Open AccessArticle

Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales

Famei Zheng

Xiaojing Wang

Xiwang Cheng

and

Bo Du

School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China

Author to whom correspondence should be addressed.

Symmetry 2023, 15(8), 1524; https://doi.org/10.3390/sym15081524

Submission received: 20 June 2023 / Revised: 29 July 2023 / Accepted: 30 July 2023 / Published: 2 August 2023

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis II)

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Abstract

Iterative differential equations provide a new idea to study functional differential equations. The study of iterative equations can provide new methods for the study of differential equations with state-dependent delays. In this paper, we are concerned with proving the existence of infinitely many positive solutions to nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii’s cone fixed point theorem in a Banach space. It is worth pointing out that in this paper, we can use the symmetry of the iterative process and Green’s function to transform the considered differential equation into an equivalent integral equation, which plays a key role in the proof of the theorem in this paper.

Keywords:

positive solution; time scales; existence; iterative system

1. Introduction

In this paper, we consider the following nonlinear first-order iterative system of singular BVPs on time scales:

\{\begin{matrix} x_{i}^{Δ} (t) + p (t) x_{i} (σ (t)) = λ (t) f_{i} (x_{i + 1} (t)), t \in {(0, a]}_{T}, 1 \leq i \leq n, \\ x_{n + 1} (t) = x_{1} (t), \\ x_{i} (0) = x_{i} (σ (a)), \end{matrix}

(1)

where

i \in N, λ (t)

has a singularity in

{(0, \frac{σ (a)}{2}]}_{T}, f_{i} : R^{+} \to R^{+}

is continuous, and

p : {[0, a]}_{T} \to R^{+}

is right-dense continuous.

Iterative differential equations belong to functional differential equations. In particular, iterative equations can be used to describe functional differential equations with state-dependent delays, see [1,2,3]. Iterative differential equations provide a new idea to study functional differential equations. Zhao and Liu [4] considered a class of iterative differential equation

c_{0} x^{''} (t) + c_{1} x^{'} (t) + c_{2} x (t) = x (p (t) + b x (t)) + h (t)

(2)

and obtained the existence, uniqueness, and stability of periodic solutions for Equation (2). Eder [5] studied the iterative functional differential equation

x^{'} (t) = x^{[2]} (t)

and obtained asymptotic properties of solutions. Fe

\overset{˘}{c}

kan [6] further studied the following nonlinear iterative equation

x^{'} (t) = f (x^{[2]} (t))

with initial condition

x (0) = 0

. In [7], Wang considered the existence of a solution for the following iterative differential equation:

x^{'} (t) = f (x (x (t))),

where f is a smooth function that maps a closed interval I into itself where this interval contains a fixed point of the unknown function x. Cheng, Si, and Wang [8] studied an iterative functional differential equation

x^{'} (t) = f (x (x \dots (x (t))))

and obtained a local existence theorem by means of the Schauder fixed point theorem.

We focus on the study of iterative equations on time scales. The dynamic equations on time scales can unify continuous and discrete systems, so studying them can obtain results with a wider applicability, see [9,10,11]. Khuddush et al. [12] studied the following second-order iterative system of boundary value problems with singularities on time scales:

\{\begin{matrix} x_{l}^{Δ \nabla} (t) + λ (t) g_{l} (x_{l + 1} (t)) = 0, t \in {(0, σ (a)]}_{T}, 1 \leq l \leq n, \\ x_{n + 1} (t) = x_{1} (t), \\ x_{l}^{Δ} (0) = 0, x_{l} (σ (a)) = \sum_{k = 1}^{n - 2} c_{k} x_{l} (ξ_{k}) \end{matrix}

(3)

and some sufficient conditions for the existence of infinitely many positive solutions were obtained by applying Krasnoselskii’s cone fixed point theorem in a Banach space. When

p (t) = 0

in system (1), then Equation (3) is similar to system (1). Therefore, the results of this article generalize the results of article [12]. In [13], the authors studied positive solutions for the iterative system of dynamic equations

\{\begin{matrix} u_{i}^{Δ^{(n)}} (t) + λ_{i} a_{i} (t) f_{i} (u_{i + 1} (σ (t))) = 0, t \in {[a, b]}_{T}, 1 \leq i \leq n, \\ u_{n + 1} (t) = u_{1} (t), \\ u_{i}^{Δ^{(m)}} (a) = 0, u_{i} (σ^{n} (b)) = 0, 0 \leq m \leq n - 2 . \end{matrix}

For an iterative system of conformable fractional order dynamic boundary value problems on time scales, see [14]; for nondecreasing and convex

C^{2}

solutions of an iterative functional differential equation, see [15]. In this paper, we will study system (1), which is a singular dynamic system, and obtain the existence of infinitely many positive solutions by applying Krasnoselskii’s cone fixed point theorem in a Banach space. To the best of our knowledge, there are few results for the existence of a positive solution to system (1).

The main contributions are summarized in the following two aspects:

(1): We study a class more extensive iterative system with singularity, which generalizes the results of article [12].
(2): We innovatively use Krasnoselskii’s cone fixed point theorem on time scales to study the existence of solutions for iterative system (1).

The following sections are organized as follows. Section 2 gives some preliminaries. The existence of positive solutions of system (1) is obtained in Section 3. In Section 4, an example is given to show the feasibility of our results. Section 4 concludes this article with a summary of our results.

2. Preliminaries

The time scales theory, which has received much more attention, was firstly introduced by Hilger [16]. A time scale

T

is a nonempty closed subset of

R .

We give the following notations, and their means can be found in [17]: the forward jump

σ (t)

, backward jump operator

ρ (t)

, regressive rd-continuous function

R

, and the delta derivative

x^{Δ} (t)

x (t)

. We also give the following notations:

{[a, b]}_{T} = {t \in T, a \leq t \leq b}

, the intervals

{[a, b)}_{T}, {(a, b]}_{T}

and

{[a, b]}_{T}

are defined similarly. Let

p, q \in R .

The exponential function is defined by

e_{p} (t, s) = exp (\int_{s}^{t} ξ_{μ (τ)} (p (τ)) Δ τ),

where

ξ_{h} (z)

is the so-called cylinder transformation. If

p, q \in R

, a circle plus addition is defined by

(p \oplus q) : = p + q + μ p q .

For

p \in R

, we defined a circle minus p by

(⊖ p) : = \frac{- p}{1 + μ p} .

Lemma 1

([17]).

[i]: $e_{0} (t, s) \equiv 1$ and $e_{p} (t, t) \equiv 1;$
[ii]: $e_{p} (ρ (t), s) = (1 - μ (t) p (t)) e_{p} (t, s);$
[iii]: $e_{p} (t, s) = \frac{1}{e_{p} (s, t)} = e_{⊖ p} (s, t);$
[iv]: $e_{p} (t, s) e_{p} (s, r) = e_{p} (t, r);$
[v]: $e_{p} (t, s) e_{q} (t, s) = e_{p \oplus q} (t, s) .$

Lemma 2

(Krasnoselskii’s [18]). Let P be a cone in a Banach space E and

Ω_{1}, Ω_{2}

are open sets with

0 \in Ω_{1}, {\bar{Ω}}_{1} \subset Ω_{2}

. Let

L : E \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) \to E

be a completely continuous operator such that

(i): $| | L u | | \leq | | u | |, u \in E \cap \partial Ω_{1}$ , and $| | L u | | \geq | | u | |, u \in E \cap \partial Ω_{2}$ or
(ii): $| | L u | | \geq | | u | |, u \in E \cap \partial Ω_{1}$ , and $| | L u | | \leq | | u | |, u \in E \cap \partial Ω_{2}$ .

Then

L

has a fixed point in

E \cap ({\bar{Ω}}_{2} ∖ Ω_{1})

Let

C = {u | u : {[0, σ (a)]}_{T} \to R is continuous}

with the norm

| | u | | = {max}_{t \in {[0, σ (a)]}_{T}} | u (t) |

Lemma 3.

For any

y \in C

, the boundary value problem

\{\begin{matrix} x_{1}^{Δ} (t) + p (t) x_{1} (σ (t)) = y (t), t \in {[0, a]}_{T}, \\ x_{1} (0) = x_{1} (σ (a)), \end{matrix}

(4)

has a unique solution

x_{1} (t) = \int_{0}^{σ (a)} G (s, t) y (s) d s,

where

G (s, t) = \{\begin{matrix} (A + 1) e_{p} (s, t), if 0 \leq s \leq t \leq σ (a) \\ A e_{p} (s, t), if 0 \leq t < s \leq σ (a), \end{matrix}

A = \frac{1}{e_{p} (σ (a), 0) - 1}

Proof.

Similar to the proof of [19], BVP (4) has a unique solution

x_{1} (t) = \frac{1}{e_{p} (t, 0)} [\int_{0}^{t} e_{p} (s, 0) y (s) d s + A \int_{0}^{σ (a)} e_{p} (s, 0) y (s) d s] .

Thus,

\begin{matrix} x_{1} (t) & = (A + 1) \int_{0}^{t} e_{p} (s, t) y (s) d s + A \int_{t}^{σ (a)} e_{p} (s, t) y (s) d s \\ = \int_{0}^{σ (a)} G (s, t) y (s) d s . \end{matrix}

□

Lemma 4.

Let

η \in {(0, \frac{σ (a)}{2})}_{T}

. The kernel

G (s, t)

has the following properties:

(i): $0 \leq G (s, t) \leq G_{L}$ for all $t, s \in [0, σ (a)]$ , where $G_{L} = min {A + 1, A e_{p} (σ (a), 0)}$ ;
(ii): $G (s, t) \geq G_{l}$ for all $s \in [η, σ (a) - η], t \in [0, σ (a)]$ , where $G_{l} = max {(A + 1) e_{p} (η - σ (a), 0), A}$ .

Proof.

We first show that case (i) holds. If

0 \leq s \leq t \leq σ (a)

, we have

G (s, t) \leq A + 1 .

(5)

0 \leq t < s \leq σ (a)

, we have

G (s, t) \leq A e_{p} (σ (a), 0) .

(6)

From (5) and (6), we obtain that

G (s, t) \leq min {A + 1, A e_{p} (σ (a), 0)} = G_{L} .

Next, we show that case (ii) holds. Let

s \in [η, σ (a) - η]

. If

s \leq t

, we have

G (s, t) \geq (A + 1) e_{p} (η - σ (a), 0) .

(7)

s > t

, we have

G (s, t) \geq A .

(8)

From (7) and (8), we obtain that

G (s, t) \leq max {(A + 1) e_{p} (η - σ (a), 0), A} = G_{l} .

□

Obviously,

(x_{1} (t), x_{2} (t), \dots, x_{n} (t))

is a solution of the iterative BVP (1) if and only if

x_{i} (t) = \int_{0}^{σ (a)} G (s, t) λ (s) f_{i} (x_{i + 1} (s)) Δ s

and

x_{n + 1} (t) = x_{1} (t), t \in {(0, a]}_{T}, 1 \leq i \leq n .

Then, we have

\begin{matrix} x_{1} (t) & = \int_{0}^{σ (a)} G (s_{1}, t) λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} . \end{matrix}

For

η \in {(0, \frac{σ (a)}{2})}_{T}

, we define a cone

P_{η} \subset C

P_{η} = \{x \in C : x (t) is nonnegative and min_{t \in {[η, σ (a) - η]}_{T}} x (t) \geq \frac{G_{l}}{G_{L}} | | x | |\},

where

G_{l}

and

G_{L}

are defined by Lemma 4. For any

x_{1} \in P_{η}

, we define an operator

L : P_{η} \to C

\begin{matrix} (L x_{1}) (t) & = \int_{0}^{σ (a)} G (s_{1}, t) λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} . \end{matrix}

Lemma 5.

For each

η \in {(0, \frac{σ (a)}{2})}_{T}

L : P_{η} \to P_{η}

is completely continuous.

Proof.

By Lemma 4, we obtain that

G (s, t) \geq 0

and

(L x_{1}) (t)

for all

s, t \in {[0, σ (a)]}_{T}

. Again by Lemma 4, for

x_{1} \in P_{η}

, we have

\begin{matrix} | | L x_{1} | | & = max_{t \in {[0, σ (a)]}_{T}} \int_{0}^{σ (a)} G (s_{1}, t) λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} \\ \leq G_{L} \int_{0}^{σ (a)} λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} \end{matrix}

and

\begin{matrix} min_{t \in {[η, σ (a) - η]}_{T}} {(L x_{1}) (t)} & \geq G_{l} \int_{0}^{σ (a)} λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} \\ \geq \frac{G_{l}}{G_{L}} | | L x_{1} | | . \end{matrix}

Hence,

L x_{1} \in P_{η}

and

L (P_{η}) \subset P_{η}

. It follows by the Arzela–Ascoli theorem that

L

is completely continuous. □

3. Main Results

Theorem 1.

Suppose that the following assumption holds:

(H₁): there exists a sequence ${t_{j}}_{j = 1}^{\infty}$ such that $0 < t_{j + 1} < t_{j} < \frac{σ (a)}{2}$ and

$lim_{j \to \infty} t_{j} = t^{*} < \frac{σ (a)}{2}, lim_{t \to t_{j}} λ (t) = + \infty .$

Let ${η_{j}}_{j = 1}^{\infty}$ with $0 < t_{j + 1} < η_{j} < t_{j}$ . Let ${Γ_{j}}_{j = 1}^{\infty}$ and ${Λ_{j}}_{j = 1}^{\infty}$ be such that

$Γ_{j + 1} < \frac{G_{l}}{G_{L}} Λ_{j} < Λ_{j} < θ Λ_{j} < Γ_{j}, j \in N,$

where $θ \geq \frac{1}{G_{l} δ (σ (a) - 2 η_{1})}$ . Assume that $f_{i}$ statisfies
(H₂): $f_{i} (u (t)) \leq Ξ Γ_{j}$ for $t \in {(0, σ (a)]}_{T}, 0 \leq u \leq Γ_{j},$ where $Ξ \leq \frac{1}{G_{L} \int_{0}^{σ (a)} λ (s) Δ s} .$
(H₃): $f_{i} (u (t)) \geq θ Λ_{j}$ for $t \in {[η_{j}, σ (a) - η_{j}]}_{T}, \frac{G_{l}}{G_{L}} Λ_{j} \leq u \leq Λ_{j} .$
Then the iterative BVP (1) has infinitely many solutions ${(x_{1}^{[j]}, x_{2}^{[j]}, \dots, x_{n}^{[j]})}_{j = 1}^{\infty}$ such that $x_{i}^{[j]} \geq 0$ on ${(0, σ (a)]}_{T}$ for $i = 1, 2, \dots, n$ .

Proof.

It follows that by (

H_{1}

), there exists

δ > 0

such that

λ (t) > δ

. Let

Ω_{1, j} = {x \in C : | | x | | < Γ_{j}}, Ω_{2, j} = {x \in C : | | x | | < Λ_{j}} .

By assumption (

H_{1}

), we have

0 < t^{*} < t_{j + 1} < η_{j} < t_{j} < \frac{σ (a)}{2} for j \in N .

For

j \in N

, we define the cone

P_{η_{j}}

P_{η_{j}} = \{x \in C : x (t) is nonnegative and min_{t \in {[η_{j}, σ (a) - η_{j}]}_{T}} x (t) \geq \frac{G_{l}}{G_{L}} | | x | |\} .

Let

x_{1} \in P_{η_{j}} \cap \partial Ω_{1, j}

. Then,

x_{1} (s) \leq Γ_{j}

for all

s \in {(0, σ (a)]}_{T}

. For

s_{n - 1} \in {(0, σ (a)]}_{T}

, by (

H_{2}

), we have

\begin{matrix} \int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n} \\ \leq G_{L} Ξ Γ_{j} \int_{0}^{σ (a)} λ (s_{n}) Δ s_{n} \\ \leq Γ_{j} . \end{matrix}

Similar to the above proof, for

s_{n - 2} \in {(0, σ (a)]}_{T}

, by (

H_{2}

), we also have

\begin{matrix} \int_{0}^{σ (a)} G (s_{n - 1}, s_{n - 2}) λ (s_{n - 1}) f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] Δ s_{n - 1} \\ \leq \int_{0}^{σ (a)} G (s_{n - 1}, s_{n - 2}) λ (s_{n - 1}) f_{n - 1} (Γ_{j}) Δ s_{n - 1} \\ \leq G_{L} Ξ Γ_{j} \int_{0}^{σ (a)} λ (s_{n - 1}) Δ s_{n - 1} \\ \leq Γ_{j} . \end{matrix}

Thus, we obtain

\begin{matrix} \int_{0}^{σ (a)} G (s_{1}, t) λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} \\ \leq Γ_{j} \end{matrix}

and

(L x_{1}) (t) \leq Γ_{j}

. By

Γ_{j} = | | x_{1} | |

for

x_{1} \in P_{η_{j}} \cap \partial Ω_{1, j}

, we have

| | L x_{1} | | \leq | | x_{1} | | .

(9)

Next, for

t \in {[η_{j}, σ (a) - η_{j}]}_{T}

, we have

Λ_{j} = | | x_{1} | | \geq x_{1} (t) \geq min_{t \in {[η_{j}, σ (a) - η_{j}]}_{T}} x_{1} (t) \geq \frac{G_{l}}{G_{L}} | | x_{1} | | .

For

s_{n - 1} \in {[η_{j}, σ (a) - η_{j}]}_{T}

, by (

H_{3}

), we have

\begin{matrix} \int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n} \\ \geq \int_{η_{j}}^{σ (a) - η_{j}} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n} \\ \geq G_{l} θ Λ_{j} δ (σ (a) - 2 η_{j}) \\ \geq G_{l} θ Λ_{j} δ (σ (a) - 2 η_{1}) \\ \geq Λ_{j} . \end{matrix}

Continuing with the bootstrapping argument, we obtain

\begin{matrix} \int_{0}^{σ (a)} G (s_{1}, t) λ (s_{1}) f_{1} [\int_{0}^{σ (a)} G (s_{2}, s_{1}) λ (s_{2}) f_{2} [\int_{0}^{σ (a)} G (s_{3}, s_{2}) \dots \\ \times f_{n - 1} [\int_{0}^{σ (a)} G (s_{n}, s_{n - 1}) λ (s_{n}) f_{n} (x_{1} (s_{n})) Δ s_{n}] \dots Δ s_{3}] Δ s_{2}] Δ s_{1} \\ \geq Λ_{j} \end{matrix}

and

(L x_{1}) (t) \geq Λ_{j}

. By

Λ_{j} = | | x_{1} | |

for

x_{1} \in P_{η_{j}} \cap \partial Ω_{2, j}

, we have

| | L x_{1} | | \geq | | | x_{1} | | .

(10)

It is easy to see that

0 \in Ω_{2, j} \subset {\bar{Ω}}_{2, j} \subset Ω_{1, j}

. In view of (9) and (10), the operator

L

has a fixed point

x_{1}^{[j]} \in P_{η_{j}} \cap ({\bar{Ω}}_{1, j} ∖) Ω_{2, j}

such that

x_{1}^{[j]} \geq 0

{(0, a]}_{T}

. Let

x_{n + 1} = x_{1}

. We obtain infinitely many positive solutions

{(x_{1}^{[j]}, x_{2}^{[j]}, \dots, x_{n}^{[j]})}_{j = 1}^{\infty}

of BVP (1) given iteratively by

x_{i} (t) = \int_{0}^{σ (a)} G (s, t) λ (s) f_{i} (x_{i + 1} (s)) Δ s, i = 1, 2, \dots, n .

□

Remark 1.

In system (1), the initial conditions are different from the ones in system (2). Obviously, the initial conditions in system (2) are more complicated. In future work, we will study the following iterative system:

\{\begin{matrix} x_{l}^{Δ \nabla} (t) + p (t) x_{l} (σ (t)) + λ (t) g_{l} (x_{l + 1} (t)) = 0, t \in {(0, σ (a)]}_{T}, 1 \leq l \leq n, \\ x_{n + 1} (t) = x_{1} (t), \\ x_{l}^{Δ} (0) = 0, x_{l} (σ (a)) = \sum_{k = 1}^{n - 2} c_{k} x_{l} (ξ_{k}), \end{matrix}

where

l \in N, c_{k} \in R^{+} with \sum_{k = 1}^{n - 2} c_{k} < 1, 0 < ξ_{k} < \frac{σ (a)}{2}, λ (t) = Π_{i = 1}^{m} λ_{i} (t)

and each

λ_{i} (t) \in L_{\nabla}^{p_{i}} ({(0, σ (a)]}_{T}) (p_{1} \geq 1)

has a singularity in

{(0, \frac{σ (a)}{2}]}_{T}, g_{l} : R^{+} \to R^{+}

is continuous, and

p : {[0, a]}_{T} \to R^{+}

is right-dense continuous.

4. Example

Let

T = R,

consider the following BVP of model (1):

\{\begin{matrix} x_{i}^{Δ} (t) + p (t) x_{i} (t) = λ (t) f_{i} (x_{i + 1} (t)), t \in (0, 1], 1 \leq i \leq 4, \\ x_{5} (t) = x_{1} (t), \\ x_{i} (0) = x_{i} (1), \end{matrix}

(11)

Choose

p (t) = 2, λ (t) = \frac{1}{{| t - 0.25 |}^{\frac{1}{2}}}

then

δ = {(\frac{4}{3})}^{\frac{1}{2}}

. For

j \in N, i = 1, 2, 3, 4

, let

f_{i} (x) = \{\begin{matrix} 0.02 \times 10^{- 4 j}, x \in [10^{- 4 j}, \infty), \\ \frac{50 \times 10^{- (4 j + 3)} - 0.02 \times 10^{- 4 j}}{10^{- (4 j + 3)} - 10^{- 4 j}} (x - 10^{- 4 j}) + 0.02 \times 10^{- 6 j}, x \in [10^{- (4 j + 3)}, 10^{- 4 j}], \\ 50 \times 10^{- (4 j + 3)}, x \in [0.01 \times 10^{- (4 j + 3)}, 10^{- (4 j + 3)}], \\ \frac{50 \times 10^{- (4 j + 3)} - 0.02 \times 10^{- 6 j}}{0.02 \times 10^{- (4 j + 3)} - 10^{- 4 j}} (x - 10^{- (4 j + 4)}) + 0.02 \times 10^{- 6 j}, x \in [10^{- (4 j + 4)}, 0.02 \times 10^{- (4 j + 3)}], \\ 0, x = 0 . \end{matrix}

Let

t_{j} = \frac{33}{70} - \sum_{k = 1}^{j} \frac{1}{4 {(k + 1)}^{4}}, η_{j} = 0.5 (t_{j} + t_{j + 1}), j \in N,

then

η_{1} = \frac{493}{1120} - \frac{1}{628} < \frac{493}{1120}

and

t_{j + 1} < η_{j} < t_{j} .

Obviously,

t_{1} = \frac{493}{1120} < 0.5, t_{j} - t_{j + 1} = \frac{1}{4 {(j + 2)}^{4}} .

Using

\sum_{j = 1}^{\infty} \frac{1}{j^{4}} = \frac{π^{4}}{90}

, we have

t^{*} = lim_{j \to \infty} t_{j} = \frac{33}{70} - \sum_{k = 1}^{\infty} \frac{1}{4 {(k + 1)}^{4}} = \frac{33}{70} + \frac{1}{64} - \frac{π^{4}}{360} \approx 0.45 .

After a simple calculation, we obtain

A = 0.18, G_{L} = min {A + 1, A e_{p} (σ (a), 0)} \approx 0.28

G_{l} = max {(A + 1) e_{p} (η - σ (a), 0), A} \approx 0.18, θ \geq \frac{1}{G_{l} δ (σ (a) - 2 η_{1})} \approx 40,

Ξ \leq \frac{1}{G_{L} \int_{0}^{σ (a)} λ (s) Δ s} \approx 1.307 .

Choose

Ξ = 1.2 .

Furthermore, if we take

Γ_{j} = 10^{- 4 j} and Λ_{j} = 10^{- (4 j + 3)},

then

\begin{matrix} Γ_{j + 1} & = 10^{- (4 j + 4)} < \frac{G_{l}}{G_{L}} Λ_{j} = 0.64 \times 10^{- (4 j + 3)} \\ < Λ_{j} = 10^{- (4 j + 3)} < θ Λ_{j} = 40 \times 10^{- (4 j + 3)} < Γ_{j} = 10^{- 4 j} . \end{matrix}

Obviously,

f_{i}, (i = 1, 2, 3, 4)

satisfies the following growth conditions:

(i): $f_{i} (x) \leq Ξ Γ_{j} = 1.2 \times 10^{- 4 j}$ for $x \in [0, 10^{- 4 j}]$ ,
(ii): $f_{i} (x) \geq θ Λ_{j} = 40 \times 10^{- (4 j + 3)}$ for $x \in [0.64 \times 10^{- (4 j + 3)}, 10^{- (4 j + 3)}]$ .

Then, all the conditions of Theorem 1 are satisfied. Therefore, by Theorem 1, the BVP (4.1) has infinitely many solutions

{(x_{1}^{[j]}, x_{2}^{[j]}, \dots, x_{n}^{[j]})}_{j = 1}^{\infty}

such that

x_{i}^{[j]} \geq 0

{(0, σ (a)]}_{T}

for

i = 1, 2, 3, 4

5. Discussion and Conclusions

In the past few years, iterative differential equations have received extensive attention. The concept of iterative equations was firstly introduced by Babbage [20] in 1815. Babbage found a function equalling its

n -

th iterate. The real development of iterative equations began in the twentieth century, for more details, see [21] and related references. Iterative equations are a particular type of delay differential equations that depend on both the time t and the state variable x, which are defined implicitly by the iterates. The theory of time scale calculus unifies the calculus of the theory of continuous equations with that of discrete equations, which gives a new way to study hybrid discrete–continuous dynamical systems and has more applications in any field that requires simultaneous modeling of discrete and continuous data. This paper considers the existence of infinitely many positive solutions of nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii’s cone fixed point theorem in a Banach space. It should be pointed out that the properties of Green’s function are the basis of this study. We believe that the properties of Green’s function obtained in this paper can be used to study other types of equations. In future work, we will study the iterative system in Remark 1. Furthermore, we will further investigate iterative systems with impulses and fractional order iterative systems.

Author Contributions

Writing—review and editing, B.D., X.W. and X.C.; Methodology, F.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11971197).

Data Availability Statement

Not applicable.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their helpful comments and valuable suggestions regarding this article.

Conflicts of Interest

The authors declare no conflict of interest.

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Zheng, F.; Wang, X.; Cheng, X.; Du, B. Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales. Symmetry 2023, 15, 1524. https://doi.org/10.3390/sym15081524

AMA Style

Zheng F, Wang X, Cheng X, Du B. Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales. Symmetry. 2023; 15(8):1524. https://doi.org/10.3390/sym15081524

Chicago/Turabian Style

Zheng, Famei, Xiaojing Wang, Xiwang Cheng, and Bo Du. 2023. "Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales" Symmetry 15, no. 8: 1524. https://doi.org/10.3390/sym15081524

APA Style

Zheng, F., Wang, X., Cheng, X., & Du, B. (2023). Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales. Symmetry, 15(8), 1524. https://doi.org/10.3390/sym15081524

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Infinitely Many Positive Solutions to Nonlinear First-Order Iterative Systems of Singular BVPs on Time Scales

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

5. Discussion and Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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