Solvability of nth-order Lipschitz equations with nonlinear three-point boundary conditions. (English) Zbl 1307.34041
Summary: We investigate the solvability of \(n\)th-order Lipschitz equations
\[
y^{(n)}= f(x,y,y',\dots, y^{(n-1)}),\quad x_1\leq x\leq x_3
\]
with nonlinear three-point boundary conditions of the form
\[
k(y(x_2), y'(x_2),\dots, y^{(n-1)}(x_2);\quad y(x_1,y'(x_1),\dots, y^{(n-1)}(x_1))= 0,
\]
\[ g_i(y^{(i)}(x_2), y^{(i+1)}(x_2,\dots, y^{(n-1)}(x_2))= 0,\quad i= 0,1,\dots, n-2, \]
\[ h(y(x_2), y'(x_2),\dots, y^{(n-1)}(x_2); \]
\[ y(x_3),y'(x_3),\dots, y^{(n-1)}(x_3))= 0, \] where \(n\geq 3\), \(x_1< x_2< x_3\). By using the matching technique together with set-valued function theory, the existence and uniqueness of solutions for the problems are obtained. An application of our results, is given.
\[ g_i(y^{(i)}(x_2), y^{(i+1)}(x_2,\dots, y^{(n-1)}(x_2))= 0,\quad i= 0,1,\dots, n-2, \]
\[ h(y(x_2), y'(x_2),\dots, y^{(n-1)}(x_2); \]
\[ y(x_3),y'(x_3),\dots, y^{(n-1)}(x_3))= 0, \] where \(n\geq 3\), \(x_1< x_2< x_3\). By using the matching technique together with set-valued function theory, the existence and uniqueness of solutions for the problems are obtained. An application of our results, is given.
MSC:
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
\(n\)th-order Lipschitz equation; nonlinear three-point boundary value problem; matching method; existence; uniquenessReferences:
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