution. Let u = (x, y)⊤, and . First note that for .

Then

On the other hand,

Clearly,

So for some R ≤ 90 and α = 5, we have

where . Now it is sufficient to find a positive constant R satisfying

(2.15)

It is easy to see that any number in satisfies (2.15). Then our conclusion follows from Corollary 2.2.

In what follows we focus on the first-order integro-differential equations of mixed type in the form of (1.2). The results presented in the following three statements are similar to Theorem 2.1, Theorem 2.2 and Corollary 2.2, respectively. So we omit all the proofs here.

Consider the following periodic boundary value problem

(2.16)

where g : [0, 1] × ℝ n × ℝ n × ℝ n → ℝ n and m : [0, 1] → ℝ are both continuous functions, with m having no zeros in [0, 1].

Theorem 2.3. Assume there are constants R > 0, α ≥ 0 such that

and

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (2.16) has at least one solution x ∈ C with ||x|| C < R.

Theorem 2.4. Suppose there are constants R > 0, α ≥ 0 such that

and

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (1.2) has at least one solution x ∈ C with ||x|| C < R.

Corollary 2.3. If there exist constants R > 0, α ≥ 0 such that

(2.17)

and

(2.18)

where M(R) is a positive constant dependent on R, B R = {x ∈ ℝ n , ||x|| ≤ R}, then PBVP (1.3) has at least one solution x ∈ C with ||x|| C < R.

Now we give an example to illustrate how to apply our theorems.

Example 2.2 2. Consider the following PBVP with n = 2.

(2.19)
Figure 2
figure 2

Figure of Example 2.2

Full size image

We prove that (2.19) has at least one solution (x(t), y(t))⊤ with , ∀t ∈ [0, 1].

First note that (2.19) has no constant solution. Let u = (x, y)⊤, and .

Since and , we obtain

On the other hand,

Clearly,

Thus,

where

Now it is sufficient to find a positive constant R satisfying

We compute directly . Then our conclusion follows from Corollary 2.3.

Notice that the conclusion of Theorem 2.1 still holds if (2.3) is replaced by

Now we modify Theorem 2.1 and Corollary 2.2 to obtain some new results.

Theorem 2.5. Let g and m be as in Lemma 2.1. Assume there exist constants R > 0, α ≥ 0 such that

and

(2.20)

where M(R) is a positive constant dependent on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (2.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. The proof is similar to that of Theorem 2.1 except choosing r(t) = - ||x(t) ||2 instead.

See that (1.1) is equivalent to the PBVP

(2.21)

Corollary 2.4. Suppose there exist constants R > 0, α ≥ 0 such that

and

(2.22)

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then PBVP (1.1) has at least one solution x ∈ C with ||x|| C < R.

Proof. Consider PVPB (2.21), which is in the form from (2.1) with m(t) ≡ 1 and g(t, x, (Kx)(t)) = f(t, x, (Kx)(t)) + x. Clearly,

Multiply both sides of (2.22) by λ ∈ [0, 1] to obtain

Then the conclusion follows from Theorem 2.5.

Remark 2.1. Corollary 2.4 and Corollary 2.2 differ in sense that Corollary 2.4 may apply to certain problems, whereas Corollary 2.2 may not apply, and vice-versa.

Example 2.3 3. Let us prove that the PBVP

Figure 3
figure 3

Figure of Example 2.3

Full size image
(2.23)

has at least one solution x(t) with |x(t)| < 1, ∀t ∈ [0, 1].

Denote . It is clearly that for all (t, x) ∈ [0, 1] × B R ,

Then for all (t, x) ∈ [0, 1] × B R ,

On the other hand,

Taking into account that

we choose

It is not difficult to check that if R ∈ [0.5, 2]. So the conclusion follows from Corollary 2.4.

Remark 2.2 Since the coefficient of x 3 is negative, it appears impossible to find two constants R > 0 and α ≥ 0 satisfying (2.12) and (2.13) at the same time.

3. Existence results for "non-periodic" conditions

In this section we study the problem of existence of solutions for BVP (1.2).

Lemma 3.1. The BVP (1.2) is equivalent to the integral equation

Proof. The result can be obtained by direct computation.

Theorem 3.1. Assume det B ≠ 0 and ||B -1 A|| ≤ 1. Suppose there exist constants R > 0, α ≥ 0 such that

(3.1)

and

(3.2)

where M(R) is a positive constant depending on R, BR = {x ∈ ℝ n , ||x|| ≤ R}. Then BVP (1.2) has at least one solution x ∈ C with ||x|| C < R.

Proof. Let C = C([0, 1], R n ) and Ω = {x(t) ∈ C, ||x(t)|| C < Rg. Define an operator by

(3.3)

Since f is continuous, we see that T is also a continuous map. It is easy to verify that the operator T is compact by the Arzela-Ascoli theorem. It is sufficient to prove

(3.4)

See that the family of problems

(3.5)

is equivalent to the family of BVPs

(3.6)

Consider function r(t) = ||x(t)||2, t ∈ [0, 1], where x(t) is a solution of (3.6). By the product rule we have

Note that ||B -1 A||| ≤ 1 implies

Let x be a solution of (3.5) with . We now show that x ∉ ∂ Ω. From (3.2) and (3.3) we obtain, for each t ∈ [0, 1] and each λ ∈ [0, 1],

Then it follows from (3.1) that x ∉ ∂ Ω. Thus, (3.4) is true and the proof is completed.

Corollary 3.1 Let f be a scalar-valued function in (1.1). and assume there exist constants R > 0, α ≥ 0 such that

and

(3.8)

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , |x| ≤ R}. Then anti-periodic boundary value problem

has at least one solution x ∈ C[0, 1] with |x(t)| < R, t ∈ [0, 1].

Proof. Since A = B = 1, we have , B -1 A = 1, . Then the conclusion follows from Lemma 3.1.

Example 3.1. Let us show that

(3.9)

has at least one solution x(t) with |x(t)| < 1, ∀t ∈ [0, 1].

Denoting , we see that, for all (t, x) ∈ [0, 1] × B R ,

On the other hand,

Since

we choose

Then

It is not difficult to check that . So, the conclusion follows from Corollary 3.1.

Now we modify Theorem 3.1 to include another class of f.

Theorem 3.2. Assume det B ≠ 0 and ||A -1 B|| ≤ 1. Suppose there exist constants R > 0, α ≥ 0 such that

and

(3.10)

where M(R) is a positive constant depending on R, B R = {x ∈ ℝ n , ||x|| ≤ R}. Then BVP (1.2) has at least one solution x ∈ C with ||x|| C < R.

Proof. Note that ||A -1 B|| ≤ 1 implies

Introducing the function r(t) = -||x(t) ||2, t ∈ [0, 1], where x(t) is a solution of (3.6), for the rest part of the proof we proceed as in the proof of Theorem 3.1.

Corollary 3.2 Let f be a scalar-valued function in (1.1). If there exist constants R > 0, α ≥ 0 such that

(3.11)

and

(3.12)

where M(R) is a positive constant dependent on R, B R = {x ∈ ℝ n , |x| ≤ R}. Then anti-periodic boundary value problem

has at least one solution x ∈ C[0, 1] with |x(t)| < R, t ∈ [0, 1].

Proof. Since A = B = 1, we have , A -1 B = 1, . Then the conclusion follows from Lemma 3.2.

In what follows, we discuss the problem of existence of solutions for (1.2) with f satisfying

where nonnegative functions p, q, r ∈ L 1[0, 1]. We denote for any function x ∈ L 1 [0, 1].

Theorem 3.3. Assume (*) is true and

(3.13)

where . Then (1.2) has at least one solution.

Proof. Let C = C([0, 1], R n ). Define an operator T : C → C by

As we discussed in the proof of Theorem 3.1, T is compact. Taking into account that the family of BVP (1.2) is equivalent to the family of problem x = Tx, our problem is reduced to show that T has a least one fixed point. For this purpose, we apply Schaefer's Theorem by showing that all potential solutions of

(3.14)

are bounded a priori, with the bound being independent of λ. With this in mind, let x be a solution of (3.14). Note that x is also a solution of (3.6). We have, for ∀t ∈ [0, 1] and ∀λ [0, 1],

Thus,

It then follows from (3.13) that

The proof is completed.

Remark 3.1. If A = B = I, then (2.13) reduces to

We can also extend the discussion to the existence of at least one solution for integro-differential equations of mixed type with "anti-periodic" conditions.

We omit it here because it is trivial.