lagrangian formalism - Is the principle of least action fully equivalent to the Euler-Lagrange equations?

I am citing from Landau and Lifschitz, this statement that will seem to you well-known, trivial, etc:

"Between these positions, (i.e. $q_1$ and $q_2$) the system moves then in such a way that the integral $S = \int_{t_1}^{t_2} L(q, \dot q, t) dt \ $ have the smallest possible value."

My question: Are the Euler-Lagrange equations for this action functional, i.e. the differential laws of Lagrangian mechanics, fully equivalent to the integral statement of the principle of least action?

That the principle of least action implies that classical solutions fulfill the Euler-Lagrange equation is easy to see. But what about the converse implication - is every solution to the E-L equations a minimum of the action?

Can somebody show whether there is equivalence between the minimization of the integral and the differential equation, i.e. that the implication is in both directions?

I saw previous related questions and answers, but I am asking here a question of equivalence.