Suppose, contrary to the result of the theorem, that the inequalities (
50) and (
51) are satisfied. Since
\(\lambda \ge 0\),
\(\lambda ^{T} e=1\), the inequalities (
50) and (
51) yield
$$\begin{aligned} \int \limits _{a}^{b}\sum \limits _{i=1}^{p}\lambda _{i}f^{i} ( t,x,\overset{\cdot }{x}) dt&\leqq \int \limits _{a}^{b}\sum _{i=1}^{p}\lambda _{i}f^{i} ( t,y,\overset{\cdot }{y}) dt +\int \limits _{a}^{b}\sum \limits _{j=1}^{q}\xi _{j}g^{j}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad +\int \limits _{a}^{b}\sum _{k=1}^{s}\zeta _{k}h^{k}( t,y, \overset{\cdot }{y}) dt\leqq 0. \end{aligned}$$
(52)
By assumption, the hypotheses (a)–(d) are fulfilled. Thus, by Definition 3, we have
$$\begin{aligned}&\!\!\int \limits _{a}^{b}f^{i}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}f^{i}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\!\!\quad >\int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}; \left( f_{y}^{i}( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ f_{\overset{\cdot }{y}}^{i} ( t,y,\overset{\cdot }{y}) \right] ,\rho _{f_{i}}\right] \right) dt,\quad i\in A, \end{aligned}$$
(53)
$$\begin{aligned}&\!\!\int \limits _{a}^{b}g^{j}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}g^{j}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\!\!\quad \geqq \int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( g_{y}^{j}( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ g_{\overset{\cdot }{y}}^{j}( t,y,\overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) \right) dt,\quad j\in J, \end{aligned}$$
(54)
$$\begin{aligned}&\!\!\int \limits _{a}^{b}h^{k}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}h^{k}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\!\!\quad \geqq \int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( h_{y}^{k} ( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ h_{\overset{\cdot }{y}}^{k}( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt,\quad k\in K^{+}(t), \nonumber \\\end{aligned}$$
(55)
$$\begin{aligned}&\!\!-\int \limits _{a}^{b}h^{k} ( t,x,\overset{\cdot }{x}) dt+\int \limits _{a}^{b}h^{k} ( t,y,\overset{\cdot }{y}) dt \nonumber \\&\!\!\quad \geqq -\int \limits _{a}^{b}\Phi \left( \! t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( \! -h_{y}^{k}( t,y,\overset{\cdot }{y}) \!-\!\frac{d}{dt} \left[ \! -h_{\overset{\cdot }{y}}^{k} ( t,y,\overset{\cdot }{y}) \!\right] ,\rho _{h_{k}}\!\right) \!\right) dt, k\!\in \! K^{-}(t).\nonumber \\ \end{aligned}$$
(56)
Using the last constraint of (
VWD) together with (
53)–(
56), we get
$$\begin{aligned}&\int \limits _{a}^{b}\lambda _{i}f^{i} ( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}\lambda _{i}f^{i}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad >\int \limits _{a}^{b}\lambda _{i}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( f_{y}^{i}( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ f_{\overset{\cdot }{y}}^{i}( t,y,\overset{\cdot }{y}) \right] ,\rho _{f_{i}}\right) \right) dt,\quad i\in A,\qquad \qquad \end{aligned}$$
(57)
$$\begin{aligned}&\int \limits _{a}^{b}\lambda _{r}f^{r}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}\lambda _{r}f^{r}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad >\int \limits _{a}^{b}\lambda _{r}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}; \left( f_{y}^{r}( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ f_{\overset{\cdot }{y}}^{r}( t,y,\overset{\cdot }{y}) \right] ,\rho _{f_{r}}\right) \right) dt \nonumber \\&\quad \text {\ for at least one }r\in A, \end{aligned}$$
(58)
$$\begin{aligned}&\int \limits _{a}^{b}\xi _{j}g^{j}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}\xi _{j}g^{j}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad \geqq \! \int \limits _{a}^{b}\xi _{j}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}; \left( g_{y}^{j} ( t,y,\overset{\cdot }{y}) \!-\!\frac{d}{dt} \left[ g_{\overset{\cdot }{y}}^{j}( t,y,\overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) \right) dt,\quad \! j\!\in \! J, \end{aligned}$$
(59)
$$\begin{aligned}&\int \limits _{a}^{b}\zeta _{k}h^{k}( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}\zeta _{k}h^{k}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad \geqq \! \int \limits _{a}^{b}\zeta _{k}\Phi \left( t,x,\overset{\cdot }{x},y, \overset{\cdot }{y}; \left( h_{y}^{k}( t,y,\overset{\cdot }{y}) \!-\!\frac{d}{dt} \left[ \! h_{\overset{\cdot }{y}}^{k} ( t,y,\overset{\cdot }{y}) \!\right] ,\rho _{h_{k}}\!\right) \!\right) dt, \quad \! k\!\in \! K^{+}(t), \end{aligned}$$
(60)
$$\begin{aligned}&\int \limits _{a}^{b}\zeta _{k}h^{k} ( t,x,\overset{\cdot }{x}) dt-\int \limits _{a}^{b}\zeta _{k}h^{k}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad \geqq \int \limits _{a}^{b}( -\zeta _{k}) \Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}; \left( -h_{y}^{k} ( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ -h_{\overset{\cdot }{y}}^{k} ( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt, \nonumber \\&\qquad k\in K^{-} (t). \end{aligned}$$
(61)
Adding both sides of the inequalities (
59)–(
61) and then, using the feasibility of
\(x\) in problem (
MVPP) and the feasibility of
\( \left( y,\lambda ,\xi ,\zeta \right) \) in problem (
VWD), we obtain, respectively,
$$\begin{aligned}&0\geqq \int \limits _{a}^{b}\sum _{j=1}^{q}\xi _{j}g^{j} ( t,y,\overset{\cdot }{y}) dt \nonumber \\&\qquad +\int \limits _{a}^{b}\sum _{j=1}^{q}\xi _{j}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}; g_{y}^{j}( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ g_{\overset{\cdot }{y}}^{j}( t,y,\overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) dt,\quad j\in J, \nonumber \\\end{aligned}$$
(62)
$$\begin{aligned}&0\geqq \int \limits _{a}^{b}\sum _{k\in K}\zeta _{k}h^{k}( t,y,\overset{\cdot }{y}) dt \nonumber \\&\quad +\int \limits _{a}^{b}\sum _{k\in K^{+}(t)}\zeta _{k}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( h_{y}^{k}( t,y,\overset{\cdot }{y}) -\frac{d}{dt} \left[ h_{\overset{\cdot }{y}}^{k}( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt \nonumber \\&\quad +\!\int \limits _{a}^{b}\sum _{k\in K^{-}(t) } ( -\zeta _{k}) \Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( \!-h_{y}^{k}( t,y,\overset{\cdot }{y}) \!-\!\frac{d}{dt}\left[ \! -h_{\overset{\cdot }{y}}^{k} ( t,y,\overset{\cdot }{y}) \!\right] ,\rho _{h_{k}}\!\right) \!\right) dt. \end{aligned}$$
(63)
Combining (
57), (
58), (
62) and, (
63), we get
$$\begin{aligned}&\int \limits _{a}^{b}\sum _{i=1}^{p}\lambda _{i}f^{i} ( t,x,\overset{\cdot }{x}) dt>\int \limits _{a}^{b}\sum _{i=1}^{p}\lambda _{i}f^{i} ( t,y,\overset{\cdot }{y}) dt\nonumber \\&\quad + \int \limits _{a}^{b}\sum _{j=1}^{q}\xi _{j}g^{j}( t,y,\overset{\cdot }{y}) dt+\int \limits _{a}^{b}\sum _{k=1}^{s}\zeta _{k}h^{k}( t,y,\overset{\cdot }{y}) dt\nonumber \\&\quad +\int \limits _{a}^{b}\sum _{i=1}^{p}\lambda _{i}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( f_{y}^{i} ( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ f_{\overset{\cdot }{y}}^{i} ( t,y,\overset{\cdot }{y}) \right] ,\rho _{f_{i}}\right) \right) dt \nonumber \\&\quad +\int \limits _{a}^{b}\sum _{j=1}^{q}\xi _{j}\Phi \left( t,x,\overset{\cdot }{x} ,y,\overset{\cdot }{y};\left( g_{y}^{j} ( t,y,\overset{\cdot }{y} ) -\frac{d}{dt}\left[ g_{\overset{\cdot }{y}}^{j} ( t,y,\overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) \right) dt \nonumber \\&\quad +\int \limits _{a}^{b}\sum _{k\in K^{+} ( t) }\zeta _{k}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( h_{y}^{k}( t,y, \overset{\cdot }{y}) -\frac{d}{dt}\left[ h_{\overset{\cdot }{y} }^{k} ( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt \nonumber \\&\quad +\!\int \limits _{a}^{b}\sum _{k\in K^{-}(t) } ( -\zeta _{k}) \Phi \left( \! t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( \!-h_{y}^{k}( t,y,\overset{\cdot }{y}) \!-\!\frac{d}{dt}\left[ \! -h_{\overset{\cdot }{y}}^{k} ( t,y,\overset{\cdot }{y}) \!\right] ,\rho _{h_{k}}\!\right) \!\right) dt.\quad \end{aligned}$$
(64)
By (
52) and (
64), it follows that
$$\begin{aligned}&\int \limits _{a}^{b}\sum _{i=1}^{p}\lambda _{i}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( f_{y}^{i}( t,y,\overset{\cdot }{y} ) -\frac{d}{dt}\left[ f_{\overset{\cdot }{y}}^{i}( t,y,\overset{\cdot }{y}) \right] ,\rho _{f_{i}}\right) \right) dt \nonumber \\&\quad +\int \limits _{a}^{b}\sum _{j=1}^{q}\xi _{j}\Phi \left( t,x,\overset{\cdot }{x},y, \overset{\cdot }{y};\left( g_{y}^{j}( t,y,\overset{\cdot }{y}) - \frac{d}{dt}\left[ g_{\overset{\cdot }{y}}^{j}( t,y,\overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) \right) dt \nonumber \\&\quad + \int \limits _{a}^{b}\sum _{k\in K^{+}(t) }\zeta _{k}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( h_{y}^{k} ( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ h_{\overset{\cdot }{y} }^{k}( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt \nonumber \\&\int \limits _{a}^{b}\!\sum _{k\in K^{-}( t) }( -\zeta _{k}) \Phi \!\left( \! t,x,\overset{\cdot }{x}, y,\overset{\cdot }{y};\left( -h_{y}^{k}( t,y,\overset{\cdot }{y}) - \frac{d}{dt}\left[ \! -h_{\overset{\cdot }{y}}^{k}( t,y,\overset{\cdot }{y}) \!\right] , \rho _{h_{k}}\!\right) \!\right) dt\!<\!0.\nonumber \\ \end{aligned}$$
(65)
We denote
$$\begin{aligned} \widetilde{\lambda }_{i}&= \frac{\lambda _{i}}{\sum _{i=1}^{p}\lambda _{i}+\sum _{j=1}^{q}\xi _{j}( t) +\sum _{k\in K^{+}( t)}\zeta _{k}(t) -\sum _{k\in K^{-}( t) }\zeta _{k}( t) },\quad i\in A, \quad \quad \end{aligned}$$
(66)
$$\begin{aligned} \widetilde{\xi }_{j} ( t)&= \frac{\xi _{j} ( t ) }{\sum _{i=1}^{p}\lambda _{i}+\sum _{j=1}^{q}\xi _{j} (t) +\sum _{k\in K^{+}(t) }\zeta _{k}( t) -\sum _{k\in K^{-}(t)}\zeta _{k}(t) },\quad j\in J, \quad \quad \end{aligned}$$
(67)
$$\begin{aligned} \widetilde{\zeta }_{k} (t )&= \frac{\zeta _{k} (t) }{\sum _{i=1}^{p}\lambda _{i}+\sum _{j=1}^{q}\xi _{j}(t) +\sum _{k\in K^{+} (t ) }\zeta _{k} (t) -\sum _{k\in K^{-} (t) }\zeta _{k} (t) },\quad k\in K^{+} (t),\nonumber \\\end{aligned}$$
(68)
$$\begin{aligned} \widetilde{\zeta }_{k}(t)&= \frac{-\zeta _{k}(t) }{\sum _{i=1}^{p}\lambda _{i}+\sum _{j=1}^{q}\xi _{j}(t) +\sum _{k\in K^{+} (t ) }\zeta _{k} (t ) -\sum _{k\in K^{-} (t) }\zeta _{k} (t) },\quad k\in K^{-} (t).\nonumber \\ \end{aligned}$$
(69)
By (
66)–(
69), it follows that
$$\begin{aligned} \sum _{i=1}^{p}\widetilde{\lambda }_{i}+\sum _{j=1}^{q}\widetilde{\xi }_{j}(t) +\sum _{k\in K^{+}(t) }\widetilde{\zeta }_{k}(t) +\sum _{k\in K^{-}(t) }\widetilde{\zeta }_{k}(t) =1. \end{aligned}$$
(70)
Combining (
65)–(
69), we get
$$\begin{aligned}&\!\!\int \limits _{a}^{b}\sum _{i=1}^{p}\widetilde{\lambda }_{i}\Phi \left( t,x, \overset{\cdot }{x},y,\overset{\cdot }{y};\left( f_{y}^{i}( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ f_{\overset{\cdot }{y}}^{i}( t,y, \overset{\cdot }{y}) \right] ,\rho _{f_{i}}\right) \right) dt \nonumber \\&\!\!\quad +\int \limits _{a}^{b}\sum _{j=1}^{q}\widetilde{\xi }_{j}\Phi \left( t,x, \overset{\cdot }{x},y,\overset{\cdot }{y};\left( g_{y}^{j} ( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ g_{\overset{\cdot }{y}}^{j} ( t,y, \overset{\cdot }{y}) \right] ,\rho _{g_{j}}\right) \right) dt \nonumber \\&\!\!\quad +\int \limits _{a}^{b}\sum _{k\in K^{+}( t) }\widetilde{\zeta }_{k}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( h_{y}^{k}( t,y,\overset{\cdot }{y}) -\frac{d}{dt}\left[ h_{\overset{\cdot }{y}}^{k}( t,y,\overset{\cdot }{y}) \right] ,\rho _{h_{k}}\right) \right) dt \nonumber \\&\!\!\quad +\int \limits _{a}^{b}\sum _{k\in K^{-}(t) }\widetilde{\zeta }_{k}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\left( -h_{y}^{k}( t,y,\overset{\cdot }{y})- \frac{d}{dt}\left[ \! -h_{\overset{\cdot }{y} }^{k}( t,y,\overset{\cdot }{y}) \!\right] ,\rho _{h_{k}}\!\right) \!\right) dt\!<\!0.\nonumber \\ \end{aligned}$$
(71)
By Definition 3, it follows that the functional
\(\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y};\cdot \right) \) is convex on
\(R^{n+1}\). Thus, by (
71) and (
70), Definition 2 implies
$$\begin{aligned}&\int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y} ;\left( \sum _{i=1}^{p}\widetilde{\lambda }_{i}f_{y}^{i} ( t,y, \overset{\cdot }{y}) +\sum _{j=1}^{q}\widetilde{\xi }_{j}g_{y}^{j}( t,y,\overset{\cdot }{y}) \right. \right. \\&\quad +\sum _{k\in K^{+}(t) }\widetilde{\zeta }_{k}h_{y}^{k} ( t,y,\overset{\cdot }{y}) +\sum _{k\in K^{-} (t) } ( -\widetilde{\zeta }_{k}) h_{y}^{k} ( t,y,\overset{\cdot }{y} ) \\&\quad -\frac{d}{dt}\left[ \sum _{i=1}^{p}\widetilde{\lambda }f_{\overset{\cdot }{y}}^{i} ( t,y,\overset{\cdot }{y}) +\sum _{j=1}^{q}\widetilde{\xi }_{j}g_{\overset{\cdot }{y}}^{j}( t,y,\overset{\cdot }{y}) + \sum _{k\in K^{+}(t) }\widetilde{\zeta }_{k}h_{\overset{\cdot }{y}}^k ( t,y,\overset{\cdot }{y})\right. \\&\quad \left. \left. \left. +\!\sum _{k\in K^{-}( t)}( -\widetilde{\zeta _{k} }) h_{\overset{\cdot }{y}}^k ( t,y,\overset{\cdot }{y})\right] \!, \sum _{i=1}^{p}\widetilde{\lambda }\rho _{f_{i}}\!+\!\sum _{j=1}^{q}\widetilde{\xi }_{j}\rho _{g_{j}}\!+\!\sum _{k\in K^{+} ( t) \cup K^{-} (t ) }\widetilde{\zeta }_{k}\rho _{h_{k}}\right) \right) dt\!<\!0. \end{aligned}$$
Hence, the first constraint of (
VWD) yields
$$\begin{aligned} \int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y} ;\left( 0, \sum _{i=1}^{p}\widetilde{\lambda }_{i}\rho _{f_{i}}\!+\!\sum _{j=1}^{q}\widetilde{\xi }_{j} ( t) \rho _{g_{j}} \!+\!\sum _{k\in K^{+}(t) \cup K^{-} (t) }\widetilde{\zeta }_{k}( t) \rho _{h_{k}}\!\right) \!\right) dt\!<\!0.\nonumber \\ \end{aligned}$$
(72)
From the hypothesis (e), it follows that
$$\begin{aligned} \sum _{i=1}^{p}\widetilde{\lambda }_{i}\rho _{f_{i}}+\sum _{j=1}^{q}\widetilde{ \xi }_{j} (t ) \rho _{g_{j}}+\sum _{k=1}^{s}\widetilde{\zeta }_{k}(t) \rho _{h_{k}}\geqq 0. \end{aligned}$$
(73)
By Definition 3, it follows that
\(\Phi ( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y}, ( 0,a)) \geqq 0\) for any
\(a\in R_{+}\). Thus, (
73) implies that the following inequality
$$\begin{aligned} \int \limits _{a}^{b}\Phi \left( t,x,\overset{\cdot }{x},y,\overset{\cdot }{y} ;\left( 0, \sum _{i=1}^{p}\widetilde{\lambda }_{i}\rho _{f_{i}}\!+\!\sum _{j=1}^{q}\widetilde{\xi }_{j}\left( t\right) \rho _{g_{j}}\!+\! \sum _{k\in K^{+} (t) \cup K^{-}(t) }\widetilde{\zeta }_{k}(t) \rho _{h_{k}}\!\right) \!\right) dt\!\geqq \! 0 \end{aligned}$$
holds, contradicting (
72).
\(\square \)