Nonlinear mappings of monotone type

Pascali, Dan; Sburlan, Silviu

doi:10.1007/978-94-009-9544-4_3

Cited by 270 publications

(126 citation statements)

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“…This makes difference in proving the existence of solution of (1.1). Specifically, if in (1.1), Ω is reduced to be a bounded interval of R + = (0, +∞), p = q = 2 and g(x) = 0, we can get the following special example of (1.1), which can be found in [5]:…”

Section: Discussion Of Equation (11)mentioning

confidence: 99%

“…Since X * is strictly convex, J is a single-valued mapping (c.f. [5,16]). A multi-valued mapping B : X → 2 X * is said to be monotone (c.f.…”

Section: Preliminariesmentioning

confidence: 97%

“…Lemma 2.1 [5] . If B : X → 2 X * is a maximal monotone operator such that D(B) = X, then B is pseudo-monotone.…”

Section: Preliminariesmentioning

confidence: 98%

“…Then A is said to be a pseudo-monotone operator (c.f. [5]) provided that (i) For each x ∈ C, the image Ax is a non-empty closed and convex subset of X * ; (ii) If {x n } is a sequence in C converging weakly to x ∈ C and if f n ∈ Ax n is such that…”

Section: Preliminariesmentioning

confidence: 99%

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Existence of solutions to nonlinear Neumann boundary value problems with p-Laplacian operator and iterative construction

Zhou

Agarwal

2011

Acta Math. Appl. Sin. Engl. Ser.

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Section: Discussion Of Equation (11)mentioning

confidence: 99%

“…Since X * is strictly convex, J is a single-valued mapping (c.f. [5,16]). A multi-valued mapping B : X → 2 X * is said to be monotone (c.f.…”

Section: Preliminariesmentioning

confidence: 97%

“…Lemma 2.1 [5] . If B : X → 2 X * is a maximal monotone operator such that D(B) = X, then B is pseudo-monotone.…”

Section: Preliminariesmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Existence of solutions to nonlinear Neumann boundary value problems with p-Laplacian operator and iterative construction

Zhou

Agarwal

2011

Acta Math. Appl. Sin. Engl. Ser.

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“…Papers which concern differential equations and variational inequality are too many to cite. We will only cite [3], [14], [17] and [26]. Readers will find many further references cited in these.…”

Section: Introductionmentioning

confidence: 99%

Generalized variational-like inequalities with pseudomonotone set-valued mappings

Ding

Tarafdar

2000

Archiv der Mathematik

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In this paper, authors introduce the concepts of hY h-pseudo-monotone and hY h-demi-monotone mappings and prove some existence results of solutions for a new class of generalized variational-like inequalities with pseudo-monotone set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces. 1. Introduction and preliminaries. Since the appearance of the papers of Minty [20] and [21], Hartman and Stampacchia [14] and Browder [3], the theory of monotone (nonlinear) operators in general and the variational inequality in particular have generated a tremendous interest amongst mathematicians. This is because of the wide applicability of the variational inequality in nonlinear elliptic boundary value problems, obstacle problems, complementarity problems, mathematical programming, mathematical economics and in many other areas. Papers which concern differential equations and variational inequality are too many to cite. We will only cite [3], [14], [17] and [26]. Readers will find many further references cited in these. In [16] Kamaradian showed that the problem of complementarity can be reduced to the variational inequality, while the relationship between mathematical programming and the variational inequality was shown by Mancino and Stampacchia [19], between the variational inequality and convex functions by Rockafeller [28] and Moreau [22], and between the variational inequality and the equilibrium point of Walrasian economy in Riesz spaces in [1]. In [32] we have given a proof of the theorem on the existence of a solution of the variational inequality in topological vector space by applying a fixed point theorem of ours and recently in [33] we have generalized this fixed point theorem and shown that this generalized fixed point theorem is equivalent to Fan-Knaster-KuratowskiMazurkiewicz theorem [10] and we have applied this fixed point theorem to obtain variational inequalities under more general conditions [34]. In [8] and [9] we have dealt with variational inequalities of set-valued mappings. The present paper can be viewed as a further contribution in this direction.Let E and F be two vector spaces over a scalar field F (either the real field or the complex field). We shall denote by 2 F the family of all subsets of F and by fE the family of all

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