Nonlinear integral inclusions of Hammerstein type

Appell, Jürgen; Pascale, Espedito De; Nguyêñ, Hôǹg Thái; Zabreiko, P. P.

doi:10.12775/tmna.1995.007

Cited by 20 publications

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References 22 publications

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“…Theorem 2.1/(2) together with its proof is crucially based on a new relation (see Lemma 2.1) between the Q-upper limit and the M-upper limit of a sequence of subsets of F in the case dim F -+oo. Note that Theorem 2.1/(2) allows immediately to refine recent existence theorems [2,3] for nonlinear inclusions with nonpolynomial / exponential nonlinearities by droping such the additional assumption of [2,3] that Nf maps an order bounded set into an order bounded set of Y.…”

Section: Introductionmentioning

confidence: 94%

On the Sequential Strong-Weak Closedness of the Nemytskij Multivalued Operator

Nguyêñ¹

2002

Demonstratio Mathematica

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Abstract. Let Q be a measure space, and E, F be separable Banach spaces. Given a multifunction / : fi x E -> 2 F , denote by Nf(x) the set of all measurable selections of the multifunction /(•, x(s)), for a function x : £1 -> E. In this note we obtain a general theorem on the sequential strong-weak closedness for the Nemytskij multivalued superposition operator Nf acting into a Banach space of measurable F-valued functions in the infinite-dimensional case dim F = +oo, via discovering a new relation between the Q-upper limit and the M-upper limit of a sequence of subsets of F. IntroductionClosedness-type theorems play important roles in many problems of the theories of differential / integral inclusions and optimal control. The first results of this kind were obtained in the work of C. Olech, A. Lasota, L. Cesari, C. Castaing, C. Castaing and M. Valadier, and others in 1960's decade (see references e.g. in [7,9]).The present note devotes the sequential strong-weak closedness problem for the Nemytskij multivalued superposition operator Nf generated by a multifunction / : f2 x E -> 2 F (fi is a measure space, and E, F are separable Banach spaces) and which acts into a Banach space Y of measurable F-valued functions. In the finite-dimensional case dim F < +oo the general sequential strong-weak closedness result for Nf, at least in the case of the L p -type space Y, can be immediately deduced from the above mentioned work. In the infinite-dimensional case dim F = +oo, various results on the sequential strong-weak closedness for Nf acting into the L p -type space Y 2000 Mathematics Subject Classification: Primary 47H04, 47H30; Secondary 28B20, 46E30.Key words and phrases: multivalued Nemytskij superposition operator, sequential strong-weak closedness, Q-upper limit / M-upper limit of subsets, Q-upper / M-upper semicontinuities, Banach space of measurable functions, Kôthe-Bochner space.

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Section: Introductionmentioning

confidence: 94%

On the Sequential Strong-Weak Closedness of the Nemytskij Multivalued Operator

Nguyêñ¹

2002

Demonstratio Mathematica

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“…Suitable choices for E are the space C of continuous function, the Hölder spaces C α , the Lebesgue spaces L p , the Orlicz spaces L ϕ , or more generally, ideal spaces (cf. [2]). If x ∈ E and y ∈ L ∞ implies that xy ∈ E and xy E ≤ x E y L∞ , i.e.…”

Section: Theorem 25 ([26]mentioning

confidence: 99%

Solvability of functional quadratic integral equations with perturbation

Metwali¹

2013

OpMath

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“…In this direction we have the works of Lyapin [11], Coffman [8], Glashoff-Sperkels [9], Papageorgiou [18], Appell et al [3] and O'Regan [14]. Most of the existence theorems proved in the above works are based on the fixed point principles of Nadler and of Kakutani-KyFan (see KleinThompson [10]).…”

Section: Introductionmentioning

confidence: 99%

“…These fixed point principles are multivalued analogs of the Banach and Schauder-Tichonov fixed point theorems respectively. Only Coffman [8], Appell et al [3] and O'Regan [14] used different approaches. Coffman studied eigenvalue problems by means of a topological characteristic (called "genus") for set-valued operators.…”

Section: Introductionmentioning

confidence: 99%

“…Coffman studied eigenvalue problems by means of a topological characteristic (called "genus") for set-valued operators. Appell et al [3] and O'Regan [14] used multivalued variants of the Leray-Schauder principle. The first of these works used the theory of ideal spaces, while the second deals with superlinear integral inclusions defined on R.…”

Section: Introductionmentioning

confidence: 99%

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Hammerstein integral inclusions in reflexive Banach spaces

Cardinali

Papageorgiou

1999

Proc. Amer. Math. Soc.

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Abstract. In this paper we examine multivalued Hammerstein integral equations defined in a separable reflexive Banach space. We prove existence theorems for both the "convex" problem (the multifunction is convex-valued) and the "nonconvex" problem (the multifunction is not necessarily convex-valued). We also show that the solution set of the latter is dense in the solution set of the former (relaxation theorem). Finally we present some examples illustrating the applicability of our abstract results.

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Nonlinear integral inclusions of Hammerstein type

Cited by 20 publications

References 22 publications

On the Sequential Strong-Weak Closedness of the Nemytskij Multivalued Operator

On the Sequential Strong-Weak Closedness of the Nemytskij Multivalued Operator

Solvability of functional quadratic integral equations with perturbation

Hammerstein integral inclusions in reflexive Banach spaces

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