We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

For the third-order differential equation y′″ = ƒ(t, y, y′, y″), where , questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) where 1<α≤2, η∈(0,1),β∈ℝ=(−∞,+∞), βηα−1≠1, Dα is the Riemann–Liouville differential operator of order α, and f:[0,1]×ℝ→ℝ is continuous, q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann–Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55