IN THIS NOTE WE DISCUSS THE EXISTENCE AND STABILITY OF AN INVERSE PROBLEM ARISING FROM THE DETERMINATION OF THE REACTION COEFFICIENTS FOR AN SIS MODEL. THE STUDY IS MOTIVATED BY A REMARK REGARDING THE FINAL DISCUSSION OF THE RECENT PAPER BY XIANG AND LIU (2015). THE WEAK ASSUMPTION OF THE WORK OF H. XIANG AND B. LIU IS THAT THE PROOFS OF EXISTENCE AND STABILITY RESULTS ARE VALID ONLY FOR THE ONE-DIMENSIONAL CASE. HERE, WE INTRODUCE AN APPROPRIATE FRAMEWORK WHICH IS ALSO VALID IN THE MULTIDIMENSIONAL CASE AND THAT GENERALIZES THE PREVIOUS RESULTS.

IN THIS SHORT NOTE THE AUTHORS GIVE ANSWERS TO THE THREE OPEN PROBLEMS FORMULATED BY WU AND SRIVASTAVA [{\IT APPL. MATH. LETT. 25 (2012), 1347--1353}]. WE DISPROVE THE PROBLEM 1, BY SHOWING THAT THERE EXISTS A TRIANGLE WHICH DOES NOT SATISFIES THE PROPOSED INEQUALITY. WE PROVE THE INEQUALITIES CONJECTURED IN PROBLEMS 2 AND 3. FURTHERMORE, WE INTRODUCE AN OPTIMAL REFINEMENT OF THE INEQUALITY CONJECTURED ON PROBLEM 3.

THIS PAPER DISCUSSES THE STUDY OF ASYMPTOTIC BEHAVIOR OF NON-OSCILLATORY SOLUTIONS FOR HIGH ORDER DIFFERENTIAL EQUATIONS OF POINCARÉ TYPE. WE PRESENT TWO NEW AND WEAKER HYPOTHESES ON THE COEFFICIENTS, WHICH IMPLIES A WELL POSEDNESS RESULT AND A CHARACTERIZATION OF ASYMPTOTIC BEHAVIOR FOR THE SOLUTION OF THE POINCARÉ EQUATION. IN OUR DISCUSSION, WE USE THE SCALAR METHOD: WE DEFINE A CHANGE OF VARIABLE TO REDUCE THE ORDER OF THE POINCARÉ EQUATION AND THUS DEMONSTRATE THAT A NEW VARIABLE CAN SATISFIES A NONLINEAR DIFFERENTIAL EQUATION; WE APPLY THE METHOD OF VARIATION OF PARAMETERS AND THE BANACH FIXED-POINT THEOREM TO OBTAIN THE WELL POSEDNESS AND ASYMPTOTIC BEHAVIOR OF THE NON-LINEAR EQUATION; AND WE ESTABLISH THE EXISTENCE OF A FUNDAMENTAL SYSTEM OF SOLUTIONS AND FORMULAS FOR THE ASYMPTOTIC BEHAVIOR OF THE POINCARÉ TYPE EQUATION BY REWRITING THE RESULTS IN TERMS OF THE ORIGINAL VARIABLE. MOREOVER WE PRESENT AN EXAMPLE TO SHOW THAT THE RESULTS INTRODUCED IN THIS PAPER CAN BE USED IN CLASS OF FUNCTIONS WHERE CLASSICAL THEOREMS FAIL TO BE APPLIED.

IN THIS PAPER WE PROVE THREE POWER-EXPONENTIAL INEQUALITIES FOR POSITIVE REAL NUMBERS. IN PARTICULAR, WE CONCLUDE THAT THIS PROOFS GIVE AFFIRMATIVELY ANSWERS TO THREE, UNTIL NOW, OPEN PROBLEMS (CONJECTURES 4.4, 2.1 AND 2.2) POSED BY CÎRTOAJE (J. INEQUAL. PURE APPL. MATH. 10:21, 2009; J. NONLINEAR SCI. APPL. 4(2):130-137, 2011). MOREOVER, WE PRESENT A NEW PROOF OF THE INEQUALITY ARA+BRB?ARB+BRA FOR ALL POSITIVE REAL NUMBERS A AND B AND R?[0,E]. IN ADDITION, THREE NEW CONJECTURES ARE PRESENTED.