IN THIS WE GIVE A UNIFICATION OF DISCRETE AND CONTINUOUS RESULTS ABOUT ASYMPTOTIC FORMULAE FOR SOLUTIONS OF DYNAMIC SYSTEMS ON TIME SCALES WITH A FUNCTIONAL PERTURBATION

ASYMPTOTIC FORMULAE ARE GIVEN FOR THE SOLUTIONS OF A KIND OF LINEAR FDE. A PARTICULAR CASE OF THIS KIND IS THE PERTURBED LINEAR NONAUTONOMOUS DELAY DIFFERENTIAL EQUATION

IN THIS ARTICLE WE GIVE SOME ASYMPTOTIC FORMULAE FOR IMPULSIVE DIFFERENTIAL SYSTEM WITH PIECEWISE CONSTANT ARGUMENT OF GENERALIZED TYPE (ABBREVIATED IDEPCAG). THESE FORMULAE ARE BASED ON CERTAIN INTEGRABILITY CONDITIONS, BY MEANS OF A GR¨ONWALL-BELLMAN TYPE INEQUALITY AND THE BANACH?S FIXED POINT THEOREM. ALSO, WE STUDY THE EXISTENCE OF AN ASYMPTOTIC EQUILIBRIUM OF NONLINEAR AND SEMILINEAR IDEPCAG SYSTEMS. WE PRESENT EXAMPLES THAT ILLUSTRATE OUR THE RESULTS.

IN THIS PAPER, WE GIVE A CHARACTERIZATION OF THE SOLUTIONS OF A SCALAR LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS OF THE FORM Y'(T)=:\0 (T) Y (T)+ B (T, Y,), T?: 0, WHERE YT:[-T, O]-T< CIS DEFINED BY Y,(S)= Y (T+ S) FORT?: 0 AND S E [-T, O],:\O:[0,+ OO [-T< C IS A LOCALLY INTEGRABLE FUNCTION,{B (T,·)},: 00 IS A FAMILY OF BOUNDED LINEAR FUNCTIONALS FROM L'([-T, OJ,< C) INTO< C WITH T> 0. ASYMPTOTIC FORMULAS UNDER SMALLNESS CONDITIONS ON B (T, EF.'+> O (S) DS) ARE OBTAINED. EXAMPLES ARE GIVEN.

WE STUDY ALMOST AUTOMORPHIC SOLUTIONS OF RECURRENCE RELATIONS WITH VALUES IN A BANACH SPACE V FOR QUASILINEAR ALMOST AUTOMORPHIC DIFFERENCE SYSTEMS. ITS LINEAR PART IS A CONSTANT BOUNDED LINEAR OPERATOR ? DEFINED ON V SATISFYING AN EXPONENTIAL DICHOTOMY. WE STUDY THE EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS OF THE NON-HOMOGENEOUS LINEAR DIFFERENCE EQUATION AND TO QUASILINEAR DIFFERENCE EQUATION. ASSUMING GLOBAL LIPSCHITZ TYPE CONDITIONS, WE OBTAIN MASSERA TYPE RESULTS FOR THESE ABSTRACT SYSTEMS. THE CASE WHERE THE EIGENVALUES ? VERIFY |?| = 1 IS ALSO TREATED. AN APPLICATION TO DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT IS GIVEN.

IN THIS ARTICLE WE INTRODUCE A CLASS OF DISCONTINUOUS ALMOST AUTOMORPHIC FUNCTIONS WHICH APPEARS NATURALLY IN THE STUDY OF ALMOST AUTOMORPHIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. THEIR FUNDAMENTAL PROPERTIES ARE USED TO PROVE THE ALMOST AUTOMORPHICITY OF BOUNDED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. DUE TO THE STRONG DISCRETE CHARACTER OF THESE EQUATIONS, THE EXISTENCE OF A UNIQUE DISCRETE ALMOST AUTOMORPHIC SOLUTION OF A NON-AUTONOMOUS ALMOST AUTOMORPHIC DIFFERENCE SYSTEM IS OBTAINED, FOR WHICH CONDITIONS OF EXPONENTIAL DICHOTOMY AND DISCRETE BI-ALMOST AUTOMORPHICITY ARE FUNDAMENTAL.

IN THIS WORK, USING DISCONTINUOUS ALMOST PERIODIC TYPE FUNCTIONS, EXPONENTIAL DICHOTOMY AND THE NOTION OF BI-ALMOST AUTOMORPHICITY WE GIVE SUFFICIENT CONDITIONS TO OBTAIN A UNIQUE ALMOST AUTOMORPHIC SOLUTION OF A QUASILINEAR SYSTEM OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS. FINALLY, AN APPLICATION TO THE LASOTA-WAZEWSKA MODEL WITH PIECEWISE CONSTANT DELAYED ARGUMENT IS GIVEN.

EN ESTE TRABAJO, SE ESTABLECE UNA CORRESPONDENCIA ENTRE LAS FECHAS DE UN CALENDARIO LUNAR CON LAS DEL CALENDARIO GREGORIANO, QUE ES EL CALENDARIO USUAL. SE USAN PRINCIPIOS BáSICOS DE ARITMéTICA, PARA CALCULAR EL NúMERO DE DíAS EXISTENTE ENTRE DOS FECHAS DEL CALENDARIO GREGORIANO Y PARA CALCULAR EL NúMERO DE DíAS EXISTENTE ENTRE DOS FECHAS DE UN CALENDARIO LUNAR DADO. SE ESCOGE EL CALENDARIO HEBREO CARAíTA POR SER EL CALENDARIO LUNAR QUE, SEGúN EL AUTOR, MáS SE AJUSTA AL CALENDARIO HEBREO BíBLICO. SE DAN RAZONES BíBLICAS Y LINGüíSTICAS, PARA RESPALDAR ESTA ELECCIóN. NO OBSTANTE, EL LECTOR PUEDE ELEGIR OTRO CALENDARIO LUNAR.

IN THIS MANUSCRIPT, WE DISCUSS THE ROUGHNESS OF NONUNIFORM EXPONENTIAL DICHOTOMIES ON A TIME SCALE IN A BANACH SPACE AND GIVE THE EXISTENCE RESULTS FOR NONUNIFORM AND UNIFORM EXPONENTIAL DICHOTOMY FOR THE TIME SCALE. WE EXTEND AND UNIFY THE PREVIOUS RESULTS BY CONSIDERING THE EQUATION ON A TIME SCALE. FOR A GIVEN LINEAR EQUATION ON A TIME SCALE, THE EXISTENCE OF EXPONENTIAL DICHOTOMY PERSISTS UNDER AN ADEQUATELY SMALL VARIABLE LINEAR PERTURBATION. TO ESTABLISH THE RESULTS, WE ACQUIRE RELATED ROUGHNESS RESULTS FOR THE CASE OF UNIFORM EXPONENTIAL CONTRACTIONS. IN THE END, A SUITABLE EXAMPLE IS GIVEN FOR ILLUSTRATION.

THIS WORK CONCERNS THE EXISTENCE OF ALMOST PERIODIC SOLUTIONS FOR CERTAIN DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS. THE COEFFICIENTS OF THESE EQUATIONS ARE ALMOST PERIODIC AND THE EQUATION CAN BE SEEN AS PERTURBATIONS OF A LINEAR EQUATION SATISFYING AN EXPONENTIAL DICHOTOMY ON A DIFFERENCE EQUATION. THE STABILITY OF THAT SOLUTION ON A SEMI-AXIS IS ALSO STUDIED.

IN THIS WORK, WE WILL GIVE A NOVEL METHOD TO CONSTRUCT A CONTINUOUS APPROXIMATION OF THE EXPONENTIAL, LOGISTIC, AND GAUSSIAN FUNCTIONS THAT ALLOW US TO DO A HANDMADE DRAWING OF THEIR GRAPHS FOR WHICH THERE IS NO ACCURACY OF DRAWING AT ELEMENTARY LEVELS (EVEN AT ADVANCED ONES!). THIS METHOD ARISES FROM SOLVING THE ELEMENTARY ORDINARY DIFFERENTIAL EQUATION X0 (T) = AX(T) COMBINED WITH A SUITABLE PIECEWISE CONSTANT ARGUMENT. THE PROPOSED APPROXIMATION WILL ALLOW US TO GENERATE SEVERAL NUMERICAL SCHEMES IN AN ELEMENTARY WAY, GENERALIZING THE CLASSICAL ONES AS, EULER?S SCHEMES. NO SOPHISTICATED MATHEMATICAL TOOLS ARE NEEDED.

IN THIS PAPER WE PRESENT IMPROVEMENT ON TWO RESULTS ABOUT ASYMPTOTIC FORMULAE: A HARTMAN WINTNER RESULT, WHERE THE NOT PERTURBED SYSTEM IS NET NECESSARILY DIAGONAL AND A POINCAR´E RESULT, WHERE THE ROOTS OF THE CHARACTERISTIC EQUATION OF THE NOT PERTURBED SYSTEM COULD HAVE THE SAME MODULI ARE PRESENTED. BOTH RESULTS INCLUDE NONLINEAR PERTURBATIONS. APPLICATIONS ARE GIVEN.

IN THIS PAPER, WE STUDY A NEW CLASS OF FUNCTIONS, WHICH WE CALL (OMEGA, C)-ASYMPTOTICALLY PERIODIC FUNCTIONS. THIS COLLECTION INCLUDES ASYMPTOTICALLY PERIODIC, ASYMPTOTICALLY ANTIPERIODIC, ASYMPTOTICALLY BLOCH-PERIODIC, AND UNBOUNDED FUNCTIONS. WE PROVE THAT THE SET CONFORMED BY THESE FUNCTIONS IS A BANACH SPACE WITH A SUITABLE NORM. FURTHERMORE, WE SHOW SEVERAL PROPERTIES OF THIS CLASS OF FUNCTIONS AS THE CONVOLUTION INVARIANCE. WE PRESENT SOME EXAMPLES AND A COMPOSITION RESULT. AS AN APPLICATION, WE PROVE THE EXISTENCE AND UNIQUENESS OF (OMEGA, C)-ASYMPTOTICALLY PERIODIC MILD SOLUTIONS TO THE FIRST-ORDER ABSTRACT CAUCHY PROBLEM ON THE REAL LINE. ALSO, WE ESTABLISH SOME SUFFICIENT CONDITIONS FOR THE EXISTENCE OF POSITIVE (OMEGA, C)-ASYMPTOTICALLY PERIODIC SOLUTIONS TO THE LASOTA-WAZEWSKA EQUATION WITH UNBOUNDED OSCILLATING PRODUCTION OF RED CELLS.

IN THIS PAPER WE STUDY A NEW CLASS OF FUNCTIONS, WHICH WE CALL (?,C )-PSEUDO PERIODICFUNCTIONS. THIS COLLECTION INCLUDES PSEUDO PERIODIC, PSEUDO ANTI-PERIODIC, PSEUDOBLOCH-PERIODIC, AND UNBOUNDED FUNCTIONS. WE PROVE THAT THE SET CONFORMED BY THESEFUNCTIONS IS A BANACH SPACE WITH A SUITABLE NORM. FURTHERMORE, WE SHOW SEVERALPROPERTIES OF THIS CLASS OF FUNCTIONS AS THE CONVOLUTION INVARIANCE. WE PRESENT SOMEEXAMPLES AND A COMPOSITION RESULT. AS AN APPLICATION, WE PROVE THE EXISTENCE ANDUNIQUENESS OF (?,C )-PSEUDO PERIODIC MILD SOLUTIONS TO THE ?RST ORDER ABSTRACT CAUCHYPROBLEM ON THE REAL LINE. ALSO, WE ESTABLISH SOME SU?CIENT CONDITIONS FOR THEEXISTENCE OF POSITIVE (?,C )-PSEUDO PERIODIC SOLUTIONS TO THE LASOTA?WAZEWSKAEQUATION WITH UNBOUNDED OSCILLATING PRODUCTION OF RED CELLS.

IN THIS PAPER, WE ANALYZE THE EXISTENCE AND UNIQUENESS OF REMOTELY ALMOST PERIODIC SOLUTIONS FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. THE EXISTENCE AND UNIQUENESS OF REMOTELY ALMOST PERIODIC SOLUTIONS ARE ACHIEVED BY USING THE RESULTS ABOUT THE EXPONENTIAL DICHOTOMY AND THE BI-ALMOST REMOTELY ALMOST PERIODICITY OF THE HOMOGENEOUS PART OF THE CORRESPONDING SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS UNDER OUR CONSIDERATION.

IN THIS PAPER WE GIVE A UNIFORM APPROXIMATION OF A CNN-HOPFIELD TYPE IMPULSIVE SYSTEM BY MEANS OF AN IDEPCA APPROXIMATING SYSTEM. AS A CONSEQUENCE OF THE UNIFORM APPROXIMATION, CERTAIN PROPERTIES LIKE BOUNDEDNESS ARE INHERITED. WE ALSO CONSIDER THE ANALYSIS OF A CONSTANT COEFFICIENTS CASE. THESE RESULTS ARE NOVEL IN THE IMPULSIVE DIFFERENTIAL EQUATIONS FRAME. EXAMPLES ARE SIMULATED, ILLUSTRATING THE EFFECTIVENESS OF OUR RESULTS.

IN THIS MANUSCRIPT, WE DISCUSS THE GENERALIZED ALMOST AUTOMORPHIC CONCEPT OF ARBITRARY BOUNDED AND UNBOUNDED TIME SCALES AND INITIATE A NEW IDEA, NAMELY THE CONCEPT OF CHANGING-PERIODIC TIME SCALES. WE STUDY WEIGHTED PSEUDO ALMOST AUTOMORPHIC FUNCTIONS UNDER THE SHIFT OPERATOR FOR TRANSLATION AND NON-TRANSLATION TIME SCALES. IMPORTANT NOVEL RESULTS AND ESSENTIAL PROPERTIES OF SUCH FUNCTIONS ARE ESTABLISHED ON IRREGULAR HYBRID DOMAINS. THE OBTAINED RESULTS ARE VALID FOR SEVERAL EQUATIONS DEFINED OVER VARIOUS TIME SCALES SUCH AS QUANTUM TIME SCALES, CANTOR SET TIME SCALES, HARMONIC TIME SCALES, AND SO ON. THE IDEA OF CHANGING-PERIODIC TIME SCALES PRESENTED IN THIS PAPER WILL HELP IN COMPREHENSION AND ELIMINATE THE GENUINE LACK THAT ARISES IN THE STUDY OF CLASSICAL FUNCTIONS ON TIME SCALES WHERE THE BOUNDEDNESS OF THE GRAININESS FUNCTION IS REQUIRED. THE RESULTS ARE NEW, NONTRIVIAL, AND COMPLEMENT THE EXISTING ONES.