Positive solutions of operator equations on ordered Banach spaces and applications

This paper is concerned with an operator equation Ax + Bx + C x = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.

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“…, then u n y n v n . Using the same proof of Theorem 1.3 in [39], we can easily obtain that there exists y * ∈ P h such that T y * = y * ,…”

Section: Fixed Point Theoremsmentioning

confidence: 87%

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

Zhai

Anderson

2011

Journal of Mathematical Analysis and Applications

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“…And then, as two special cases of (1.1), we consider the existence and uniqueness of positive solutions for (1.1) with an h-concave operator and a generalized β-concave operator. The corresponding results in [7,10,11,13,21,[26][27][28]32,33] turn out to be special cases of our main results. As applications, we first utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for first-order initial value problems and two-point boundary value problems.…”

Section: Introductionmentioning

confidence: 69%

“…Remark 2.1 In many papers, the condition that operator A : P h → P h or A :P →P was always required (for example, see [7,10,21,28,[31][32][33]). It is clear that Ah ∈ P h is automatically satisfied.…”

Section: Lemma 22 Assume That Operator a Satisfiesmentioning

confidence: 99%

“…In [10] D.J.Guo widens the conditions and removes the hypotheses of continuation for operators, and then extends the results of fixed points, eigenvectors for β-concave operators. Here we shall study a class of generalized β-concave and extend the corresponding results of [10,21,26,27,32,33] for β-concave operators.…”

Section: Equation (11) With Generalized β-Concave Operatormentioning

confidence: 99%

“…In [32] the authors studied the existence and uniqueness of positive solutions for the following operator equation on ordered Banach spaces…”

Section: Introductionmentioning

confidence: 99%

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Positive solutions for nonlinear operator equations and several classes of applications

2009

Self Cite

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In this paper, we study a class of nonlinear operator equations x = Ax + x 0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.

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Positive fixed points for a class of nonlinear operators and applications

Berzig

Samet

2012

Positivity

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Positive solutions of operator equations on ordered Banach spaces and applications

Cited by 19 publications

References 10 publications

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

Positive solutions for nonlinear operator equations and several classes of applications

Positive fixed points for a class of nonlinear operators and applications

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