Weighted space method for the stability of some nonlinear equations

Cădariu, Liviu; Gǎvruţa, Laura; Găvruţa, P.

doi:10.2298/aadm120309007c

Cited by 41 publications

(32 citation statements)

References 30 publications

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“…Namely, taking a positive constant p >0, we will be using the space C p ([ a , b ]) of continuous functions

u : false[a, b false] \to double-struckC

endowed with the Bielecki metric

d_{p} false(u, v false) = \sup_{x \in false[a, b false]} \frac{false| u false(x false) - v false(x false) false|}{e^{p false(x - a false)}} .

Anyway, in a more global sense, we will also consider the space C ([ a , b ]) of continuous functions on [ a , b ], endowed with a generalization of the Bielecki metric

d false(u, v false) = \sup_{x \in false[a, b false]} \frac{false| u false(x false) - v false(x false) false|}{σ false(x false)},

where σ is a nondecreasing continuous function σ :[ a , b ]→(0, ∞ ). We recall that

((), C_{p} false(false[a, b false] false), d_{p})

and

((), C false(false[a, b false] false), d)

are complete metric spaces (cf previous studies).…”

Section: Hyers‐ulam‐rassias Stability In the Finite Interval Casementioning

confidence: 99%

“…where σ is a nondecreasing continuous function σ:[a, b]→(0,∞). We recall that C p ð½a; bÞ; d p À Á and Cð½a; bÞ; d ð Þare complete metric spaces (cf previous studies 39,40 Under the present conditions, we will deduce that the operator T is strictly contractive with respect to the metric (7). Indeed, for all u, v ∈ C([a, b]), we have Due to the fact that M 2 þ Lη ð Þ<1, it follows that T is strictly contractive.…”

Section: Hyers-ulam-rassias Stability In the Finite Interval Casementioning

confidence: 99%

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Hyers‐Ulam‐Rassias stability of nonlinear integral equations through the Bielecki metric

Castro

Simões

2018

Math Methods in App Sciences

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We analyse different kinds of stabilities for classes of nonlinear integral equations of Fredholm and Volterra type. Sufficient conditions are obtained in order to guarantee Hyers‐Ulam‐Rassias, σ‐semi‐Hyers‐Ulam and Hyers‐Ulam stabilities for those integral equations. Finite and infinite intervals are considered as integration domains. Those sufficient conditions are obtained based on the use of fixed point arguments within the framework of the Bielecki metric and its generalizations. The results are illustrated by concrete examples.

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“…Namely, taking a positive constant p >0, we will be using the space C p ([ a , b ]) of continuous functions

u : false[a, b false] \to double-struckC

endowed with the Bielecki metric

d_{p} false(u, v false) = \sup_{x \in false[a, b false]} \frac{false| u false(x false) - v false(x false) false|}{e^{p false(x - a false)}} .

Anyway, in a more global sense, we will also consider the space C ([ a , b ]) of continuous functions on [ a , b ], endowed with a generalization of the Bielecki metric

d false(u, v false) = \sup_{x \in false[a, b false]} \frac{false| u false(x false) - v false(x false) false|}{σ false(x false)},

where σ is a nondecreasing continuous function σ :[ a , b ]→(0, ∞ ). We recall that

((), C_{p} false(false[a, b false] false), d_{p})

and

((), C false(false[a, b false] false), d)

are complete metric spaces (cf previous studies).…”

Section: Hyers‐ulam‐rassias Stability In the Finite Interval Casementioning

confidence: 99%

Section: Hyers-ulam-rassias Stability In the Finite Interval Casementioning

confidence: 99%

Hyers‐Ulam‐Rassias stability of nonlinear integral equations through the Bielecki metric

Castro

Simões

2018

Math Methods in App Sciences

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show abstract

“…Instead of these techniques, we may simply use the classical Banach fixed point theorem if combined with an appropriate weighted metric framework. We should mention that this type of weighted spaces goes back to [12] and [4], where it was applied to other types of functional and integral equations. Here, it is also significant to recall the seminal paper by A. Bielecki [3], where specific exponential weighted metrics were introduced with the aim of obtaining global existence of solutions of certain functional equations.…”

Section: Theorem 11 ([9]mentioning

confidence: 99%

“…We recall that (C(I), d) is a complete metric space (cf., e.g., [4]). The next theorem is the main result of this note and exhibits weaker conditions than those of [20], under which the Volterra integral equation introduced in (1.2) is Hyers-Ulam-Rassias stable.…”

Section: Hyers-ulam-rassias Stability Of Volterra Integral Equations mentioning

confidence: 99%

Hyers-Ulam-Rassias stability of Volterra integral equations with delay within weighted spaces

Castro¹,

Guerra²

2014

Libertas Math. new ser.

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We obtain weak conditions to guarantee the Hyers-UlamRassias stability of (nonlinear) Volterra integral equations with delay. In particular, this leads to a generalization of some results previously known. Basically, this is done by using certain weight functions in the framework of the space of continuous functions. Indeed, the method consists in a convenient combination of the classical Banach fixed point theorem together with a consideration of a weighted metric. Therefore, we avoid the use of the strict successive approximation method and also the consideration of generalized metrics (which need to be typically combined with a consequent fixed point alternative theorem). Some concrete examples are presented at the end of the paper.

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“…This technique, known as the perturbation method (see [2]), has many applications in the theory of fractional differentiation operators (see [3]), in reaction-diffusion equations, stochastic stability, and asymptotic stability (see [4][5][6][7][8][9]), and for some numerical considerations (see, for example, [10][11][12]). …”

Section: Introductionmentioning

confidence: 99%