EFFICIENT CONDITIONS GUARANTEEING THE EXISTENCE OF A T - PERIODIC SOLUTION TO THE SECOND ORDER DIFFERENTIAL EQUATION U = H( T) U.,.. ( 0, 1), ARE ESTABLISHED. HERE, H. L( R/ TZ) IS A RATHER GENERAL SIGN- CHANGING FUNCTION WITH H < 0. IN CONTRAST WITH THE RESULTS IN GODOY AND ZAMORA ( PROC R SOC EDINB SECT A MATH) AND HAKL AND ZAMORA ( J DIFFER EQU 263: 451- 469, 2017), THE KEY INGREDIENT TO SOLVE THE AFOREMENTIONED PROBLEM SEEMS TO BE CONNECTED MORE WITH THE OSCILLATION AND THE SYMMETRY ASPECTS OF THE WEIGHT FUNCTION H THAN WITH THE MULTIPLICITY OF ITS ZEROES. ROUGHLY SPEAKING, THE SOLVABILITY FOR THE ABOVE- MENTIONED PROBLEM CAN BE GUARANTEED WHEN H+ H- AND H+ IS LARGE ENOUGH.

BY MEANS OF A NEW CHANGE OF VARIABLE WE PROVE THE EXISTENCE OF A POSITIVE 2?-PERIODIC SOLUTION FOR THE MATHIEU?DUFFING TYPE EQUATIONS HAVING ITS NONLINEARITY A SUPER-LINEAR GROWTH. AS RESULT WE CAN GUARANTEE THE EXISTENCE OF 2?-PERIODIC SOLUTIONS EVEN ASSUMING THAT THE PARAMETER OF THE ASSOCIATED MATHIEU EQUATION IS IN THE CONTENTIOUS ZONE OF RESONANCE.

A FREDHOLM-TYPE THEOREM FOR BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS IS ESTABLISHED. THE THEOREM GENERALIZES RESULTS KNOWN FOR THE SYSTEMS WITH LINEAR OR HOMOGENEOUS OPERATORS TO THE CASE OF SYSTEMS WITH POSITIVELY HOMOGENEOUS OPERATORS.

WE PRESENT NEW CRITERIA FOR UNIQUENESS AND ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS OF A SECOND ORDER DIFFERENTIAL EQUATION BASED ON TOPOLOGICAL DEGREE THEORY. AS AN APPLICATION, WE WILL STUDY SOME WELL KNOWN EQUATIONS AND SOME ILLUSTRATIVE EXAMPLES.

SUFFICIENT CONDITIONS ARE ESTABLISHED IN ORDER TO GUARANTEE THE EXISTENCE OF POSITIVE PERIODIC SOLUTIONS TO CERTAIN TYPE OF SINGULAR DIFFERENTIAL EQUATIONS INVOLVING SINGULAR \PHI-LAPLACIAN OPERATOR.

AS A CONSEQUENCE OF THE MAIN RESULT OF THIS PAPER EFFICIENT CONDITIONS GUARANTEEING THE EXISTENCE OF A T -PERIODIC SOLUTION TO THE SECOND-ORDER DIFFERENTIAL EQUATION{ U} = {{H((T))}/U-LAMBDA ARE ESTABLISHED. HERE, H IS AN ELEMENT OF L(DOUBLE-STRUCK CAPITAL R/TDOUBLE-STRUCK CAPITAL Z) IS A PIECEWISE-CONSTANT SIGN-CHANGING FUNCTION AND THE NON-LINEAR TERM PRESENTS A WEAK SINGULARITY AT 0 (I.E. LAMBDA IS AN ELEMENT OF (0, 1)).

WE STUDY A SECOND-ORDER ORDINARY DIFFERENTIAL EQUATION COMING FROM THE KEPLER PROBLEM ON S-2. THE FORCING TERM UNDER CONSIDERATION IS A PIECEWISE CONSTANT WITH SINGULAR NONLINEARITY THAT CHANGES SIGN. WE ESTABLISH NECESSARY AND SUFFICIENT CONDITIONS TO THE EXISTENCE AND MULTIPLICITY OF T-PERIODIC SOLUTIONS.

WE STUDY A NONLINEAR SINGULAR BOUNDARY VALUE PROBLEM AND PROVE THAT, DEPENDING ON A RELATIONSHIP BETWEEN EXPONENTS OF POWER TERMS, THE PROBLEM HAS EITHER SOLUTIONS OF DIRICHLET TYPE OR HOMOCLINIC SOLUTIONS. WE MAKE USE OF SHOOTING TECHNIQUES AND LOWER AND UPPER SOLUTIONS.