Existence results for first derivative dependent ϕ-Laplacian boundary value problems

Our main concern in this article is to investigate the existence of solution for the boundary-value problem (ϕ(x′(t))′=g1(t,x(t),x′(t)),∀t∈[0,1],ϒ1(x(0),x(1),x′(0))=0,ϒ2(x(0),x(1),x′(1))=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ \end{document} where g1:[0,1]×R2→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$\end{document} is an L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{1}$\end{document}-Carathéodory function, ϒi:R3→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $\end{document} are continuous functions, i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1,2$\end{document}, and ϕ:(−a,a)→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi :(-a,a)\rightarrow \mathbb{R}$\end{document} is an increasing homeomorphism such that ϕ(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi (0)=0$\end{document}, for 0<a<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< a< \infty $\end{document}. We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

The study of φ-Laplacian equations is a classical topic having applications in glaciology, population biology, nonlinear flow laws, non-Newtonian mechanics, and combustion theory; see for example [28][29][30][31], and the references therein. Moreover, the φ-Laplacian operators are involved in some models, e.g., in non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity, and theory of capillary surfaces [32].
A lower and upper solutions (LUSs) approach is widely investigated to develop the existence and solvability results for classical and fractional order differential equations with φ-Laplacian operator; see for example [33][34][35][36][37] and the references therein.
Motivated by the above-mentioned work on φ-Laplacian differential equations, we develop an existence criterion for the boundary value problem (BVP), (1)- (2). To the best of our knowledge, (1) is still an untreated problem with BCs (2). The approach we use in our study is simpler than the approaches used in [1,3,33,34]. We prove the existence of solutions of (1)-(2) by using new conditions given on the BCs (2). These conditions allow us to obtain a solution in the sector defined by well-ordered functions. These ordered functions do not require one to check the definitions of LUSs. Moreover, the requirement to impose the monotonicity assumptions on the arguments of the BCs is not necessary in our case. The arguments we use in our study are the Arzelà-Ascoli and Schauder's fixed point theorems.
The rest of the article is organized as follows: in Sect. 2, we present preliminary definitions and auxiliary results, in Sect. 3, we prove the existence of solutions of the problem (1)-(2), in Sect. 4, we consider an example to verify the results of Sect. 3, and in Sect. 5, the conclusion is given.
The following lemmas are very useful for obtaining our main results.
Proof First, the uniqueness and existence is shown, then continuity. Let, for (k, For some s ∈ [0, 1], we have Since φ -1 is injective function, (5) implies that Equation (6) further implies hence, is well defined, decreasing, and continuous. Also Equation (9) implies the existence of a unique d = D φ (k, g). Now, it remains to show that (-a, a). Now let D φ (k n , g n ) be a subsequence converges to d 0 , then by the application of the dominated convergence theorem, we deduce that (-a, a) − → R is a continuous function.
Step 1: The solution of the problem (11)-(12) is equivalent to find a fixed points of the operator, : Firstly, we ensure the existence of λ x ∈ R. For this, we claim that for each x ∈ C 1 [0, 1], there exists a unique λ x ∈ R, such that Let and Obviously, Consequently, λ x ∈ R exists by Lemma 2.1.
Since G(x) is bounded and continuous on [0, 1], and the integral is a continuous function on [0, s]. Furthermore, λ x exists, φ is a homomorphism and its inverse exists. Also, is continuous, and its integral exists. Therefore (x) is continuous on [0, 1]. Further, the class { (x) : x ∈ C 1 [0, 1]} is uniformly bounded and equicontinuous. Therefore in view of the Arzelà-Ascoli theorem { (x) : x ∈ C 1 [0, 1]} is relatively compact. Consequently is a compact map. Now the Schauder fixed point theorem guarantees the existence of at least a fixed point since is continuous and compact.

Example
Consider the following nonlinear BVP: subject to following nonlinear BCs: This problem is a particular case of the problem (1)-(2) with and a = 1.

Conclusion
We studied the existence of solutions of φ-Laplacian boundary value problems by employing the topological approach and the new conditions on the boundary functions. The conditions on the boundary functions allowed us to obtain a solution in the sector defined by two well-ordered functions which did not require satisfying the differential inequalities to ensure the existence of lower and upper solutions. This way to deal with the φ-Laplacian boundary value problems made our approach simpler than the lower and upper solutions approach. Moreover, in our approach the monotonicity assumptions on the arguments of the boundary functions are not required. We considered an example to check the applicability of the developed theoretical results.