- Research
- Open Access
- Published:

# 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

- 1.
Lepin LA: Generalized solutions and solvability of boundary value problems for the second order differential equations.

*Differ. Equ.*1982, 18: 1323-1330. - 2.
Lepin AY, Lepin LA:

*Boundary Value Problems for Ordinary Differential Equations*. Zinatne, Riga; 1988. (Russian) - 3.
Lepin, AY, Lepin, LA: Generalized lower and upper functions for

*φ*-Laplacian equations. Differ. Equ. (2014, in press) - 4.
Nagumo M: Über die Differentialgleichung ${y}^{\u2033}=f(x,y,{y}^{\prime})$.

*Proc. Phys. Math. Soc. Jpn.*1937, 19(3):861-866. - 5.
Schrader KW: Existence theorems for second order boundary value problems.

*J. Differ. Equ.*1969, 5(3):572-584. 10.1016/0022-0396(69)90094-1 - 6.
Lepin AY: Compactness of generalized solutions between the generalized lower and upper functions.

*Proc. LUMII Math. Differ. Equ.*2011, 11: 22-24.

## 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.

## Author information

## Additional information

### Competing interests

The author declares that he has no competing interests.

### Authors’ contributions

The author participated in drafting, revising and commenting on the manuscript. The author read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- boundary value problems
- upper and lower functions