J. Nonl. Evol. Equ. Appl. 2017 (7), pp. 95-108, published on November 23, 2017:

Existence of solutions for a class of semilinear evolution equations with impulses and delays


H. Leiva

School of Mathematical Sciences and Information Technology, Department of Mathematics, San Miguel de Urcuqui-100119, Imbabura-Ecuador

P. Sundar

Louisiana State University, Department of Mathematics, Baton Rouge, LA -70803, USA

Received on November 12, 2016
Accepted on March 26, 2017

Communicated by Toka Diagana

Abstract.  We prove the existence and uniqueness of the solutions for the followingclass of semilinear evolution equations with impulses and delays
z ́ = −Az + F(t, zt), z ∈ Z, t ∈ (0, τ ], t ≠ tk,
z(s) = φ(s), s ∈ [−r, 0],
z(tk+) = z(tk) + Jk(tk, z(tk)), k = 1, 2, 3, . . . , p.
where 0 < t1 < · · · < tp < τ , Z a Banach space,  zt defined as a function from [−r, 0] to Z by zt(s) = z(t + s), −r ≤ s ≤ 0,  and Jk : [0, τ ] × Zα → Zα, F : [0, τ ] × C(−r, 0;Zα) → Z. In the above problem,  A : D(A) ⊂ Z → Z is a sectorial operator in Z with −A being the generator of a strongly continuous compact semigroup {T(t)}t≥0, and Zα = D(Aα). The novelty of this work is that our class of evolution equations contain nonlinear terms that involve spatial derivatives. Our framework includes several important partial differential equations such as Burgers Equation with impulses and delays.
Keywords:Sectorial operator, Fractional power spaces, Semigroups, Semilinear evolution equations, Impulses, Delays, Karakostas fixed point theorem.
2010 Mathematics Subject Classification:   34K30, 34k35, 35R10; secondary: 93B05, 93C10.

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