Abstract
We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general n th problem in time scales with linear dependence on the i th Δ-derivatives for i = 1,2,…,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f(t,x) which is rd-continuous in t and continuous in x uniformly in t. To do that, we obtain the expression of the Green's function of a related linear operator in the space of the antiperiodic functions.
Highlights
The theory of dynamic equations has been introduced by Stefan Hilger in his Ph.D. thesis [12]
We study the existence and uniqueness of solutions of the following nthorder dynamic equation with antiperiodic boundary value conditions: (Ln) n−1 u∆n (t) + Mju∆j (t) = f t, u(t), ∀t ∈ I = [a, b], j=1 u∆i (a) = −u∆i σ(b), 0 ≤ i ≤ n − 1
To study the existence and uniqueness of solutions of problem (Ln) in an arbitrary bounded time scale T ⊂ R, we use the technique developed in [3, 4], based on the concept of coupled lower and upper solutions, similar to the definition given in [10] for operators defined in abstract spaces and in [11] for antiperiodic boundary first-order differential equations
Summary
The theory of dynamic equations has been introduced by Stefan Hilger in his Ph.D. thesis [12]. This formula is analogous to the one given in [9] for nth-order dynamic equations with periodic boundary value conditions. To study the existence and uniqueness of solutions of problem (Ln) in an arbitrary bounded time scale T ⊂ R, we use the technique developed in [3, 4], based on the concept of coupled lower and upper solutions, similar to the definition given in [10] for operators defined in abstract spaces and in [11] for antiperiodic boundary first-order differential equations.
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