Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Ngoc, Le Thi Phuong; Long, Nguyễn Thành

doi:10.1002/mana.201300065

Cited by 4 publications

(11 citation statements)

References 10 publications

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“…In order to continue, we need the following preliminary. Its proof is similar to that in [, Lemma ]. Lemma Let w ,

a \in C ({[0, b]}^{N}; R_{+})

and

r \in C (Δ \cap {[0, b]}^{N}; R_{+}),

r (x, y) \leq r (x, 0) \leq r (0, 0),

for all

y \in B_{x},

x \in {[0, b]}^{N} .

\begin{matrix} true \\ right w (x) \leq a (x) + \int_{B_{x}} r (x, y) w (y) d y, \end{matrix}

for all

x \in {[0, b]}^{N},

then

\begin{matrix} true \\ right (i) & w (x) \leq a (x) + r (x, 0) \sum_{k = 0}^{\infty} \frac{{(r (0, 0) x_{1} \dots x_{N})}^{k}}{{(k!)}^{N}} \int_{B_{x}} a (y) d y, \\ right (i i) & w (x) \leq a (x) + r (x, 0) exp (r (0, 0) x_{1} \dots x_{N}) \end{matrix}

…”

Section: The Connectivity and Compactness Of Solution Setmentioning

confidence: 74%

“…Proof The details of the proof can be found in , let us sum up the main points of this proof as follows. Eq.…”

Section: The Existence Resultsmentioning

confidence: 99%

“…Let ( A 1 ), ( A 2 ),

(A_{3}^{'})

(A_{4}^{'})

hold. We have that has a solution in

{double-struckR}_{+}^{N},

see . On the other hand, if u is a solution of then

\begin{matrix} true \\ right (|| v (x)) \leq \int_{B_{x}} r (x, y) (|| v (y)) d y + a (x), \end{matrix}

where

\begin{matrix} true \\ right a (x) = \frac{1}{1 - L} \int_{{double-struckR}_{+}^{N}} ω_{2} (x, y) d y, r (x, y) = \frac{1}{1 - L} ω_{1} (x, y) . \end{matrix}

…”

Section: An Rδ‐structurementioning

confidence: 99%

“…In , by applying Theorem , the existence of asymptotically stable solutions for the following Volterra–Hammerstein functional integral equation in N variables is investigated

u (x) = q (x) + f (x, u (x)) + \int_{B_{x}} V (x, y, u (y)) d y + \int_{R_{+}^{N}} F (x, y, u (y)) d y,

x \in {double-struckR}_{+}^{N} = \{(x_{1}, \dots, x_{N}) \in R^{N} : x_{1} \geq 0, \dots, x_{N} \geq 0\},

where

q : {double-struckR}_{+}^{N} \to E;

f : {double-struckR}_{+}^{N} \times E \to E;

F : {double-struckR}_{+}^{2 N} \times E \to E;

V : Δ \times E \to E

are continuous, E is a Banach space with norm

(|| \cdot)

and

Δ = \{(x, y) \in R_{+}^{2 N} : y \in B_{x}\},

B_{x} = [0, x_{1}] \times \dots \times [0, x_{N}] .

…”

Section: Introductionmentioning

confidence: 99%

“…In [16], by applying Theorem 1.1, the existence of asymptotically stable solutions for the following Volterra-Hammerstein functional integral equation in N variables is investigated…”

Section: Introductionmentioning

confidence: 99%

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A continuum of solutions in a Fréchet space of a nonlinear functional integral equation in N variables

Ngoc

Long

2016

Mathematische Nachrichten

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Key wordsThe fixed point theorem of Krasnosel'skii type, a structure theorem of Krasnosel'skii and Perov, contraction mapping, completely continuous, a continuum, Hukuhara-Kneser property, a compact R δ -set MSC(2010) 45N05, 47H10In this paper, we investigate the set of solutions of a nonlinear functional integral equation in N variables in a Fréchet space. Applying a fixed point theorem of Krasnosel'skii type and a structure theorem of Krasnosel'skii and Perov, a sufficient condition is established such that the set of solutions is a continuum, that is, nonempty, compact and connected. Furthermore, based on Aronszajn type results and a theorem proved by Vidossich, we show that this solutions set is also a compact R δ . This is also true with solutions set of a nonlinear VolterraHammerstein integral equation.

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“…In order to continue, we need the following preliminary. Its proof is similar to that in [, Lemma ]. Lemma Let w ,

a \in C ({[0, b]}^{N}; R_{+})

and

r \in C (Δ \cap {[0, b]}^{N}; R_{+}),

r (x, y) \leq r (x, 0) \leq r (0, 0),

for all

y \in B_{x},

x \in {[0, b]}^{N} .

\begin{matrix} true \\ right w (x) \leq a (x) + \int_{B_{x}} r (x, y) w (y) d y, \end{matrix}

for all

x \in {[0, b]}^{N},

then

\begin{matrix} true \\ right (i) & w (x) \leq a (x) + r (x, 0) \sum_{k = 0}^{\infty} \frac{{(r (0, 0) x_{1} \dots x_{N})}^{k}}{{(k!)}^{N}} \int_{B_{x}} a (y) d y, \\ right (i i) & w (x) \leq a (x) + r (x, 0) exp (r (0, 0) x_{1} \dots x_{N}) \end{matrix}

…”

Section: The Connectivity and Compactness Of Solution Setmentioning

confidence: 74%

“…Proof The details of the proof can be found in , let us sum up the main points of this proof as follows. Eq.…”

Section: The Existence Resultsmentioning

confidence: 99%

“…Let ( A 1 ), ( A 2 ),

(A_{3}^{'})

(A_{4}^{'})

hold. We have that has a solution in

{double-struckR}_{+}^{N},

see . On the other hand, if u is a solution of then

\begin{matrix} true \\ right (|| v (x)) \leq \int_{B_{x}} r (x, y) (|| v (y)) d y + a (x), \end{matrix}

where

\begin{matrix} true \\ right a (x) = \frac{1}{1 - L} \int_{{double-struckR}_{+}^{N}} ω_{2} (x, y) d y, r (x, y) = \frac{1}{1 - L} ω_{1} (x, y) . \end{matrix}

…”

Section: An Rδ‐structurementioning

confidence: 99%

“…In , by applying Theorem , the existence of asymptotically stable solutions for the following Volterra–Hammerstein functional integral equation in N variables is investigated

u (x) = q (x) + f (x, u (x)) + \int_{B_{x}} V (x, y, u (y)) d y + \int_{R_{+}^{N}} F (x, y, u (y)) d y,

x \in {double-struckR}_{+}^{N} = \{(x_{1}, \dots, x_{N}) \in R^{N} : x_{1} \geq 0, \dots, x_{N} \geq 0\},

where

q : {double-struckR}_{+}^{N} \to E;

f : {double-struckR}_{+}^{N} \times E \to E;

F : {double-struckR}_{+}^{2 N} \times E \to E;

V : Δ \times E \to E

are continuous, E is a Banach space with norm

(|| \cdot)

and

Δ = \{(x, y) \in R_{+}^{2 N} : y \in B_{x}\},

B_{x} = [0, x_{1}] \times \dots \times [0, x_{N}] .

…”

Section: Introductionmentioning

confidence: 99%

“…In [16], by applying Theorem 1.1, the existence of asymptotically stable solutions for the following Volterra-Hammerstein functional integral equation in N variables is investigated…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations