Bulletin of the American Mathematical Society

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Book Information:

Author: Svatopluk Fučík
Title: Solvability of nonlinear equations and boundary value problems
Additional book information: Mathematics and its Applications, vol. 4, D. Reidel Publishing Company, Dordrecht, Holland/Boston, USA, London, England, 1980, 400 pp., $29.95.

S. Ahmad, A. C. Lazer, and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), no. 10, 933–944. MR 427825, DOI 10.1512/iumj.1976.25.25074

Herbert Amann and Peter Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 1-2, 145–151. MR 549877, DOI 10.1017/S0308210500017017

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 4, 539–603. MR 600524

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4) 93 (1972), 231–246. MR 320844, DOI 10.1007/BF02412022

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7

Antonio Ambrosetti and Giovanni Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), no. 5, 635–645. MR 541874, DOI 10.1016/0362-546X(79)90092-0

Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32. MR 621969, DOI 10.1090/S0002-9947-1981-0621969-9

J. Bebernes, R. Gaines, and K. Schmitt, Existence of periodic solutions for third and fourth order ordinary differential equations via coincidence degree, Ann. Soc. Sci. Bruxelles Sér. I 88 (1974), 25–36. MR 340729

Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, DOI 10.1007/BF01389883

M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1974/75), 837–846. MR 377274, DOI 10.1512/iumj.1975.24.24066

11.

M. S. Berger, Review of "Nonlinear differential equations", by S. Fučík and A. Kufner, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 362-368.

H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326. MR 513090

Alfonso Castro B., A two-point boundary value problem with jumping nonlinearities, Proc. Amer. Math. Soc. 79 (1980), no. 2, 207–211. MR 565340, DOI 10.1090/S0002-9939-1980-0565340-1

Lamberto Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations 1 (1963), 149–187. MR 151678

Lamberto Cesari, Functional analysis and Galerkin’s method, Michigan Math. J. 11 (1964), 385–414. MR 173839

Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693–712. MR 622618, DOI 10.1002/cpa.3160340503

Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633

David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 296777, DOI 10.1512/iumj.1972.22.22008

David C. Clark, On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations, Proc. Amer. Math. Soc. 39 (1973), 579–584. MR 315217, DOI 10.1090/S0002-9939-1973-0315217-8

D. G. Costa and J. V. A. Gonçalves, Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance, J. Math. Anal. Appl. 84 (1981), no. 2, 328–337. MR 639667, DOI 10.1016/0022-247X(81)90171-2

E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. (9) 57 (1978), no. 4, 351–366. MR 524624

E. N. Dancer, Nonuniqueness for nonlinear boundary-value problems, Rocky Mountain J. Math. 13 (1983), no. 3, 401–412. MR 715763, DOI 10.1216/RMJ-1983-13-3-401

C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289–307. MR 32923, DOI 10.1090/S0002-9947-1949-0032923-4

Hans Ehrmann, Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung, Math. Ann. 134 (1957), 167–194 (German). MR 92056, DOI 10.1007/BF01342795

Djairo G. de Figueiredo and Jean-Pierre Gossez, Nonlinear perturbations of a linear elliptic problem near its first eigenvalue, J. Differential Equations 30 (1978), no. 1, 1–19. MR 511471, DOI 10.1016/0022-0396(78)90019-0

Djairo Guedes de Figueiredo and Wei Ming Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Anal. 3 (1979), no. 5, 629–634. MR 541873, DOI 10.1016/0362-546X(79)90091-9

Svatopluk Fučík, Nonlinear equations with noninvertible linear part, Czechoslovak Math. J. 24(99) (1974), 467–495. MR 348568

Svatopluk Fučík and Vladimír Lovicar, Periodic solutions of the equation $x^{^{\prime \prime }}(t)+g(x(t))=p(t)$, Časopis Pěst. Mat. 100 (1975), no. 2, 160–175. MR 0385239

Svatopluk Fučík, Milan Kučera, and Jindřich Nečas, Ranges of nonlinear asymptotically linear operators, J. Differential Equations 17 (1975), 375–394. MR 372696, DOI 10.1016/0022-0396(75)90050-9

Svatopluk Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat. 101 (1976), no. 1, 69–87 (English, with Loose Russian summary). MR 0447688

Robert E. Gaines and Jean L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR 0637067

A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), no. 1, 117–176 (German). MR 1555304, DOI 10.1007/BF02547519

Alain Haraux, Nonlinear evolution equations—global behavior of solutions, Lecture Notes in Mathematics, vol. 841, Springer-Verlag, Berlin-New York, 1981. MR 610796

Joachim A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1970/71), 983–996. MR 279423, DOI 10.1512/iumj.1971.20.20094

Helmut Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493–514. MR 682663, DOI 10.1007/BF01457453

Jerry L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), no. 5, 567–597. MR 477445, DOI 10.1002/cpa.3160280502

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609–623. MR 0267269

A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), no. 1, 282–294. MR 639539, DOI 10.1016/0022-247X(81)90166-9

39.

A. C. Lazer and P. J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem (to appear).

40.

L. A. Lyustemik, On a class of nonlinear operators in Hilbert space, Izv. Akad. Nauk SSSR Ser. Mat. 3 (1939), 257-264. (Russian)

J. Mawhin, $L_{2}$-estimates and periodic solutions of some nonlinear differential equations, Boll. Un. Mat. Ital. (4) 10 (1974), 341–354 (English, with Italian summary). MR 0369823

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52 (1984), no. 2, 264–287. MR 741271, DOI 10.1016/0022-0396(84)90180-3

P. J. McKenna and J. Rauch, Strongly nonlinear perturbations of nonnegative boundary value problems with kernel, J. Differential Equations 28 (1978), no. 2, 253–265. MR 491053, DOI 10.1016/0022-0396(78)90070-0

Jindřich Nečas, The range of nonlinear operators with linear asymptotes which are not invertible, Comment. Math. Univ. Carolinae 14 (1973), 63–72. MR 318995

L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 267–302. MR 609039, DOI 10.1090/S0273-0979-1981-14888-6

E. Podolak, On the range of operator equations with an asymptotically nonlinear term, Indiana Univ. Math. J. 25 (1976), no. 12, 1127–1137. MR 425698, DOI 10.1512/iumj.1976.25.25089

Paul H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 161–177. MR 0501092

Bernhard Ruf, On nonlinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4) 128 (1981), 133–151. MR 640779, DOI 10.1007/BF01789470

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079

50.

S. Solimini, Existence of a third solution for a class of boundary value problems with jumping nonlinearities, preprint.

Michael Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), no. 3-4, 335–364. MR 595426, DOI 10.1007/BF01299609

1.

S. Ahmad, A. Lazer, and J. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), 933-944. MR 0427825

2.

H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 145-151. MR 549877

3.

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa 7 (1980), 539-603. MR 600524

4.

A. Ambrosetti and G. Prodi, On the inversion of some differentiate mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4) 93 (1972), 231-247. MR 320844

5.

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional. Anal. 14 (1973), 349-381. MR 370183

6.

A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. Theory, Applications, Methods 3 (1979), 635-645. MR 541874

7.

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1-32. MR 621969

8.

J. Bebernes, R. Gaines, and K. Schmitt, Existence of periodic solutions of third and fourth-order ordinary differential equations via coincidence degree, Ann. Soc. Sci. Bruxelles, Ser. I 88 (1974), 25-36. MR 340729

9.

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273. MR 537061

10.

M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846. MR 377274

11.

M. S. Berger, Review of "Nonlinear differential equations", by S. Fučík and A. Kufner, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 362-368.

12.

H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa 5 (1978), 225-326. MR 513090

13.

A Castro, A two point boundary value problem with jumping nonlinearities, Proc. Amer. Math. Soc. 79 (1980), 207-211. MR 565340

14.

L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contr. Diff. Equations 1 (1963), 149-187. MR 151678

15.

L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J. 11 (1964), 385-414. MR 173839

16.

C. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), 693-712. MR 622618

17.

S-N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982. MR 660633

18.

D. C. Clark, A variant of Lusternik Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65-74. MR 296777

19.

D. C. Clark, On periodic solutions of autonomous Hamiltonian system of ordinary differential equations, Proc. Amer. Math. Soc. 39 (1973), 579-584. MR 315217

20.

D. G. Costa and J. V. A. Gonçalves, Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance, J. Math. Anal. Appl. 84 (1981), 328-337. MR 639667

21.

E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. 57 (1978), 351-366. MR 524624

22.

E. N. Dancer, Nonuniqueness for nonlinear boundary-value problems (to appear). MR 715763

23.

C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 60 (1949), 289-307. MR 32923

24.

H. Ehrmann, Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlicher nichtlinear differentialgleichungen zweiter ordnung, Math. Ann. 134 (1957), 167-194. MR 92056

25.

D. G. de Figueiredo and J.-P. Gossez, Nonlinear perturbations of a linear elliptic problem near its first eigenvalue, J. Differential Equations 30 (1978), 1-19. MR 511471

26.

D. G. de Figueiredo and W-M Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Analysis, Theory, Applications, Methods 3 (1979), 629-634. MR 541873

27.

S. Fučík, Nonlinear equations with noninvertible linear part, Czechoslovak Math. J. 24 (1974), 259-271. MR 348568

28.

S. Fučík and V. Lovicar, Periodic solutions of the equation x"(t) + g(x(t)) = p(t), Časopis Pest. Mat. 100 (1975), 160-175. MR 385239

29.

S. Fučík, J. Neč, and M. Kučera, Ranges of nonlinear asymptotically linear operators, J. Differential Equations 17 (1975), 375-394. MR 372696

30.

S. Fučík, Boundary value problems with jumping nonlinearities, Časopis. Pěst. Mat. 101 (1976), 69-87. MR 447688

31.

R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations, Lecture Notes in Math., vol. 568, Springer-Verlag, Berlin and New York, 1977. MR 637067

32.

A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta. Math. 54 (1930), 117-176. MR 1555304

33.

A. Haraux, Nonlinear evolution equations--global behavior of solutions, Lecture Notes in Math., vol 841, Springer-Verlag, Berlin and New York, 1981. MR 610796

34.

J. A. Hemple, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1971), 983-996. MR 279423

35.

H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. (to appear). MR 682663

36.

J. L. Kazdan and F. Warner, Remarks on quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. MR 477445

37.

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. MR 267269

38.

A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282-294. MR 639539

39.

A. C. Lazer and P. J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem (to appear).

40.

L. A. Lyustemik, On a class of nonlinear operators in Hilbert space, Izv. Akad. Nauk SSSR Ser. Mat. 3 (1939), 257-264. (Russian)

41.

J. Mawhin, L, Boll. Un. Mat. Ital. 10 (1974), 343-354. MR 369823

42.

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations (to appear). MR 741271

43.

P. J. McKenna and J. Rauch, Strongly nonlinear perturbations of nonnegative boundary value problems with kernel, J. Differential Equations 29 (1978), 253-265. MR 491053

44.

J. Necas, On the range of nonlinear operators with linear asymptotes which are not invertible, Comment. Math. Univ. Carolinae 14 (1973), 63-72. MR 318995

45.

L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 267-302. MR 609039

46.

E. Podolak, On the range of operator equations with an asymptotically nonlinear term, Indiana Univ. Math. J. 25 (1976), 1127-1137. MR 425698

47.

P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Analysis: A collection of Paper in Honor of Erich H. Rothe (L. Cesari, R. Kannan, and H. F. Weinberger, eds.), Academic Press, New York, 1978, pp. 161-177. MR 501092

48.

B. Ruf, On nonlinear problems with jumping nonlinearities, Ann. Mat. Pura Appl. 128 (1981), 133-151. MR 640779

49.

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000. MR 299921

50.

S. Solimini, Existence of a third solution for a class of boundary value problems with jumping nonlinearities, preprint.

51.

M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. (1980), 335-364. MR 595426