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Boundary value problems for quasilinear second order elliptic equations. (English) Zbl 0378.35022


MSC:

35J25 Boundary value problems for second-order elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI

References:

[1] Nagumo, M., Über die Differentialgleichung \(y″=ƒ(x,y,y′)\), Proc. phys.-math. Soc. Japan, 19, 861-866 (1937) · JFM 63.1021.04
[2] Nagumo, M., Über das Randwertproblem der nichtlinearen gewöhnlichen Differentialgleichung zweiter Ordnung, Proc. phys.-math. Soc. Japan, 24, 845-851 (1942) · Zbl 0061.17205
[3] Nagumo, M., On principally linear elliptic differential equations of the second order, Osaka math. J., 6, 207-229 (1954) · Zbl 0057.08201
[4] Akǒ, K., On the Dirichlet problem for quasilinear elliptic differential equations of the second order, J. math. Soc. Japan, 13, 45-62 (1961) · Zbl 0102.09303
[5] Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Ind. Univ. Math. J., 21, 125-146 (1971) · Zbl 0219.35037
[6] Erbe, L., Nonlinear boundary value problems for second order differential equations, J. diff. Eqns, 7, 459-472 (1970) · Zbl 0284.34017
[7] GainesMawhinJ. diff. Eqns; GainesMawhinJ. diff. Eqns
[8] Keller, H. B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Archs ration. Mech. Analysis, 5, 363-381 (1969) · Zbl 0188.17102
[9] Bandle, C., Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems, Archs ration. Mech. Analysis, 58, 219-238 (1975) · Zbl 0335.35046
[10] Keller, H. B.; Cohen, D. S., Some position problems suggested by nonlinear heat generation, J. Math. Mech., 16, 1361-1376 (1967) · Zbl 0152.10401
[11] Simpson, R. B.; Cohen, D. S., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, 895-910 (1970) · Zbl 0195.39502
[12] Rabinowitz, P., Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Communs. pure appl. Math., 23, 939-961 (1970) · Zbl 0202.08902
[13] Crandall, G.; Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19, 1083-1102 (1970) · Zbl 0206.09705
[14] Turner, R. E.L., Superlinear Sturm-Liouville problems, J. diff. Eqns., 13, 157-171 (1973) · Zbl 0272.34031
[15] Cushing, J., Global branches of solutions to nonlinear elliptic eigenvalue problems, Ind. Univ. Math. J., 201, 1035-1045 (1971) · Zbl 0214.36403
[16] Wolkowisky, J. H., Nonlinear Sturm-Liouville problems, Archs ration. Mech. Analysis, 35, 299-320 (1969) · Zbl 0186.41203
[17] Wolkowisky, J. H., Branches of periodic solutions of nonlinear Hill’s equations, J. diff. Eqns., 11, 385-400 (1972) · Zbl 0217.12101
[18] KnoblochSchmittProc. R. phys. Soc., Edinb.,; KnoblochSchmittProc. R. phys. Soc., Edinb.,
[19] Gelfand, I. M., Some problems in the theory of quasilinear equations, Am. math. Soc., Transl., 29, 295-381 (1963) · Zbl 0127.04901
[20] Joseph, D. D.; Sparrow, E. M., Nonlinear diffusion induced by nonlinear sources, Q.J. appl. Math., 28, 327-342 (1970) · Zbl 0208.13006
[21] Markus, L.; Amundson, N., Nonlinear boundary value problems arising in chemical reactor theory, J. diff. Eqns, 4, 102-113 (1968) · Zbl 0155.13301
[22] Pazy, A.; Rabinowitz, P., Corrigendum, Iibd., 35, 409-410 (1969)
[23] Bandle, C., Existence theorems and a priori bounds for a class of nonlinear Dirichlet problems with mixed boundary conditions, J. diff. Eqns, 19, 36-45 (1975) · Zbl 0305.35046
[24] Jackson, L., Subfunctions and boundary value problems for second order ordinary differential equations, Adv. Math., 2, 307-363 (1968) · Zbl 0197.06401
[25] Schmitt, K., Randwertaufgaben für gewöhnliche Differentialgleichungen, Proc. Steiermark. Math. Symposium, Vol. VII, 1-55 (1973), Graz, Austria · Zbl 0276.34012
[26] Amann, H., Fixed point theorems and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044
[27] Mawhin, J., Nonlinear perturbations of Fredholm mappings in normed spaces and applications to differential equations, (Trabalho de Matem. No. 61 (May 1974), Univ. de Brasilia) · Zbl 0334.34007
[28] Mawhin; Mawhin
[29] Mawhin, J., Recent results on periodic solutions of differential equations, (International Conference on Differential Equations (1975), Acad. Press: Acad. Press New York), 537-556 · Zbl 0321.34030
[30] Mawhin, J., Recent trends in nonlinear boundary value problems, Proc. Int. Conf. Nonl. Oscillations (1975), Berlin, DDR · Zbl 0362.34015
[31] Mawhin; Mawhin · Zbl 0463.47039
[32] MawhinRocky Mount. J. Math.; MawhinRocky Mount. J. Math.
[33] Bailey, R. B.; Shampine, L. F.; Waltman, P. E., Nonlinear Two Point Boundary Value Problems (1968), Acad. Press: Acad. Press New York · Zbl 0169.10502
[34] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[35] Rouche, N.; Mawhin, J., Equations Differentielles Ordinaries, Vol. 2 (1973), Masson: Masson Paris · Zbl 0289.34001
[36] Bernfeld, S. R.; Lakshmkantham, V., Introduction to Nonlinear Boundary Value Problems (1974), Acad. Press: Acad. Press New York · Zbl 0286.34018
[37] Gaines, R. E.; Mawhin, J., Coincidence Degree and Nonlinear Differential Equations, (Lectures Notes in Math., Vol. 568 (1977), Springer Verlag: Springer Verlag Berlin) · Zbl 0326.34020
[38] Gudkov, V.; Klokov, Yu; Lepin, A.; Ponomariev, V., Two Point Boundary Value Problems For Ordinary Differential Equations (1973), Izdat. Zinatnei: Izdat. Zinatnei Riga
[39] Sattinger, D. H., Topics in Stability and Bifurcation Theory, (Lecture Notes in Mathematics, Vol. 309 (1973), Springer Verlag: Springer Verlag Berlin) · Zbl 0466.58016
[40] Gustafson, G. B.; Schmitt, K., Nonzero solutions of boundary value problems for second order ordinary and delay—differential equations, J. diff. Eqns, 12, 129-147 (1972) · Zbl 0227.34017
[41] Amann, H., Lectures on Some Fixed Point Theorems, (Monografias de Matematica (1974), IMPA: IMPA Rio de Janeiro)
[42] Cronin, J., Fixed Points and Topological Degree in Nonlinear Analysis (1964), Am. math. Soc: Am. math. Soc Providence, Math. Survey No. 11 · Zbl 0117.34803
[43] Deimling, K., Nichtlineare Gleichungen und Abbildungsgrade (1974), Springer Verlag: Springer Verlag Berlin · Zbl 0281.47033
[44] Fučik, S.; Nečas, J.; Souček, J.; Souček, V., Spectral Analysis of Nonlinear Operators, (Lecture Notes in Math., Vol. 346 (1973), Springer Verlag: Springer Verlag Berlin) · Zbl 0268.47056
[45] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen, Netherlands · Zbl 0121.10604
[46] Krasnosel’skii, M. A., Topological Methods in The Theory of Nonlinear Integral Equations (1964), Pergamon: Pergamon New York · Zbl 0111.30303
[47] Krasnosel’skii, M. A., Translation Along Trajectories of Differential Equations, Vol. 19 (1968), Am. Math. Soc: Am. Math. Soc Providence, Transl. Math. Monographs. · Zbl 1398.34003
[48] Nirenberg, L., Topics In Nonlinear Functional Analysis, (Courant Institute Lecture Notes (1974), New York University: New York University New York) · Zbl 0980.58005
[49] Schwartz, J., Nonlinear Functional Analysis (1969), Gordon and Breach: Gordon and Breach New York · Zbl 0203.14501
[50] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations (1968), Acad. Press: Acad. Press New York · Zbl 0164.13002
[51] Schauder, J., Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Annln, 38, 257-282 (1938) · JFM 60.0422.01
[52] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Communs pure appl. Math., 12, 623-727 (1959) · Zbl 0093.10401
[53] Agmon, S., Lecture on Elliptic Boundary Value Problems, Van Nostrand Math. Studies No. 2 (1965), Princeton · Zbl 0151.20203
[54] Courant, R.; Hilbert, D., Methods of Mathematical Physics, Vol. II (1962), Interscience New York · Zbl 0729.00007
[55] Bers, L.; John, F.; Schechter, M., Partial Differential Equations (1964), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0514.35001
[56] Adams, R. A., Sobolev Spaces (1975), Acad. Press: Acad. Press New York · Zbl 0186.19101
[57] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice Hall: Prentice Hall Englewood Cliff, NJ · Zbl 0153.13602
[58] Bernstein, S., Sur certaines equations differentielles ordinaires du second ordre, C.r. Acad. Sci., Paris, 138, 950-951 (1904) · JFM 35.0341.02
[59] Tomi, F., Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111, 350-366 (1969) · Zbl 0179.43602
[60] Schmitt, K.; Thompson, R., Boundary value problems for infinite systems of second order differential equations, J. diff. Eqns, 18, 277-295 (1974) · Zbl 0302.34081
[61] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’tseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Vol. 23 (1968), Am. Math. Soc: Am. Math. Soc Providence, R.I, Translation Math. Monographs. · Zbl 0174.15403
[62] Heinz, E., On certain nonlinear elliptic differential equations and univalent mappings, J. Analyse math., 5, 197-272 (1956/7) · Zbl 0085.08701
[63] Eastham, M. S.P., The Spectral Theory of Periodic Differential Operators (1973), Scottish Acad. Press: Scottish Acad. Press Edinburgh, Scotland · Zbl 0287.34016
[64] Bebernes, J. W.; Schmitt, K., An existence theorem for periodic boundary value problems for systems of second order differential equations, Arch. Mat. Brno., 7, 173-176 (1972) · Zbl 0284.34016
[65] Bebernes, J. W.; Schmitt, K., Periodic boundary value problems for systems of second order differential equations, J. diff. Eqns, 13, 32-47 (1973) · Zbl 0253.34020
[66] Gustafson, G. B.; Schmitt, K., Periodic solutions of hereditary differential systems, J. diff. Eqns, 13, 567-587 (1973) · Zbl 0254.34070
[67] SchmittThompsonWalter; SchmittThompsonWalter
[68] Schmitt, K.; Volkmann, P., Boundary value problems for second order differential equations in convex subsets of a Banach space, Trans. Am. math. Soc., 220, 3976405 (1976)
[69] Tsai, L. Y., Nonlinear boundary value problems for systems of second order elliptic differential equations (1975), National Chengchi University: National Chengchi University Taiwan, preprint
[70] Schmitt, K.; Tsai, L.-Y., Existence of solutions of nonlinear Schrödinger equations, Proc. R. phys. Soc. Edinb., 76, 227-232 (1977) · Zbl 0346.35050
[71] LemmertVolkmannJ. diff. Eqns; LemmertVolkmannJ. diff. Eqns
[72] HessIndiana Univ. Math. J.; HessIndiana Univ. Math. J.
[73] Tsai, L. Y., Periodic solutions of nonlinear parabolic systems, (Ph.D. Thesis (1975), University of Utah) · Zbl 0993.35050
[74] Weinberger, H., Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8, 295-310 (1975) · Zbl 0312.35043
[75] BebernesSchmittRocky Mount. J. Math.; BebernesSchmittRocky Mount. J. Math.
[76] ChuehConleySmoller; ChuehConleySmoller
[77] ConwaySmoller; ConwaySmoller
[78] ConwaySmoller; ConwaySmoller · Zbl 0368.35021
[79] Hartman, P., On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. Am. math. Soc., 96, 493-507 (1969) · Zbl 0098.06101
[80] Knobloch, H. W., On the existence of periodic solutions of second order vector differential equations, J. diff. Eqns, 9, 67-85 (1971) · Zbl 0211.11801
[81] Bebernes, J. W., A simple alternative problem for finding periodic solutions of second order ordinary differential systems, Proc. Am. math. Soc., 42, 121-127 (1974) · Zbl 0286.34055
[82] Gaines, R., A priori bounds and upper and lower solutions for nonlinear second order boundary value problems, J. diff. Eqns, 12, 291-312 (1972) · Zbl 0227.34016
[83] Bernfeld, S. R.; Ladde, G.; Lakshmikantham, V., Existence of solutions of two point boundary value problems for nonlinear systems, J. diff. Eqns, 18, 103-110 (1975) · Zbl 0305.34029
[84] Akǒ, K., Subfunctions for ordinary differential equations, J. Fac. Sci. Univ. Tokyo, 12, 17-43 (1965) · Zbl 0136.08002
[85] Akǒ, K., Subfunctions for ordinary differential equations II, Funkcialaj. Ekvacioj., 10, 145-162 (1967) · Zbl 0162.11601
[86] Akǒ, K., Subfunctions for ordinary differential equations III, Funkcialaj. Ekvacioj., 11, 111-129 (1968) · Zbl 0224.34015
[87] Akǒ, K., Subfunctions for ordinary differential equations IV, Funkcialaj. Ekvacioj., 11, 185-195 (1968) · Zbl 0193.04801
[88] Akǒ, K., Subfunctions for ordinary differential equations VI, J. Fac. Sci. Univ. Tokyo, 16, 149-156 (1969) · Zbl 0215.44002
[89] Jackson, L.; Schrader, K., Comparison theorems for nonlinear differential equations, J. diff. Eqns, 3, 248-255 (1967) · Zbl 0149.29701
[90] Schmitt, K., A nonlinear boundary value problem, J. diff. Eqns, 7, 527-537 (1969) · Zbl 0198.12301
[91] Schmitt, K., Applications of variational equations to ordinary and partial differential equations—nonzero solutions of boundary value problems, J. diff. Eqns, 17, 154-186 (1975) · Zbl 0278.34018
[92] Shampine, L. F.; Wing, G. M., Existence and uniqueness of solutions of a class of nonlinear elliptic boundary value problems, J. Math. Mech., 19, 971-979 (1970) · Zbl 0197.37401
[93] Amann, H., Existence of multiple solutions for nonlinear elliptic boundary value problems, Ind. Univ. Math. J., 21, 925-935 (1972) · Zbl 0222.35023
[94] Amann, H., Existence and multiplicity theorems for semilinear elliptic boundary value problems, Math. Annln, 150, 281-295 (1976) · Zbl 0331.35026
[95] Knobloch, H. W., Eine neue Methode zur Approximation periodischer Lösungen nichtlinearer Differentialgleichungen zweiter Ordnung, Math. Z., 82, 177-197 (1963) · Zbl 0117.05404
[96] Schmitt, K., Periodic solutions of nonlinear second order differential equations, Math. Z., 98, 200-207 (1967) · Zbl 0153.12501
[97] Hukuhara, M., Familles Kneseriennes el le problème aux limites, Publ. Res. Inst. Math. Sci., 4, 131-147 (1968) · Zbl 0237.34035
[98] Schmitt, K., Periodic solutions of systems of second order differential equations, J. diff. Eqns, 11, 180-182 (1972) · Zbl 0215.44402
[99] Carmignani, R.; Scharder, K., Subfunctions and distributional inequalities, SIAM J. Math. Anal., 8, 52-68 (1977) · Zbl 0358.34015
[100] Hess, P., On strongly nonlinear elliptic problems, (Proc. Conf. Analysis. (1976), Campinas: Campinas Brasil) · Zbl 0341.35038
[101] Hess; Hess
[102] Hess; Hess
[103] SchraderJ. diff. Eqns; SchraderJ. diff. Eqns
[104] Schmitt, K., Periodic solutions of a small period of systems of \(n\) th order differential equations, Proc. Am. math. Soc., 36, 459-463 (1972) · Zbl 0257.34045
[105] Klaasen, G., Differential inequalities and existence theorems for second and third order boundary value problems, J. diff. Eqns, 10, 529-537 (1971) · Zbl 0218.34024
[106] Schmitt, K., Boundary value problems and comparison theorems for ordinary differential equations, SIAM J. Appl. Math., 26, 670-678 (1974) · Zbl 0279.34027
[107] Lan, C. C., Boundary value problems for second and third order differential equations, J. diff. Eqns, 18, 258-274 (1975) · Zbl 0288.34010
[108] Umamaheswaram, S., Boundary value problems for higher order differential equations, J. diff. Eqns, 18, 188-201 (1975) · Zbl 0274.34016
[109] Grimm, L.; Schmitt, K., Boundary value problems for differential equations with deviating arguments, Aequat. Math., 4, 176-190 (1970) · Zbl 0198.13201
[110] De Nevers, K.; Schmitt, K., An application of the shooting method to boundary value problems for second order delay equations, J. math. Analysis Applic., 36, 588-597 (1971) · Zbl 0219.34050
[111] Schmitt, K., Periodic solutions of linear second order differential equations with deviating arguments, Proc. Am. math. Soc., 26, 282-285 (1970) · Zbl 0207.08902
[112] Schmitt, K., Intermediate value theorems for nonlinear periodic functional differential operators, (Equations Différentielles et Fonctionelles Nonlinéaires (1973), Hermann: Hermann Paris), 66-78 · Zbl 0268.34077
[113] MuldowneyWilletCan. J. Math.; MuldowneyWilletCan. J. Math.
[114] Schröder, J., Upper and lower bounds for solutions of generalized two-point boundary value problems, Num. Math., 23, 433-457 (1975) · Zbl 0293.65058
[115] Stüben, K., Einschliessungsaussagen bei Differentialgleichungen mit periodischen Randbedingungen, (Rep. 75-6 (1975), Universität Köln)
[116] Heidel, J. W., A counterexample in nonlinear boundary value problems, Proc. Am. math. Soc., 34, 133-137 (1972) · Zbl 0243.34023
[117] HeidelJ. math. Analysis Applic.; HeidelJ. math. Analysis Applic.
[118] BebernesSIAM J. Math. Anal.; BebernesSIAM J. Math. Anal. · Zbl 0237.34101
[119] Jackson, L. K.; Klaasen, G., A variation of the topological method of Ważewski, SIAM J. Appl. Math., 20, 124-130 (1971) · Zbl 0231.34007
[120] Kaplan, J. L.; Lasota, A.; Yorke, J., An application of the Ważewski retract method to boundary value problems, Zest. Nauk. Univ. Jagel, 16, 7-14 (1974) · Zbl 0301.34018
[121] Kelley; Kelley
[122] Knobloch, H. W., Comparison theorems for nonlinear second order differential equations, J. diff. Eqns, 1, 1-26 (1965) · Zbl 0134.07005
[123] Schmitt, K., Boundary value problems for nonlinear second order differential equations, Monatsh. Math., 72, 347-354 (1968) · Zbl 0162.11502
[124] Shampine, L., Existence for certain nonlinear boundary value problems, J. math. Phys., 10, 1177-1178 (1969) · Zbl 0179.12902
[125] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Ind. Univ. Math. J., 21, 979-1000 (1972) · Zbl 0223.35038
[126] Bernfeld, S.; Chandra, J., Minimal and maximal solutions of nonlinear boundary value problems (1976), University of Texas: University of Texas Arlington, preprint
[127] Chandra, J.; Davis, P. W., A monotone method for quasilinear boundary value problems, Archs ration. Mech. Analysis, 54, 257-266 (1974) · Zbl 0317.34010
[128] Krasnosel’skii, M.; Sobolevskii, P., Structure of the set of solutions of an equation of parabolic type, Ukrain. Math. J., 16, 319-333 (1964)
[129] Erbe, L., A uniqueness theorem for second order differential equations, Math. Z., 109, 92-96 (1969) · Zbl 0172.10902
[130] Schmitt, K., Existence and multiplicity of solutions of some nonlinear equations, Proc. Conf. Ord. Partial Diff. Eqns. Proc. Conf. Ord. Partial Diff. Eqns, Springer Lecture Notes, Vol. 415, 277-295 (1974) · Zbl 0313.34019
[131] Amann, H., On the number of solutions of nonlinear equations in ordered Banach spaces, J. funct. Analysis, 11, 346-384 (1972) · Zbl 0244.47046
[132] Bebernes, J.; Wilhelmson, R., A general boundary value problem technique, J. diff. Eqns, 8, 404-415 (1970) · Zbl 0211.39904
[133] Kelley; Kelley
[134] Nussbaum, R. E., Periodic solutions of some nonlinear integral equations, Proc. Int. Conf. Diff. Eqns (1976), Gainesville, Florida · Zbl 0538.45005
[135] Krein, M.; Rutman, M., Linear operators leaving invariant a cone in a Banach space, Am. math. Soc. Transl., 26, 1-128 (1950)
[136] Dohen, D. S., Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech., 17, 209-216 (1967) · Zbl 0154.12701
[137] Amann, H.; Laetsch, T., Positive solutions of convex nonlinear eigenvalue problems, Ind. Univ. Math. J., 25, 259-270 (1976) · Zbl 0329.35028
[138] Coffman, C., Spectral theory of monotone Hammerstein operators, Pacif. J. Math., 36, 303-322 (1971) · Zbl 0212.46803
[139] CoffmanProc. Am. math. Soc.31; CoffmanProc. Am. math. Soc.31 · Zbl 0229.35067
[140] Pohozaev, N., On the eigenfunctions of the equation Δ \(u\) + λ \(f(u) = 0\), Sov. Math. Dokl., 6, 1408-1411 (1965) · Zbl 0141.30202
[141] Coffman, C., On the positive solutions of boundary value problems for a class of nonlinear differential equations, J. diff. Eqns, 3, 92-111 (1967) · Zbl 0152.08603
[142] Sprekels, J., Existenz und erste Einschliessung positiver Lösungen bei superlinearen Randwertaufgaben zweiter Ordnung, Num. Math., 26, 421-428 (1976) · Zbl 0345.65044
[143] Jacobowitz, H., Periodic solutions of \(x\)″ + \(f\)(t, x) = 0 via the Poincaré-Birkhoff Theorem, J. diff. Eqns, 20, 37-52 (1976) · Zbl 0285.34028
[144] Butler, G. J., Rapid oscillation, nonextendability and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. diff. Eqns, 22, 467-477 (1976) · Zbl 0299.34050
[145] Macki, J. W.; Waltman, P., A nonlinear boundary value problem of Sturm-Liouville type for a twodimensional system of ordinary differential equations, SIAM J. Appl. Math., 21, 225-231 (1971) · Zbl 0248.34031
[146] Macki, J. W.; Waltman, P., A nonlinear Sturm-Liouville problem, Ind. Univ. Math. J., 22, 217-225 (1972) · Zbl 0246.34034
[147] Lasota, A.; Yorke, J., Existence of solutions of two point boundary value problems for nonlinear systems, J. diff. Eqns, 11, 509-518 (1972) · Zbl 0263.34016
[148] Knobloch, H. W., Second order differential inequalities and a nonlinear boundary value problem, J. diff. Eqns, 5, 55-71 (1969) · Zbl 0165.41401
[149] ThompsonSIAM J. Num. Analysis; ThompsonSIAM J. Num. Analysis
[150] Akǒ, K., Two topics on second order elliptic equations, Jap. J. Math., 34, 71-81 (1964) · Zbl 0139.05703
[151] Amann, H., On the existence of solutions of nonlinear equations in ordered Banach spaces, J. funct. Analysis, 11, 346-383 (1972) · Zbl 0244.47046
[152] Amann; Amann
[153] Bazley, N.; Zwahlen, B., A branch of positive solutions of nonlinear eigenvalue problems, Man. Math., 2, 365-374 (1970) · Zbl 0192.49501
[154] Bebernes, J.; Gaines, R.; Schmitt, K., Existence of periodic solutions for third and fourth order ordinary differential equations via coincidence degree, Annls. Soc. scient. Brux., 88, 25-36 (1974) · Zbl 0286.34056
[155] Bobylev, N. A., A two point boundary value problem, Diff. Urav., 11, 1573-1582 (1975) · Zbl 0359.34016
[156] Choquet-Bruhat, Y.; Leray, J., Sur le probleme de Dirichlet quasi-lineaire d’ordre 2, C.r. Acad. Sci. Paris, 274, 81-85 (1972) · Zbl 0227.35045
[157] Cohen, D. S., Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory, SIAM J. Appl. Math., 20, 1-13 (1971) · Zbl 0219.34027
[158] Cohen, D. S.; Hansen, E. B., Inverse problems connected with two-point boundary value problems, SIAM J. Appl. Math., 20, 413-425 (1971) · Zbl 0219.34028
[159] Crandall, M.; Rabinowitz, P., Bifurcation from simple eigenvalues, J. funct. Analysis, 8, 321-340 (1971) · Zbl 0219.46015
[160] Cronin, J., Equations with bounded nonlinearities, J. diff. Eqns, 14, 581-596 (1973) · Zbl 0251.34049
[161] Cronin, J., Quasilinear equations and equations with large nonlinearities, Rocky Mount. Math. J., 4, 41-63 (1974) · Zbl 0287.47042
[162] Dancer, E., Global solution branches for positive mappings, Archs. ration. Mech. Analysis, 52, 181-192 (1973) · Zbl 0275.47043
[163] Dancer, E., Solution branches for mappings in cones and application, Bull. Aust. Math. Soc., 11, 131-143 (1974)
[164] Dancer, E., On a nonlinear elliptic boundary value problem, Bull. Aust. Math. Soc., 12, 399-405 (1975) · Zbl 0297.35031
[165] Dolph, C. L., Nonlinear integral equations of the Hammerstein type, Trans. Am. Math. Soc., 289-307 (1948) · Zbl 0036.20202
[166] Durkovicova, M., On the existence of solutions of a singular boundary value problem, Mat. Cas., 23, 34-39 (1974) · Zbl 0256.34017
[167] Epheser, H., Über die Existenz der Lösungen von Randwettaufgaben mit gewöhnlichen nichtlinearen Differentialglicheungen zweiter Ordnung, Math. Z., 61, 435-454 (1955) · Zbl 0064.08602
[168] Gaines, R., Existence of periodic solutions to second order nonlinear ordinary differential equations, 16, 186-199 (1974) · Zbl 0281.34033
[169] GainesSIAM J. Num. Analysis,; GainesSIAM J. Num. Analysis,
[170] George, J. H.; Sutton, W. G., Application of Liapunov theory to boundary value problems, Proc. Am. Math. Soc., 25, 666-671 (1970) · Zbl 0277.34023
[171] George, J. H.; Yorke, R. J., Application of Liapunov theory to boundary value problems II, Proc. Am. Math. Soc., 37, 207-212 (1973) · Zbl 0256.34020
[172] Granas, A., The theory of compact vector fields and some of its applications to topology of functional spaces, Rozprawy Mat. xxx (1962), Warsaw · Zbl 0111.11001
[173] Gustafson, G. B.; Schmitt, K., Methods of Nonlinear Analysis in the Theory of Differential Equations (1975), University of Utah Lecture Notes
[174] Heimes, K. A., Boundary value problems for ordinary nonlinear second order systems, J. diff. Eqns, 21, 449-463 (1966) · Zbl 0178.43103
[175] Hess, P., On uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z., 154, 17-18 (1977) · Zbl 0352.35046
[176] Keller, H. B., Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Am. math. Soc., 74, 887-991 (1968) · Zbl 0162.16501
[177] Keller, H. B., Positive solutions of some nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, 279-296 (1969) · Zbl 0188.17103
[178] Kiguradze, I.; Lazhava, N., On a nonlinear boundary value problem, Funct. Theory. Meth. diff. Eqns, 259-276 (1976) · Zbl 0346.34008
[179] LaetschJ. Math. Mech.; LaetschJ. Math. Mech.
[180] Levinson, N., Positive eigenfunctions for \(Δu + λƒ(u) = 0\), Arch. ration. Mech. Analysis, 11, 258-272 (1962) · Zbl 0108.28902
[181] Levinson, N., Dirichlet problem for \(Δu = ƒ(p, u)\), J. Math. Mech., 12, 567-575 (1962) · Zbl 0124.30901
[182] Mawhin, J., Degré topologique et solutions periodiques des systemes differentiels non lineaires, Bull. Soc. r. Sci. Liege, 38, 308-398 (1969) · Zbl 0186.41704
[183] Mawhin, J., Existence of periodic solutions for higher order differential systems that are not of class D, J. diff. Eqns, 8, 523-530 (1970) · Zbl 0224.34034
[184] Mawhin, J., Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. diff. Eqns, 12, 610-636 (1972) · Zbl 0244.47049
[185] Mawhin, J., Boundary value problems for nonlinear second order vector differential equations, J. diff. Eqns, 16, 257-269 (1974) · Zbl 0301.34019
[186] Mlak, W., Parabolic differential inequalities and Chaplighin’s method, Annls pol. math., 8, 139-152 (1960) · Zbl 0093.10301
[187] Pao, C. V., Positive solutions of a nonlinear boundary value problem of parabolic type, J. diff. Eqns, 22, 145-163 (1976) · Zbl 0326.35041
[188] Puel, J. D., Existence, comportement a l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre 2, Annali Scu. norm. sup., Pisa, 3, 89-119 (1976) · Zbl 0331.35027
[189] Rabinowitz, P., A note on a nonlinear elliptic equation, Ind. Univ. Math. J., 22, 43-49 (1972) · Zbl 0241.35028
[190] Redheffer R. & WalterAchs. ration. Mech.; Redheffer R. & WalterAchs. ration. Mech.
[191] Schmitt, K., Bounded solutions of nonlinear second order differential equations, Duke Math. J., 36, 237-244 (1969) · Zbl 0182.12001
[192] Schmitt, K., A note on periodic solutions of second order ordinary differential equations, SIAM J. Appl. Math., 21, 491-494 (1971) · Zbl 0238.34073
[193] Schrader, K.; Umamaheswaram, S., Existence theorems for boundary value problems, Proc. Am. Math. Soc., 47, 89-97 (1975) · Zbl 0272.34024
[194] Scorza-Dragoni, C., Sul problema dei valori ai limit per i sistemi di equasioni differenzali del secondo ordine, Bull. Un mat. ital., 14, 225-230 (1935) · Zbl 0012.25702
[195] Scorza-Dragoni, C., Sui sistemi di equazioni integrali non lineari, Rc. Semin. mat. Univ. Padova, 7, 1-35 (1936) · Zbl 0015.35202
[196] Stampacchia, G., Problemi al contorno e-littici, con dati discontinui, dotati di soluzioni holderiane, Annali Mat. pura appl., 51, 1-37 (1960) · Zbl 0204.42001
[197] Thompson, R. C., An invariance property of solutions to second order differential inequalities in ordered Banach spaces, SIAM J. Math. Anal., 8, 592-603 (1977) · Zbl 0376.34050
[198] Tsai, L. Y., The existence of solutions of nonlinear elliptic boundary value problems (1975), National Chengchi University: National Chengchi University Taiwan, preprint
[199] Turner, R. E.L., Transversality and cone maps, Archs ration. Mech. Analysis, 58, 151-179 (1975) · Zbl 0321.47044
[200] Williams, S. A., A nonlinear elliptic boundary value problem, Pacif. J. Math., 44, 767-774 (1973) · Zbl 0244.35049
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