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basic monotonicity methods with some applications

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R.P. wireless publishers functions matrix monotone communications 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). We know that B = is bounded and continuous. In the following sequence of lemmas we show that the above are satisfied. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Surveys 23, 117165 (1968) 32. J. Exercise 9.18 Using Theorem 6.5 examine the existence of a weak solution to (9.30) for a < . R.A. Adams, Sobolev Spaces (Academic Press, London, 1975) 2. monotonic monotonicity Then functional J is sequentially weakly lower semicontinuous on H01 (0, 1). 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. Since W0 (0, 1) is uniformly convex (and therefore strictly convex) we are able to conclude by Remark 3.2 that A1 is strictly monotone. We may at last study problem corresponding to (1.1) with a nonlinear term as well. 0 0 Moreover, A1 is coercive and dmonotone with respect to (x) = x p1 . Appl. It remains to comment that the uniqueness is reached in case functional J has exactly one critical point. Proof Recall that A1 is strictly monotone and continuous. S. Reich, Book review: geometry of banach spaces, duality mappings and nonlinear problems. scheduling computer hierarchy example figure J.L. R.R. 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. Math. Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox. Graduate Studies in Mathematics, vol. 26, 275298 (2019) 29. 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 . Mathematics 8, 1538 (2020). 26, 367370 (1992) 50. S. Boyd, E.K. monotonic (9.23) 0 Proof We write Eq.

Repov, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (CRC Press/Taylor and Francis Group, Boca Raton, 2015) 48. 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. Bauschke, P.L. 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). We have: Lemma 9.6 Assume that A11 holds. (McGraw-Hill Book Co., New York, 1964) 53. 0 We see that with assumption A6 any solution is non-zero which we prove by a direct calculation assuming to the contrary. 6 (North-Holland Publishing Co., Amsterdam, 1979), xii+460 pp 28. Exercise 9.12 Prove that g defined above belongs to W 1,q (0, 1), i.e. that g is a linear 1,p and continuous functional over W0 (0, 1). Then problem p1 d u (t)p2 dtd t, dtd u (t) dt d dt u (t) = g (t) , for a.e.

monotonic decreasing monotonically increasing monotonicity 10, 289300 (2021) 23. 18 (Springer, New York, 2009) 37. Birkhuser Advanced Texts. 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. 2 (Elsevier Scientific Publishing Company, Amsterdam, 1980) 19. 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. As announced above we begin with the solvability of a problem with a fixed right hand side. Radulescu, D.D. t is a Carathodory function as well. 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). (9.22) Concerning equations involving the above introduced operator we will follow the scheme developed for (9.12), i.e. Uspekhi Mat. USA 50, 10381041 (1963) 40. 162 (Cambridge University, Cambridge, 2016) 42. 0 (9.36) 0 Summing up we have J (u) 1 1 1 |g (t)| uH 1 c1 1 2 a1 u2H 1 + b1 0 2 0 0 for any u H01 (0, 1). From Example 2.11 we see that J2 is sequentially weakly continuous. Hint: consult Example 3.7 in showing that T1 is strongly monotone. 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. t [0, 1] and all x R 0 d F (t, x) = f (t, x) for a.e. H. Brzis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. flowchart 168 9 Some Selected Applications Lemma 9.5 Assume that conditions A9, A10 are satisfied and that operator is defined by (9.31). From Example 2.5 it follows that J1 is C 1 as well. G. Dinca, P. Jeblean, Some existence results for a class of nonlinear equations involving a duality mapping. J. fuzzification dataset Port. 0 By a direct calculation we see that operator B is coercive. Soc. 0 Exercise 9.17 When a is constant show that operator T1 (u) , v = 0 1 1 u (t) v (t) dt + 0 a u (t) v (t) dt for u, v H01 (0, 1) is strongly monotone and next apply Theorem 6.5 in order to examine the existence of a weak solution to (9.30). Anal. Z. Denkowski, S. Migrski, N.S. Reasoning by a contradiction we see that this solution is necessarily nontrivial. By Lemma 3.3 it follows that operator A satisfies property (S). Our partners will collect data and use cookies for ad personalization and measurement. Encyclopedia of Mathematics and its Applications, vol. Translated from the Russian by Karol Makowski. 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. The first part of condition (ii) follows from Lemma 9.4. explanation monotone stack Note that in considering the existence of solutions to (9.32) we will look for weak solutions which are critical points (9.34). 72(3), 389394 (2002) 27. Aplikace Matematiky 35(4), 257301 (1990) 18. ISBN 978-83-66287-37-2 22. Anal. Studies in Mathematics and its Applications, vol. Math. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (Kluwer Academic Publishers, Dordrecht, 1990) 10. 156 (AMS, New York, 2014) 26. Figueredo, Lectures on the Ekeland Variational Principle with Applications and Detours. We finally prove that condition (iv) holds. (N.S.) Proof Put a1 = aL , b1 = 1 0 1 |b (t)| dt, c1 = 0 |c(t)| dt. J. Mahwin, Problemes de Dirichlet Variationnels Non Linaires. monotonic monotonically decreasing monotonicity This provides that condition (iii) holds. (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 . 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. Exercise 3.25 provides that it is coercive. 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. t (0, 1) , u (0) = u (1) = 0 1,p has a unique weak solution u W0 1 1,p (0, 1), i.e. Repov, Nonlinear analysistheory and methods, in Springer Monographs in Mathematics (Springer, Cham, 2019) References 177 45. D.G. G. Kassay, V.D. 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. 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). Anal. Therefore we have the assertion of the lemma satisfied. Numer. 13 (Springer, Heidelberg, 2004) 51. 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. This is why the condition of monotonicity in the principal part holds and we have the assumption (ii) satisfied.

t [0, 1] function x F (t, x) is convex on R and therefore functional J2 is convex as well. Acad. Optim. Proof By a direct calculation we see that (u, u) = T (u) for every u H01 (0, 1), so (i) holds. monotonic monotonicity

Papageorgiou, V.D. Appl. Z. Denkowski, S. Migrski, N.S. N. Iusem, D. Reem, S. Reich, Fixed points of Legendre-Fenchel type transforms. Exercise 9.19 Let p > 2. 1 Thus by Theorem 6.4 operator p is continuous which means that p defines a 1,p homeomorphism between W0 (0, 1) and W 1,q (0, 1). t [0, 1]. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Marcel Dekker, New York, 1984) 25. Papageorgiou, An Introduction to Nonlinear Analysis: Theory (Kluwer Academic/Plenum Publishers, New York, 2003) 11. G.J. Uspekhi Mat. Texts in Applied Mathematics, vol. t [0, 1] . J. Aust. This finishes the proof. Adv. marginal distributional uta J. Necas, Introduction to the theory of nonlinear elliptic equations, in Teubner-Texte zur Math, vol. A. Matei, M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems. Natl. Radulescu, Equilibrium Problems and Applications (Academic Press, Oxford, 2019) 33. Math. 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. Izdat., Nauka, Moscow (1974); see also: Theory of extremal problems. M. Galewski, On variational nonlinear equations with monotone operators. t [0, 1] and all x R it holds F (t, x) 1 a (t) x 2 + b (t) x + c(t). 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 . we will start from the problem with fixed right hand side and next we will proceed with nonlinear problems: 162 9 Some Selected Applications Proposition 9.2 Assume that A5 holds. Copyright 2022 EBIN.PUB. J. Jahn, Introduction to the Theory of Nonlinear Optimization. T. Tao, An introduction to measure theory, in Graduate Studies in Mathematics, vol. Then operator T is bounded, continuous and coercive. t [0, 1] and assume that g Lq (0, 1), g = 0. P. Drbek, J. Milota, Methods of Nonlinear Analysis. 233(11), 29852993 (2010) 24.

Then functional J is differentiable in the sense of Gteaux on H01 (0, 1). Exercise 9.15 Assume that f : R R is continuous and nondecreasing. t [0, 1] operator x f (t, x) is monotone on RN . But this leads the strong continuity of A2 . Comput. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, in Encyclopedia of Mathematics and its Applications, vol. C. Canuto, A. Tabacco, Mathematical Analysis I&II (Springer, Berlin, 2008) 7. R.E. A.D. Ioffe, V.M. (9.32) Here (apart from some further growth conditions): A11 g L2 (0, 1) and f : [0, 1] R R is an L2 Carathodory function. Thus operator p is uniformly monotone. We easily observe that T is coercive. R. Chiappinelli, D.E. Proof Since J3 is linear and bounded, it is obviously C 1 . Edmunds, Remarks on Surjectivity of Gradient operators. Basler Lehrbcher (Springer, Basel, 2013) 15. By the same arguments it follows that G is bounded. J. Chabrowski, Variational Methods for Potential Operator Equations (De Gruyter, Berlin, 1997) 8. 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. , 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. colored pencils drawing tonal applications techniques methods draw blending lessons pencil Tikhomirov, Theory of Extremal Problems (in Russian). Preliminary Lecture Notes (SISSA) (1988) 17. 160 (University of Wisconsin, Madison, 1960) 58. (N.S.) application composition scheduler issues examples t3 t1 t2 belong guarantee example same per threads figure 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 . Sminaire de Mathmatiques Suprieures, Montreal, vol. M. Galewski, J. Smejda, On variational methods for nonlinear difference equations. Radulescu, V.D. Assume that function f : R R is continuous and nondecreasing. Adv. Proof Observe that T (u) , v = B (u) + G (u) , v for all u, v H01 (0, 1). monotonic 3rd edn. Hence we need one additional assumption: A13 for a.e. From Proposition 9.2 we see that A is strictly monotone and coercive. S. Migrski, M. Sofonea, VariationalHemivariational inequalities with applications, in Chapman & Hall/CRC Monographs and Research Notes in Mathematics, Boca Raton, FL (2018) 38. S. Fucik, A. Kufner, Nonlinear differential equations, in Studies in Applied Mechanics, vol. T.L. W. Rudin, Principles of Mathematical Analysis, 2nd edn. The application of Theorem 6.4 finishes the proof of the existence and the uniqueness of the solution. Nonl. photoluminescence With the above Lemmas 9.4 and 9.5 we have all assumptions of Theorem 6.9 satisfied. By Proposition 9.2 we see that Theorem 6.4 can be applied to problem (9.24). Bull. D. Idczak, A. Rogowski, On the Krasnoselskij theorema short proof and some generalization. Papageorgiou, An Introduction to Nonlinear Analysis: Applications (Kluwer Academic/Plenum Publishers, New York, 2003) 12. dataset distribution right figure weibull kumaraswamy geometric applications cdf ttt plot second left wg kw estimated histogram displays fitted Hence we must properly define mapping and demonstrate that all assumptions (i)(iv) of Theorem 6.9 are satisfied. Nonlinear Anal. Then problem (9.30) has at least one weak solution. We have the following result: Theorem 9.11 Assume that conditions A5, A6, A8 are satisfied. 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). Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces. G. Molica Bisci, V.D. (9.34) 0 Observe that J = J1 + J2 + J3 , all considered on H01 (0, 1), where J1 (u) = 1 2 0 1 |u (t)|2 dt, J2 (u) = 0 1 1 F (t, u (t)) dt, J3 (u) = g (t) u (t) dt.

J. Comput. 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) . pid monotonicity Lecture Notes in Mathematics, vol. H.H. Since T1 is not strongly monotone when a is some function satisfying (9.29) we will apply Theorem 6.9 in order to reach the existence result. t [0, 1] and all x R; A10 for a.e. t [0, 1] . All rights reserved. Minty, Monotone (nonlinear) operators in Hilbert space. By the strict monotonicity of p , we see that this operator is invertible. immunoassay advances pharmaceutical methodology N.S. 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. Funct. Additionally, operator A1 is invertible and its inverse A1 1 is continuous. Therefore by Theorem 6.4 operator A1 is invertible and its inverse A1 1 is continuous. The Reader is invited to provide detailed calculations as an exercise. hsien kaohsiung 126 (2011) 56. Nonlinear Anal. 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. Observe that F : [0, 1] R R given by x F (t, x) = f (t, s) ds for a.e. Lemma 9.4 Assume that conditions (9.29) and A9, A10 are satisfied. 58(3), 339378 (2001) 14. Proc. 46, 347-363 (2001) 13. J. Francu, Monotone operators: a survey directed to applications to differential equations. (International Series of Numerical Mathematics, vol. Hence in order to apply Theorem 6.5 we need to prove that A is coercive. csharp t [0, 1] . 0 But this means that A1 is well defined. T. Roubcek, Nonlinear Partial Differential Equations with Applications. 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.

Then Dirichlet Problem (9.32) has exactly one solution u H01 (0, 1) H 2 (0, 1) . Nauk 15, 213 215 (1960) 31. We start with lemma summarizing some obvious properties of operator T . it sends points from W0 1,p functionals working on W0 (0, 1). As for the continuity of G we argue using the generalized Krasnoselskii Theorem, see Theorem 2.12. Troutman, Variational calculus and optimal control, in Optimization with Elementary Convexity, Undergraduate Texts in Mathematics (Springer, New York, 1996) 57. Ryu, A primer on monotone operator methods (survey). Then problem (9.26) has at least one nontrivial solution. Lemma 9.9 Under assumptions A11, A12 functional J is coercive over H01 (0, 1). G. Molica Bisci, D.D. 0 References 1. Let F : R R be defined by x F (x) = f (s) ds. Radulescu, T. Andreescu, Problems in Real Analysis: Advanced Calculus on the Real Axis (Springer, New York, 2008) 47. Nauk 23, 121168 (1968); English translation: Russian Math. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. M. Renardy, R.C. Indeed, note by A10 that G (u) , u 0 for all u H01 (0, 1). 2(2), 127146 (2013) 41. Exercise 9.14 Consider the following Dirichlet problem: p1 d d d u (t)p2 t, dt dt u (t) dt d dt u (t) + f (u (t)) = g (t) , for a.e. Now we proceed to consider the Dirichlet problem related to the pLaplacian. R.I. Kacurovski, Nonlinear monotone operators in Banach spaces. Am. Therefore it is also coercive and satisfies the property (S). Math. W. Rudin, Functional analysis, in McGraw-Hill Series in Higher Mathematics (McGraw-Hill Book Co., New York, 1973) 54. Radulescu, Variational and Nonvariational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Its Applications (Springer, Berlin, 2003) 43.

Therefore the coercivity of A follows and by Theorem 6.5 we obtain the assertion. Soc. t (0, 1) (9.26) 1,p We say that a function u W0 1,p W0 (0, 1) it holds 1 (0, 1) is a weak solution of (9.26) if for all v t, |u (t)|p1 |u (t)|p2 u (t) v (t) dt + 0 1 f (t, u (t)) v (t) dt = 0 1 g (t) v (t) dt.

(9.29) 166 9 Some Selected Applications Assume that f : [0, 1] R R is a Carathodory function. uta monotonic G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian. M. Galewski, On the application of monotonicity methods to the boundary value problems on the Sierpinski gasket. In order to consider a problem with a nonlinear right hand side, we need to make some assumptions: A6 f : [0, 1] RN RN is an L1 Carathodory function with f (t, 0) = 0 for a.e. H. Gajewski, K. Grger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974) 20. Applications to Differential Equations, 2nd edn. Fix u H01 (0, 1). Functional J1 is convex and continuous which by Theorem 2.20 implies that it is sequentially weakly lower semicontinuous. Sci. 467, 1208 1232 (2018) 49. Proof We see that J3 is linear and bounded, and therefore sequentially weakly continuous. 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). t (0, 1) u(0) = u(1) = 0. (Springer, Berlin, 2010) 6. This in turn complies with the definition of the weak solution. Radulescu, D.D. Using Theorem 6.5 prove that (9.28) has at least one weak solution. (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). 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 . Theorem 9.12 Assume that A9, A10 are satisfied. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976) 16. 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. 52 (Teubner, Leipzig, 1983) 44. t [0, 1] and x [d, d] . In order to prove that operator A2 is continuous we use Theorem 2.12. Math. 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.

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basic monotonicity methods with some applications