A Krasnoselskii type compression-expansion fixed point theorem is adapted for the treatment of systems of semi-Unear equations. The compression-expansion conditions are given componentwise which allows the nonlinear term of a system to have different behaviors both in components and in variables. Applications to boundary value problems for systems of second order differential equations are included.
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
nonlinear boundary value problem; differential system; positive solution; fixed point; cone.
R. Precup, Compression-expansion critical point theory in conical shells, Nonlinear Analysis and Variational Problems, in P.M. Pardos, Th.M. Rassias, A.A. Khan eds., Springer, New York, 2009, pp 135-146, https://doi.org/10.1007/978-1-4419-0158-3_12
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Nonlinear Analysis and Variational Problems
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 R.P Agarwal, M. Meehan, D. O’Regan and R. Precup, “Location of nonnegative solutions for differential equations on finite and semi-infinite intervals”, Dynam. Syst Appl. 12 (2003), 323-332.
 H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces”, SIAMRev. 18(1976), 620-709.
 A. Canada and A. Zertiti, “Fixed point theorems for systems of equations in ordered Banach spaces with applications to differential and integral equations”. Nonlinear Anal. 27 (1996), 397-411.
 X. Cheng, “Existence of positive solutions for a class of second-order ordinary differential systems”. Nonlinear Anal. 69 (2008), 3042-3049.
 J-F. Couchouron and R. Precup, “Homotopy method for positive solutions of p-Laplace inclusions”, Topol. Methods Nonlinear Anal 30 (2007), 157-169.
 L.H. Erbe, S. Hu and H. Wang, “Multiple positive solutions of some boundary value problems”, /. Math. Anal. Appl 184 (1994), 640-648.
 L.H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations”, Proc. Amer Math. Soc. 120 (1994), 743-748.
 D. Franco, E. Liz and P.J. Torres, “Existence of periodic solutions for population models with periodic delay”,/nd/an/. Pure Appl. Math. 38 (2007), 143-152.
 D. Franco and J.R.L. Webb, “CoUisionless orbits of singular and nonsingular dynamical systems”. Discrete Contin. Dyn. Syst. 15 (2006), 747-757.
 A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
 D.J. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer, Dordrecht, 1996.
 J. Henderson and H. Wang, “Positive solutions for nonlinear eigenvalue problems”, /. Math. Anal. Appl. 208 (1997), 252-259.
 M.A. Krasnoselsldi, “Fixed points of cone-compressing and cone-expanding operators”, Soviet. Math. Dokl. 1 (1960), 1285-1288.
 M.A. Krasnoselsldi, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
 K. Lan and J.R.L. Webb, “Positive solutions of semiUnear differential equations with singularities”,/. Differential Equations 148 (1998), 407-421.
 W-C. Lian, F-H. Wong and C-C. Yeh, “On the existence of positive solutions of nonlinear second order differential equations”, Proc. Amer Math. Soc. 124 (1996), 1117-1126.
 D. O’ Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
 D. O’Regan and R. Precup, “Positive solutions of nonUnear systems with p-Laplacian on finite and semi-infinite intervals”, Positivity 11 (2007), 537-548.
 R. Precup, “Positive solutions of semi-linear elliptic problems via Krasnoselsldi type theorems in cones and Harnack’s inequaUty”, Mathematical Analysis and Applications, AIP Conference Proceedings 835 (2006), 125-132.
 R. Precup, “A vector version of Krasnoselskii’s fixed point theorem in cones and positive periodic solutions of nonUnear systems”, /. Fixed Point Theory Appl. 2 (2007), 141-151.
 R. Precup, “Positive solutions of nonlinear systems via the vector version of Krasnoselskii’s fixed point theorem in cones”, Ann. Tiberiu Popoviciu Sem. 5 (2007), 129-138.
 R. Precup, “Existence, locaUzation and multipUcity results for positive radial solutions of semUinear elliptic systems”, /. Math. Anal. Appl. 352 (2009), 48-56.
 P.J. Torres, “Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselsldi fixed point theorem”, /. Differential Equations 190 (2003), 643-662.
 H. Wang, “Positive periodic solutions of functional differential equations”, /. Differential Equations 202 (2004), 354-366.