@ Appl. Gen. Topol. 22, no. 2 (2021), 259-294doi:10.4995/agt.2021.13248 © AGT, UPV, 2021 On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs Svetlin Georgiev Georgiev a and Karima Mebarki b a Department of Differential Equations, Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria. (svetlingeorgiev1@gmail.com) b Laboratory of Applied Mathematics, Faculty of Exact Sciences,University of Bejaia, 06000 Bejaia, Algeria. (mebarqi karima@hotmail.fr, karima.mebarki@univ-bejaia.dz) Communicated by E. A. Sánchez-Pérez Abstract The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a k-set contraction obtained in [3, 6], to the case of the sum T + F , where T is a mapping such that (I − T ) is Lipschitz invert- ible and F is a k-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of non-negative solutions for two classes of differential equations, covering a class of first-order ordi- nary differential equations (ODEs for short) posed on the non-negative half-line as well as a class of partial differential equations (PDEs for short). 2010 MSC: 37C25; 58J20; 47J35. Keywords: positive solution; fixed point index; cone; sum of operators; ODEs; PDEs. 1. Preliminaries Many problems in science lead to nonlinear equations T x + Fx = x posed in some closed convex subset of a Banach space. In particular, ordinary, frac- tional, partial differential equations and integral equations can be formulated Received 10 March 2020 – Accepted 09 April 2021 http://dx.doi.org/10.4995/agt.2021.13248 S. G. Georgiev and K. Mebarki like these abstract equations. It is the reason for which it becomes desirable to develop fixed point theorems for such equations. When T is compact and F is a contraction there are many classical tools to deal with such problems (see [2], [5], [9], [11] and references therein). The main aim of this paper is to give some recent results for existence of fixed points for some operators that are of the form T + F , where T is an expansive operator and F is a k-set contraction. The positivity of solutions of nonlinear equations, especially ordinary, partial differential equations, and integral equations is a very important issue in ap- plications, where a positive solution may represent a density, a temperature, a velocity, etc. In this paper we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a k-set contraction, obtained in [1, 3, 4, 8, 6, 7], to the case when T is a mapping such that (I − T ) is Lipschitz invertible and F is a k-set contraction. We illustrate some of our theoretical results. More precisely, we study the existence of non-negative solutions for the following IVP x′ = f(t, x), t > 0, x(0) = x0, where x0 ∈ R is a given constant, f : [0, ∞) × R → R is a continuous function satisfying a general polynomial growth condition. Moreover, we consider an application for an IVP subject to Burgers-Fisher equation: ut − uxx + α(t)uux = β(t)u(1 − u), t > 0, x ≥ 0, u(0, x) = u0(x), x ≥ 0, where u0 ∈ C2([0, ∞)) and α, β ∈ C([0, ∞)) with α < 0, β ≥ 0 on [0, ∞). The paper is organized as follows. In the next section, we give some auxiliary results. In sections 3 and 4, we will present our contribution in fixed point index theory for the sum of two operators of the form T + F , where T is a mapping such that (I − T ) is Lipschitz invertible with constant γ > 0 and F is a k-set contraction when 0 ≤ k < γ−1. We will consider separately two cases: firstly the computation of fixed point index on cones is treated in Section 3. Then in Section 4, we will discuss the computation of fixed point index on translates of cones. Applications are given in sections 4 and 5. 2. Auxiliary results Let X be a linear normed space and I be the identity map of X. The following Lemmas give sufficient conditions for I − T to be Lipschitz invertible. Lemma 2.1 ([12, Lemma 2.1]). Let (X, ‖.‖) be a normed linear space, D ⊂ X. If a mapping T : D → X is expansive with a constant h > 1, then the mapping © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 260 Fixed point index theory for the sum of operators I − T : D → (I − T )(D) is invertible and ‖(I − T )−1x − (I − T )−1y‖ ≤ (h − 1)−1‖x − y‖ for all x, y ∈ (I − T )(D). Lemma 2.2 ([13, Lemma 2.3]). Let (E, ‖.‖) be a Banach space and T : E → E be Lipschitzian map with constant β > 0. Assume that for each z ∈ E, the map Tz : E → E defined by Tzx = T x + z satisfies that T pz is expansive and onto for some p ∈ N. Then (I − T ) maps E onto E, the inverse of I − T : E → E exists, and ‖(I − T )−1x − (I − T )−1y‖ ≤ γp‖x − y‖ for all x, y ∈ E, where γp = βp − 1 (β − 1)(lip(T p) − 1) · Lemma 2.3 ([13, Lemma 2.5]). Let (X, ‖.‖) be a linear normed space, M ⊂ X. Assume that the mapping T : M → X is contractive with a constant k < 1, then the inverse of I − T : M → (I − T )(M) exist, and ‖(I − T )−1x − (I − T )−1y‖ ≤ (1 − k)−1‖x − y‖ for all x, y ∈ (I − T )(M). Lemma 2.4 ([13, Lemma 2.6]). Let (E, ‖.‖) be a Banach space and T : E → E be Lipschitzian map with constant β ≥ 0. Assume that for each z ∈ E, the map Tz : E → E defined by Tzx = T x + z satisfies that T pz is contractive for some p ∈ N. Then (I − T ) maps E onto E, the inverse of I − T : E → E exists, and ‖(I − T )−1x − (I − T )−1y‖ ≤ ρp‖x − y‖ for all x, y ∈ E, where ρp =      p 1−Lip(T p) , if β = 1; 1 1−β , if β < 1; β p −1 (β−1)(1−Lip(T p)) , if β > 1. 3. Fixed point index on cones In all what follows, P will refer to a cone in a Banach space (E, ‖.‖), Ω is a subset of P, and U is a bounded open subset of P. For r > 0 define the conical shell Pr = P ⋂ {x ∈ E : ‖x‖ < r}. Assume that T : Ω → E is a mapping such that (I−T ) is Lipschitz invertible with constant γ > 0 and F : U → E is a k-set contraction. Suppose that (3.1) 0 ≤ k < γ−1, (3.2) F(U) ⊂ (I − T )(Ω), and (3.3) x 6= T x + Fx, for all x ∈ ∂U ⋂ Ω. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 261 S. G. Georgiev and K. Mebarki Then x 6= (I − T )−1Fx, for all x ∈ ∂U and the mapping (I − T )−1F : U → P is a strict γk-set contraction. Indeed, (I − T )−1F is continuous and bounded; and for any bounded set B in U, we have α(((I − T )−1F)(B)) ≤ γ α(F(B)) ≤ γkα(B). The fixed point index i ((I − T )−1F, U, P) is so well defined. Thus we put (3.4) i∗ (T + F, U ⋂ Ω, P) = i ((I − T )−1F, U, P). Proposition 3.1. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and tF(U) ⊂ (I − T )(Ω) for all t ∈ [0, 1]. If (I − T )−10 ∈ U, and (3.5) x − T x 6= λFx for all x ∈ ∂U ⋂ Ω and 0 ≤ λ ≤ 1, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. Proof. Consider the homotopic deformation H : [0, 1] × U → P defined by H(t, x) = (I − T )−1tFx. The operator H is continuous and uniformly continuous in t for each x. More- over, H(t, .) is a strict k-set contraction for each t and the mapping H(t, .) has no fixed point on ∂U. Otherwise, there would exist some x0 ∈ ∂U ⋂ Ω and t0 ∈ [0, 1] such that x0 − T x0 = t0Fx0, which contradicts our assumption. From the invariance under homotopy and the normalization property of the index fixed point, we deduce that i∗ ((I − T )−1F, U, P) = i∗ ((I − T )−10, U, P) = 1. Consequently, from (3.4), we deduce that i∗ (T + F, U ⋂ Ω, P) = 1, which completes the proof. � As a consequence of Proposition 3.1 , we have the two following results. Corollary 3.2. Assume that the mapping T : Ω ⊂ P → E be such that (I −T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and tF(U) ⊂ (I − T )(Ω) for all t ∈ [0, 1]. If (I − T )−10 ∈ U, and ‖Fx‖ ≤ ‖x − T x‖ and T x + Fx 6= x for all x ∈ ∂U ⋂ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. Proof. It is sufficient to prove that Assumption (3.5) is satisfied. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 262 Fixed point index theory for the sum of operators Corollary 3.3. Assume that the mapping T : Ω ⊂ P → E be such that (I −T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and tF(U) ⊂ (I − T )(Ω) for all t ∈ [0, 1]. If (I − T )−10 ∈ U, Fx ∈ P for all x ∈ ∂U ⋂ Ω, and Fx � x − T x for all x ∈ ∂U ⋂ Ω, then the fixed point index i∗ (T + F, ⋂ Ω, P) = 1. Proof. It is easy to see that Assumption (3.5) is satisfied. � Proposition 3.4. Let U be a bounded open subset of P with 0 ∈ U. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If Fx 6= (I − T )(λx) for all x ∈ ∂U, λ ≥ 1 and λx ∈ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. Proof. The mapping (I − T )−1F : U → P is a strict γk-set contraction and it is readily seen that the following condition of Leray-Schauder type is satisfied (I − T )−1Fx 6= λx, for all x ∈ ∂U and λ ≥ 1. In fact, if there exist x0 ∈ ∂U and λ0 ≥ 1 such that (I − T )−1Fx0 = λ0x0. Then Fx0 = (I − T )(λ0x0), which contradicts our assumption. The claim then follows from (3.4) and [8, Theorem 1.3.7]. � Proposition 3.5. Let U be a bounded open subset of P with 0 ∈ U ⋂ Ω. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If (3.6) γ‖Fx + T 0‖ ≤ ‖x‖ and T x + Fx 6= x for all x ∈ ∂U ⋂ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. Proof. The mapping (I − T )−1F : U → P is a strict γk-set contraction. (I − T ) being Lipschitz invertible with constant γ > 0, for each x ∈ U (3.7) ‖(I − T )−1Fx‖ = ‖(I − T )−1Fx − (I − T )−1(I − T )0‖ ≤ γ‖Fx + T 0‖. Therefor, from (3.7) and Assumption (3.6), we conclude that for all x ∈ ∂U, ‖(I − T )−1Fx‖ ≤ γ‖Fx + T 0‖ ≤ ‖x‖. Our claim then follows from (3.4) and [8, Theorem 1.3.7]. � The following result is as straightforward consequence of Proposition [8, Corollary 1.3.1]. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 263 S. G. Georgiev and K. Mebarki Proposition 3.6. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If further (I − T )−1F(U) ⊂ U, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. As a particular case, we obtain Corollary 3.7. Assume that the mapping T : Ω ⊂ P → E be such that (I −T ) is Lipschitz invertible with constant γ > 0, F : Pr → E is a k-set contraction with 0 ≤ k < γ−1, and F(Pr) ⊂ (I − T )(Ω). If 0 ∈ Ω and (3.8) γ‖Fx + T 0‖ < r, for all x ∈ Pr, then the fixed point index i∗ (T + F, Pr ⋂ Ω, P) = 1. Proof. From (3.7) and Assumption (3.8), for any x ∈ Pr, we conclude that ‖(I − T )−1Fx‖ ≤ γ‖Fx + T 0‖ < r, which implies that (I − T )−1F(Pr) ⊂ Pr. � Taking r > γ 1−γ ‖T 0‖, we get Corollary 3.8. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant 0 < γ < 1, F : Pr → E is a k-set contraction with 0 ≤ k < γ−1, and F(Pr) ⊂ (I − T )(Ω). If 0 ∈ Ω and (3.9) ‖Fx‖ ≤ ‖x‖, for all x ∈ Pr, then T + F has at least one fixed point in Pr ⋂ Ω. Proposition 3.9. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If there exists u0 ∈ P∗ such that (3.10) Fx 6= (I − T )(x − λu0), for all λ ≥ 0 and x ∈ ∂U ⋂ (Ω + λu0), then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 0. Proof. The mapping (I − T )−1F : U → P is a strict γk-set contraction and for some u0 ∈ P∗ this operator satisfies x − (I − T )−1Fx 6= λu0, ∀ x ∈ ∂U, ∀ λ ≥ 0. By (3.4) and [8, Theorem 1.3.8], we deduce that i∗ (T + F, U ⋂ Ω, P) = i ((I − T )−1F, U, P) = 0. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 264 Fixed point index theory for the sum of operators Proposition 3.10. Assume that the mapping T : Ω ⊂ P → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). Suppose further that there exists u0 ∈ P∗ such that T (x − λu0) ∈ P, for all λ ≥ 0 and x ∈ ∂U ⋂ (Ω + λu0), and one of the following conditions holds: (a) Fx x, ∀ x ∈ ∂U. (b) Fx ∈ P, ‖Fx‖ > N‖x‖, ∀ x ∈ ∂U, and the cone P is normal with constant N. Then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 0. Proof. We show that conditions (a) or (b) imply that Fx 6= (I − T )(x − λu0), for all λ ≥ 0 and x ∈ ∂U ⋂ (Ω + λu0). On the contrary, assume the existence of λ0 ≥ 0 and x0 ∈ ∂U ⋂ (Ω + λ0u0) such that Fx0 = (I − T )(x0 − λ0u0). Then x0 − Fx0 = T (x0 − λ0u0) + λ0u0 ∈ P. If condition (a) holds, then a contradiction is achieved. Otherwise, we deduce that Fx0 ≤ x0. Since P is normal, we deduce that ‖Fx0‖ ≤ N‖x0‖, contradicting condition (b) and ending the proof of our Proposition. � 4. Fixed point index on translates of cones In this section, let E be a Banach space, P (P 6= {0}) be a cone in it. Given θ ∈ E, we consider the translate of P, namely K = P + θ = {x + θ, x ∈ P}. Then K is a closed convex of E, so it is a retract of E. Let Ω be any subset of K and U be a bounded open of K such that U ⋂ Ω 6= ∅. We denote by U and ∂U the closure and the boundary of U relative to K. The fixed point index i∗ (T + F, U ⋂ Ω, K) defined by (4.1) i∗ (T + F, U ⋂ Ω, K) = i ((I − T )−1F, U, K). is well defined whenever T : Ω → E is a mapping such that (I − T ) is Lipschitz invertible with constant γ > 0 and F : U → E is a k-set contraction, 0 ≤ k < γ−1 and F(U) ⊂ (I − T )(Ω). Proposition 4.1. Let U be a bounded open subset of K with θ ∈ U. Assume that the mapping T : Ω ⊂ K → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If (4.2) Fx 6= (I−T )(λx+(1−λ)θ) for all x ∈ ∂U, λ ≥ 1 and λx+(1−λ)θ ∈ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 265 S. G. Georgiev and K. Mebarki Proof. Define the homotopic deformation H : [0, 1] × U → K by H(t, x) = t(I − T )−1Fx + (1 − t)θ. Then, the operator H is continuous and uniformly continuous in t for each x, and the mapping H(t, .) is a strict γk-set contraction for each t. Moreover, H(t, .) has no fixed point on ∂U. Otherwise, there would exist some x0 ∈ ∂U and t0 ∈ [0, 1] such that 1t0 x0 + (1 − 1 t0 )θ ∈ Ω for t0 6= 0, and t0(I − T )−1Fx0 + (1 − t0)θ = x0. We may distinguish between two cases: (i) If t0 = 0, then x0 = θ, which is a contradiction. (ii) If t0 ∈ (0, 1], then Fx0 = (I − T )( 1t0 x0 + (1 − 1 t0 )θ), which contradicts our assumption. The properties of invariance by homotopy and normalization of the fixed point index guarantee that i ((I − T )−1F, U, K) = i (θ, U, K). Consequently, by (4.1), we deduce that i∗ (T + F, U ⋂ Ω, K) = 1. � Proposition 4.2. Let U be a bounded open subset of K with θ ∈ U. Assume that the mapping T : Ω ⊂ K → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If (4.3) ‖Fx − T θ − θ‖ ≤ ‖x − θ‖ and T x + Fx 6= x, for all x ∈ ∂U ⋂ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, P) = 1. Proof. The mapping (I − T )−1F : U → P is a strict γk-set contraction. Since (I − T ) is Lipschitz invertible with constant γ > 0, for each x ∈ U (4.4) ‖(I − T )−1Fx − θ‖ = ‖(I − T )−1Fx − (I − T )−1(I − T )θ‖ ≤ γ‖Fx + T θ − θ‖. Therefor, from (4.4) and Assumption (4.3), we conclude that for all x ∈ ∂U, ‖(I − T )−1Fx − θ‖ ≤ γ‖Fx + T θ − θ‖ ≤ ‖x − θ‖, which implies the condition (4.5) in Proposition 4.1. This completes the proof. � Remark 4.3. Propositions 4.1,4.2 can be proven directly by appealing to [4, proposition 2.2], and [4, Corollary 2.2], respectively. Proposition 4.4. Let U be a bounded open subset of K. Assume that the mapping T : Ω ⊂ K → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and ( tF(U) + (1 − t)θ ) ⊂ (I − T )(Ω) for all t ∈ [0, 1]. If (I − T )−1θ ∈ U, and (4.5) x − T x 6= λFx + (1 − λ)θ for all x ∈ ∂U ⋂ Ω and 0 ≤ λ ≤ 1, then the fixed point index i∗ (T + F, U ⋂ Ω, K) = 1. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 266 Fixed point index theory for the sum of operators Proof. Define the homotopic deformation H : [0, 1] × U → E by H(t, x) = tFx + (1 − t)θ. Then, the operator H is continuous and uniformly continuous in t for each x, and the mapping H(t, .) is a k-set contraction for each t. Moreover, T + H(t, .) has no fixed point on ∂U ⋂ Ω. Otherwise, there would exist some x0 ∈ ∂U ⋂ Ω and t0 ∈ [0, 1] such that T x0 + t0Fx0 + (1 − t0)θ = x0, then x0−T x0 = t0Fx0+(1−t0)θ, leading to a contradiction with the hypothesis. By (4.1), property (c) in [3, Theorem 2.3] and the normalization property of the fixed point index, we conclude that i∗ (T + F, U ⋂ Ω, K) = i∗ (T + θ, Kr ⋂ Ω, K) = ((I − T )−1θ, U ⋂ Ω, K) = 1. � Corollary 4.5. Let U be a bounded open subset of K. Assume that the mapping T : Ω ⊂ K → E be such that (I −T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and ( tF(U) + (1 − t)θ ) ⊂ (I − T )(Ω) for all t ∈ [0, 1]. If (I − T )−1θ ∈ U, Fx ∈ K for all x ∈ Ω ⋂ ∂U, and (4.6) Fx � x − T x for all x ∈ ∂U ⋂ Ω, then the fixed point index i∗ (T + F, U ⋂ Ω, K) = 1. Proof. It is easy to see that Assumption (4.5) is satisfied. Otherwise, there exist some x0 ∈ ∂U ⋂ Ω and 0 ≤ λ0 ≤ 1 such that x0 − T x0 = λ0Fx0 + (1 − λ0)θ. Then Fx0 − x0 + T x0 = (1 − λ0)(Fx0 − θ) ∈ P, which leads us to a contradiction with (4.6). � Proposition 4.6. Let U be a bounded open subset of K. Assume that the mapping T : Ω ⊂ K → E be such that (I − T ) is Lipschitz invertible with constant γ > 0, F : U → E is a k-set contraction with 0 ≤ k < γ−1, and F(U) ⊂ (I − T )(Ω). If there exists u0 ∈ P∗ such that (4.7) Fx 6= (I − T )(x − λu0), for all λ ≥ 0 and x ∈ ∂U ⋂ (Ω + λu0), then the fixed point index i∗ (T + F, U ⋂ Ω, K) = 0. Proof. The mapping (I − T )−1F : U → K is a strict γk-set contraction. Suppose that i∗ (T + F, U ⋂ Ω, K) 6= 0. Then, i ((I − T )−1F, U, P) 6= 0. For each r > 0, define the homotopy: H(t, x) = (I − T )−1Fx + tru0, for x ∈ U and t ∈ [0, 1]. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 267 S. G. Georgiev and K. Mebarki The operator H is continuous and uniformly continuous in t for each x. More- over, H(t, .) is a strict k-set contraction for each t and H([0, 1] × U) = (I − T )−1F(U) + tru0 ⊂ K. We check that H(t, x) 6= x, for all (t, x) ∈ [0, 1]×∂U. If H(t0, x0) = x0 for some (t0, x0) ∈ [0, 1] × ∂U, then x0 − t0ru0 = (I − T )−1Fx0, and so x0 − t0ru0 ∈ Ω. Hence (I − T )(x0 − t0ru0) = Fx0, for x0 ∈ ∂U ⋂ (Ω + t0ru0), contradicting Assumption (4.7). By homotopy invariance property of the fixed point index, we deduce that i ((I − T )−1F + ru0, U ⋂ Ω, P) = i ((I − T )−1F, U, P) 6= 0. Thus the existence property of the fixed point index, for each r > 0, there exists xr ∈ U such that (4.8) xr − (I − T )−1 Fxr = ru0. Letting r → +∞ in (4.8), the left-hand side of (4.8) is bounded while the right-hand side is not, which is a contradiction. Therefore i∗ (T + F, U ⋂ Ω, P) = 0, which completes the proof. � 5. Applications to ODE In this section we investigate the IVP (5.1) x′ = f(t, x), t > 0, x(0) = x0, where x0 ∈ R is a given constant, f : [0, ∞) × R → R is a given function. Let l ∈ N and x0, s, r, Aj, j ∈ {0, 1, . . . , l}, are positive constants such that (H1): x0 + l ∑ j=0 ( r 2 )j Aj < r 2 , (H2): f ∈ C([0, ∞) × R) and 0 ≤ f(y, x) ≤ l ∑ j=0 aj(y)|x|j, y ∈ [0, ∞), x ∈ R, where aj ∈ C([0, ∞)), aj ≥ 0 on [0, ∞) and ∫ ∞ 0 aj(y)dy ≤ Aj, j ∈ {0, 1, . . . , l}. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 268 Fixed point index theory for the sum of operators Theorem 5.1. Assume that (H1)-(H2) hold. Then the IVP (5.1) has a solu- tion x ∈ C1([0, ∞)) such that 0 ≤ x(t) < r 2 , t ∈ [0, ∞). Proof. Case 1.: Let t ∈ [0, 1]. Consider the IVP (5.2) x′ = f(t, x), t ∈ (0, 1], x(0) = x0. Take ǫ > 0 arbitrarily. Let E1 = C([0, 1]) be endowed with the maxi- mum norm and P1 = {x ∈ E1 : x(t) ≥ 0, t ∈ [0, 1]}, Ω1 = P1r = {x ∈ P1 : ‖x‖ < r} , U1 = P1 r 2 = { x ∈ P1 : ‖x‖ < r 2 } . For x ∈ E1, define the operators T1x(t) = (1 + ǫ)x(t), F1x(t) = −ǫ ( x0 + ∫ t 0 f(y, x(y))dy ) , t ∈ [0, 1]. Note that for any fixed point x ∈ E1 of the operator T1 + F1 we have that x ∈ C1([0, 1]) and it is a solution of the IVP (5.2). (1) For x, y ∈ E1, we have ‖(I − T1)−1x − (I − T1)−1y‖ = 1 ǫ ‖x − y‖, i.e., (I − T1) : E1 → E1 is Lipschitz invertible with constant 1ǫ . (2) For x ∈ U1 and t ∈ [0, 1], we have |F1x(t)| = ǫ ( x0 + ∫ t 0 f(y, x(y))dy ) ≤ ǫ  x0 + ∫ t 0 l ∑ j=0 aj(y)(x(y)) jdy   ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j ∫ t 0 aj(y)dy   ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j Aj   © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 269 S. G. Georgiev and K. Mebarki and |(F1x)′(t)| = ǫ f(t, x(t)) ≤ ǫ l ∑ j=0 aj(y)(x(y)) j ≤ ǫ l ∑ j=0 ( r 2 )j aj(y) ≤ ǫ l ∑ j=0 ( r 2 )j Bj Thus, ‖F1x‖ ≤ ǫ  x0 + l ∑ j=0 ( r 2 )j Aj   and ‖(F1x)′‖ ≤ ǫ l ∑ j=0 ( r 2 )j Bj. Hence, using the Arzela-Ascoli theorem, we conclude that F1 : U1 → E is a completely continuous mapping. Therefore F1 : U1 → E is a 0-set contraction. (3) Let λ ∈ [0, 1] and x ∈ U1 be arbitrarily chosen. Then z(t) = λ ( x0 + ∫ t 0 f(y, x(y))ds ) ∈ E1 and z(t) ≤ λ ( x0 + ∫ ∞ 0 f(y, x(y))dy ) ≤ λ  x0 + l ∑ j=0 ∫ ∞ 0 aj(y)(x(y)) j dy   ≤ λ  x0 + l ∑ j=0 ( r 2 )j Aj   < λ r 2 ≤ r 2 , t ∈ [0, 1], © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 270 Fixed point index theory for the sum of operators i.e., z ∈ Ω1. Next, λF1x(t) = −λǫ ( x0 + ∫ t 0 f(y, x(y))dy ) = −ǫz(t) = (I − T1)z(t), t ∈ [0, 1]. Thus, λF1(U1) ⊂ (I − T1)(Ω1). (4) Note that (I − T1)−10 = 0 ∈ U1. (5) Assume that there are x ∈ ∂U1 ⋂ Ω1 and λ ∈ [0, 1] such that x − T1x = λF1x. If λ = 0, then 0 = x − T1x = −ǫx on [0, 1], whereupon x(t) = 0, t ∈ [0, 1]. This is a contradiction because x ∈ ∂U1. Therefore λ ∈ (0, 1]. Let t1 ∈ [0, 1] be such that x(t1) = r2. Then (I − T1)x(t1) = −ǫx(t1) = −ǫ r 2 = −λǫ ( x0 + ∫ t1 0 f(y, x(y))dy ) , © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 271 S. G. Georgiev and K. Mebarki whereupon r 2 = λ ( x0 + ∫ t1 0 f(y, x(y))dy ) ≤ λ ( x0 + ∫ ∞ 0 f(y, x(y))dy ) ≤ λ  x0 + l ∑ j=0 ∫ ∞ 0 aj(y)(x(y)) jdy   ≤ λ  x0 + l ∑ j=0 Aj ( r 2 )j   < λ r 2 ≤ r 2 , i.e., r 2 < r 2 , which is a contradiction. By 1, 2, 3, 4, 5 and Proposition 3.1, it follows that the operator T1 +F1 has a fixed point in U1. Denote it by x1. We have 0 ≤ x1(t) < r 2 , t ∈ [0, 1], and x1 ∈ C1([0, 1]) is a solution of the IVP (5.2). Case 2.: Let t ∈ [1, 2]. Consider the IVP (5.3) x′ = f(t, x), t ∈ (1, 2], x(1) = x1(1). Take ǫ > 0 arbitrarily. Let E2 = C([1, 2]) be endowed with the maxi- mum norm and P2 = {x ∈ E2 : x(t) ≥ 0, t ∈ [1, 2]}, Ω2 = P2r = {x ∈ P2 : ‖x‖ < r} , U2 = P2 r 2 = { x ∈ P2 : ‖x‖ < r 2 } . For x ∈ E2 define the operators T2x(t) = (1 + ǫ)x(t), F2x(t) = −ǫ ( x1(1) + ∫ t 1 f(s, x(s))ds ) , t ∈ [1, 2]. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 272 Fixed point index theory for the sum of operators Note that for x ∈ U2, we have x1(1) + ∫ t 1 f(s, x(s))ds = x0 + ∫ t 0 f(y, x(y))dy ≤ x0 + ∫ ∞ 0 f(y, x(y))dy ≤ x0 + l ∑ j=0 aj(y)(x(y)) j dy ≤ x0 + l ∑ j=0 Ajr j < r 2 , t ∈ [1, 2]. As in Case 1 we prove that the operator T2 + F2 has a fixed point x2 ∈ U2. We have that 0 ≤ x2(t) < r 2 , t ∈ [1, 2], x2 ∈ C1([1, 2]). Note that x1(1) = x2(1), x′1(1) = f(1, x1(1)) = f(1, x2(1)) = x′2(1). Thus, x(t) =    x1(t) t ∈ [0, 1] x2(t) t ∈ [1, 2] is a solution to the IVP x′ = f(t, x), t ∈ (0, 2], x(0) = x0. Case 3.: Consider the IVP x′ = f(t, x), t ∈ (2, 3], x(2) = x2(2). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 273 S. G. Georgiev and K. Mebarki And so on, the function x(t) =                            x1(t) t ∈ [0, 1] x2(t) t ∈ [1, 2] x3(t) t ∈ [2, 3] x4(t) t ∈ [3, 4] . . . is a solution to the IVP (5.1). This completes the proof. � 6. Applications to PDE In this section we consider the IVP for Burgers-Fisher equation (6.1) ut − uxx + α(t)uux = β(t)u(1 − u), t > 0, x ≥ 0, (6.2) u(0, x) = u0(x), x ≥ 0, where (A1): u0 ∈ C2([0, ∞)), r1 ≥ u0 ≥ r12 on [0, ∞), where r1 ∈ ( 0, 1 2 ) is a given constant, (A2): α, β ∈ C([0, ∞)), α < 0, β ≥ 0 on [0, ∞) , A ∈ (0, 1) is a constant and g is a positive continuous function on [0, ∞) × [0, ∞) such that 1 − (1 + 2r1)A > 0, ( 4 + 3 2 r1 ) A < 1 2 , and 120 ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g(t1, x1) ( 1 + ∫ t1 0 (β(t2) − α(t2))dt2 ) dx1dt1 ≤ A, t ≥ 0, x ≥ 0. Let E = C1([0, ∞), C2([0, ∞))) be endowed with the norm ‖u‖ = { sup (t,x)∈[0,∞)×[0,∞) |u(t, x)|, sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂ ∂t u(t, x) ∣ ∣ ∣ ∣ , sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂ ∂x u(t, x) ∣ ∣ ∣ ∣ , sup (t,x)∈[0,∞)×[0,∞) ∣ ∣ ∣ ∣ ∂2 ∂x2 u(t, x) ∣ ∣ ∣ ∣ } , provided it exists. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 274 Fixed point index theory for the sum of operators Lemma 6.1. Suppose (A1) and (A2). If a function u ∈ E is a solution of the integral equation 0 = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×u(t2, x2)(1 − u(t2, x2))dx2dt2dx1dt1 − 1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 u(t2, x1)dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×(u0(x2) − u(t1, x2))dx2dx1dt1, (t, x) ∈ [0, ∞) × [0, ∞), then it is a solution to the IVP (6.1)-(6.2). Proof. We differentiate the considered integral equation five times in t and five times in x and using that g > 0 on [0, ∞) × [0, ∞), we get 0 = g(t, x) ∫ t 0 ∫ x 0 ∫ x1 0 β(t1)u(t1, x2)(1 − u(t1, x2))dx2dx1dt1 − 1 2 g(t, x) ∫ t 0 ∫ x 0 α(t1)(u(t1, x1)) 2 dx1dt1 +g(t, x) ∫ t 0 u(t1, x)dt1 +g(t, x) ∫ x 0 ∫ x1 0 (u0(x2) − u(t1, x2))dx2dx1, (t, x) ∈ [0, ∞) × [0, ∞), © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 275 S. G. Georgiev and K. Mebarki whereupon 0 = ∫ t 0 ∫ x 0 ∫ x1 0 β(t1)u(t1, x2)(1 − u(t1, x2))dx2dx1dt1 −1 2 ∫ t 0 ∫ x 0 α(t1)(u(t1, x1)) 2dx1dt1 + ∫ t 0 u(t1, x)dt1 + ∫ x 0 ∫ x1 0 (u0(x2) − u(t1, x2))dx2dx1, (t, x) ∈ [0, ∞) × [0, ∞). The last equation we differentiate twice in x and we get (6.3) 0 = ∫ t 0 β(t1)u(t1, x)(1 − u(t1, x))dt1 − ∫ t 0 α(t1)u(t1, x)ux(t1, x)dt1 + ∫ t 0 uxx(t1, x)dt1 +u0(x) − u(t, x), (t, x) ∈ [0, ∞) × [0, ∞), which we differentiate in t and we obtain 0 = β(t)u(t, x)(1 − u(t, x)) − α(t)u(t, x)ux(t, x) +uxx(t, x) − ut(t, x), (t, x) ∈ [0, ∞) × [0, ∞), i.e., u satisfies (6.1). Now we put t = 0 in (6.3) and we get u(0, x) = u0(x), x ∈ [0, ∞). This completes the proof. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 276 Fixed point index theory for the sum of operators For u ∈ E, define the operators F1u(t, x) = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u(t1, x2)dx2dx1dt1, F2u(t, x) = ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×u(t2, x2)dx2dt2dx1dt1 −1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 u(t2, x1)dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1, (t, x) ∈ [0, ∞) × [0, ∞). Note that if u ∈ E is a fixed point of the operator F2 − F1, then it is a solution of the IVP (6.1)-(6.2). Lemma 6.2. Suppose (A1), (A2) and r > 0. If u ∈ E and ‖u‖ ≤ r, then ‖F1u‖ ≤ (1 + r)A‖u‖, ‖F2u‖ ≤ ( 3 + r 2 ) rA and F2 : {u ∈ E : ‖u‖ ≤ r} → E is a completely continuous operator. More- over, ‖F1u1 − F1u2‖ ≤ (2r + 1)A‖u1 − u2‖ for any u1, u2 ∈ {u ∈ E : ‖u‖ ≤ r}. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 277 S. G. Georgiev and K. Mebarki Proof. Take u ∈ {E : ‖u‖ ≤ r} arbitrarily. Then |F1u(t, x)| ≤ ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1)dx1dt1 ≤ r‖u‖t4x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +‖u‖t4x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)A‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t F1u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ 4r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1)dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 278 Fixed point index theory for the sum of operators ≤ 4r‖u‖t3x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖t3x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)A‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x F1u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 ≤ 4r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1)dx1dt1 ≤ 4r‖u‖t4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +4‖u‖t4x5 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)A‖u‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 F1u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2) ×|u(t1, x2)|dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 279 S. G. Georgiev and K. Mebarki ≤ 12r‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +12‖u‖ ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1)dx1dt1 ≤ 12r‖u‖t4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 +12‖u‖t4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ (1 + r)A‖u‖, t ≥ 0, x ≥ 0, Consequently ‖F1u‖ ≤ (1 + r)A‖u‖. Next, |F2u(t, x)| ≤ ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −1 2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 280 Fixed point index theory for the sum of operators ≤ r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −1 2 r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)4g(t1, x1)dx1dt1 +r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)4g(t1, x1)dx1dt1 ≤ rt4x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −1 2 r2t4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +rt5x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +rt4x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t F2u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −2 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 281 S. G. Georgiev and K. Mebarki ≤ 4r ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2 ∫ t 0 ∫ x 0 x1(t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4r ∫ t 0 ∫ x 0 t1(t − t1)3(x − x1)4g(t1, x1)dx1dt1 +4r ∫ t 0 ∫ x 0 x21(t − t1)3(x − x1)4g(t1, x1)dx1dt1 ≤ 4rt3x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2t3x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4rt4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +4rt3x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂t2 F2u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −6 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)2(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 282 Fixed point index theory for the sum of operators ≤ 12r ∫ t 0 ∫ x 0 x21(t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2 ∫ t 0 ∫ x 0 x1(t − t1)2(x − x1)4g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12r ∫ t 0 ∫ x 0 t1(t − t1)2(x − x1)4g(t1, x1)dx1dt1 +12r ∫ t 0 ∫ x 0 x21(t − t1)2(x − x1)4g(t1, x1)dx1dt1 ≤ 12rt2x6 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2t2x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12rt3x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +12rt2x6 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x F2u(t, x) ∣ ∣ ∣ ∣ ≤ 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −2 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 283 S. G. Georgiev and K. Mebarki ≤ 4r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)3g(t1, x1)dx1dt1 +4r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)3g(t1, x1)dx1dt1 ≤ 4rt4x5 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −2r2t4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +4rt5x3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +4rt4x5 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 F2u(t, x) ∣ ∣ ∣ ∣ ≤ 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −6 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 284 Fixed point index theory for the sum of operators ≤ 12r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)2g(t1, x1)dx1dt1 +12r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)2g(t1, x1)dx1dt1 ≤ 12rt4x4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −6r2t4x3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +12rt5x2 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +12rt4x4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂3 ∂x3 F2u(t, x) ∣ ∣ ∣ ∣ ≤ 24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×|u(t2, x2)|dx2dt2dx1dt1 −12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 ∫ x1 0 α(t2)(u(t2, x2)) 2 ×dx2dt2dx1dt1 +24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ t1 0 |u(t2, x1)|dt2dx1dt1 +24 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 285 S. G. Georgiev and K. Mebarki ≤ 24r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −12r2 ∫ t 0 ∫ x 0 x1(t − t1)4(x − x1)g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +24r ∫ t 0 ∫ x 0 t1(t − t1)4(x − x1)g(t1, x1)dx1dt1 +24r ∫ t 0 ∫ x 0 x21(t − t1)4(x − x1)g(t1, x1)dx1dt1 ≤ 24rt4x3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dx1dt1 −12r2t4x2 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 α(t2)dt2dx1dt1 +24rt5x ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 +24rt4x3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ ( 3 + r 2 ) rA, t ≥ 0, x ≥ 0. Consequently ‖F2u‖ ≤ ( 3 + r 2 ) rA, ∥ ∥ ∥ ∥ ∂2 ∂t2 F2u ∥ ∥ ∥ ∥ C0 ≤ ( 3 + r 2 ) rA, ∥ ∥ ∥ ∥ ∂3 ∂x3 F2u ∥ ∥ ∥ ∥ C0 ≤ ( 3 + r 2 ) rA. By the Arzela-Ascoli theorem, it follows that the operator F2 : {u ∈ E : ‖u‖ ≤ r} → E is a completely continuous operator. Let now, u1, u2 ∈ {u ∈ E : ‖u‖ ≤ r}. Then |F1u1(t, x) − F1u2(t, x)| ≤ ( ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 286 Fixed point index theory for the sum of operators + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 2rx6t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +x6t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)A‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂t F1u1(t, x) − ∂ ∂t F1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 +4 ∫ t 0 ∫ x 0 (t − t1)3(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 8rx6t3 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +4x6t3 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)A‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂ ∂x F1u1(t, x) − ∂ ∂x F1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 287 S. G. Georgiev and K. Mebarki +4 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)3g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 8rx5t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +4x5t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)A‖u1 − u2‖, t ≥ 0, x ≥ 0, and ∣ ∣ ∣ ∣ ∂2 ∂x2 F1u1(t, x) − ∂2 ∂x2 F1u2(t, x) ∣ ∣ ∣ ∣ ≤ ( 12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2) ×β(t2) (|u1(t2, x2)| + |u2(t2, x2)|) dx2dt2dx1dt1 +12 ∫ t 0 ∫ x 0 (t − t1)4(x − x1)2g(t1, x1) ∫ x1 0 (x1 − x2)dx2dx1dt1 ) ‖u1 − u2‖ ≤ ( 24rx4t4 ∫ t 0 ∫ x 0 g(t1, x1) ∫ t1 0 β(t2)dt2dt1 +12x4t4 ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ) ‖u1 − u2‖ ≤ (2r + 1)A‖u1 − u2‖, t ≥ 0, x ≥ 0. Therefore ‖F1u1 − F1u2‖ ≤ (2r + 1)A‖u1 − u2‖. This completes the proof. � Theorem 6.3. Suppose (A1) and (A2). Then the IVP (6.1)-(6.2) has at least one non-negative solution u ∈ C1([0, ∞), C2([0, ∞))). Proof. Set P = {u ∈ E : u(t, x) ≥ 0, t ≥ 0, x ≥ 0}, Ω = {u ∈ P : ‖u‖ ≤ r1, u(t, x) ≤ u0(x), t ≥ 0, x ≥ 0}, U = {u ∈ P : ‖u‖ ≤ r1, 1 2 u0(x) ≤ u(t, x) ≤ u0(x), t ≥ 0, x ≥ 0}. For u ∈ E, define the operators T u(t, x) = −F1u(t, x), Su(t, x) = F2u(t, x), t ≥ 0, x ≥ 0. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 288 Fixed point index theory for the sum of operators (1) Let u, v ∈ Ω. Then (I − T )(u − v) = (I + F1)(u − v) and using Lemma 6.2, we get ‖(I − T )(u − v)‖ ≥ ‖u − v‖ − ‖F1(u − v)‖ ≥ (1 − (1 + 2r1)A) ‖u − v‖. Thus, I − T : Ω → E is Lipschitz invertible with γ = 1 1−(1+2r1)A . (2) By Lemma 6.2, we have that S : U → E is a completely continuous operator. Therefore S : U → E is 0-set contraction. (3) Let v ∈ U be arbitrarily chosen. For u ∈ Ω, we have −F1u(t, x) + F2v(t, x) ≥ − ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×(u(t2, x2))2dx2dt2dx1dt1 − ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u(t1, x2)dx2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×v(t2, x2)dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×u0(x2)dx2dx1dt1 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 289 S. G. Georgiev and K. Mebarki ≥ ( r1 2 − r21 ) ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ t1 0 ∫ x1 0 (x1 − x2)β(t2) ×dx2dt2dx1dt1 + ∫ t 0 ∫ x 0 (t − t1)4(x − x1)4g(t1, x1) ∫ x1 0 (x1 − x2) ×(u0(x2) − u(t1, x2))dx2dx1dt1 ≥ 0, t ≥ 0, x ≥ 0, and −F1u(t, x) + F2v(t, x) ≤ ‖F1u‖ + ‖F2v‖ ≤ (1 + r1)r1A + ( 3 + r1 2 ) r1A = ( 4 + 3 2 r1 ) r1A < r1 2 ≤ u0(x), t ≥ 0, x ≥ 0. For u ∈ Ω, define the operator Lu(t, x) = −F1u(t, x) + F2v(t, x), t ≥ 0, x ≥ 0. Then, using Lemma 6.2, we get ‖Lu‖ ≤ ‖F1u‖ + ‖F2v‖ ≤ r1(1 + r1)A + ( 3 + r1 2 ) r1A = ( 4 + 3 2 r1 ) r1A ≤ r1 2 . Consequently L : Ω → Ω. Again, applying Lemma 6.2, we obtain ‖Lu1 − Lu2‖ ≤ (2r1 + 1)A‖u1 − u2‖. Therefore L : Ω → Ω is a contraction operator and there exists a unique u ∈ Ω so that u = Lu or (I − T )u = Sv. Then S(U) ⊂ (I − T )(Ω). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 290 Fixed point index theory for the sum of operators (4) Assume that there are an u ∈ ∂U and λ ≥ 1 so that Su = (I − T )(λu) and λu ∈ Ω. Then Su = (I + F1)(λu) and applying Lemma 6.2, we obtain ( 3 + r1 2 ) r1A ≥ ‖Su‖ ≥ λ‖u‖ − ‖F1(λu)‖ ≥ λ‖u‖ − (1 + r1)A‖λu‖ = (1 − (1 + r1)A) λ‖u‖ ≥ (1 − (1 + r1)A)‖u‖ = r1(1 − (1 + r1)A), whereupon ( 3 + r1 2 ) A ≥ 1 − (1 + r1)A or ( 4 + 3 2 r1 ) A ≥ 1, which is a contradiction. Hence and Proposition 3.4, it follows that the operator T + S has at least one fixed point in U ⋂ Ω, which is a nontrivial nonnegative solution of the IVP (6.1)-(6.2). This completes the proof. 6.1. Example. Below, we will illustrate our main result. Let h(x) = log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 , l(s) = arctan s11 √ 2 1 − s22 , s ∈ R. Then h′(s) = 22 √ 2s10(1 − s22) (1 − s11 √ 2 + s22)(1 + s11 √ 2 + s22) , l′(s) = 11 √ 2s10(1 + s20) 1 + s40 , s ∈ R. Therefore −∞ < lim s→±∞ (1 + s + · · · + s9)h(s) < ∞, −∞ < lim s→±∞ (1 + s + · · · + s9)l(s) < ∞. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 291 S. G. Georgiev and K. Mebarki Hence, there exists a positive constant C1 so that (1 + s + · · · + s9) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 − s22 ) ≤ C1, (1 + s + · + s9) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 − s22 ) ≤ C1, s ∈ [0, ∞). Note that by [10](pp. 707, Integral 79), we have ∫ dz 1 + z4 = 1 4 √ 2 log 1 + z √ 2 + z2 1 − z √ 2 + z2 + 1 2 √ 2 arctan z √ 2 1 − z2 . Let Q(s) = s10 (1 + s44) (1 + (1 + s + · · · + s9)2)28 , s ∈ [0, ∞), and g1(t, x) = Q(t)Q(x), t, x ∈ [0, ∞). Then there exists a positive constant A1 such that 720(1 + t + · · · + t6)(1 + x + · · · + x6) ∫ t 0 ∫ x 0 g1(t1, x1)dx1dt1 ≤ A1, t, x ≥ 0. Take g(t, x) = g1(t,x) 280A1 , A = 1 50 , r1 = 1 4 . Consider the IVP ut − uxx − uux = u(1 − u), t > 0, x ≥ 0, u(0, x) = 1 8 + 1 8(1 + x2) , x ≥ 0. Here α = −1, β = 1 on [0, ∞), r1 2 ≤ u0(x) = 1 8 + 1 8(1 + x2) ≤ r1, x ≥ 0, 1 − (1 + r1)A = 39 40 > 0, ( 4 + 3 2 r1 ) A = 7 80 < 1 2 , and 120 ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 292 Fixed point index theory for the sum of operators × ∫ t 0 ∫ x 0 g(t1, x1) ( 1 + ∫ t1 0 (β(t2) − α(t2))dt2 ) dx1dt1 ≤ 240(1 + t) ( 1 + t + t2 + t3 + t4 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g(t1, x1)dx1dt1 ≤ 720 280A1 ( 1 + t + t2 + t3 + t4 + t5 )( 1 + x + x2 + x3 + x4 + x5 + x6 ) × ∫ t 0 ∫ x 0 g1(t1, x1)dx1dt1 ≤ 1 280 ≤ A. Therefore the considered IVP has at least one non-negative solution u ∈ C1([0, ∞), C2([0, ∞))). � Acknowledgements. The second author was supported by: Direction Générale de la Recherche Scientifique et du Développement Technologique DGRSDT. MESRS Algeria. Projet PRFU : C00L03UN060120180009. References [1] S. Benslimane, S. G. Georgiev and K. Mebarki, Expansion-compression fixed point theo- rem of Leggett-Williams type for the sum of two operators and application in three-point BVPs, Studia UBB Math, to appear. [2] G. Cain and M. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581–592. [3] S. Djebali and K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contraction, Topol. Meth. in Nonlinear Anal. 54, no. 2 (2019), 613– 640. [4] S. Djebali and K. Mebarki, Fixed point index on translates of cones and applications, Nonlinear Studies 21, no. 4 (2014), 579–589. [5] D. Edmunds, Remarks on nonlinear functional equations, Math. Ann. 174 (1967), 233– 239. [6] S. G. Georgiev and K. Mebarki, Existence of positive solutions for a class ODEs, FDEs and PDEs via fixed point index theory for the sum of operators, Commun. on Appl. Nonlinear Anal. 26, no. 4 (2019), 16–40. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 293 S. G. Georgiev and K. Mebarki [7] S. G. Georgiev and K. Mebarki, Existence of solutions for a class of IBVP for nonlinear parabolic equations via the fixed point index theory for the sum of two operators, New Trends in Nonlinear Analysis and Applications, to appear. [8] D. Guo, Y. J. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Shang- don Science and Technology Publishing Press, Shangdon, 1985. [9] M. Nashed and J. Wong, Some variants of a fixed point theorem Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech. 18 (1969), 767–777. [10] A. Polyanin and A. Manzhirov, Handbook of integral equations, CRC Press, 1998. [11] V. Sehgal and S. Singh, A fixed point theorem for the sum of two mappings, Math. Japonica 23 (1978), 71–75. [12] T. Xiang and R. Yuan, A class of expansive-type Krasnosel’skii fixed point theorems, Nonlinear Anal. 71, no. 7-8 (2009), 3229–3239. [13] T. Xiang and S. G. Georgiev, Noncompact-type Krasnoselskii fixed-point theorems and their applications, Math. Methods Appl. Sci. 39, no. 4 (2016), 833–863. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 294