Geodesic
On the theory of upper and lower solutions and the solvability of quasilinear integro-differential equations
Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 499-507 
V. P. Polityukov. On the theory of upper and lower solutions and the solvability of quasilinear integro-differential equations. Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 499-507. http://geodesic.mathdoc.fr/item/SM_1979_35_4_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper nonlinear integro-differential equations of the type $\partial u/\partial t-\Delta u=\Phi(x,t,u,F(u))$ are considered, where $F(u)$ is a nonlinear integral operator. The concepts of upper and lower solutions of problems for nonlinear integro-differential equations are defined, and based on this new solvability criteria for these problems are obtained. Nonstationary, as well as stationary, integro-differential equations are considered. Bibliography: 10 titles.

[1] Matematicheskaya entsiklopediya, izd-vo “Sovetskaya entsiklopediya”, Moskva, 1977

[2] M. Nagumo, “On principally linear elliptic differential equations of the second order”, Osaka Math. J., 6:2 (1954), 207–230 | MR

[3] S. I. Khudyaev, “Kriterii razreshimosti zadachi Dirikhle dlya ellipticheskikh uravnenii”, DAN SSSR, 148:1 (1963), 44–46 | Zbl

[4] S. I. Khudyaev, “O kraevykh zadachakh dlya nekotorykh kvazilineinykh ellipticheskikh uravnenii”, DAN SSSR, 154:4 (1964), 787–790 | Zbl

[5] H. Amann, “On the existence of positive solutions of nonlinear elliptic boundary value problems”, Indiana Univ. Math. J., 21 (1971), 125–146 | DOI | MR

[6] H. Amann, “On the number of solutions of nonlinear equations in ordered Banach spaces”, J. Func. Analysis, 11 (1972), 346–384 | DOI | MR | Zbl

[7] D. S. Cohen, “Positive solutions of nonlinear eigenvalue problems. Applications to nonlinear reactor dynamics”, Arch. Rat. Mach. Anal., 26 (1967), 305–315 | DOI | MR | Zbl

[8] D. H. Sattinger, “Topics in stability and bifurcation theory”, Lecture Notes in Math., 309, Springer-Verlag, New York–Berlin, 1973 | MR | Zbl

[9] D. H. Sattinger, “Monotone methods in nonlinear elliptic and parabolic boundary value problems”, Indiana Univ. Math. J., 21 (1972), 979–1000 | DOI | MR | Zbl

[10] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, izd-vo “Mir”, Moskva, 1968