Differential equations involving homeomorphism with nonlinear boundary conditions

We establish existence results for differential equations of the form: (φ(u′))′=f(t,u,u′),$$ {\left(\varphi \left({u}^{\prime}\right)\right)}^{\prime }=f\left(t,u,{u}^{\prime}\right), $$ where f$$ f $$ is continuous and φ$$ \varphi $$ is a homeomorphism, with nonlinear boundary conditions: g(u(0),G(u′))=h(u′(1),H(u))=0;g(u′(0),G(u))=h(u(1),H(u′))=0;g(u(0),G(u))=h(u′(1),H(u′))=0org(u′(0),G(u′))=h(u(1),H(u))=0,$$ {\displaystyle \begin{array}{cc}\hfill g\left(u(0),G\left({u}^{\prime}\right)\right)=h\left({u}^{\prime }(1),H(u)\right)=& 0;\kern6.05pt g\left({u}^{\prime }(0),G(u)\right)=h\left(u(1),H\left({u}^{\prime}\right)\right)=0;\hfill \\ {}\hfill g\left(u(0),G(u)\right)=h\left({u}^{\prime }(1),H\left({u}^{\prime}\right)\right)=& 0\kern6.05pt \mathrm{or}\kern6.05pt g\left({u}^{\prime }(0),G\left({u}^{\prime}\right)\right)=h\left(u(1),H(u)\right)=0,\hfill \end{array}} $$ where g,h$$ g,h $$ are continuous and G,H$$ G,H $$ are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators.

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