In the present paper we are going to consider in a one dimension bounded domain a transmission system with a varying delay. Under suitable assumptions on the weights of the damping and the delay terms, we prove the well-possedness and the uniqueness of solution using the semigroup theory. Also we show the exponential stability by introducing an appropriate Lyaponov functional.2000 Mathematics Subject Classification. Primary: 435B37; Secondary: 35L55.
In this second part of the paper, we state some general conditions that guarantee that the C 0 -semigroups generated by special classes of one-dimensional second-order differential operators acting on weighted spaces of continuous functions on an arbitrary real interval can be represented as limits of iterates of the positive linear operators constructively defined according to the method developed in the first part. The particular case where the interval is compact is treated in full detail. Finally, an application to the Black-Scholes equation, which appears in some models arising from mathematical finance, is discussed as well.
PurposeA generalization of Ascoli–Arzelá theorem in Banach spaces is established. Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order. The authors’ results are obtained under, rather, general assumptions.Design/methodology/approachFirst, a generalization of Ascoli–Arzelá theorem in Banach spaces in Cn is established. Second, this new generalization with Schauder's fixed point theorem to prove the existence of a solution for a boundary value problem of higher order is used. Finally, an illustrated example is given.FindingsThere is no funding.Originality/valueIn this work, a new generalization of Ascoli–Arzelá theorem in Banach spaces in Cn is established. To the best of the authors’ knowledge, Ascoli–Arzelá theorem is given only in Banach spaces of continuous functions. In the second part, this new generalization with Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order, where the derivatives appear in the non-linear terms.
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