Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 25, 1-10; http://www.math.u-szeged.hu/ejqtde/

Integrable and continuous solutions of a nonlinear quadratic integral equation

El-Sayed A.M.A and Hashem H.H.G Faculty of Science, Alexandria University, Alexandria, Egypt e.mails : amasayed@hotmail.com, amasayed5@yahoo.com

Abstract We are concerned here with a nonlinear quadratic integral equation of Volterra type. The existence of at least one L_{1}− positive solution will be proved under the Carathèodory condition. Secondly we will make a link between Peano condition and Carathèodory condition to prove the existence of at least one positive continuous solution. Finally the existence of the maximal and minimal solutions will be proved.

Keywords: Quadratic integral equation; Positive integrable solution; Continuous solution; Maximal and minimal solutions.

1 Introduction

Quadratic integral equations are often applicable in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport and in the traffic theory. Especially, the so-called quadratic integral equation of Chandraskher type can be very often encountered in many applications (see[1]-[4]). Let I = [0, T], L_{1} = L_{1}[0, T] be the space of Lebesgue integrable functions on I and C[0, T] be the space of continuous functions defined on I. Consider now the following integral equation

x(t) = x(t_{0}) + ∫_{t_{0}}^{t} f(s, x(s)) ds.  (1)

It is proved that ( [5], Caratheodory Theorem ) if f is measurable in t ∈ [t_{0}, T] for each fixed x ∈ B ⊂ R and continuous in x ∈ B for each fixed t ∈ [t_{0}, T] and there exists a function m ∈ L_{1} such that |f(t, x)| ≤ m(t) then, there exists a solution x ∈ C[t_{0}, T] of the integral equation (1) and the equivalent initial value problem

\frac{d}{dt}x(t) = f(t, x(t)), t > t_{0}; x(t_{0}) = x_{0}.

This result has been generalized (by the authors [7]) for the fractional-order integral equation

x(t) = x(t_{0}) + ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} f(s, x(s)) ds.  (2)

In this work, we generalize this result for the nonlinear quadratic integral equation

x(t) = a(t) + g(t, x(t)) ∫_{0}^{t} k(t, s) f(s, x(s)) ds, t ∈ [0, 1].  (3)

Firstly we prove the existence of at least one positive solution x ∈ L_{1} of the quadratic integral equation (3) where the functions f and g satisfy Carathèodory condition. Secondly we prove the existence of at least one positive continuous solution for equation (3) where g satisfies Peano condition and f satisfies Carathèodory condition. Also the existence of the maximal and minimal ( continuous ) solutions will be proved

2 Preliminaries

The following theorems will be needed in our work ( see [5], [6] and [9]). Tychonov‘s Theorem. Suppose B is a complete, locally convex linear space and S is a closed convex subset of B. Let the mapping T : B → B be continuous and T (S) ⊂ S. If the closure of T (S) is compact, then T has a fixed-point in S.

Schauder fixed-point Theorem. Let S be a convex subset of a Banach space B, let the mapping T : S → S be compact, continuous. Then T has at least one fixed-point in S.

Arzelà-Ascoli Theorem. Let E be a compact metric space and C(E) the Banach space of real or complex valued continuous functions normed by

∥ f(t) ∥ = max_{t ∈ E} | f(t) |.

If A = {f_{n}} is a sequence in C(E) such that f_{n} is uniformly bounded and equicontinuous. Then the closure of A is compact.

Lusin Theorem. Let m : [0, 1] → R be a measurable function. For any ϵ > 0 there exists a closed subset A_{ϵ} of [0, 1], meas.(A_{ϵ}^{c}) < ϵ, such that m restricted to A_{ϵ} is continuous.

Scorza Dragoni Theorem. Let k : [0, 1] × [0, 1] → R be a function satisfying Carathèodory condition (i.e. it is measurable in t for all x ∈ R and continuous in x for all t ∈ [0, 1] ). For any ϵ > 0 there exists a closed subset A_{ϵ} of [0, 1], meas.(A_{ϵ}^{c}) < ϵ, such that k restricted to A_{ϵ} × [0, 1] is continuous.

3 L_{1}−positive solution

Let I = [0, 1], and consider the assumptions:

(i) a : I → R_{+} = [0, + ∞) is integrable on I; (ii) f, g : I × R_{+} → R_{+} satisfy Caratheodory condition (i.e. measurable in t for

all x ∈ R_{+} and continuous in x for all t ∈ [0, 1] ) and there exist two functions m_{1}, m_{2} ∈ L_{1} such that

g(t, x) ≤ m_{1}(t), f(t, x) ≤ m_{2}(t) for all (t, x) ∈ I × R_{+};

(iii) k : [0, 1] × [0, 1] → R satisfies Carathèodory condition (i.e. it is measurable in t for all s and continuous in s for all t ).

For the existence of at least one L_{1}−positive solution of the quadratic integral equation (3) we have the following theorem.

Theorem 3.1 Let the assumptions (i)-(iii) be satisfied. Then the nonlinear quadratic integral equation (3) has at least one L_{1}−positive solution. Proof. Consider the set Q ⊂ L_{1} such that

Q = { x ∈ L_{1}, | x(t) | ≤ x_{0}(t) a.e. }

where

x_{0}(t) = a(t) + m_{1}(t) ∫_{0}^{t} k(t, s) m_{2}(s) ds.  (4)

The set Q can be shown to be nonempty, bounded, closed and convex in L_{1}. Let H be the operator defined by the formula

(Hx)(t) = a(t) + g(t, x(t)) ∫_{0}^{t} k(t, s) f(s, x(s)) ds, t ∈ I.  (5)

We shall prove that H : Q → Q. For that let x ∈ Q, then

(Hx)(t) ≤ | a(t) | + m_{1}(t) ∫_{0}^{t} k(t, s) m_{2}(s) ds

≤ | a(t) | + m_{1}(t) ∫_{0}^{t} k(t, s) m_{2}(s) ds = x_{0}(t),

so Hx ∈ Q and hence HQ ⊂ Q. To apply Schauder fixed-point Theorem, we shall prove that HQ is relatively compact in L_{1}. By using Lusin Theorem and Scorza Dragoni Theorem , we can find a closed subset A_{n} of [0, 1], with meas.(A_{n}^{c}) < \frac{1}{n} such that a|_{A_{n}}, m_{1}|_{A_{n}}, k|_{A_{n}×[0,1]} and g|_{A_{n}×Q} are uniformly continuous. Assume that x_{h} is any sequence in Q, then for t_{1}, t_{2} ∈ A_{n}, we have

(Hx_{h})(t_{2}) − (Hx_{h})(t_{1}) = a(t_{2}) − a(t_{1})

+ g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{2}} k(t_{2}, s) f(s, x_{h}(s)) ds − g(t_{1}, x_{h}(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s)) ds

= a(t_{2}) − a(t_{1}) + g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{2}} k(t_{2}, s) f(s, x_{h}(s)) ds

− g(t_{1}, x_{h}(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s)) ds + g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s)) ds

− g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s)) ds
= a(t_{2}) − a(t_{1}) + g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} k(t_{2}, s) f(s, x_{h}(s))ds

+ g(t_{2}, x_{h}(t_{2})) ∫_{t_{1}}^{t_{2}} k(t_{2}, s) f(s, x_{h}(s))ds − g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s))ds

+ g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s))ds − g(t_{1}, x_{h}(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s))ds

= a(t_{2}) − a(t_{1}) + g(t_{2}, x_{h}(t_{2})) ∫_{0}^{t_{1}} [ k(t_{2}, s) − k(t_{1}, s) ] f(s, x_{h}(s)) ds
+ g(t_{2}, x_{h}(t_{2})) ∫_{t_{1}}^{t_{2}} k(t_{2}, s) f(s, x_{h}(s)) ds
+ [ g(t_{2}, x_{h}(t_{2})) − g(t_{1}, x_{h}(t_{1})) ] ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x_{h}(s)) ds

Then we get

| (Hx_{h})(t_{2}) − (Hx_{h})(t_{1}) | ≤ | a(t_{2}) − a(t_{1}) | + m_{1}(t_{2}) ∫_{t_{1}}^{t_{2}} k(t_{2}, s) m_{2}(s) ds

+ m_{1}(t_{2}) ∫_{0}^{t_{1}} | k(t_{2}, s) − k(t_{1}, s) | m_{2}(s) ds
+ | g(t_{2}, x_{h}(t_{2})) − g(t_{1}, x_{h}(t_{1})) | ∫_{0}^{t_{1}} k(t_{1}, s) m_{2}(s) ds

This means that the sequence {Hx_{h}} is sequence of equi-continuous functions on A_{n} and we can prove that this sequence is uniformly bounded. Now

| (Hx_{h})(t) | = | a(t) | + m_{1}(t) ∫_{0}^{t} k(t, s) m_{2}(s) ds
≤ M_{1} + M_{2} K ∫_{0}^{t} m_{2}(s) ds
≤ M_{1} + M_{2} K ∫_{0}^{1} m_{2}(s) ds = M

where | a(t) |_{A_{n}} ≤ M_{1}, | m_{1}(t) |_{A_{n}} ≤ M_{2} and k|_{A_{n}×[0,1]} ≤ K. Hence by Arzelà-Ascoli Theorem Hx_{h} is relatively compact subset of C(A_{n}) and this can be done for each n ∈ N. This implies the existence of convergent subsequence {x_{h_{j}}} of {x_{h}} in each C(A_{n}). Given ϵ > 0 and choose n_{1} ∈ N so that meas(A_{n_{1}}) < ϵ, then

∫_{0}^{1} | Hx_{h_{j}} − Hx_{h_{l}} | dt = ∫_{A_{n_{1}}^{c}} | Hx_{h_{j}} − Hx_{h_{l}} | dt

+ ∫_{A_{n_{1}}} | Hx_{h_{j}} − Hx_{h_{l}} | dt.

Since C(A_{n}) is complete metric space, hence this subsequence is a Cauchy sequence in each C(A_{n}), n = 1, 2, 3, ... That is for given ϵ > 0 and j, l are arbitrary large we have

|| Hx_{h_{j}} − Hx_{h_{l}} ||_{C(A_{n})} < ϵ.  (6)

But we want to prove that the set HQ is relatively compact in L_{1}, that is \overline{HQ} is compact in L_{1}. To do this, we will prove that the sequence { Hx_{h} } is convergent in L_{1}, since L_{1} is complete metric space, then it is sufficient to prove that the subsequence { Hx_{h_{j}} } is a Cauchy sequence in L_{1}. i.e. ∀ η > 0, ∃ N(η) and ∫_{A_{n}} x_{0}(t) dt < η/4 such that

|| Hx_{h_{j}} − Hx_{h_{l}} ||_{L_{1}} < η, j, l > N(η).

Now from (4) and (5) we have

∫_{0}^{1} | Hx_{h_{j}} − Hx_{h_{l}} | dt = ∫_{A_{n}^{c}} | Hx_{h_{j}} − Hx_{h_{l}} | dt

+ ∫_{A_{n}} | Hx_{h_{j}} − Hx_{h_{l}} | dt.
≤ ∫_{A_{n}^{c}} { | Hx_{h_{j}} | + | Hx_{h_{l}} | } dt + || Hx_{h_{j}} − Hx_{h_{l}} ||_{C(A_{n})}
≤ η/4 + η/4 + || Hx_{h_{j}} − Hx_{h_{l}} ||_{C(A_{n})}.

Choose N such that l, j > N, then (6) implies that || Hx_{h_{j}} − Hx_{h_{l}} ||_{C(A_{n})} ≤ η/2. This means that the subsequence { Hx_{h_{j}} } is a Cauchy sequence in L_{1} which implies that HQ is relatively compact in L_{1}. Then H has at least one fixed point. Hence there exists at least one solution x ∈ L_{1} of (1). Since all conditions of Shauder’s fixed-point Theorem hold, then H has a fixed point in Q.

4 Continuous solutions

Let I = [0, 1], and consider the assumptions:

(i) a : I → R_{+} = [0, + ∞) is continuous on I; (ii) f : I × R_{+} → R_{+} satisfies Carathéodory condition (i.e. measurable in t for all

x ∈ R_{+} and continuous in x for all t ∈ [0, 1] ) and there exists function m ∈ L_{1} such that

f(t, x(t)) ≤ m(t) ∀ (t, x) ∈ I × R_{+};

(iii) g : I × R_{+} → R_{+} is continuous in t, x and | g(t, x) | ≤ M;

(iv) k : [0, 1] × [0, 1] → R satisfies Carathèodory condition (i.e. measurable in t for all s and continuous in s for all t.)

Now for the existence of at least one positive continuous solution of the nonlinear quadratic integral equation (3) we have the following theorem. Theorem 4.1 Let the assumptions (i)-(iv) be satisfied. Then the equation (3) has at least one positive solution x ∈ C(I). Proof. We shall use Tychonov‘s fixed point Theorem to prove this theorem It can be verified that [6] C is complete locally convex linear space. Define a subset S of C by

S = { x ∈ C : | x(t) | ≤ M_{2} }, t ∈ [0, 1],

where M_{2} is a positive constant. It is clear that the set S is closed and convex. Let H be the operator defined by the formula

(Hx)(t) = a(t) + g(t, x(t)) ∫_{0}^{t} k(t, s) f(s, x(s)) ds, ∀ x ∈ S.

Assumptions (ii) and (iii) imply that H : S → C is continuous operator in x . We shall prove that HS ⊂ S. For every x ∈ S we have

| (Hx)(t)| ≤ | a(t) | + M ∫_{0}^{t} k(t, s) m(s) ds t ∈ [0, 1]
≤ | a(t) | + M ∫_{0}^{1} k(t, s) m(s) ds
= M_{2},

where | a(t) | ≤ M_{1}. Then, Hx ∈ S and hence HS ⊂ S. Also for x ∈ S and t_{1} and t_{2} ∈ [0, 1] we can have

(Hx)(t_{2}) − (Hx)(t_{1}) = a(t_{2}) − a(t_{1})
+ g(t_{2}, x(t_{2})) ∫_{0}^{t_{2}} k(t_{2}, s) f(s, x(s)) ds − g(t_{1}, x(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds.
+ g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds − g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds
= a(t_{2}) − a(t_{1}) + g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} k(t_{2}, s) f(s, x(s)) ds

+ g(t_{2}, x(t_{2})) ∫_{t_{1}}^{t_{2}} k(t_{2}, s) f(s, x(s)) ds − g(t_{1}, x(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds.
+ g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds − g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds
= a(t_{2}) − a(t_{1}) + g(t_{2}, x(t_{2})) ∫_{0}^{t_{1}} [k(t_{2}, s) − k(t_{1}, s)] f(s, x(s)) ds

+ g(t_{2}, x(t_{2})) ∫_{t_{1}}^{t_{2}} k(t_{2}, s) f(s, x(s)) ds + [g(t_{2}, x(t_{2})) − g(t_{1}, x(t_{1}))] ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds.

Then we have

| (Hx)(t_{2}) − (Hx)(t_{1}) | ≤ | a(t_{2}) − a(t_{1}) | + M ∫_{t_{1}}^{t_{2}} k(t_{2}, s) m(s) ds

+ M ∫_{0}^{t_{1}} | k(t_{2}, s) − k(t_{1}, s) | m(s) ds + |g(t_{2}, x(t_{2})) − g(t_{1}, x(t_{1}))| ∫_{0}^{t_{1}} k(t_{1}, s) m(s) ds.

This means that the functions of HS are equi-continuous on [0, 1], then by Arzelá-Ascoli Theorem the closure of HS is compact . Hence, all conditions of Tychonov fixed-point Theorem hold, then H has a fixed point in S.

5 Maximal and minimal solutions

Definition 5.1 [8] Let q(t) be a solution of the nonlinear quadratic integral equation (3). Then q(t) is said to be a maximal solution of (3) if every solution of (3) satisfies the inequality x(t) < q(t), ∀t ∈ I. A minimal solution s(t) can be defined by similar way by reversing the above inequality i.e. x(t) > s(t), ∀t ∈ I. We shall use the following lemma to prove the existence of the maximal and minimal solutions. Lemma 5.1 Let a(t) be a continuous function on I and k(t, s) satisfying the assumption (iv) of Theorem 4.1. Let f(t, x), g(t, x) ∈ L_{1} and x(t), y(t) be continuous functions on [0, 1] satisfying

x(t) ≤ a(t) + g(t, x(t)) ∫_{0}^{t} k(t, s) f(s, x(s)) ds,
y(t) ≥ a(t) + g(t, y(t)) ∫_{0}^{t} k(t, s) f(s, y(s)) ds

and one of them is strict. If f(t, x), g(t, x) are monotonic nondecreasing in x , then

x(t) < y(t), t > 0.  (7)

Proof. Let the conclusion (7) be false, then there exists t_{1} such that

x(t_{1}) = y(t_{1}), t_{1} > 0

and

x(t) < y(t), 0 < t < t_{1}.

From the monotonicity of f(t, x), g(t, x) in x , we get

x(t_{1}) ≤ a(t_{1}) + g(t_{1}, x(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, x(s)) ds,
< a(t_{1}) + g(t_{1}, y(t_{1})) ∫_{0}^{t_{1}} k(t_{1}, s) f(s, y(s)) ds
< y(t_{1}),

which contradicts the fact that x(t_{1}) = y(t_{1}) , then x(t) < y(t) .

Theorem 5.1 Let the assumptions of Theorem 4.1 be satisfied and f(t, x), g(t, x) are nondecreasing in x on I . Then there exist maximal and minimal solutions of equation (3). Proof. Firstly we shall prove the existence of the maximal solution of (3). Let ϵ > 0 be given. Now consider the quadratic integral equation

x_{ϵ} (t) = a(t) + g_{ϵ}(t, x_{ϵ}(t)) ∫_{0}^{t} k(t, s) f_{ϵ}(s, x_{ϵ}(s)) ds t ∈ [0, 1],  (8)

where

f_{ϵ}(t, x_{ϵ}(t)) = f(t, x_{ϵ}(t)) + ϵ,
g_{ϵ}(t, x_{ϵ}(t)) = g(t, x_{ϵ}(t)) + ϵ,

Clearly the functions f_{ϵ}(t, x_{ϵ}) and g_{ϵ}(t, x_{ϵ}) satisfy assumptions (ii) and (iii) and therefore equation (8) has at least a positive solution x_{ϵ}(t) ∈ C(I). Let ϵ_{1} and ϵ_{2} be such that 0 < ϵ_{2} < ϵ_{1} < ϵ. Then

x_{ϵ_{2}} (t) = a(t) + g_{ϵ_{2}}(t, x_{ϵ_{2}}(t)) ∫_{0}^{t} k(t, s) f_{ϵ_{2}}(s, x_{ϵ_{2}}(s)) ds

x_{ϵ_{2}} (t) = a(t) + g(t, x_{ϵ_{2}}(t)) + ϵ_{2}) ∫_{0}^{t} k(t, s) (f(s, x_{ϵ_{2}}(s)) + ϵ_{2}) ds  (9)
x_{ϵ_{1}} (t) = a(t) + g_{ϵ_{1}}(t, x_{ϵ_{1}}(t)) ∫_{0}^{t} k(t, s) f_{ϵ_{1}}(s, x_{ϵ_{1}}(s)) ds
= a(t) + (g(t, x_{ϵ_{1}}(t)) + ϵ_{1}) ∫_{0}^{t} k(t, s) (f(s, x_{ϵ_{1}}(s)) + ϵ_{1}) ds
x_{ϵ_{1}} (t) > a(t) + g(t, x_{ϵ_{1}}(t)) + ϵ_{2}) ∫_{0}^{t} k(t, s) (f(s, x_{ϵ_{1}}(s)) + ϵ_{2}) ds.  (10)

Applying Lemma 5.1 on (9) and (10), we have

x_{ϵ_{2}}(t) < x_{ϵ_{1}}(t) for t ∈ I.

As shown before the family of functions x_{ϵ}(t) is equi-continuous and uniformly bounded. Hence by Arzelá-Ascoli Theorem, there exists a decreasing sequence ϵ_{n} such that ϵ → 0 as n → ∞ and lim_{n→∞} x_{ϵ_{n}} (t) exists uniformly in I and denote this limit by q(t) . From the continuity of the functions f_{ϵ}(t, x_{ϵ}) and g_{ϵ}(t, x_{ϵ}) in the second argument, we get

q(t) = lim_{n→∞} x_{ϵ_{n}}(t) = a(t) + g(t, q(t)) ∫_{0}^{t} k(t, s) f(s, q(s)) ds

which implies that q(t) as a solution of (3). Finally, we shall show that q(t) is the maximal solution of (3). To do this, let x(t) be any solution of (3). Then

x_{ϵ} (t) = a(t) + g_{ϵ}(t, x_{ϵ}(t)) ∫_{0}^{t} k(t, s) f_{ϵ}(s, x_{ϵ}(s)) ds

= a(t) + (g(t, x_{ϵ}(t)) + ϵ) ∫_{0}^{t} k(t, s) (f(s, x_{ϵ}(s)) + ϵ) ds
> a(t) + g(t, x_{ϵ}(t)) ∫_{0}^{t} k(t, s) f(s, x_{ϵ}(s)) ds.

Also

x(t) = a(t) + g(t, x(t)) ∫_{0}^{t} k(t, s) f(s, x(s)) ds

implies that

x(t) < x_{ϵ}(t) for t ∈ I

from the uniqueness of the maximal solution (see [8]), it is clear that x_{ϵ}(t) tends to q(t) uniformly in t ∈ I as ϵ → 0. By similar way as done above we can prove the existence of the minimal solution.

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(Received April 30, 2008)