A fixed point theorem for function spaces

Vidossich, Giovanni

doi:10.1016/0022-247x(71)90040-0

Cited by 20 publications

(8 citation statements)

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“…In § 3, we apply the Schauder Fixed Point Theorem in order to discuss the existence of solutions of (1.1). We also show that the set of solutions of equation 1.1is an R δ by using the Vidossich Theorem stated in [14]. Here, R δ is the intersection of a decreasing sequence of compact absolute retracts [3].…”

Section: Introductionmentioning

confidence: 85%

The Darboux problem involving the distributional Henstock–Kurzweil integral

Lu¹,

Ye²,

Wang³

et al. 2012

Proceedings of the Edinburgh Mathematical Society

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Section: Introductionmentioning

confidence: 85%

The Darboux problem involving the distributional Henstock–Kurzweil integral

Lu¹,

Ye²,

Wang³

et al. 2012

Proceedings of the Edinburgh Mathematical Society

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“…It can be shown that F satisfies the assumptions of Theorem 1.3 from [6] (see also Vidossich [7] ). By this theorem we conclude that under the assumptions of the Theorem, the set of all solutions of (1) − (2) defined on J is a compact R δ in C(J, E), i.e.…”

Section: Ifmentioning

confidence: 99%

An existence theorem for solutions of an integro-differential equation in Banach spaces

Dutkiewicz

2012

Demonstratio Mathematica

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Abstract. The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.Consider the following Cauchy problemin a Banach space E, where m ≥ 1 is a natural number. Throughout this paper we shall assume that D = [0, a] is a compact interval in R, B = {x ∈ E : x ≤ b}, f : D × B × E → E is a continuous function, and g : D 2 × B → E is a bounded continuous function. Moreover, we suppose that f (t, x, z) ≤ M for t ∈ D, x ∈ B, z ∈ W , whereDenote by α the Kuratowski measure of noncompactness in E (cf.[1]). Main resultIn this section we shall prove an existence theorem for local solutions of the above initial value problem for the nonlinear integro-differential equation in Banach spaces.2000 Mathematics Subject Classification: 47G20, 45J05.

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“…Recall that if a set is an R δ , it is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts. Furthermore, G. Vidossich [31] pointed out that R δ is a nonempty, compact and connected set.…”

Section: Darboux Problemmentioning

confidence: 99%

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

Liu¹

2016

JMS

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This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.

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A fixed point theorem for function spaces

Cited by 20 publications

References 4 publications

The Darboux problem involving the distributional Henstock–Kurzweil integral

The Darboux problem involving the distributional Henstock–Kurzweil integral

An existence theorem for solutions of an integro-differential equation in Banach spaces

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

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