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. t [0, 1] . We may at last study problem corresponding to (1.1) with a nonlinear term as well. M. Galewski, J. Smejda, On variational methods for nonlinear difference equations. Soc. M. Haase, Functional Analysis: An Elementary Introduction. Acad. J.L. for any v W0 t, |u (t)|p1 |u (t)|p2 u (t) v (t) dt = 0 (0, 1) it holds 1 g (t) v (t) dt. (Harcourt/Academic Press, San Diego, 2001) 34. V.D. D.G. 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. G. Molica Bisci, D.D. A.D. Ioffe, V.M. We start with lemma summarizing some obvious properties of operator T .
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.
Using Lemmas 9.8, 9.6, and 9.9 we can apply Theorem 2.22 to reach the following result: Theorem 9.13 Assume conditions A11, A12. 13 (Springer, Heidelberg, 2004) 51.
J. Chabrowski, Variational Methods for Potential Operator Equations (De Gruyter, Berlin, 1997) 8.
Basler Lehrbcher (Springer, Basel, 2013) 15.
P. Drbek, J. Milota, Methods of Nonlinear Analysis. Then operator T is bounded, continuous and coercive. As for the continuity of G we argue using the generalized Krasnoselskii Theorem, see Theorem 2.12.
E.H. Zarantonello, Solving functional equations by contractive averaging, in Mathematical Research Center Technical Summary Report no. Hence in order to apply Theorem 6.5 we need to prove that A is coercive. Proposition 9.3 Assume that g Lq (0, 1) is fixed and that condition A5 holds. 136 (Cambridge University, Cambridge, 2010) 35.
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26, 367370 (1992) 50. 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 . Thus it remains to consider functional J2 .
application composition scheduler issues examples t3 t1 t2 belong guarantee example same per threads figure A. Matei, M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems. 233(11), 29852993 (2010) 24. Exercise 9.18 Using Theorem 6.5 examine the existence of a weak solution to (9.30) for a < . Repov, Nonlinear analysistheory and methods, in Springer Monographs in Mathematics (Springer, Cham, 2019) References 177 45. 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) .
colored pencils drawing tonal applications techniques methods draw blending lessons pencil Then problem (9.30) has at least one weak solution.
Appl. Optim. N. Iusem, D. Reem, S. Reich, Fixed points of Legendre-Fenchel type transforms. t [0, 1] . 0 References 1. As announced above we begin with the solvability of a problem with a fixed right hand side. 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. Proof We see that J3 is linear and bounded, and therefore sequentially weakly continuous. A7 for a.e. Math. Math. Aplikace Matematiky 35(4), 257301 (1990) 18. From Example 2.11 we see that J2 is sequentially weakly continuous.
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. 26, 275298 (2019) 29.
aimspress monotonic uta 
Then conditions (i)(iv) from Theorem 6.9 are satisfied. Repov, On some variational algebraic problems. The above relations and the monotonicity of B imply that (u, u) (u, v), u v 0 for all u, v H01 (0, 1). Series in Nonlinear Analysis and its Applications. Math.
bhuvaneswari sambandham Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, in Encyclopedia of Mathematics and its Applications, vol. Translated from the Russian by Karol Makowski. Then functional J is sequentially weakly lower semicontinuous on H01 (0, 1).
This provides that condition (iii) holds. 3rd edn. Lemma 9.9 Under assumptions A11, A12 functional J is coercive over H01 (0, 1). Proof Recall that A1 is strictly monotone and continuous. Using Theorem 6.5 prove that (9.28) has at least one weak solution.
Uspekhi Mat.
marginal distributional uta 162 (Cambridge University, Cambridge, 2016) 42. S. Fucik, A. Kufner, Nonlinear differential equations, in Studies in Applied Mechanics, vol. Radulescu, Variational and Nonvariational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Its Applications (Springer, Berlin, 2003) 43. t [0, 1]. All rights reserved.
This finishes the proof. 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. Natl. 52 (Teubner, Leipzig, 1983) 44. 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). Papageorgiou, V.D. 6 (North-Holland Publishing Co., Amsterdam, 1979), xii+460 pp 28. Repov, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (CRC Press/Taylor and Francis Group, Boca Raton, 2015) 48. Adv. T.L. Exercise 9.12 Prove that g defined above belongs to W 1,q (0, 1), i.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. 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). 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). 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. Hence we need one additional assumption: A13 for a.e. J. Convex Anal.
The Reader is invited to provide detailed calculations as an exercise. t [0, 1] and all x R 0 d F (t, x) = f (t, x) for a.e. 29, 341346 (1962) 39. We have: Lemma 9.6 Assume that A11 holds. Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox. By Lemma 3.3 it follows that operator A satisfies property (S).
photoluminescence Radulescu, D.D. t [0, 1] and all x R it holds xf (t, x) 0. 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). Ryu, A primer on monotone operator methods (survey). Adv.
immunoassay advances pharmaceutical methodology Studies in Mathematics and its Applications, vol. G. Dinca, P. Jeblean, Some existence results for a class of nonlinear equations involving a duality mapping. Math. Bull. 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. 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. Math. USA 50, 10381041 (1963) 40.
monotonic decreasing monotonically increasing monotonicity Graduate Studies in Mathematics, vol. R.E. 104 (1987) 36. (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. Soc. Radulescu, V.D. Anal. We know that B = is bounded and continuous. Preliminary Lecture Notes (SISSA) (1988) 17. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 2000) 3.
csharp Surveys 23, 117165 (1968) 32. 2 (Elsevier Scientific Publishing Company, Amsterdam, 1980) 19. Proof From A13 it follows that for a.e. Hint: consult Example 3.7 in showing that T1 is strongly monotone. The application of Theorem 6.4 finishes the proof of the existence and the uniqueness of the solution. Encyclopedia of Mathematics and its Applications, vol. D. Idczak, A. Rogowski, On the Krasnoselskij theorema short proof and some generalization. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd edn. We look for weak solutions of (9.30), i.e. Numer. 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. Applications to Differential Equations, 2nd edn. Texts in Applied Mathematics, vol. D. Motreanu, V.D. We finally prove that condition (iv) holds. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Tikhomirov, Theory of Extremal Problems (in Russian). 0 0 Moreover, A1 is coercive and dmonotone with respect to (x) = x p1 . Port.
Nauk 23, 121168 (1968); English translation: Russian Math.
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. Prove that (9.28) has exactly one weak solution. Lemma 9.4 Assume that conditions (9.29) and A9, A10 are satisfied. J. Aust. Then problem (9.26) has exactly one nontrivial solution. R.I. Kacurovski, Nonlinear monotone operators in Banach spaces. C. Canuto, A. Tabacco, Mathematical Analysis I&II (Springer, Berlin, 2008) 7.
156 (AMS, New York, 2014) 26. Let us fix g L2 (0, 1) and a L (0, 1) such that there is a constant a1 < a (t) a1 > 0 for a.e. 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.
dataset distribution right figure weibull kumaraswamy geometric applications cdf ttt plot second left wg kw estimated histogram displays fitted 9.7 Applications of the LerayLions Theorem The LerayLions Theorem is about the existence of solutions to second order nonlinear problems involving also first order derivatives. S. Reich, Book review: geometry of banach spaces, duality mappings and nonlinear problems. M. Galewski, On variational nonlinear equations with monotone operators.
t (0, 1) u(0) = u(1) = 0. 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). Then Dirichlet Problem (9.32) has exactly one solution u H01 (0, 1) H 2 (0, 1) . J. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Marcel Dekker, New York, 1984) 25. We have the following result: Theorem 9.10 Assume that conditions A5A7 are satisfied. Rogers, An Introduction to Partial Differential Equations, 2nd edn. 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). Minty, Monotone (nonlinear) operators in Hilbert space. 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. Uspekhi Mat. Learn how we and our ad partner Google, collect and use data.
monotonic monotonically https://doi.org/10.3390/math8091538 9. 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. 126 (2011) 56. t [0, 1] and assume that g Lq (0, 1), g = 0. 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 . 0 But this means that A1 is well defined. M. Galewski, Wprowadzenie do metod wariacyjnych (Wydawnictwo Politechniki dzkiej, dz, 2020). By Proposition 9.2 we see that Theorem 6.4 can be applied to problem (9.24). Proof Since J3 is linear and bounded, it is obviously C 1 . Advances in Mechanics and Mathematics, vol. Then problem p1 d u (t)p2 dtd t, dtd u (t) dt d dt u (t) = g (t) , for a.e. Lecture Notes in Mathematics, vol. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (Kluwer Academic Publishers, Dordrecht, 1990) 10.
Therefore the coercivity of A follows and by Theorem 6.5 we obtain the assertion. 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). Note that in considering the existence of solutions to (9.32) we will look for weak solutions which are critical points (9.34). Radulescu, D.D. 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. t [0, 1] operator x f (t, x) is monotone on RN . 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 . Exercise 9.15 Assume that f : R R is continuous and nondecreasing. 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 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. 72(3), 389394 (2002) 27. Additionally, operator A1 is invertible and its inverse A1 1 is continuous.
(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). Lemma 9.8 Assume that A11 holds. Birkhuser Advanced Texts. t [0, 1] function x F (t, x) is convex on R and therefore functional J2 is convex as well. 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. (9.23) 0 Proof We write Eq. Hence we must properly define mapping and demonstrate that all assumptions (i)(iv) of Theorem 6.9 are satisfied. Papageorgiou, An Introduction to Nonlinear Analysis: Applications (Kluwer Academic/Plenum Publishers, New York, 2003) 12.
Applying Theorem 2.12 we see that A1 is continuous. H.H. The coercivity is obtained directly. Math. 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) . Proc. A. Kristly, V.D.
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M. Renardy, R.C. But this leads the strong continuity of A2 . 15(1), 343 (2016) 5. Therefore we have the assertion of the lemma satisfied.
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