Sturm~--- Liouville problem for an equation with a discontinuous nonlinearity

V. N. Pavlenko ; E. Yu. Postnikova

Čelâbinskij fiziko-matematičeskij žurnal, Tome 4 (2019) no. 2, pp. 142-154.

Résumé

On the segment $ [0, 1] $, we consider the Sturm — Liouville problem with a discontinuous nonlinearity on the right-hand side multiplied by a positive parameter. For nonnegative values of the phase variable $u$ the nonlinearity is zero, and for negative values it coincides with a continuous function on $ [0,1] \times (- \infty; 0] $. The boundary conditions are $ u (0) = a $, $ u (1) = b $, where $ a, b $ are positive numbers. The initial problem is converted to an equivalent homogeneous one, which for all positive values of the parameter has a zero solution. Its spectrum consists of those parameter values for which the boundary value problem has a nonzero solution. Assuming sublinear growth of nonlinearity at infinity for each positive value of the parameter we construct an iterative process that converges monotonically to the minimal solution. It is proved that the spectrum of the problem is of the form $ [C; + \infty) $, where $ C> 0 $, if it is non-empty.

Détail

Keywords: nonlinear spectral problem, Sturm — Liouville equation, discontinuous nonlinearity, monotone iterations.

Bibliographie

[1] H. Herbert, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces”, SIAM Rev., 18:4 (1976), 620–709 | DOI | MR | Zbl

[2] Shragin I.E., “Conditions for measurability of superpositions”, Reports of USSR Academy of Sciences, 197:2 (1971), 295–298 (In Russ.) | MR | Zbl

[3] Goldshtik A.V., “A mathematical model of discontinuous incompressible fluid flows”, Reports of USSR Academy of Sciences, 147:6 (1962), 1310–1313 (In Russ.)

[4] P. Nistri, “Positive solutions of a non-linear eigenvalue problem with discontinuous non-linearity”, Proc. of the Royal Soc. of Edinburgh. Section A: Mathematics, 83:1–2 (1979), 133–145 | DOI | MR | Zbl

[5] Potapov D.K., “Sturm — Liouville’s problem with discontinuous nonlinearity”, Differential Equations, 50:9 (2014), 1272–1274 | DOI | DOI | MR | Zbl

[6] Potapov D.K., “Existence of solutions, estimates for the differential operator, and a “separating” set in a boundary value problem for a second-order differential equation with a discontinuous nonlinearity”, Differential Equations, 51:7 (2015), 967–972 | DOI | DOI | MR | Zbl

[7] S. Bensid, J. Diaz, “Stability results for discontinuous nonlinear elliptic and parabolic problems with a $S$-shaped bifurcation branch of stationary solutions”, Discrete and Continuous Dynamical Systems. Ser. B., 22:5 (2017), 1757–1778 | DOI | MR | Zbl

[8] Pavlenko V.N., Potapov D.K., “Existence of two nontrivial solutions for sufficiently large values of the spectral parameter in eigenvalue problems for equations with discontinuous right-hand sides”, Sbornik: Mathematics, 208:1 (2017), 157–172 | DOI | DOI | MR | Zbl

[9] Krasnosel'skii M.A., Pokrovskii A.V., Systems with Hysteresis, Springer-Verlag, Berlin, Heidelberg, 1989, 410 pp. | MR

[10] Pavlenko V.N., “The existence of solutions for nonlinear equations with discontinuous monotone operators”, Bulletin of Moscow State University. Series 1. Mathematics. Mechanics, 1973, no. 6, 21–29 (In Russ.) | Zbl

[11] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984, 513 pp. | MR

[12] Pavlenko V.N., Ul'yanova O.V., “The method of upper and lower solutions for elliptic-type equations with discontinuous nonlinearities”, Russian Mathematics, 42:11 (1998), 65–72 | MR | Zbl

[13] Krasnosel'skii M.A., Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964, 381 pp. | MR | Zbl