In this paper, some new extensions of Schauder\u27s and Darbo\u27s fixed point theorems are given. As applications of the main results, the existence of global solutions for first-order nonlinear integro-differential equations of mixed type in a real Banach space is investigated

Bohner, Martin

Hao, Zhaocai

Wang, Junjun

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine Mathematics and Statistics Faculty Research & Creative Works Mathematics and Statistics 01 Jan 2019 Extensions of Schauder's and Darbo's Fixed Point Theorems Zhaocai Hao Martin Bohner Missouri University of Science and Technology, bohner@mst.edu Junjun Wang Follow this and additional works at: https://scholarsmine.mst.edu/math_stat_facwork  Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Z. Hao et al., "Extensions of Schauder's and Darbo's Fixed Point Theorems," Nonlinear Dynamics and Systems Theory, vol. 19, no. 3, pp. 396-404, InforMath Publishing Group, Jan 2019. This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Mathematics and Statistics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. Nonlinear Dynamics and Systems Theory, 19 (3) (2019) 396–404Extensions of Schauder’s andDarbo’s Fixed Point TheoremsZhaocai Hao 1,2, Martin Bohner 1∗ and Junjun Wang 21 Missouri S&T, Rolla, MO 65409, USA2 Qufu Normal University, Qufu 273165, Shandong, ChinaReceived: February 28, 2019; Revised: July 22, 2019Abstract: In this paper, some new extensions of Schauder’s and Darbo’s fixedpoint theorems are given. As applications of the main results, the existence of globalsolutions for first-order nonlinear integro-differential equations of mixed type in a realBanach space is investigated.Keywords: nonlinear integro-differential equation; Darbo fixed point theorem;Schauder fixed point theorem; Kuratowksi measure of noncompactness.Mathematics Subject Classification (2010): 34K30, 34L30, 45J05, 47G20,58J20.1 IntroductionIt is well known that the following two fixed points are very important.Theorem 1.1 (Schauder’s fixed point theorem) Let Ω be a nonempty, bounded,closed, and convex subset of a Banach space E. Then each continuous and compact mapT : Ω→ Ω has at least one fixed point in Ω.The Schauder fixed point theorem plays an important role in nonlinear analysis. In1955, Darbo [9] proved a fixed point property for set-contraction on a closed, boundedand convex subset of Banach spaces in terms of the measure of noncompactness, whichwas first defined by Kuratowski [17].∗ Corresponding author: mailto:bohner@mst.educ© 2019 InforMath Publishing Group/1562-8353 (print)/1813-7385 (online)/http://e-ndst.kiev.ua396NONLINEAR DYNAMICS AND SYSTEMS THEORY, 19 (3) (2019) 396–404 397Theorem 1.2 (Darbo’s fixed point theorem) Let Ω 6= ∅ be a bounded, closed,and convex subset of a Banach space E and let T : Ω → Ω be a continuous mapping.Assume that there exists a constant k ∈ [0, 1) such thatα(TX) ≤ kα(X) (1)for any nonempty subset X ⊂ Ω, where α is a measure of noncompactness defined in E.Then T has a fixed point in Ω.Darbo’s fixed point theorem is a significant extension of the Schauder fixed pointtheorem, and it also plays a key role in nonlinear analysis, especially in proving theexistence of solutions for many classes of nonlinear equations. Since then, some general-izations of Darbo’s fixed point theorem have appeared. For example, we refer the readerto [1–3,6, 13,23] and the references therein.Recently, the authors of [21] established the following new fixed point theorem, whichis a generalization of Darbo’s fixed point theorem.Theorem 1.3 (See [21, Lemma 2.4]) Let F be a closed and convex subset of areal Banach space E, A : F → F be a continuous operator, and A(F ) be bounded. Forany bounded subset B ⊂ F , putÃ1(B) = A(B) and Ãn+1(B) = A(co(Ãn(B))), n ∈ N.If there exist a constant 0 ≤ k < 1 and n0 ∈ N such that for any bounded subset B ⊂ F ,α(Ãn0(B)) ≤ kα(B), (2)then A has a fixed point in F .As an application of their result, the authors in [21] investigated the existence ofglobal solutions of the Volterra type integral equationu(t) = h(t) +∫ t0G(t, s)f(s, u(s), (Tu)(s), (Su)(s))ds, t ∈ J, (3)where J = [0, a], a > 0, f ∈ C(J × E × E × E,E),(Tu)(t) =∫ t0k(t, s)u(s)ds, (Su)(t) =∫ a0h(t, s)u(s)ds, t ∈ J,k ∈ C(D,R), h ∈ C(D0,R),D = {(t, s) ∈ R2 : 0 ≤ s ≤ t ≤ a}, D0 = {(t, s) ∈ R2 : 0 ≤ t, s ≤ a},and R denotes the set of real numbers. The main results of [21] extend and improve re-lated results in [11,12,20–22]. For other results concerning integro-differential equations,we refer to [4, 5, 7, 8, 10,14–16].Motivated by the above works, in this paper, we first establish a new fixed pointtheorem (Theorem 2.1), which is an extension of Schauder’s fixed point theorem. Then,by using this extended Schauder fixed point theorem, we get a new extension of Darbo’sfixed point theorem (Theorem 2.2). As an application of the new extended Darbo fixedpoint theorem, we obtain the existence of global solutions of (3). The existence result(Theorem 3.1) includes and extends and improves related results in [11,12,20–22].This paper is organized as follows. In Section 2, we present our main results, theextensions of Schauder’s and Darbo’s fixed point theorems. In Section 3, in order todemonstrate the applicability of our main results, we obtain the existence of globalsolutions of (3).398 Z. HAO, M. BOHNER AND J. WANG2 Main ResultsThroughout this paper, let C(J,E) denote the Banach space of all continuous mappingsu : J → E with norm ‖u‖c = maxt∈J ‖u(t)‖, while C1(J,E) denotes the Banach space ofall u ∈ C(J,E) such that u′ is continuous on J with norm ‖u‖c1 = max{‖u‖c , ‖u′‖c}. Letα denote the Kuratowski measure of noncompactness in E and C(J,E). Please, see [18]for more details on the Kuratowski measure of noncompactness. For any B ∈ C(J,E),t ∈ J , letB(t) = {u(t) : u ∈ B} ⊂ E,(TB)(t) ={∫ t0k(t, s)u(s)ds : u ∈ B},(SB)(t) ={∫ a0h(t, s)u(s)ds : u ∈ B}.For any R > 0, letTR = {x ∈ E : ‖x‖ ≤ R} and BR = {u ∈ C(J,E) : ‖u‖c ≤ R}.Lemma 2.1 (See [12]) If B ⊂ C(J,E) is bounded and equicontinuous, thenco(B) ⊂ C(J,E) is also bounded and equicontinuous.Lemma 2.2 (See [12]) If B ⊂ C(J,E) is bounded and equicontinuous, then α(B(t))is continuous on J andα(∫JB(s)ds)≤∫Jα(B(s))ds.Lemma 2.3 (See [19]) If f is bounded and uniformly continuous on J × TR ×TR × TR for all R > 0 and B ⊂ C(J,E) is bounded and equicontinuous, then{f(t, u(t), (Tu)(t), (Su)(t)) : u ∈ B} is bounded and equicontinuous in C(J,E).First, we give the extension of Schauder’s fixed point theorem.Theorem 2.1 Let D be a closed and convex subset of a real Banach space E. Supposethat the operator A : D → D is continuous. If there exists n0 ∈ N such that Ãn0−1(D)is bounded and α(Ãn0(D)) = 0, whereÃ0(D) = D and Ãn(D) = co(A(Ãn−1(D))), n ∈ N,then A has a fixed point in D.Proof. Since A(D) ⊂ D and D is a closed convex subset, we haveÃ1(D) = co(A(D)) ⊂ co(D) = D = Ã0(D).Hence,Ã2(D) = co(A(Ã1(D))) ⊂ co(A(D)) = Ã1(D).By the method of mathematical induction, we can deduce thatÃn(D) ⊂ Ãn−1(D), n ∈ N.NONLINEAR DYNAMICS AND SYSTEMS THEORY, 19 (3) (2019) 396–404 399Thus,A(Ãn0−1(D)) ⊂ co(A(Ãn0−1(D))) = Ãn0(D) ⊂ Ãn0−1(D).Consequently, A : Ãn0−1(D) → Ãn0−1(D) is continuous. Moreover, for any boundedsubset S ⊂ Ãn0−1(D), we getA(S) ⊂ A(Ãn0−1(D)) ⊂ Ãn0(D),and hence,α(A(S)) ≤ α(Ãn0(D)) = 0.Noting that Ãn0−1(D) is a closed bounded convex subset of E, we know from Schauder’sfixed point theorem that A has a fixed point in Ãn0−1(D) ⊂ D.Remark 2.1 The well-known Schauder fixed point theorem is the special case n0 = 1of Theorem 2.1.By using Theorem 2.1, we now present a new extension of Darbo’s fixed point theorem.Theorem 2.2 Let D be a closed and convex subset of a real Banach space E. Supposethat the operator A : D → D is continuous. For any bounded subset B ⊂ E, putÃ0(B) = B and Ãn(B) = A(co(Ãn−1(B))), n ∈ N. (4)If there exists n0 ∈ N such that Ãn0−1(D) is bounded and for any decreasing sequence ofsets {Bn} ⊂ D, n ∈ N,α(Ãn0( ∞⋂n=1Bn))= 0, (5)then A has a fixed point in D.Proof. LetB0 = D and Bn = co(Ãn0(Bn−1)), n ∈ N. (6)Then (6) and A : D → D imply thatB1 = co(Ãn0(B0)) ⊂ D = B0.Hence, Ãn0(B1) ⊂ Ãn0(B0). Therefore,B2 = co(Ãn0(B1)) ⊂ co(Ãn0(B0)) = B1.By the method of mathematical induction, we can proveBn ⊂ Bn−1, n ∈ N. (7)If we setB̂ =∞⋂n=0Bn, (8)where {Bn} is defined as in (6), then B̂ is a nonempty and convex subset in D. Hence,(5), (7), and (8) implyα(Ãn0(B̂)) = 0. (9)400 Z. HAO, M. BOHNER AND J. WANGSince Ãn0−1(D) is bounded, we get thatÃn0−1(B̂) is bounded. (10)Next, we shall proveA(B̂) ⊂ B̂. (11)In fact, from B1 ⊂ co(Ãn0−1(B0)), we haveA(B1) ⊂ A(co(Ãn0−1(B0))) = Ãn0(B0) ⊂ co(Ãn0(B0)) = B1.By the same method, we can prove A(Bn) ⊂ Bn, n ∈ N. Hence, we getA(B̂) =∞⋂n=0A(Bn) ⊂∞⋂n=0Bn = B̂.Then (11) holds. From (9), (10), (11), and Theorem 2.1, we deduce that A has a fixedpoint in B̂ ⊂ D.Remark 2.2 WhenBn ≡ D, n ∈ Nin Theorem 2.2, then Theorem 2.1 is obtained.Remark 2.3 Whenn0 = 1, Bn ≡ D, n ∈ Nin Theorem 2.2, then Theorem 1.2, i.e., Darbo’s fixed point theorem is obtained. SoTheorem 2.2 includes and extends Darbo’s fixed point theorem.Remark 2.4 Comparing with [21, Lemma 2.4], i.e., Theorem 1.3, the conclusionof Theorem 2.2 is the same. But the conditions are different. First, the assumptionthat Ãn0−1(D) is bounded is weaker than that A(D) is bounded in [21, Lemma 2.4].After that, we only need to consider the decreasing sequences without the boundedness{Bn} ⊂ D in (5), while [21, Lemma 2.4] needs to consider all bounded sets B ⊂ F in(2). Finally, (5) and (2) cannot be deduced from each other. Above all, Theorem 2.2 isa good supplement to the extension of Darbo’s fixed point theorem.3 ApplicationsNow, as an application of Theorem 2.2, we give an existence theorem for global solutionsof (3).Theorem 3.1 Let E be a real Banach space. Assume(H1) For any R > 0, f is bounded and uniformly continuous on J × TR × TR × TR, andlim supR→∞M(R)R<1aa0b, (12)wherea0 = max{1, ak0, ah0}, k0 = max{|k(t, s)| : (t, s) ∈ D},h0 = max{|h(t, s)| : (t, s) ∈ D0}, b = max{G(t, s) : (t, s) ∈ D},M(R) = sup{‖f(t, x, y, z)‖ : (t, x, y, z) ∈ J × TR × TR × TR, t ∈ J}.NONLINEAR DYNAMICS AND SYSTEMS THEORY, 19 (3) (2019) 396–404 401(H2) There exist nonnegative Lebesgue integrable functions Li ∈ L(J,R+) such that forany decreasing sequences of bounded sets {Din} ⊂ E, α(Din) → 0, n → ∞, i =1, 2, 3 and t ∈ J ,α(f(t,D1n, D2n, D3n)) ≤3∑i=1Li(t)α(Din). (13)Then (3) has at least one global solution in C1(J,E).Proof. First, we define an operator A : C(J,E)→ C(J,E) by(Au)(t) = h(t) +∫ t0G(t, s)f(s, u(s), (Tu)(s), (Su)(s))ds, u ∈ C(J,E). (14)Note that since u ∈ C1(J,E) is a solution of (3) if and only if u ∈ C(J,E) is a solutionof the integral equationu(t) = (Au)(t), t ∈ J,we only need to prove that A has a fixed point. Since f is uniformly continuous onJ × TR × TR × TR, we can easily see that A : C(J,E) → C(J,E) is continuous andbounded. On account of (12), there exist 0 < r < (aa0b)−1 and R0 > 0 such that forany R ≥ a0R0,M(R)R< r. (15)LetR∗ = max{R0, ‖h‖c (1− aa0b)−1 }. (16)Then, by using (14) and (15), it is not difficult to verify that A(BR∗) ⊂ C(J,E) isequicontinuous and bounded, and A : BR∗ → BR∗ is bounded and continuous. SetD = co(A(BR∗)). Then, from Lemma 2.1, we get that D ⊂ BR∗ is bounded andequicontinuous andA : D → D is continuous and bounded. (17)For any decreasing sequence of bounded sets {Bm} ⊂ D, m ∈ N, by (H1) and (14), wehave that A(Bm) is bounded and equicontinuous. Hence, from Lemma 2.1, Lemma 2.3,and (4), we get for any n ∈ N that Ãn(Bm) is bounded and equicontinuous on J , and soα(Ãn(Bm)) = maxt∈Jα((Ãn(Bm))(t)), m ∈ N.Next, we show that for any n0 ∈ N, we haveÃn0−1(D) is bounded, (18)and for any decreasing sequence of sets {Bm} ⊂ D, α(Bm)→ 0, m ∈ N,α(Ãn0( ∞⋂m=1Bm))= 0. (19)402 Z. HAO, M. BOHNER AND J. WANGIndeed, (18) follows from the fact that D ⊂ BR∗ is bounded. Furthermore, from (13),(14), and (15), we getα((Ã1(Bm))(t))= α(∫ t0G(t, s)f(s, (coBm)(s), (T (coBm))(s), (S(coBm))(s))ds)≤ b∫ t0[L1(s)α((coBm)(s)) + L2(s)α((T (coBm))(s)) + L3(s)α((S(coBm))(s))]ds≤ b∫ t0[L1(s)α((coBm)(s)) + L2(s)k0α((coBm)(s)) + L3(s)h0α((coBm)(s))]ds= b∫ t0[L1(s)α((Bm(s)) + L2(s)k0α(Bm(s)) + L3(s)h0α(Bm(s))]ds→ 0, m→∞.Assumeα((Ãk(Bm))(t))→ 0, m→∞, k ∈ N \ {1}.Thenα((Ãk+1(Bm))(t))= α(∫ t0G(t, s)f(s, (coÃk(Bm))(s), T (coÃk(Bm))(s), S(coÃk(Bm))(s))ds)≤ b∫ t0[L1(s)α((coÃk(Bm))(s)) + L2(s)α(T (coÃk(Bm))(s))+L3(s)α(S(coÃk(Bm)(s)))]ds≤ b∫ t0[L1(s)α((coÃk(Bm))(s)) + L2(s)k0α((coÃk(Bm))(s))+L3(s)h0α((coÃk(Bm))(s))]ds= b∫ t0[L1(s)α((Ãk(Bm))(s)) + L2(s)k0α((Ãk(Bm))(s))+L3(s)h0α((Ãk(Bm))(s))]ds→ 0, m→∞.Consequently,α(Ãn(Bm))→ 0, m→∞, n ∈ N,and so α(Ãn0(Bm))→ 0, m→∞. Thus, we haveα(Ãn0( ∞⋂m=1Bm))≤ α(Ãn0(Bm))→ 0, m→∞.Hence, (19) holds. It follows from Theorem 2.2, (17), (18), and (19) that A has a fixedpoint in D. Thus, (3) has at least one global solution in C1(J,E).Remark 3.1 The main result of [21], i.e., [21, Theorem 3.1] is as follows: Let E bea real Banach space. AssumeNONLINEAR DYNAMICS AND SYSTEMS THEORY, 19 (3) (2019) 396–404 403(H3) For any R > 0, f is bounded and uniformly continuous on J × TR × TR × TR, andlim supR→∞M(R)R<1aa0b,wherea0 = max{1, ak0, ah0}, b = max{G(t, s) : (t, s) ∈ D},M(R) = sup{‖f(t, x, y, z)‖ : (t, x, y, z) ∈ J × TR × TR × TR, t ∈ J}.(H4) There exist nonnegative Lebesgue integrable functions Li ∈ L(J,R+) such that forany bounded sets {Di} ⊂ E and t ∈ J ,α(f(t,D1, D2, D3)) ≤3∑i=1Li(t)α(Di). (20)Then (3) has at least one global solution in C1(J,E). Comparing [21, Theorem 3.1] withTheorem 3.1 above, we can see that the only difference is between (13) and (20). Forall bounded sets {Di} ⊂ E, (20) should hold. For only those bounded and decreasingsequences {Din} ⊂ E, α(Din)→ 0, i = 1, 2, 3, we need that (13) holds. So (13) is weakerthan (20). Moreover, (20) is a special case of (13) when Din ≡ Di, i = 1, 2, 3, n ∈ N.Thus, Theorem 3.1 includes and extends [21, Theorem 3.1], which extended and improvedthe main results of [11,12,20,22]. Consequently, Theorem 3.1 in this paper extends andimproves not only the main results of [21], but also related results of [11,12,20,22].AcknowledgementsThis research was supported financially by the NSFC (11571296, 11371221), the Fund ofthe Natural Science of Shandong Province (ZR2014AM034), and colleges and universi-ties of Shandong province science and technology plan projects (J13LI01), and Universityoutstanding scientific research innovation team of Shandong province (Modeling, opti-mization and control of complex systems) and Qufu Normal University Fund (XJ201126).References[1] Asadollah Aghajani and M. Aliaskari. 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Extensions of Schauder\u27s and Darbo\u27s Fixed Point Theorems

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Extensions of Schauder\u27s and Darbo\u27s Fixed Point Theorems

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