Cauchy problem in the class of monotone increasing functions (Q944134)

The flight along a given trajectory, when the gravity, reaction force, environment resistance and breaking force are taken into account is described by the two following equalities and one inequality: \[ u'(x) = a(x) u(x) + w(x),\;0 \leq x \leq x_0, \quad u(0)= m_0 > 0, \;\text{and} \;u'(x) \geq 0,\;0 \leq x \leq x_0, \] where \(m_0\) and \(x_0\) are given positive constants, \(u(x)\) is the unknown function, \(a(x)\) and \(w(x)\) are given function continuously differentiable in \([0,x_0]\). Without the inequality this Cauchy problem has always a unique solution but the determination whether the solution satisfies the given inequality may be difficult. In the paper, the problem of solvability is divided into four cases and solvability conditions are determined. It follows from these conditions that the flight along the trajectory \(y=f(x)\) is always possible if \(f''(x) < 0\) and \(f'''(x) > 0\). The possibility of a flight along the trajectory \(y\) when the sign of \(f'''(x)\) is alternating will be a subject of separate investigation.