We will give an algorithm for obtaining an approximate solution of the boundary value problem for a nonlinear differential equation with ‘maxima’ (1)-(3).

**Theorem 1** *Let the following conditions be fulfilled*:

- 1.
*The functions* ${\alpha}_{0},{\beta}_{0}\in P(h,T)$ *form a couple of quasi*-*lower and quasi*-*upper solutions of* (1)-(3) *such that* ${\alpha}_{0}(t)\le {\beta}_{0}(t)$ *for* $t\in [-h,T]$.

- 2.
*The function* $g\in C(W({\alpha}_{0},{\beta}_{0}),\mathbb{R})$ *is quasi*-*nondecreasing in* $W({\alpha}_{0},{\beta}_{0})$ *and* $g\in L(\gamma ,{\alpha}_{0},{\beta}_{0})$.

- 3.
*The function* $f\in C(\mathrm{\Omega}({\alpha}_{0},{\beta}_{0}),\mathbb{R})$ *and for* $(t,{x}_{1},{y}_{1}),(t,{x}_{2},{y}_{2})\in \mathrm{\Omega}({\alpha}_{0},{\beta}_{0})$ *such that* ${x}_{1}\le {x}_{2}$,

${y}_{1}\le {y}_{2}$ *the inequality* $f(t,{x}_{1},{y}_{1})-f(t,{x}_{2},{y}_{2})\le -M(t)[{x}_{1}-{x}_{2}]-L(t)[{y}_{1}-{y}_{2}]$

*holds*, *where the functions* $M,L\in C([0,T],{\mathbb{R}}_{+})$ *satisfy inequality* (6).

*Then there exist two sequences*
${\{{\alpha}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$
*and*
${\{{\beta}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$
*such that*
- (a)
*The functions* ${\alpha}_{n},{\beta}_{n}\in P(h,T)$ ($n=1,2,\dots $) *and* $({\alpha}_{n},{\beta}_{n})$ *is a couple of quasi*-*lower and quasi*-*upper solutions of boundary value problem* (1)-(3).

- (b)
*The sequence* ${\{{\alpha}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ *is nondecreasing*.

- (c)
*The sequence* ${\{{\beta}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ *is nonincreasing*.

- (d)
*For*
$t\in [-h,T]$
*the inequalities*

*hold*.

- (e)
*Both sequences are uniformly convergent on* $[-h,T]$, *and* $(V,W)$ *is a couple of quasi*-*solutions of boundary value problem* (1)-(3) *in* $S({\alpha}_{0},{\beta}_{0})$.

- (f)
*If additionally the function* $f(t,x,y)$ *is Lipschitz in* $\mathrm{\Omega}({\alpha}_{0},{\beta}_{0})$, *then there exists a unique solution* $u(t)$ *of boundary value problem* (1)-(3) *and* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}(t)={lim}_{n\to \mathrm{\infty}}{\beta}_{n}(t)=V(t)=W(t)=u(t)$ *for* $t\in [-h,T]$.

*Proof* We will give an algorithm for construction of successive approximations to the unknown exact solution of nonlinear boundary value problem (1)-(3).

Assume the functions

${\alpha}_{j}(t)$ and

${\beta}_{j}(t)$,

$j=0,1,\dots ,n$, are constructed. Then consider both initial value problems for the linear differential equations with ‘maxima’

where

${Q}_{n+1}(t)=f(t,{\alpha}_{n}(t),\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s))+M(t){\alpha}_{n}(t)+L(t)\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s)$

and

${P}_{n+1}(t)=f(t,{\beta}_{n}(t),\underset{s\in [t-h,t]}{max}{\beta}_{n}(s))+M(t){\beta}_{n}(t)+L(t)\underset{s\in [t-h,t]}{max}{\beta}_{n}(s).$

According to Lemma 2, initial value problems (12), (13) and (14), (15) have unique solutions ${\alpha}_{n+1},{\beta}_{n+1}\in P(h,T)$.

So, step by step we can construct two sequences of functions ${\{{\alpha}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ and ${\{{\beta}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$.

Now, we will prove by induction that for $j=0,1,2,\dots $ ,

(H1) ${\alpha}_{j+1}(t)\ge {\alpha}_{j}(t)$ and ${\beta}_{j+1}(t)\le {\beta}_{j}(t)$ for $t\in [-h,T]$;

(H2) ${\alpha}_{j+1}(t)\le {\beta}_{j+1}(t)$ for $t\in [-h,T]$;

(H3) $({\alpha}_{j+1},{\beta}_{j+1})$ is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).

Assume the claims (H1)-(H3) are satisfied for $j=0,1,\dots ,n-1$.

We will prove (H1) for $j=n$.

Define the function ${p}_{1}\in P(h,T)$ by the equality ${p}_{1}(t)={\alpha}_{n}(t)-{\alpha}_{n+1}(t)$.

Let

$t\in [-h,0]$. Then according to condition 2 of Theorem 1, the inductive assumption and the definition of the functions

${\alpha}_{n}(t)$,

${\alpha}_{n+1}(t)$, we have

$\begin{array}{rcl}{p}_{1}(t)& =& {\alpha}_{n-1}(0)-{\alpha}_{n}(0)\\ +\frac{1}{\gamma}[g({\alpha}_{n}(0),{\beta}_{n}(T))-g({\alpha}_{n-1}(0),{\beta}_{n-1}(T))]\\ =& {\alpha}_{n-1}(0)-{\alpha}_{n}(0)\\ +\frac{1}{\gamma}[g({\alpha}_{n}(0),{\beta}_{n}(T))-g({\alpha}_{n-1}(0),{\beta}_{n}(T))]\\ +\frac{1}{\gamma}[g({\alpha}_{n-1}(0),{\beta}_{n}(T))-g({\alpha}_{n-1}(0),{\beta}_{n-1}(T))]\\ \le & 0.\end{array}$

(16)

Let

$t\in [0,T]$. From (H1) for

$j=n-1$, condition 3 of Theorem 1, the definition of the functions

${\alpha}_{n}(t)$,

${\alpha}_{n+1}(t)$ and (12), we get

$\begin{array}{rcl}{p}_{1}^{\prime}(t)& \le & -M(t)[{\alpha}_{n}(t)-{\alpha}_{n+1}(t)]\\ -L(t)[\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s)-\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s)].\end{array}$

(17)

Note that for any

$t\in [0,T]$ the following inequality holds:

$\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s)-\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s)\ge \underset{s\in [t-h,t]}{min}[{\alpha}_{n}(s)-{\alpha}_{n+1}(s)].$

(18)

From inequalities (17) and (18) it follows

${p}_{1}^{\prime}(t)\le -M(t){p}_{1}(t)-L(t)\underset{s\in [t-h,t]}{min}{p}_{1}(s).$

According to Lemma 1, we get ${p}_{1}(t)\le 0$ for $t\in [-h,T]$. Thus, ${\alpha}_{n}(t)\le {\alpha}_{n+1}(t)$ for $t\in [-h,T]$.

Define the function

${p}_{2}\in P(h,T)$ by the equality

${p}_{2}(t)={\beta}_{n+1}(t)-{\beta}_{n}(t)$. Then for

$t\in [-h,0]$ we have

$\begin{array}{rcl}{p}_{2}(t)& =& {\beta}_{n}(0)-{\beta}_{n-1}(0)\\ +\frac{1}{\gamma}[g({\beta}_{n-1}(0),{\alpha}_{n-1}(T))-g({\beta}_{n}(0),{\alpha}_{n}(T))]\\ =& {\beta}_{n}(0)-{\beta}_{n-1}(0)\\ +\frac{1}{\gamma}[g({\beta}_{n-1}(0),{\alpha}_{n-1}(T))-g({\beta}_{n}(0),{\alpha}_{n-1}(T))]\\ +\frac{1}{\gamma}[g({\beta}_{n}(0),{\alpha}_{n-1}(T))-g({\beta}_{n}(0),{\alpha}_{n}(T))]\\ \le & 0.\end{array}$

(19)

From equation (

14), the inductive assumption, the definition of the functions

${\beta}_{n}(t)$,

${\beta}_{n+1}(t)$ and condition 3 of Theorem 1, it follows the validity of the inequality

${p}_{2}^{\prime}(t)\le -M(t){p}_{2}(t)-L(t)\underset{s\in [t-h,t]}{min}{p}_{2}(s).$

According to Lemma 1, we get ${p}_{2}(t)\le 0$ for $t\in [-h,T]$, *i.e.*, the claim (H1) is true for $j=n$.

Define the function ${p}_{3}\in P(h,T)$ by the equality ${p}_{3}(t)={\alpha}_{n+1}(t)-{\beta}_{n+1}(t)$.

Let

$t\in [-h,0]$. From condition 2 of Theorem 1, the inductive assumption and the definition of the functions

${\alpha}_{n+1}(t)$,

${\beta}_{n+1}(t)$, we obtain

$\begin{array}{rcl}{p}_{3}(t)& =& {\alpha}_{n}(0)-{\beta}_{n}(0)+\frac{1}{\gamma}[g({\beta}_{n}(0),{\alpha}_{n}(T))-g({\alpha}_{n}(0),{\alpha}_{n}(T))]\\ +\frac{1}{\gamma}[g({\alpha}_{n}(0),{\alpha}_{n}(T))-g({\alpha}_{n}(0),{\beta}_{n}(T))]\\ \le & 0.\end{array}$

Let

$t\in [0,T]$. According to the choice of the functions

${\alpha}_{n+1}(t)$,

${\beta}_{n+1}(t)$, condition 3 of Theorem 1 and inequality

${max}_{s\in [t-h,t]}{\alpha}_{n+1}(s)-{max}_{s\in [t-h,t]}{\beta}_{n+1}(s)\ge {min}_{s\in [t-h,t]}[{\alpha}_{n+1}(s)-{\beta}_{n+1}(s)]$, we get

${p}_{3}^{\prime}(t)\le -M(t){p}_{3}(t)-L(t)\underset{s\in [t-h,t]}{min}{p}_{3}(s).$

According to Lemma 1, it follows ${p}_{3}(t)\le 0$ for $t\in [-h,T]$. Therefore, the claim (H2) is satisfied for $j=n$.

Now, we will prove the claim (H3) for $j=n$.

Let

$t\in [-h,0]$. Then from (13) we get

${\alpha}_{n+1}(t)={\alpha}_{n}(0)-\frac{1}{\gamma}g({\alpha}_{n}(0),{\beta}_{n}(T))={\alpha}_{n+1}(0).$

(20)

From (H1) for

$j=n$, condition 2 of Theorem 1 and the choice of the function

${\alpha}_{n+1}(t)$, we obtain

Let

$t\in [0,T]$. From condition 3 of Theorem 1, inequalities (18) and (H1), we get

$\begin{array}{rcl}{\alpha}_{n+1}^{\prime}(t)& =& -M(t)[{\alpha}_{n+1}(t)-{\alpha}_{n}(t)]\\ -L(t)[\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s)-\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s)]\\ +f(t,{\alpha}_{n+1}(t),\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s))\\ +[f(t,{\alpha}_{n}(t),\underset{s\in [t-h,t]}{max}{\alpha}_{n}(s))\\ -f(t,{\alpha}_{n+1}(t),\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s))]\\ \le & f(t,{\alpha}_{n+1}(t),\underset{s\in [t-h,t]}{max}{\alpha}_{n+1}(s)).\end{array}$

(22)

Similarly, we prove the function ${\beta}_{n+1}(t)$ satisfies inequalities (5). Therefore, the claim (H3) is true for $j=n$. Furthermore, the functions ${\alpha}_{n+1}(t),{\beta}_{n+1}(t)\in S({\alpha}_{n},{\beta}_{n})$.

For any fixed $t\in [-h,T]$, the sequences ${\{{\alpha}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ and ${\{{\beta}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ are nondecreasing and nonincreasing, respectively, and they are bounded by ${\alpha}_{0}(t)$ and ${\beta}_{0}(t)$.

Therefore, both sequences converge pointwisely and monotonically. Let ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}(t)=V(t)$ and ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}(t)=W(t)$ for $t\in [-h,T]$. According to Dini’s theorem, both sequences converge uniformly and the functions $V(t)$, $W(t)$ are continuous. Additionally, the claims (H1), (H2) prove $V,W\in S({\alpha}_{0},{\beta}_{0})$.

Now, we will prove that for any

$t\in [0,T]$ the following equality holds:

$\underset{n\to \mathrm{\infty}}{lim}[\underset{\xi \in [t-h,t]}{max}{\alpha}_{n}(\xi )]=\underset{\xi \in [t-h,t]}{max}[\underset{n\to \mathrm{\infty}}{lim}{\alpha}_{n}(\xi )].$

(23)

For any $t\in [0,T]$, we introduce the notation ${max}_{{\xi}_{t}\in [t-h,t]}{\alpha}_{n}({\xi}_{t})={A}_{n}(t)$. From condition (H1) it follows that for any ${\xi}_{t}\in [t-h,t]$ the inequalities ${\alpha}_{n-1}({\xi}_{t})\le {\alpha}_{n}({\xi}_{t})\le {A}_{n}(t)$ hold and thus, ${A}_{n-1}(t)\le {A}_{n}(t)$, $n=1,2,\dots $ , *i.e.*, the sequence ${\{{A}_{n}(t)\}}_{n=0}^{\mathrm{\infty}}$ is monotone nondecreasing and bounded from above by ${\beta}_{0}(t)$ for any $t\in [-h,T]$. Therefore, there exists the limit $A(t)={lim}_{n\to \mathrm{\infty}}{A}_{n}(t)$.

From the monotonicity of the sequence of the quasi-lower solutions ${\alpha}_{n}(t)$, we get that for ${\xi}_{t}\in [t-h,t]$ the inequality ${\alpha}_{n}({\xi}_{t})\le V({\xi}_{t})$ holds. Let ${\eta}_{t}\in [t-h,t]$ be such that ${max}_{{\xi}_{t}\in [t-h,t]}V({\xi}_{t})=V({\eta}_{t})$.

Assume $V({\eta}_{t})<A({\eta}_{t})$. Then there exists a natural number *N* such that the inequalities $V({\eta}_{t})<{A}_{N}({\eta}_{t})\le A({\eta}_{t})$ hold. Therefore, there exists ${\xi}_{t}\in [{\eta}_{t}-h,{\eta}_{t}]$ such that ${\alpha}_{N}({\xi}_{t})={max}_{{\xi}_{t}\in [{\eta}_{t}-h,{\eta}_{t}]}{\alpha}_{N}({\xi}_{t})={A}_{N}({\eta}_{t})$ or $V({\eta}_{t})<{\alpha}_{N}({\xi}_{t})\le V({\xi}_{t})$. The obtained contradiction proves the assumption is not valid.

Assume $V({\eta}_{t})>A({\eta}_{t})$. According to the definition of the function $V(t)$, it follows that for the fixed number ${\eta}_{t}$, we have ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}({\eta}_{t})=V({\eta}_{t})$. Then there exists a natural number *N* such that $A({\eta}_{t})<{\alpha}_{N}({\eta}_{t})\le V({\eta}_{t})$ and ${max}_{{\eta}_{t}\in [t-h,t]}{\alpha}_{N}({\eta}_{t})={A}_{N}({\eta}_{t})$. Therefore, ${\alpha}_{N}({\eta}_{t})\le {max}_{{\eta}_{t}\in [t-h,t]}{\alpha}_{N}({\eta}_{t})\le A({\eta}_{t})$. The obtained contradiction proves the assumption is not valid.

Therefore, the required equality (23) is fulfilled.

In a similar way, we can prove that for any

$t\in [0,T]$ the equality

$\underset{n\to \mathrm{\infty}}{lim}[\underset{\xi \in [t-h,t]}{max}{\beta}_{n}(\xi )]=\underset{\xi \in [t-h,t]}{max}[\underset{n\to \mathrm{\infty}}{lim}{\beta}_{n}(\xi )]$

(24)

holds.

Take a limit as

$n\to \mathrm{\infty}$ in (13) and get

$V(t)=V(0)-\frac{1}{\gamma}g(V(0),W(T))\phantom{\rule{1em}{0ex}}\text{for}t\in [-h,0].$

(25)

From (25) for $t=0$, we get $g(V(0),W(T))=0$.

Taking a limit in the integral equation equivalent to (12), we obtain the function $V(t)$ satisfies equation (1) for $t\in [0,T]$.

In a similar way, we can prove that $W(t)$ satisfies equation (1) for $t\in [0,T]$ and $g(W(0),V(T))=0$. Therefore, the couple $(V,W)$ is a couple of quasi-solutions of (1)-(3) in $S({\alpha}_{0},{\beta}_{0})$ such that $V(t)\le W(t)$ for $t\in [-h,T]$.

Let the function

$f(t,x,y)$ be Lipschitz. Then if (1) has a solution

$u(t)$, it is unique (see [

11]). In this case,

$V(t)\equiv W(t)$ and for

$t\in [-h,T]$,

$\underset{n\to \mathrm{\infty}}{lim}{\alpha}_{n}(t)=\underset{n\to \mathrm{\infty}}{lim}{\beta}_{n}(t)=V(t)=W(t)=u(t).$

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