On the existence of solutions for quadratic integral equations in Orlicz spaces

Cichoń, Mieczysław; Metwali, Mohamed M. A.

doi:10.1515/ms-2016-0233

Cited by 9 publications

(10 citation statements)

References 24 publications

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“…nondecreasing on I. Similarly, as claimed in [10] this set is nonempty, bounded, closed and convex in ( )…”

Section: Resultssupporting

confidence: 57%

“…and K 0 is continuous with a norm (b) By assumption (iv), the operator K 0 is continuous and the norm of ( ) K x 0 is estimated by (cf. [10])…”

Section: Resultsmentioning

confidence: 99%

“…In [9,10], the authors extend the results in [6] to arbitrary Orlicz spaces which are not necessary Banach algebras using different sets of assumptions controlling the intermediate spaces. The authors in [11] discussed the quadratic functional-integral equations in Orlicz spaces L φ for φ satisfying Δ 2 -condition.…”

Section: mentioning

confidence: 99%

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On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

Metwali

2020

Demonstratio Mathematica

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“…nondecreasing on I. Similarly, as claimed in [10] this set is nonempty, bounded, closed and convex in ( )…”

Section: Resultssupporting

confidence: 57%

“…and K 0 is continuous with a norm (b) By assumption (iv), the operator K 0 is continuous and the norm of ( ) K x 0 is estimated by (cf. [10])…”

Section: Resultsmentioning

confidence: 99%

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On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

Metwali

2020

Demonstratio Mathematica

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“…It can be easily checked that under some typical assumptions this problem is equivalent to the integral equation [9] x(t) = g(t) + f 1 (t, x(t)) · t 0 f 2 (s, x(s)) ds − g(0) f 1 (0, 0) .…”

Section: Remarksmentioning

confidence: 99%

On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

Metwali

2017

JMA

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The paper discusses the existence of solutions for Cauchytype problem of fractional order in the space of Lebesgue integrable functions on bounded interval. Some qualitative properties of solutions are presented such as monotonicity, uniqueness and continuous dependence on the initial data. The main tools used are measure of weak (strong) noncompactness, Darbo fixed point theorem and fractional calculus.Let R be the field of real numbers, J be the interval [0, 1] and L 1 (J) be the space of Lebesgue integrable functions (equivalence classes of functions) on a measurable subset J of R, with the standard normBy L ∞ (J) we denote the Banach space of essentially bounded measurable functions with the essential supremum norm (denoted by x L∞ ). We will write L 1 and L ∞ instead of L 1 (J) and L ∞ (J) respectively.Definition 2.1.[2] Assume that a function f : J × R → R satisfies the Carathéodory conditions i.e. it is measurable in t for any x ∈ R and continuous in x for almost all t ∈ J. Then to every function x(t) being measurable on J we assign (F x)(t) = f (t, x(t)), t ∈ J.The operator F is called the superposition (Nemytskii) operator.

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“…Let us stress that our results are new even in the context of single quadratic fractional integral equations, in particular with the Hadamard fractional integrals. For an approach allowing to generalize growth conditions with the use of Orlicz spaces, see [14,Theorem 4.16] or [15,Remark 2], or [13,Theorem 5.1], but only for the case of quadratic integral equations with Riemann-Liouville integrals, or [16] for basics about Hadamard integral operators on Orlicz spaces. We say that the pair p, q ∈ [1, ∞] is of "conjugate exponents" if p, q are connected by the relation 1/p + 1/q = 1 for 1 < p < ∞ with the convention that 1/∞ = 0.…”

Section: Introductionmentioning

confidence: 99%

On positive solutions of a system of equations generated by Hadamard fractional operators

2020

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This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann-Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

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On the existence of solutions for quadratic integral equations in Orlicz spaces

Abstract: We study quadratic integral equations in Orlicz spaces on the interval [

Cited by 9 publications

References 24 publications

On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

On positive solutions of a system of equations generated by Hadamard fractional operators

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