Existence of nonnegative solutions of a semilinear equation at resonance with linear growth

Author:
Jairo Santanilla

Journal:
Proc. Amer. Math. Soc. **105** (1989), 963-971

MSC:
Primary 34C15; Secondary 34B15, 47H15

DOI:
https://doi.org/10.1090/S0002-9939-1989-0964462-9

MathSciNet review:
964462

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Abstract | References | Similar Articles | Additional Information

Abstract: A coincidence degree result is established to study sufficient conditions for the existence of nonnegative solutions of a semilinear equation at resonance in which the nonlinearity has at most linear growth. Nonnegative solutions to some boundary value problems are obtained to illustrate the theory.

**[1]**A. Cañada and P. Martinez-Amores,*Solvability of some operator equations and periodic solutions of nonlinear functional differential equations*, J. Differential Equations**49**(1983), 415-429. MR**715694 (86d:47069)****[2]**A. Castro and R. Shivaji,*Nonnegative solution for a class of nonpositone problems*, preprint. MR**943804 (90m:34040)****[3]**L. Cesari and R. Kannan,*Solutions of nonlinear hyperbolic equations at resonance*, Nonlinear Analysis, TMA**6**(1982), 751-805. MR**671720 (84a:35176)****[4]**L. Cesari, R. Kannan and J. Schuur Edit.,*Nonlinear functional analysis and differential equations*, Dekker, New York, 1976. MR**0477808 (57:17314)****[5]**L. Cesari and R. Kannan,*Qualitative study of a class of nonlinear boundary value problems at resonance*, J. Differential Equations**56**(1985), 63-81. MR**772121 (86g:47083)****[6]**-,*An abstract theorem at resonance*, Proc. Amer. Math. Soc.**63**(1977), 221-225. MR**0448180 (56:6489)****[7]**-,*Existence of solutions of a nonlinear differential equation*, Proc. Amer. Math. Soc.**4**(1983), 605-613. MR**702284 (85d:34017)****[8]**R. E. Gaines and J. Mawhin,*Coincidence degree and nonlinear differential equations*, Lecture Notes in Math., vol. 568, Springer-Verlag, Berlin, 1977. MR**0637067 (58:30551)****[9]**R. Gaines and J. Santanilla,*A coincidence theorem in convex sets with applications to periodic soltuions of ordinary differential equations*, Rocky Mountain J. Math.**12**(1982), 669-678. MR**683861 (84c:47061)****[10]**J. A. Gatica and H. Smith,*Fixed point techniques in a cone with application*, J. Math. Anal. Appl.**61**(1977), 58-71. MR**0513057 (58:23797)****[11]**A. Granas,*On a certain class of nonlinear mappings in Banach spaces*, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.**9**(1957), 867-871.**[12]**G. B. Gustafson and K. Schmitt,*Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations*, J. Math. Anal. Appl.**12**(1972), 129-147. MR**0346234 (49:10959)****[13]**R. Kannan and V. Lakshmikantham,*Periodic solutions of nonlinear boundary value problems*, Nonlinear Analysis, TMA**6**(1982), 1-10. MR**647584 (83d:34031)****[14]**R. Kannan and J. Locker,*Nonlinear boundary value problems and operators*, J. Differential Equations**28**(1978), 60-103. MR**0477918 (57:17418)****[15]**R. Kannan, J. J. Nieto and M. B. Ray,*A class of nonlinear boundary value problems without Landesman-Lazer condition*, J. Math. Anal. Appl.**105**(1985), 1-11. MR**773569 (86c:34031)****[16]**R. Kannan, J. J. Nieto and V. Lakshmikantham,*Sufficient conditions for existence of solutions of nonlinear boundary value problems at renonance*, Nonlinear Anal. TMA**7**(1983), 1013-1020. MR**713210 (85j:47064a)****[17]**M. A. Krasnosel' skii,*Positive solutions of operator equations*, Noordhoff, Groningen, 1964.**[18]**J. Mawhin,*Topological degree methods in nonlinear boundary value problems*, CBMS Regional Conference Series in Math. no. 40, Amer. Math. Soc., Providence, R.I. 1979, 1981. MR**525202 (80c:47055)****[19]**J. Mawhin and K. P. Rybakowski,*Continuation theorems for semi-linear equations in Banach spaces*, preprint.**[20]**J. Nieto,*Existence of solutions in a convex set for nonlinear alternative problems*, Proc. Amer. Math. Soc.**94**(1985), 433-436. MR**787888 (86d:47006)****[21]**E. Rothe,*Zur Theorie der topologischer Ordnung und dere Vektorfelder in Banachsen Raumen*, Compositio Math.**5**(1937), 177-197.**[22]**J. Santanilla,*Some coincidence theorem in wedges, cones and convex sets*, J. Math. Anal. Appl.**105**(1985), 357-371. MR**778471 (86h:47092)****[23]**-,*Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations*, J. Math. Anal. Appl.**126**(1987), 397-408. MR**900756 (88h:34014)****[24]**J. Mawhin, Landesman-Lazer's type problems for nonlinear equations, Conf. Sem. Mat. Univ. Bari, (1977), No. 147. MR**0477923 (57:17423)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0964462-9

Keywords:
Coincidence degree,
semilinear equations at resonance,
boundary value problems

Article copyright:
© Copyright 1989
American Mathematical Society