On the singular generalized Fisher-like equation with derivative depending nonlinearity

Djebali, Smaïl; Mebarki, Karima

doi:10.1016/j.amc.2008.08.009

Cited by 8 publications

(3 citation statements)

References 13 publications

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“…Remark To discuss the validity of Assumption

false(A_{2} false)

, see Djebali and Mebarki, remark 3.2). By

false(A_{1} false)

and the properties of Green function, the integrals M 0 , M 1 , M 2 are convergent. …”

Section: Applicationmentioning

confidence: 98%

Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

Benzenati

Mebarki

2019

Math Methods in App Sciences

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In this work, we are concerned with the existence of multiple positive fixed points for the sum of an expansive mapping with constant h > 1 and a k-set contraction when 0 ≤ k < h − 1. In particular, the case of the sum of an expansive mapping with constant h > 1 and an e-concave operator and an e-convex operator is considered. Two examples of application illustrate some of the theoretical results. KEYWORDSexpansive mapping, sum of operators, e-concave operator, e-convex operator, k-set contraction INTRODUCTIONMany problems in science, when modeled under the mathematical point of view, lead to the nonlinear equation Tx + Fx = x, posed in some closed convex subset of a Banach space. In particular, ordinary and partial differential equations and integral equations can be formulated like abstract equations. It is the reason for which it becomes desirable to develop fixed point theorems for such situations. When T is compact continuous and F is a contraction, a classical tool generally used to deal with such problems is the well-known Krasnosel'skii fixed point theorem. 1 For contributions which develop this capturing result, see previous studies. [2][3][4] The study of positive solutions to nonlinear equations, especially ordinary, partial differential equations, and integral equations, draws great attention. Positivity can be developed by arbitrary cones, that is, nonempty closed convex subsets  in some Banach space E satisfying  ⊂  for all real positive number and  ∩ (−) = {0}.On the other hand, many researchers have been interested in the extension of the fixed point theory for positive nonlinear operators with convexity and concavity, which is applied to the study of various nonlinear differential equations defined on a cone in a Banach space. Note that the definitions and properties of e-concave operators and e-convex operators were first given by Krasnosel'skii 5 and the concepts of -concavity and -convexity for nonlinear operators were introduced by Potter. 6 Zhao 7 discussed the existence of multiple positive fixed points for the sum of e-concave and e-convex operators by using the cone expansion fixed point theorem for strict k-set contraction. Djebali and Mebarki 8 developed a generalized fixed point index for the sum of an expansive mapping and a k-set contraction.Motivated by the results obtained in Zhao 7 and Djebali and Mebarki, 8 the main goal of this work is the study of the existence of multiple positive fixed points for some operators that are of the form T + F, where T is an expansive operator and F is a k-set contraction. For goal, we use the generalized fixed point index introduced in Djebali and Mebarki. 8 As corollaries of the main theorem, we obtain some fixed point results in the case where the k-set contraction F is a sum of e-concave and e-convex operators, or the sum of -concave and -convex operators. Our results generalize and improve the corresponding ones in Zhao 7 and Sang et al. 9 4412

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“…Remark To discuss the validity of Assumption

false(A_{2} false)

, see Djebali and Mebarki, remark 3.2). By

false(A_{1} false)

and the properties of Green function, the integrals M 0 , M 1 , M 2 are convergent. …”

Section: Applicationmentioning

confidence: 98%

Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

Benzenati

Mebarki

2019

Math Methods in App Sciences

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“…In this paper, we are concerned with the existence of positive solutions to the following third-order boundary value problem for a φ-Laplacian operator: Many applied problems modeling various phenomena in physics, epidemiology, combustion theory, mechanics (see, e.g., [2] and the references therein) are governed by boundary value problems (bvps for short) posed on the half-axis [0, +∞); we quote for instance the propagation of a flame in a long tube. A large amount of research papers have been devoted to these problems, in particular for the second-order boundary value problems; we refer the reader to [4], [5], [6], [7], [8], [9], [12], [15], [16], [17], and the references therein. However problems with higher-order differential equations on [0, +∞) have not been so extensively investigated; we can only cite [13], [14], [18], and [19].…”

Section: Introductionmentioning

confidence: 99%

Singular $\phi $-Laplacian third-order BVPs with derivative dependance

Djebali¹,

Saifi²

2016

Arch. Math. (Brno)

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This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a φ-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed. An example of application illustrates the applicability of the existence result.

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“…Many applications such as the propagation of nonadiabatic flames inside long tubes are governed by boundary value problems of the form (1.1). Also, the propagation of an epidemic through a population confined within a given region is governed by the so-called autonomous generalized Fisher equation (see [13,16]):…”

Section: Introductionmentioning

confidence: 99%

Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

Djebali¹,

Mebarki

2008

Acta Appl Math

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This work is concerned with the existence of unbounded positive solutions for a second-order nonlinear three-point boundary value problem on the positive half-line. The interesting points of the results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. New existence results of nontrivial single and multiple positive solutions are proved using recent fixed point theorems on cones in a special Banach space.

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On the singular generalized Fisher-like equation with derivative depending nonlinearity

Cited by 8 publications

References 13 publications

Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

Singular $\phi $-Laplacian third-order BVPs with derivative dependance

Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

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