Existence theorems for semilinear equations in cones. (English) Zbl 1015.47042

This article is a continuation of the author’s previous article [C. T. Cremins, Nonlinear Anal., Theory Methods Appl. 46 A, 789–806 (2001; Zbl 1015.47041)]. It deals with positive and non-negative solutions for operator equations of type \(Lx=Nx\) with Fredholm linear operator \(L\) of zero index in a Banach space \(X\) ordered by a cone \(K\). Based on the fixed-point index theory introduced by the author and the notion of a quasi-normal cone, the author presents a series of solvability results for semilinear equations with \(P_\gamma\)-compact maps. These results are close to some results by K. Deimling, R. E. Gaines and J. Santanilla, Guo Dajun B. Lafferriere and W. V. Petryshyn, J. Mawhin, J. Mawhin and K. P. Rybakovski, J. J. Nieto, W. V. Petryshyn, J. Santanilla, J. R. L. Webbs. As an application, the Picard boundary value problem \[ -x''(t)= f\bigl(t,x(t), x'(t),x''(t) \bigr),\quad x(0)= x(1)= 0 \] in the cone \(K=\{x\in X=C^2 [0,1]: -x''(t)\geq 0\), \(x(0)= x(1)=0\}\) is considered.


47H11 Degree theory for nonlinear operators
47J05 Equations involving nonlinear operators (general)
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47J25 Iterative procedures involving nonlinear operators
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations


Zbl 1015.47041
Full Text: DOI


[1] C. T. Cremins, A fixed point index and existence theorems for semilinear equations in cones, Nonlinear Anal. Theory, Methods, Appl, to appear. · Zbl 1015.47041
[2] Dancer, E.N.; Nussbaum, R.D.; Staurt, C., Quasinormal cones in Banach spaces, Nonlinear anal., 7, 539-553, (1983) · Zbl 0514.47043
[3] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[4] Fitzpatrick, P.M.; Petryshyn, W.V., On the nonlinear eigenvalue problem T(u)=λC(u), involving noncompact abstract and differential operators, Boll. un. mat. ital. B, 15, 80-107, (1978) · Zbl 0386.47033
[5] Gaines, R.E.; Santanilla, J., A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky mountain J. math., 12, 669-678, (1982) · Zbl 0508.34030
[6] Guo, Dajun, Some fixed point theorems on cone maps, Kexue tongbao, 29, 575-578, (1984) · Zbl 0553.47022
[7] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations, International series in pure and applied mathematics, (1964), Macmillan New York
[8] Lafferriere, B.; Petryshyn, W.V., New positive fixed point and eigenvalue results for Pγ-compact maps and applications, Nonlinear anal., 13, 1427-1440, (1989) · Zbl 0702.47036
[9] Lami-Dozo, E., Quasinormality of cones in Hilbert spaces, Acad. R. belg. bull. cl. sci., 67, 536-541, (1981) · Zbl 0494.47033
[10] J. Mawhin, and, K. P. Rybakowkski, Continuation theorems for semilinear equations in Banach spaces, preprint.
[11] Nieto, J.J., Existence of solutions in a convex set for nonlinear alternative problems, Proc. amer. math. soc., 94, 433-436, (1985) · Zbl 0585.47050
[12] Petryshyn, W.V., Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, Nonlinear anal. theory, methods, appl., 4, 259-281, (1980) · Zbl 0444.47046
[13] Peryshyn, W.V., On the solvability of x∈tx+λfx in quasinormal cones with T and F k-set contractive, Nonlinear anal., 5, 585-591, (1981) · Zbl 0474.47028
[14] Petryshyn, W.V., Generalized topological degree and semilinear equations, (1995), Cambridge Univ. Press Cambridge · Zbl 0834.47053
[15] Santanilla, J., Existence of nonnegative solutions of a semilinear equation at resonance with linear growth, Proc. amer. math. soc., 105, 963-971, (1989) · Zbl 0687.47045
[16] Webb, J.R.L., Solutions of semilinear equations in cones and wedges, Proceedings, WCNA, 1992, (1996), de Gruyter Berlin/New York, p. 137-147 · Zbl 0858.47030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.