Existence for a class of partial functional differential equations with infinite delay

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille-Yosida operator whereas Carathèodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.Mathematics Subject Classification (2010). Primary 34G25; Secondary 34A60, 47D62, 47H04, 47H09, 47H10, 49J24.

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“…Then, by applying Proposition 1(iii) and using the Lipschitz continuity of V (·), one may verify the following relation (see [1]):…”

Section: Definition 3 a Linear Operatormentioning

confidence: 99%

“…Recently the fixed point approach was applied in [1] and [11] to obtain the existence results for some classes of functional differential equations and inclusions in Banach spaces. In both papers, it was assumed that the linear part A generates an integrated semigroup whose derivative is a compact semigroup.…”

Section: Introductionmentioning

confidence: 99%

On semilinear differential inclusions in Banach spaces with nondensely defined operators

Obukhovskiĭ

Zecca

2011

J. Fixed Point Theory Appl.

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“…With regard to the infinite delay case, the situation is di¤erent, since properties of solutions depend on the choice of the phase space. The fundamental theory related to functional di¤erential equations with infinite delay can be found in [25] and we refer the reader to works in [1,2,3,6,10,11,12,13,14,17,24] and the references therein, in which the results are about some quantitative and qualitative aspects of study.…”

Section: àYmentioning

confidence: 99%

Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

Bouzahir

You

Yuan

2011

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“…Since such equations are often more realistic to describe natural phenomena than those without delay, they have been investigated in variant aspects by many authors(see, e.g., [14][15][16][17] and references therein). Pavel and Iacob [18] discussed viability problem of semilinear differential equations of retarded type.…”

Section: §1 Introductionmentioning

confidence: 99%

Viability for a class of semilinear differential equations of retarded type

Dong

2009

Appl. Math. J. Chin. Univ.

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Let X be a Banach space, A : D(A) ⊂ X → X the generator of a compact C0-semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b)×X → X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded typeX → X, t ≥ 0, D a nonempty subset in X. Let q and T be positive numbers and −∞ ≤ a < b ≤ +∞. Given t 0 ∈ (a, b), a function x : [t 0 − q, t 0 + T ] → X and t ∈ [t 0 , t 0 + T ], define x t : [−q, 0] → X by x t (θ) = x(t + θ) for all θ ∈ [−q, 0]. In this paper, we discuss the semilinear differential equation of retarded type,2) where C([−q, 0]; X) denotes the Banach space of continuous X-valued functions on [−q, 0] with supermum norm, f : (a, b) × X → X and t 0 ∈ (a, b).We say that D is a viable domain for (1.1) if for each t 0 ∈ (a, b), φ ∈ C([−q, 0]; D) , there exists at least one mild solution u :We recall that by mild solution to (1.1) and (1.2) we mean a continuous function u : [t 0 − q, t 0 + T ] → X, satisfying u(t 0 + θ) = φ(θ)

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Existence for a class of partial functional differential equations with infinite delay

Cited by 58 publications

References 17 publications

On semilinear differential inclusions in Banach spaces with nondensely defined operators

On semilinear differential inclusions in Banach spaces with nondensely defined operators

Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

Viability for a class of semilinear differential equations of retarded type

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