Abstract
An algorithm for constructing two sequences of successive approximations of a solution of the nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’ is given. The case of a boundary condition of anti-periodic type is investigated. This algorithm is based on the monotone iterative technique. Two sequences of successive approximations are constructed. It is proved both sequences are monotonically convergent. Each term of the constructed sequences is a solution of an initial value problem for a linear differential equation with ‘maxima’ and it is a lower/upper solution of the given problem. A computer realization of the algorithm is suggested and it is illustrated on a particular example.
Highlights
Differential equations with ‘maxima’ are adequate models of real world problems, in which the present state depends significantly on its maximum value on a past time interval.Note that usually differential equations with ‘maxima’ are not possible to be solved in an explicit form and that requires the application of approximate methods
The case when the nonlinear boundary function is a nondecreasing one with respect to its second argument is studied. This type of the boundary function covers the case of an anti-periodic boundary condition
We study boundary condition ( ) in the case when the function g(x, y) is nondecreasing with respect to its second argument y
Summary
Differential equations with ‘maxima’ are adequate models of real world problems, in which the present state depends significantly on its maximum value on a past time interval (see [ – ], monograph [ ]).Note that usually differential equations with ‘maxima’ are not possible to be solved in an explicit form and that requires the application of approximate methods. Definition We will say that the functions α, β ∈ P(h, T) form a couple of quasisolutions of boundary value problem ( )-( ), if they satisfy the equations g(α( ), β(T)) = g(β( ), α(T)) = , ( ) and ( ). Definition We will say that the functions α, β ∈ P(h, T) form a couple of quasi-lower and quasi-upper solutions of boundary value problem ( )-( ), if α (t) ≤ f t, α(t), max α(s) for t ∈ [ , T], s∈[t–h,t]
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