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E-BOOK
Title Solution sets for differential equations and inclusions / Smaïl Djebali, Lech Gorniewicz, Abdelghani Ouahab.
Imprint Berlin : De Gruyter, [2013]
©2013

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Description 1 online resource (xix, 453 pages)
Series De Gruyter series in nonlinear analysis and applications, 0941-813X ; 18
De Gruyter series in nonlinear analysis and applications ; 18. 0941-813X
Bibliog. Includes bibliographical references and index.
Note Available only to authorized UTEP users.
English.
Print version record.
Subject Differential equations -- Numerical solutions.
Differential inclusions.
Fixed point theory.
Boundary value problems.
Contents Topological structure of fixed point sets -- Existence theory for differential equations and inclusions -- Solution sets for differential equations and inclusions -- Impulsive differential inclusions : existence and solution sets -- Preliminary notions of topology and homology -- Background in multi-valued analysis.
Machine-generated contents note: 1. Topological structure of fixed point sets -- 1.1. Case of single-valued mappings -- 1.1.1. Fundamental fixed point theorems -- 1.1.1.1. Banach's fixed point theorem -- 1.1.1.2. Brouwer's fixed point theorem -- 1.1.1.3. Schauder's fixed point theorem -- 1.1.2. Approximation theorems -- 1.1.3. Browder-Gupta theorems -- 1.1.4. Acyclicity of the solution sets of operator equations -- 1.1.5. Non-expansive maps -- 1.1.5.1. Existence theory -- 1.1.5.2. Solution sets -- 1.2. The case of multi-valued mappings -- 1.2.1. Approximation of multi-valued maps -- 1.2.2. Fixed point theorems -- 1.2.3. Multi-valued contractions -- 1.2.4. Fixed point sets of multi-valued contractions -- 1.2.5. Fixed point sets of multi-valued nonexpansive maps -- 1.2.6. Fixed point sets of multi-valued condensing maps -- 1.2.6.1. Measure of non-compactness -- 1.2.6.2. Condensing maps -- 1.3. Admissible maps -- 1.3.1. Generalities -- 1.3.2. Fixed point theorems for admissible multi-valued maps -- 1.3.3. The general Brouwer fixed point theorem -- 1.3.4. Browder-Gupta type results for admissible mappings -- 1.3.5. Topological dimensions of solution sets -- 1.4. Topological structure of fixed point sets of inverse limit maps -- 1.4.1. Definition -- 1.4.2. Basic properties -- 1.4.3. Multi-maps of inverse systems -- 2. Existence theory for differential equations and inclusions -- 2.1. Fundamental theorems -- 2.1.1. Existence and uniqueness results -- 2.1.2. Picard-Lindelof theorem -- 2.1.2.1. Maximal solutions -- 2.1.3. Peano and Caratheodory theorems -- 2.1.3.1. Peano theorem -- 2.2. The extendability problem -- 2.2.1. Global existence theorems -- 2.2.2. Existence results on non-compact intervals -- 2.2.2.1. The Lipschitz case -- 2.2.2.2. The Lipschitz-Nagumo case -- 2.2.2.3. The Nagumo case -- 2.2.3. A boundary value problem on the half-line -- 2.3. The case of differential inclusions -- 2.3.1. Initial value problems -- 2.3.1.1. A Nagumo type non-linearity -- 2.3.1.2. A Lipschitz non-convex non-linearity -- 2.3.2. Boundary value problems -- 2.3.2.1. The convex case -- 2.3.2.2. The non-convex case -- 3. Solution sets for differential equations and inclusions -- 3.1. General results -- 3.1.1. Kneser-Hukuhara theorem -- 3.1.2. Problems on bounded intervals -- 3.1.3. Problems on unbounded intervals -- 3.1.4. Second-order differential equations -- 3.1.5. Abstract Volterra equations -- 3.1.6. Aronszajn-type results for differential inclusions -- 3.2. Second-order differential inclusions -- 3.2.1. The convex case -- 3.2.2. The non-convex case -- 3.2.3. Solution sets -- 3.3. Higher-order differential inclusions -- 3.4. Neutral differential inclusions -- 3.4.1. The convex case -- 3.4.2. The non-convex case -- 3.4.3. Solutions sets -- 3.5. Non-local problems -- 3.5.1. Main results -- 3.5.2. A viability problem -- 3.6. Hyperbolic differential inclusions -- 3.6.1. Existence results -- 3.6.1.1. The convex case -- 3.6.1.2. The non-convex case -- 3.6.2. Solution sets -- 4. Impulsive differential inclusions: existence and solution sets -- 4.1. Motivation -- 4.1.1. Ecological model with impulsive control strategy -- 4.1.2. Leslie predator-prey system -- 4.1.3. Pulse vaccination model -- 4.2. Semi-linear impulsive differential inclusions -- 4.2.1. Existence results -- 4.2.1.1. The convex case -- 4.2.1.2. The non-convex case -- 4.2.2. Structure of solution sets -- 4.3. A periodic problem -- 4.3.1. Existence results: 1 ε ρϞβ 4.3.2. The convex case: a direct approach -- 4.3.3. The convex case: an MNC approach -- 4.3.4. The non-convex case -- 4.3.5. The parameter-dependant case -- 4.3.5.1. The convex case -- 4.3.5.2. The non-convex case -- 4.3.6. Filippov's Theorem -- 4.3.7. Existence of solutions: 1 ρϞβ 4.3.7.1. A non-linear alternative -- 4.3.7.2. A Poincaré translation operator -- 4.3.7.3. The MNC approach -- 4.4. Impulsive functional differential inclusions -- 4.4.1. Introduction -- 4.4.2. Existence results -- 4.4.3. Structure of the solution set -- 4.5. Impulsive differential inclusions on the half-line -- 4.5.1. Existence results and compactness of solution sets -- 4.5.1.1. The convex u.s.c. case -- 4.5.1.2. The non-convex Lipschitz case -- 4.5.1.3. The non-convex l.s.c. case -- 4.5.2. Topological structure via the projective limit -- 4.5.2.1. The non-convex case -- 4.5.2.2. The convex case -- 4.5.2.3. The terminal problem -- 4.5.3. Using solution sets to prove existence results -- 5. Preliminary notions of topology and homology -- 5.1. Retracts, extension and embedding properties -- 5.2. Absolute retracts -- 5.3. Homotopical properties of spaces -- 5.4. Cech homology (cohomology) functor -- 5.5. Maps of spaces of finite type -- 5.6. Cech homology functor with compact carriers -- 5.7. Acyclic sets and Vietoris maps -- 5.8. Homology of open subsets of Euclidean spaces -- 5.9. Lefschetz number -- 5.10. The coincidence problem -- 6. Background in multi-valued analysis -- 6.1. Continuity of multi-valued mappings -- 6.1.1. Basic notions -- 6.1.2. Upper semi-continuity -- 6.1.2.1. Generalities -- 6.1.2.2. ε δ ρπcκαννζλϛπ 6.1.2.3. U.s.c. maps and closed graphs -- 6.1.3. Lower semi-continuity -- 6.1.3.1. Generalities -- 6.1.3.2. ε δ ιπcκαννζλϛπ 6.1.4. Hausdorff continuity -- 6.2. The selection problem -- 6.2.1. Michael's selection theorem -- 6.2.2. Michael's family of subsets -- 6.2.3. σπδιδcϟζμλαβιδ καννζλϛπ 6.2.4. The Kuratowski-Ryll-Nardzewski selection theorem -- 6.2.5. Aumann and Filippov theorems -- 6.2.6. Hausdorff measurable multi-valued maps -- 6.2.7. Product-measurability and the Scorza-Dragoni property -- 6.3. Decomposable sets -- 6.3.1. The Bressan-Colombo-Fryszkowski selection theorem -- 6.3.2. Decomposability in L1(T, E) -- 6.3.3. Integration of multi-valued maps -- 6.3.4. Nemytskii operators -- Appendix -- A.1. Axioms of the Cech homology theory -- A.2. The Bochner integral -- A.3. Absolutely continuous functions -- A.4. Compactness criteria in C([a, b], E), Cb([0, E), and PC([a, b], E) -- A.5. Weak-compactness in L1 -- A.6. Proper maps and vector fields -- A.7. Fundamental theorems in functional analysis -- A.8. Co-Semigroups.
Summary "This monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. Several examples of applications relating to initial and boundary value problems are discussed in detail."--Publisher's website.
Other Author Górniewicz, Lech, author.
Ouahab, Abdelghani, author.
Other Title Print version: Djebali, Smaïl. Solution sets for differential equations and inclusions. Berlin : De Gruyter, ©2013 9783110293449