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 quasilower and quasiupper 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 quasinondecreasing 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 quasilower and quasiupper 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 quasisolutions 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’
and
where
{Q}_{n+1}(t)=f(t,{\alpha}_{n}(t),\underset{s\in [th,t]}{max}{\alpha}_{n}(s))+M(t){\alpha}_{n}(t)+L(t)\underset{s\in [th,t]}{max}{\alpha}_{n}(s)
and
{P}_{n+1}(t)=f(t,{\beta}_{n}(t),\underset{s\in [th,t]}{max}{\beta}_{n}(s))+M(t){\beta}_{n}(t)+L(t)\underset{s\in [th,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 quasilower and quasiupper solutions of boundary value problem (1)(3).
Assume the claims (H1)(H3) are satisfied for j=0,1,\dots ,n1.
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}_{n1}(0){\alpha}_{n}(0)\\ +\frac{1}{\gamma}[g({\alpha}_{n}(0),{\beta}_{n}(T))g({\alpha}_{n1}(0),{\beta}_{n1}(T))]\\ =& {\alpha}_{n1}(0){\alpha}_{n}(0)\\ +\frac{1}{\gamma}[g({\alpha}_{n}(0),{\beta}_{n}(T))g({\alpha}_{n1}(0),{\beta}_{n}(T))]\\ +\frac{1}{\gamma}[g({\alpha}_{n1}(0),{\beta}_{n}(T))g({\alpha}_{n1}(0),{\beta}_{n1}(T))]\\ \le & 0.\end{array}
(16)
Let t\in [0,T]. From (H1) for j=n1, 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 [th,t]}{max}{\alpha}_{n}(s)\underset{s\in [th,t]}{max}{\alpha}_{n+1}(s)].\end{array}
(17)
Note that for any t\in [0,T] the following inequality holds:
\underset{s\in [th,t]}{max}{\alpha}_{n}(s)\underset{s\in [th,t]}{max}{\alpha}_{n+1}(s)\ge \underset{s\in [th,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 [th,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}_{n1}(0)\\ +\frac{1}{\gamma}[g({\beta}_{n1}(0),{\alpha}_{n1}(T))g({\beta}_{n}(0),{\alpha}_{n}(T))]\\ =& {\beta}_{n}(0){\beta}_{n1}(0)\\ +\frac{1}{\gamma}[g({\beta}_{n1}(0),{\alpha}_{n1}(T))g({\beta}_{n}(0),{\alpha}_{n1}(T))]\\ +\frac{1}{\gamma}[g({\beta}_{n}(0),{\alpha}_{n1}(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 [th,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 [th,t]}{\alpha}_{n+1}(s){max}_{s\in [th,t]}{\beta}_{n+1}(s)\ge {min}_{s\in [th,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 [th,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 [th,t]}{max}{\alpha}_{n+1}(s)\underset{s\in [th,t]}{max}{\alpha}_{n}(s)]\\ +f(t,{\alpha}_{n+1}(t),\underset{s\in [th,t]}{max}{\alpha}_{n+1}(s))\\ +[f(t,{\alpha}_{n}(t),\underset{s\in [th,t]}{max}{\alpha}_{n}(s))\\ f(t,{\alpha}_{n+1}(t),\underset{s\in [th,t]}{max}{\alpha}_{n+1}(s))]\\ \le & f(t,{\alpha}_{n+1}(t),\underset{s\in [th,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 [th,t]}{max}{\alpha}_{n}(\xi )]=\underset{\xi \in [th,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 [th,t]}{\alpha}_{n}({\xi}_{t})={A}_{n}(t). From condition (H1) it follows that for any {\xi}_{t}\in [th,t] the inequalities {\alpha}_{n1}({\xi}_{t})\le {\alpha}_{n}({\xi}_{t})\le {A}_{n}(t) hold and thus, {A}_{n1}(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 quasilower solutions {\alpha}_{n}(t), we get that for {\xi}_{t}\in [th,t] the inequality {\alpha}_{n}({\xi}_{t})\le V({\xi}_{t}) holds. Let {\eta}_{t}\in [th,t] be such that {max}_{{\xi}_{t}\in [th,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 [th,t]}{\alpha}_{N}({\eta}_{t})={A}_{N}({\eta}_{t}). Therefore, {\alpha}_{N}({\eta}_{t})\le {max}_{{\eta}_{t}\in [th,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 [th,t]}{max}{\beta}_{n}(\xi )]=\underset{\xi \in [th,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 quasisolutions 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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