Abstract. In this paper we establish a fixed point result of Krasnoselskii type for the sum A+B, ... more Abstract. In this paper we establish a fixed point result of Krasnoselskii type for the sum A+B, where A and B are continuous maps acting on locally convex spaces. Our results extend previous ones. We apply such results to obtain strong solutions for some quasi-linear elliptic equations with lack of compactness. We also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a Banach space. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in L p (E). 1.
We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly nu... more We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unit sphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.
In the present paper we establish a fixed point result of Krasnoselskii type for the sum A+B, whe... more In the present paper we establish a fixed point result of Krasnoselskii type for the sum A+B, where A and B are sequentially weakly continuous, and B is a nonexpansive mapping. As an application, we study the existence of solutions for an nonlinear integral equation in Banach spaces. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used as the main tool to provide an existence theory for integral equations in L(E).
In this paper, we study a class of Banach spaces, called \phi-spaces. In a natural way, we associ... more In this paper, we study a class of Banach spaces, called \phi-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.
Let X be a Banach space with separable dual. It is proved that for every e ∈ ( 0 , 1 ) , X embeds... more Let X be a Banach space with separable dual. It is proved that for every e ∈ ( 0 , 1 ) , X embeds isometrically into a Banach space W with a shrinking basis ( w n ) which is ( 1 + e ) -monotone. Moreover, if X has further an FDD ( E n ) whose strong bimonotonicity projection constant is not larger than D , then ( w n ) has strong bimonotonicity projection constant not exceeding D ( 1 + e ) . Further, if ( E n ) is C -unconditional then ( w n ) is C ( 1 + e ) -unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking D -unconditional basis with D 6 − 1 , has the weak fixed point property.
Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hau... more Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hausdorff vector topology in E. We establish an approximate fixed point result for sequentially continuous maps f: (C,σ) → (C,τ). As application, we obtain the weak-approximate fixed point property for demicontinuous self-mapping weakly compact convex sets in general Banach spaces and use this to prove new results in asymptotic fixed point theory. These results are also applied to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces.
In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces.... more In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fr\'echet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe{\l}czy\'nski on Banach spaces containing $L_1(\mu)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence...
Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
We establish the existence of solutions for a differential equation in Banach spaces. Our analysi... more We establish the existence of solutions for a differential equation in Banach spaces. Our analysis relies on two approaches, one using the notion of φ-space together with a fixed point theorem for weakly sequentially continuous maps which are φ-condensing and the other using the Banach contraction principle.
We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional su... more We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symme... more We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.
ABSTRACT In this paper, we obtain new results for the weak-AFPP in abstract spaces by exploiting ... more ABSTRACT In this paper, we obtain new results for the weak-AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong-AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak-AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ1-sequences in terms of strongly-equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak-AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non-locally convex) Hausdorff vector topology which is complete, non-metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the -AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non-locally convex F-space with an absolute basis, then the weak-AFPP is equivalent to the fact that every bounded convex subset of X is compact.
In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\s... more In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with $C\sp{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.
ABSTRACT In every infinite-dimensional vector space we build a complete, non-locally convex linea... more ABSTRACT In every infinite-dimensional vector space we build a complete, non-locally convex linear Hausdorff topology under the prescribed condition that it shall be compatible with a bounded algebraic structure for the space, that is, it admits a bounded Hamel basis whose coefficient linear functionals are continuous. As a byproduct, we obtain the existence of non-locally convex topologies having the approximate and fixed point property as prescribed conditions. The obstruction of bounded Hamel bases by complete vector topologies on linear space of countable dimension is also discussed.
ABSTRACT For a finite and positive measure space (Omega, Sigma, mu) and any weakly compact convex... more ABSTRACT For a finite and positive measure space (Omega, Sigma, mu) and any weakly compact convex subset of L-infinity(Omega, Sigma, mu) a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space C(Omega). Also, for a bounded domain with C-1 boundary Omega subset of R-N, we construct a family of Banach spaces h(p) (Omega), p > 1, which are continuously embedded in L-infinity (Omega) and possess the fixed point property. (c) 2006 Elsevier Ltd. All rights reserved.
Abstract. In this paper we establish a fixed point result of Krasnoselskii type for the sum A+B, ... more Abstract. In this paper we establish a fixed point result of Krasnoselskii type for the sum A+B, where A and B are continuous maps acting on locally convex spaces. Our results extend previous ones. We apply such results to obtain strong solutions for some quasi-linear elliptic equations with lack of compactness. We also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a Banach space. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in L p (E). 1.
We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly nu... more We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unit sphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.
In the present paper we establish a fixed point result of Krasnoselskii type for the sum A+B, whe... more In the present paper we establish a fixed point result of Krasnoselskii type for the sum A+B, where A and B are sequentially weakly continuous, and B is a nonexpansive mapping. As an application, we study the existence of solutions for an nonlinear integral equation in Banach spaces. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used as the main tool to provide an existence theory for integral equations in L(E).
In this paper, we study a class of Banach spaces, called \phi-spaces. In a natural way, we associ... more In this paper, we study a class of Banach spaces, called \phi-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.
Let X be a Banach space with separable dual. It is proved that for every e ∈ ( 0 , 1 ) , X embeds... more Let X be a Banach space with separable dual. It is proved that for every e ∈ ( 0 , 1 ) , X embeds isometrically into a Banach space W with a shrinking basis ( w n ) which is ( 1 + e ) -monotone. Moreover, if X has further an FDD ( E n ) whose strong bimonotonicity projection constant is not larger than D , then ( w n ) has strong bimonotonicity projection constant not exceeding D ( 1 + e ) . Further, if ( E n ) is C -unconditional then ( w n ) is C ( 1 + e ) -unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking D -unconditional basis with D 6 − 1 , has the weak fixed point property.
Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hau... more Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hausdorff vector topology in E. We establish an approximate fixed point result for sequentially continuous maps f: (C,σ) → (C,τ). As application, we obtain the weak-approximate fixed point property for demicontinuous self-mapping weakly compact convex sets in general Banach spaces and use this to prove new results in asymptotic fixed point theory. These results are also applied to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces.
In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces.... more In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fr\'echet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe{\l}czy\'nski on Banach spaces containing $L_1(\mu)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence...
Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
We establish the existence of solutions for a differential equation in Banach spaces. Our analysi... more We establish the existence of solutions for a differential equation in Banach spaces. Our analysis relies on two approaches, one using the notion of φ-space together with a fixed point theorem for weakly sequentially continuous maps which are φ-condensing and the other using the Banach contraction principle.
We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional su... more We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symme... more We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.
ABSTRACT In this paper, we obtain new results for the weak-AFPP in abstract spaces by exploiting ... more ABSTRACT In this paper, we obtain new results for the weak-AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong-AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak-AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ1-sequences in terms of strongly-equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak-AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non-locally convex) Hausdorff vector topology which is complete, non-metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the -AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non-locally convex F-space with an absolute basis, then the weak-AFPP is equivalent to the fact that every bounded convex subset of X is compact.
In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\s... more In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with $C\sp{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.
ABSTRACT In every infinite-dimensional vector space we build a complete, non-locally convex linea... more ABSTRACT In every infinite-dimensional vector space we build a complete, non-locally convex linear Hausdorff topology under the prescribed condition that it shall be compatible with a bounded algebraic structure for the space, that is, it admits a bounded Hamel basis whose coefficient linear functionals are continuous. As a byproduct, we obtain the existence of non-locally convex topologies having the approximate and fixed point property as prescribed conditions. The obstruction of bounded Hamel bases by complete vector topologies on linear space of countable dimension is also discussed.
ABSTRACT For a finite and positive measure space (Omega, Sigma, mu) and any weakly compact convex... more ABSTRACT For a finite and positive measure space (Omega, Sigma, mu) and any weakly compact convex subset of L-infinity(Omega, Sigma, mu) a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space C(Omega). Also, for a bounded domain with C-1 boundary Omega subset of R-N, we construct a family of Banach spaces h(p) (Omega), p > 1, which are continuously embedded in L-infinity (Omega) and possess the fixed point property. (c) 2006 Elsevier Ltd. All rights reserved.
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