Abstract
Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.
Highlights
Maximum principles are a well known tool for studying differential equations, which can be used to receive prior information about solutions of differential inequalities and to obtain lower and upper solutions of differential equations and so on
Hilger [16] established the theory of time scales calculus to unify the continuous and discrete calculus in 1990
Ordinary dynamic equations and partial dynamic equations on time scales have been extensively studied by some authors
Summary
Maximum principles are a well known tool for studying differential equations, which can be used to receive prior information about solutions of differential inequalities and to obtain lower and upper solutions of differential equations and so on. It is well known that there are many results and applications for continuous and discrete maximum principles About these theories and applications, we can refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and the references therein.
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