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 form a couple of quasi-lower and quasi-upper solutions of (1)-(3) such that for .
-
2.
The function is quasi-nondecreasing in and .
-
3.
The function and for such that , the inequality
holds, where the functions satisfy inequality (6).
Then there exist two sequences
and
such that
-
(a)
The functions () and is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).
-
(b)
The sequence is nondecreasing.
-
(c)
The sequence is nonincreasing.
-
(d)
For
the inequalities
hold.
-
(e)
Both sequences are uniformly convergent on , and is a couple of quasi-solutions of boundary value problem (1)-(3) in .
-
(f)
If additionally the function is Lipschitz in , then there exists a unique solution of boundary value problem (1)-(3) and for .
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 and , , are constructed. Then consider both initial value problems for the linear differential equations with ‘maxima’
and
where
and
According to Lemma 2, initial value problems (12), (13) and (14), (15) have unique solutions .
So, step by step we can construct two sequences of functions and .
Now, we will prove by induction that for ,
(H1) and for ;
(H2) for ;
(H3) is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).
Assume the claims (H1)-(H3) are satisfied for .
We will prove (H1) for .
Define the function by the equality .
Let . Then according to condition 2 of Theorem 1, the inductive assumption and the definition of the functions , , we have
(16)
Let . From (H1) for , condition 3 of Theorem 1, the definition of the functions , and (12), we get
(17)
Note that for any the following inequality holds:
(18)
From inequalities (17) and (18) it follows
According to Lemma 1, we get for . Thus, for .
Define the function by the equality . Then for we have
(19)
From equation (14), the inductive assumption, the definition of the functions , and condition 3 of Theorem 1, it follows the validity of the inequality
According to Lemma 1, we get for , i.e., the claim (H1) is true for .
Define the function by the equality .
Let . From condition 2 of Theorem 1, the inductive assumption and the definition of the functions , , we obtain
Let . According to the choice of the functions , , condition 3 of Theorem 1 and inequality , we get
According to Lemma 1, it follows for . Therefore, the claim (H2) is satisfied for .
Now, we will prove the claim (H3) for .
Let . Then from (13) we get
(20)
From (H1) for , condition 2 of Theorem 1 and the choice of the function , we obtain
Let . From condition 3 of Theorem 1, inequalities (18) and (H1), we get
(22)
Similarly, we prove the function satisfies inequalities (5). Therefore, the claim (H3) is true for . Furthermore, the functions .
For any fixed , the sequences and are nondecreasing and nonincreasing, respectively, and they are bounded by and .
Therefore, both sequences converge pointwisely and monotonically. Let and for . According to Dini’s theorem, both sequences converge uniformly and the functions , are continuous. Additionally, the claims (H1), (H2) prove .
Now, we will prove that for any the following equality holds:
(23)
For any , we introduce the notation . From condition (H1) it follows that for any the inequalities hold and thus, , , i.e., the sequence is monotone nondecreasing and bounded from above by for any . Therefore, there exists the limit .
From the monotonicity of the sequence of the quasi-lower solutions , we get that for the inequality holds. Let be such that .
Assume . Then there exists a natural number N such that the inequalities hold. Therefore, there exists such that or . The obtained contradiction proves the assumption is not valid.
Assume . According to the definition of the function , it follows that for the fixed number , we have . Then there exists a natural number N such that and . Therefore, . 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 the equality
(24)
holds.
Take a limit as in (13) and get
(25)
From (25) for , we get .
Taking a limit in the integral equation equivalent to (12), we obtain the function satisfies equation (1) for .
In a similar way, we can prove that satisfies equation (1) for and . Therefore, the couple is a couple of quasi-solutions of (1)-(3) in such that for .
Let the function be Lipschitz. Then if (1) has a solution , it is unique (see [11]). In this case, and for ,
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