| Summary: | In this paper, we consider the second order discontinuous differential equation in the real line, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mi>a</mi> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>,</mo> <mi>u</mi> </mfenced> <mspace width="4.pt"></mspace> <mi>ϕ</mi> <mfenced separators="" open="(" close=")"> <msup> <mi>u</mi> <mo>′</mo> </msup> </mfenced> </mfenced> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> </mfenced> <mo>,</mo> <mspace width="4.pt"></mspace> <mi>a</mi> <mo>.</mo> <mi>e</mi> <mo>.</mo> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mi>u</mi> <mrow> <mo>(</mo> <mo>−</mo> <mo>∞</mo> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>ν</mi> <mo>−</mo> </msup> <mo>,</mo> <mspace width="4.pt"></mspace> <mspace width="4.pt"></mspace> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mo>∞</mo> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>ν</mi> <mo>+</mo> </msup> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> an increasing homeomorphism such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mi mathvariant="double-struck">R</mi> <mo>)</mo> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mi>C</mi> <mo>(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> <mo>→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> a <inline-formula> <math display="inline"> <semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics> </math> </inline-formula>-Carathéodory function and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ν</mi> <mo>−</mo> </msup> <mo>,</mo> <msup> <mi>ν</mi> <mo>+</mo> </msup> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> such that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ν</mi> <mo>−</mo> </msup> <mo><</mo> <msup> <mi>ν</mi> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>. The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> </mrow> </semantics> </math> </inline-formula>. To the best of our knowledge, this result is even new when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo> <mo>=</mo> <mi>y</mi> </mrow> </semantics> </math> </inline-formula>, that is for equation <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mi>a</mi> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mfenced> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mo>,</mo> <mspace width="4.pt"></mspace> <mi>a</mi> <mo>.</mo> <mi>e</mi> <mo>.</mo> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>. Moreover, these results can be applied to classical and singular <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula>-Laplacian equations and to the mean curvature operator.
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