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Existence results for boundary value problems associated with singular strongly nonlinear equations

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Abstract

We consider a strongly nonlinear differential equation of the following general type:

$$\begin{aligned} (\Phi (a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text {a.e. on }[0,T], \end{aligned}$$

where f is a Carathédory function, \(\Phi \) is a strictly increasing homeomorphism (the \(\Phi \)-Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that \(1/h \in L^p(0,T)\) for some \(p> 1\), and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.

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References

  1. Bereanu, C., Mawhin, J.: Periodic solutions of nonlinear perturbations of \(\Phi \)-Laplacians with possibly bounded \(\Phi \). Nonlinear Anal. TMA 68, 1668–1681 (2008)

    Article  MathSciNet  Google Scholar 

  2. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science and Business Media, New York (2010)

    Book  Google Scholar 

  3. Cabada, A., Pouso, R.L.: Existence results for the problem \((\phi (u^\prime ))^\prime =f(t, u, u^\prime )\) with periodic and Neumann boundary conditions. Nonlinear Anal. TMA 30, 1733–1742 (1997)

    Article  MathSciNet  Google Scholar 

  4. Cabada, A., Pouso, R.L.: Existence results for the problem \((\phi (u^\prime ))^\prime =f(t, u, u^\prime )\) with nonlinear boundary conditions. Nonlinear Anal. TMA 35, 221–231 (1999)

    Article  Google Scholar 

  5. Cabada, A., O’Regan, D., Pouso, R.L.: Second order problems with functional conditions including Sturm–Liouville and multipoint conditions. Math. Nachr. 281, 1254–1263 (2008)

    Article  MathSciNet  Google Scholar 

  6. Calamai, A.: Heteroclinic solutions of boundary value problems on the real line involving singular \(\Phi \)-Laplacian operators. J. Math. Anal. Appl. 378, 667–679 (2011)

    Article  MathSciNet  Google Scholar 

  7. Calamai, A., Marcelli, C., Papalini, F.: Boundary value problems for singular second order equations. Fixed Point Theory Appl. 2018, Paper No. 20

  8. Cupini, G., Marcelli, C., Papalini, F.: Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators. Differ. Int. Equ. 24, 619–644 (2011)

    MathSciNet  MATH  Google Scholar 

  9. El Khattabi, N., Frigon, M., Ayyadi, N.: Multiple solutions of boundary value problems with \(\phi \)-Laplacian operators and under a Wintner–Nagumo growth condition. Bound. Value Probl. 2013, 236, 21 (2013)

    Article  MathSciNet  Google Scholar 

  10. Ferracuti, L., Papalini, F.: Boundary value problems for strongly nonlinear multivalued equations involving different \(\Phi \)-Laplacians. Adv. Differ. Equ. 14, 541–566 (2009)

    MATH  Google Scholar 

  11. Liu, Y.: Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line. Nonlinear Anal. Model. Control 17, 460–480 (2012)

    Article  MathSciNet  Google Scholar 

  12. Liu, Y., Yang, P.: Existence and non-existence of positive solutions of BVPs for singular ODEs on whole lines. Kyungpook Math. J. 55, 997–1030 (2015)

    Article  MathSciNet  Google Scholar 

  13. Marcelli, C.: Existence of solutions to boundary-value problems governed by general non-autonomous nonlinear differential operators. Electron. J. Differ. Equ. 171 (2012)

  14. Marcelli, C.: The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators. Bound. Value Probl. 252 (2013)

  15. Marcelli, C., Papalini, F.: Boundary value problems for strongly nonlinear equations under a Wintner–Nagumo growth condition. Bound. Value Probl. 2017 Paper No. 183 (2017)

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131, Ancona, Italy

    Stefano Biagi

  2. Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131, Ancona, Italy

    Alessandro Calamai

  3. Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131, Ancona, Italy

    Francesca Papalini

Authors
  1. Stefano Biagi
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  2. Alessandro Calamai
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  3. Francesca Papalini
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Correspondence to Stefano Biagi.

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Biagi, S., Calamai, A. & Papalini, F. Existence results for boundary value problems associated with singular strongly nonlinear equations. J. Fixed Point Theory Appl. 22, 53 (2020). https://doi.org/10.1007/s11784-020-00784-7

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  • DOI: https://doi.org/10.1007/s11784-020-00784-7

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