Abstract
Existence of solutions of nonlinear third-order two-point boundary value problems
Highlights
We are concerned with boundary value problems (BVPs) for the differential equation x = f (t, x, x, x ), t ∈ (0, 1), (1.1)
We study the existence of C3[0, 1]-solutions to the above problems which do not change their sign, are monotone and do not change their curvature
We will say that for some of the BVPs (1.1),(1.k), k = 2, 3, 4, 5, 6 (k = 2, 6 for short), the condition (H2) holds for constants mi ≤ Mi, i = 0, 2, if: (H2) [m0 − σ, M0 + σ] ⊆ Dx, [m1 − σ, M1 + σ] ⊆ Dp, [m2 − σ, M2 + σ] ⊆ Dq, where σ is as in (H1), and f (t, x, p, q) is continuous on [0, 1] × J, where J = [m0 − σ, M0 + σ] × [m1 − σ, M1 + σ] × [m2 − σ, M2 + σ]. Such type of conditions have been used for studying the solvability of various problems for first and second order differential equations, see P
Summary
Brighi [2] have considered the equation x + xx + g(x ) = 0, t ∈ [0, ∞), with boundary conditions similar to (1.3), and Z. (H2) [m0 − σ, M0 + σ] ⊆ Dx, [m1 − σ, M1 + σ] ⊆ Dp, [m2 − σ, M2 + σ] ⊆ Dq, where σ is as in (H1), and f (t, x, p, q) is continuous on [0, 1] × J, where J = [m0 − σ, M0 + σ] × [m1 − σ, M1 + σ] × [m2 − σ, M2 + σ] Such type of conditions have been used for studying the solvability of various problems for first and second order differential equations, see P.
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