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. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, in Encyclopedia of Mathematics and its Applications, vol. A7 for a.e. Funct. Adv. 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. M. Galewski, Wprowadzenie do metod wariacyjnych (Wydawnictwo Politechniki dzkiej, dz, 2020). 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. 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. 26, 367370 (1992) 50. 
Nauk 23, 121168 (1968); English translation: Russian Math. Fix u H01 (0, 1). N.S. Anal. Hence we must properly define mapping and demonstrate that all assumptions (i)(iv) of Theorem 6.9 are satisfied. 
We have the following result: Theorem 9.10 Assume that conditions A5A7 are satisfied. W.G. Math. 6 (North-Holland Publishing Co., Amsterdam, 1979), xii+460 pp 28. 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.L. Then conditions (i)(iv) from Theorem 6.9 are satisfied. 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). , 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. 26, 275298 (2019) 29. Radulescu, T. Andreescu, Problems in Real Analysis: Advanced Calculus on the Real Axis (Springer, New York, 2008) 47. Troutman, Variational calculus and optimal control, in Optimization with Elementary Convexity, Undergraduate Texts in Mathematics (Springer, New York, 1996) 57. Anal. 
M. Galewski, On variational nonlinear equations with monotone operators. 162 (Cambridge University, Cambridge, 2016) 42. 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. CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC (Springer, Cham, 2017) 4. (Springer, Berlin, 2010) 6. Studies in Mathematics and its Applications, vol. Assume that function f : R R is continuous and nondecreasing. T. Tao, An introduction to measure theory, in Graduate Studies in Mathematics, vol. Birkhuser Advanced Texts. Adv. 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. t [0, 1] operator x f (t, x) is monotone on RN . P. Drbek, J. Milota, Methods of Nonlinear Analysis. Repov, On some variational algebraic problems. Series in Nonlinear Analysis and its Applications. 233(11), 29852993 (2010) 24. 
From Example 2.5 it follows that J1 is C 1 as well. Therefore it is also coercive and satisfies the property (S). As for the continuity of G we argue using the generalized Krasnoselskii Theorem, see Theorem 2.12. 29, 341346 (1962) 39. Proof Recall that A1 is strictly monotone and continuous. S. Reich, Book review: geometry of banach spaces, duality mappings and nonlinear problems. Exercise 9.15 Assume that f : R R is continuous and nondecreasing. The application of Theorem 6.4 finishes the proof of the existence and the uniqueness of the solution. D.G. Uspekhi Mat. (9.23) 0 Proof We write Eq. By the same arguments it follows that G is bounded. Applying similar calculations and the fact operator B has the property (S), we see that if un u0 and if lim (un , un ) (un , u0 ), un u0 = B(un ) B(u0 ), un u0 = 0, n+ it follows that un u0 . 18 (Springer, New York, 2009) 37. G. Molica Bisci, D.D. The first part of condition (ii) follows from Lemma 9.4. W. Rudin, Principles of Mathematical Analysis, 2nd edn. M. Galewski, J. Smejda, On variational methods for nonlinear difference equations. Our partners will collect data and use cookies for ad personalization and measurement. Z. Denkowski, S. Migrski, N.S. Theorem 9.12 Assume that A9, A10 are satisfied. Bull. E.H. Zarantonello, Solving functional equations by contractive averaging, in Mathematical Research Center Technical Summary Report no.
Bauschke, P.L. D. Idczak, A. Rogowski, On the Krasnoselskij theorema short proof and some generalization. V.D. A.D. Ioffe, V.M. t [0, 1]. 3rd edn. Minty, Monotone (nonlinear) operators in Hilbert space. Indeed, note by A10 that G (u) , u 0 for all u H01 (0, 1). Exercise 9.19 Let p > 2. We have: Lemma 9.6 Assume that A11 holds. 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). J. Comput. 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). 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. This provides that condition (iii) holds. 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) . 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. Comput. 1,p Exercise 9.13 Show that A2 is well defined, i.e. Appl. Exercise 9.16 Check whether assumption A8 can be replaced with the following: there exists a constant a1 < and a function b1 Lq (0, 1) such that (f (t, x) , x) a1 |x|p1 + b (t) |x| for all x RN and for a.e.
ISBN 978-83-66287-37-2 22. R.P. Radulescu, Variational and Nonvariational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Its Applications (Springer, Berlin, 2003) 43. 467, 1208 1232 (2018) 49. Proof Observe that T (u) , v = B (u) + G (u) , v for all u, v H01 (0, 1). W. Rudin, Functional analysis, in McGraw-Hill Series in Higher Mathematics (McGraw-Hill Book Co., New York, 1973) 54. 2 (Elsevier Scientific Publishing Company, Amsterdam, 1980) 19. R.I. Kacurovski, Monotone operators and convex functionals. We say that u H01 (0, 1) is a weak solution to (9.32) if 1 1 u(t) v (t) dt + 0 1 f (t, u(t))v (t) dt = 0 g (t) v (t) dt (9.35) 0 for all v H01 (0, 1). With the above Lemmas 9.4 and 9.5 we have all assumptions of Theorem 6.9 satisfied. 1364 (Springer, Berlin, 1993) 46. Copyright 2022 EBIN.PUB. Math. 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 have the following result: Theorem 9.11 Assume that conditions A5, A6, A8 are satisfied. J. Convex Anal. Basler Lehrbcher (Springer, Basel, 2013) 15. Hint: consult Example 3.7 in showing that T1 is strongly monotone. (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 . Proof By a direct calculation we see that (u, u) = T (u) for every u H01 (0, 1), so (i) holds. 0 0 Moreover, A1 is coercive and dmonotone with respect to (x) = x p1 . G. Kassay, V.D. Appl. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 2000) 3. Therefore we have the assertion of the lemma satisfied. G. Dinca, P. Jeblean, Some existence results for a class of nonlinear equations involving a duality mapping. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Marcel Dekker, New York, 1984) 25. D. Reem, S. Reich, Fixed points of polarity type operators. 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. 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] and all x R 0 d F (t, x) = f (t, x) for a.e. G.J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces. Proc. Applications to Differential Equations, 2nd edn. But this leads the strong continuity of A2 .
Then Dirichlet Problem (9.32) has exactly one solution u H01 (0, 1) H 2 (0, 1) . Then problem (9.26) has at least one nontrivial solution. Radulescu, Equilibrium Problems and Applications (Academic Press, Oxford, 2019) 33. 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. 10, 289300 (2021) 23. Appl. M. Galewski, On the application of monotonicity methods to the boundary value problems on the Sierpinski gasket. Am. Lemma 9.9 Under assumptions A11, A12 functional J is coercive over H01 (0, 1). I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976) 16. Hence we need one additional assumption: A13 for a.e. 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. Francu, Monotone operators: a survey directed to applications to differential equations. (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. Then functional J is sequentially weakly lower semicontinuous on H01 (0, 1). Soc. We may at last study problem corresponding to (1.1) with a nonlinear term as well.
Math. 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. Thus operator p is uniformly monotone. Translated from the Russian by Karol Makowski. G.J. R.E. M. Renardy, R.C. 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 . 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. 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. 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. Lecture Notes in Mathematics, vol. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/BNonlinear Monotone Operators (Springer, New York, 1990) Index A Adjoint operator, 64 Algebraic equation, 12 ArzelaAscoli Theorem, 31 B Banach Contraction Principle, 107 BanachSteinhaus Theorem, 62 Bounded bilinear form, 143 Brouwer Fixed Point Theorem, 9 BrowderMinty Theorem, 112 C Carathodory function, 37 Chain Rule, 42 Clarkson Inequality, 33 Condition (M), 116 (S), 75 (S)+ , 76 (S)0 , 76 (S)2 , 76 Continuous demicontinuous, 66 hemicontinuous, 66 Lipschitz, 66 radially, 66 strongly, 66 uniformly, 66 weakly, 66 Convexity criteria, 44 D Direct method of the calculus of variation, 51 Dirichlet Problem, 1 for the generalized plaplacian, 163 for the laplacian, 156 Duality mapping, 127 Duality mapping relative to a function, 134 du Bois-Reymond Lemma, 35 E Effective domain, 81 Ekeland Variational Principle, 152 Epigraph, 44 F Fenchel-Moreau Theorem, 89 Fenchel-Young dual, 85 Fenchel-Young Inequality, 86 Fermat Rule, 40 Finite Dimensional Existence Theorem, 10 Frchet derivative, 40 Fubini Theorem, 27 G Galerkin method, 139 Galerkin solution, 138 Gteaux derivative, 40 Gteaux variation, 43 Generalized BrowderMinty Theorem, 117 Generalized Krasnoselskii Theorem, 38 H Hahn-Banach Theorem, 125 Hlder Inequality, 23 Hyperplane closed, 84 supporting , 85 I Invertibility of a monotone operator, 114 Iteration method, 109 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/10.1007/978-3-030-75308-5 179 180 K Krasnoselskii Theorem, 39 L Lax-Milgram Theorem -nonlinear , 145 Lebesgue Dominated Convergence Theorem, 37 Lemma du Bois-Reymond Lemma, 35 fundamental in operator theory, 78 LerayLions Theorem, 121 M Mean Value Theorem, 46 Minty Lemma, 77 finite dimensional , 7 Monotone map, 6 N Nemytskii operator, 37 Normalized duality mapping, 126 O Operator bounded, 63 coercive, 69 d-monotone, 56 locally bounded, 62 monotone, 55 potential, 91 pseudomonotone, 118 Riesz, 17 strictly monotone, 55 strongly monotone , 56 uniformly monotone , 55 weakly coercive, 69 Index P Poincar Inequality, 31 Pseudomonotone operator, 118 R Riesz Representation in Lp , 22 Riesz Representation Theorem, 17 Ritz solution, 140 S Separation Theorem, 83 Sequentially weakly closed, 17 Sequential weak lower semicontinuity, 47 Sobolev Inequality, 31 Strictly convex space, 21 Strongly Monotone Principle -general version, 118 Strongly Monotone Principle in a Banach space, 133 Sufficient conditions for monotonicity, 60 Sufficient convexity condition, 46 T Theorem convexity and monotonicity, 59 existence result for a pseudomonotone operator, 120 sufficient condition for sequential weak lower semicontinuoity, 48 U Uniformly convex space, 21 W Weak convergence, 16 Weak derivative, 24 Weak sequential compactness of a ball, 18 Weierstrass Theorem, 47. (Harcourt/Academic Press, San Diego, 2001) 34.
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 . Repov, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (CRC Press/Taylor and Francis Group, Boca Raton, 2015) 48. Texts in Applied Mathematics, vol. 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. Rogers, An Introduction to Partial Differential Equations, 2nd edn. Exercise 9.18 Using Theorem 6.5 examine the existence of a weak solution to (9.30) for a < . 0 But this means that A1 is well defined. We start with lemma summarizing some obvious properties of operator T . 136 (Cambridge University, Cambridge, 2010) 35. Advances in Mechanics and Mathematics, vol. (N.S.) 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. Proposition 9.3 Assume that g Lq (0, 1) is fixed and that condition A5 holds. Tikhomirov, Theory of Extremal Problems (in Russian).
Then functional J is differentiable in the sense of Gteaux on H01 (0, 1). We finally prove that condition (iv) holds. 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.
Nauk 23, 121168 (1968); English translation: Russian Math. Fix u H01 (0, 1). N.S. Anal. Hence we must properly define mapping and demonstrate that all assumptions (i)(iv) of Theorem 6.9 are satisfied. We have the following result: Theorem 9.10 Assume that conditions A5A7 are satisfied. W.G. Math. 6 (North-Holland Publishing Co., Amsterdam, 1979), xii+460 pp 28. 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.L. Then conditions (i)(iv) from Theorem 6.9 are satisfied. 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). , 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. 26, 275298 (2019) 29. Radulescu, T. Andreescu, Problems in Real Analysis: Advanced Calculus on the Real Axis (Springer, New York, 2008) 47. Troutman, Variational calculus and optimal control, in Optimization with Elementary Convexity, Undergraduate Texts in Mathematics (Springer, New York, 1996) 57. Anal. M. Galewski, On variational nonlinear equations with monotone operators. 162 (Cambridge University, Cambridge, 2016) 42. 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. CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC (Springer, Cham, 2017) 4. (Springer, Berlin, 2010) 6. Studies in Mathematics and its Applications, vol. Assume that function f : R R is continuous and nondecreasing. T. Tao, An introduction to measure theory, in Graduate Studies in Mathematics, vol. Birkhuser Advanced Texts. Adv. 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. t [0, 1] operator x f (t, x) is monotone on RN . P. Drbek, J. Milota, Methods of Nonlinear Analysis. Repov, On some variational algebraic problems. Series in Nonlinear Analysis and its Applications. 233(11), 29852993 (2010) 24. From Example 2.5 it follows that J1 is C 1 as well. Therefore it is also coercive and satisfies the property (S). As for the continuity of G we argue using the generalized Krasnoselskii Theorem, see Theorem 2.12. 29, 341346 (1962) 39. Proof Recall that A1 is strictly monotone and continuous. S. Reich, Book review: geometry of banach spaces, duality mappings and nonlinear problems. Exercise 9.15 Assume that f : R R is continuous and nondecreasing. The application of Theorem 6.4 finishes the proof of the existence and the uniqueness of the solution. D.G. Uspekhi Mat. (9.23) 0 Proof We write Eq. By the same arguments it follows that G is bounded. Applying similar calculations and the fact operator B has the property (S), we see that if un u0 and if lim (un , un ) (un , u0 ), un u0 = B(un ) B(u0 ), un u0 = 0, n+ it follows that un u0 . 18 (Springer, New York, 2009) 37. G. Molica Bisci, D.D. The first part of condition (ii) follows from Lemma 9.4. W. Rudin, Principles of Mathematical Analysis, 2nd edn. M. Galewski, J. Smejda, On variational methods for nonlinear difference equations. Our partners will collect data and use cookies for ad personalization and measurement. Z. Denkowski, S. Migrski, N.S. Theorem 9.12 Assume that A9, A10 are satisfied. Bull. E.H. Zarantonello, Solving functional equations by contractive averaging, in Mathematical Research Center Technical Summary Report no. Bauschke, P.L. D. Idczak, A. Rogowski, On the Krasnoselskij theorema short proof and some generalization. V.D. A.D. Ioffe, V.M. t [0, 1]. 3rd edn. Minty, Monotone (nonlinear) operators in Hilbert space. Indeed, note by A10 that G (u) , u 0 for all u H01 (0, 1). Exercise 9.19 Let p > 2. We have: Lemma 9.6 Assume that A11 holds. 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). J. Comput. 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). 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. This provides that condition (iii) holds. 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) . 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. Comput. 1,p Exercise 9.13 Show that A2 is well defined, i.e. Appl. Exercise 9.16 Check whether assumption A8 can be replaced with the following: there exists a constant a1 < and a function b1 Lq (0, 1) such that (f (t, x) , x) a1 |x|p1 + b (t) |x| for all x RN and for a.e.
ISBN 978-83-66287-37-2 22. R.P. Radulescu, Variational and Nonvariational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Its Applications (Springer, Berlin, 2003) 43. 467, 1208 1232 (2018) 49. Proof Observe that T (u) , v = B (u) + G (u) , v for all u, v H01 (0, 1). W. Rudin, Functional analysis, in McGraw-Hill Series in Higher Mathematics (McGraw-Hill Book Co., New York, 1973) 54. 2 (Elsevier Scientific Publishing Company, Amsterdam, 1980) 19. R.I. Kacurovski, Monotone operators and convex functionals. We say that u H01 (0, 1) is a weak solution to (9.32) if 1 1 u(t) v (t) dt + 0 1 f (t, u(t))v (t) dt = 0 g (t) v (t) dt (9.35) 0 for all v H01 (0, 1). With the above Lemmas 9.4 and 9.5 we have all assumptions of Theorem 6.9 satisfied. 1364 (Springer, Berlin, 1993) 46. Copyright 2022 EBIN.PUB. Math. 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 have the following result: Theorem 9.11 Assume that conditions A5, A6, A8 are satisfied. J. Convex Anal. Basler Lehrbcher (Springer, Basel, 2013) 15. Hint: consult Example 3.7 in showing that T1 is strongly monotone. (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 . Proof By a direct calculation we see that (u, u) = T (u) for every u H01 (0, 1), so (i) holds. 0 0 Moreover, A1 is coercive and dmonotone with respect to (x) = x p1 . G. Kassay, V.D. Appl. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 2000) 3. Therefore we have the assertion of the lemma satisfied. G. Dinca, P. Jeblean, Some existence results for a class of nonlinear equations involving a duality mapping. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Marcel Dekker, New York, 1984) 25. D. Reem, S. Reich, Fixed points of polarity type operators. 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. 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] and all x R 0 d F (t, x) = f (t, x) for a.e. G.J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces. Proc. Applications to Differential Equations, 2nd edn. But this leads the strong continuity of A2 .
Then Dirichlet Problem (9.32) has exactly one solution u H01 (0, 1) H 2 (0, 1) . Then problem (9.26) has at least one nontrivial solution. Radulescu, Equilibrium Problems and Applications (Academic Press, Oxford, 2019) 33. 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. 10, 289300 (2021) 23. Appl. M. Galewski, On the application of monotonicity methods to the boundary value problems on the Sierpinski gasket. Am. Lemma 9.9 Under assumptions A11, A12 functional J is coercive over H01 (0, 1). I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976) 16. Hence we need one additional assumption: A13 for a.e. 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. Francu, Monotone operators: a survey directed to applications to differential equations. (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. Then functional J is sequentially weakly lower semicontinuous on H01 (0, 1). Soc. We may at last study problem corresponding to (1.1) with a nonlinear term as well.
Math. 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. Thus operator p is uniformly monotone. Translated from the Russian by Karol Makowski. G.J. R.E. M. Renardy, R.C. 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 . 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. 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. 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. Lecture Notes in Mathematics, vol. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/BNonlinear Monotone Operators (Springer, New York, 1990) Index A Adjoint operator, 64 Algebraic equation, 12 ArzelaAscoli Theorem, 31 B Banach Contraction Principle, 107 BanachSteinhaus Theorem, 62 Bounded bilinear form, 143 Brouwer Fixed Point Theorem, 9 BrowderMinty Theorem, 112 C Carathodory function, 37 Chain Rule, 42 Clarkson Inequality, 33 Condition (M), 116 (S), 75 (S)+ , 76 (S)0 , 76 (S)2 , 76 Continuous demicontinuous, 66 hemicontinuous, 66 Lipschitz, 66 radially, 66 strongly, 66 uniformly, 66 weakly, 66 Convexity criteria, 44 D Direct method of the calculus of variation, 51 Dirichlet Problem, 1 for the generalized plaplacian, 163 for the laplacian, 156 Duality mapping, 127 Duality mapping relative to a function, 134 du Bois-Reymond Lemma, 35 E Effective domain, 81 Ekeland Variational Principle, 152 Epigraph, 44 F Fenchel-Moreau Theorem, 89 Fenchel-Young dual, 85 Fenchel-Young Inequality, 86 Fermat Rule, 40 Finite Dimensional Existence Theorem, 10 Frchet derivative, 40 Fubini Theorem, 27 G Galerkin method, 139 Galerkin solution, 138 Gteaux derivative, 40 Gteaux variation, 43 Generalized BrowderMinty Theorem, 117 Generalized Krasnoselskii Theorem, 38 H Hahn-Banach Theorem, 125 Hlder Inequality, 23 Hyperplane closed, 84 supporting , 85 I Invertibility of a monotone operator, 114 Iteration method, 109 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/10.1007/978-3-030-75308-5 179 180 K Krasnoselskii Theorem, 39 L Lax-Milgram Theorem -nonlinear , 145 Lebesgue Dominated Convergence Theorem, 37 Lemma du Bois-Reymond Lemma, 35 fundamental in operator theory, 78 LerayLions Theorem, 121 M Mean Value Theorem, 46 Minty Lemma, 77 finite dimensional , 7 Monotone map, 6 N Nemytskii operator, 37 Normalized duality mapping, 126 O Operator bounded, 63 coercive, 69 d-monotone, 56 locally bounded, 62 monotone, 55 potential, 91 pseudomonotone, 118 Riesz, 17 strictly monotone, 55 strongly monotone , 56 uniformly monotone , 55 weakly coercive, 69 Index P Poincar Inequality, 31 Pseudomonotone operator, 118 R Riesz Representation in Lp , 22 Riesz Representation Theorem, 17 Ritz solution, 140 S Separation Theorem, 83 Sequentially weakly closed, 17 Sequential weak lower semicontinuity, 47 Sobolev Inequality, 31 Strictly convex space, 21 Strongly Monotone Principle -general version, 118 Strongly Monotone Principle in a Banach space, 133 Sufficient conditions for monotonicity, 60 Sufficient convexity condition, 46 T Theorem convexity and monotonicity, 59 existence result for a pseudomonotone operator, 120 sufficient condition for sequential weak lower semicontinuoity, 48 U Uniformly convex space, 21 W Weak convergence, 16 Weak derivative, 24 Weak sequential compactness of a ball, 18 Weierstrass Theorem, 47. (Harcourt/Academic Press, San Diego, 2001) 34.
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 . Repov, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (CRC Press/Taylor and Francis Group, Boca Raton, 2015) 48. Texts in Applied Mathematics, vol. 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. Rogers, An Introduction to Partial Differential Equations, 2nd edn. Exercise 9.18 Using Theorem 6.5 examine the existence of a weak solution to (9.30) for a < . 0 But this means that A1 is well defined. We start with lemma summarizing some obvious properties of operator T . 136 (Cambridge University, Cambridge, 2010) 35. Advances in Mechanics and Mathematics, vol. (N.S.) 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. Proposition 9.3 Assume that g Lq (0, 1) is fixed and that condition A5 holds. Tikhomirov, Theory of Extremal Problems (in Russian).
Then functional J is differentiable in the sense of Gteaux on H01 (0, 1). We finally prove that condition (iv) holds. 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.