## Abstract

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.

## Authors

**Radu Precup**

Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

nonlinear boundary value problem; differential system; positive solution; fixed point; cone.

## Paper coordinates

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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## About this paper

##### Journal

Nonlinear Analysis and Variational Problems

##### Publisher Name

Springer

##### Print ISBN

978-1-4419-0157-6

##### Online ISSN

google scholar link

[1] 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.

[2] H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces”, SIAMRev. 18(1976), 620-709.

[3] 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.

[4] X. Cheng, “Existence of positive solutions for a class of second-order ordinary differential systems”. Nonlinear Anal. 69 (2008), 3042-3049.

[5] J-F. Couchouron and R. Precup, “Homotopy method for positive solutions of p-Laplace inclusions”, Topol. Methods Nonlinear Anal 30 (2007), 157-169.

[6] L.H. Erbe, S. Hu and H. Wang, “Multiple positive solutions of some boundary value problems”, /. Math. Anal. Appl 184 (1994), 640-648.

[7] L.H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations”, Proc. Amer Math. Soc. 120 (1994), 743-748.

[8] 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.

[9] D. Franco and J.R.L. Webb, “CoUisionless orbits of singular and nonsingular dynamical systems”. Discrete Contin. Dyn. Syst. 15 (2006), 747-757.

[10] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.

[11] D.J. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer, Dordrecht, 1996.

[12] J. Henderson and H. Wang, “Positive solutions for nonlinear eigenvalue problems”, /. Math. Anal. Appl. 208 (1997), 252-259.

[13] M.A. Krasnoselsldi, “Fixed points of cone-compressing and cone-expanding operators”, Soviet. Math. Dokl. 1 (1960), 1285-1288.

[14] M.A. Krasnoselsldi, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

[15] K. Lan and J.R.L. Webb, “Positive solutions of semiUnear differential equations with singularities”,/. Differential Equations 148 (1998), 407-421.

[16] 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.

[17] D. O’ Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.

[18] 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.

[19] 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.

[20] 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.

[21] 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.

[22] R. Precup, “Existence, locaUzation and multipUcity results for positive radial solutions of semUinear elliptic systems”, /. Math. Anal. Appl. 352 (2009), 48-56.

[23] 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.

[24] H. Wang, “Positive periodic solutions of functional differential equations”, /. Differential Equations 202 (2004), 354-366.