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On a certain way of proving the solvability for boundary value problems

Abstract

A certain way of replacing a given boundary value problem by another one, a solution of which solves also the original problem, is considered.

MSC:34B15.

Consider the solvability of the boundary value problem (BVP)

( φ ( t , x , x ) ) =f ( t , x , x ) ,tI=[a,b],
(1)
H 1 x= h 1 , H 2 x= h 2 ,αxβ,
(2)

where φC(I× R 2 ,R) is strictly increasing in x for fixed t and x, f:I× R 2 R satisfies the Caratheodory conditions, that is, f(t,,) is measurable in I for fixed x, x R, f(,x, x ) is continuous on R 2 for fixed tI, and for any compact set P R 2 there exists function gL(I,R) such, that for any (t,x, x )I×P, the estimate |f(t,x, x )|g(t) holds, H 1 , H 2 C( C 1 (I,R),R), h 1 , h 2 R, α is the lower function, β the upper function.

This boundary value problem is replaced by another one, which is dependent on the parameter M( M 0 ,+), M 0 >0,

( φ M ( t , x , x ) ) = f M ( t , x , x ) , t I = [ a , b ] , H 1 x = h 1 , H 2 x = h 2 , α x β ,
(3)

where φ M C(I× R 2 ,R) is strictly increasing in x for fixed t and x, and f:I× R 2 R satisfies the Caratheodory conditions.

Definition 1 A function x C 1 (I,R) is a solution of (1), if φ(t,x(t), x (t)) is absolutely continuous on I and (1) is satisfied almost everywhere on I.

We provide below definitions of generalized upper and lower functions and the generalized solution along with Theorem 1 from [13]. This is needed to prove the main result.

Definition 2 The class B B + (I,R) consists of functions α:IR, which possess the property: for any t(a,b] there exist the left derivative α l (t) and the limit lim τ t α l (τ), and α l (t) lim τ t α l (τ); for any t[a,b) there exist the right derivative α r (t) and the limit lim τ t + α r (τ), and α r (t) lim τ t + α r (τ), and, for any t(a,b), α l (t) α r (t).

The class B B (I,R) consists of functions β:IR, which possess the following property: for any t(a,b] there exist the left derivative β l (t) and the limit lim τ t β l (τ), and β l (t) lim τ t β l (τ); for any t[a,b) there exist the right derivative β r (t) and the limit lim τ t + β r (τ), and β r (t) lim τ t + β r (τ), and, for any t(a,b), β l (t) β r (t).

Definition 3 We call a bounded function αB B + (I,R) a generalized lower function and write αAG(I,R), if in any interval [c,d]I, where this function satisfies the Lipschitz condition, for any t 1 (c,d) and t 2 ( t 1 ,d) where the derivative exists, the inequality

φ ( t 2 , α ( t 2 ) , α ( t 2 ) ) φ ( t 1 , α ( t 1 ) , α ( t 1 ) ) t 1 t 2 f ( s , α ( s ) , α ( s ) ) ds

holds. We will call a bounded function βB B (I,R) a generalized upper function and write βBG(I,R), if in any interval [c,d]I, where this function satisfies the Lipschitz condition, for any t 1 (c,d) and t 2 ( t 1 ,d) where the derivative exists, the inequality

φ ( t 2 , β ( t 2 ) , β ( t 2 ) ) φ ( t 1 , β ( t 1 ) , β ( t 1 ) ) t 1 t 2 f ( s , β ( s ) , β ( s ) ) ds

holds.

A function x:IR will be called a generalized solution, if xAG(I,R)BG(I,R).

A generalized solution has a derivative at any point, possibly infinite, either −∞ or +∞, and x is continuous on [,+]; if in some interval the derivative x does not attain the values −∞ or +∞, then x is a solution of (1) in this interval.

Theorem 1 Let αAG(I,R), βBG(I,R) and αβ. Then for any A[α(a),β(a)] and B[α(b),β(b)] there exists a generalized solution of the Dirichlet problem

( φ ( t , x , x ) ) =f ( t , x , x ) ,x(a)=A,x(b)=B,αxβ.
(4)

In addition to conditions on α and β the compactness conditions are needed for solvability of the boundary value problem (1)-(2). The Nagumo condition [4] for φ-Laplacian and the Schrader condition [5] are sufficient conditions for compactness of a set of solutions. We accept the following compactness conditions.

Definition 4 We say that the compactness condition is fulfilled, if for all A[α(a),β(a)] and B[α(b),β(b)] any generalized solution of the Dirichlet problem (4) is a solution.

It is clear that this condition is weaker than the Schrader condition.

A set of solutions of the Dirichlet problem (4) will be denoted by S.

Remark 1 If αAG(I,R), βBG(I,R), αβ and the compactness condition is fulfilled, then the Dirichlet problem (4) has a solution.

Theorem 2 Let αAG(I,R), βBG(I,R) and the compactness condition be fulfilled. If the boundary value problem (3) has a solution u M for all M( M 0 ,+) and for tI

φ M ( t , x , x ) =φ ( t , x , x ) , f M ( t , x , x ) =f ( t , x , x ) ,αxβ,| x |M,

then there exists M 1 ( M 0 ,+) such that u M 1 solves the boundary value problem (1)-(2).

Proof Notice that the results in [6] imply that sup{ x C :xS}= M 0 <+. Suppose the contrary. Let the sequence { M i }, where M i ( M 0 ,+), i=1,2, tend to infinity. Consider the sequence { u i }, where u i = u M i , i=1,2, . We can assume, without loss of generality, that it converges in any rational points of the interval (a,b) to the function u, located between α and β. Notice that without loss of generality for any interval ( a 1 , b 1 )(a,b) it follows from the boundedness of u and the Mean Value Formula that there exists an interval [c,d]( a 1 , b 1 ) such that

sup { | u i ( t ) | : i { 1 , 2 , } , t [ c , d ] } =L<+.

It is clear that u i , i{1,2,}, and u satisfy the Lipschitz condition with constant L in [c,d]. The u can be extended by continuity to the entire interval [c,d], and thus we obtain a function u that satisfies the Lipschitz condition. It follows from the Lipschitz condition that { u i (t)} converges to u(t) for any t[c,d]. It is clear that the derivatives { u i (t)} converge to the derivative u (t) for any t[c,d]. Therefore, u(t) is a solution of (1) in the interval [c,d]. Continuing the construction of u(t) on both sides, one gets a solution of (1) on the maximal interval ( c 1 , d 1 ). If c 1 >a, then lim t c 1 + u (t) is either −∞ or +∞. Similarly, if d 1 <b, then lim t d 1 u (t) is either −∞ or +∞. If c 1 =a and lim t a + u (t) is not −∞ or +∞, then u(t) can be continued to a. Similarly, if d 1 =b and lim t b u (t) is not −∞ nor +∞, then u(t) can be continued to b. By repeating this construction, find an open set I 1 in I, where the function u(t) is defined and u(t) is a solution of (1) on intervals from I 1 . A set I 2 =I I 1 is closed and nowhere dense. For t I 2 the limit lim i u i (t) is equal to −∞ or +∞. Indeed, assuming the contrary and acting as above, we get t I 1 . Extend u(t) to irrational points of I 2 . If a I 2 , then u(a)= lim t a + u(t), and in the remaining cases u(τ)= lim t τ u(t). The above limits exist since u(t) is monotone in neighborhood of any point from I 2 . Similarly we get for t I 2 , u (t)= lim i u i (t) and lim τ t u (τ)= u (t). Therefore u(t) is a generalized solution of (1). It follows from the compactness condition that u(t) is a solution of (1). Let us show that the sequence { u i (t)} uniformly converges to u (t). Suppose the contrary is true. We assume, without loss of generality, that there exist ε>0 and a sequence { t i }, where t i I, i=1,2, such that | u ( t i ) u i ( t i )|>ε, i=1,2, and lim i t i = t 0 . Consider the case u i ( t i )> u ( t i )+ε, i=1,2, . We can assume, without loss of generality, that u i ( t 0 )> u ( t 0 )+ε/2, i=1,2, , and this contradicts the equality lim i u i ( t 0 )= u ( t 0 ). The uniform convergence is proved. We can conclude now that all u i (t) are the solutions of the boundary value problem (1)-(2). □

Remark 2 Theorem 2 gives the possibility to prove the solvability of boundary value problems if the solvability of more simple boundary value problems is known.

Remark 3 If α (a) β (a) and the inequalities α (a) x (a) β (a) hold for a solution x of the boundary value problem (1)-(2), then the compactness condition (Definition 4) can be weakened.

Definition 5 We will say that the compactness condition holds if for any A 1 [ β (a), α (a)] and B[α(b),β(b)] all generalized solutions of the problem

( φ ( t , x , x ) ) =f ( t , x , x ) , x (a)= A 1 ,x(b)=B,αxβ,

are classical solutions.

Example One way to use Theorem 2 is to verify that for all tI, x, x R and M( M 0 ,+), M 0 >0, the following conditions are satisfied:

φ M ( t , x , x ) = φ ( t , x , x ) , f M ( t , x , x ) = f ( t , x , δ ( M , x , M ) ) ,

where δ(u,v,w)=u if v<u, δ(u,v,w)=v if uvw, δ(u,v,w)=w if w<v.

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Acknowledgements

The author sincerely thanks the reviewers for their valuable suggestions and useful comments. This research was supported by the Institute of Mathematics and Computer Science, University of Latvia.

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Lepin, A.Y. On a certain way of proving the solvability for boundary value problems. Bound Value Probl 2014, 111 (2014). https://doi.org/10.1186/1687-2770-2014-111

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