# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Anti-periodic solutions for evolution equations associated with maximal monotone mappings. (English) Zbl 1215.34069
The authors consider the existence of anti-periodic solutions for differential inclusions in a real Hilbert space $H$. The first result concerns the problem $$x^{\prime }(t)\in -Ax(t)+f(t)\text{ a.e. on }\Bbb{R},$$ $$x(t)=-x(t+T)\text{ for }t\in \Bbb{R}.$$ $A$ is assumed to be an odd maximal monotone mapping, $D(A)$ is symmetric and convex, $f$ is $L^{2}$ and satisfies $f(t)=-f(t+T)$ for $t\in \Bbb{R}$, $\|g\| \leq M\| x\|$ for all $x\in D(A)$ and $g\in Ax$, and $M>0$ is a constant such that $MT<2$. The other theorem concerns the problem $$x^{\prime }(t)\in -Ax(t)+\partial G(x(t))+f(t)\text{ a.e. on }\Bbb{R},$$ $$x(t)=-x(t+T)\text{ for }t\in \Bbb{R}.$$ In addition to the hypotheses of the first result, it is assumed that $G$ is continuously differentiable and even, $\partial G$ maps bounded sets to bounded sets, and $D(A)$ is compactly embedded in $H$. In the case in which $A$ is the subdifferential of a lower semi-continuous convex function, the authors obtained similar results as in [{\it Y. Q. Chen, D. O’Regan} and {\it J. J. Nieto}, Math. Comput. Modelling 46, No. 9--10, 1183--1190 (2007; Zbl 1142.34313)]. The authors conclude the paper by applying the first of these theorems to a boundary value problem for a partial differential equation.

##### MSC:
 34G25 Evolution inclusions 34C25 Periodic solutions of ODE
Full Text:
##### References:
 [1] Nakao, M.: Existence of anti-periodic solution for the quasilinear wave equation with viscosity, J. math. Anal. appl. 204, 754-764 (1996) · Zbl 0873.35051 · doi:10.1006/jmaa.1996.0465 [2] Aftabizadeh, A. R.; Aizicovici, S.; Pavel, N. H.: On a class of second-order anti-periodic boundary value problems, J. math. Anal. appl. 171, 301-320 (1992) · Zbl 0767.34047 · doi:10.1016/0022-247X(92)90345-E [3] Aftabizadeh, A. R.; Aizicovici, S.; Pavel, N. H.: Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces, Nonlinear anal. 18, 253-267 (1992) · Zbl 0779.34054 · doi:10.1016/0362-546X(92)90063-K [4] Aizicovici, S.; Mckibben, M.; Reich, S.: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear anal. 43, 233-251 (2001) · Zbl 0977.34061 · doi:10.1016/S0362-546X(99)00192-3 [5] Aizicovici, S.; Pavel, N. H.: Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space, J. funct. Anal. 99, 387-408 (1991) · Zbl 0743.34067 · doi:10.1016/0022-1236(91)90046-8 [6] Cabada, A.; Vivero, D. R.: Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations, Adv. differential equations 4, 291-310 (2004) · Zbl 1083.39017 · doi:10.1155/S1687183904310022 [7] Chen, H. L.: Antiperiodic wavelets, J. comput. Math. 14, 32-39 (1996) · Zbl 0839.42014 [8] Chen, Y. Q.: Note on massera’s theorem on anti-periodic solution, Adv. math. Sci. appl. 9, 125-128 (1999) · Zbl 0924.34037 [9] Chen, Y. Q.; Wang, X. D.; Xu, H. X.: Anti-periodic solutions for semilinear evolution equations, J. math. Anal. appl. 273, 627-636 (2002) · Zbl 1055.34113 · doi:10.1016/S0022-247X(02)00288-3 [10] Chen, Y. Q.; Cho, Y. J.; Jung, J. S.: Anti-periodic solutions for evolution equations, Math. comput. Modelling 40, 1123-1130 (2004) · Zbl 1074.34058 · doi:10.1016/j.mcm.2003.06.007 [11] Chen, Y. Q.; Cho, Y. J.; O’regan, D.: Anti-periodic solutions for evolution equations, Math. nachr. 278, 356-362 (2005) · Zbl 1068.34059 · doi:10.1002/mana.200410245 [12] Chen, Y. Q.: Anti-periodic solutions for semilinear evolution equations, J. math. Anal. appl. 315, 337-348 (2006) · Zbl 1100.34046 · doi:10.1016/j.jmaa.2005.08.001 [13] Chen, Y. Q.; O’regan, D.; Nieto, J. J.: Anti-periodic solutions for fully nonlinear first-order differential equations, Math. comput. Modelling 46, 1183-1190 (2007) · Zbl 1142.34313 · doi:10.1016/j.mcm.2006.12.006 [14] Franco, D.; Nieto, J. J.: First order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear anal. 42, 163-173 (2000) · Zbl 0966.34025 · doi:10.1016/S0362-546X(98)00337-X [15] Franco, D.; Nieto, J. J.; O’regan, D.: Anti-periodic boundary value problem for nonlinear first order ordinary differential equations, Math. inequal. Appl. 6, 477-485 (2003) · Zbl 1097.34015 [16] Franco, D.; Nieto, J. J.; O’regan, D.: Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions, Appl. math. Comput. 153, 793-802 (2004) · Zbl 1058.34015 · doi:10.1016/S0096-3003(03)00678-7 [17] Haraux, A.: Anti-periodic solutions of some nonlinear evolution equations, Manuscripta math. 63, 479-505 (1989) · Zbl 0684.35010 · doi:10.1007/BF01171760 [18] Luo, Z. G.; Shen, J. H.; Nieto, J. J.: Antiperiodic boundary value problem for first-order impulsive ordinary differential equations, Comput. math. Appl. 49, 253-261 (2005) · Zbl 1084.34018 · doi:10.1016/j.camwa.2004.08.010 [19] Okochi, H.: On the existence of anti-periodic solutions to a nonlinear evolution equation associated with odd sub-differential operators, J. funct. Anal. 91, 246-258 (1990) · Zbl 0735.35071 · doi:10.1016/0022-1236(90)90143-9 [20] Okochi, H.: On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains, Nonlinear anal. 14, 771-783 (1990) · Zbl 0715.35091 · doi:10.1016/0362-546X(90)90105-P [21] Souplet, P.: Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations, Nonlinear anal. 26, 1511-1525 (1996) · Zbl 0855.34075 · doi:10.1016/0362-546X(95)00012-K [22] Souplet, P.: Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations, Nonlinear anal. 32, 279-286 (1998) · Zbl 0892.35078 · doi:10.1016/S0362-546X(97)00477-X [23] Yin, Y.: Monotone iterative technique and quasilinearization for some anti-periodic problem, Nonlinear world 3, 253-266 (1996) · Zbl 1013.34015 [24] Okochi, H.: On the existence of periodic solutions to nonlinear abstract parabolic equations, J. math. Soc. Japan 40, No. 3, 541-553 (1988) · Zbl 0679.35046 · doi:10.2969/jmsj/04030541 [25] Aubin, J. P.; Cellina, A.: Differential inclusions, (1984) · Zbl 0538.34007