Serdica Mathematical Journal

Volume 23, Number 1, 1997

C O N T E N T S

• Elabbasy, E. M. Oscillation theorems for perturbed second order nonlinear differential equations with damping (pp. 1-14)
• González, M., M. Molina Some theoretical results on the progeny of a bisexual Galton-Watson branching process (pp. 15-24)
• Koppitz, J. M-solid subvarieties of some varieties of commutative semigroups (pp. 25-34)
• Ivanov, M., N. Zlateva. Abstract subdifferential calculus and semi-convex functions (pp. 35-58)
• Ohbuchi, A. On the difference of 4-gonal linear systems on some curves (pp. 59-68)
• Ceppitelli, R., L. Faina. Extremal solutions for a class of functional differential equations (pp. 69-94)

A B S T R A C T S

OSCILLATION THEOREMS FOR PERTURBED SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DAMPING
E. M. Elabbasy

1991 Mathematics Subject Classification: 34C10, 34K15.
Key words: oscillatory solutions, second order differential equations.

Some oscillation criteria for solutions of a general perturbed second order ordinary differential equation with damping


(r(t)x¢(t))¢+ h(t)f(x)x¢(t) + y(t,x) = H(t,x(t),x¢(t))

with alternating coefficients are given. The results obtained improve and extend some existing results in the literature.

SOME THEORETICAL RESULTS ON THE PROGENY OF A BISEXUAL GALTON-WATSON BRANCHING PROCESS
M. González, M. Molina

1991 Mathematics Subject Classification: 60J80.
Key words: Bisexual Galton-Watson branching process, stochastic monotony, total progeny, limit behaviour.

A Superadditive Bisexual Galton-Watson Branching Process is considered and the total number of mating units, females and males, until the n-th generation, are studied. In particular some results about the stochastic monotony, probability generating functions and moments are obtained. Finally, the limit behaviour of those variables suitably normed is investigated.

M-SOLID SUBVARIETIES OF SOME VARIETIES OF COMMUTATIVE SEMIGROUPS
J. Koppitz

1991 Mathematics Subject Classification: 20M07, 08B15.
Key words: hypersubstitutions, M-hyperidentity, M-solid subvarieties of semigroups.

The basic concepts are M-hyperidentities, where M is a monoid of hypersubstitutions. The set of all M-solid varieties of semigroups forms a complete sublattice of the lattice of all varieties of semigroups. We fix some specific varieties V of commutative semigroups and study the set of all M-solid subvarieties of V, in particular, if V is nilpotent.

ABSTRACT SUBDIFFERENTIAL CALCULUS AND SEMI-CONVEX FUNCTIONS
Milen Ivanov, Nadia Zlateva

1991 Mathematics Subject Classification: Primary: 49J52; Secondary 47H04, 47H05.
Key words: nonsmooth calculus, subdifferential, integrability.

We develop an abstract subdifferential calculus for lower semicontinuous functions and investigate functions similar to convex functions. As application we give sufficient conditions for the integrability of a lower semicontinuous function.

ON THE DIFFERENCE OF 4-GONAL LINEAR SYSTEMS ON SOME CURVES
Akira Ohbuchi

1991 Mathematics Subject Classification: 14H45, 14H10, 14C20.
Key words: curve theory, algebraic geometry.

Let C=(C,g₄¹) be a tetragonal curve. We consider the scrollar invariants e₁, e₂, e₃ of g₄¹. We prove that if W₄¹(C) is a non-singular variety, then every g₄¹Î W₄¹(C) has the same scrollar invariants.

EXTREMAL SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS
Rita Ceppitelli, Loris Faina

1991 Mathematics Subject Classification: 34K05, 34A12. Key words: extremal solutions, hereditary setting, Carath\'eodory assumptions

We study, in Carathéodory assumptions, existence, continuation and continuous dependence of extremal solutions for an abstract and rather general class of hereditary differential equations. By some examples we prove that, unlike the nonfunctional case, solved Cauchy problems for hereditary differential equations may not have local extremal solutions.