An investigation of the existence of solutions of the nonlinear boundary value problem
x
′
=
f
(
t
,
x
,
y
)
,
y
′
=
g
(
t
,
x
,
y
)
,
A
V
(
a
,
x
(
a
)
,
y
(
a
)
)
+
B
W
(
a
,
x
(
a
)
,
y
(
a
)
)
=
C
1
,
C
V
(
b
,
x
(
b
)
,
y
(
b
)
)
+
D
W
(
b
,
x
(
b
)
,
y
(
b
)
)
=
C
2
x’ = f(t,x,y),y’ = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}
, is made. Here we assume
g
,
f
:
[
a
,
b
]
×
R
p
×
R
q
→
R
p
g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}
are continuous, and
V
,
W
:
[
a
,
b
]
×
R
p
×
R
q
→
R
V,W:[a,b] \times {R^p} \times {R^q} \to R
are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.