An Alternative Way of Utilizing Fixed Point Theory
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
An Alternative Way of Utilizing Fixed Point Theory Dapeng DU1
Abstract The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations. As an example, a nonlocal ordinary differential equation is considered. The idea is to solve homogeneous equations in the linearization. One feature of this method is that it does not need the equation to have special structures, for instance, variational structures, maximum principle, etc. Roughly speaking, the existence comes from good properties of the suitably linearized equation. The idea may have wider application. Keywords Existence, Semi-linear equations, Fixed point theory, Homogeneous linearization 2000 MR Subject Classification 35A99
1 Introduction The method of studying multiple solutions using fixed point theory has long time history. For instance, Schauder linearization (see [1, p. 593]). Leray [6] also suggested the possible connection between multiple solution and fixed point theory. Comparing with the rich development of fixed point theory, the application in multiple solutions for equations without variational structures and maximum principle seems not very fruitful (see [1, p. 594]). In this paper, we give an idea which might help the application of this point theory. Roughly speaking, the idea is to solve homogeneous linear equations in the reduction from finding solution to fixed point problem. One thing about this idea is that somehow it is helpful to keep the structures of the original nonlinear equations. We use the following model to introduce the idea. Theorem 1.1 Assume that V is a continuous operator from Lp (0, 1) to L1 (0, 1), 1 ≤ p < ∞, and it holds that (1) V (u) ≥ 0, ∀u ∈ Lp (0, 1), (2) V (θu) = |θ|p V (u), θ ∈ R, (3) CkukLp ≥ kV (u)kL1 ≥ c0 kukLp , where c0 > 0. Then the problem ( u′′ − u + V (u)u = 0, x ∈ (0, 1) (1.1) u′ (0) = u′ (1) = 0, has non-trivial solution u ∈ W 2,1 (0, 1) and u > 0 for x ∈ (0, 1). Manuscript received April 22, 2019. Revised October 31, 2019. of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China. E-mail: [email protected]
1 Department
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D. P. Du
Theorem 1.2 Assume that V is a continuous operator from L1 (0, 1) to Lq (0, 1), 1 < q < 2, and it holds that (1) V (u) ≥ 0, ∀u ∈ Lq (0, 1), 1 (2) V (θu) = |θ| q V (u), θ ∈ R, (3) C0 kukL1 ≤ kV (u)kqLq ≤ CkukL1 . Then the problem ( u′′ − u + V (u)u = 0, x ∈ (0, 1) (1.2) u′ (0) = u′ (1) = 0, has non-trivial solution u ∈ H 2 (0, 1) and u > 0 for x ∈ (0, 1). Remark 1.1 The nonlinear potential is the generalization of the standard nonlinear term |u| . For specific nonlinear terms, the improvement of regularity of solution usually is standard and simple. These two results could be known. Our focus is on the method. R1 One example of potential is V (u)(x) = |u(x)| + | 0 k(x, y)u(y)dy|, k ∈ L1 . Then the potential V is non-local and satisfies the assumptions in Theorem 1.1, and the solution
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