Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 30, 1–15; http://www.math.u-szeged.hu/ejqtde/

SOLUTIONS FOR SINGULAR VOLTERRA INTEGRAL EQUATIONS

Patricia J. Y. Wong

School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore email: ejywong@ntu.edu.sg

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

We consider the system of Volterra integral equations

\begin{matrix}u_{i}(t) = ∫_{0}^{t} g_{i}(t, s)[P_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s)) \\ + Q_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s))]ds, t ∈ [0, T], 1 ≤ i ≤ n \end{matrix}

where T > 0 is fixed and the nonlinearities P_{i}(t, u_{1}, u_{2}, ⋯, u_{n}) can be singular at t = 0 and u_{j} = 0 where j ∈ {1, 2, ⋯, n}. Criteria are offered for the existence of fixed-sign solutions (u_{1}^{∗}, u_{2}^{∗}, ⋯, u_{n}^{∗}) to the system of Volterra integral equations, i.e., θ_{i}u_{i}^{∗}(t) ≥ 0 for t ∈ [0, 1] and 1 ≤ i ≤ n, where θ_{i} ∈ {1, −1} is fixed. We also include an example to illustrate the usefulness of the results obtained.

Key words and phrases: Fixed-sign solutions, singularities, Volterra integral equations. AMS (MOS) Subject Classifications: 45B05

1 Introduction

In this paper we shall consider the system of Volterra integral equations

\begin{matrix}u_{i}(t) = ∫_{0}^{t} g_{i}(t, s)[P_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s)) + Q_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s))]ds, \\ t ∈ [0, T], 1 ≤ i ≤ n \end{matrix}

(1.1)

where T > 0 is fixed. The nonlinearities P_{i}(t, u_{1}, u_{2}, ⋯, u_{n}) can be singular at t = 0 and u_{j} = 0 where j ∈ {1, 2, ⋯, n}.

Throughout, let u = (u_{1}, u_{2}, ⋯, u_{n}). We are interested in establishing the existence of solutions u of the system (1.1) in (C[0, T])^{n} = C[0, T] × C[0, T] × ⋯ × C[0, T] (n times). Moreover, we are concerned with fixed-sign solutions u, by which we mean θ_{i}u_{i}(t) ≥ 0 for all t ∈ [0, T] and 1 ≤ i ≤ n, where θ_{i} ∈ {1, −1} is fixed. Note that positive solution is a special case of fixed-sign solution when θ_{i} = 1 for 1 ≤ i ≤ n.

The system (1.1) when P_{i} = 0, 1 ≤ i ≤ n reduces to

u_{i}(t) = ∫_{0}^{t} g_{i}(t, s)Q_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s))ds, t ∈ [0, T], 1 ≤ i ≤ n.  (1.2)

This equation when n = 1 has received a lot of attention in the literature [12, 13, 14, 16, 17, 18, 19], since it arises in real-world problems. For example, astrophysical problems (e.g., the study of the density of stars) give rise to the Emden differential equation

\{ \begin{matrix}y′′ − t^{p}y^{q} = 0, t ∈ [0, T] \\ y(0) = y′(0) = 0, p ≥ 0, 0 < q < 1 \end{matrix}

which reduces to (1.2)|_{n=1} when g_{1}(t, s) = (t − s)s^{p} and Q_{1}(t, y) = y^{q}. Other examples occur in nonlinear diffusion and percolation problems (see [13, 14] and the references cited therein), and here we get (1.2) where g_{i} is a convolution kernel, i.e.,

u_{i}(t) = ∫_{0}^{t} g_{i}(t − s)Q_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s))ds, t ∈ [0, T], 1 ≤ i ≤ n.

In particular, Bushell and Okrasiński [13] investigated a special case of the above system given by

y(t) = ∫_{0}^{t}(t − s)^{γ−1}Q(y(s))ds, t ∈ [0, T]

where γ > 1.

In the literature, the conditions placed on the kernels g_{i} are not natural. A new approach is thus employed in this paper to present new results for (1.1). In particular, new “lower type inequalities” on the solutions are presented. Our results extend, improve and complement the existing theory in the literature [1, 2, 3, 4, 11, 15, 20, 21, 22]. We have generalized the problems to (i) systems, (ii) general form of nonlinearities P_{i}, 1 ≤ i ≤ n that can be singular in both independent and dependent variables, (iii) existence of fixed-sign solutions, which include positive solutions as special case. Other related work on systems of integral equations can be found in [5, 6, 7, 8, 9, 10, 23]. Note that the technique employed in singular integral equations [10] is entirely different from the present work.

2 Main Results

Let the real Banach space B = (C[0, T])^{n} be equipped with the norm

∥u∥ = max_{1≤i≤n} sup_{t∈[0,T]} |u_{i}(t)|.

Our main tool is the following theorem.

Theorem 2.1 Consider the system

u_{i}(t) = c_{i}(t) + ∫_{0}^{t} g_{i}(t, s)f_{i}(s, u(s))ds, t ∈ [0, T], 1 ≤ i ≤ n  (2.1)

where T > 0 is fixed. Let 1 ≤ p ≤ ∞ be an integer and q be such that \frac{1}{p} + \frac{1}{q} = 1. Assume the following conditions hold for each 1 ≤ i ≤ n:

(C_{1}) c_{i} ∈ C[0, T];

(C_{2}) f_{i} : [0, T] × ℝ^{n} → ℝ is a L^{q}-Carathéodory function, i.e., (i) the map u ↦ f_{i}(t, u) is continuous for almost all t ∈ [0, T], (ii) the map t ↦ f_{i}(t, u) is measurable for all u ∈ ℝ^{n}, (iii) for any r > 0, there exists μ_{r,i} ∈ L^{q}[0, T] such that ∥u∥ ≤ r (∥u∥ denotes the norm in ℝ^{n}) implies |f_{i}(t, u)| ≤ μ_{r,i}(t) for almost all t ∈ [0, T];

(C_{3}) g_{i}(t, s) : Δ → ℝ, where Δ = {(t, s) ∈ ℝ^{2} : 0 ≤ s ≤ t ≤ T }, g_{i}^{t}(s) = g_{i}(t, s) ∈ L^{p}[0, t] for each t ∈ [0, T], and

sup_{t∈[0,T]} ∫_{0}^{t} |g_{i}^{t}(s)|^{p} ds < ∞, 1 ≤ p < ∞
sup_{t∈[0,T]} ess sup_{s∈[0,t]} |g_{i}^{t}(s)| < ∞, p = ∞;

and

(C_{4}) for any t, t′ ∈ [0, T] with t^{∗} = min{t, t′}, we have

∫_{0}^{t^{∗}} |g_{i}^{t}(s) − g_{i}^{t′}(s)|^{p} ds → 0 as t → t′, 1 ≤ p < ∞

ess sup_{s∈[0,t^{∗}]} |g_{i}^{t}(s) − g_{i}^{t′}(s)|^{p} → 0 as t → t′, p = ∞.

In addition, suppose there is a constant M > 0, independent of λ, with ∥u∥ ≠ M for any solution u ∈ (C[0, T])^{n} to

u_{i}(t) = c_{i}(t) + λ ∫_{0}^{t} g_{i}(t, s)f_{i}(s, u(s))ds, t ∈ [0, T], 1 ≤ i ≤ n  (2.2)_{λ}

for each λ ∈ (0, 1). Then, (2.1) has at least one solution in (C[0, T])^{n}.

Proof. For each 1 ≤ i ≤ n, define

g_{i}^{∗}(t, s) = \{ \begin{matrix}g_{i}(t, s), & 0 ≤ s ≤ t ≤ T \\ 0, & 0 ≤ t ≤ s ≤ T. \end{matrix}

Then, (2.1) is equivalent to

u_{i}(t) = c_{i}(t) + ∫_{0}^{T} g_{i}^{∗}(t, s)f_{i}(s, u(s))ds, t ∈ [0, T], 1 ≤ i ≤ n.  (2.3)

Now, the system (2.3) (or equivalently (2.1)) has at least one solution in (C[0, T])^{n} by Theorem 2.1 in [23], which is stated as follows: Consider the system below

u_{i}(t) = c_{i}(t) + ∫_{0}^{T} g_{i}(t, s)f_{i}(s, u(s))ds, t ∈ [0, T], 1 ≤ i ≤ n  (∗)

where the following conditions hold for each 1 ≤ i ≤ n and for some integers p, q such that 1 ≤ p ≤ ∞ and \frac{1}{p} + \frac{1}{q} = 1: (C_{1}), (C_{2}), g_{i}(t, s) ∈ [0, T]^{2} → ℝ, and g_{i}^{t}(s) = g_{i}(t, s) ∈ L^{p}[0, T] for each t ∈ [0, T]. Further, suppose there is a constant M > 0, independent of λ, with ∥u∥ ≠ M for any solution u ∈ (C[0, T])^{n} to

u_{i}(t) = c_{i}(t) + λ ∫_{0}^{T} g_{i}(t, s)f_{i}(s, u(s))ds, t ∈ [0, T], 1 ≤ i ≤ n

for each λ ∈ (0, 1). Then, (∗) has at least one solution in (C[0, T])^{n}. □

Remark 2.1 If (C_{4}) is changed to

(C_{4})′ for any t, t′ ∈ [0, T] with t^{∗} = min{t, t′} and t^{∗∗} = max{t, t′}, we have for 1 ≤ p < ∞,

∫_{0}^{t^{∗}} |g_{i}(t, s) − g_{i}(t′, s)|^{p} ds + ∫_{t^{∗}}^{t^{∗∗}} |g_{i}(t^{∗∗}, s)|^{p} ds → 0

as t → t′, and for p = ∞,

ess sup_{s∈[0,t^{∗}]} |g_{i}(t, s) − g_{i}(t′, s)| + ess sup_{s∈[t^{∗},t^{∗∗}]} |g_{i}(t^{∗∗}, s)| → 0

as t → t′,

then automatically we have the inequalities in (C_{3}).

We shall now apply Theorem 2.1 to obtain an existence result for (1.1). Let θ_{i} ∈ {−1, 1}, 1 ≤ i ≤ n be fixed. For each 1 ≤ j ≤ n, we define

[0, ∞)_{j} = \{ \begin{matrix}[0, ∞), & θ_{j} = 1 \\ (−∞, 0], & θ_{j} = −1 \end{matrix}

and (0, ∞)_{j} is similarly defined.

Theorem 2.2 Let θ_{i} ∈ {−1, 1}, 1 ≤ i ≤ n be fixed and let the following conditions be satisfied for each 1 ≤ i ≤ n:

(I_{1}) P_{i} : (0, T] × (ℝ\{0})^{n} → ℝ, θ_{i}P_{i}(t, u) > 0 and is continuous for (t, u) ∈ (0, T] × ∏_{j=1}^{n}(0, ∞)_{j}, Q_{i} : [0, T] × ℝ^{n} → ℝ, θ_{i}Q_{i}(t, u) ≥ 0 and is continuous for (t, u) ∈

[0, T] × ∏_{j=1}^{n}[0, ∞)_{j};

(I_{2}) θ_{i}P_{i} is ‘nonincreasing’ in u, i.e., if θ_{j}u_{j} ≥ θ_{j}v_{j} for some j ∈ {1, 2, ⋯, n}, then

θ_{i}P_{i}(t, u_{1}, ⋯, u_{j}, ⋯, u_{n}) ≤ θ_{i}P_{i}(t, u_{1}, ⋯, v_{j}, ⋯, u_{n}), t ∈ (0, T];

(I_{3}) there exist nonnegative r_{i} and γ_{i} such that r_{i} ∈ C(0, T], γ_{i} ∈ C(0, ∞), γ_{i} > 0 is nonincreasing, and

θ_{i}P_{i}(t, u) ≥ r_{i}(t)γ_{i}(|u_{i}|), (t, u) ∈ (0, T] × ∏_{j=1}^{n}(0, ∞)_{j};

(I_{4}) there exist nonnegative d_{i} and h_{ij}, 1 ≤ j ≤ n such that d_{i} ∈ C[0, T], h_{ij} ∈ C(0, ∞), h_{ij} is nondecreasing, and

\frac{Q_{i}(t, u)}{P_{i}(t, u)} ≤ d_{i}(t)h_{i1}(|u_{1}|)h_{i2}(|u_{2}|) ⋯ h_{in}(|u_{n}|), (t, u) ∈ (0, T] × ∏_{j=1}^{n}(0, ∞)_{j};

(I_{5}) g_{i}(t, s) : Δ → ℝ, g_{i}^{t}(s) = g_{i}(t, s) ∈ L^{1}[0, t] for each t ∈ [0, T], and

sup_{t∈[0,T]} ∫_{0}^{t} |g_{i}^{t}(s)|ds < ∞;

(I_{6}) for any t, t′ ∈ [0, T] with t^{∗} = min{t, t′}, we have ∫_{0}^{t^{∗}} |g_{i}^{t}(s) − g_{i}^{t′}(s)|ds → 0 as t → t′;

(I_{7}) for each t ∈ [0, T], g_{i}(t, s) ≥ 0 for a.e. s ∈ [0, t];

(I_{8}) for any t_{1}, t_{2} ∈ (0, T] with t_{1} < t_{2}, we have

g_{i}(t_{1}, s) ≤ g_{i}(t_{2}, s), a.e. s ∈ [0, t_{1}];

(I_{9}) for any k_{j} ∈ {1, 2, ...}, 1 ≤ j ≤ n, we have

sup_{t∈[0,T]} ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i} (s, \frac{θ_{1}}{k_{1}} , ⋯, \frac{θ_{n}}{k_{n}} ) ds < ∞,

sup_{s∈[0,T]} ∫_{0}^{s} g_{i}(s, x)r_{i}(x)dx < ∞,

∫_{0}^{s} g_{i}(s, x)r_{i}(x)dx > 0, a.e. s ∈ [0, T],

sup_{t∈[0,T]} ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i}(s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s))ds < ∞

where

β_{i}(s) = G_{i}^{−1} (∫_{0}^{s} g_{i}(s, x)r_{i}(x)dx)

for s ∈ [0, T] and

G_{i}(z) = \frac{z}{γ_{i}(z)}

for z > 0, with G_{i}(0) = 0 = G_{i}^{−1}(0);

(I_{10}) there exists ρ_{i} ∈ C[0, T] such that for t, x ∈ [0, T] with t < x, we have

∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)]θ_{i}P_{i}(s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s))ds ≤ |ρ_{i}(x) − ρ_{i}(t)|;

and

(I_{11}) if z > 0 satisfies

z ≤ K + L {1 + max_{1≤i≤n} [ sup_{t∈[0,T]} d_{i}(t)] [∏_{j=1}^{n} h_{ij}(z)]}

for some constants K, L ≥ 0, then there exists a constant M (which may depend on K and L) such that z ≤ M.

Then, (1.1) has a fixed-sign solution u ∈ (C[0, T])^{n} with

θ_{i}u_{i}(t) ≥ β_{i}(t)

for t ∈ [0, T] and 1 ≤ i ≤ n (β_{i} is defined in (I_{9})).

Proof. Let N = {1, 2, ⋯} and k = (k_{1}, k_{2}, ⋯, k_{n}) ∈ N^{n}. First, we shall show that the nonsingular system

u_{i}(t) = \frac{θ_{i}}{k_{i}} + ∫_{0}^{t} g_{i}(t, s) [P_{i}^{∗}(s, u(s)) + Q_{i}^{∗}(s, u(s))] ds, t ∈ [0, T], 1 ≤ i ≤ n (2.4)^{k}

has a solution for each k ∈ N^{n}, where

P_{i}^{∗}(t, u_{1}, ⋯, u_{n}) = \{ \begin{matrix}P_{i}(t, v_{1}, ⋯, v_{n}), & t ∈ (0, T] \\ 0, & t = 0 \end{matrix}

with

v_{j} = \{ \begin{matrix}u_{j}, & θ_{j}u_{j} ≥ \frac{1}{k_{j}} \\ \frac{θ_{j}}{k_{j}} , & θ_{j}u_{j} ≤ \frac{1}{k_{j}} \end{matrix}

and

Q_{i}^{∗}(t, u_{1}, ⋯, u_{n}) = Q_{i}(t, w_{1}, ⋯, w_{n}), t ∈ [0, T]

with

w_{j} = \{ \begin{matrix}u_{j}, & θ_{j}u_{j} ≥ 0 \\ 0, & θ_{j}u_{j} ≤ 0. \end{matrix}

Let k ∈ N^{n} be fixed. We shall use Theorem 2.1 to show that (2.4)^{k} has a solution. Note that conditions (C_{1})–(C_{4}) are satisfied with p = 1 and q = ∞. We need to consider the family of problems

u_{i}(t) = \frac{θ_{i}}{k_{i}} + λ ∫_{0}^{t} g_{i}(t, s) [P_{i}^{∗}(s, u(s)) + Q_{i}^{∗}(s, u(s))] ds, t ∈ [0, T], 1 ≤ i ≤ n (2.5)_{λ}^{k}

where λ ∈ (0, 1). Let u ∈ (C[0, T])^{n} be any solution of (2.5)_{λ}^{k}. Clearly, for each 1 ≤ i ≤ n,

θ_{i}u_{i}(t) ≥ \frac{1}{k_{i}} > 0, t ∈ [0, T]

and so P_{i}^{∗}(t, u(t)) = P_{i}(t, u(t)) for t ∈ (0, T] and Q_{i}^{∗}(t, u(t)) = Q_{i}(t, u(t)) for t ∈ [0, T]. Applying (I_{2}) and (I_{4}), we find for t ∈ [0, T] and 1 ≤ i ≤ n,

|u_{i}(t)| = θ_{i}u_{i}(t)

= \frac{1}{k_{i}} + λ ∫_{0}^{t} g_{i}(t, s) [θ_{i}P_{i}^{∗}(s, u(s)) + θ_{i}Q_{i}^{∗}(s, u(s))] ds

= \frac{1}{k_{i}} + λ ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i}(s, u(s)) [1 + \frac{Q_{i}(s, u(s))}{P_{i}(s, u(s))}] ds

≤ 1 + ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i} (s, \frac{θ_{1}}{k_{1}} , ⋯, \frac{θ_{n}}{k_{n}} ) [1 + d_{i}(s) ∏_{j=1}^{n} h_{ij}(|u_{j}(s)|)] ds

≤ 1 + C_{i}(1 + D_{i})

where

C_{i} = sup_{t∈[0,T]} ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i} (s, \frac{θ_{1}}{k_{1}} , ⋯, \frac{θ_{n}}{k_{n}} ) ds

and

D_{i} = [ sup_{s∈[0,T]} d_{i}(s)] ∏_{j=1}^{n} h_{ij}(∥u∥).

Thus,

∥u∥ ≤ 1 + ( max_{1≤i≤n} C_{i}) (1 + max_{1≤i≤n} D_{i})

and so by (I_{11}) there exists a constant M_{k} with ∥u∥ ≤ M_{k}. Theorem 2.1 now guarantees that (2.4)^{k} has a solution u^{k} ∈ (C[0, T])^{n} with θ_{i}u_{i}^{k}(t) ≥ \frac{1}{k_{i}} for t ∈ [0, T] and 1 ≤ i ≤ n.

Consequently, P_{i}^{∗}(t, u^{k}(t)) = P_{i}(t, u^{k}(t)), Q_{i}^{∗}(t, u^{k}(t)) = Q_{i}(t, u^{k}(t)) and u^{k} is a solution of the system

u_{i}(t) = \frac{θ_{i}}{k_{i}} + ∫_{0}^{t} g_{i}(t, s)[P_{i}(s, u(s)) + Q_{i}(s, u(s))]ds, t ∈ [0, T], 1 ≤ i ≤ n. (2.6)

Moreover, θ_{i}u_{i}^{k} is nondecreasing on (0, T), since for t, x ∈ (0, T) with t < x,

θ_{i}u_{i}^{k}(x) − θ_{i}u_{i}^{k}(t)
= ∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)] [θ_{i}P_{i}(s, u^{k}(s)) + θ_{i}Q_{i}(s, u^{k}(s))] ds
+ ∫_{t}^{x} g_{i}(x, s) [θ_{i}P_{i}(s, u^{k}(s)) + θ_{i}Q_{i}(s, u^{k}(s))] ds
≥ 0

where we have made use of (I_{1}), (I_{7}) and (I_{8}).

Next, we shall obtain a solution to (1.1) by means of the Arzéla-Ascoli theorem, as a limit of solutions of (2.4)^{k} (as k_{i} → ∞, 1 ≤ i ≤ n). For this we shall show that

{u^{k}}_{k∈N^{n}} is a bounded and equicontinuous family on [0, T].  (2.7)

To proceed we need to obtain a lower bound for θ_{i}u_{i}^{k}(t), t ∈ [0, T], 1 ≤ i ≤ n. Using (I_{3}) and the fact that θ_{i}u_{i}^{k} = |u_{i}^{k}| is nondecreasing on (0, T), we get

|u_{i}^{k}(t)| = θ_{i}u_{i}^{k}(t)

= \frac{1}{k_{i}} + ∫_{0}^{t} g_{i}(t, s)[θ_{i}P_{i}(s, u^{k}(s)) + θ_{i}Q_{i}(s, u^{k}(s))]ds

≥ ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i}(s, u^{k}(s))ds

≥ ∫_{0}^{t} g_{i}(t, s)r_{i}(s)γ_{i}(|u_{i}^{k}(s)|)ds

≥ γ_{i}(|u_{i}^{k}(t)|) ∫_{0}^{t} g_{i}(t, s)r_{i}(s)ds

or equivalently

G_{i}(|u_{i}^{k}(t)|) = \frac{|u_{i}^{k}(t)|}{γ_{i}(|u_{i}^{k}(t)|)} ≥ ∫_{0}^{t} g_{i}(t, s)r_{i}(s)ds.

Noting that G_{i} is an increasing function (since γ_{i} is nonincreasing), we have

θ_{i}u_{i}^{k}(t) = |u_{i}^{k}(t)| ≥ G_{i}^{−1} (∫_{0}^{t} g_{i}(t, s)r_{i}(s)ds) = β_{i}(t), t ∈ [0, T], 1 ≤ i ≤ n (2.8)

for each k ∈ N^{n}.

We shall now show that {u^{k}}_{k∈N^{n}} is a bounded family on [0, T]. Fix k ∈ N^{n}. Using (I_{2}), (2.8) and (I_{4}), we obtain for t ∈ [0, T] and 1 ≤ i ≤ n,

|u_{i}^{k}(t)| = θ_{i}u_{i}^{k}(t)

= \frac{1}{k_{i}} + ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i}(s, u^{k}(s)) [1 + \frac{Q_{i}(s, u^{k}(s))}{P_{i}(s, u^{k}(s))}] ds

≤ 1 + ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) [1 + d_{i}(s) ∏_{j=1}^{n} h_{ij}(|u_{j}^{k}(s)|)] ds

≤ 1 + E_{i}(1 + D_{i})

where

E_{i} = sup_{t∈[0,T]} ∫_{0}^{t} g_{i}(t, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds.

It follows that

∥u^{k}∥ ≤ 1 + ( max_{1≤i≤n} E_{i}) (1 + max_{1≤i≤n} D_{i})

and by (I_{11}) there exists a constant M (independent of k) with ∥u^{k}∥ ≤ M. Thus, {u^{k}}_{k∈N^{n}} is bounded.

Next, we shall show that {u^{k}}_{k∈N^{n}} is equicontinuous. Let k ∈ N^{n} be fixed. For t, x ∈ [0, T] with t < x, using the fact that θ_{i}u_{i}^{k} is nondecreasing and an earlier technique, we obtain for each 1 ≤ i ≤ n,

|u_{i}^{k}(x) − u_{i}^{k}(t)| = θ_{i}u_{i}^{k}(x) − θ_{i}u_{i}^{k}(t)

= ∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)]θ_{i}P_{i}(s, u^{k}(s)) [1 + \frac{Q_{i}(s, u^{k}(s))}{P_{i}(s, u^{k}(s))}] ds

+ ∫_{t}^{x} g_{i}(x, s)θ_{i}P_{i}(s, u^{k}(s)) [1 + \frac{Q_{i}(s, u^{k}(s))}{P_{i}(s, u^{k}(s))}] ds

≤ {∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)]θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds

+ ∫_{t}^{x} g_{i}(x, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds}

× {1 + [ sup_{s∈[0,T]} d_{i}(s)] ∏_{j=1}^{n} h_{ij}(M)}

≤ [|ρ_{i}(x) − ρ_{i}(t)| + ∫_{t}^{x} g_{i}(T, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds]

× {1 + [ sup_{s∈[0,T]} d_{i}(s)] ∏_{j=1}^{n} h_{ij}(M)}

where we have used (I_{10}) and (I_{8}) in the last inequality. This shows that {u^{k}}_{k∈N^{n}} is an equicontinuous family on [0, T].

Now, the Arzéla-Ascoli theorem guarantees the existence of a subsequence N^{∗} of N, and a function u^{∗} ∈ (C[0, T])^{n} with u^{k} converging uniformly on [0, T] to u^{∗} as k_{i} → ∞, 1 ≤ i ≤ n through N^{∗}. Further,

β_{i}(t) ≤ θ_{i}u_{i}^{∗}(t) ≤ M, t ∈ [0, T], 1 ≤ i ≤ n.  (2.9)

It remains to show that u^{∗} is indeed a solution of (1.1). Fix t ∈ [0, T]. Then, from (2.6) we have for each 1 ≤ i ≤ n,

u_{i}^{k}(t) = \frac{θ_{i}}{k_{i}} + ∫_{0}^{t} g_{i}(t, s)[P_{i}(s, u^{k}(s)) + Q_{i}(s, u^{k}(s))]ds.

Let k_{i} → ∞ through N^{∗}, and use the Lebesgue dominated convergence theorem with (I_{9}), to obtain for each 1 ≤ i ≤ n,

u_{i}^{∗}(t) = ∫_{0}^{t} g_{i}(t, s)[P_{i}(s, u^{∗}(s)) + Q_{i}(s, u^{∗}(s))]ds.

This argument holds for each t ∈ [0, T], hence u^{∗} is indeed a solution of (1.1). □

Remark 2.2 If (I_{6}) is changed to

(I_{6})′ for any t, t′ ∈ [0, T] with t^{∗} = min{t, t′} and t^{∗∗} = max{t, t′}, we have

∫_{0}^{t^{∗}} |g_{i}(t, s) − g_{i}(t′, s)|ds + ∫_{t^{∗}}^{t^{∗∗}} |g_{i}(t^{∗∗}, s)|ds → 0
as t → t′,

then automatically we have sup_{t∈[0,T]} ∫_{0}^{t} |g_{i}^{t}(s)|ds < ∞ which appears in (I_{5}).

Remark 2.3 If Q_{i} ≡ 0, then we can pick d_{i} = 0 in (I_{4}), and trivially (I_{11}) is satisfied with M = K + L.

Remark 2.4 Let p and q be as in Theorem 2.1. Suppose (C_{4}) and

(C_{5}) ∫_{0}^{T} [θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s))]^{q} ds < ∞

are satisfied. Then, (I_{10}) is not required in Theorem 2.2. In fact, (I_{10}) is only needed to show that {u^{k}}_{k∈N^{n}} is equicontinuous. Let k ∈ N^{n} be fixed. For t, x ∈ [0, T] with t < x, from the proof of Theorem 2.2 we have for each 1 ≤ i ≤ n,

|u_{i}^{k}(x) − u_{i}^{k}(t)| = θ_{i}u_{i}^{k}(x) − θ_{i}u_{i}^{k}(t)

≤ {∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)]θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds

+ ∫_{t}^{x} g_{i}(x, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds}

× {1 + [ sup_{s∈[0,T]} d_{i}(s)] ∏_{j=1}^{n} h_{ij}(M)}

≤ \{ (∫_{0}^{t}[g_{i}(x, s) − g_{i}(t, s)]^{p} ds)^{\frac{1}{p}} (∫_{0}^{T} [θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s))]^{q} ds)^{\frac{1}{q}}

+ ∫_{t}^{x} g_{i}(T, s)θ_{i}P_{i} (s, θ_{1}β_{1}(s), ⋯, θ_{n}β_{n}(s)) ds}

× {1 + [ sup_{s∈[0,T]} d_{i}(s)] ∏_{j=1}^{n} h_{ij}(M)} .

Hence, in view of (C_{4}) and (C_{5}), we see that {u^{k}}_{k∈N^{n}} is an equicontinuous family on [0, T].

3 Example

Consider the system of singular Volterra integral equations

\{ \begin{matrix}u_{1}(t) = ∫_{0}^{t}(t − s) \{[u_{1}(s)]^{−a_{1}} + [u_{2}(s)]^{−a_{2}} + [u_{1}(s)]^{a_{3}}[u_{2}(s)]^{a_{4}}\} ds, t ∈ [0, T] \\ u_{2}(t) = ∫_{0}^{t}(t − s) \{[u_{1}(s)]^{−b_{1}} + [u_{2}(s)]^{−b_{2}} + [u_{1}(s)]^{b_{3}}[u_{2}(s)]^{b_{4}}\} ds, t ∈ [0, T] \end{matrix}

(3.1)

where a_{i}, b_{i} > 0, i = 1, 2, 3, 4 and T > 0 are fixed with

\begin{matrix}a_{1} < 1, b_{2} < 1, 2a_{2} < b_{2} + 1, \\ 2b_{1} < a_{1} + 1, a_{1} + a_{3} + a_{4} = b_{1} + b_{3} + b_{4} = \frac{1}{3}. \end{matrix}  (3.2)

(Many a_{i} and b_{i}, i = 1, 2, 3, 4 fulfill (3.2), for instance a_{1} = \frac{1}{6}, a_{2} < \frac{7}{12}, a_{3} = \frac{1}{8}, a_{4} = \frac{1}{24}, b_{1} = \frac{1}{24}, b_{2} = b_{3} = \frac{1}{6}, b_{4} = \frac{1}{8}.)

Here, (3.1) is of the from (1.1) with

g_{1}(t, s) = g_{2}(t, s) = t − s,
P_{1}(t, u_{1}, u_{2}) = u_{1}^{−a_{1}} + u_{2}^{−a_{2}}, Q_{1}(t, u_{1}, u_{2}) = u_{1}^{a_{3}} u_{2}^{a_{4}},
P_{2}(t, u_{1}, u_{2}) = u_{1}^{−b_{1}} + u_{2}^{−b_{2}}, Q_{2}(t, u_{1}, u_{2}) = u_{1}^{b_{3}} u_{2}^{b_{4}}.

It is clear that g_{i}, i = 1, 2 fulfill (I_{5})–(I_{8}). Suppose we are interested in positive solutions of (3.1), i.e., when θ_{1} = θ_{2} = 1. Clearly, (I_{1}) and (I_{2}) are satisfied. Further, (I_{3}) and (I_{4}) are fulfilled if we choose

r_{1} = r_{2} = d_{1} = d_{2} = 1,
γ_{1}(z) = z^{−a_{1}}, γ_{2}(z) = z^{−b_{2}},
h_{11}(z) = z^{a_{1}+a_{3}}, h_{12}(z) = z^{a_{4}},
h_{21}(z) = z^{b_{1}+b_{3}}, h_{22}(z) = z^{b_{4}}.

Hence, we have

\begin{matrix}G_{1}(z) = \frac{z}{γ_{1}(z)} = z^{a_{1}+1}, G_{2}(z) = \frac{z}{γ_{2}(z)} = z^{b_{2}+1}, \\ G_{1}^{−1}(z) = z^{\frac{1}{a_{1}+1}} , G_{2}^{−1}(z) = z^{\frac{1}{b_{2}+1}} \end{matrix}

and subsequently

\begin{aligned}β_{1}(t) & = & G_{1}^{−1} (∫_{0}^{t} g_{1}(t, x)r_{1}(x)dx) \\ & = & (∫_{0}^{t}(t − x)dx)^{\frac{1}{a_{1}+1}} = (\frac{t^{2}}{2} )^{\frac{1}{a_{1}+1}} , \\ β_{2}(t) & = & G_{2}^{−1} (∫_{0}^{t} g_{2}(t, x)r_{2}(x)dx) \\ & = & (∫_{0}^{t}(t − x)dx)^{\frac{1}{b_{2}+1}} = (\frac{t^{2}}{2} )^{\frac{1}{b_{2}+1}} . \end{aligned}  (3.3)

Now, noting (3.2) we see that

\begin{matrix}∫_{0}^{T} P_{1}(s, β_{1}(s), β_{2}(s))ds \\ = ∫_{0}^{T} [(\frac{s^{2}}{2} )^{\frac{−a_{1}}{a_{1}+1}} + (\frac{s^{2}}{2} )^{\frac{−a_{2}}{b_{2}+1}} ] ds < ∞ \end{matrix}  (3.4)

and

\begin{matrix}∫_{0}^{T} P_{2}(s, β_{1}(s), β_{2}(s))ds \\ = ∫_{0}^{T} [(\frac{s^{2}}{2} )^{\frac{−b_{1}}{a_{1}+1}} + (\frac{s^{2}}{2} )^{\frac{−b_{2}}{b_{2}+1}} ] ds < ∞. \end{matrix}  (3.5)

Applying (3.4) and (3.5), we find for i = 1, 2,

\begin{matrix}sup_{t∈[0,T]} ∫_{0}^{t} g_{i}(t, s)P_{i}(s, β_{1}(s), β_{2}(s))ds \\ ≤ T ∫_{0}^{T} P_{i}(s, β_{1}(s), β_{2}(s))ds < ∞. \end{matrix}

Thus, the condition (I_{9}) is satisfied.

Next, to check condition (I_{10}), we note that for t, x ∈ [0, T] with t < x, on using (3.4) and (3.5) we have

\begin{matrix}∫_{0}^{t}[g_{1}(x, s) − g_{1}(t, s)]P_{1}(s, β_{1}(s), β_{2}(s))ds \\ ≤ (x − t) ∫_{0}^{T} P_{1}(s, β_{1}(s), β_{2}(s))ds ≤ (x − t)K_{1} \end{matrix}

and

\begin{matrix}∫_{0}^{t}[g_{2}(x, s) − g_{2}(t, s)]P_{2}(s, β_{1}(s), β_{2}(s))ds \\ ≤ (x − t) ∫_{0}^{T} P_{2}(s, β_{1}(s), β_{2}(s))ds ≤ (x − t)K_{2} \end{matrix}

where K_{1} and K_{2} are some finite constants. Hence, condition (I_{10}) is satisfied. Finally, the condition (I_{11}) is equivalent to

\{ \begin{matrix}if z > 0 satisfies z ≤ K + L (1 + z^{\frac{1}{3}} ) \\ for some constants K, L ≥ 0, then there exists \\ a constant M(which may depend on K and L) \\ such that z ≤ M, \end{matrix}  (3.6)

which is true since if z is unbounded, then obviously z > K + L (1 + z^{\frac{1}{3}} ) for any K, L ≥ 0. As an illustration, pick K = L = 1, then the inequality in (3.6) becomes

z ≤ 1 + (1 + z^{\frac{1}{3}} )

which can be solved to obtain

0 < z ≤ 3.5213 = M.

It now follows from Theorem 2.2 that the system (3.1), (3.2) has a positive solution u ∈ (C[0, T])^{2} with u_{i}(t) ≥ β_{i}(t) for t ∈ [0, T] and i = 1, 2, where β_{i}(t) is given by (3.3).

References

[1] M. A. Abdou, Fredholm-Volterra integral equation with singular kernel, Appl. Math. Comput. 137 (2003), 231-243.

[2] R. P. Agarwal and D. O’Regan, Singular Volterra integral equations, Appl. Math. Lett. 13 (2000), 115-120.

[3] R. P. Agarwal and D. O’Regan, Volterra integral equations: the singular case, Hokkaido Math. J. 32 (2003), 371-381.

[4] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999.

[5] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Constant-sign solutions of a system of Fredholm integral equations, Acta Appl. Math. 80 (2004), 57-94.

[6] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Eigenvalues of a system of Fredholm integral equations, Math. Comput. Modelling 39 (2004), 1113-1150.

[7] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Triple solutions of constant sign for a system of Fredholm integral equations, Cubo 6 (2004), 1-45.

[8] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Constant-sign solutions of a system of integral equations: The semipositone and singular case, Asymptotic Analysis 43 (2005), 47-74.

[9] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Constant-sign solutions of a system of integral equations with integrable singularities, J. Integral Equations Appl. 19 (2007), 117-142.

[10] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Constant-sign solutions of a system of Volterra integral equations, Comput. Math. Applic. 54 (2007), 58-75.

[11] C. C. Aranda and E. Lami Dozo, Multiple solutions to a singular Lane-Emden-Fowler equation with convection term, Electron. J. Differential Equations, 2008, No. 5.

[12] P. J. Bushell, On a class of Volterra and Fredholm non-linear integral equations, Math. Proc. Cambridge Philos. Soc. 79 (1976), 329-335.

[13] P. J. Bushell and W. Okrasiński, Uniqueness of solutions for a class of nonlinear Volterra integral equations with convolution kernel, Math. Proc. Cambridge Philos. Soc. 106 (1989), 547-552.

[14] P. J. Bushell and W. Okrasiński, Nonlinear Volterra integral equations with convolution kernel, J. London Math. Soc. 41 (1990), 503-510.

[15] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, New York, 1990.

[16] W. Dong, Uniqueness of solutions for a class of non-linear Volterra integral equations without continuity, Appl. Math. Mech. (English Ed.) 18 (1997), 1191-1196.

[17] G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981), 59-67.

[18] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications 34, Cambridge University Press, Cambridge, 1990.

[19] M. Meehan and D. O’Regan, Positive solutions of Volterra integral equations using integral inequalities, J. Inequal. Appl. 7 (2002), 285-307.

[20] D. O’Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer, Dordrecht, 1998.

[21] D. W. Reynolds, On linear singular Volterra integral equations of the second kind, J. Math. Anal. Appl. 103 (1984), 230-262.

[22] J. Wang, W. Gao and Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl. 233 (1999), 246-256.

[23] P. J. Y. Wong, Existence of solutions to singular integral equations, preprint.