On a fixed point theorem of Krasnosel'skii type and application to integral equations

Ngoc, Le Thi Phuong; Long, Nguyễn Thành

doi:10.1155/fpta/2006/30847

Cited by 17 publications

(14 citation statements)

References 7 publications

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“…For each

n \in N

, let

X_{n} = C ({[0, n]}^{N}; E)

be the Banach space of all continuous functions

u : {[0, n]}^{N} \to E

with the norm

{(||, u)}_{n} = {trueprefixsup}_{x \in {[0, n]}^{N}} |u (x)|

and

A_{n} = {u |_{{[0, n]}^{N}} : u \in A}

. The set A in X is relatively compact if and only if for each

n \in N, A_{n}

is equicontinuous in

X_{n}

and for every

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

, the set

A_{n} (x) = {u (x) : u \in A_{n}}

is relatively compact in E . Proof The proof of this lemma is similar to that in the Appendix of , it follows from the Ascoli‐Arzela's Theorem (see , p. 211).

□

Let Ω be a bounded subset of X .…”

Section: Existence Resultsmentioning

confidence: 99%

“…Recently, in case the Banach space E is arbitrary, the existence of asymptotically stable solutions to the following integral equations, in one variable (see )

x (t) = q (t) + f (t, x (t)) + \int_{0}^{t} V (t, s, x (s)) d s + \int_{0}^{\infty} G (t, s, x (s)) d s, t \in {double-struckR}_{+},

and in two variables (see )

\begin{matrix} true \\ right u (x, y) & = & left q (x, y) + f (x, y, u (x, y)) + \int_{0}^{x} \int_{0}^{y} V (x, y, s, t, u (s, t)) d s d t \\ left + \int_{0}^{\infty} \int_{0}^{\infty} F (x, y, s, t, u (s, t)) d s d t, (x, y) \in {double-struckR}_{+}^{2}, \end{matrix}

also have been proved by using the fixed point theorem of Krasnosels'kiĭ type as follows. Theorem Let

(X, {(|| \cdot)}_{n})

be a Fréchet space and let

U, C : X \to X

be two operators.…”

Section: Introductionmentioning

confidence: 99%

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

Ngoc¹,

Long

2014

Math. Nachr.

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Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'kiĭ type. In order to illustrate the results obtained here, an example is given.

show abstract

“…For each

n \in N

, let

X_{n} = C ({[0, n]}^{N}; E)

be the Banach space of all continuous functions

u : {[0, n]}^{N} \to E

with the norm

{(||, u)}_{n} = {trueprefixsup}_{x \in {[0, n]}^{N}} |u (x)|

and

A_{n} = {u |_{{[0, n]}^{N}} : u \in A}

. The set A in X is relatively compact if and only if for each

n \in N, A_{n}

is equicontinuous in

X_{n}

and for every

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

, the set

A_{n} (x) = {u (x) : u \in A_{n}}

is relatively compact in E . Proof The proof of this lemma is similar to that in the Appendix of , it follows from the Ascoli‐Arzela's Theorem (see , p. 211).

□

Let Ω be a bounded subset of X .…”

Section: Existence Resultsmentioning

confidence: 99%

“…Recently, in case the Banach space E is arbitrary, the existence of asymptotically stable solutions to the following integral equations, in one variable (see )

x (t) = q (t) + f (t, x (t)) + \int_{0}^{t} V (t, s, x (s)) d s + \int_{0}^{\infty} G (t, s, x (s)) d s, t \in {double-struckR}_{+},

and in two variables (see )

\begin{matrix} true \\ right u (x, y) & = & left q (x, y) + f (x, y, u (x, y)) + \int_{0}^{x} \int_{0}^{y} V (x, y, s, t, u (s, t)) d s d t \\ left + \int_{0}^{\infty} \int_{0}^{\infty} F (x, y, s, t, u (s, t)) d s d t, (x, y) \in {double-struckR}_{+}^{2}, \end{matrix}

also have been proved by using the fixed point theorem of Krasnosels'kiĭ type as follows. Theorem Let

(X, {(|| \cdot)}_{n})

be a Fréchet space and let

U, C : X \to X

be two operators.…”

Section: Introductionmentioning

confidence: 99%

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

Ngoc¹,

Long

2014

Math. Nachr.

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show abstract

“…Theorem 1 [8] Let (X, |·| n ) be a Fréchet space and U, C : X → X be two operators. Assume that U is a k−contraction operator, k ∈ [0, 1) (depending on n) with respect to a family of seminorms · n equivalent with the family |·| n and C is completely continuous such that lim |x| n →∞ |Cx| n |x| n = 0 ∀n ∈ N. Then, U + C has a fixed point.…”

Section: S)f (S X(θ (S)))ds + σ (T)mentioning

confidence: 99%

“…First, the following lemma for the relative compactness of a subset in X is useful to prove the main results. Lemma 2 was proved in detail in appendix [8], by applying the Ascoli-Arzela's Theorem, see [6, p. 211].…”

Section: Existence Of Solutionsmentioning

confidence: 99%

Asymptotically Stable Solutions for a Nonlinear Functional Integral Equation

Ngoc

Long

2015

Acta Math Vietnam

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“…More recently, the Krasnosel'skij fixed point theorem is proved in the framework of Fréchet spaces in [30] with application to a general equation involving the sum of two nonlinear integrals.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in L1 spaces

Djebali

Sahnoun

2010

Journal of Differential Equations

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MSC: 47H09 47H10 47H30 Keywords: Weak topology Measure of weak noncompactness Fixed point theorem Nonlinear alternative Hammerstein equation This paper is devoted to establishing new variants of some nonlinear alternatives of Leray-Schauder and Krasnosel'skij type involving the weak topology of Banach spaces. The De Blasi measure of weak noncompactness is used. An application to solving a nonlinear Hammerstein integral equation in L 1 spaces is given. Our results complement recent ones in [K. Latrach, M.A. Taoudi, A. Zeghal, Some fixed point theorems of the Schauder and the Krasnosel'skij type and application to nonlinear transport equations, J. Differential Equations 221 (2006) 256-2710] and [K. Latrach, M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L 1 spaces, Nonlinear Anal. 66 (2007) 2325-2333].

show abstract

On a fixed point theorem of Krasnosel'skii type and application to integral equations

Abstract: This paper presents a remark on a fixed point theorem of Krasnosel'skii type. This result is applied to prove the existence of asymptotically stable solutions of nonlinear integral equations.

Cited by 17 publications

References 7 publications

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

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

Asymptotically Stable Solutions for a Nonlinear Functional Integral Equation

Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in L1 spaces

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