basic monotonicity methods with some applications

T. Tao, An introduction to measure theory, in Graduate Studies in Mathematics, vol. t (0, 1) u (0) = u (1) = 0, (9.28) where g Lq (0, 1) and where f : RN RN is a continuous function such that (f (x) , x) 0 for all x RN . Moreover the following estimation holds due to (9.29) and the Poincar Inequality 1 0 a (t) u (t) u (t) dt a1 u2H 1 for all u H01 (0, 1). csharp

9.8 On Some Application of a Direct Method 9.8 169 On Some Application of a Direct Method Finally we remark on the variational solvability of (1.1) containing a nonlinear term in the special case when a = 0 and N = 1. 2(2), 127146 (2013) 41. 136 (Cambridge University, Cambridge, 2010) 35. By Lemma 3.3 it follows that operator A satisfies property (S). Additionally, operator A1 is invertible and its inverse A1 1 is continuous. Anal. 72(3), 389394 (2002) 27. Therefore it is also coercive and satisfies the property (S). (Springer, Berlin, 2010) 6. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Since un u0 in H01 (0, 1) implies that un u0 in C [0, 1], we see by Theorem 2.12 that (iv) is satisfied. J.L. (9.23) as follows: A1 (u) = g , (9.24) 9.6 Applications to Problems with the Generalized pLaplacian where the linear and bounded functional g : W0 1,p g (v) = 163 (0, 1) R is given by 1 (9.25) g (t) v (t) dt, 0 and where A1 is defined by (9.22). M. Galewski, J. Smejda, On variational methods for nonlinear difference equations. G.J. Therefore we have the assertion of the lemma satisfied. Math. Proof Using (3.8) and relation 1/p + 1/q = 1 we obtain that q q t, |u (t)|p1 |u (t)|p2 |u (t)| dt M q 1 0 1 |u (t)|p . 6 (North-Holland Publishing Co., Amsterdam, 1979), xii+460 pp 28. J. Mahwin, Problemes de Dirichlet Variationnels Non Linaires. Anal. From Example 2.11 we see that J2 is sequentially weakly continuous. 9.7 Applications of the LerayLions Theorem 167 We proceed now with the following definitions which are introduced in order to separate the effects of higher and lower derivatives: g : H01 (0, 1) 1 R, g (u) = g (t) w (t) dt, 0 B : H01 (0, 1) H 1 (0, 1), B (v) , w = G : H01 (0, 1) H 1 (0, 1), G (u) , w = 1 1 v (t) w (t) dt, 0 (f (t, u (t)) + a (t) u (t)) w (t) dt, 0 for u, v, w H01 (0, 1). (N.S.) Functional J1 is convex and continuous which by Theorem 2.20 implies that it is sequentially weakly lower semicontinuous. S. Reich, Book review: geometry of banach spaces, duality mappings and nonlinear problems. Math. Minty, Monotone (nonlinear) operators in Hilbert space. Now we can consider the existence and also uniqueness result for the following problem: d t, d u (t)p1 d u (t)p2 dt dt dt u (0) = u (1) = 0. d dt u (t) + f (t, u (t)) = g (t) , for a.e. Then problem (9.32) has at least one solution u H01 (0, 1) H 2 (0, 1) . But this leads the strong continuity of A2 . Theorem 9.12 Assume that A9, A10 are satisfied. From Theorem 3.3 we know that A1 is dmonotone with respect to (x) = x p1 and 1,p by Theorem 5.1 it is potential. J. Comput. it sends points from W0 1,p functionals working on W0 (0, 1). Appl. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd edn. Learn how we and our ad partner Google, collect and use data. M. Haase, Functional Analysis: An Elementary Introduction. Prove that (9.28) has exactly one weak solution. t [0, 1] and all x R; A10 for a.e. With the above Lemmas 9.4 and 9.5 we have all assumptions of Theorem 6.9 satisfied. Since a1 < 2 , we see that J is coercive over H01 (0, 1). Nonlinear Anal. Repov, On some variational algebraic problems. 0 But this means that A1 is well defined. Appl. t [0, 1] function x F (t, x) is convex on R and therefore functional J2 is convex as well. https://doi.org/10.3390/math8091538 9. Rogers, An Introduction to Partial Differential Equations, 2nd edn.

2 (Elsevier Scientific Publishing Company, Amsterdam, 1980) 19. Since J1 is strictly convex and J3 convex, we see that now J is strictly convex and therefore its critical point, which exists by Theorem 9.13, is unique. H. Gajewski, K. Grger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974) 20. t (0, 1) , u (0) = u (1) = 0 (9.30) under the assumptions: A9 there are constants c > 0, m > 1 and a function f0 L1 (0, 1) that such that |f (t, x)| c f0 (t) + |x|m for a.e. This finishes the proof. Anal. Sci. Indeed, 1,p for any u W0 (0, 1) we obtain 1 A (u) , u 1 |u (t)|p dt a1 0 |u (t)|p dt ( a1 ) u 0 p 1,p W0 . Proc. J. Jahn, Introduction to the Theory of Nonlinear Optimization. By Proposition 9.2 we see that Theorem 6.4 can be applied to problem (9.24). 104 (1987) 36. 160 (University of Wisconsin, Madison, 1960) 58. 0 9.8 On Some Application of a Direct Method 173 Prove that the Dirichlet problem p2 d d dt dt u (t) d dt u (t) + f (u (t)) = 0, for a.e. Our partners will collect data and use cookies for ad personalization and measurement. This is why the condition of monotonicity in the principal part holds and we have the assumption (ii) satisfied. N. Iusem, D. Reem, S. Reich, Fixed points of Legendre-Fenchel type transforms. (9.22) Concerning equations involving the above introduced operator we will follow the scheme developed for (9.12), i.e.

Proof Put a1 = aL , b1 = 1 0 1 |b (t)| dt, c1 = 0 |c(t)| dt.

0 (continued) 170 9 Some Selected Applications Remark 9.4 (continued) The following 1 F (t, x) = 2 tx 2 + (sin t) x 4 serves as an example of a function satisfying A12 and the associated nonlinear term is 1 f (t, x) = 2 tx + sin t. 2 In order to apply the Direct Method, Theorem 2.22, we need to demonstrate for functional J the following properties: sequential weak lower semicontinuity; coercivity; Gteaux differentiability; strict convexity (if one wishes to obtain uniqueness). 0 We see that with assumption A6 any solution is non-zero which we prove by a direct calculation assuming to the contrary. Advances in Mechanics and Mathematics, vol. 58(3), 339378 (2001) 14. M. Galewski, On the application of monotonicity methods to the boundary value problems on the Sierpinski gasket. We apply Theorem 6.9 in order to consider the existence of a solution for the following problem: u (t) + f (t, u (t)) + a (t) u (t) = g (t) , for a.e. Lemma 9.8 Assume that A11 holds. Optim. 0 By a direct calculation we see that operator B is coercive. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (American Mathematical Society, Providence, RI, 1997) 55. Encyclopedia of Mathematics and its Applications, vol. such functions u H01 (0, 1) that 1 0 1 u (t) v (t) dt+ 1 f (t, u (t)) v (t) dt+ 0 1 a (t) u (t) v (t) dt = 0 g (t) v (t) dt 0 for all v H01 (0, 1). We see that dx [0, 1] and all x R. Additionally we will assume that: A12 there exist a L (0, 1), b, c L1 (0, 1) such that aL < 2 and that for a.e. Namely, basing on some exposition from [21], we consider the following Dirichlet Problem: find a function u H01 (0, 1) such that the following equation is satisfied: u(t) + f (t, u(t)) = g (t) , for a.e. (Harcourt/Academic Press, San Diego, 2001) 34. T. Roubcek, Nonlinear Partial Differential Equations with Applications. 172 9 Some Selected Applications Observe that it holds by the Sobolev and the Poincar Inequality 1 1 1 J2 (u) 12 0 a (t) |u (t)|2 dt + 0 b (t) u (t) dt + 0 c(t)dt 21 2 a1 u2H 1 b1 uH 1 c1 for any u H01 (0, 1) . 0 0 Moreover, A1 is coercive and dmonotone with respect to (x) = x p1 . 1,p Exercise 9.13 Show that A2 is well defined, i.e. A7 for a.e. We look for weak solutions of (9.30), i.e. 168 9 Some Selected Applications Lemma 9.5 Assume that conditions A9, A10 are satisfied and that operator is defined by (9.31). D. Motreanu, V.D. For a given v H01 (0, 1) we investigate the existence of the following limit: lim 0 0 1 F (t, u (t) + v (t)) F (t, u (t)) dt. Assume that function f : R R is continuous and nondecreasing.

, RN , we get for u, v W0 161 (0, 1) : (u) p (v) , u v = 1 p (t)|p2 u (t) |v (t)|p2 v (t) , u (t) v (t) dt 0 |u 1 (1/2)p 0 |u (t) v (t)|p dt = u vW 1,p u vW 1,p , 0 0 where (x) = (1/2)p x p1 for x 0. t [0, 1]. 29, 341346 (1962) 39. Radulescu, D.D. Using Theorem 6.5 prove that (9.28) has at least one weak solution. Z. Denkowski, S. Migrski, N.S. t [0, 1] F (t, u (t) + v (t)) F (t, u (t)) max |f (t, s)| d. s[d,d] Hence we can apply the Lebesgue Dominated Convergence Theorem. G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976) 16. 2 (9.33) Remark 9.4 AssumptionaL < 2 is connected with the Poincar Inequality and is required in order to prove the coercivity of the corresponding Euler action functional J : H01 (0, 1) R given by 1 2 J (u) = 0 1 1 |u (t)|2 dt + 1 F (t, u (t)) dt 0 g (t) u (t) dt. J. Francu, Monotone operators: a survey directed to applications to differential equations. t (0, 1) u (0) = u (1) = 0 1,p has exactly one weak solution u W0 (0, 1) which is a minimizer to the following action 1,p functional J : W0 (0, 1) R given by the formula J (u) = 1 p 1 0 |u (t)|p dt + 1 F (u (t)) dt. Exercise 9.18 Using Theorem 6.5 examine the existence of a weak solution to (9.30) for a < . Math. R.R. Soc. Using the fact that both u, v are continuous (and so are bounded by some constant, say d > 0) and the Lagrange Mean Value Theorem we obtain that for a.e. Therefore the coercivity of A follows and by Theorem 6.5 we obtain the assertion. G. Molica Bisci, V.D. P. Drbek, J. Milota, Methods of Nonlinear Analysis. Therefore by Theorem 6.4 operator A1 is invertible and its inverse A1 1 is continuous. W. Rudin, Functional analysis, in McGraw-Hill Series in Higher Mathematics (McGraw-Hill Book Co., New York, 1973) 54. R.I. Kacurovski, Monotone operators and convex functionals.

We have the following result: Theorem 9.11 Assume that conditions A5, A6, A8 are satisfied. M. Galewski, Wprowadzenie do metod wariacyjnych (Wydawnictwo Politechniki dzkiej, dz, 2020). Radulescu, Equilibrium Problems and Applications (Academic Press, Oxford, 2019) 33. We may at last study problem corresponding to (1.1) with a nonlinear term as well. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces. t [0, 1] function x f (t, x) is nondecreasing on R. Theorem 9.14 Assume that A11, A13 hold. Natl. Bauschke, P.L. R. Chiappinelli, D.E. J. Necas, Introduction to the theory of nonlinear elliptic equations, in Teubner-Texte zur Math, vol. For all u, v H01 (0, 1) we directly calculate that (u, u) , u v = B (u) , u v + G (u) , u v , (u, v) , u v = B (v) , u v + G (u) , u v . H.H. H. Brzis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. 26, 367370 (1992) 50. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, in Encyclopedia of Mathematics and its Applications, vol. (0, 1) into 164 9 Some Selected Applications With g given by (9.25), we see that problem (9.26) is equivalent to the following abstract equation: A (u) = g . (Springer, Berlin, 2007) 30. In order to prove that operator A2 is continuous we use Theorem 2.12. Indeed, note by A10 that G (u) , u 0 for all u H01 (0, 1). We finally prove that condition (iv) holds. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Surveys 23, 117165 (1968) 32. Series in Nonlinear Analysis and its Applications. W. Rudin, Principles of Mathematical Analysis, 2nd edn. Adv. Duke Math. Birkhuser Advanced Texts. D.G. that g is a linear 1,p and continuous functional over W0 (0, 1). Math. Then problem (9.26) has exactly one nontrivial solution. Radulescu, V.D. 153) (Birkhuser, Basel, 2013) 52. Aplikace Matematiky 35(4), 257301 (1990) 18. Exercise 9.12 Prove that g defined above belongs to W 1,q (0, 1), i.e. By the strict monotonicity of p , we see that this operator is invertible. Proof We see that J3 is linear and bounded, and therefore sequentially weakly continuous.

Troutman, Variational calculus and optimal control, in Optimization with Elementary Convexity, Undergraduate Texts in Mathematics (Springer, New York, 1996) 57. 46, 347-363 (2001) 13. Repov, Nonlinear analysistheory and methods, in Springer Monographs in Mathematics (Springer, Cham, 2019) References 177 45. Then functional J is differentiable in the sense of Gteaux on H01 (0, 1). This provides that condition (iii) holds. Thus operator p is uniformly monotone. Indeed, 1,p since a convergent sequence (un ) n=1 W0 (0, 1) is uniformly bounded by some d > 0, 1 then by A6 there is some function fd L (0, 1) such that |f (t, x)| fd (t) for a.e. t [0, 1] and for all x R+ ; there exists a constant > 0 such that (t, x) x (t, y) y (x y) for all x y 0 and for a.e. Then functional J is sequentially weakly lower semicontinuous on H01 (0, 1). 1,p Let us define operators A, A2 : W0 (0, 1) W 1,q (0, 1) as follows: 1 A2 (u) , v = (9.27) f (t, u(t))v (t) dt, 0 A (u) , v = A1 (u) , v + A2 (u) , v 1,p for u, v W0 (0, 1). From Example 2.5 it follows that J1 is C 1 as well. t [0, 1] . USA 50, 10381041 (1963) 40. Ryu, A primer on monotone operator methods (survey). Nauk 15, 213 215 (1960) 31. R.A. Adams, Sobolev Spaces (Academic Press, London, 1975) 2. Copyright 2022 EBIN.PUB. All rights reserved. Note that in considering the existence of solutions to (9.32) we will look for weak solutions which are critical points (9.34). Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox. Figueredo, Lectures on the Ekeland Variational Principle with Applications and Detours. 9.8 On Some Application of a Direct Method 171 Obviously lim 0 F (t, u (t) + v (t)) F (t, u (t)) = f (t, u (t)) v (t) for a.e. R.E.

J. Convex Anal.

Nonl. We have: Lemma 9.6 Assume that A11 holds. Then problem p1 d u (t)p2 dtd t, dtd u (t) dt d dt u (t) = g (t) , for a.e. V.D. Math. Now we put : H01 (0, 1) H01 (0, 1) H 1 (0, 1) by the following formula: (u, v) , w = B (v) , w + G (u) , w (9.31) for u, v, w H01 (0, 1). (9.23) 0 Proof We write Eq. 18 (Springer, New York, 2009) 37. (N.S.) Apart from Theorem 6.4 we may apply Theorem 6.5 for which require some growth condition on f instead of assumption A7: A8 there exists a constant a1 < such that (f (t, x) , x) a1 |x|p1 for all x RN and for a.e.

We have the following result: Theorem 9.10 Assume that conditions A5A7 are satisfied. Then conditions (i)(iv) from Theorem 6.9 are satisfied. Nonlinear Anal. G. Kassay, V.D. Proof By a direct calculation we see that (u, u) = T (u) for every u H01 (0, 1), so (i) holds. Papageorgiou, V.D. R.I. Kacurovski, Nonlinear monotone operators in Banach spaces. 126 (2011) 56. 3rd edn. Studies in Mathematics and its Applications, vol. Lemma 9.9 Under assumptions A11, A12 functional J is coercive over H01 (0, 1). t [0, 1]. Exercise 9.15 Assume that f : R R is continuous and nondecreasing. Hence we need one additional assumption: A13 for a.e. J. Aust. We assume that A5 : [0, 1] R+ R+ is a Carathodory function and there exists constant M > 0 such that (t, x) M for a.e. Applying Theorem 2.12 we see that A1 is continuous. Nauk 23, 121168 (1968); English translation: Russian Math. t [0, 1] and x [d, d] . Radulescu, D.D. C. Canuto, A. Tabacco, Mathematical Analysis I&II (Springer, Berlin, 2008) 7. J. Take v H01 (0, 1), un u0 and assume that (un , v) z which means that B (v) , un + G (un ) , w 0 for all w H01 (0, 1). Then operator A1 defined by (9.22) is continuous and potential 1,p with the potential F : W0 (0, 1) R defined by F (u) = 1 |u(t)| t, s p1 s p1 dsdt. Sminaire de Mathmatiques Suprieures, Montreal, vol. t is a Carathodory function as well. Then u H01 (0, 1) is a critical point to functional (9.34) if and only if it satisfies (9.35).

Fix u H01 (0, 1). J. Chabrowski, Variational Methods for Potential Operator Equations (De Gruyter, Berlin, 1997) 8. This formula suggests as usual that we should consider operator T : H01 (0, 1) H 1 (0, 1) given by the following formula for u, v H01 (0, 1) : T (u) , v = 1 1 u (t) v (t) dt + 0 1 f (t, u (t)) v (t) dt + 0 a (t) u (t) v (t) dt. The first part of condition (ii) follows from Lemma 9.4. It remains to comment that the uniqueness is reached in case functional J has exactly one critical point. Z. Denkowski, S. Migrski, N.S. We consider a more general nonlinear operator 1,p A1 : W0 (0, 1) W 1,q (0, 1) , given by 1 A1 (u) , v = 0 1,p t, |u (t)|p1 |u (t)|p2 u (t) v (t) dt for u, v W0 (0, 1) . Proof Recall that A1 is strictly monotone and continuous. 156 (AMS, New York, 2014) 26. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 2000) 3. 40(11), 13441354 (2019) The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-030-75308-5 175 176 References 21. From Proposition 9.2 we see that A is strictly monotone and coercive. T.L. G. Dinca, P. Jeblean, Some existence results for a class of nonlinear equations involving a duality mapping. G.J. Am. M. Renardy, R.C. Repov, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (CRC Press/Taylor and Francis Group, Boca Raton, 2015) 48. Proof Observe that T (u) , v = B (u) + G (u) , v for all u, v H01 (0, 1). Observe that F : [0, 1] R R given by x F (t, x) = f (t, s) ds for a.e. Funct. In the following sequence of lemmas we show that the above are satisfied.

From formula (9.35) defining the weak solution and from Lemma 9.6 we obtain at once the following result connecting solutions to (9.32) with critical point to functional (9.34): Lemma 9.7 Assume that A11 holds. t (0, 1) u(0) = u(1) = 0.

t (0, 1) , u (0) = u (1) = 0 1,p has a unique weak solution u W0 1 1,p (0, 1), i.e. Hint: consult Example 3.7 in showing that T1 is strongly monotone. t [0, 1] . A. Kristly, V.D. 9.7 Applications of the LerayLions Theorem 165 Proof As in the proof of Theorem 9.10 we see that operator A2 defined by (9.27) is strongly continuous. Then Dirichlet Problem (9.32) has exactly one solution u H01 (0, 1) H 2 (0, 1) . Preliminary Lecture Notes (SISSA) (1988) 17.

E.H. Zarantonello, Solving functional equations by contractive averaging, in Mathematical Research Center Technical Summary Report no. Then problem (9.30) has at least one weak solution. 15(1), 343 (2016) 5. Note that for any u, v 1,p W0 (0, 1) 1 (f (t, u(t)) f (t, v(t))) (u(t) v (t)) dt 0 0 which implies that A2 is monotone. Exercise 9.19 Let p > 2. Radulescu, T. Andreescu, Problems in Real Analysis: Advanced Calculus on the Real Axis (Springer, New York, 2008) 47. 162 (Cambridge University, Cambridge, 2016) 42.

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