EXISTENCE AND APPROXIMATE SOLUTIONS OF NONLINEAR INTEGRAL EQUATIONS

We investigate the existence of continuous solutions on compact intervals of some nonlinear integral equations. The existence of such solutions is based on some well-known fixed point theorems in Banach spaces such as Schaefer fixed point theorem, Schauder fixed point theorem, and Leray-Schauder principle. A special interest is devoted to the study of nonlinear Volterra equations and to the numerical treatment of these equations.


Introduction
In the first part of this work, we study the existence of a solution of the following functional integral equation: Note that the previous integral equation can be considered as a nonlinear Fredholm equation expressed as a perturbed linear equation. A Krasnoselkii-Schafer fixed point theorem [4] is used to prove the existence of a solution of some special cases of (1.1), see [8]. The general nonlinear integral equation has the following form: (1.2) We should mention that an extensive work has been done in the study of the solutions of various types of (1.2), see, for example, [1,2,5,7,11,13,15,16,17,19]. Usually the existence of a solution of (1.2) starts with some conditions on the function g(t,s,x) as well as the integration bounds a, b and the function f (·). Based on these conditions, a Banach space is chosen in such a way that the existence problem is converted to a fixed point problem of an operator over this Banach space.
To prove the existence of a continuous solution of the integral equation (1.1), we use some conditions on the function f (·), the kernels K(t,s), V (t,s) as well as on the function g(t,x). By using these conditions, we define a completely continuous operator T over the Banach space C( [a,b]) whose fixed points are solutions of (1.1). The well-known fixed point theorem of Schaefer [20] is used to prove the existence of a fixed point of the operator T. Also, by introducing a convenient new norm · µ on the space C( [a,b]), we study the existence of continuous solutions of the general nonlinear equation (1.2) with finite bounds a and b.
In the second part of this work, we study the existence of continuous solution of the following nonlinear Volterra equation: (1.3) where f (·) ∈ C( [a,b]). The main tool in the proof of the existence of a solution of (1.3) is the Leray-Schauder principle combined with a general version of Gronwall's inequality. Moreover, we prove the uniqueness of the solution of (1.3) by showing that there exists n ∈ N such that T n is a contraction on some closed ball containing all possible continuous solutions of (1.3). This paper is organized as follows. In Section 2, we prove the existence of the solutions of some special cases of (1.1) and (1.2). In Section 3, we investigate the existence and the uniqueness of a solution of the nonlinear Volterra equation (1.3). Finally in Section 4, we provide the reader with a numerical scheme for solving nonlinear Volterra equations.

Existence of a solution of nonlinear integral equations
In the first part of this paragraph, we show that under some conditions on the kernels K(t,s), V (t,s) and the function g(s,x), the functional integral equation (1.1) has a solution in C( [a,b]). The following theorem ensures the existence of such a solution. Note that the proof of this theorem is based on the well-known Schaefer fixed point theorem that can be easily found in the literature, see for example [8,20].
Theorem 2.1. Consider the functional integral equation: b]). Assume that the function g(s,x) satisfies the following conditions:

2)
for some measurable function G(·) and bounded function φ(·). Assume that the kernels K(t,s), V (t,s) satisfy the following conditions:

3)
for some continuous functions K 1 (·), V 1 (·), and L 1 ([a,b]) function K 2 (·). Also, we assume Abderrazek Karoui 571 that the function G(·)V 2 (·) ∈ L 1 ([a,b]). Finally, we assume that one of the following two conditions is satisfied: Proof. We first define the operator T by: x(s))ds. By using (2.3) and by applying the dominated convergence theorem, one concludes that then lim n→+∞ Tx n − Tx| ∞ = 0, or equivalently, T is continuous over C( [a,b]). Next, we prove that T is completely continuous on E = C([a,b]) or equivalently, it maps an arbitrary bounded set of the Banach space E into a compact set of E. By using Arzèla theorem [12], the complete continuity of T is ensured if {Tx n ; n ∈ N} is equicontinuous and uniformly bounded for every uniformly bounded sequence (x n ) n of C( [a,b]). This is done as follows: Since (x n ) n is uniformly bounded, then by applying the dominated convergence theorem to the right-hand side of the previous inequality, one concludes that lim t→τ |Tx n (t) − Tx n (τ)| = 0 independently of n or equivalently, (Tx n ) n is equicontinuous. Moreover, it is easy to see that (Tx n ) n is uniformly bounded whenever (x n ) n is a uniformly bounded sequence of C([a,b]). Hence, T is completely continuous. Finally, we prove the existence of a solution of (2.1). Since T is completly continuous, then by Schaefer fixed point theorem, we know that either: (i) x = λTx has a solution for λ = 1, or (ii) the set Ᏹ = {u ∈ C([a,b]); ∃λ ∈]0, 1[, u = λTu} is unbounded. We prove that (ii) is not possible. Two cases are to be considered.
First case. We assume that (c 1 ) is satisfied and take u ∈ Ᏹ satisfying u = λTu for some 0 < λ < 1. Since b a K 2 (s)|u(s)|ds = |u(s * )| b a K 2 (s)ds, for some s * ∈ [a,b], then it is easy to see that there exists a positive real number M such that (2.7) By using (c 1 ) and by taking t = s * in the previous inequality, one gets By substituting (2.8) in (2.7), one concludes that |u(·)| is bounded and consequently Ᏹ is bounded.
Second case. We assume that condition (c 2 ) is satisfied. In this case, it is easy to see that for An extension of the result of the previous theorem to a more general nonlinear integral equation is given by the following theorem. We skip the proof of this theorem because its techniques are similar to the techniques of the previous proof.
Theorem 2.2. Consider the nonlinear integral equation: . Assume that the function g(t,s,x) satisfies the following conditions: Condition (2.10) with bounded φ(·) is a limitation of the previous theorem. Nonetheless, by using a convenient new norm · µ and the Schauder fixed point theorem, one can prove the existence of continuous solutions of more general nonlinear integral equations with some weaker conditions. This is the subject of the next theorem.
Abderrazek Karoui 573 Theorem 2.3. Consider the nonlinear integral equation Assume that f (·) is bounded and g(t,s,x) is continuous w.r.t. t and satisfies the following conditions: where V 1 (·) is a measurable and bounded positive function, φ(·) is a positive and measurable function satisfying the condition and where ψ(·) is a positive and continuous function over [0,+∞[. Moreover, assume that there exists a continuous, positive and bounded away from zero function µ(·) satisying the following condition: (2.14)

Under the above conditions, the nonlinear integral equation (2.11) has a solution in C([a,b]).
Proof. We first mention that the function · µ defined on Next let r ≥ 0 be a positive real number that will be fixed later on and define the subset B r of X by B r = {x ∈ C([a,b]); x µ ≤ r}. It is clear that B r is a closed and convex subset of X. Let T be the operator defined on B r by s,x(s))ds. It is easy to check that T maps bounded sets of B r into relatively compact sets. By Schauder fixed point theorem see [20], to prove the existence of a solution of (2.11), it suffices to check that T ∈ C(B r ,B r ). We first prove that where M φ,|x| is a constant depending only on φ(·) and |x(·)|, then by applying the dominated convergence theorem, one concludes that Consequently, Tx(·) ∈ C([a,b]). Next, we prove that T is continuous over B r w.r.t. · µ norm. Let (x n ) n be a sequence of B r converging to x in the · µ norm. Since (B r , · µ ) is complete, then x ∈ B r . Moreover, we have (2.17) Since µ(·) is continuous and bounded away from zero, then it is clear that convergence of (x n ) n to x in the · µ norm implies also the uniform convergence over Hence, the condition T(B r ) ⊂ B r is satisfied for any positive real number r satisfying

Existence and uniqueness results for a nonlinear integral equation
If in the Fredholm integral equation (2.11), we replace the integration bound b by the variable t, we obtain a nonlinear Volterra equation. We should mention that an extensive Abderrazek Karoui 575 amount of work has been done in the existence and uniqueness of solutions of some special cases of Volterra integral equations, see for example [3,6,10,18]. Under some conditions on the function g(t,s,x) and by using the following Leray-Schauder principle, Theorem 3.2 ensures the existence of a solution of a nonlinear Volterra equation.
Theorem 3.1 (Leray-Schauder principle). Let (X,| · |) be a Banach space and suppose that T ∈ C(X,X) and compact. Suppose that any solution x of x = λTx, 0 ≤ λ ≤ 1 satisfies the a priori bound |x| ≤ M for some constant M > 0, then T has a fixed point.
Theorem 3.2. Consider the nonlinear Volterra integral equation where f is continuous over [a,b]. Assume that g(t,s,x) satisfies the following conditions: Proof. Let X = (C([a,b]), · ∞ ) denotes the Banach space of continuous functions over [a,b] and define the operator T over X by Tx(t) = f (t) + t a g(t,s,x(s))ds. By using the conditions of the theorem, it is easy to check that TX ⊂ X and T is compact. From Leray-Schauder principle, to prove the result of the theorem, it suffices to prove that T is continuous over X and any solution of x = λTx, 0 ≤ λ ≤ 1 is bounded by the same constant M > 0. To prove the continuity of T over C ([a,b]), it suffices to replace µ(t) by 1 in the proof of the continuity of the operator T of the previous theorem and follow the different steps of this proof. Next, we note that the condition lim y→+∞ (φ(y)/y) = L < +∞ implies the existence of a positive real number A > 0 such that |φ(u)| ≤ (3/2)L = L , for all u ≥ A. Let x ∈ C([a,b]) be a solution of x = λTx, for some 0 ≤ λ ≤ 1, then we have Hence, to prove the uniqueness of the solution of (3.1), it suffices to check that there exists n 0 ∈ N such that T n0 is a contraction in B M . By using the notations of the proof of Theorem 3.2, one can easily check that for all x, Similarly, one shows that Continuing in this manner, one can easily show that Hence Since lim n→+∞ [ n−1 i=1 (1/(q + i))]C n (b − a) n−1+1/q = 0, then there exists n 0 ∈ N such that T n0 is a contraction over B M . Consequently, the fixed point of T n0 is unique. Since a fixed point of T is also a fixed point of T n0 , then one concludes that the fixed point of T is also unique and consequently, the solution of (3.1) is unique.

Approximate solution of Volterra integral equation
In this last paragraph, we are interested in finding an approximate solution of Volterra integral equation of the type Note that the natural approach for finding an approximate solution of (4.1) is to use a quadrature scheme for the approximation of the integral term of (4.1), see [9,14,21].
In this section, we provide a new approach for approximating the solution of (4.1). It is described as follows. We first assume that (4.1) has a solution in C α ([a,b]) for some ([a,b]), the function g(t,t,x) is continuous with respect to t and Lipschitzian w.r.t. x. Moreover, if (t n ) n is a sequence in [a,b], then we assume that the functions (∂g/∂t)(t n ,s,x) is equicontinuous w.r.t. s and Lipschitzian w.r.t. x. By using the above conditions and the standard existence proof for ordinary differential equation (O.D.E.) which is based on the successive approximations technique, one can easily check that the solution of (4.1) coincides with the unique solution of the following initial value problem obtained by differentiating (4.1): Hence the problem of finding an approximate solution of (4.1) is converted to the approximation of the solution of the integro-differential equation (4.2). Note that finding an approximate solution of the second problem is easier than for the first problem. This is due to the possibility of adapting existent approximation schemes from O.D.E. Our approximation scheme for solving (4.2) is described as follows. We first choose a uniform subdivision of [a,b] denoted by a = t 0 < t 1 < ··· < t N = b and let h = t n+1 − t n , 0 ≤ n ≤ N − 1 be the stepsize of this subdivision. For t n ≤ t < t n + 1, we define a quadrature scheme Q(t,x) for the approximation of the integral t a (∂g/∂t)(t,s,x(s))ds as follows: where Q 1 (t,x) is a qth order composite quadrature scheme for the approximation of tn a (∂g∂t)(t,s,x(s))ds constructed from a qth degree Lagrange interpolation polynomial obtained by the use of the grid points t i ,...,t i−q+1 at the integration subinterval [t i−1 ,t i ] for 1 ≤ i ≤ n. Moreover, Q 2 (t,x) is a qth order quadrature scheme for the approximation of t tn (∂g/∂t)(t,s,x(s))ds constructed from a qth degree Lagrange extrapolation polynomial obtained by the use of the grid points t n ,...,t n−q+1 . Then, we consider a stable p-step method for solving the initial value problem y (t) = F(t, y(t)), y(a) = y a given by If x(t n+1 ) denotes the solution at t = t n+1 of the following problem: then an approximation x n+1 of x(t n+1 ) is given by: In the sequel, we will denote by x n , the approximation obtained via (4.6) of x(t n ), where x(t n ) denotes the exact value of the solution of (4.2) at t = t n . The aim of the remaining of this paragraph is to find a global bound of the approximation error | x n − x(t n )|, n = 1,...,N. To this end, we first look for a bound of the local approximation error at the integration step [t n ,t n+1 ] and under the assumption that x k = x(t k ) for all k = 0,...,n. The order of this local error is given by the following proposition. Under the above conditions, we have ) and x n+1 is an approximation of x(t n+1 ) obtained by the use of the p-step method (4.4), then we have Here µ n+1 ∈]t n ,t n+1 [ and M n+1 = c sup tn≤t≤tn+1 | x (p+1) (µ n+1 )|. It remains to bound the quantity |x(t n+1 ) − x(t n+1 )|, this is done as follows. Since By combining (4.12) and (4.13), one concludes that If e(t) = |x(t) − x(t)|, then the previous inequality is written as follows: Finally, by removing the condition x(t i ) = x(t i ) for i = 0,...,n, we obtain a global approximation error bound given by the following proposition. Proof. We first note that since (∂g/∂t)(t,s,x) is Lipschitzian w.r.t. x, then the quatrature scheme Q(t,x,h) for the approximation of t a (∂g/∂t)(t,s,x(s))ds is also Lipschitzian w.r.t. x. Hence, there exists a constant L Q > 0 such sup t∈ [a,b] |Q(t, x,h) − Q(t, y,h)| ≤ L Q max i≤n |x i − y i |. Next, let F(t,x) = f (t) + Q(t,x,h) + g(t,t,x) and note that