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Fractional Differential Equations And Inclusions: Classical And Advanced Topics [Kõva köide]

(Universidade De Santiago De Compostela, Spain), (Tahar Moulay University Of Saida, Algeria), (Macau Univ Of Science And ), (Djillali Liabes University Of Sidi Bel-abbes, Algeria), (Djillali Liabes University Of Sidi Bel-abbes, Algeria)
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Fractional Differential Equations And Inclusions: Classical And Advanced TopicsSuurem pilt
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This Monograph Is Devoted To The Existence And Stability (Ulam – Hyers – Rassias Stability And Asymptotic Stability) Of Solutions For Various Classes Of Functional Differential Equations Or Inclusions Involving The Hadamard Or Hilfer Fractional Derivative. Some Equations Present Delay Which May Be Finite, Infinite, Or State-Dependent. Others Are Subject To Impulsive Effect Which May Be Fixed Or Non-Instantaneous. Readers Will Find The Book Self-Contained And Unified In Presentation. It Provides The Necessary Background Material Required To Go Further Into The Subject And Explores The Rich Research Literature In Detail. Each Chapter Concludes With A Section Devoted To Notes And Bibliographical Remarks And All Abstract Results Are Illustrated By Examples. The Tools Used Include Many Classical And Modern Nonlinear Analysis Methods Such As Fixed-Point Theorems, As Well As Some Notions Of Ulam Stability, Attractivity And The Measure Of Non-Compactness As Well As The Measure Of Weak Noncompactness. It Is Useful For Researchers And Graduate Students For Research, Seminars, And Advanced Graduate Courses, In Pure And Applied Mathematics, Physics, Mechanics, Engineering, Biology, And All Other Applied Sciences.



"This monograph is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stability) of solutions for various classes of functional differential equations or inclusions involving the Hadamard or Hilfer fractional derivative. Some equations present delay which may be finite, infinite, or state-dependent. Others are subject to impulsive effect which may be fixed or non-instantaneous. Readers will find the book self-contained and unified in presentation. It provides the necessary background material required to go further into the subject and explores the rich research literature in detail. Each chapter concludes with a section devoted to notes and bibliographical remarks and all abstract results are illustrated by examples. Thetools used include many classical and modern nonlinear analysis methods such as fixed-point theorems, as well as some notions of Ulam stability, attractivity and the measure of non-compactness as well as the measure of weak noncompactness. It is useful for researchers and graduate students for research, seminars, and advanced graduate courses, in pure and applied mathematics, physics, mechanics, engineering, biology, and all other applied sciences"--
Preface vii
About the Authors ix
Introduction xxi
1 Preliminary Background
1(24)
1.1 Notations and Definitions
1(1)
1.2 Fractional Calculus
2(5)
1.3 Multivalued Analysis
7(3)
1.4 Measure of Noncompactness
10(3)
1.5 Measure of Weak Noncompactness
13(2)
1.6 Some Attractivity Concepts
15(1)
1.7 Some Ulam Stability Concepts
16(1)
1.8 Some Fixed-Point Theorems
17(3)
1.9 Auxiliary Lemmas
20(5)
2 Hadamard and Hilfer Fractional Differential Equations and Inclusions in Banach Spaces
25(28)
2.1 Introduction
25(1)
2.2 Caputo-Hadamard Fractional and Partial Fractional Differential Equations in Banach Spaces
25(12)
2.2.1 Introduction
25(1)
2.2.2 Existence results for Caputo-Hadamard fractional differential equations
26(4)
2.2.3 Caputo-Hadamard partial fractional differential equations
30(6)
2.2.4 Examples
36(1)
2.3 Hilfer and Hilfer-Hadamard Fractional Differential Equations in Banach Spaces
37(8)
2.3.1 Introduction
37(1)
2.3.2 Hilfer fractional differential equations
38(4)
2.3.3 Hilfer-Hadamard fractional differential equations
42(2)
2.3.4 An example
44(1)
2.4 Hilfer and Hilfer-Hadamard Fractional Differential Inclusions in Banach Spaces
45(7)
2.4.1 Introduction
45(1)
2.4.2 Hilfer fractional differential inclusions
45(5)
2.4.3 Hilfer-Hadamard fractional differential inclusions
50(1)
2.4.4 An example
51(1)
2.5 Notes and Remarks
52(1)
3 Attractivity Results for Hilfer Fractional Differential Equations
53(14)
3.1 Introduction
53(1)
3.2 Asymptotic Stability for Implicit Hilfer Fractional Differential Equations
54(5)
3.2.1 Existence of solutions
54(4)
3.2.2 An example
58(1)
3.3 Global Stability for Implicit Hilfer-Hadamard Fractional Differential Equations
59(6)
3.3.1 Introduction and motivations
59(1)
3.3.2 Existence of solutions
59(5)
3.3.3 An example
64(1)
3.4 Notes and Remarks
65(2)
4 Ulam Stability Results for Hilfer Fractional Differential Equations
67(34)
4.1 Introduction
67(1)
4.2 Dynamics and Ulam Stability for Hilfer Fractional Differential Equations
67(6)
4.2.1 Introduction
67(1)
4.2.2 Existence and Ulam stability results
68(5)
4.2.3 Example
73(1)
4.3 Ulam Stability for Hilfer Fractional Differential Inclusions via Picard Operators
73(9)
4.3.1 Introduction
73(1)
4.3.2 Existence and Ulam stability results
74(7)
4.3.3 Example
81(1)
4.4 Ulam Stability for Hilfer-Hadamard Fractional Differential Equations
82(7)
4.4.1 Introduction
82(1)
4.4.2 Existence and Ulam-Hyers-Rassias stability results
83(5)
4.4.3 Example
88(1)
4.5 Ulam Stabilities for Hilfer Fractional Differential Equations in Banach Spaces
89(7)
4.5.1 Introduction
89(1)
4.5.2 Existence and Ulam stability results
90(5)
4.5.3 Example
95(1)
4.6 Ulam-Hyers Stability for Fractional Differential Equations with Maxima via Picard Operators
96(4)
4.6.1 Introduction
96(1)
4.6.2 Uniqueness and stability results
97(2)
4.6.3 Example
99(1)
4.7 Notes and Remarks
100(1)
5 Random Hilfer Fractional Differential Equations and Inclusions
101(32)
5.1 Introduction
101(1)
5.2 Random Hilfer and Hilfer-Hadamard Fractional Differential Equations
102(9)
5.2.1 Introduction
102(1)
5.2.2 Hilfer fractional random differential equations
102(7)
5.2.3 Hilfer-Hadamard fractional random differential equations
109(1)
5.2.4 Example
110(1)
5.3 Random Hilfer and Hilfer-Hadamard Fractional Differential Inclusions
111(14)
5.3.1 Introduction
111(1)
5.3.2 Existence of random solutions
112(12)
5.3.3 Examples
124(1)
5.4 Random Hilfer Fractional Differential Equations in Frechet Spaces
125(7)
5.4.1 Introduction
125(1)
5.4.2 Hilfer fractional random differential equations
126(4)
5.4.3 Hilfer-Hadamard fractional random differential equations
130(1)
5.4.4 Example
131(1)
5.5 Notes and Remarks
132(1)
6 Nonlinear Hadamard-Pettis Fractional Integral Equations
133(48)
6.1 Introduction
133(1)
6.2 Hadamard-Pettis Fractional Integral Equations
133(6)
6.2.1 Introduction
133(1)
6.2.2 Existence of weak solutions
134(4)
6.2.3 Example
138(1)
6.3 Partial Hadamard-Pettis Fractional Integral Inclusions
139(7)
6.3.1 Introduction
139(1)
6.3.2 Existence of weak solutions
139(6)
6.3.3 Example
145(1)
6.4 Fredholm-Type Partial Hadamard-Pettis Fractional Integral Equations
146(12)
6.4.1 Introduction
146(1)
6.4.2 Existence of weak solutions
146(10)
6.4.3 Example
156(2)
6.5 Partial Hadamard-Stieltjes-Pettis Fractional Integral Equations
158(7)
6.5.1 Introduction
158(1)
6.5.2 Existence of weak solutions
158(6)
6.5.3 Example
164(1)
6.6 Partial Random Hadamard-Pettis Fractional Integral Equations
165(7)
6.6.1 Introduction
165(1)
6.6.2 Existence of weak solutions
165(6)
6.6.3 Example
171(1)
6.7 Partial Random Hadamard-Pettis Fractional Integral Equations with Multiple Delay
172(8)
6.7.1 Introduction
172(1)
6.7.2 Existence of weak solutions
173(5)
6.7.3 Example
178(2)
6.8 Notes and Remarks
180(1)
7 Nonlinear Implicit Hadamard-Pettis Fractional Differential Equations
181(40)
7.1 Introduction
181(1)
7.2 Implicit Hadamard-Pettis Fractional Differential Equations
181(6)
7.2.1 Introduction
181(1)
7.2.2 Existence of weak solutions
182(4)
7.2.3 Example
186(1)
7.3 Implicit Hadamard-Pettis Fractional Differential Equations with Delay
187(5)
7.3.1 Introduction
187(1)
7.3.2 Existence of weak solutions
187(4)
7.3.3 Example
191(1)
7.4 Successive Approximations for Implicit Hadamard-Pettis Fractional Differential Equations
192(10)
7.4.1 Introduction
192(1)
7.4.2 Successive approximations and uniqueness results
193(7)
7.4.3 Example
200(2)
7.5 Impulsive Implicit Hadamard-Pettis Fractional Differential Equations
202(10)
7.5.1 Introduction
202(3)
7.5.2 Existence of weak solutions
205(6)
7.5.3 Example
211(1)
7.6 Implicit Hadamard-Pettis Fractional Differential Equations with Not Instantaneous Impulses
212(8)
7.6.1 Introduction
212(1)
7.6.2 Existence of weak solutions
213(6)
7.6.3 Example
219(1)
7.7 Notes and Remarks
220(1)
8 Hilfer-Pettis Fractional Differential Equations and Inclusions
221(30)
8.1 Introduction
221(1)
8.2 Hilfer-Pettis Fractional Differential Equations
221(5)
8.2.1 Introduction
221(1)
8.2.2 Existence of weak solutions
221(4)
8.2.3 Example
225(1)
8.3 Hilfer-Pettis Fractional Differential Inclusions
226(6)
8.3.1 Introduction
226(1)
8.3.2 Existence of weak solutions
226(5)
8.3.3 Example
231(1)
8.4 Hilfer-Hadamard-Pettis Fractional Differential Equations and Inclusions
232(12)
8.4.1 Introduction
232(1)
8.4.2 Existence of weak solutions for Hilfer-Hadamard fractional differential equations
233(4)
8.4.3 Existence of weak solutions for Hilfer-Hadamard fractional differential inclusions
237(5)
8.4.4 Examples
242(2)
8.5 Hilfer-Pettis Fractional Differential Equations with Maxima
244(5)
8.5.1 Introduction
244(1)
8.5.2 Existence of weak solutions
245(3)
8.5.3 Example
248(1)
8.6 Notes and Remarks
249(2)
9 Implicit Hilfer-Pettis Fractional Differential Equations
251(26)
9.1 Introduction
251(1)
9.2 Implicit Hilfer-Pettis Fractional Differential Equations
251(5)
9.2.1 Introduction
251(1)
9.2.2 Existence of weak solutions
251(4)
9.2.3 Example
255(1)
9.3 Implicit Hilfer-Hadamard-Pettis Fractional Differential Equations
256(6)
9.3.1 Introduction
256(1)
9.3.2 Existence of weak solutions
257(4)
9.3.3 Example
261(1)
9.4 Implicit Hilfer-Pettis Fractional Differential Equations with Not Instantaneous Impulses
262(7)
9.4.1 Introduction
262(1)
9.4.2 Existence of weak solutions
263(5)
9.4.3 Example
268(1)
9.5 Implicit Hilfer-Pettis Fractional Differential Equations with Retarded and Advanced Arguments
269(6)
9.5.1 Introduction
269(1)
9.5.2 Existence of weak solutions
270(4)
9.5.3 Example
274(1)
9.6 Notes and Remarks
275(2)
Bibliography 277(22)
Index 299