EXISTENCE OF SOLUTIONS TO SOME BOUNDARY VALUE PROBLEMS WITH ϕ-LAPLACIAN OPERATORS USING MULTIPLE SIGN CONDITIONS

Abstract

We study the nonlinear third order differential equation $\left(\phi\left(v^{\prime \prime}\right)\right)^{\prime}= g\left(t, v, v^{\prime}, v^{\prime \prime}\right)$ with $g$ an $L^1$-Carathéodory function, under the nonlinear boundary conditions

$$
\begin{aligned}
& \phi\left(v^{\prime \prime}(0)\right)=f_1\left(v(0), v^{\prime}(0)\right) \\
& \phi\left(v^{\prime \prime}(T)\right)=f_2\left(v(T), v^{\prime}(T)\right)
\end{aligned}
$$

where $T>0$. The existence of solutions is proven by applying topological methods. Furthermore, we use some sign conditions that involve more than one variable of $g$. Our results can be extended to equation $\left(\phi\left(v^{(n-1)}\right)\right)^{\prime}=g\left(t, v, v^{\prime}, v^{\prime \prime}, \ldots, v^{(n-1)}\right)$ with $g$ an $L^1-$ Carathéodory function, under the nonlinear boundary conditions

$$
\begin{aligned}
& \phi\left(v^{(n-1)}(0)\right)=f_1\left(v(0), v^{\prime}(0), \ldots, v^{(n-1)}(0)\right), \\
& \phi\left(v^{(n-1)}(T)\right)=f_2\left(v(T), v^{\prime}(T), \ldots, v^{(n-1)}(T)\right), \text { for } n \geq 3 .
\end{aligned}
$$

Received: July 28, 2025
Revised: August 2, 2025
Accepted: September 9, 2025