1 Introduction

Integro-differential equations are the mathematical models of many phenomena of physics, biology, hydromechanics, chemistry, etc. In general, it is impossible to find the exact solutions of the problems involving these equations, especially when they are nonlinear. Therefore, many analytical approximate methods and numerical methods have been developed for these equations (see, e.g., [1, 2, 11,12,13,14,15,16,17,18]).

Below, we mention some works concerning the solution methods for integro-differential equations. First, it is worthy to mention the recent work of Tahernezhad and Jalilian in 2020 [15]. In this work the authors consider the second-order linear problem

$$ \begin{array}{@{}rcl@{}} u^{\prime\prime}(x)&+&p(x)u^{\prime}(x)+q(x)u(x)=f(x)+ {{\int}_{a}^{b}} k(x,t)u(t) dt , \quad a<x<b,\\ u(a)&=&\alpha , u(b)=\beta, \end{array} $$

where p(x),q(x),k(x,t) are sufficiently smooth functions.

Using non-polynomial spline functions, namely, the exponential spline functions, the authors constructed the numerical solution of the problem and proved that the error of the approximate solution is O(h2), where h is the grid size on [a,b]. Before [15] there are interesting works of Chen et al. [2, 3], where the authors used a multiscale Galerkin method for constructing a approximate solution of the above second-order problem, for which the computed convergence rate is two.

Besides the researches evolving the second-order integro-differential equations, recently many authors have been interested in fourth-order integro-differential equations due to their wide applications. We first mention the work of Singh and Wazwaz [13]. In this work the authors developed a technique based on the Adomian decomposition method with the Green’s function for constructing a series solution of the nonlinear Voltera equation associated with the Dirichlet boundary conditions

$$ \begin{array}{@{}rcl@{}} y^{(4)}(x)&=&g(x)+ {{\int}_{0}^{x}} k(x,t)f(y(t)) dt , \quad 0<x<b, \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} y(0)&=&\alpha_{1} , y^{\prime}(0)=\alpha_{2}, y(b)=\alpha_{3} , y^{\prime}(b)=\alpha_{4}. \end{array} $$
(2)

Under some conditions it was proved that the series solution converges as a geometric progression.

For the linear Fredholm IDE [1]

$$ \begin{array}{@{}rcl@{}} y^{(4)}(x)+\alpha y^{\prime\prime}(x) +\beta y (x)- {{\int}_{a}^{b}} K(x,t)y(t) dt = f(x) , \quad a<x<b, \end{array} $$

with the Navier boundary conditions, the difference method and the trapezium rule are used to design the corresponding linear system of algebraic equations. A new variant called the Modified Arithmetic Mean iterative method is proposed for solving the latter system, but the error estimate of the method is not obtained. The boundary value problem for the nonlinear IDE

$$ \begin{array}{@{}rcl@{}} y^{(4)}(x)&-&\varepsilon y^{\prime\prime}(x)-\frac{2}{\pi}\Big ({\int}_{0}^{\pi} |y^{\prime}(t)|^{2} dt \Big )y^{\prime\prime}(x) =p(x) , \quad 0<x<\pi, \\ y(0)&=&0 , y(\pi)=0, y^{\prime\prime}(0)=0 , y^{\prime\prime}(\pi)=0 \end{array} $$

was considered in [10, 18], where the authors constructed approximate solutions by the iterative and spectral methods, respectively. Recently, Dang and Nguyen [6] studied the existence and uniqueness of solution and constructed iterative method for finding the solution for the IDE

$$ \begin{array}{@{}rcl@{}} u^{(4)}(x)&-&M\Big ({{\int}_{0}^{L}} |u^{\prime}(t)|^{2} dt \Big )u^{\prime\prime}(x) = f(x,u,u^{\prime},u^{\prime\prime},u^{\prime\prime\prime}) , \quad 0<x<L, \\ u(0)&=&0 , u(L)=0, u^{\prime\prime}(0)=0 , u^{\prime\prime}(L)=0, \end{array} $$

where M is a continuous non-negative function.

Very recently, Wang [17] considered the problem

$$ \begin{array}{@{}rcl@{}} y^{(4)}(x)& =&f(x,y(x), {{\int}_{0}^{1}} k(x,t) y(t) dt ), 0<x<1,\\ y(0)&=&0 , y(1)=0, y^{\prime\prime}(0)=0 , y^{\prime\prime}(1)=0. \end{array} $$
(3)

This problem can be seen as a generalization of the linear fourth-order problem

$$ \begin{array}{@{}rcl@{}} u^{(4)}(x)&+&Mu(x) -N {{\int}_{0}^{1}} k(x,t) u(t) dt )=p(x), 0<x<1,\\ u(0)&=&0 , u(1)=0, u^{\prime\prime}(0)=0 , u^{\prime\prime}(1)=0, \end{array} $$

where M,N are constants, p ∈ C[0,1]. The latter problem arises from the models for suspension bridges [19, 20], quantum theory [21].

Using the monotone method and a maximum principle, Wang constructed the sequences of functions, which converge to the extremal solutions of the problem (3).

Motivated by the above facts, in this paper we consider an extension of the above problem, namely, the problem

$$ \begin{array}{@{}rcl@{}} u^{(4)}(x)& =&f(x,u(x),u^{\prime}(x), {{\int}_{0}^{1}} k(x,t) u(t) dt ),\\ u(0)&=&0 , u(1)=0, u^{\prime\prime}(0)=0 , u^{\prime\prime}(1)=0, \end{array} $$
(4)

where the function f(x,u,v,z) and k(x,t) are assumed to be continuous. The presence of an extra \(u^{\prime }\) in the right hand side function does not allow to use the argument in [17] to study the existence of solutions of the problem. Here, using the method developed in our previous papers [4,5,6,7,8,9] we establish the existence and uniqueness of the solution and propose an iterative method at both continuous and discrete levels for finding the solution. The second-order convergence of the method is proved. The theoretical results are illustrated by some examples.

2 Existence results

Using the methodology in [4,5,6,7,8,9] we introduce the operator A defined in the space of continuous functions C[0,1] by the formula

$$ (A\varphi ) (x)= f(x,u(x),u^{\prime}(x),{{\int}_{0}^{1}} k(x,t) u(t) dt ), $$
(5)

where u(x) is the solution of the boundary value problem

$$ \begin{array}{@{}rcl@{}} u^{\prime\prime\prime\prime}&=&\varphi (x), 0<x<1,\\ u(0)&=&u^{\prime\prime}(0)=u(1)=u^{\prime\prime}(1)=0 . \end{array} $$
(6)

It is easy to verify the following lemma.

Lemma 1

If the function φ is a fixed point of the operator A, i.e., φ is the solution of the operator equation

$$ A\varphi = \varphi , $$
(7)

where A is defined by (5)–(6) then the function u(x) determined from the BVP (6) is a solution of the BVP (4). Conversely, if the function u(x) is the solution of the BVP (4) then the function

$$ \varphi (x)= f(x,u(x),u^{\prime}(x),{{\int}_{0}^{1}} k(x,t) u(t) dt) $$

satisfies the operator (7).

Due to the above lemma we shall study the original BVP (4) via the operator (7). Before doing this we notice that the BVP (6) has a unique solution representable in the form

$$ u(x)={{\int}_{0}^{1}} G_{0}(x,s)\varphi(s)ds, \quad 0<t<1, $$
(8)

where

$$ \begin{array}{@{}rcl@{}} G_{0}(x,s)=\frac{1}{6} \begin{cases} s(x-1)(x^{2}-x+s^{2}), \quad & 0\leq s\leq x\leq 1 \\ x(s-1)(s^{2}-s+x^{2}), & 0\leq x\leq s\leq 1 \end{cases} \end{array} $$
(9)

is the Green’s function of the operator \(u^{\prime \prime \prime \prime }(t)=0\) associated with the homogeneous boundary conditions \(u(0)=u^{\prime \prime }(0)= u(1)=u^{\prime \prime }(1)=0\). Differentiating both sides of (8) gives

$$ \begin{array}{@{}rcl@{}} u^{\prime}(x) & ={{\int}_{0}^{1}} G_{1}(x,s)\varphi(s)ds, \end{array} $$
(10)

where

$$ G_{1}(x,s) =\frac{1}{6} \begin{cases} s(3x^{2}-6x+s^{2}+2), \quad & 0\leq s\leq x\leq 1, \\ (s-1)(3x^{2}-2s+s^{2}),& 0\leq x\leq s\leq 1 . \end{cases} $$
(11)

Set

$$ \begin{array}{@{}rcl@{}} M_{0} & =&\max_{0\leq x\leq 1} {{\int}_{0}^{1}} |G_{0}(x,s)|ds ,\\ M_{1} & =&\max_{0\leq x\leq 1} {{\int}_{0}^{1}} |G_{1}(x,s)|ds ,\\ M_{2} & =&\max_{0\leq x\leq 1} {{\int}_{0}^{1}} |k(x,s)|ds \end{array} $$
(12)

It is easy to obtain

$$ M_{0}= \frac{5}{384} , M_{1}= \frac{1}{24}. $$
(13)

Now for any positive number M, we define the domain

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} \mathcal{D}_{M} = \{(x,u,v,z) | &0\leq x\leq 1, |u|\leq M_{0}M,\\ &|v|\leq M_{1}M, |z|\leq M_{0}M_{2}M \}. \end{array} \end{array} $$
(14)

As usual, we denote by B[0,M] the closed ball centered at 0 with radius M in the space C[0,1], i.e.,

$$ \begin{array}{@{}rcl@{}} B[0, M] =\lbrace u\in C[0,1] \vert \|u\| \le M \rbrace , \end{array} $$

where \(\|u\| =\max \limits _{0\le x \le 1} |u(x)|\).

Theorem 1 (Existence and uniqueness)

Suppose that the function k(x,t) is continuous in the square [0,1] × [0,1] and there exist numbers M > 0, L0,L1,L2 ≥ 0 such that:

  1. (i)

    The function f(x,u,v,z) is continuous in the domain \(\mathcal {D}_{M}\) and |f(x,u,v,z)|≤ M, \( \forall (x,u,v,z)\in \mathcal {D}_{M}\).

  2. (ii)

    \(|f(x_{2},u_{2},v_{2},z_{2})-f(x_{1},u_{1},v_{1},z_{1})| \leq L_{0} |u_{2}-u_{1}|+ L_{1} |v_{2}-v_{1}|+ L_{2} |z_{2}-z_{1}|, \forall (x_{i},u_{i},v_{i},z_{i})\in \mathcal {D}_{M}, i=1,2.\)

  3. (iii)

    q = L0M0 + L1M1 + L2M0M2 < 1.

Then the problem (4) has a unique solution u ∈ C4[0,1] satisfying |u(x)|≤ M0M, \( |u^{\prime }(x)| \leq M_{1}M\) for any 0 ≤ x ≤ 1.

Proof

Under the assumptions of the theorem we shall prove that the operator A is a contraction mapping in the closed ball B[O,M]. Then the operator (7) has a unique solution u ∈ C(4)[0,1] and this implies the existence and uniqueness of solution of the BVP (4).

Indeed, take φ ∈ B[O,M]. Then the problem (6) has a unique solution of the form (8). From there and (12) we obtain |u(x)|≤ M0∥φ∥ for all x ∈ [0,1]. Analogously, we have \(\| u^{\prime }(x)\| \le M_{1} \|\varphi \|\) for all x ∈ [0,1]. Denote by K the integral operator defined by

$$ (Ku)(x) = {{\int}_{0}^{1}} k(x,t) u(t) dt.$$

Then from the last equation in (12) we have the estimate |(Ku)(x)|≤ M0M2∥φ∥, x ∈ [0,1]. Thus, if φ ∈ B[O,M], i.e., ∥φ∥≤ M then for any x ∈ [0,1] we have

$$ |u(x)| \le M_{0}M, |u^{\prime}(x)| \le M_{1}M, |(Ku) (x)|\le M_{0}M_{2}M. $$

Therefore, \((x, u(x),u^{\prime }(x), (Ku) (x))\in \mathcal {D}_{M}\). By the assumption (i) there is

$$ |f(x, u(x),u^{\prime}(x), (Ku) (x))| \le M \quad \forall x\in [0,1]. $$

Hence, |(Aφ)(x)|≤ M,∀x ∈ [0,1] and ∥Aφ∥≤ M. It means that A maps B[O,M] into itself.

Next, take φ1,φ2 ∈ B[O,M]. Using the assumption (ii) and (iii) it is easy to obtain

$$ \begin{array}{@{}rcl@{}} \|A\varphi_{2} - A\varphi_{1}\| \le (L_{0}M_{0}+L_{1}M_{1}+L_{2}M_{0}M_{2})\|\varphi_{2} -\varphi_{1} \| =q \|\varphi_{2} -\varphi_{1} \|. \end{array} $$

Since q < 1 the operator A is a contraction in B[O,M]. This completes the proof of the theorem. □

Now, in order to study positive solutions of the BVP (4) we introduce the domain

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} \mathcal{D}_{M}^{+} = \{(x,u,v,z) | &0\leq x\leq 1, 0\le u\leq M_{0}M,\\ &|v|\leq M_{1}M, |z|\leq M_{0}M_{2}M \}. \end{array} \end{array} $$
(15)

and denote

$$ S_{M} = \{\varphi \in C[0,1], 0 \le \varphi (x) \le M \}. $$

Theorem 2 (Positivity of solution)

Suppose that the function k(x,t) is continuous in the square [0,1] × [0,1] and there exist numbers M > 0, L0,L1,L2 ≥ 0 such that:

  1. (i)

    The function f(x,u,v,z) is continuous in the domain \(\mathcal {D}_{M}^{+}\) and \(0\le f(x,u,v,z)\leq M, \forall (x,u,v,z)\in \mathcal {D}_{M}^{+}\) and f(x,0,0,0)≢0.

  2. (ii)

    |f(x2,u2,v2,z2)−f(x1,u1,v1,z1)|≤ L0|u2 −u1|+L1|v2 −v1|+L2|z2 −z1|, \(\forall (x_{i},u_{i},v_{i},z_{i})\in \mathcal {D}_{M}^{+}, i=1,2.\)

  3. (iii)

    q = L0M0 + L1M1 + L2M0M2 < 1.

Then the problem (4) has a unique positive solution u ∈ C4[0,1] satisfying 0 ≤ u(x) ≤ M0M, \( |u^{\prime }(x)| \leq M_{1}M\) for any 0 ≤ x ≤ 1.

Proof

Similarly to the proof of Theorem 1, where instead of \(\mathcal {D}_{M}\) and B[O;M] there stand \(\mathcal {D}_{M}^{+}\) and SM, we conclude that the problem has a non-negative solution. Due to the condition f(x,0,0,0)≢0, this solution must be positive. □

3 Numerical method

In this section we suppose that all the conditions of Theorem 1 are satisfied. Then the problem (4) has a unique solution. For finding this solution consider the following iterative method:

  1. 1.

    Given

    $$ \varphi_{0}(x)=f(x,0,0,0). $$
    (16)
  2. 2.

    Knowing φm(x) (m = 0,1,...) compute

    $$ \begin{array}{@{}rcl@{}} u_{m}(x) &=& {{\int}_{0}^{1}} G_{0}(x,t)\varphi_{m}(t)dt ,\\ v_{m}(x) &=& {{\int}_{0}^{1}} G_{1}(x,t)\varphi_{m}(t)dt ,\\ z_{m}(x) &=& {{\int}_{0}^{1}} k(x,t)u_{m}(t) dt . \end{array} $$
    (17)
  3. 3.

    Update

    $$ \varphi_{m+1}(x) = f(x,u_{m}(x),v_{m}(x),z_{m}(x)). $$
    (18)

This iterative method indeed is the successive iterative method for finding the fixed point of operator A. Therefore, it converges with the rate of geometric progression and there holds the estimate

$$ \|\varphi_{m} -\varphi\| \leq \frac{q^{m}}{1-q}\|\varphi_{1} -\varphi_{0}\| = p_{m} d, $$

where φ is the fixed point of the operator A and

$$ p_{m}=\tfrac{q^{m}}{1-q}, d=\|\varphi_{1} -\varphi_{0}\|. $$
(19)

This estimate implies the following result of the convergence of the iterative method (16)–(18).

Theorem 3

Under the conditions of Theorem 1 the iterative method (16)–(18) converges and for the approximate solution uk(t) there hold estimates

$$ \begin{array}{@{}rcl@{}} \|u_{m}-u\| &\leq M_{0}p_{m}d, \|u^{\prime}_{m}-u^{\prime}\| \leq M_{1}p_{m}d, \end{array} $$

where u is the exact solution of the problem (4), pm and d are defined by (19).

To numerically realize the above iterative method we construct a corresponding discrete iterative method. For this purpose cover the interval [0,1] by the uniform grid \(\bar {\omega }_{h}=\{x_{i}=ih, h=1/N, i=0,1,...,N \}\) and denote by Φm(x),Um(x),Vm(x),Zm(x) the grid functions, which are defined on the grid \(\bar {\omega }_{h}\) and approximate the functions φm(x),um(x),vm(x),zm(x) on this grid.

Consider now the following discrete iterative method:

  1. 1.

    Given

    $$ {{\varPhi}}_{0}(x_{i})=f(x_{i},0,0,0),\ i=0,...,N. $$
    (20)
  2. 2.

    Knowing Φm(xi),m = 0,1,...;i = 0,...,N, compute approximately the definite integrals (17) by the trapezium formulas

    $$ \begin{array}{@{}rcl@{}} U_{m}(x_{i}) &=& \sum\limits_{j=0}^{N} h\rho_{j} G_{0}(x_{i},x_{j}){{\varPhi}}_{m}(x_{j}) ,\\ V_{m}(x_{i}) &=& \sum\limits_{j=0}^{N} h\rho_{j} G_{1}(x_{i},x_{j}){{\varPhi}}_{m}(x_{j}) ,\\ Z_{m}(x_{i}) &=& {\sum}_{j=0}^{N} h\rho_{j} k(x_{i},x_{j})U_{m}(x_{j}) , i=0,...,N, \end{array} $$
    (21)

    where ρj is the weight of the trapezium formula, namely

    $$ \rho_{j} = \begin{cases} 1/2, j=0,N\\ 1, j=1,2,...,N-1. \end{cases} $$
  3. 3.

    Update

    $$ {{\varPhi}}_{m+1}(x_{i}) = f(x_{i},U_{m}(x_{i}),V_{m}(x_{i}),Z_{m}(x_{i})). $$
    (22)

In order to get the error estimates for the approximate solution for u(t) and its derivatives on the grid we need some following auxiliary results.

Proposition 1

Assume that the function f(t,u,v,z) has all continuous partial derivatives up to second order in the domain \(\mathcal {D}_{M}\) and the kernel function k(x,t) also has all continuous partial derivatives up to second order in the square [0,1] × [0,1]. Then for the functions φm(x),um(x),vm(x),zm(x),m = 0,1,..., constructed by the iterative method (16)–(18) we have φm(x) ∈ C2[0,1], um(x) ∈ C6[0,1], vm(x) ∈ C5[0,1],zm(x) ∈ C2[0,1].

Proof

We prove the proposition by induction. For k = 0, by the assumption on the function f we have φ0(t) ∈ C2[0,1] since φ0(x) = f(x,0,0,0). Taking into account

$$ \begin{array}{@{}rcl@{}} u_{0}(x) = {{\int}_{0}^{1}} G_{0}(x,t)\varphi_{0}(t)dt \end{array} $$

we deduce that the function u0(x) is the solution of the BVP

$$ \begin{array}{@{}rcl@{}} u_{0}^{(4)}(x) &=& \varphi_{0} (x),\quad x\in (0,1),\\ u_{0}(0)&=&u_{0}(1)= u_{0}^{\prime\prime}(0)=u_{0}^{\prime\prime}(1)=0. \end{array} $$

Therefore, u0(x) ∈ C6[0,1]. It implies that v0(x) ∈ C5[0,1] because \(v_{0}(x)= u_{0}^{\prime }(x)\). Since by assumptions k(x,t) has all continuous derivatives up to second order, the function \(z_{0}(x) ={{\int \limits }_{0}^{1}} k(x,t) u_{0}(t)dt\) belongs to C2[0,1].

Now suppose φm(x) ∈ C2[0,1], um(x) ∈ C6[0,1], vm(x) ∈ C5[0,1],zm(x) ∈ C2[0,1]. Then, because φm+ 1(x) = f(x,um(x),vm(x),zm(x)) and the functions f by the assumption has continuous derivative in all variables up to order 2, it follows that φm+ 1(x) ∈ C2[0,1]. Repeating the same argument as for φ0(x) above we obtain that um+ 1(x) ∈ C6[0,1], vm+ 1(x) ∈ C5[0,1],zm+ 1(x) ∈ C2[0,1] Thus, the proposition is proved. □

Proposition 2

For any function φ(x) ∈ C2[0,1] there holds the estimate

$$ {{\int}_{0}^{1}} G_{n} (x_{i},t) \varphi (t) dt = \sum\limits_{j=0}^{N} h\rho_{j} G_{n}(x_{i},t_{j})\varphi(t_{j}) +O(h^{2}) \quad (n=0,1). $$
(23)

Proof

The above estimate is obvious in view of the error estimate of the compound trapezium formula because the functions Gn(xi,t)(n = 0,1) are continuous at tj and are polynomials in the intervals [0,tj] and [tj,1]. □

Proposition 3

Under the assumptions of Proposition 1, for any m = 0,1,... there hold the estimates

$$ \|{{\varPhi}}_{m} -\varphi_{m} \|= O(h^{2}),\quad \|U_{m} -u_{m} \|=O(h^{2}), $$
(24)
$$ \|V_{m}-v_{m} \|=O(h^{2}), \|Z_{m} -z_{m} \|=O(h^{2}). $$
(25)

where \( \|.\|= \|.\|_{\bar {\omega }_{h}}\) is the max-norm of function on the grid \(\bar {\omega }_{h}\).

Proof

We prove the proposition by induction. For m = 0 we have immediately ∥Φ0 − φ0∥ = 0. Next, by the first equation in (17) and Proposition 2 we have

$$ u_{0}(x_{i})={{\int}_{0}^{1}} G_{0} (x_{i},t) \varphi_{0} (t) dt = \sum\limits_{j=0}^{N} h\rho_{j} G_{0}(x_{i},t_{j})\varphi_{0}(t_{j})+O(h^{2}) $$
(26)

for any i = 0,...,N. On the other hand, in view of the first equation in (21) we have

$$ U_{0}(x_{i})= \sum\limits_{j=0}^{N} h\rho_{j} G_{0}(x_{i},t_{j}){{\varPhi}}_{0}(t_{j}). $$
(27)

Therefore, |U0(ti) − u0(ti)| = O(h2) because Φ0(tj) = φ0(tj) = f(tj,0,0,0). Consequently, ∥U0 − u0∥ = O(h2). Similarly, we have

$$ \|V_{0} -v_{0} \|=O(h^{2}). $$
(28)

Next, by the trapezium formula we have

$$ \begin{array}{@{}rcl@{}} z_{0}(x_{i})= {{\int}_{0}^{1}} k(x_{i},t)u_{0}(t) dt = \sum\limits_{j=0}^{N} h\rho_{j} k(x_{i},t_{j})u_{0}(t_{j}) +O(h^{2}), \end{array} $$

while by the third equation in (21) we have

$$ \begin{array}{@{}rcl@{}} Z_{0}(x_{i}) = \sum\limits_{j=0}^{N} h\rho_{j} k(x_{i},t_{j})U_{0}(t_{j}) , i=0,...,N. \end{array} $$

Therefore,

$$ \begin{array}{@{}rcl@{}} \Big | Z_{0}(x_{i}) - z_{0}(x_{i}) \Big | &=& \Big | \sum\limits_{j=0}^{N} h\rho_{j} k(x_{i},t_{j}) (U_{0}(t_{j}) -u_{0}(t_{j}) ) \Big | +O(h^{2}) \\ &\le& \sum\limits_{j=0}^{N} h\rho_{j} |k(x_{i},t_{j})| |U_{0}(t_{j}) -u_{0}(t_{j}) | +O(h^{2}) \\ &\le &C h^{2} \sum\limits_{j=0}^{N} h\rho_{j} |k(x_{i},t_{j})| + O(h^{2}) \\ &\le &CC_{1} h^{2} \sum\limits_{j=0}^{N} h\rho_{j} + O(h^{2}) = O(h^{2}) \end{array} $$

because |U0(tj) − u0(tj)|≤ Ch2,|k(xi,tj)|≤ C1, where C,C1 are some constants.

Now suppose that (24) and (25) are valid for m ≥ 0. We shall show that these estimates are valid for m + 1. By the Lipschitz condition of the function f and the estimates (24) and (25) it is easy to obtain the estimate

$$\|{{\varPhi}}_{m+1} -\varphi_{m+1} \|= O(h^{2}).$$

Now from the first equation in (17) by Proposition 2 we have

$$ u_{m+1}(x_{i})={{\int}_{0}^{1}} G_{0} (x_{i},t) \varphi_{m+1} (t) dt = {\sum}_{j=0}^{N} h\rho_{j} G_{0}(x_{i},x_{j})\varphi_{m+1}(x_{j})+O(h^{2}). $$

On the other hand by the first formula in (21) we have

$$ U_{m+1}(x_{i}) = {\sum}_{j=0}^{N} h\rho_{j} G_{0}(x_{i},x_{j}){{\varPhi}}_{m+1}(x_{j}). $$

From this equality and the above estimates we obtain the estimate

$$\|U_{m+1} -u_{m+1} \|=O(h^{2}). $$

Similarly, we obtain

$$\|V_{m+1} -v_{m+1}\|=O(h^{2}), \|Z_{m+1} -z_{k+1}\|=O(h^{2}). $$

Thus, by induction we have proved the proposition. □

Now combining Proposition 3 and Theorem 3 results in the following theorem.

Theorem 4

Assume that all the conditions of Theorem 1 and Proposition 1 are satisfied. Then, for the approximate solution of the problem (4) obtained by the discrete iterative method on the uniform grid with grid size h there hold the estimates

$$ \|U_{m}-u\| \leq M_{0} p_{m}d +O(h^{2}), \|V_{m}-u^{\prime}\| \leq M_{2}p_{m}d +O(h^{2}). $$
(29)

Proof

The first above estimate is easily obtained if representing

$$ U_{m}(t_{i})-u(t_{i})= (u_{m}(t_{i})-u(t_{i}))+(U_{m}(t_{i})-u_{m}(t_{i})) $$

and using the first estimate in Theorem 3 and the second estimate in (24). The remaining estimate is obtained in the same way. Thus, the theorem is proved. □

4 Examples

Example 1

Consider the problem (4) with

$$ \begin{array}{@{}rcl@{}} &&k(x,t)= e^{x} \sin (\pi t), \quad (x,t)\in [0,1] \times [0,1],\\ &&f(x,u(x),u^{\prime}(x) , {{\int}_{0}^{1}} k(x,t) u(t) dt) = u^{2}(x) {{\int}_{0}^{1}} k(x,t)u(t) dt + u(x) u^{\prime}(x) \\ &&-\frac{1}{2} e^{x} \sin^{2} (\pi x) + \pi^{4} \sin (\pi x) - \frac{\pi}{2} \sin (2 \pi x). \end{array} $$

In this case

$$ f(x,u,v,z)=u^{2}z+uv-\frac{1}{2} e^{x} \sin^{2} (\pi x) + \pi^{4} \sin (\pi x) - \frac{\pi}{2} \sin (2 \pi x) $$

and \(M_{2} = \frac {2 e}{\pi } \). It is possible to verify that the function \(u= {\sin \limits } (\pi x)\) is the exact solution of the problem. In the domain \(\mathcal {D}_{M}\) defined by

$$ \begin{array}{@{}rcl@{}} \mathcal{D}_{M} = \{(x,u,v,z) | &0\leq x\leq 1, |u|\leq M_{0} M, |u^{\prime}| \le M_{1}M, |z|\leq M_{0}M_{2} M \} \end{array} $$

we have

$$ \begin{array}{@{}rcl@{}} |f(x,u,v,z)| \le {M_{0}^{3}}M_{2} M^{3} + M_{0}M_{1} M^{2} + \pi^{4} + \frac{\pi}{2}+\frac{e}{2}. \end{array} $$

It is possible to verify that for M = 113 all the conditions of Theorem 1 are satisfied with L0 = 12.2010,L1 = 1.4714,L2 = 2.1649,q = 0.2690. Therefore, the problem has a unique solution u(x) satisfying the estimates \(|u(x)| \le 1.4714 , |u^{\prime }(x)| \le 4.7083 \). These theoretical estimates are somewhat greater than the exact estimates \(|u(x)| \le 1 , |u^{\prime }(x)| \le \pi \).

Below we report the numerical results by the discrete iterative method (20)–(22) for the problem. In Tables 1 and 2 we use the notation Error = ∥Um − u∥, where u is the exact solution of the problem.

Table 1 The convergence in Example 1 for the stopping criterion ∥Um − u∥≤ h2
Full size table
Table 2 The convergence in Example 1 for the stopping criterion ∥Φm −Φm− 1∥≤ 10− 10
Full size table

It is interesting to notice that if taking the stopping criterion ∥Φm −Φm− 1∥≤ 10− 10 instead of ∥Um − u∥≤ h2 then we obtain better accuracy of the approximate solution with more iterations. See Table 2.

From Table 2 we see that the accuracy of the approximate solution is near O(h4) although by the proved theory it is only O(h2).

Example 2

(Example 4.2 in [17]) Consider the nonlinear fourth-order BVP

$$ \begin{array}{ll} u^{(4)}(x)& =\sin (\pi x)[(2-u^{2}(x)) {{\int}_{0}^{1}} tu(t) dt +1], x\in (0,1)\\ u(0)&=0 , u(1)=0, u^{\prime\prime}(0)=0 , u^{\prime\prime}(1)=0. \end{array} $$
(30)

This is the problem (4) with

$$ \begin{array}{@{}rcl@{}} &k(x,t)=\sin (\pi x) t, \quad (x,t)\in [0,1] \times [0,1],\\ &f(x,u(x), u^{\prime}(x), {{\int}_{0}^{1}} k(x,t) u(t) dt) = (2-u^{2}(x)) {{\int}_{0}^{1}} \sin (\pi x)tu(t) dt +\sin (\pi x). \end{array} $$

So, \(f(x,u,v,z) = (2-u^{2})z+ {\sin \limits } (\pi x)\). It is easy to see that \(M_{2} =\max \limits _{0\leq x\leq 1} {{\int \limits }_{0}^{1}} |k(x,t)|dt = \frac {1}{2}\). Since M0 and M1 are given by (13) we define

$$ \begin{array}{@{}rcl@{}} \mathcal{D}_{M} = \{(x,u,v,z) | &0\leq x\leq 1, |u|\leq \frac{5}{384} M, |v| \le \frac{1}{24}M, |z|\leq \frac{5}{768} M \}. \end{array} $$
(31)

It is possible to verify that for M = 1.1 all the assumptions of Theorem 1 are satisfied with L0 = 2.0515e − 04,L1 = 0,L2 = 2,q = 0.0130. Therefore, the problem (30) has a unique solution satisfying \(|u(x)| \le 0.0143, |u^{\prime }(x)|\le 0.0458\).

It is worth emphasizing that in [ 17 ] by the monotone method the author could only prove the convergence of the iterative sequences to extremal solutions of the problem but not the existence and uniqueness of solution.

Using the discrete iterative method (20)–(22) on the grid with grid step h = 0.01 and the stopping criterion ∥Φm −Φm− 1∥≤ 10− 10 we found an approximate solution after 7 iterations. The graph of this approximate solution is depicted in Fig. 1.

Fig. 1
figure 1

The graph of the approximate solution in Example 2

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5 Conclusion

In this paper we have established the existence and uniqueness of the solution for a fourth-order nonlinear integro-differential equation with the Navier boundary conditions and proposed an iterative method at both continuous and discrete levels for finding the solution. The second order of accuracy of the discrete method has been proved. Some examples, where the exact solution is known and is not known, demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method. It should be emphasized that for the example of Wang in [17] we have established the existence and uniqueness of solution and found it numerically but Wang could prove only the convergence of the iterative sequences constructed by the monotone method to extremal solutions.

The method used in this paper with appropriate modifications can be applied to nonlinear integro-differential equations of any order with other boundary conditions and more complicated nonlinear terms. This is the direction of our research in the future.