Positive solutions for discontinuous problems with applications to ϕ-Laplacian equations

[1] Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984) Book MATH Google Scholar

[2] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidian and Minkowski spaces. Proc. Am. Math. Soc. 137, 171–178 (2009) MATH Google Scholar

[3] Bereanu, C., Jebelean, P., Şerban, C.: The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space. Electron. J. Qual. Theory Differ. Equ. 35, 1–7 (2015) MathSciNet Article MATH Google Scholar

[4] Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian. J. Differ. Equ. 243, 536–557 (2007) MathSciNet Article Google Scholar

[5] Biles, D.C., Federson, M., Pouso, R.L.: A survey of recent results for the generalizations of ordinary differential equations. Abstr. Appl. Anal. 2014, 1–9 (2014) MathSciNet Article Google Scholar

[6] Bonanno, G., Bisci, G.M.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 670675 (2009) MathSciNet Article MATH Google Scholar

[7] Bonanno, G., Buccellato, S.M.: Two point boundary value problems for the Sturm–Liouville equation with highly discontinuous nonlinearities. Taiwan. J. Math. 14(5), 2059–2072 (2010)  MathSciNet Article MATH Google Scholar

[8] Bonanno, G., Giovannelli, N.: An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities. J. Math. Anal. Appl. 308, 596–604 (2005) MathSciNet Article Google Scholar

[9] Bonanno, G., Jebelean, P., Şerban, C.: Three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator. Proc. R. Soc. Edinb. A 147, 673–681 (2017) MathSciNet Article Google Scholar

[10] Cabada, A., Heikkilä, S.: Implicit nonlinear discontinuous functional boundary value $\phi$-Laplacian problems: extremality results. Appl. Math. Comput. 129, 537–549 (2002) MathSciNet Google Scholar

[11] Cabada, A., Pouso, R.L.: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions. Nonlinear Anal. 42, 1377–1396 (2000) MathSciNet Article MATH Google Scholar

[12] Cellina, A., Fryszkowsk, A., Rzezuchowsk, T.: Upper semicontinuity of Nemytskij operators. Ann. Mat. Pura Appl. 160(4), 321–330 (1991) MathSciNet Article Google Scholar

[13] Cid, J.A., Franco, D., Minhós, F.: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc. 41, 72–78 (2009) MathSciNet Article MATH Google Scholar

[14] Cid, J.A., Pouso, R.L.: Ordinary differential equations and systems with time-dependent discontinuity sets. Proc. R. Soc. Edinb. A 134, 617–637 (2004) MathSciNet Article MATH Google Scholar

[15] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) Book MATH Google Scholar

[16] Figueroa, R., Pouso, R.L.: Discontinuous first-order functional boundary value problems. Nonlinear Anal. 69, 2142–2149 (2008) MathSciNet Article MATH Google Scholar

[17] Figueroa, R., Pouso, R.L., Rodríguez-López, J.: A version of Krasnosel’skiĭ’s compression-expansion fixed point theorem in cones for discontinuous operators with applications. Topol. Methods Nonlinear Anal. 51, 493–510 (2018) MathSciNet MATH Google Scholar

[18] Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988) Book Google Scholar

[19] Frigon, M., O’Regan, D.: Fixed points of cone-compressing and cone-extending operators in Fréchet spaces. Bull. Lond. Math. Soc. 35(5), 672–680 (2003) Article MATH Google Scholar

[20] Heikkilä, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Marcel Dekker, New York (1994) MATH Google Scholar

[21] Heikkilä, S., Seikkala, S.: On singular, functional, nonsmooth and implicit $\phi$-Laplacian initial and boundary value problems. J. Math. Anal. Appl. 308, 513–531 (2005) MathSciNet Article Google Scholar

[22] Herlea, D.-R.: Existence, localization and multiplicity of positive solutions for the Dirichlet BVP with $\phi$-Laplacian. Fixed Point Theory 18(1), 237–246 (2017) MathSciNet Article Google Scholar

[23] Herlea, D.-R.: Harnack type inequalities and multiple positive solutions of nonlinear problems. Ph.D. Thesis, Babeş -Bolyai University, Cluj–Napoca (2016)

[24] Herlea, D.-R.: Positive solutions for second-order boundary-value problems with $\phi$-Laplacian. Electron. J. Differ. Equ. 18, 1–8 (2016) MathSciNet Google Scholar

[25] Herlea, D.-R., Precup, R.: Existence, localization and multiplicity of positive solutions to $\phi$-Laplace equations and systems. Taiwan. J. Math. 20, 77–89 (2016) MathSciNet Article Google Scholar

[26] Hu, S.: Differential equations with discontinuous right-hand sides. J. Math. Anal. Appl. 154, 377–390 (1991) MathSciNet Article MATH Google Scholar

[27] Pouso, R.L.: Schauder’s fixed–point theorem: new applications and a new version for discontinuous operators. Bound. Value Probl. 2012, 7 (2012) MathSciNet Article MATH Google Scholar

[28] McShane, E.J.: Integration. Princeton University Press, Princeton (1967) MATH Google Scholar

[29] O’Regan, D., Perán, J.: One dimensional $\phi$-Laplacian functional equations. J. Math. Anal. Appl. 371, 177–183 (2010) MathSciNet Article Google Scholar

[30] Precup, R.: Fixed point theorems for decomposable multi-valued maps and applications. Z. Anal. ihre. Anwend. 22(4), 843–861 (2003) MathSciNet Article MATH Google Scholar

[31] Precup, R.: Moser–Harnack inequality, Krasnosel’ski ĭ type fixed point theorems in cones and elliptic problems. Topol. Methods Nonlinear Anal. 40, 301–313 (2012) MathSciNet MATH Google Scholar

[32] Rachunková, I., Tvrdý, M.: Periodic problems with $\phi$-Laplacian involving non-ordered lower and upper functions. Fixed Point Theory 6, 99–112 (2005) MathSciNet Google Scholar

[33] Stromberg, K.R.: An Introduction to Classical Real Analysis. Wadsworth Inc., Belmon (1981) MATH Google Scholar

[34] Webb, J.R.L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions. Proc. R. Soc. Edinb. A 148, 427–446 (2008)