Existence and asymptotic behaviour of solutions to second-order evolution equations of monotone type

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Existence and asymptotic behaviour of solutions to second-order evolution equations of monotone type El Paso, Nov 1 st, 2014 15 th Joint UTEP/NMSU Workshop on Mathematics, Computer Science, and Computational Sciences Klara Loos

AGENDA 1. The second order differential equation 2. Recent results on existence of solutions 3. Recent results on asymptotic behaviour of solutions 4. Discussion 2

AGENDA 1. The second order differential equation 2. Recent results on existence of solutions 3. Recent results on asymptotic behaviour of solutions 4. Discussion 3

The second-order differential equation second-order positive half-line p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) incomplete evolution problem 4

The second-order differential equation second-order positive half-line p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) initial data (B) (H1) conditions for A (H2) conditions for p, q Monotone type: A maximal monotone operator Homogenous: f(t) = 0 p,q are constants p,q are real functions, 5

The second-order differential equation second-order positive half-line p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) initial data (B) (H1) conditions for A (H2) conditions for p, q A in Hilbert space H is monotone: Ax 1 Ax 2, x 1 x 2 0 x 1, x 2 D A A is maximal monotone: A is maximal monotone, if it is maximal in the set of monotone operators. R I + λa = H, λ > 0 [Brézis, 2010] 6

AGENDA 1. The second order differential equation 2. Recent results on existence of solutions 3. Recent results on asymptotic behaviour of solutions 4. Discussion 7

Recent Results: Existence and uniqueness of a bounded solution p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) (B) Where A is a maximal monotone operator in a real Hilbert space H. p,q are real valued functions defined on [0,). (H1) A: D A H H, A maximal monotone operator (H2) p, q L R +, ess inf p > 0, q + L 1 R +, q + = max{q t, 0} [E1] G. Moroşanu, Existence results for second-order monotone differential inclusions on the positive half-line, J.Math. Appl. 419 (2014) 94-113 8

Recent Results: Existence and uniqueness of a bounded solution p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) (B) Where A is a maximal monotone operator in a real Hilbert space H. p,q are real valued functions defined on [0,). (H1) A: D A H H, A maximal monotone operator (H2) p, q L R +, ess inf p > 0, q + L 1 R +, q + = max{q t, 0} Existence and uniqueness of bounded solution non-homogenous f t 0 Non-constant functions q,p & mild condition p(t) α > 0 L R + space, of essential bounded functions [E1] G. Moroşanu, Existence results for second-order monotone differential inclusions on the positive half-line, J.Math. Appl. 419 (2014) 94-113 9

Existence of a unique bounded solution Case: p 1, q 0, f 0 u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) (B) Where A is a maximal monotone operator in a real Hilbert space H. (H1) A: D A H H, A maximal monotone operator [E2] V. Barbu, Sur un problème aux limites pour une classe d équations differentielles nonlinéaires abstraites du deuxièmes ordre en t, C.R. Accad. Sci. Paris 27 (1972) 459-462 [E3] V. Barbu, A clase of boundary problems for second-order abstract differential equations, J. Fae. Sci. Univ. Tokyo, Sect. I 19 (1972) 295-319 10

Existence of a unique bounded solution Homogenous case: f 0 p u t + q u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) (B) Where A is a m-accretive operator in a real Banach space. (H1) p, q R + are constants [E4] H.Brezis, Équations d évolution du second ordre associées à des opérateurs monotones, Isreal J. Math. 12 (1972) 51-60. [E5] N. Pavel, Boundary value problems on [0, +] for second-order differential equations associated to monotone operators in Hilber spaces, in: Proceedings of the Institute of Mathematics Iasi (1974), Editura Acad. R. S. R., Bucharest, 1976, pp.145-154. [E6] L. Véron, Problèmes d évolution du second ordre associées à des opérateurs monotones, C.R. Acad. Sci. Paris 278 (1974) 1099-1101. [E7] L. Véron, Equations d evolution du second ordre associées à des opérateurs maximaux monotones, Proc. Roy. Soc. Edinburgh Sect. A 75 (2) (1975/1976) 131-147. [E8] E.I. Poffald, S.Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 (2) (1986) 514-543. 11

Recent Results: Existence and uniqueness of a bounded solution p t u t + q t u t Au t + f t for a. a. t R + [0, ) (E) with the condition u 0 = x D(A) (B) Where A is a maximal monotone operator in a real Hilbert space H. p,q are real valued functions defined on [0,). (H1) A: D A H H, A maximal monotone operator (H2) p, q L R +, ess inf p > 0, q + L 1 R +, q + = max{q t, 0} Existence and uniqueness of bounded solution non-homogenous f t 0 Non-constant functions q,p & mild condition [1] G. Moroşanu, Existence results for second-order monotone differential inclusions on the positive halfline, J.Math. Appl. 419 (2014) 94-113 12

Development: Existence and uniqueness of a bounded solution [E1, E2] [E4, E5, E6, E7, E8] [1] 1972 72 74 75/76 82 2014 13

AGENDA 1. The second order differential equation 2. Recent results on existence of solutions 3. Recent results on asymptotic behaviour of solutions 4. Discussion 14

RECENT RESULTS: Asymptotic behaviour Nonlinear second order evolution equation: p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < + (B) Where A is a maximal monotone operator in a real Hilbert space H. [3] B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. 15

RECENT RESULTS: Asymptotic behaviour Nonlinear second order evolution equation: f 0 p, r time dependant p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < + bounded solution (B) Where A is a maximal monotone operator in a real Hilbert space H. [3] B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. 16

RECENT RESULTS: Asymptotic behaviour Nonlinear second order evolution equation: p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < + (B) Where A is a maximal monotone operator in a real Hilbert space H. - Strong convergence for t : u t p A 1 (0) - rate of convergence: u t p = Ο( t e 0 s r τ 2p τ dτ ds) 0 A p H - u t p A 1 0 A 1 0 (not assumed, now a consequence) [3] B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. 17

RECENT RESULTS: Asymptotic behaviour Theorem 2.1. [3] u(t) issolution of (E), (B) 0 e 0 s r τ 2p τ dτ ds < r t 0 conditions I. u t p A 1 0 II. u t p = Ο( t e 0 s r τ 2p τ dτ ds) consequences [3] B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. 18

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Au = 3u 5 3 Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) 19

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) Verify solution u t = 1 t+1 3: u t = 3 t + 1 4 u t = 12 t + 1 5 20

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) Verify solution u t = 1 t+1 3: u t = 3 t + 1 4 u t = 12 t + 1 5 u + 3 t + 1 u = 12 t + 1 5 3 (t + 1) 3 t + 1 4 = 3 (t + 1) 5 = 3u5 3 21

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) Verify condition I: r (t) = 3 t + 1 2 0 22

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) Verify condition II: 0 e s 3 02(τ+1) dτ ds = 0 s e 3 2 ln(τ+1) 0 ds = 0 e 3 2 ln(s+1) ds 23

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) Verify condition II: 0 = e s 3 02(τ+1) dτ ds = 0 0 s + 1 3 2ds = lim β s e 3 2 ln(τ+1) 0 2 ds = β s + 1 1 2 0 0 e 3 2 ln(s+1) ds = 0 + 2 < 24

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) r (t) = 0 3 t + 1 2 0 e s 3 02(τ+1) dτ ds = 2 < conditions 25

Example Ordinary differential equation p t = 1, r t = 3 t + 1 u + 3 t + 1 u = 3u 5 3, t R + Theorem 2.1, [3] u(t) is solution of (E), (B) 0 e s r τ 02p τ dτ ds < r t 0 conditions u t p A 1 0 u t p = Ο( consequences t e s r τ 02p τ dτ ds) r (t) = 0 3 t + 1 2 0 e s 3 02(τ+1) dτ ds = 2 < conditions u t = u t p = Ο( consequences 1 t + 1 3 t 0 0 = A 0 t e s 3 02(τ+1) dτ ds) 26

OUTLOOK: Further research & development Existence of solutions Time asymptotics 27

OUTLOOK: Further research & development Existence of solutions Time asymptotics More generality Higher convergence Milder conditions on p(t), r(t) Less assumptions 2014 28

REFERENCES (1) G. Moroşanu, Existence results for second-order monotone differential inclusions on the positive half-line, J. Math. Anal. Appl. 419 (2014) 94-113. (2) B. Djafari-Rouhani, H. Khatibzadeh, A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl. 363 (2010) 648 654. (3) B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. (4) B. Djafari-Rouhani, H. Khatibzadeh, Asymptotic Behavior for a General Class of Homogeneous Second Order Evolution Equations in a Hilbert Space, To appear in Dynamic Systems and Applications 29

THANK YOU FOR YOUR ATTENTION Please give a comment in order to discuss about the topic! 30

BACKUP: Development: Asymptotic behaviour 31

Development Overview timeline Strong convergence [A1, A2] [A3] [A4, A5] p t u t + r t u t Au t u 0 = u 0, sup u(t) < + + Convergence rate Existence and uniqueness of a bounded solution 74/75 76 85/86 75/76 80 82 Asymptotic behaviour 85 88 98 2007 09 10 [3] 2013 [A6] [A7] [A6,A8, A9, A10, A11, A12] [A13, A14, A15] [A17] 32

Development: Asymptotic behavior 1. Véron proved the existence of solutions to the second order evolution equation: p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < +, t 0 (B) where A is a maximal monotone operator, following conditions hold (C1) p W 2, 0, +, r W 1, 0, + (C2) α > 0, such that t 0, p t α Additional results from Véron for solution u: u H 1 0, ; H u is unique if 0 e 0 tr(s) p(s) ds dt = + [A1] L. Véron, Problèmes d evolution du second ordre associés à opérateurs monotones, C. R. Acad. Sci. Paris Sér. A 278 (1974) 1099-1101. [A2] L. Véron, Equations d évolution du second ordre associés à opérateurs maximaux monotones, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975-1976) 131-147. 33

Development: Asymptotic behavior homogenous 1. Véron proved the existence of solutions to the second order evolution equation: p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < +, t 0 (B) where A is a maximal monotone operator, following conditions hold (C1) p W 2, 0, +, r W 1, 0, + Sobolev spaces (C2) α > 0, such that t 0, p t α p(t) α > 0 Additional results from Véron for solution u: u H 1 0, ; H u, u L 2 Sobolev space H 1 Ω = W 1,2 (Ω) u is unique if 0 e 0 tr(s) p(s) ds dt = + [A1] L. Véron, Problèmes d evolution du second ordre associés à opérateurs monotones, C. R. Acad. Sci. Paris Sér. A 278 (1974) 1099-1101. [A2] L. Véron, Equations d évolution du second ordre associées à opérateurs maximaux monotones, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975-1976) 131-147. 34

Development: Asymptotic behavior 2. Barbu proved the existence of solutions to the second order evolution equation: p 1, r 0 with the condition u t Au t for a. a. t R + [0, ) (E) u 0 = u 0, sup u(t) < +, t 0 (B) Where A is a maximal monotone operator in Hilbert spaces. [A3] V.Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976 35

Development: Asymptotic behavior 3. Poffald and Reich proved the existence of solutions to the second order evolution equation: p 1, r 0 with the condition u t Au t for a. a. t R + [0, ) (E) u 0 = u 0, sup u(t) < +, t 0 (B) Where A is a maximal monotone operator in Banach spaces. [A4] E.I. Poffald, S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 (1986) 514-543 [A5] E.I. Poffald, S. Reich, A quasi-autonomous second-order differential inclusion, in: Trends in the Theory and Practice of Nonlinear Analysis (Arlington, Tex., 1984), in: North-Holland Math. Stud., vol. 110, North-Holland, Amsterdam, 1985, pp. 387-392. 36

Development: : Asymptotic behavior 3. Poffald and Reich proved the existence of solutions to the second order evolution equation: p = 1, r = 0 with the condition u t Au t for a. a. t R + [0, ) (E) u 0 = u 0, sup u(t) < +, t 0 (B) Where A is a maximal monotone operator in Banach spaces. 4. Bruck proved weak convergence in this special case. [A6] R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal. 18 (1975) 15-26 37

Development: : Asymptotic behavior 3. Poffald and Reich proved the existence of solutions to the second order evolution equation: p = 1, r = 0 with the condition u t Au t for a. a. t R + [0, ) (E) u 0 = u 0, sup u(t) < +, t 0 (B) Where A is a maximal monotone operator in Banach spaces. 4. Bruck proved weak convergence in this special case. 5. Véron showed that the strong convergence does not hold in general for a maximal monotone operator A, for p t 1, r t 0. [A7] L. Véron, Un exemple concernant le comportement asymptotique de la solution bornée de l équation d2 dt2 φ(u), Monatsh. Math. 89 (1980) 57-67. 38

Development: : Asymptotic behavior 6. Bruck, Mitidieri, Morosanu, Apreutesei: Strong convergence of solutions with additional assumptions on the maximal monotone operator A [A6] R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal. 18 (1975) 15-26 [A8] E. Mitidieri, Asymptotic behaviour of some second order evolution equations, Nonlinear Anal. 6 (1982) 1245 1252. [A9] E. Mitidieri, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985) 211 221. [A10] G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane, Bucharest, (1988), (and D. Reidel publishing Company). [A11] N.C. Apreutesei, Nonlinear Second Order Evolution Equation of Monotone Type, Pushpa publishing House, Allahabad, India, (2007). [A12] N.C. Apreutesei, Second-order differential equations on half-line associated with monotone operators, J. Math. Anal. Appl. 223 (1998) 472 493. 39

Development: Asymptotic behavior 6. Bruck, Mitidieri, Morosanu, Apreutesei: Strong convergence of solutions with additional assumptions on the maximal monotone operator A 7. Djafari-Rouhani, Khatibzadeh: Strong convergence of solutions to (E), (B) when p t 1, r t c 0 + extend previous results to nonhomogeneous case without assumptions A 1 0 or A as maximal monotone operator [A13] B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal. 70 (2009) 4369 4376. [A14] B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type, Nonlinear Anal. 71 (2009) e147 e152. [A15] B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math. 40 (2010) 1289 1311. [A16] B. Djafari Rouhani, H. Khatibzadeh, A strong convergence theorem for solutions to a nonhomogenous second order evolution equation, J. Math. Anal. Appl. 363 (2010) 648-654. 40

Development: Asymptotic behavior 6. Bruck, Mitidieri, Morosanu, Apreutesei: Strong convergence of solutions with additional assumptions on the maximal monotone operator A 7. Djafari-Rouhani, Khatibzadeh: Asymptotic behaviour of solutions to (E), (B) when p t 1, r t c 0 + extend previous results to nonhomogeneous case without assumptions A 1 0 or A as maximal monotone operator 8. Djafari-Rouhani, Khatibzadeh: A = φ is the subdifferential of a proper, convex and lower semicontinuous function φ + proof of an ergodic theorem + weak and strong convergence theorems to (E),(B) with additional assumptions on r(t) and A [A17] B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type, J. Inequal. Appl. (2007) 8. Art. ID 72931. 41

RECENT RESULTS: Asymptotic behavior Strong convergence of solutions to the nonlinear second order evolution equation: p t u t + r t u t Au t for a. a. t R + [0, ) (E) with the condition u 0 = u 0, sup u(t) < + (B) Where A is a maximal monotone operator in a real Hilbert space H. A 1 0 u(t) zero of A - homogenous; f(t) = 0 (H1) A: D A H H, A maximal monotone operator (H2) p, q L R, ess inf p > 0, q + L 1 R +, q + = max{q t, 0} [3] B. Djafari-Rouhani, H. Khatibzadeh, A note on the strong convergence of solutions to a second order evolution equation, J. Math. Anal. Appl. 401 (2013) 963 966. 42

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