A bisection-type method based on \(\alpha\) -dense curves

<p>Let <span class="math inline">\((X,d)\)</span> be a metric space, <span class="math inline">\(D\subset (X,d)\)</span> and <span class="math inline">\(f:D \longrightarrow \mathbb{R}\)</span> a continuous function (with respect to the metric <span class="math inline">\(d\)</span> and the Euclidean metric on <span class="math inline">\(\mathbb{R}\)</span>) . Under suitable conditions on the function <span class="math inline">\(f\)</span> and the set <span class="math inline">\(D\)</span>, we prove the existence of zeros of the function <span class="math inline">\(f\)</span> on <span class="math inline">\(D\)</span> by using the so-called <span class="math inline">\(\alpha\)</span>-dense curves. To be more precise, we prove that if <span class="math inline">\(f(x_{1})f(x_{2})<0\)</span> for some <span class="math inline">\(x_{1},x_{2}\in D\)</span> and <span class="math inline">\(D\)</span> is densifiable then <span class="math inline">\(f\)</span> has some zero in <span class="math inline">\(D\)</span>. Moreover, from this result we derive a numerical method to approximate, with arbitrarily small error, at least one of these zeros. In particular, as a compact interval <span class="math inline">\([a,b]\)</span> is densifiable, our method generalizes the well known bisection method. The feasibility and reliability of the proposed method is illustrated by several numerical examples.</p>