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In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation 			   $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &amp;x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$  { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , where  λ  and  L  are positive parameters,    $f\in C[0,\infty ) \cap C^{2}(0,\infty )$  f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , and    $f(u)&gt;0$  f ( u ) &gt; 0 for    $0&lt; u&lt; L$  0 &lt; u &lt; L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when  f  satisfies    $f''(u)&gt;0$  f ″ ( u ) &gt; 0 and    $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$  u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) for    $0&lt; u&lt; L$  0 &lt; u &lt; L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of  λ . The arguments are based upon a detailed analysis of the time map.

Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation