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Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 50 (1975), 143-149
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1975-0380527-7
MathSciNet review: 0380527
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Abstract: Let $ K$ be a bounded closed convex subset of a Banach space $ X$ with $ \operatorname{int} K \ne \emptyset $, and suppose $ K$ has the fixed point property with respect to nonexpansive self-mappings (i.e., mappings $ U:K \to K$ such that $ \vert\vert U(x) - U(y)\vert\vert \leq \vert\vert x - y\vert\vert,x,y \in K)$. Let $ T:K \to X$ be nonexpansive and satisfy

$\displaystyle \inf \{ \vert\vert x - T(x)\vert\vert:x \in {\text{ boundary }}K,T(x) \notin K\} > 0.$

It is shown that if in addition, either (i) $ T$ satisfies the Leray-Schauder boundary condition: there exists $ z \in \operatorname{int} K$ such that $ T(x) - z \ne \lambda (x - z)$ for all $ x \in {\text{ boundary }}K,\lambda < 1$, or (ii) $ \inf \{ \vert\vert x - T(x)\vert\vert:x \in K\} = 0$, is satisfied, then $ T$ has a fixed point in $ K$.

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  • [1] N. A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math. 43 (1972), 553-562. MR 0341459 (49:6210)
  • [2] M. S. Brodskiĭ and D. P. Mil'man, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948), 837-840. (Russian) MR 9, 448. MR 0024073 (9:448f)
  • [3] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad Sci. U.S.A. 54 (1965), 1041-1044. MR 32 #4574. MR 0187120 (32:4574)
  • [4] -, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665. MR 37 #5742. MR 0230179 (37:5742)
  • [5] R. E. Bruck, Jr., Private communication.
  • [6] M. G. Crandall, Differential equations on convex sets, J. Math. Soc. Japan 22 (1970), 443-455. MR 42 #3388. MR 0268491 (42:3388)
  • [7] J. A. Gatica and W. A. Kirk, A fixed point theorem for $ k$-set-contractions defined in a cone, Pacific J. Math. 53 (1974), 131-136. MR 0353071 (50:5557)
  • [8] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258. MR 32 #8129. MR 0190718 (32:8129)
  • [9] B. Halpern and G. Bergman, A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353-358. MR 36 #4397. MR 0221345 (36:4397)
  • [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 32 #6436. MR 0189009 (32:6436)
  • [11] -, Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings, Pacific J. Math. 38 (1971), 89-94. MR 46 #4290. MR 0305160 (46:4290)
  • [12] R. D. Nussbaum, The fixed point index for local condensing mappings, Ann. Mat. Pura Appl. (4) 89 (1971), 217-258. MR 47 #903. MR 0312341 (47:903)
  • [13] Z. Opial, Nonexpansive and monotone mappings in Banach spaces, Lecture Notes 67-1, Brown University, Providence, R. I., 1967.
  • [14] W. V. Petryshyn, Structure of the fixed point sets of $ k$-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312-328. MR 42 #8358. MR 0273480 (42:8358)
  • [15] -, Remarks on condensing and $ k$-set-contractive mappings, J. Math. Anal. Appl. 39 (1972), 717-741. MR 0328687 (48:7029)
  • [16] S. Reich, Remarks on fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 52 (1972), 690-697. MR 0331139 (48:9473)
  • [17] -, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973), 460-467. MR 0322609 (48:971)
  • [18] G. Vidossich, Nonexistence of periodic solutions of differential equations and applications to zeros of nonlinear operators (preprint).
  • [19] -, Applications of topology to analysis: On the topological properties of the set of fixed points of nonlinear operators, Confer. Sem. Mat. Univ. Bari 126 (1971), 1-62.
  • [20] J. R. L. Webb, A fixed point theorem and applications to functional equations in Banach spaces, Boll. Un. Mat. Ital. (4) 4 (1971), 775-778. MR 0377631 (51:13802)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0380527-7
Keywords: Nonexpansive mapping, fixed point theorem, Leray-Schauder boundary condition
Article copyright: © Copyright 1975 American Mathematical Society

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