MMN-4481

# Two-point boundary value problems for 4th order ordinary differential equations

*Mariam Manjikashvili*;

*Sulkhan Mukhigulashvili*;

## Abstract

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations
$$
u^{(4)}(t)= p(t)u(t)+ q(t) \quad \text{for} \quad t\in [a,\,b],
$$
$$
u^{(4)}(t)= p(t)u(t)+ f(t,u(t)) \quad \text{for} \quad t\in [a,\,b],
$$
under the following two-point boundary conditions
$$
u^{(i)}(a)=0,\quad u^{(i)}(b)=0 \;\; (i=0,1),
$$
and
$$
u^{(i)}(a)=0 \;\; (i=0,1,2),\quad u(b)=0,
$$
where $p\in L([a,\,b];\,R)$ is a nonconstant sign function and $f\in K([a,\,b]\times R; R).$

Vol. 25 (2024), No. 1, pp. 399-409