Nonlocal boundary value problems with even gaps in boundary conditions for third order differential equations. (English) Zbl 1312.34053
Summary: We use solution matching to study the uniqueness and existence of solutions for the nonlocal boundary value problem for the third-order differential equation,
\[
y'''(x)= f(x, y(x))\quad\text{on }[a, c]
\]
satisfying
\[
y(a)- \int^b_a y(x)d\alpha(x)= y_1,\;y'(b)= y_2,\;\int^c_b y(x) d\beta(x)- y)c)= y_3,
\]
where
\[
\int^b_a y(x)d\alpha(x)\quad\text{and}\quad \int^c_b y(x)d\beta(x)
\]
are Riemann-Stieltjes integrals with positive measures \(d\alpha(x)\) and \(d\beta(x)\), respectively. We match solutions on \([a,b]\) with solutions on \([b,c]\). Monotonicity conditions and some growth conditions on \(f\) are imposed.
MSC:
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |