Claudio Morales - Academia.edu (original) (raw)
Papers by Claudio Morales
Proceedings of the American Mathematical Society, Nov 1, 1995
Springer eBooks, 2001
Boundary and inwardness conditions have been particularly useful in extending fixed point theory ... more Boundary and inwardness conditions have been particularly useful in extending fixed point theory for nonexpansive mappings to broader classes of mappings, particularly to mappings satisfying local contractive and pseudocontractive assumptions. At the same time these conditions often enable one to relax the assumption that the mapping takes values in its own domain.
Commentationes Mathematicae Universitatis Carolinae, 1986
Let D be an open subset of a Banach space X, and let B(X) denote the family of all nonempty, boun... more Let D be an open subset of a Banach space X, and let B(X) denote the family of all nonempty, bounded and closed subsets of X. Suppose T:D-*-B(X) is a continuous (with respect to the Hausdorff metric) and strongly accretive mapping. It is shown that if for some zeD:t(x-z)^T(x) for x in the boundary of D and t<0, it is sufficient to guarantee that T has a zero in 0. Several implications of this result are considered, particularly on a localized version of it.
Linear Algebra and its Applications, 1990
Let Z s R and D c C, where R and 4: are the fields of real and compiex numbers, respectively. Let... more Let Z s R and D c C, where R and 4: are the fields of real and compiex numbers, respectively. Let C" x" be the space of square matrices of order tz over C. A matrix-valued function F: Z-) 62" xn is said to be proper on I if F(t) =f(t, A), where A E C'lx" and f:ZXD+C is a scalar function, and F is said to be semiproper on Z if F(t)F(t) = F(T)F(~) for all t, T E 1. The main results presented here are: (1) a characterization together with some potentially useful results for proper matrix functions; (2) a characterization of semiproper matrix functions in terms of proper ones; (3) a new and syskmatic procedure for decomposing a semiproper matrix function into a finite sum of mutually commutative proper ones. Some important applications of these new results in control engineering, linear systems theory, and the theory of linear differential equations are also included.
Linear Algebra and its Applications, 1992
It is known that a semiproper system has a closed-form fundamental solution matrix X,(t) = expj'A... more It is known that a semiproper system has a closed-form fundamental solution matrix X,(t) = expj'A(r)dr, where the matrix exponential is defined by the power series exp(.) = XT= i(.)k/ k!. Therefore the problem of solving a semiproper system amounts to that of finding a finite-form expression for the matrix exponential. Based on some recent results obtained by the authors for decomposing semiproper matrix functions, a systematic approach is developed for finding a finite-form analytical solution for the entire family of semiproper systems. This solution is then used to derive a number of important and practical stability criteria for semiproper systems. Applications of the new results are also discussed.
Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contr... more Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contractive if for every x, y in K, u ∈ T (x), v ∈ T (y) and all r > 0, x−y ≤ (1+r)(x−y)−r(u−v). Denote by G(z, w) the set {u ∈ K : u−w ≤ u−z }. Suppose every bounded closed and convex subset of X has the fixed point property with respect to nonexpansive selfmappings. Now if T is a Lipschitzian and pseudo-contractive mapping from K into the family of closed and bounded subsets of K so that the set G(z, w) is bounded for some z ∈ K and some w ∈ T (z), then T has a fixed point in K.
Two commonly used boundary conditions which imply existence of fixed points for local strong pseu... more Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions. Throughout this paper we suppose A' is a Banach space and D an open subset of X. We use B(x; r) to denote the closed ball centered ai x G X with radius r > 0 and D and 9Z> to denote, respectively, the closure and boundary of D. An operator T: D -»• X is said to be a local k-pseudocontraction (k > 0) (see (6)) if each point x G D has a neighborhood N for which (X - *)||u - o|| k. (1) For k < 1 (k = 1) such mappings are said to be local strong pseudocontractions (resp., local pseudocontractions). It is easy to verify that the condition (1) is implied by the assumption that T is a local contraction with (uniform) Lipschitz constant k < 1 (|| T(u) - T(v)\\ < k\\u - v\\, u,v G N), a...
Journal of Mathematical Analysis and Applications, 2016
This paper is a contribution to the theory of monotone operators in topological vector spaces. On... more This paper is a contribution to the theory of monotone operators in topological vector spaces. On the one hand, we provide new results concerning topological and geometric properties of monotone operators satisfying mild continuity assumptions. In particular, we give fairly general conditions on the operator to become single-valued, to be closed and maximal. A fundamental tool is a generalization of the well-known Minty's Lemma that is interesting in its own right and, surprisingly, remains true for general topological vector spaces. As a consequence of Minty's, we obtain an extension of a rather remarkable theorem of Kato for multi-valued mappings defined on general locally convex spaces. 1991 Mathematics Subject Classification. 47H05.
Proceedings of the American Mathematical Society
Let X be a (real) Banach space, let D be an open subset of X, and let B(X) denote the collection ... more Let X be a (real) Banach space, let D be an open subset of X, and let B(X) denote the collection of all nonempty bounded and closed subsets of X. Suppose T is continuous from D̄ into B(X) with respect to the Hausdorff metric and strongly pseudo-contractive, while g is compact from D̄ into X. Then T+g has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of D.
Journal of Applied Mathematics, 2015
Proceedings of the American Mathematical Society, 2005
This paper continues a discussion that arose twenty years ago, concerning the perturbation of an ... more This paper continues a discussion that arose twenty years ago, concerning the perturbation of an m m -accretive operator by a compact mapping in Banach spaces. Indeed, if A A is m m -accretive and g g is compact, then the boundary condition t x ∉ A ( x ) g ( x ) tx \notin A(x)g(x) for x ∈ ∂ G ∩ D ( A ) x \in \partial G \cap D(A) and t > 0 t>0 implies that 0 0 is in the closure of the range of A + g A+g . Perhaps the most interesting aspect of this result is the proof itself, which does not appeal to the classical degree theory argument used for this type of problem.
Let X be a Banach space and D a bounded closed subset of X with 0 6 intCD). A mapping T:D—> X ... more Let X be a Banach space and D a bounded closed subset of X with 0 6 intCD). A mapping T:D—> X such that (&-1) ilx-y IU |i(A,I-T)(x)-(AJ-T)(y)ll for all xfy eDf r>0 f is called pseudo-contractive, while T is said to be nonexpansive if l|T(x)-T(y) I) * il x-y )) f xfy €D. It is well-known that if T is nonexpansive, then the Leray-Schauder condition: T(x)^#x for x e 3D, X > lf is sufficient to guarantee that inf 4iix-T(x)li :x€D} » 0. This result is extended here te the wider class of continuous pseudo-contractive mappings under the weaker Leray-Schauder condition: T(x) s ^ x for x e 3 D, %> 1 ==» T(y) a <u,y for some y 6 D and /cte Clf X ). Several related results are also obtained.
ABSTRACT. Let X be a Banach space, D an open bounded subset of X and T: D-•X an accretive mapping... more ABSTRACT. Let X be a Banach space, D an open bounded subset of X and T: D-•X an accretive mapping (i.e., (X- 1)11u- vii < • II(X- l)(u- v) + T(u)- T(v)ll for all u,v G D and X> 1). For the class of spaces whose dual is uniformly convex, it is shown here that if T is demicontinuous and accretive and if there exist z G D and e> 0 for which IIT(z)11 < • IIT(x)11- e for all x • OD, then the open ball B(0;r) is contained in the range of T for r = IIT(z)ll + •. Several related results are also included. The purpose of this paper is to study the solvability of nonlinear functional equations of the form T(x) = z where T is a demicontinuous accretive mapping and z is contained in a ball centered at the origin. It is known that for certain applications the continuity assumption becomes a rather strong condition. In view of this, our results are obtained for demicontinuous mappings. In the present note we shall study accretive operators which are, in fact, closely related with the ...
This paper is a contribution to the theory of monotone operators in topological vector spaces. On... more This paper is a contribution to the theory of monotone operators in topological vector spaces. On the one hand, we provide new results concerning topological and geometric properties of monotone operators satisfying mild continuity assumptions. In particular, we give fairly general conditions on the operator to become single-valued, to be closed and maximal. A fundamental tool is a generalization of the well-known Minty’s Lemma that is interesting in its own right and, surprisingly, remains true for general topological vector spaces. As a consequence of Minty’s, we obtain an extension of a rather remarkable theorem of Kato for multi-valued mappings defined on general locally convex spaces.
Nonlinear Analysis Theory Methods Applications, Jul 1, 1992
The Journal of Physical Chemistry, 1985
Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence... more Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence of Random Fixed Point Theorems. As a consequence of this, we have been able to obtain various generalizations for pseudo-contractive mappings with rather simple proofs. In addition, while we were deriving these extensions for random operators, some deterministic results arose, which also appear to be new.
Nonlinear Anal Theor Meth App, 2006
Proc Amer Math Soc, 1981
Two commonly used boundary conditions which imply existence of fixed points for local strong pseu... more Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions. Throughout this paper we suppose A' is a Banach space and D an open subset of X. We use B(x; r) to denote the closed ball centered ai x G X with radius r > 0 and D and 9Z> to denote, respectively, the closure and boundary of D. An operator T: D-»• X is said to be a local k-pseudocontraction (k > 0) (see [6]) if each point x G D has a neighborhood N for which (X-*)||u-o|| < \\(XI-T)(u)-(XI-T)(v)\\, u,vGN,X>k. (1) For k < 1 (k = 1) such mappings are said to be local strong pseudocontractions (resp., local pseudocontractions). It is easy to verify that the condition (1) is implied by the assumption that T is a local contraction with (uniform) Lipschitz constant k < 1 (|| T(u)-T(v)\\ < k\\u-v\\, u,v G N), and in fact all of our results appear to be new even for this more restricted class of mappings. Our main objective in this paper is to clarify the relationship between two conditions which have been prominently used in the recent development of fixed-point theory for the local and global strong pseudocontractions. These conditions are the standard Leray-Schauder condition (see, e.g., [2], [6]) which asserts that the mapping T: D-» X satisfies for some z G D: T(x)-z ^ X(x-z) for aU x G 9Z> and X > 1, and the assumption (used in [1], [3], [4]) that the mapping IT not assume its infimum on 9Z). We show in fact, that for local strong pseudocontractions these conditions are in some sense equivalent. In the process we are able to sharpen Theorem 1 of [2] by showing that boundedness of the domain D is not an essential assumption. This fact is a consequence of the imphcation (iii) => (ii) in Theorem 1 below. We also obtain (Theorem 2) a localization of a result of [3].
Proceedings of the American Mathematical Society, Nov 1, 1995
Springer eBooks, 2001
Boundary and inwardness conditions have been particularly useful in extending fixed point theory ... more Boundary and inwardness conditions have been particularly useful in extending fixed point theory for nonexpansive mappings to broader classes of mappings, particularly to mappings satisfying local contractive and pseudocontractive assumptions. At the same time these conditions often enable one to relax the assumption that the mapping takes values in its own domain.
Commentationes Mathematicae Universitatis Carolinae, 1986
Let D be an open subset of a Banach space X, and let B(X) denote the family of all nonempty, boun... more Let D be an open subset of a Banach space X, and let B(X) denote the family of all nonempty, bounded and closed subsets of X. Suppose T:D-*-B(X) is a continuous (with respect to the Hausdorff metric) and strongly accretive mapping. It is shown that if for some zeD:t(x-z)^T(x) for x in the boundary of D and t<0, it is sufficient to guarantee that T has a zero in 0. Several implications of this result are considered, particularly on a localized version of it.
Linear Algebra and its Applications, 1990
Let Z s R and D c C, where R and 4: are the fields of real and compiex numbers, respectively. Let... more Let Z s R and D c C, where R and 4: are the fields of real and compiex numbers, respectively. Let C" x" be the space of square matrices of order tz over C. A matrix-valued function F: Z-) 62" xn is said to be proper on I if F(t) =f(t, A), where A E C'lx" and f:ZXD+C is a scalar function, and F is said to be semiproper on Z if F(t)F(t) = F(T)F(~) for all t, T E 1. The main results presented here are: (1) a characterization together with some potentially useful results for proper matrix functions; (2) a characterization of semiproper matrix functions in terms of proper ones; (3) a new and syskmatic procedure for decomposing a semiproper matrix function into a finite sum of mutually commutative proper ones. Some important applications of these new results in control engineering, linear systems theory, and the theory of linear differential equations are also included.
Linear Algebra and its Applications, 1992
It is known that a semiproper system has a closed-form fundamental solution matrix X,(t) = expj'A... more It is known that a semiproper system has a closed-form fundamental solution matrix X,(t) = expj'A(r)dr, where the matrix exponential is defined by the power series exp(.) = XT= i(.)k/ k!. Therefore the problem of solving a semiproper system amounts to that of finding a finite-form expression for the matrix exponential. Based on some recent results obtained by the authors for decomposing semiproper matrix functions, a systematic approach is developed for finding a finite-form analytical solution for the entire family of semiproper systems. This solution is then used to derive a number of important and practical stability criteria for semiproper systems. Applications of the new results are also discussed.
Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contr... more Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contractive if for every x, y in K, u ∈ T (x), v ∈ T (y) and all r > 0, x−y ≤ (1+r)(x−y)−r(u−v). Denote by G(z, w) the set {u ∈ K : u−w ≤ u−z }. Suppose every bounded closed and convex subset of X has the fixed point property with respect to nonexpansive selfmappings. Now if T is a Lipschitzian and pseudo-contractive mapping from K into the family of closed and bounded subsets of K so that the set G(z, w) is bounded for some z ∈ K and some w ∈ T (z), then T has a fixed point in K.
Two commonly used boundary conditions which imply existence of fixed points for local strong pseu... more Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions. Throughout this paper we suppose A' is a Banach space and D an open subset of X. We use B(x; r) to denote the closed ball centered ai x G X with radius r > 0 and D and 9Z> to denote, respectively, the closure and boundary of D. An operator T: D -»• X is said to be a local k-pseudocontraction (k > 0) (see (6)) if each point x G D has a neighborhood N for which (X - *)||u - o|| k. (1) For k < 1 (k = 1) such mappings are said to be local strong pseudocontractions (resp., local pseudocontractions). It is easy to verify that the condition (1) is implied by the assumption that T is a local contraction with (uniform) Lipschitz constant k < 1 (|| T(u) - T(v)\\ < k\\u - v\\, u,v G N), a...
Journal of Mathematical Analysis and Applications, 2016
This paper is a contribution to the theory of monotone operators in topological vector spaces. On... more This paper is a contribution to the theory of monotone operators in topological vector spaces. On the one hand, we provide new results concerning topological and geometric properties of monotone operators satisfying mild continuity assumptions. In particular, we give fairly general conditions on the operator to become single-valued, to be closed and maximal. A fundamental tool is a generalization of the well-known Minty's Lemma that is interesting in its own right and, surprisingly, remains true for general topological vector spaces. As a consequence of Minty's, we obtain an extension of a rather remarkable theorem of Kato for multi-valued mappings defined on general locally convex spaces. 1991 Mathematics Subject Classification. 47H05.
Proceedings of the American Mathematical Society
Let X be a (real) Banach space, let D be an open subset of X, and let B(X) denote the collection ... more Let X be a (real) Banach space, let D be an open subset of X, and let B(X) denote the collection of all nonempty bounded and closed subsets of X. Suppose T is continuous from D̄ into B(X) with respect to the Hausdorff metric and strongly pseudo-contractive, while g is compact from D̄ into X. Then T+g has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of D.
Journal of Applied Mathematics, 2015
Proceedings of the American Mathematical Society, 2005
This paper continues a discussion that arose twenty years ago, concerning the perturbation of an ... more This paper continues a discussion that arose twenty years ago, concerning the perturbation of an m m -accretive operator by a compact mapping in Banach spaces. Indeed, if A A is m m -accretive and g g is compact, then the boundary condition t x ∉ A ( x ) g ( x ) tx \notin A(x)g(x) for x ∈ ∂ G ∩ D ( A ) x \in \partial G \cap D(A) and t > 0 t>0 implies that 0 0 is in the closure of the range of A + g A+g . Perhaps the most interesting aspect of this result is the proof itself, which does not appeal to the classical degree theory argument used for this type of problem.
Let X be a Banach space and D a bounded closed subset of X with 0 6 intCD). A mapping T:D—> X ... more Let X be a Banach space and D a bounded closed subset of X with 0 6 intCD). A mapping T:D—> X such that (&-1) ilx-y IU |i(A,I-T)(x)-(AJ-T)(y)ll for all xfy eDf r>0 f is called pseudo-contractive, while T is said to be nonexpansive if l|T(x)-T(y) I) * il x-y )) f xfy €D. It is well-known that if T is nonexpansive, then the Leray-Schauder condition: T(x)^#x for x e 3D, X > lf is sufficient to guarantee that inf 4iix-T(x)li :x€D} » 0. This result is extended here te the wider class of continuous pseudo-contractive mappings under the weaker Leray-Schauder condition: T(x) s ^ x for x e 3 D, %> 1 ==» T(y) a <u,y for some y 6 D and /cte Clf X ). Several related results are also obtained.
ABSTRACT. Let X be a Banach space, D an open bounded subset of X and T: D-•X an accretive mapping... more ABSTRACT. Let X be a Banach space, D an open bounded subset of X and T: D-•X an accretive mapping (i.e., (X- 1)11u- vii < • II(X- l)(u- v) + T(u)- T(v)ll for all u,v G D and X> 1). For the class of spaces whose dual is uniformly convex, it is shown here that if T is demicontinuous and accretive and if there exist z G D and e> 0 for which IIT(z)11 < • IIT(x)11- e for all x • OD, then the open ball B(0;r) is contained in the range of T for r = IIT(z)ll + •. Several related results are also included. The purpose of this paper is to study the solvability of nonlinear functional equations of the form T(x) = z where T is a demicontinuous accretive mapping and z is contained in a ball centered at the origin. It is known that for certain applications the continuity assumption becomes a rather strong condition. In view of this, our results are obtained for demicontinuous mappings. In the present note we shall study accretive operators which are, in fact, closely related with the ...
This paper is a contribution to the theory of monotone operators in topological vector spaces. On... more This paper is a contribution to the theory of monotone operators in topological vector spaces. On the one hand, we provide new results concerning topological and geometric properties of monotone operators satisfying mild continuity assumptions. In particular, we give fairly general conditions on the operator to become single-valued, to be closed and maximal. A fundamental tool is a generalization of the well-known Minty’s Lemma that is interesting in its own right and, surprisingly, remains true for general topological vector spaces. As a consequence of Minty’s, we obtain an extension of a rather remarkable theorem of Kato for multi-valued mappings defined on general locally convex spaces.
Nonlinear Analysis Theory Methods Applications, Jul 1, 1992
The Journal of Physical Chemistry, 1985
Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence... more Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence of Random Fixed Point Theorems. As a consequence of this, we have been able to obtain various generalizations for pseudo-contractive mappings with rather simple proofs. In addition, while we were deriving these extensions for random operators, some deterministic results arose, which also appear to be new.
Nonlinear Anal Theor Meth App, 2006
Proc Amer Math Soc, 1981
Two commonly used boundary conditions which imply existence of fixed points for local strong pseu... more Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions. Throughout this paper we suppose A' is a Banach space and D an open subset of X. We use B(x; r) to denote the closed ball centered ai x G X with radius r > 0 and D and 9Z> to denote, respectively, the closure and boundary of D. An operator T: D-»• X is said to be a local k-pseudocontraction (k > 0) (see [6]) if each point x G D has a neighborhood N for which (X-*)||u-o|| < \\(XI-T)(u)-(XI-T)(v)\\, u,vGN,X>k. (1) For k < 1 (k = 1) such mappings are said to be local strong pseudocontractions (resp., local pseudocontractions). It is easy to verify that the condition (1) is implied by the assumption that T is a local contraction with (uniform) Lipschitz constant k < 1 (|| T(u)-T(v)\\ < k\\u-v\\, u,v G N), and in fact all of our results appear to be new even for this more restricted class of mappings. Our main objective in this paper is to clarify the relationship between two conditions which have been prominently used in the recent development of fixed-point theory for the local and global strong pseudocontractions. These conditions are the standard Leray-Schauder condition (see, e.g., [2], [6]) which asserts that the mapping T: D-» X satisfies for some z G D: T(x)-z ^ X(x-z) for aU x G 9Z> and X > 1, and the assumption (used in [1], [3], [4]) that the mapping IT not assume its infimum on 9Z). We show in fact, that for local strong pseudocontractions these conditions are in some sense equivalent. In the process we are able to sharpen Theorem 1 of [2] by showing that boundedness of the domain D is not an essential assumption. This fact is a consequence of the imphcation (iii) => (ii) in Theorem 1 below. We also obtain (Theorem 2) a localization of a result of [3].