Given a topological space
X
X
, an interval
I\subseteq {\bf R}
I
⊆
R
and five continuous functions
\varphi, \psi, \omega: X\to {\bf R}
φ
,
ψ
,
ω
:
X
→
R
,
\alpha, \beta:I\to {\bf R}
α
,
β
:
I
→
R
, we are interested in the infimum of the function
\Phi:X\to ]-\infty,+\infty]
Φ
:
X
→
]
−
∞
,
+
∞
]
defined by
\Phi(x)=\sup_{\lambda\in I}(\alpha(\lambda)\varphi(x)+\beta(\lambda)\psi(x))+\omega(x)\,.
Φ
(
x
)
=
sup
λ
∈
I
(
α
(
λ
)
φ
(
x
)
+
β
(
λ
)
ψ
(
x
)
)
+
ω
(
x
)
.
Using a recent minimax theorem of the author [see Minimax theorems in a fully non-convex setting, J. Nonlinear Var. Analysis 3 (2019) 45-52], we build a general scheme which provides the exact value of
\inf_X\Phi
inf
X
Φ
for a large class of functions
\Phi
Φ
. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions
\gamma, \eta:X\to {\bf R}
γ
,
η
:
X
→
R
such that, for some
\tilde x\in X
x
~
∈
X
, one has
\gamma(\tilde x)\varphi(\tilde x)+\eta(\tilde x)\psi(\tilde x)+\omega(\tilde x) = \inf_{x\in X}(\gamma(\tilde x)\varphi(x)+\eta(\tilde x)\psi(x)+\omega(x))\,.
γ
(
x
~
)
φ
(
x
~
)
+
η
(
x
~
)
ψ
(
x
~
)
+
ω
(
x
~
)
=
inf
x
∈
X
(
γ
(
x
~
)
φ
(
x
)
+
η
(
x
~
)
ψ
(
x
)
+
ω
(
x
)
)
.