Abstract
We investigate the solvability of a fully fourth-order periodic boundary value problem of the formx(4)=f(t,x,x′,x′′,x′′′), x(i)(0)=x(i)(T), i=0,1,2,3,wheref:[0,T]×R4→Rsatisfies Carathéodory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. Meanwhile, as applications, some examples are given to illustrate our results.
Highlights
We investigate the solvability of a fully fourth-order periodic boundary value problem of the form x(4) = f(t, x, x, x, x), x(i)(0) = x(i)(T), i = 0, 1, 2, 3, where f : [0, T] × R4 → R satisfies Caratheodory conditions
Let λ ∈ (0, 1) and let xλ ∈ Ω be a solution of the following PBVP: x(4) = λf (t, x, x, x, x) + (1 − λ) μεx, (44)
Assume that all conditions in Theorem 5 hold with the exception of (H1), which is replaced by the following: (H1) there exists a constant r∗ ∈ (0, r) such that if x0 > −r∗, |x1| ≤ r1, |x2| ≤ r2, |x3| ≤ r3, μf (t, x0, x1, x2, x3) ≥ 0 for a.e. t ∈ [0, T], (50)
Summary
We investigate the solvability of a fully fourth-order periodic boundary value problem of the form x(4) = f(t, x, x, x, x), x(i)(0) = x(i)(T), i = 0, 1, 2, 3, where f : [0, T] × R4 → R satisfies Caratheodory conditions. We consider a fully nonlinear fourth-order periodic boundary value problem of the form x(4) = f (t, x, x, x, x) , (1) Let L : dom L ⊂ X → Z be a Fredholm mapping of index zero; there exist continuous projectors P : X → X and Q : Z → Z such that
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