A NEW CONDITION IS INTRODUCED BY GENERALIZING THE RITT AND KREISS OPERATORS, NAMED THE (?, ?)-RK CONDITION. GEOMETRICAL PROPERTIES OF THE SPECTRUM FOR ? < 1 ARE STUDIED AND MOREOVER IT IS SHOWN THAT THEN IF ?+? = 1 THE OPERATOR IS RITT. ESTIMATES FOR THE POWER AND POWER DIFFERENCES NORMS FOR THIS TYPE OF OPERATORS ARE ALSO STUDIED. LASTLY, WE APPLY THIS THEORY TO OBTAIN AN INTERPOLATION RESULT FOR RITT AND KREISS OPERATORS ON LP SPACES.

THIS WORK PRIMARILY FOCUSES ON (N, ?)-PERIODIC SEQUENCES AND THEIR APPLICATIONS. TO BEGIN, WE PROVIDE A BRIEF OVERVIEW OF (N, ?)-PERIODIC SEQUENCES AND INTRODUCE SEVERAL RESULTS. SECONDLY, IN TERMS OF APPLICATIONS AND MAIN OBJECTIVE, WE ESTABLISH SUFFICIENT CRITERIA FOR BOTH THE EXISTENCE AND UNIQUENESS OF (N, ?)-PERIODIC MILD SOLUTIONS FOR ABSTRACT DIFFERENCE EQUATIONS OF CONVOLUTION TYPE. FURTHERMORE, WE PRESENT ILLUSTRATIVE EXAMPLES TO HIGHLIGHT OUR KEY FINDINGS.

IN THIS ARTICLE, WE STUDY THE ASYMPTOTIC BEHAVIOR AND DECAY OF THE SOLUTION OF THE FULLY DISCRETE HEAT PROBLEM. WE SHOW BASIC PROPERTIES OF ITS SOLUTIONS, SUCH AS THE MASS CONSERVATION PRINCIPLE AND THEIR MOMENTS, AND WE COMPARE THEM TO THE KNOWN ONES FOR THE CONTINUOUS ANALOGUE PROBLEMS. WE PRESENT THE FUNDAMENTAL SOLUTION, WHICH IS GIVEN IN TERMS OF SPHERICAL HARMONICS, AND WE STATE POINTWISE AND ?P ESTIMATES FOR THAT. SUCH CONSIDERATIONS ALLOW TO PROVE DECAY AND LARGE-TIME BEHAVIOR RESULTS FOR THE SOLUTIONS OF THE FULLY DISCRETE HEAT PROBLEM, GIVING THE CORRESPONDING RATES OF CONVERGENCE ON ?P SPACES.