# 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)

${\left(\phi \left(t,x,{x}^{\prime }\right)\right)}^{\prime }=f\left(t,x,{x}^{\prime }\right),\phantom{\rule{1em}{0ex}}t\in I=\left[a,b\right],$
(1)
${H}_{1}x={h}_{1},\phantom{\rule{2em}{0ex}}{H}_{2}x={h}_{2},\phantom{\rule{1em}{0ex}}\alpha \le x\le \beta ,$
(2)

where $\phi \in C\left(I×{R}^{2},R\right)$ is strictly increasing in ${x}^{\prime }$ for fixed t and x, $f:I×{R}^{2}\to R$ satisfies the Caratheodory conditions, that is, $f\left(t,\cdot ,\cdot \right)$ is measurable in I for fixed $x,{x}^{\prime }\in R$, $f\left(\cdot ,x,{x}^{\prime }\right)$ is continuous on ${R}^{2}$ for fixed $t\in I$, and for any compact set $P\subset {R}^{2}$ there exists function $g\in L\left(I,R\right)$ such, that for any $\left(t,x,{x}^{\prime }\right)\in I×P$, the estimate $|f\left(t,x,{x}^{\prime }\right)|\le g\left(t\right)$ holds, ${H}_{1},{H}_{2}\in C\left({C}^{1}\left(I,R\right),R\right)$, ${h}_{1},{h}_{2}\in R$, α is the lower function, β the upper function.

This boundary value problem is replaced by another one, which is dependent on the parameter $M\in \left({M}_{0},+\mathrm{\infty }\right)$, ${M}_{0}>0$,

$\begin{array}{r}{\left({\phi }_{M}\left(t,x,{x}^{\prime }\right)\right)}^{\prime }={f}_{M}\left(t,x,{x}^{\prime }\right),\phantom{\rule{1em}{0ex}}t\in I=\left[a,b\right],\\ {H}_{1}x={h}_{1},\phantom{\rule{2em}{0ex}}{H}_{2}x={h}_{2},\phantom{\rule{1em}{0ex}}\alpha \le x\le \beta ,\end{array}$
(3)

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

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

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

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

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

$\phi \left({t}_{2},\alpha \left({t}_{2}\right),{\alpha }^{\prime }\left({t}_{2}\right)\right)-\phi \left({t}_{1},\alpha \left({t}_{1}\right),{\alpha }^{\prime }\left({t}_{1}\right)\right)\ge {\int }_{{t}_{1}}^{{t}_{2}}f\left(s,\alpha \left(s\right),{\alpha }^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$

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

$\phi \left({t}_{2},\beta \left({t}_{2}\right),{\beta }^{\prime }\left({t}_{2}\right)\right)-\phi \left({t}_{1},\beta \left({t}_{1}\right),{\beta }^{\prime }\left({t}_{1}\right)\right)\le {\int }_{{t}_{1}}^{{t}_{2}}f\left(s,\beta \left(s\right),{\beta }^{\prime }\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$

holds.

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

A generalized solution has a derivative at any point, possibly infinite, either −∞ or +∞, and ${x}^{\prime }$ is continuous on $\left[-\mathrm{\infty },+\mathrm{\infty }\right]$; 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 $\alpha \in AG\left(I,R\right)$, $\beta \in BG\left(I,R\right)$ and $\alpha \le \beta$. Then for any $A\in \left[\alpha \left(a\right),\beta \left(a\right)\right]$ and $B\in \left[\alpha \left(b\right),\beta \left(b\right)\right]$ there exists a generalized solution of the Dirichlet problem

${\left(\phi \left(t,x,{x}^{\prime }\right)\right)}^{\prime }=f\left(t,x,{x}^{\prime }\right),\phantom{\rule{2em}{0ex}}x\left(a\right)=A,\phantom{\rule{2em}{0ex}}x\left(b\right)=B,\phantom{\rule{1em}{0ex}}\alpha \le x\le \beta .$
(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  for φ-Laplacian and the Schrader condition  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 $A\in \left[\alpha \left(a\right),\beta \left(a\right)\right]$ and $B\in \left[\alpha \left(b\right),\beta \left(b\right)\right]$ 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 $\alpha \in AG\left(I,R\right)$, $\beta \in BG\left(I,R\right)$, $\alpha \le \beta$ and the compactness condition is fulfilled, then the Dirichlet problem (4) has a solution.

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

${\phi }_{M}\left(t,x,{x}^{\prime }\right)=\phi \left(t,x,{x}^{\prime }\right),\phantom{\rule{2em}{0ex}}{f}_{M}\left(t,x,{x}^{\prime }\right)=f\left(t,x,{x}^{\prime }\right),\phantom{\rule{1em}{0ex}}\alpha \le x\le \beta ,|{x}^{\prime }|\le M,$

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

Proof Notice that the results in  imply that $sup\left\{{\parallel {x}^{\prime }\parallel }_{C}:x\in S\right\}={M}_{0}<+\mathrm{\infty }$. Suppose the contrary. Let the sequence $\left\{{M}_{i}\right\}$, where ${M}_{i}\in \left({M}_{0},+\mathrm{\infty }\right)$, $i=1,2,\dots$ tend to infinity. Consider the sequence $\left\{{u}_{i}\right\}$, 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 $\left(a,b\right)$ to the function u, located between α and β. Notice that without loss of generality for any interval $\left({a}_{1},{b}_{1}\right)\subset \left(a,b\right)$ it follows from the boundedness of u and the Mean Value Formula that there exists an interval $\left[c,d\right]\subset \left({a}_{1},{b}_{1}\right)$ such that

$sup\left\{|{u}_{i}^{\prime }\left(t\right)|:i\in \left\{1,2,\dots \right\},t\in \left[c,d\right]\right\}=L<+\mathrm{\infty }.$

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

${\left(\phi \left(t,x,{x}^{\prime }\right)\right)}^{\prime }=f\left(t,x,{x}^{\prime }\right),\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(a\right)={A}_{1},\phantom{\rule{2em}{0ex}}x\left(b\right)=B,\phantom{\rule{1em}{0ex}}\alpha \le x\le \beta ,$

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 $M\in \left({M}_{0},+\mathrm{\infty }\right)$, ${M}_{0}>0$, the following conditions are satisfied:

$\begin{array}{c}{\phi }_{M}\left(t,x,{x}^{\prime }\right)=\phi \left(t,x,{x}^{\prime }\right),\hfill \\ {f}_{M}\left(t,x,{x}^{\prime }\right)=f\left(t,x,\delta \left(-M,{x}^{\prime },M\right)\right),\hfill \end{array}$

where $\delta \left(u,v,w\right)=u$ if $v, $\delta \left(u,v,w\right)=v$ if $u\le v\le w$, $\delta \left(u,v,w\right)=w$ if $w.

## 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 ${y}^{″}=f\left(x,y,{y}^{\prime }\right)$. 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

Authors

### Corresponding author

Correspondence to Arnold Y Lepin.

### 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 