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

*Boundary Value Problems*
**volume 2014**, Article number: 111 (2014)

## 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)

where \phi \in C(I\times {R}^{2},R) is strictly increasing in {x}^{\prime} for fixed *t* and *x*, f:I\times {R}^{2}\to R satisfies the Caratheodory conditions, that is, f(t,\cdot ,\cdot ) is measurable in *I* for fixed x,{x}^{\prime}\in R, f(\cdot ,x,{x}^{\prime}) is continuous on {R}^{2} for fixed t\in I, and for any compact set P\subset {R}^{2} there exists function g\in L(I,R) such, that for any (t,x,{x}^{\prime})\in I\times P, the estimate |f(t,x,{x}^{\prime})|\le g(t) holds, {H}_{1},{H}_{2}\in C({C}^{1}(I,R),R), {h}_{1},{h}_{2}\in 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\in ({M}_{0},+\mathrm{\infty}), {M}_{0}>0,

where {\phi}_{M}\in C(I\times {R}^{2},R) is strictly increasing in {x}^{\prime} for fixed *t* and *x*, and f:I\times {R}^{2}\to R satisfies the Caratheodory conditions.

**Definition 1** A function x\in {C}^{1}(I,R) is a solution of (1), if \phi (t,x(t),{x}^{\prime}(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 [1–3]. This is needed to prove the main result.

**Definition 2** The class B{B}^{+}(I,R) consists of functions \alpha :I\to R, which possess the property: for any t\in (a,b] there exist the left derivative {\alpha}_{l}^{\prime}(t) and the limit {lim}_{\tau \to t-}{\alpha}_{l}^{\prime}(\tau ), and {\alpha}_{l}^{\prime}(t)\ge {lim}_{\tau \to t-}{\alpha}_{l}^{\prime}(\tau ); for any t\in [a,b) there exist the right derivative {\alpha}_{r}^{\prime}(t) and the limit {lim}_{\tau \to t+}{\alpha}_{r}^{\prime}(\tau ), and {\alpha}_{r}^{\prime}(t)\le {lim}_{\tau \to t+}{\alpha}_{r}^{\prime}(\tau ), and, for any t\in (a,b), {\alpha}_{l}^{\prime}(t)\le {\alpha}_{r}^{\prime}(t).

The class B{B}^{-}(I,R) consists of functions \beta :I\to R, which possess the following property: for any t\in (a,b] there exist the left derivative {\beta}_{l}^{\prime}(t) and the limit {lim}_{\tau \to t-}{\beta}_{l}^{\prime}(\tau ), and {\beta}_{l}^{\prime}(t)\le {lim}_{\tau \to t-}{\beta}_{l}^{\prime}(\tau ); for any t\in [a,b) there exist the right derivative {\beta}_{r}^{\prime}(t) and the limit {lim}_{\tau \to t+}{\beta}_{r}^{\prime}(\tau ), and {\beta}_{r}^{\prime}(t)\ge {lim}_{\tau \to t+}{\beta}_{r}^{\prime}(\tau ), and, for any t\in (a,b), {\beta}_{l}^{\prime}(t)\ge {\beta}_{r}^{\prime}(t).

**Definition 3** We call a bounded function \alpha \in B{B}^{+}(I,R) a *generalized lower function* and write \alpha \in AG(I,R), if in any interval [c,d]\in I, where this function satisfies the Lipschitz condition, for any {t}_{1}\in (c,d) and {t}_{2}\in ({t}_{1},d) where the derivative exists, the inequality

holds. We will call a bounded function \beta \in B{B}^{-}(I,R) a *generalized upper function* and write \beta \in BG(I,R), if in any interval [c,d]\in I, where this function satisfies the Lipschitz condition, for any {t}_{1}\in (c,d) and {t}_{2}\in ({t}_{1},d) where the derivative exists, the inequality

holds.

A function x:I\to R will be called a *generalized solution*, if x\in AG(I,R)\cap BG(I,R).

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

**Theorem 1** *Let* \alpha \in AG(I,R), \beta \in BG(I,R) *and* \alpha \le \beta. *Then for any* A\in [\alpha (a),\beta (a)] *and* B\in [\alpha (b),\beta (b)] *there exists a generalized solution of the Dirichlet problem*

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\in [\alpha (a),\beta (a)] and B\in [\alpha (b),\beta (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 \alpha \in AG(I,R), \beta \in BG(I,R), \alpha \le \beta and the compactness condition is fulfilled, then the Dirichlet problem (4) has a solution.

**Theorem 2** *Let* \alpha \in AG(I,R), \beta \in BG(I,R) *and the compactness condition be fulfilled*. *If the boundary value problem* (3) *has a solution* {u}_{M} *for all* M\in ({M}_{0},+\mathrm{\infty}) *and for* t\in I

*then there exists* {M}_{1}\in ({M}_{0},+\mathrm{\infty}) *such that* {u}_{{M}_{1}} *solves the boundary value problem* (1)-(2).

*Proof* Notice that the results in [6] imply that sup\{{\parallel {x}^{\prime}\parallel}_{C}:x\in S\}={M}_{0}<+\mathrm{\infty}. Suppose the contrary. Let the sequence \{{M}_{i}\}, where {M}_{i}\in ({M}_{0},+\mathrm{\infty}), i=1,2,\dots tend to infinity. Consider the sequence \{{u}_{i}\}, where {u}_{i}={u}_{{M}_{i}}, i=1,2,\dots . 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})\subset (a,b) it follows from the boundedness of *u* and the Mean Value Formula that there exists an interval [c,d]\subset ({a}_{1},{b}_{1}) such that

It is clear that {u}_{i}, i\in \{1,2,\dots \}, 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\in [c,d]. It is clear that the derivatives \{{u}_{i}^{\prime}(t)\} converge to the derivative {u}^{\prime}(t) for any t\in [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\to {c}_{1}+}{u}^{\prime}(t) is either −∞ or +∞. Similarly, if {d}_{1}<b, then {lim}_{t\to {d}_{1}-}{u}^{\prime}(t) is either −∞ or +∞. If {c}_{1}=a and {lim}_{t\to a+}{u}^{\prime}(t) is not −∞ or +∞, then u(t) can be continued to *a*. Similarly, if {d}_{1}=b and {lim}_{t\to b-}{u}^{\prime}(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\mathrm{\setminus}{I}_{1} is closed and nowhere dense. For t\in {I}_{2} the limit {lim}_{i\to \mathrm{\infty}}{u}_{i}^{\prime}(t) is equal to −∞ or +∞. Indeed, assuming the contrary and acting as above, we get t\in {I}_{1}. Extend u(t) to irrational points of {I}_{2}. If a\in {I}_{2}, then u(a)={lim}_{t\to a+}u(t), and in the remaining cases u(\tau )={lim}_{t\to \tau -}u(t). The above limits exist since u(t) is monotone in neighborhood of any point from {I}_{2}. Similarly we get for t\in {I}_{2}, {u}^{\prime}(t)={lim}_{i\to \mathrm{\infty}}{u}_{i}^{\prime}(t) and {lim}_{\tau \to t}{u}^{\prime}(\tau )={u}^{\prime}(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}^{\prime}(t)\} uniformly converges to {u}^{\prime}(t). Suppose the contrary is true. We assume, without loss of generality, that there exist \epsilon >0 and a sequence \{{t}_{i}\}, where {t}_{i}\in I, i=1,2,\dots such that |{u}^{\prime}({t}_{i})-{u}_{i}^{\prime}({t}_{i})|>\epsilon, i=1,2,\dots and {lim}_{i\to \mathrm{\infty}}{t}_{i}={t}_{0}. Consider the case {u}_{i}^{\prime}({t}_{i})>{u}^{\prime}({t}_{i})+\epsilon, i=1,2,\dots . We can assume, without loss of generality, that {u}_{i}^{\prime}({t}_{0})>{u}^{\prime}({t}_{0})+\epsilon /2, i=1,2,\dots , and this contradicts the equality {lim}_{i\to \mathrm{\infty}}{u}_{i}^{\prime}({t}_{0})={u}^{\prime}({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 {\alpha}^{\prime}(a)\ge {\beta}^{\prime}(a) and the inequalities {\alpha}^{\prime}(a)\ge {x}^{\prime}(a)\ge {\beta}^{\prime}(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}\in [{\beta}^{\prime}(a),{\alpha}^{\prime}(a)] and B\in [\alpha (b),\beta (b)] all generalized solutions of the problem

are classical solutions.

**Example** One way to use Theorem 2 is to verify that for all t\in I, x,{x}^{\prime}\in R and M\in ({M}_{0},+\mathrm{\infty}), {M}_{0}>0, the following conditions are satisfied:

where \delta (u,v,w)=u if v<u, \delta (u,v,w)=v if u\le v\le w, \delta (u,v,w)=w if w<v.

## References

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*Differ. Equ.*1982, 18: 1323-1330.Lepin AY, Lepin LA:

*Boundary Value Problems for Ordinary Differential Equations*. Zinatne, Riga; 1988. (Russian)Lepin, AY, Lepin, LA: Generalized lower and upper functions for

*φ*-Laplacian equations. Differ. Equ. (2014, in press)Nagumo M: Über die Differentialgleichung {y}^{\u2033}=f(x,y,{y}^{\prime}).

*Proc. Phys. Math. Soc. Jpn.*1937, 19(3):861-866.Schrader KW: Existence theorems for second order boundary value problems.

*J. Differ. Equ.*1969, 5(3):572-584. 10.1016/0022-0396(69)90094-1Lepin AY: Compactness of generalized solutions between the generalized lower and upper functions.

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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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DOI: https://doi.org/10.1186/1687-2770-2014-111